On some nonlinear partial
differential equations for classical
and quantum many body systems
Daniel Marahrens
Clare Hall
Department of Applied Mathematics and Theoretical Physics
University of Cambridge
A thesis submitted for the degree of
Doctor of Philosophy
July 2012
2
Declaration
This dissertation is the result of my own work and includes nothing which is
the outcome of work done in collaboration except where specifically
indicated in the text.
Daniel Marahrens
I dedicate this thesis to my parents Sigrid and Friedrich Marahrens for their
limitless love, encouragement, and support
Acknowledgements
I would not have been able to write this thesis without the help and support
of many kind people around me, some of whom I would like to mention here
in particular.
First and foremost, I would like to thank my supervisors Profs. Peter A.
Markowich and Christof Sparber. I am very grateful to Peter Markowich
for introducing me to many of the problems treated in this thesis and finding
excellent collaborators with whom I could work on these problems. I want to
thank Christof Sparber for his constant support, hospitality, and generosity
with his time and mathematical advice.
I am deeply indebted to my excellent collaborators from whom I have learnt
more than I ever thought possible in these last three years. I have profited
immensely from Cle´ment Mouhot’s keen mathematical intuition as well as op-
timism. I thank Michael Hintermu¨ller for his kind support and hospitality. I
thank Paolo Antonelli for his cheerfulness, even when trying to explain to me
the finer points of existence theory for nonlinear Schro¨dinger equations.
I profited from the wider circle of present and former Cambridge mathemati-
cians, in particular I owe a debt of gratitude to to Jonathan Ben-Artzi, Dan
Brinkman, Clemens Heitzinger, Alexander Lorz, Jan-Frederik Pietschmann,
Carola-Bibiane Scho¨nlieb, and Marie-Therese Wolfram.
Over the last three years, I have also had several very enjoyable stays with
the research group of Martin Burger in Mu¨nster, who I would like to thank
for the kind hospitality and interesting discussions.
I would like to thank my friends in the Germany, the UK, and elsewhere
for their invaluable friendship, encouragement, and support. In particular, I
would like to mention Husna Jamal, Patrick Kamus, and Alexander Sauter,
who - it is not an exaggeration to say - made me who I am. (I hope you like
the result!)
Lastly I want to thank my family, in particular Mama, Papa, and Robin for
their unwavering love and support.
All the work presented in this thesis has been supported by the Cambridge
European Trust, the EPSRC, and Award No. KUK-I1-007-43, made by the
King Abdullah University of Science and Technology (KAUST).
Abstract
This thesis deals with problems arising in the study of nonlinear partial dif-
ferential equations arising from many-body problems. It is divided into two
parts: The first part concerns the derivation of a nonlinear diffusion equation
from a microscopic stochastic process. We give a new method to show that in
the hydrodynamic limit, the particle densities of a one-dimensional zero range
process on a periodic lattice converge to the solution of a nonlinear diffusion
equation. This method allows for the first time an explicit uniform-in-time
bound on the rate of convergence in the hydrodynamic limit. We also discuss
how to extend this method to the multi-dimensional case. Furthermore we
present an argument, which seems to be new in the context of hydrodynamic
limits, how to deduce the convergence of the microscopic entropy and Fisher
information towards the corresponding macroscopic quantities from the valid-
ity of the hydrodynamic limit and the initial convergence of the entropy.
The second part deals with problems arising in the analysis of nonlinear
Schro¨dinger equations of Gross–Pitaevskii type. First, we consider the Cauchy
problem for (energy-subcritical) nonlinear Schro¨dinger equations with sub-
quadratic external potentials and an additional angular momentum rotation
term. This equation is a well-known model for superfluid quantum gases in
rotating traps. We prove global existence (in the energy space) for defocusing
nonlinearities without any restriction on the rotation frequency, generalizing
earlier results given in the literature. Moreover, we find that the rotation term
has a considerable influence in proving finite time blow-up in the focusing case.
Finally, a mathematical framework for optimal bilinear control of nonlinear
Schro¨dinger equations arising in the description of Bose–Einstein condensates
is presented. The obtained results generalize earlier efforts found in the liter-
ature in several aspects. In particular, the cost induced by the physical work
load over the control process is taken into account rather then often used L2–
or H1–norms for the cost of the control action. We prove well-posedness of
the problem and existence of an optimal control. In addition, the first order
optimality system is rigorously derived. Also a numerical solution method is
proposed, which is based on a Newton type iteration, and used to solve several
coherent quantum control problems.
Contents
Contents 7
List of Figures 11
Nomenclature 15
1 Introduction 17
1.1 Quantitative uniform hydrodynamic limits . . . . . . . . . . . . . . . . . . 19
1.2 The nonlinear Schro¨dinger equation . . . . . . . . . . . . . . . . . . . . . . 24
1.2.1 Existence theory with an angular momentum rotation term . . . . . 26
1.2.2 Optimal control of nonlinear Schro¨dinger equations . . . . . . . . . 27
I Quantitative uniform in time hydrodynamic limits 29
2 A quantitative perturbative approach to hydrodynamic limits 31
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Previous results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 A toy model: independent random walks . . . . . . . . . . . . . . . . . . . 38
2.4 Hydrodynamic limit for zero range processes . . . . . . . . . . . . . . . . . 49
2.4.1 The hydrodynamic limit . . . . . . . . . . . . . . . . . . . . . . . . 53
2.4.2 Regularity of the limit equation . . . . . . . . . . . . . . . . . . . . 59
2.4.3 Consistency and stability . . . . . . . . . . . . . . . . . . . . . . . . 62
2.4.4 Proof of the hydrodynamic limit . . . . . . . . . . . . . . . . . . . . 79
2.5 Convergence of the entropy . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.6 The replacement lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7
2.6.1 Attractivity and moment bounds . . . . . . . . . . . . . . . . . . . 92
2.6.2 Proof of the replacement lemma from the block estimates . . . . . . 94
2.6.3 Equivalence of ensembles . . . . . . . . . . . . . . . . . . . . . . . . 98
2.6.4 Restriction to bounded particle configurations . . . . . . . . . . . . 104
2.6.5 Proof of the one block estimate . . . . . . . . . . . . . . . . . . . . 106
2.6.6 Proof of the two blocks estimate . . . . . . . . . . . . . . . . . . . . 111
2.7 Perspective: the case of d ≥ 2 dimensions . . . . . . . . . . . . . . . . . . . 117
II Analysis of nonlinear Schro¨dinger equations 121
3 On the Cauchy-problem of nonlinear Schro¨dinger equations with angular
momentum rotation term 123
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.2 Mathematical setting and main result . . . . . . . . . . . . . . . . . . . . . 125
3.3 Local and global existence . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.4 Finite time blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
3.5 Numerical simulations of a rotating Bose-Einstein condensate . . . . . . . . 142
4 Optimal bilinear control of Gross–Pitaevskii equations 149
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4.1.1 Physics background . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4.1.2 Mathematical setting . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.1.3 Relation to other works and organization of the chapter . . . . . . . 154
4.2 Existence of minimizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.2.1 Convergence of infimizing sequences . . . . . . . . . . . . . . . . . . 156
4.2.2 Minimizers as mild solutions . . . . . . . . . . . . . . . . . . . . . . 159
4.2.3 Lower semicontinuity of objective functional . . . . . . . . . . . . . 162
4.3 Derivation and analysis of the adjoint equation . . . . . . . . . . . . . . . . 164
4.3.1 Identification of the derivative of J(ψ, α) . . . . . . . . . . . . . . . 164
4.3.2 Local and global existence theory for solutions of higher regularity . 167
4.3.3 Existence of solutions to the adjoint equation . . . . . . . . . . . . 171
8
4.4 Rigorous characterization of critical points . . . . . . . . . . . . . . . . . . 174
4.4.1 Lipschitz continuity with respect to the control . . . . . . . . . . . 174
4.4.2 Proof of differentiability and characterization of critical points . . . 177
4.5 Numerical simulation of the optimal control problem . . . . . . . . . . . . 182
4.5.1 Gradient-related descent method . . . . . . . . . . . . . . . . . . . 183
4.5.2 Newton method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
4.5.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . 186
5 Concluding remarks and future work 193
5.1 On the Cauchy-problem of nonlinear Schro¨dinger equations with angular
momentum rotation term . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5.2 Optimal bilinear control of Gross-Pitaevskii equations . . . . . . . . . . . . 194
5.3 Uniform quantitative hydrodynamic limits . . . . . . . . . . . . . . . . . . 195
References 197
9
10
List of Figures
1.1 The microscopic model. A particle jumps from a site x with η(x) particles
to a randomly chosen neighbouring site at a rate g(η(x)). . . . . . . . . . . 20
1.2 Embedding TdN into T
d yields lattice sites of distance 1/N . In the limit, we
obtain a limit density ft. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 The two boxes Λ1l and Λ
2
l . Opposite faces have been chosen for Γ, defined
below, and it holds (x∗, z∗) ∈ Γ. . . . . . . . . . . . . . . . . . . . . . . . . 113
3.1 Contour plots of the density function |ψ(t, x)|2 for dynamics of a vortex
lattice at different times . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
3.2 Contour plots of the density function |ψ(t, x)|2 for dynamics of a vortex
lattice as found in [6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.1 Shifting a linear wave paket . . . . . . . . . . . . . . . . . . . . . . . . . . 188
4.2 Splitting a linear wave paket with γ1 = 0 . . . . . . . . . . . . . . . . . . . 189
4.3 Splitting a linear wave paket with γ1 > 0 . . . . . . . . . . . . . . . . . . . 189
4.4 Direct comparison between results . . . . . . . . . . . . . . . . . . . . . . . 190
4.5 The weight factor ω =
∫
V |ψ|2dx over time . . . . . . . . . . . . . . . . . . 190
4.6 Value of J(ψ, α) over number of iterations . . . . . . . . . . . . . . . . . . 191
4.7 Splitting a condensate with γ1 > 0 . . . . . . . . . . . . . . . . . . . . . . . 191
4.8 Splitting a focusing condensate with γ1 > 0 . . . . . . . . . . . . . . . . . . 192
11
12
Nomenclature
General Notation
xn ⇀ x Weak convergence of the sequence (xn)n∈N to x
xn → x Strong convergence of the sequence (xn)n∈N to x
Lp(Rd) The space of Lebesgue–integrable functions whose absolute value raised to the
p–th power has finite integral
Hk(Rd) The Sobolev space W k,2(Rd)
W k,p(Rd) The Sobolev space of k-times weakly differentiable functions with derivatives
in Lp(Rd)
W˙ k,p(Rd) The homogeneous Sobolev space of degree n
LptL
q
x Short-hand for L
p(0, T ;Lq(Rd)), the space of functions that are Lp–integrable in t
and Lq–integrable in x
〈·, ·〉 An inner product, or an integration of a function with respect to a measure. More
generally, a duality pairing
Hydrodynamic Limits
TdN The discrete torus
x, y, z Microscopic variables, i.e. sites in TdN
x ∼ y Denotes the fact that x, y ∈ TdN are neighbours, i.e. |x− y| = 1
Td The continuous torus Rd/Zd
u Macroscopic variables, i.e. positions u ∈ Td
XN The state space of the discrete particle system
η, ξ Particle configurations in XN
13
αNη The empirical measure
δx/N The Dirac distribution centered at x/N ∈ Td
H The function space of the limit equation
αN,η The mollified empirical measure
δ()x/N The mollified Dirac distribution centered at x/N ∈ T
d (a Dirac sequence with
respect to )
s, r Multi-indices in Nd
P (XN) Space of probability measures on XN
Ckb (R) Uniformly bounded k-times differentiable functions on R with uniformly bounded
derivatives
Cb(XN) Space of bounded continuous functions on XN with supremum norm
M+(Td) The set of positive Radon measures on the torus
fN , fN0 , f
N
t Functions in Cb(XN); usually densities with respect to some measure in
P (XN)
µ, ν Probability measures
µN , νN Probability measures on the state space XN
f, f0, ft Functions in (a subset of) H
HN(µ|ν) Microscopic relative entropy of two probability measures µ and ν
DN(µ|ν) Microscopic Fisher information corresponding to HN(µ|ν)
H∞ Entropy of the limit equation
D∞ Fisher information of the limit equation
νNρ The Gibbs measure with average density ρ
Ψ,Φ Observables in C(H)
ϕ A test function in C(Td)
ρ, % Different densities ρ, % ∈ [0,∞)
S∞t Semigroup of the limit equation
SNt Semigroup of the particle process
14
TNt Dual semigroup of S
N
t of the particle process on Cb(XN)
T∞t Pullback semigroup of S
∞
t on the level of observables
DS∞t , DT
∞
t , DΨ Derivatives in the sense of linearizations, cf. Def. 2.4.16
Schro¨dinger Equations
ψ A wave function, solution to a (nonlinear) Schro¨dinger equation
U(x) An external potential
V (x) A control potential
W (t, x) A time-dependent potential
λ The proportionality factor in front of nonlinearity
σ The exponent of nonlinearity of Gross-Pitaevskii type
α(t) A time-dependent control parameter
Ω The angular velocity vector of a rotation
L The quantum angular momentum operator
E0 The energy without angular momentum
EΩ The energy of the NLS with rotation
LΩ The ensemble-average of the angular momentum
Σ The energy space of solutions
Σm The higher-order energy space
S(t) The unitary semigroup of a linear Scho¨dinger equation
A An observable
J The objective functional
γ1, γ2 Cost parameters appearing in J
J The reduced objective functional
P (ψ, α) The operator of a nonlinear Schro¨dinger equation
ϕ The solution to the adjoint equation
J′(α) The (Gaˆteaux-)derivative of J(α) with respect to α
F (t, s) The propagator of the adjoint equation
15
16
Chapter 1
Introduction
The theory of partial differential equations (PDE) is one of the main research areas in
mathematics and has applications in many disciplines, among them physics, engineering,
economics, and chemistry. While the fundamental laws of nature as discovered by Isaac
Newton, Albert Einstein, Erwin Schro¨dinger, Werner Heisenberg, and many others, can
be expressed as differential equations, there is a plethora of models describing nature in
terms of a partial differential equation, which cannot yet be justified from first principles.
Often these models are obtained by considering systems of many interacting entities (in
gases on the order of 1026 particles), where the entities can be comprised of anything from
classical particles in gases to pedestrians within crowds. These systems inherently possess
two scales, a microscopic one and a macroscopic one. While it is usually fairly simple to
establish the microscopic laws, it is impossible to solve them for most many body systems,
making a purely microscopic description infeasible. Fortunately, we are not interested in
the properties of every single entity (henceforth called particle). Instead it often suffices to
know certain macroscopic, measurable quantities like the density, the temperature, or the
velocity. One of the important challenges of science lies in the derivation of macroscopic,
effective PDE models for these macroscopic quantities when only the microscopic laws
of interaction are known. In the many body systems under consideration in this thesis,
this effective description is accurate in the limit of infinitely many particles. An impor-
tant step in the study of scaling limits of interacting particle systems usually consists in
showing that correlations (between particles or occupation numbers) vanish in the limit.
This allows one to replace the interactions between different particles by a self-consistent
field according to a mean-field theory. These self-consistent fields imply that the limit
equation is in general nonlinear. The analysis of nonlinear partial differential equations
is in and of itself an important field of study. In this thesis we will show how to obtain
a limit description from a class of microscopic dynamics and investigate two problems
related to a particular limit system, the nonlinear Schro¨dinger equation.
Part I of this thesis deals with the rigorous derivation of a macroscopic PDE descrip-
17
Introduction
tion from a microscopic stochastic particle dynamics. The derivation of limit descriptions
from stochastic interacting particle systems has a long history that can be traced back to
Ludwig Boltzmann. Since a rigorous approach is so far only feasible for simple models,
we concentrate on a well-studied interacting particle system, the zero range process on a
domain with periodic boundary conditions. The zero range process is a stochastic jump
process consisting of discrete particles on a lattice, where particles only interact if they
occupy the same lattice site. The macroscopic limit for the zero range process in the
hydrodynamic scaling, i.e. large time and space scales, is well-known and has been shown
to hold in [36] and [86], using different methods. Our contribution to the hydrodynamic
limit, presented in Chapter 2, is a new approach that allows us to obtain an explicit rate
of convergence which holds uniform in time. This approach is based on a Duhamel–type
formula in the space of observables, here taken to be the space of continuous functions of
the state space. Furthermore, we present an argument which seems to be new in the con-
text of hydrodynamic limits, which allows to establish the convergence of the microscopic
entropy and Fisher information to their macroscopic versions.
In Part II, we consider the nonlinear Schro¨dinger equation, which in a special (cubic)
case is obtained in a scaling limit of the Schro¨dinger equation for infinitely many Bosonic
particles (Bose-Einstein condensation) [28]. Here the microscopic scale is given by the
scattering length of the interaction potential, which must be macroscopically small for
the particle correlations to vanish in the limit of infintely many particles. The nonlinear
Schro¨dinger equation has received a lot of attention in mathematical physics not only
as a model for ultra-cold dilute atomic gases, but also for nonlinear optics and shallow
water waves. From the mathematical point of view, it is a dispersive semilinear equa-
tion exhibiting many interesting phenomena such as the existence of solitons, scattering,
blow-up, and global existence in different parameter ranges. In this thesis we concentrate
on two problems for the nonlinear Schro¨dinger equation. First we consider in Chapter 3
a nonlinear Schr ”odinger equation with an angular momentum rotation term which has
been used in the physics literature as a model for Bose-Einstein condensates in a rotating
trap. We investigate local and global existence in the usual parameter ranges, i.e. local
existence for energy-subcritical nonlinearities, and global existence for mass-subcritical or
defocusing nonlinearities. Furthermore we deduce conditions on the existence of blow-up
solutions by a virial argument (following Glassey’s approach [32]).
As a second application, in Chapter 4 we consider the optimal control problem corre-
sponding to the nonlinear Schro¨dinger equation. The experimental control of quantum
systems described by linear and nonlinear Schro¨dinger equations has applications in mi-
croscopic magnetic-field imaging, atom interferometry, and quantum computing, to name
but a few examples. In many applications, one is not interested in controlling the whole
state of the system but only a few observables. In order to guarantee well-posedness of
the control problem, we restrict ourselves to minimizing a certain objective functional
18
1.1. Quantitative uniform hydrodynamic limits
consisting of the observable quantity we want to minimize and a “regularizing” cost term.
Thus we propose a mathematical framework for optimal control of nonlinear Schro¨dinger
equations through an external potential, where the objective functional is given by the
expected value of the observable with the H1–norm of the energy as regularizing term.
In our example, the external potential depends on the control parameter only through its
amplitude, whereas its general shape is fixed. The choice of the H1–norm of the energy
has the advantage that it penalizes large oscillations which are typically found for cost
terms involving the L2–norm. Furthermore it has a direct physical interpretation as the
L2–norm of the power, i.e. the time–derivative of the energy.
In the rest of this introduction, I will give a brief mathematical exposition of the results
obtained in this thesis.
1.1 Quantitative uniform hydrodynamic limits
The zero range process on the discrete torus is an stochastic interacting particle system
on the lattice TdN = {1, . . . , N}
d with state space XN = NT
d
N , i.e. η ∈ XN is the particle
configuration with η(x) particles at each site x ∈ TdN . Particles are randomly distributed
over the lattice and perform a jump process, jumping to neighbouring sites at a rate
that only depends on the number of particles at the original site, see Figure 1.1. The
distribution of particle configurations at each time t > 0 is a probability measure µNt ∈
P (XN), where P (XN) is the set of probability measures on XN . Let us denote by Cb(XN)
the space of uniformly bounded, continuous functions and the integral of a function fN ∈
Cb(XN) with respect to the measure µN ∈ P (XN) by
〈µN , fN〉 =
∫
XN
fN(η); dµN(η).
Then the evolution of the state µNt ∈ P (XN) of the particle process, given an initial
distribution µN0 ∈ P (XN), is determined by
d
dt
〈µNt , f
N〉 = 〈µNt , G
NfN〉 for all fN ∈ Cb(XN),
where the generator GN : Cb(XN)→ Cb(XN) satisfies
(1.1) GNfN(η) = N2
∑
x∼y∈TdN
g(η(x))
(
fN(ηx,y)− fN(η)
)
.
Here the sum over x ∼ y is over all sites x, y ∈ TdN that are neighbours, i.e. |x − y| = 1,
and ηx,y is the state of the particle system after one particle has jumped from x to y.
19
Introduction
Explicitly it holds that
ηx,y(z) =



η(x)− 1 if z = x,
η(y) + 1 if z = y,
η(z) otherwise.
In this thesis, we shall assume that the jump rates are not degenerate and satisfy a
monotonicity condition which corresponds to uniform ellipticity of the limit equation. A
x ∈ TdN
g(η(x))
Figure 1.1: The microscopic model. A particle jumps from a site x with η(x) particles to
a randomly chosen neighbouring site at a rate g(η(x)).
measure νN ∈ P (XN) is invariant under the evolution of the zero range process if
〈νN , GNfN〉 = 0 for all fN ∈ Cb(XN).
We shall see in Chapter 2 that there exists a family of invariant (and translation-invariant)
measures νNρ , indexed by their mean density ρ ≥ 0, for which the occupation numbers
η(x), x ∈ TdN , are mutually independent. The average jump rate under the law of ν
N
ρ is
a smooth function σ : [0,∞)→ [0,∞), i.e.
〈νNρ , g(η(x))〉 = σ(ρ).
If we assume the process to equilibrate locally, we can expect the densities to converge
locally by a law of large numbers and the densities ft(u) at macroscopic points u ∈ Td
to change according to a partial differential equation. The limit equation is the filtration
equation
(1.2) ∂tft(u) = ∆σ(ft(u)) for all t > 0, u ∈ Td,
where ft : Td → [0,∞) denotes the particle density at time t. Here we shall only consider
uniformly elliptic limit equations, i.e. σ′(ρ) ≥ δ > 0 for all ρ ≥ 0.
The scaling factor N2 in the definition (1.1) of GN yields a (macroscopic) time scale and
is related to the fact that the limit equation is a second order PDE. In order to get a
continuum description via a partial differential equation, we also need to scale space by
embedding the discrete torus into the continuous (macroscopic) torus TdN ⊂ T
d = Rd/Zd
20
1.1. Quantitative uniform hydrodynamic limits
via x 7→ x/N ∈ Td. Thus the microscopic spatial scale is N−1, whereas the microscopic
time scale is N2. This implies a macroscopically visible displacement of particles through
the non-zero variance of the jumps, since by symmetry the mean displacement of the
particles vanishes. If the number of particles remains roughly constant with respect to
N , we expect that the average density N−1
∑
x∈TdN
η(x) does not scale with N . The zero
range process has only one conserved quantity, the total number of particles, and hence
the particle density is the only macroscopic information which we can expect to retain in
the limit as N →∞. We measure particle densities via the empirical measure
αNη (du) =
1
Nd
∑
x∈TdN
η(x)δ x
N
(du),
where δu is the Dirac mass at u ∈ Td. The convergence of local particle densities can be
quantified in terms of weak convergence of the empirical measure. Thus we test αNη with
a function ϕ ∈ C(Td) and expect convergence of the resulting random variable 〈αNη , ϕ〉
under the law µNt . Assuming that the initial data are compatible and ft solves (1.2), it
u ∈ Td
ft(u)
1
N
x
N
Figure 1.2: Embedding TdN into T
d yields lattice sites of distance 1/N . In the limit, we
obtain a limit density ft.
holds that
(1.3) lim
N→∞
PµNt
(
|〈αNη , ϕ〉 − 〈ϕ, ft〉| > δ
)
= 0,
where PµNt (A) denotes the probability of the event A under µ
N
t , i.e. 〈µ
N
t , χA〉, where χA
is the characteristic function of A. Figure 1.2 gives an idea of the microscopic scaling (in
space) and the local particle densities. The proof of this result was given in [36] for the
Ginzburg-Landau model with Kawasaki dynamics, which is a closely related model. Their
proof holds for the zero range process with only minor modifications, see for instance [47].
Further results are available in the literature, e.g. the relative entropy method [86] which
shows convergence of the relative entropy with respect to local equilibrium states and can
probably be extended to give an explicit rate of convergence - however there is no reason
to expect the convergence to be uniform in time. Let us also mention that there exists a
result [34] for the Ginzburg-Landau model with Kawasaki dynamics with (almost) explicit
rate of convergence which can be extended to hold uniformly in time - however it is not
yet clear how to extend this result to the zero range process. Our contribution to the
theory is a new approach to quantify the rate of convergence and make it uniform in time.
21
Introduction
Specifically, letting F ∈ C2b (R), we prove the convergence
(1.4)
∣
∣
∣〈µNt , F
(
〈αNη , ϕ
)
〉 − F
(
〈ft, ϕ〉L2
)∣∣
∣ ≤ CN−β
for some β > 0, uniformly in t > 0. This implies the convergence in probability in (1.3)
upon choosing F to be an approximation of an indicator function with support on a trans-
lation of (−, ). Even though the particle trajectories diverge as t→∞, the dissipative
properties of the system motivate the uniform convergence: As t→∞, the system relaxes
to an invariant measure, for which the hydrodynamic limit holds. Our approach seems to
be fairly flexible and we hope that it can be applied to more complex problems. It was
originally developed by Mischler and Mouhot [67] to derive an explicit uniform rate of
convergence of a jump process towards the (homogeneous in space) Boltzmann equation.
Let us now present the idea of this approach. First, without worrying about regularity,
we set
Ψ(f) = F (〈f, ϕ〉)
for any function or distribution f on Td. The function Ψ can be turned into a function
of the empirical measure by employing the map piN defined by
(piNΨ)(η) = Ψ(µNη ) for all η ∈ XN .
Furthermore we can define a limit semigroup on functions on Td via the pushforward
T∞t Ψ(f) = Ψ(ft),
where ft solves the filtration equation (1.2) with initial datum f0. Employing this notation,
we can estimate the hydrodynamic limit (1.4) by
∣
∣
∣〈µN0 , (T
N
t pi
NΨ)(η)− (piNT∞t Ψ)(η)〉
∣
∣
∣+
∣
∣
∣〈µN0 , T
∞
t Ψ(α
N
η )− T
∞
t Ψ(f0)〉
∣
∣
∣.
If the initial data converge in an appropriate sense, we can estimate the second term by
a stability estimate on the limit equation corresponding to contractivity of the semigroup
T∞t . The idea that allows us to estimate the first term is as follows. Just like T
N
t has
a generator GN , so, too, the limit semigroup T∞t has a generator (time-derivative) G
∞.
Then we obtain that
piN(T∞t Ψ)− T
N
t (pi
NΨ) =
∫ t
0
d
ds
(
TNt−spi
NT∞s Ψ
)
ds
=
∫ t
0
TNt−s
(
piNG∞ −GNpiN
)
T∞s Ψ ds.
22
1.1. Quantitative uniform hydrodynamic limits
We will provide a consistency estimate for the difference piNG∞−GNpiN and use stability
results to transport this convergence along the evolution of the limit equation which is
given by T∞s . This will allow us to obtain a uniform explicit rate of convergence.
Finally we shall present a new argument which allows one to prove convergence of the
microscopic entropy to a macroscopic entropy. Let f∞ =
∫
Td f0(u) du be the constant to-
wards which the macroscopic density equilibrates due to the dissipation. The microscopic
entropy with respect to the invariant measure νNf∞ is given by
HN(µNt |ν
N
f∞) =
∫
XN
log
( dµNt
dνNf∞
(η)
)
dµNt (η).
It holds that
(1.5)
1
Nd
HN(µNt |ν
N
f∞) +
∫ t
0
4N2−dDN(µNs |ν
N
f∞) ds ≤
1
Nd
HN(µN0 |ν
N
f∞),
where
DN(µNs |ν
N
f∞) = 〈ν
N
f∞ ,
√
dµNt
νNf∞
GN
√
dµNt
νNf∞
〉
is the microscopic Fisher information. Variational formulae are readily available for both
the entropy and the Fisher information which allow us to prove that
lim inf
N→∞
1
Nd
HN(µNt |ν
N
f∞) ≥ H∞(ft), where(1.6)
H∞(ft) =
∫
Td
∫ ft(u)
f∞
log σ(ρ) dρ du
is the macroscopic entropy and
lim inf
N→∞
4
Nd−2
DN(µNt |ν
N
f∞) ≥ D∞(ft), where(1.7)
D∞(ft) =
∫
Td
|∇σ(ft(u))|2
σ(ft(u))
du
is the macroscopic Fisher information. Differentiation of H∞(ft) in time yields
(1.8) H∞(ft) +
∫ t
0
D∞(fs) ds = H∞(f0)
Now, let us assume that the initial microscopic entropy converges to the initial macroscopic
entropy, i.e.
(1.9) lim
N→∞
1
Nd
HN(µNt |ν
N
f∞) = H∞(ft).
23
Introduction
Collecting the relations (1.5)-(1.9), we obtain that
lim
N→∞
1
Nd
HN(µNt |ν
N
f∞) = H∞(ft) and
lim
N→∞
∫ t
0
4N2−dDN(µNs |ν
N
f∞) ds =
∫ t
0
D∞(fs) ds
Note that in Chapter 2 we shall use an equivalent formulation of H∞(ft) in order to prove
the above inequality on the limit inferior of the microscopic entropy.
1.2 The nonlinear Schro¨dinger equation
In Part II, we shall consider the nonlinear Schro¨dinger equation (NLS), by which we mean
the partial differential equation
i∂tψ(t, x) = −
1
2
∆ψ(t, x) + U(x)ψ(t, x) + λ|ψ(t, x)|2σψ(t, x)
for all (t, x) ∈ R × Rd. Here i =
√
−1, λ ∈ R and σ ≥ 0 are two parameters describing
the nonlinearity, and V : Rd → R is an external potential. As mentioned before, at least
in the cubic case σ = 1, this equation can be obtained from the usual (linear) Schro¨dinger
equation in the limit of infinitely many bosons. In the cubic case, the equation is also
known as Gross-Pitaevskii equation. The NLS conserves mass and energy
M = ‖ψ‖2L2(Rd), E =
∫
Rd
(1
2
|∇ψ|2 +
λ
σ + 2
|ψ|2σ+2 + U |ψ|2
)
dx.
One of the interesting features of this equation is the occurrence of blow-up in certain
parameter regimes. Let us quickly discuss some aspects of the existence theory for the
NLS without external potential, i.e. U = 0. In this thesis, we will look for solutions in
the energy space, i.e. the space where the energy is finite. Since it holds that H1(Rd) ⊂
L2σ+2(Rd) if σ < 2/(d − 2), a good choice for the energy space is H1(Rd), where the
contribution of the nonlinearity to the energy can be estimated by the contribution of
the linear part. This is further motivated by the observation that the NLS is semilinear,
i.e. the nonlinear part only involves derivatives of the solution ψ which are of lower order
than the whole PDE. Hence it is convenient to treat the nonlinearity as a perturbation of
the linear part. To this end, let S(t) = e
i
2 ∆t denote the semigroup of the free Schro¨dinger
equation
i∂tψ(t, x) = −
1
2
∆ψ(t, x).
24
1.2. The nonlinear Schro¨dinger equation
The mild form of the NLS with initial datum ψ(t = 0, x) = ψ0(x) then becomes
ψ(t) = S(t)ψ0 − iλ
∫ t
0
S(t− s)|ψ(s)|2σψ(s) ds
where we have set ψ(t) = ψ(t, ·) for ease of notation. If σ = 0, there is of course global
existence. Similar to ordinary differential equations, we can look for mild solutions as
fixed points of an appropriately defined map. Local existence is then shown by smoothing
properties of the linear part of the equation combined with (Sobolev) embeddings in order
to control the nonlinear part. The free Schro¨dinger equation is not dissipative and does
not possess the strong smoothing effects of the heat equation, but its dispersive nature
accords us with weaker decay estimates, the so-called Strichartz estimates. These can
indeed be used to prove local existence in the energy space H1(Rd) if σ < 2/(d − 2).
These local solutions can be continued until the H1(Rd)–norm of the solution diverges.
In the special case of a positive nonlinearity λ ≥ 0, the energy is the sum of two positive
quantities. Since the energy is also conserved, both quantities are indeed bounded and in
particular, the L2–norm of the gradient of the solution is uniformly bounded in time. Since
the mass ‖ψ‖L2(Rd) is conserved as well, we conclude that there exists a global solution if
λ ≥ 0. Let us now consider the case λ < 0. A quick calculation yields the virial identity
d2
dt2
∫
Rd
x2|ψ(t, x)|2 dx =
∫
Rd
(
|∇ψ(t, x)|2 + λ
dσ
σ + 1
|ψ(t, x)|2σ+2
)
dx
= 2E +
∫
Rd
λ
dσ − 2
σ + 1
|ψ(t, x)|2σ+2 dx.
Hence if σ ≥ d/2 and E < 0, the (positive) integral on the left hand side is bounded by an
inverted parabola, which is not possible for all times. Note that it can also be shown that
the solution is global if σ < d/2, whatever the sign of λ. To summarize, local existence
holds in the following case:
• initial datum in the energy space ψ ∈ H1(Rd) and energy-subcritical nonlinearity
σ < 2/(d− 2)
and global existence holds if additionally
• defocusing nonlinearity λ ≥ 0, or
• mass-subcritical nonlinearity σ < d/2.
Otherwise, blow-up solutions exist. The described situation generally remains the same in
the presence of an external potential U , provided this potential satisfies certain regularity
properties, e.g. |U(x)| ≤ C|x|2 is subquadratic. In Chapter 3, we shall investigate the NLS
with an additional angular momentum term, which describes a rotating Bose-Einstein
25
Introduction
condensate. In Chapter 4, we consider an optimal control problem for the NLS. Our
results are summarized in the next two subsections.
1.2.1 Existence theory with an angular momentum rotation term
The NLS with angular momentum rotation term is the partial differential equation
i∂tψ = −
1
2
∆ψ + U(x)ψ + λ|ψ|2σψ − Ω · Lψ, (t, x) ∈ R× R3,
where λ ∈ R, σ ≥ 0, and U subquadratic. The angular momentum rotation term is given
by a angular velocity vector Ω ∈ R3 and the angular momentum L = −ix ∧ ∇, where ∧
denotes the cross product in R3. This equation also makes sense in two dimensions, where
the plane of rotation is the space R2. In fact, the two-dimensional NLS is usually obtained
as an approximation of the three-dimensional NLS in the case of a strongly confining
potential (a disc-shaped condensate). Hence we shall consider both cases d = 2, 3. So
far this equation has only been considered in [37, 38] for the special case where U is a
harmonic trapping potential with frequency exactly equal to |Ω|. In the present setting,
the energy space is Σ = {ψ ∈ H1(Rd) : xψ ∈ L2(Rd)}. Using Strichartz estimates for
the linear part including the rotation term, we find that local existence in Σ holds in the
usual case σ < 2/(d− 2) (i.e. sigma <∞ if d = 2). The same methods also allow us to
deduce global existence if σ < 2/d is mass-subcritical. In order to obtain global existence
in the defocusing case λ > 0, we change into a rotating coordinate system. Let X(t, x)
denote the vector obtained from rotating x around the axis Ω by an angle of −|Ω|t, then
the wave function ψ˜ in the new coordinates X(t, x) solves
i∂tψ˜ = −
1
2
∆ψ˜ + U(X(t, x))ψ˜ + λ|ψ˜|2σψ˜.
This is a NLS with time-dependent potential and as such can be treated as in [18], where
global existence has been shown to hold in the presence of time-dependent potentials if
λ > 0. Furthermore we present two variants of Glassey’s virial identity to deduce existence
of blow-up if λ < 0. These conditions are stricter than the above conditions due to Glassey,
but coincide in the symmetric or rotation-less case. We finish the chapter on the NLS with
angular momentum rotation term with a discussion of numerical simulations, where we
emphasize how the change of coordinates X(t,X) can be employed to simplify numerical
treatment of the equation.
26
1.2. The nonlinear Schro¨dinger equation
1.2.2 Optimal control of nonlinear Schro¨dinger equations
The optimal control problem we shall consider is given by
J∗ = inf
ψ˜,α˜
J(ψ˜, α˜) where
i∂tψ = −
1
2
∆ψ + U(x)ψ + λ|ψ|2σψ + α(t)V (x)ψ, x ∈ Rd, t ∈ R, and(1.10)
J(ψ, α) = 〈ψ(T, ·), Aψ(T, ·)〉2L2(Rd) + γ1
∫ T
0
(E˙(t))2 dt+ γ2
∫ T
0
(α˙(t))2 dt,
with λ ≥ 0, σ < 2/(d−2), external potential U , control potential V , and subject to initial
data
ψ(0, ·) = ψ0 ∈ Σ, α(0) = α0 ∈ R.
Here A is an observable, i.e. an operator with domain in L2(Rd), γ1 > 0 and γ2 ≥ 0
are two cost parameters, and E(t) is the energy corresponding to the NLS (1.10). In
this problem, the energy is not constant due to the presence of variations in the control
parameter α(t). The above optimal control problem models an experimenter trying to
achieve a certain value for an observable of a condensate (without loss of generality, this
value is set to zero) by manipulating the amplitude of an external field, e.g. a field induced
by a laser. The cost term models the cost of absorbing variations in total energy stored
in the condensate. Note that the underlying NLS implies the following expression for the
cost term involving γ1:
γ1
∫ T
0
(E˙(t))2 dt = γ1
∫ T
0
(α˙(t))2
(∫
Rd
V (x)|ψ(t, x)|2 dx
)2
dt.
The optimal control problem is well-posed with γ2 = 0 if the potential V (x) is strictly
bounded away from zero. On the other hand if this is not the case, it becomes necessary
to set γ2 > 0. Even in the case λ = 0, this is a bilinear control problem since the term
α(t)V (x)ψ(t, x) is linear in both the control and the wave function, making this optimal
control problem highly nonlinear. In our analysis of this optimal control problem, we
first show the existence of at least one minimizer (α∗, ψ∗) ∈ H1(0, T ) ×W (0, T ), where
W (0, T ) denotes an appropriate function space for the solutions to the NLS. We prove
this result by the direct method, i.e. we consider a minimizing sequence (αn, ψn) and
use boundedness of the functional J(ψn, αn) in order to obtain bounds on the sequence
in order to obtain a (weak) limit point (α∗, ψ∗). By going to the limit in the NLS, we
then show that (α∗, ψ∗) is itself a solution to the NLS and finally conclude by lower-
semicontinuity of J that (α∗, ψ∗) is indeed a minimizer.
In a next step, we characterize minimizers by deriving a system of equations that must
27
Introduction
be satisfied at critical points. This system can be formally derived using the Lagrangian
L(ψ, α, ϕ) = J(ψ, α)− 〈ϕ, P (ψ, α)〉L2tL2x ,
where P (ψ, α) = 0 denotes the NLS, i.e.
P (ψ, α) = i∂tψ +
1
2
∆ψ − U(x)ψ − λ|ψ|2σψ − α(t)V (x)ψ.
Formally, a minimum of J over (ψ, α) under the constraint P (ψ, α) = 0 is a minimum of
the unconstrained Lagrangian over (ψ, α, ϕ) and we expect
(
DψL(ψ, α, ϕ), DαL(ψ, α, ϕ), DϕL(ψ, α, ϕ)
)
= 0.
This is a system of three equations, the third one DϕL = 0 one being the NLS, and the
other two identifying a critical α and ϕ. Of course, there is not reason to expect any
solution to this critical system to be unique, corresponding to the lack of convexity of our
optimal control problem. The usual way to make this argument rigorous requires the use
of an implicit function theorem. However, due to the lack of regularity properties of the
Schro¨dinger operator, we could not make this approach work in the fully nonlinear case
λ > 0 and instead we derive the derivative of a reduced functional J(α) = J(ψ(α), α)
directly, where ψ(α) is the solution to the NLS with control α. In order to handle this
differentiability problem, we restrict ourselve to cases where σ ∈ N. The condition σ <
2/(d − 2) then implies d ≤ 3, which are of course the most relevant cases in physics.
Finally, we present some numerical experiments based on a Newton–type method in order
to illustrate our optimal control problem.
28
Part I
Quantitative uniform in time
hydrodynamic limits
29

Chapter 2
A quantitative perturbative
approach to hydrodynamic limits
The work in this part has been carried out in collaboration with Cle´ment Mouhot.
2.1 Introduction
We shall consider the problem of hydrodynamic limits for interacting particle systems
on a lattice. The problem is to show that under an appropriate scaling of time and
space, the local particle densities of a stochastic lattice gas converge to the solution
of a partial differential equation. The goal of this work is to provide a fairly general
framework allowing us to prove a hydrodynamic limit with an explicit uniform in time
rate of convergence. We will present our method using as an example the zero range
process for which a hydrodynamic limit is well-known and the limit equation is given by
a nonlinear diffusion equation. Our method is inspired by the work [67] on propagation
of chaos for the Boltzmann equation, see also [66] for an announcement and summary of
the work.
To make our notions precise, we need to introduce some notation. We consider a particle
process on the discrete torus
TdN = {1, . . . , N}
d
and consider particle configurations as elements in
XN := NT
d
N ,
31
Quantitative hydrodynamic limits
the state space for the zero range process. The lattice TdN can be thought of as a discrete
approximation of the d-dimensional Torus
Td = Rd/Zd
with periodic boundary conditions x + e ≡ x for all x ∈ Td and e ∈ Zd. Variables in the
discrete torus TdN are called microscopic and denoted by x, y, z, whereas variables in the
continuous torus Td are called macroscopic and denoted by u. In fact, we embed TdN in
Td via
TdN 3 x 7→
x
N ∈ T
d.
This embeds the microscopic variables x ∈ TdN to the macroscopic variables u ∈ T
d.
Hence the macroscopic distance between sites of the lattice is N−1. In general, we will
denote particle configurations in XN by the letters η or ξ. The interacting particle system
is given by a stochastic process and we let P (XN) be the set of probability (Radon)
measures over the state space. For any initial measure µN0 ∈ P (XN) we obtain a unique
measure µNt ∈ P (XN) describing the state of the process at a later time t. This also
yields a semigroup (SNt )t≥0 on P (XN), which is given by µ
N
t = S
N
t µ
N
0 for all t ≥ 0. The
semigroup SNt is a Feller-semigroup uniquely determined by its generator, see [54]. The
generator is a map GN : Cb(XN)→ Cb(XN) and satisfies
(2.1)
d
dt
〈µNt , f
N〉 = 〈µNt , G
NfN〉,
where we have denoted by 〈·, ·〉 the integral of a continuous function with respect to a
measure. Equivalently, this is the duality pairing between (Radon) measures and contin-
uous functions. Thus GN can also be thought of as the generator of the dual semigroup
on Cb(XN). Here we consider
(2.2) GNfN(η) = N2
∑
x,y∼x
g(η(x))
[
fN(ηx,y)− fN(η)
]
for each fN ∈ Cb(XN), where ηx,y is the configuration of the particle system after one
particle has jumped from site x to y and where y ∼ x whenever x and y are neighbours.
To be precise, ηx,y is given by
ηx,y(z) =



η(x)− 1 if z = x,
η(y) + 1 if z = y,
η(z) otherwise.
In order for the process to remain in the state space XN , we always demand g(0) = 0.
The jump rate g : N→ [0,∞) can be thought of as describing the interactions of particles
32
2.1. Introduction
occupying the same site. Since the jump rate on a given site only depends on the number
of particles at that particular site, this process is called zero range process. A special
case is the case of linear g, where the particles perform independent random walks on
the lattice. The factor N2 in the definition of the generator GN corresponds to a time
scale. Thus we consider a hydrodynamic limit under diffusive scaling, i.e. the microscopic
spatial variables scale with N and time with N2.
We will show that, under diffusive scaling, the zero range process is well approximated by
the solution ft : Td → [0,∞) to the filtration equation
(2.3) ∂tft(u) = ∆σ(ft(u)) t ∈ [0,∞), u ∈ Td.
We shall have to specify the space H of solutions to our limit partial differential equation.
Throughout this chapter, H will be a subspace of the space M+(Td) of positive Radon
measures on the torus. Recall that the space of Radon measures can be defined as the
dual space of continuous functions. Let us make precise the notion of convergence of the
particle process. Given a particle configuration η ∈ XN , the particle densities are given
by the empirical measure
(2.4) αNη :=
1
Nd
∑
x∈TdN
η(x)δ x
N
∈M+(Td).
Thus we have defined an embedding
αN : XN →M+(Td), η 7→ αNη ,
which allows us to compare solutions to the particle system with the solutions ft ∈ H ⊆
M+(Td) to the partial differential equation. Furthermore let ft be the solution to the
filtration equation given an initial density f0. The goal is to show that the empirical
measure (2.4) possesses an asymptotic density profile ft(·). By this we mean that for any
smooth function ϕ : Td → R, it holds that
(2.5) lim
N→∞
PµNt
(
|〈αNη , ϕ〉 − 〈ft, ϕ〉| > 
)
= 0
for all t ≥ 0 and  > 0. Furthermore we want to specify the rate of convergence explicitly.
Here PµN (A) denotes the probability corresponding to the (measurable) set A under the
probability measure µN ∈ P (XN). In other words,
PµN (A) =
∫
χA(η) dµ
N(η),
where χA denotes the characteristic function of the set A. In the following, we shall
denote the expectation of a measurable function fN with respect to a probability measure
33
Quantitative hydrodynamic limits
µN ∈ P (XN) by
EµN [f
N(η)] = 〈µN , fN〉 =
∫
fN(η) dµN(η).
A measure µN is called invariant (or equilibrium) measure, if
〈µN , GNfN〉 = 0 for all fN ∈ Cb(XN),
cf. equation (2.1). A convenient family of invariant measures is given by the grand-
canonical (or Gibbs) measures, i.e. the measures
(2.6) νNρ (η) =
∏
x∈TdN
σ(ρ)η(x)
g(η(x))! Z(σ(ρ))
,
where Z is the partition function of the zero range process, ρ ≥ 0, and
g(n)! := g(1)g(2) · · · g(n) with g(0)! := 1.
The partition function is defined as
(2.7) Z(ρ) =
∞∑
n=0
ρn
g(n)!
and the function σ(ρ) is chosen such that
〈νNρ , η(0)〉 = ρ.
We shall elaborate on the construction of σ in Section 2.4. Since the number of particles
is conserved and the process has no other conserved quantities, another important set of
invariant measure is given by the canonical measures
(2.8) νN,K(η) = νNρ
(
η
∣
∣
∑
x η(x) = K
)
,
which are the grand-canonical measures conditioned on hyperplanes of constant number
of particles. Note that this definition is independent of ρ > 0. Since the equilibrium νNρ
is made up of independent random variables, we expect the convergence (2.5) to hold if
we can show that the process is in equilibrium νNft(u) locally around u ∈ T
d with average
density ft(u).
The organization of this chapter is as follows. First we present some previous results in
Section 2.2. In Section 2.3, we present our method with the help of the particularly easy
case of independent random walks. Section 2.4 contains our main result, the hydrody-
34
2.2. Previous results
namic limit for the zero range process in one dimension with an explicit estimate on the
rate of convergence. Section 2.5 contains an argument, which is new in the context of
hydrodynamic limits, that allows us to prove convergence of the microscopic entropy if
the entropy converges initially. In Section 2.6 we prove an important ingredient in our
proof, the so-called replacement lemma. The replacement lemma is not new, see [36], but
we include its proof for the sake of completeness and in order to derive an explicit bound
on the rate of convergence in the replacement lemma. We also mention that our slightly
modified version of the replacement lemma shows convergence with respect to an L2–
norm instead of the usual L1–norm. Finally, in Section 2.7, we discuss a strategy, which
is work in progress, how to extend our result to the multi-dimensional case. Throughout
this chapter, any constant C should be understood to be generic, i.e. it can change from
line to line and only depends on the “general” parameters of the problem - this should be
clear from context.
2.2 Previous results
Using the notation introduced in Section 2.1, we now consider the general zero range
process on TdN given by generator (2.2). Let us make the following assumptions on the
rate function g : N→ [0,∞).
Assumption 1. (i) Non-degeneracy: Assume that g satisfies g(0) = 0 and g(n) > 0 for
all n > 0.
(ii) Lipschitz-property: We require that g is Lipschitz continuous with
0 ≤ |g(n+ 1)− g(n)| ≤ g∗ < +∞
for all n ∈ N.
(iii) Spectral gap: We also assume that there exist n0 > 0 and δ > 0 such that
g(n)− g(j) ≥ δ
for any j ∈ N and n ≥ j + n0.
(iv) Attractivity: Let the jump rate g be monotonously increasing, i.e.
g(n+ 1) ≥ g(n)
for all n ∈ N.
Remark 2.2.1. Let us comment on the different parts of Assumption 1: Part (i) is
essential to avoid degeneracies of the particle system. We need the spectral gap property
35
Quantitative hydrodynamic limits
(iii) in order to prove an explicit uniform in time rate of convergence, since it allows us
to quantify the local relaxation to equilibrium of the particle system as well as the global
convergence to equilibrium on the level of the limit equation. It implies in particular that
g(n) ≥ g0n with g0 > 0. The attractivity (iv) is important to obtain moment–bounds on
the particle system, see Subsection 2.6.1. The question of moment bounds is still an open
problem in its absence. Assumption (ii) is used at several points in the proof, but could
possibly be replaced using the uniform moment bounds originating from assumption (iv)
- however, it would affect our strategy to prove the regularity result in several dimensions,
see Section 2.7.
In the context of the zero range process with diffusive scaling, two very well-known meth-
ods of proving a hydrodynamic limit are the entropy method due to Guo, Papanicolaou,
and Varadhan [36], see Theorem 2.2.2, and the relative entropy method due to Yau [86],
see Theorem 2.2.3. For an extensive account of these methods in the context of zero range
process, see [47].
In order to proceed, we need one more definition. Let µ, ν ∈ P (XN) be two probability
measures. Then the relative entropy of µ relative to ν is defined as
(2.9) HN(µ|ν) =
∫
XN
log
(dµ
dν
)
dµ
whenever µ is absolutely continuous with respect to ν. The relative entropy is connected
to the Fisher information
(2.10) DN(µ|ν) =
∫
XN
√
dµ
dνG
N
√
dµ
dν dν.
The entropy method can be summarized in the following theorem. Note that we have not
taken great care to optimize the assumptions. The proofs under the assumptions given
below can be found in [47].
Theorem 2.2.2 (Guo, Papanicolaou, Varadhan). Assume (i) and (ii) of Assumption 1
as well as g(n) ≥ g0n for some g0 > 0 and let µN0 ∈ P (XN) and f0 ∈ L
∞(Td) such that
lim
N→∞
PµN0
(
|〈αNη , ϕ〉 − 〈f0, ϕ〉| > 
)
= 0,
for every continuous function ϕ ∈ C(Td) and every  > 0. Furthermore we assume that
the initial data satisfy the bounds
HN(µN0 |ν
N
ρ ) ≤ CN
d and
〈
µN0 ,
1
Nd
∑
x∈TdN
η(x)2
〉
≤ C
for some ρ > 0 and a constant C < +∞.
36
2.2. Previous results
Then, for every t ≥ 0, every continuous function ϕ ∈ C(Td), and every  > 0, it holds
that
lim
N→∞
PµNt
(
|〈αNη , ϕ〉 − 〈ft, ϕ〉| > 
)
= 0,
where ft is the unique weak solution to (2.3) and µNt solves (2.1) with Cauchy datum µ
N
0 .
Thus the entropy method yields propagation of the hydrodynamic profile. The relative
entropy method by Yau, on the other hand, concerns the conservation of a stronger notion.
In analogy to (2.6), we define a local Gibbs measure with macroscopic profile ft ∈ C(Td)
by
(2.11) νNft(·)(η) =
∏
x∈TdN
σ(ft( xN ))
η(x)
g(η(x))! Z(σ(ft( xN )))
.
This measure has the property that it is locally (in infinitesimal macroscopic neighbour-
hoods where ft is constant) in equilibrium with a macroscopic non-equilibrium profile ft
as N →∞. The relative entropy method then yields the following theorem.
Theorem 2.2.3 (Yau). Assume (i) and (ii) of Assumption 1 as well as that the partition
function Z(·) is finite on all [0,∞), e.g. g(n) ≥ g0n for some g0 > 0. Furthermore,
assume that the solution ft to (2.3) satisfies ft ∈ C2(Td) and let µNt ∈ P (XN) solve (2.1).
Finally assume that initially at t = 0, the relative entropy HN(µN0 |ν
N
f0(·)
) vanishes in the
limit, i.e.
lim
N→∞
1
Nd
HN
(
µN0 |ν
N
f0(·)
)
= 0.
Then it holds that
(2.12) lim
N→∞
1
Nd
HN
(
µNt |ν
N
ft(·)
)
= 0
for every t ≥ 0.
Note that the convergence of the relative entropy (2.12) implies that µNt has profile ft,
i.e.
lim
N→∞
PµNt
(
|〈αNη , ϕ〉 − 〈ϕ, ft〉| > 
)
= 0.
Thus the convergence of the relative entropy can be thought of as a stronger notion of
the hydrodynamic limit. Yau’s relative entropy method shows that this stronger notion
is conserved by the evolution.
Remark 2.2.4. It appears that using a quantitative replacement lemma, see Section 2.6,
this result can be translated to a quantitative result of the form
HN
(
µNt |ν
N
ft(·)
)
≤ Ceγ
−1tHN
(
µN0 |ν
N
f0(·)
)
+ tr(N),
37
Quantitative hydrodynamic limits
where limN→∞ r(N) = 0 if γ is sufficiently small, and r(N) can be made explicit (al-
though, to our knowledge, such a result has never been published). Thus it seems that a
quantitative estimate on the rate of convergence is available in the stronger form of the
hydrodynamic limit given by the convergence of the entropy relative to the local Gibbs
state. However, this convergence is not uniform in time, since γ might be very small.
Therefore even if one manages to prove exponential decay in time of the relative entropy
HN
(
µNt |ν
N
ft(·)
)
≤ Ce−λt, e.g. by employing a logarithmic Sobolev inequality, it is still not
possible to conclude uniform in time convergence if λ < γ−1. In the context of the zero
range process, the following logarithmic Sobolev inequality holds [27, 59]:
(2.13) HN(µ|νN,K) ≤ CN2DN(µ|νN,K)
uniformly in N , K, and µ ∈ P (XN), where we recall that νN,K denotes the canoni-
cal measure (2.8). Logarithmic Sobolev inequalities are very effective tools to describe
concentration of measure and have been employed widely starting with the works [5, 33].
For a related model, the Ginzburg-Landau model with Kawasaki dynamics, there exists an
additional method due to Grunewald, Otto, Villani, and Westdickenberg [34], who prove
a logarithmic Sobolev inequality and hydrodynamic limit based on a coarse-graining of
the state-space. In principle, it should be possible to extend their method to obtain a
uniform rate of convergence. On the other hand, it is not clear how to extend the method
to the zero range process and how to obtain uniform-in-time convergence.
2.3 A toy model: independent random walks
In order to demonstrate our method, let us consider the especially simple case where
g(n) = n. Then all the particles perform random walks independently of each other. The
invariant measures νNρ are now given by the Poisson distribution
νNρ (η) =
∏
x∈TdN
e−ρ
ρη(x)
η(x)!
.
Note that 〈νNρ , η(x)〉 = ρ and hence σ(ρ) = ρ in (2.6) in this case. We want to show
that in the sense of Theorem 2.2.2, as the number N of sites in the lattice TdN approaches
infinity, the particle system is approximated by the heat equation
(2.14) ∂tft = ∆ft.
38
2.3. A toy model: independent random walks
Here we consider the heat equation in the space of positive Radon measures
H = M+(Td).
It is clear that we have a well-established theory for strong solutions for the heat equation
with continuous initial data ω ∈ C(Td). Denote the corresponding semigroup on C(Td)
by S∞t ω. We define solutions in H via
(2.15) 〈S∞t f, ω〉H,C(Td) = 〈f, S
∞
t ω〉H,C(Td)
for all f ∈ H,ω ∈ C(Td). Note that indeed S∞t f stays a positive measure by the maximum
principle. Thus the solution ft to the heat equation (2.14) with initial datum f ∈ H is
given by ft = S∞t f . Let us denote by C
k
b (R) the space of uniformly bounded and k times
continuously differentiable functions on R with uniformly bounded derivatives. The main
result of this section is the following theorem, detailing a hydrodynamic limit with explicit
and uniform-in-time rate of convergence.
Theorem 2.3.1 (Hydrodynamic limit for independent random walks). Let F ∈ C2b (R),
ϕ ∈ C3(Td), and M1 be given. There exists a constant C < +∞ depending only on the
dimension d, such that for all N ∈ N, f0 ∈ H, fN0 ∈ P (XN) such that the average density
is bounded, i.e.
〈
µN0 ,
1
Nd
∑
x∈TdN
η(x)
〉
≤M1,
it holds that
(2.16)
〈
µNt , F
(
〈αNη , ϕ〉
)
− F
(
〈ft, ϕ〉
)〉
≤
C
N
M1‖F
′‖C1
(
‖∇ϕ‖C2 + ‖S
∞
1 ∇ϕ‖Hn
)
+ ‖F ′‖L∞‖ϕ‖C3 sup
ω∈C3(Td)
‖ω‖C3≤1
〈
fN0 , 〈α
N
η − f0, ω〉
〉
uniformly for all t ≥ 0. Here n denotes the smallest integer greater than 2 + d/2, ft is
given by the solution to the heat equation ∂tft = ∆ft, and µNt is given by the evolution of
the particle process, i.e. a system of independent random walks.
Theorem 2.3.1 yields convergence to the hydrodynamic limit under the condition that the
initial data are compatible. If f0 is continuous, it is straightforward to construct an initial
particle distribution µN0 for which the initial convergence holds. Indeed let f0 ∈ C(T
d),
f0 ≥ 0, be given. Then we consider the product measure νNf0(·) introduced in (2.11), i.e.
νNf0(·)(η) =
∏
x∈TdN
e−f0(
x
N )
f0( xN )
η(x)
η(x)!
39
Quantitative hydrodynamic limits
for all x ∈ TdN . Under the law µ
N
0 := ν
N
f0(·)
, the term 〈αNη , ω〉 converges in mean to 〈f0, ω〉
and one can show that
(2.17) sup
ω∈C3(Td)
‖ω‖C3≤1
〈
fN0 , 〈α
N
η − f0, ω〉
〉
≤
C
√
N
.
This can be deduced from an application of the law of large numbers and the central limit
theorem, also see the proof of Corollary 2.4.9. Under µN0 , the average density is bounded
by ‖f0‖L∞ . Hence we arrive at the following corollary.
Corollary 2.3.2. Let F ∈ C2b (R), ϕ ∈ C
3(Td), and f0 ∈ C(Td) be given. Then there
exists a constant C < +∞ and a fN0 ∈ P (XN) for all N ∈ N, such that
〈
µNt , F
(
〈αNη , ϕ〉
)
− F
(
〈ft, ϕ〉
)〉
≤
C
√
N
for all N ∈ N, where ft is given by the solution to the heat equation ∂tft = ∆ft and µNt
is given by a system of independent random walks.
In other words, αNη ⇀
∗ ft in distribution (in law) as N →∞ where the symbol ⇀∗ denotes
weak-* convergence for measures in P (XN).
Let us make several comments.
Remarks 2.3.3. (1) The rate of convergence O(1/
√
N) is the optimal rate appearing in
the law of large numbers for sums of independent random variables, similarly to the esti-
mate (2.17). Hence we see that the largest contribution to the error in the hydrodynamic
limit for independent random walks comes from the approximation of the initial datum
f0 by the initial particle distribution fN0 , since this error is typically O(1/
√
N).
(2) As we have seen, the restriction that 〈µN0 , N
−d
∑
η(x)〉 ≤ M1 in Theorem 2.3.1 is
hardly any restriction and usually follows from the convergence of the initial data.
(3) The function
Ψ ∈ Cb(H) given by Ψ(f) = F (〈f, ϕ〉) for all f ∈ H
thus satisfies 〈µNt ,Ψ(α
N
η )〉 → Ψ(ft) as N →∞ under the assumptions of Corollary 2.3.2.
(4) Here we have the convergence of the empirical measures in distribution. This sense of
convergence is the same as found in the literature on hydrodynamic limits for interacting
particle systems, but it is different in spirit from the convergence result found in [67],
from where our method of proof was inspired. In contrast to ours, the result in [67]
pertains convergence of the marginals of µNt . An analogous result in our setting would be
the convergence of marginals of µNt to the corresponding marginals of ν
N
ft(·)
, where νNft(·)
denotes as above the local Gibbs measure (2.11). In [47], a result of this kind is called
40
2.3. A toy model: independent random walks
strong conservation of local equilibrium. Indeed it seems likely that a result like this could
be deduced using a similar approach.
(5) Convergence in distribution of the random variable JN := 〈αNη , ϕ〉 to the deterministic
result J := 〈ft, ϕ〉 implies convergence in probability, cf. Theorem 2.2.2, as follows. Let
 > 0 be arbitrary and note that
PµNt (|JN − J | > ) ≤ PµNt (JN > J + ) + PµNt (JN < J − )
≤ EµNt [F(JN)] + EµNt [F˜(JN)]
where F and F˜ are smooth approximations from above of the indicator functions of
[J + ,+∞) and (−∞, J − ], respectively, such that F(J) = 0 = F˜(J). Under the
assumptions of Corollary 2.3.2, the right hand side converges as N →∞ with an explicit,
-dependent rate.
(6) Consider the embedding
piNP : P (XN)→ P (H)
given by 〈piNP f
N ,Φ〉 = 〈fN ,Φ(αNη )〉 for all Φ ∈ Cb(H). Returning to the map Ψ ∈ Cb(H),
defined in (3), let us mention that the convergence
〈µNt ,Ψ(α
N
η )〉 −Ψ(ft)→ 0
can also be rewritten as
〈piNp (µ
N
t ),Ψ〉 − 〈δft ,Ψ〉 → 0,
or, equivalently, piNP (µ
N
t ) ⇀
∗ δft in P (H), at least insofar as Ψ given in (3) can represent
all of Cb(H).
(7) The choice of a norm for the convergence of the initial data, here given by the dual of
the C3–norm, is rather flexible. Since the heat equation is a contraction in many spaces,
e.g. in Ck or Hk, for k ∈ N, we could use (the dual space of) any of these spaces, provided
ϕ is regular enough and the Dirac distribution δ lies in the dual space. Later, in Section
2.4, we shall use the H−1–norm, dual to the H1–norm, to measure the convergence of the
initial data. Here this is not possible since the Dirac delta is not an element of H−1(Td)
if d > 1.
Before proceeding with the proof of Theorem 2.3.1, let us collect some semigroups related
to the evolution of the particle system and the limit equation. We cannot compare the
semigroups of the particle system on P (XN) and of the limit equation on H directly.
Instead we will compare them on the level of observables by considering dual spaces.
Particle system: We already defined the semigroup SNt on P (XN). Let T
N
t denote the
41
Quantitative hydrodynamic limits
semigroup on Cb(XN) that is dual to SNt , i.e. the semigroup given by
〈µN , TNt f
N〉 = 〈SNt µ
N , fN〉
for all µN ∈ P (XN), fN ∈ Cb(XN). Thus the generator of TNt in Cb(XN) is given by G
N
in (2.2) with g = idN.
Limit equation: Recall the definition (2.15) of the limit semigroup S∞t on H. Next we
define a pullback semigroup T∞t on Cb(H) corresponding to the solution of the limit
partial differential equation S∞t via
T∞t Ψ(f) = Ψ(S
∞
t f)
for all Ψ ∈ Cb(H), f ∈ H, and t ≥ 0.
Thus we get a collection of semigroups
SNt : P (XN)→ P (XN) with dual T
N
t : Cb(XN)→ Cb(XN),
S∞t : H → H with pullback T
∞
t : Cb(H)→ Cb(H).
Note that for a general nonlinear limit equation, the operator S∞t will not be linear but
the semigroup T∞t will be linear.
Instead of comparing the semigroup SNt on P (XN) and the semigroup S
∞
t on H, we shall
compare the two semigroups TNt on Cb(XN) and T
∞
t on Cb(H). To this end let us define
an embedding
(2.18) piN : Cb(H)→ Cb(XN), Ψ 7→ pi
NΨ = (η 7→ Ψ(αNη )).
We shall need to identify the time-derivative of T∞t for a special class of functions in
Cb(H) which are of special importance for our version of the hydrodynamic limit, see
Theorem 2.3.1. These functions are all functions Ψ ∈ Cb(H) such that
(2.19) Ψ(f) = F (〈f, ϕ〉)
for some F ∈ C1b (R) and ϕ ∈ C
2(Td), c.f. Remark 2.3.3 (3). Here we speak of time-
derivative instead of generator, because even though it is possible to prove that T∞t
induces a C0–semigroup of contractions on a suitable subspace of Cb(H), we shall not do
so here.
Before we prove the hydrodynamic limit, Theorem 2.3.1, let us state and prove two lemmas
on the discrete particle system and the limit equation. Both lemmas will be used in the
proof of the hydrodynamic limit. The first lemma concerns the stability of the limit
42
2.3. A toy model: independent random walks
partial differential equation.
Lemma 2.3.4 (Stability). Let F ∈ C1b (R) and ϕ ∈ C
3(Td), and define Ψ as in equation
(2.19).
(i) Then for all t ≥ 0, there exists ϕt ∈ C3(Td) such that
(2.20) T∞t Ψ(f) = F (〈f, ϕt〉)
for all f ∈ H. Furthermore ϕt satisfies
(2.21) ‖ϕt‖C3 ≤ ‖ϕ‖C3 .
(ii) For any t ≥ 0, it holds
|T∞t Ψ(f2)− T
∞
t Ψ(f1)| ≤ ‖F
′‖L∞‖ϕ‖C3 sup
ω∈C3(Td)
‖ω‖C3≤1
〈f2 − f1, ω〉
for all f1, f2 ∈ H.
Proof. (i) Since S∞t was constructed on H by duality, it holds that 〈S
∞
t f, ϕ〉 = 〈f, S
∞
t ϕ〉.
Hence the choice ϕt = S∞t ϕ yields (2.20). Since ϕ ∈ C
3(Td), the function Dsϕt, where
s is any multi-index such that |s| ≤ 3, solves the heat equation with initial datum Dsϕ.
Now the maximum principle yields ϕt ∈ C3(Td) with estimate (2.21).
(ii) Using the notation and results of part (i) of this lemma, it is not difficult to see that
|Ψ(S∞t f2)−Ψ(S
∞
t f1)| = |F (〈f2, ϕt〉)− F (〈f1, ϕt〉)| ≤ ‖F
′‖L∞
∣
∣〈f2 − f1, ϕt〉
∣
∣,
which implies the result.
The stability result yields that the form (2.19) is preserved by the flow T∞t , since
T∞t Ψ(f) = F
(
〈f, S∞t ϕ〉
)
〈f, S∞t ϕ〉,
where S∞t ϕ ∈ C
3(Td). For such functions Ψ ∈ Cb(H), its time-derivative is given in the
next lemma.
Lemma 2.3.5. Let F ∈ C1b (R) and ϕ ∈ C
2(Td), and define Ψ as in equation (2.19).
Then its derivative G∞T∞t at time t ≥ 0 is given by
G∞T∞t Ψ(f) :=
d
dt
T∞t Ψ(f) = F
′ (〈f, ϕt〉) 〈f,∆ϕt〉
for all f ∈ H, where ϕt = S∞t ϕ. Note that at time t = 0, the derivative is to be understood
as the right-hand derivative (as t↘ 0) only.
43
Quantitative hydrodynamic limits
Proof. By definition of weak solutions to the heat equation, see (2.15), it holds that
d
dt
〈S∞t f, ϕ〉 = 〈S
∞
t f,∆ϕ〉.
The chain rule yields
d
dt
T∞t Ψ(f) =
d
dt
Ψ(S∞t f) = F
′(〈S∞t f, ϕ〉)〈S
∞
t f,∆ϕ〉.
Hence
d
dt
T∞t Ψ(f) = F
′(〈f, ϕt〉)〈f,∆ϕt〉
by the stability result, Lemma 2.3.4.
Remark 2.3.6. (1) Note that we can write formally
F ′ (〈f, ϕ〉) 〈f˜ , ϕ〉 = DΨ(f)(f˜),
where DΨ(f) : H → R denotes the derivative of Ψ : H → R with respect to f ∈ H.
Indeed, it holds that
|Ψ(f2)−Ψ(f1)−DΨ(f1)(f2 − f1)| ≤ ‖F
′′‖L∞(R)|〈f2 − f1, ϕ〉|
2,
which can be understood as differentiability in H, if H is equipped with weak-* conver-
gence, cf. Section 2.4.
(2) Note that it is possible to prove that the semigroup T∞t is a C0–semigroup of con-
tractions with generator G∞ on an appropriate subspace of Cb(H), but we just need its
time-derivative as obtained in Lemma 2.3.5 for the particular maps Ψ given by (2.19),
since the form of Ψ is conserved in the special case of random walks. In order to keep the
function spaces involved simple, we shall not prove the C0–semigroup property.
In order to prove a uniform in time hydrodynamic limit, we need some results on the
decay of solutions to the heat equation in the spirit of a spectral gap. Let H˙n(Td) denote
the homogeneous Sobolev space of degree n, i.e. the space of functions such that
‖f‖2H˙n :=
∑
|s|=n
∫
Td
|Dsf |2 du
is finite.
Lemma 2.3.7 (Spectral gap). For all ϕ ∈ C(Td) and t > 0, the solution S∞t ϕ to the heat
equation is in C∞(Td). Furthermore there exist positive, finite constants c and C such
that
‖∇S∞t+1ϕ‖C2 ≤ C‖∇S
∞
t+1ϕ‖H˙n ≤ C‖∇S
∞
1 ϕ‖H˙ne
−t
44
2.3. A toy model: independent random walks
for all t ≥ 0 and n > 2 + d/2.
Proof. The regularization property of the heat equation is classical. Hence h := ∇S∞1 ϕ ∈
H˙n(Td) for n > 2 + d/2 and in Fourier space it holds that
‖∇S∞t+1ϕ‖
2
H˙n =
∑
ζ∈Zd
|hˆ(ζ)|2|ζ|2ne−2ζ
2t ≤
∑
ζ∈Zd
|hˆ(ζ)|2|ζ|2ne−2t ≤ ‖∇S∞1 ϕ‖
2
H˙ne
−2t.
Furthermore a standard Sobolev embedding and Poincare´’s inequality yield
‖∇S∞t+1ϕ‖C2 ≤ C‖∇S
∞
t+1ϕ‖H˙n
since n > 2 + d/2 and the integral of gradient over the torus vanishes.
The next lemma yields closeness of the time-derivatives GN and G∞.
Lemma 2.3.8 (Consistency). Let F ∈ C2b (R) and ϕ ∈ C
3(Td), and define Ψ as in
equation (2.19). Furthermore assume that the average density is bounded, i.e.
〈
µN0 , N
−d∑
xη(x)
〉
≤M1.
Then there exists a constant C < +∞ depending only on the dimension d such that
∣
∣
〈
µNt ,
(
GNpiN − piNG∞
)
Ψ
〉∣
∣ ≤ CM1‖∇ϕ‖C2‖F
′‖C1
1
N
for all t ≥ 0.
Proof. Let I :=
〈
µNt , G
NpiNΨ
〉
denote one of the two terms we want to estimate. In view
of the expression for the particle generator (2.2) with g = idN, it holds that
I =
〈
µNt , N
2
∑
x,y∼x
η(x){(piNΨ)(ηx,y)− (piNΨ)(η)}
〉
.
By definition (2.18) of piN , we obtain that
I =
〈
µNt , N
2
∑
x,y∼x
η(x){Ψ(αNηx,y)−Ψ(α
N
η )}
〉
.
Let R1 be the error term given by
R1 =
〈
µNt , N
2
∑
x,y∼x
η(x)
(
Ψ(αNηx,y)−Ψ(α
N
η )−DΨ(α
N
η )(α
N
ηx,y − α
N
η )
)
〉
,
45
Quantitative hydrodynamic limits
where DΨ is defined in Remark 2.3.6. Then it holds that
I = R1 +
〈
µNt , N
2
∑
x,y∼x
η(x)F ′
(
〈αNη , ϕ〉
)
〈αNηx,y − α
N
η , ϕ〉
〉
.
Definition (2.4) of the empirical measure then yields
I = R1 +
〈
µNt , F
′
(
〈αNη , ϕ〉
)
N2−d
∑
x,y∼x
η(x)〈δ y
N
− δ x
N
, ϕ〉
〉
= R1 +
〈
µNt , F
′
(
〈αNη , ϕ〉
)
N−d
∑
x∈TdN
η(x)∆Nϕ( xN )
〉
where
(2.22) ∆Nf(x) = N
2
∑
e∈Zd:|e|=1
(f(x+ eN )− f(x))
denotes the discrete Laplacian. Replacing the discrete Laplacian with its continuous
version yields that
I = R1 + R2 +
〈
µNt , F
′
(
〈αNη , ϕ〉
)
〈αNη ,∆ϕ〉
〉
with an error term
R2 =
〈
µNt , F
′
(
〈αNη , ϕ〉
)
〈αNη ,∆Nϕ−∆ϕ〉
〉
.
Hence the expression for G∞, Lemma 2.3.5, yields
I =
〈
µNt , G
NpiNΨ
〉
= R1 + R2 +
〈
µNt , F
′
(
〈αNη , ϕ〉
)
〈αNη ,∆ϕ〉
〉
= R1 + R2 +
〈
µNt , pi
NG∞Ψ(αNη )
〉
.
In order to finish the proof of Lemma 2.3.8, we just need to bound the error terms R1
and R2.
First we bound R1. Setting
G(τ) = Ψ
(
αNη + τ(α
N
ηx,y − α
N
η ))
)
yields
G′(τ) = F ′
(
〈αNη + τ(α
N
ηx,y − α
N
η ), ϕ〉
)
〈αNηx,y − α
N
η , ϕ〉
and
G′′(τ) = F ′′
(
〈αNη + τ(α
N
ηx,y − α
N
η ), ϕ〉
)
〈αNηx,y − α
N
η , ϕ〉
2.
46
2.3. A toy model: independent random walks
The mean value theorem yields
G(1)−G(0)−G′(0) = 1/2G′′(ξ)
for some ξ ∈ (0, 1). Consequently |R1| is bounded by
|R1| ≤
1
2
‖F ′′‖L∞
〈
µNt , N
2
∑
x∼y
η(x)
〈
αNηx,y − α
N
η , ϕ
〉2
〉
.
For each fixed x ∈ TdN it holds that
N2
∑
y s.t.y∼x
〈αNηx,y − α
N
η , ϕ〉
2 =
N2
N2d
∑
y s.t.y∼x
(
ϕ( yN )− ϕ(
x
N )
)2
≤
2d
N2d
‖∇ϕ‖2L∞ ,
where the factor 2d stems from the number of neighbours y of x. Hence the error is
bounded by
R1 ≤
2d−1
Nd
‖F ′′‖L∞ ‖∇ϕ‖L∞
〈
µNt ,
1
Nd
∑
x∈TdN
η(x)
〉
.
Since the number of particles is conserved under the evolution, the last of these factors
yields
〈
µNt ,
1
Nd
∑
x∈TdN
η(x)
〉
=
〈
µN0 ,
1
Nd
∑
x∈TdN
η(x)
〉
≤M1,
and hence |R1| ≤ CM1‖F ′‖C1‖∇ϕ‖L∞N−1. Finally, a Taylor expansion yields
‖∆Nϕ−∆ϕ‖L∞ ≤ ‖∆ϕ‖C1
1
N
and therefore |R2| ≤ CM1‖F ′‖L∞‖∆ϕ‖C1N−1.
Using the Lemmas 2.3.4 to 2.3.8, we can prove the hydrodynamic limit.
Proof of Theorem 2.3.1. Let Ψ be as defined in equation (2.19). Then the term to be
estimated is
〈
µNt , F
(
〈αNη , ϕ〉
)〉
− F (〈ft, ϕ〉) =
〈
SNt µ
N
0 ,Ψ
(
αNη
)
−Ψ
(
S∞t f0
)〉
.
Consequently the definitions of TNt and T
∞
t yield
〈
SNt µ
N
0 ,Ψ
(
αNη
)
−Ψ (S∞t f0)
〉
=
〈
µN0 , T
N
t Ψ
(
αNη
)
− T∞t Ψ(f0)
〉
.
Note here that TNt acts on Ψ(α
N
η ) through η. Recalling the definition (2.18) of pi
N , we
need to bound the term ∣
∣
∣
〈
µN0 , T
N
t pi
NΨ− T∞t Ψ(f0)
〉∣∣
∣.
47
Quantitative hydrodynamic limits
The triangle inequality and (2.18) yield
∣
∣
∣
〈
µN0 , T
N
t pi
NΨ− T∞t Ψ(f0)
〉∣∣
∣
≤
∣
∣
〈
µN0 , T
N
t (pi
NΨ)− piN(T∞t Ψ)
〉∣
∣+
∣
∣
〈
µN0 , (T
∞
t Ψ)
(
αNη
)
− (T∞t Ψ)(f0)
〉∣
∣
=: T1 + T2.
Since Ψ as defined in (2.19) satisfies
d
dt
T∞t Ψ(f) = G
∞T∞t Ψ(f),
it follows
d
ds
(
TNs pi
NT∞t−s(η)
)
= TNs G
NpiNT∞t−sΨ(η)− T
N
s pi
NG∞T∞t−sΨ(η).
Consequently, it holds that
T1 =
∣
∣
〈
µN0 , T
N
t pi
NΨ− piNT∞t Ψ
〉∣
∣
≤
∫ t
0
∣
∣〈SNs µ
N
0 , (G
NpiN − piNG∞)(T∞t−sΨ)〉
∣
∣ ds.
Note that T∞t−sΨ is of the form (2.19) with F and ϕt−s as given in Lemma 2.3.4 on stability.
Hence Lemma 2.3.8, which regards consistency, yields
T1 ≤
CM1
N
‖F ′‖C1
∫ t
0
‖∇ϕt−s‖C2 ds.
If we split up the above integral into the contributions t ≤ 1 and t > 1, Lemma 2.3.7
yields
T1 ≤
CM1
N
‖F ′‖C1
(
‖∇ϕ‖C2 + ‖S
∞
1 ∇ϕ‖Hn
)
.
For the second term T2, Lemma 2.3.4, (ii) yields
T2 ≤ ‖F ′‖C1‖ϕ‖C3 sup
ω∈C3(Td)
‖ω‖C3≤1
〈
µN0 ,
∣
∣〈αNη − f0, ω〉
∣
∣
〉
,
which completes the proof of the hydrodynamic limit.
Remark. The term T1 is a measure for the difference between the discrete and the con-
tinuous semigroups along the empirical measure, whereas T2 measures the evolution under
the limit equation of the distance between the initial empirical measure and the initial
macroscopic datum.
This is reminiscent of the results in numerical analysis, where the convergence of a nu-
merical scheme usually requires a consistency and a stability estimate.
48
2.4. Hydrodynamic limit for zero range processes
2.4 Hydrodynamic limit for zero range processes
In this section we shall apply the method outlined in Section 2.3 using the example of
independent random walks to prove a hydrodynamic limit for the zero range process with
an explicit bound on the rate of convergence. Throughout we require Assumption 1 to
hold. Furthermore, in this section we demand d = 1. This restriction on the dimension of
the problem is necessary only for the propagation of the regularity, Lemma 2.4.13. In or-
der to emphasize the generality of our approach, we shall still denote the dimension by d,
and understand d = 1. We shall see that indeed Lemma 2.4.13 is the only result where we
explicitly need d = 1 - assuming a corresponding result in higher dimensions, our method
implies the hydrodynamic limit with an explicit, uniform-in-time rate of convergence in
d dimensions. In Section 2.7, we shall discuss work in progress on how to achieve this.
We shall show that in the sense of Theorem 2.3.1, as the number N of sites in the lattice
TdN approaches infinity, the empirical measure converges to the solution of the limit partial
differential equation, specifically the filtration equation
∂tft = ∆σ(ft)
for the nonlinearity σ : [0,∞)→ [0,∞) appearing in (2.6). The nonlinearity is explicitly
given as follows: let Z : [0, λ∗) → R be the partition function of the zero range process
given by (2.7), with λ∗ denoting the radius of convergence of Z(·). Assumption 1 (iii) in
fact yields λ∗ = +∞, since the assumption implies g(n) ≥ δ˜n for some δ˜ ≤ δ/n0.
The density function as a function of the fugacity λ is given by
(2.23) R(λ) = λ∂λ log(Z(λ)) =
1
Z(λ)
∑
n≥0
nλn
g(n)!
.
This is a smooth function R : [0,∞) → R. It is not hard to prove, see [47], that R
strictly monotonously increasing with limλ→∞R(λ) = ∞. Then σ : [0,∞) → [0,∞) is
well-defined as its inverse function. Thus νNρ , as defined in (2.6), is an invariant and
translation-invariant product measure with density
〈νNρ , η(x)〉 = ρ.
Furthermore its average jump rate satisfies
〈νNρ , g(η(x))〉 = σ(ρ).
The Lipschitz continuity of the rate function implies that σ(ρ) is also Lipschitz continuous
with constant g∗, see [47]. The second assumption implies that g(n) ≥ δ˜n and hence
49
Quantitative hydrodynamic limits
infρ σ(ρ)/ρ > 0. Therefore it is impossible to obtain a porous medium equation, e.g.
σ(ρ) = ρm, m > 1, in this limit. Indeed Assumption 1 yields
Lemma 2.4.1. It holds that
(2.24) 0 < inf
ρ≥0
σ′(ρ) ≤ sup
ρ≥0
σ′(ρ) < +∞.
For the higher derivatives of order j ≥ 2, there exist constants Cj < +∞ such that
(2.25) σ(j)(ρ) ≤ Cj(1 + ρ
j−1)
for all ρ ≥ 0 and j ≥ 2.
Proof. The upper bound in (2.24) is well-known and can be proved by coupling two
measures νNρ and ν
N
ρ˜ , cf. [47]. We obtain the lower bound corresponding to ellipticity of
(2.3) by a formula expressing σ′ through the variation of the number of particles, see [52].
Let us start by proving (2.25) in the case j = 2. This second order bound can be seen
from
(2.26) σ′′(ρ) = −
R′′(σ(ρ))σ′(ρ)
R′(σ(ρ))2
= −R′′(σ(ρ))σ′(ρ)3,
since σ and R are inverse to each other, i.e. R(σ(ρ)) = ρ. Recall that
R(λ) = λ∂λ logZ(λ).
Assumption 1 yields that
δj ≤ Z(j)(λ)/Z(λ) ≤ (g∗)j
is bounded for all j–th order derivatives with j ≥ 0. Therefore it holds that
dj
dλj
R(λ) = λ
dj
dλj
logZ(λ) + jλ
dj−1
dλj−1
logZ(λ) ≤ C(1 + λ)
and consequently setting λ = σ(ρ) ≤ g∗ρ yields σ′′(ρ) ≤ C(1 + ρ). Iterating, we see that
with each derivative of (2.26), we pick up another power of ρ.
Remark 2.4.2. The rather naive estimates on the higher derivatives can be significantly
improved using the estimates found in [52].
Since the limit partial differential equation does not allow measure-valued weak solutions,
we consider solutions in
H := L∞(Td).
Note that in particular H ⊂ L2(Td). We shall also work a lot in the weak space H−1(Td).
Let us recall its definition and a few basic facts in the following remark.
50
2.4. Hydrodynamic limit for zero range processes
Remark 2.4.3 (The weak space H−1(Td)). Set
H1(Td) :=
{
f ∈ H1(Td) |
∫
Td f(u) du = 0
}
and let H−1(Td) denote its dual space, i.e. the space of linear bounded maps H1(Td)→ R.
It follows that
−∆ : H1(Td)→ H−1(Td), f 7→ −∆f
is an isomorphism, where (−∆)−1f corresponds to the weak solution to the Poisson equa-
tion on Td, i.e.
〈∇(−∆)−1f,∇f˜〉 = 〈f, f˜〉
for all f˜ ∈ H1(Td), f ∈ H−1(Td). Thus we set
‖f‖2H−1 := 〈f, (−∆)
−1f〉
for all f ∈ H−1(Td). In Fourier space, this norm is given by
‖f‖2H−1 =
∑
ζ∈Zd\{0}
1
|ζ|2
|fˆ(ζ)|2.
This norm is indeed equivalent to the usual H−1–operator norm, e.g. it holds that
(2.27) 〈f˜ , f〉L2 ≤ ‖∇f‖L2‖f˜‖H−1
for all f ∈ H1(Td) and f˜ ∈ L2(Td). A slightly complicating factor is the fact that the
H−1–norm vanishes for constants, i.e. members of H−1(Td) are only uniquely identified
up to constants. It directly follows that (2.27) also holds for all f ∈ H1(Td) as soon as
f˜ ∈ L2(Td) satisfies
∫
Td f˜(u) du = 0. It also holds that
‖f‖H−1 ≤ C‖f‖L2
for all f ∈ H−1(Td), which can be seen as a variant of the Poincare´ inequality.
In the almost linear case, when (2.24) holds, the theory of weak solutions to equation
(2.3) is stated in the following lemma.
Lemma 2.4.4 (Weak solutions to the filtration equation). For every f0 ∈ H, the filtration
equation (2.3) possesses a unique weak solution ft ∈ H, t ∈ [0,∞), in the sense that
∫ ∞
0
∫
Td
(
ft(u)∂tω(t, u) + σ(ft(u))∆ω(t, u)
)
dudt+
∫
Td
f0(u)ω(0, u) du = 0
for all ω ∈ C1([0,∞);C2(Td)) with compact support in [0,∞) × Td. The solution f =
51
Quantitative hydrodynamic limits
(ft)t∈[0,∞) also satisfies
f ∈ L2(0,∞;H1(Td)) ∩H1(0,∞;H−1(Td)) ⊂ C([0,∞);L2(Td)).
In particular
(2.28)
d
dt
〈ft, ϕ〉 = 〈σ(ft),∆ϕ〉
for all t ≥ 0 and all ϕ ∈ C2(Td).
Proof. Starting from smooth solutions in C∞(Td), cf. Ladyzhenskaya [51], it is classical
to construct a weak solutions, see for example [83] or [47, Appendix 2]. The maximum
principle shows that the semigroup S∞t conserves the L
∞(Td)–norm, i.e.
‖S∞t f0‖L∞ ≤ ‖f0‖L∞ .
Therefore it holds that S∞t : H → H. The solution is unique, cf. the stability estimates
of Lemma 2.4.12. Furthermore it holds that
d
dt
‖ft‖
2
L2 = −2
∫
Td
σ′(ft)|∇ft|
2 du,
whence S∞t conserves the L
2(Td)–norm and
∫ T
0
∫
Td
|∇ft|
2 dudt ≤ C
∫ T
0
∫
Td
σ′(ft)|∇ft|
2 dudt ≤ C‖f0‖
2
L2 .
is bounded. It follows that f = (ft)t≥0 ∈ L2(0,∞;H1(Td)). The filtration equation
∂tft = ∆σ(ft) consequently yields (∂tft)t≥0 ∈ L2(0,∞;H−1(Td)) and therefore
f ∈ L2(0,∞;H1(Td)) ∩H1(0,∞;H−1(Td)).
Interpolation, see Theorem 3 in §5.9.2 of [29], then yields f ∈ C([0,∞);L2(Td)). Now
the weak form of the filtration equation yields equation (2.28) for all ϕ ∈ C2(Td) and
almost all t ≥ 0. Since f is continuous in time with values in L2(Td), this equation indeed
extends to all t ≥ 0.
Before we can proceed with the statement of the hydrodynamic limit and its proof, we
need to introduce a last bit of notation.
Definition 2.4.5. We say that two configurations η, ζ ∈ XN satisfy
η ≤ ζ
52
2.4. Hydrodynamic limit for zero range processes
if η(x) ≤ ζ(x) for all x ∈ TdN . A function f
N ∈ Cb(XN) is said to be monotonous if
fN(η) ≤ fN(ζ)
for all η ≤ ζ. Given two probability measures µ, ν ∈ P (XN), we say that µ is bounded by
ν, written
µ ≤ ν,
if it holds that 〈µ, fN〉 ≤ 〈ν, fN〉 for all monotonous fN ∈ Cb(XN).
Furthermore the empirical measure as defined in (2.4) is not regular enough to give sense
to ∆σ(αNη ). Hence we shall measure particle densities of the configuration η ∈ XN of the
discrete system via the mollified empirical measure
(2.29) αN,η :=
1
Nd
∑
x∈TdN
η(x)δ()x
N
∈ H,
where
(2.30) δ()0 =
1
d
χ( ·) ∈ H
is an approximation of the dirac distribution and δ()u its translation by u ∈ Td. We let
χ ∈ C∞0 (R
d) have compact support in, for example, (−1/2, 1/2)d. Since we are working on
the torus, χ(u/) should be understood in Td, e.g. by taking its periodization
∑
z∈Zd χ((u+
z)/).
2.4.1 The hydrodynamic limit
The following theorem is our main result, detailing a hydrodynamic limit with an explicit
rate of convergence in one dimension.
Theorem 2.4.6 (Hydrodynamic limit for zero range processes). Assume d = 1 and let
F ∈ C2b (R), ϕ ∈ C
3(Td), k > (d+ 2)/2, CH > 0, and ρ > 0 be given. Then there exists a
rate of convergence rHL(,N) with polynomial dependence on  and N , whose exponents
only depend on k (and d), such that the following hydrodynamic limit holds. For all t ≥ 0,
N ∈ N, 1/N <  < 1, f0 ∈ H, and µN0 ∈ P (XN) such that the entropy and the measure
itself are bounded relative to the grand-canonical measure νNρ , i.e.
HN
(
µN0 |ν
N
ρ
)
≤ CHN
d and µN0 ≤ ν
N
ρ ,
53
Quantitative hydrodynamic limits
it holds that there exists a rate of convergence rHL(,N) such that
(2.31)
〈
µNt , F
(
〈αNη , ϕ〉L2
)
− F
(
〈ft, ϕ〉L2
)〉
≤ rHL(,N)
+ ‖F ′‖L∞‖ϕ‖L∞
〈
µN0 , ‖α
N,
η − f0‖L1
〉
where ft is given by the solution to the filtration equation (2.3) and µNt is given by the
zero range process, as detailed above. For all T > 0, % > 0, and 0 ≤ 2l + 1 ≤ N , the rate
of convergence rHL(,N) is bounded by
C‖F‖C2(1 + ‖ϕ‖C3 + ‖ϕ‖
2
H1)
(
T−(d+4)((1+
d
2 )
2k2+k
2k+2 +kd)−d(1−
θ
2 )N1−θ(d+1)
+ T
1
2 −
4+d
2 N−2 + + e−cTN2+d−2d + T
1
2 rRL(%, l, , N)
)
for some finite, positive constants c and C depending only on d, k, CH , and ρ. Here
θ = θ(k) = (2k−d−2)/(2k+2) and the additional rate function rRL(%, l, , N) is bounded
by
(
N−
1
2 l
1
2 + 
1
2 l
1
4
)
%l
d
4 + %l−
d
4 + %−
1
4 +
l
N
.
It stems from the replacement lemma, Lemma 2.4.19.
Remark 2.4.7. Thus we see that the contributions to the hydrodynamic limit can be
divided into three parts. The first term describes the propagation of the initial error due
to the approximation of f0 by the discrete particle configurations at time t = 0 and the
other two are errors coming from stability properties of the limit equation and an error
term due to the replacement lemma, respectively. The replacement lemma is an older
result, appearing first in [36].
Note that the choice of the density ρ in the assumption regarding the Gibbs measure νNρ
is a slightly arbitrary. Indeed, if the relative entropy
1
Nd
HN(µN0 |ν
N
ρ ) ≤ C
is bounded, then so is
1
Nd
HN(µN0 |ν
N
ρ˜ ) ≤ C(ρ˜)
for any ρ˜ > 0. Furthermore, if ‖f0‖L∞ ≤ ρ, the local Gibbs measure (2.11) satisfies
νNf0(·) ≤ ν
N
ρ
and this choice for µN0 satisfies the bound on the entropy as well. The rate here is much
slower than the rate for independent random walks, given in Theorem 2.3.1. This is
mainly due to the fact that we shall need to replace the rate function g with its local
54
2.4. Hydrodynamic limit for zero range processes
average via a replacement lemma.
Remark 2.4.8. We can replace the term estimating the compatibility of the initial data
by a slightly modified version. Under the assumptions of Theorem 2.4.6, its proof also
yields
〈
µNt , F
(
〈αNη , ϕ〉L2
)
− F
(
〈ft, ϕ〉L2
)〉
≤ rHL(,N) + ‖F
′‖L∞‖ϕ‖H1
〈
µN0 , ‖α
N,
η − f0‖H−1
〉
+ ‖F ′‖L∞ |
∫
Td ϕ(u) du|
〈
µN0 , |N
−d
∑
x η(x)−
∫
Td f0(u) du|
〉
.
This is a consequence of the following variant of the basic H−1–stability estimate in
Lemma 2.4.12:
‖f˜t − f˜∞ − ft + f∞‖H−1 ≤ ‖f˜0 − f˜∞ − f0 + f∞‖H−1 + C|f˜∞ − f∞|,
where f∞ =
∫
Td ft(u)du is the (constant) integral of the solution ft and likewise for f˜∞.
The above estimate of the hydrodynamic limit can be useful if we only have information
about the H−1–convergence of the initial data and some information on the number of
particles of the zero range process. For example, we might require that there exists some
N0 ∈ N such that
1
Nd
∑
x∈TdN
η(x) =
∫
Td
f0(u) du
holds µN0 –almost surely for all N ≥ N0. Of course, this is only possible if the integral
∫
Td f0(u)du is an integer. The convergence of α
N,
η in H
−1(Td) is more favourable in
terms of powers of  than the convergence in L1(Td). On the other hand, the initial data
converges much faster in general than the bounds we can give for rHL(,N), so it will not
matter much in which norm we measure the convergence of the initial data, see the proof
of Corollar 2.4.9 below.
As before Theorem 2.4.6 yields convergence to the hydrodynamic limit, conditional on
convergence of the initial data.
Corollary 2.4.9. Assume d = 1 and let F ∈ C2b (R), and ϕ ∈ C
3(Td) be given. Then for
all N ∈ N and f0 ∈ C1(Td), there exists a µN0 ∈ P (XN) such that it holds
〈
µNt , F
(
〈αNη , ϕ〉L2
)
− F
(
〈ft, ϕ〉L2
)〉
≤ CN−κ
for some κ > 0, e.g. any κ < 1/6700 works. Here µNt is given by the zero range process
and ft by the solution to the corresponding filtration equation ∂tft = ∆σ(ft).
In other words, if η is distributed according to law µNt , then α
N
η ⇀
∗ ft in distribution as
N →∞.
55
Quantitative hydrodynamic limits
Remarks 2.4.10. (1) The statements (3)-(6) of Remark 2.3.3 remain valid and relevant
to this case.
(2) The condition HN(µN0 |ν
N
ρ ) ≤ CN
d is quite useful in connection with the so-called
entropy inequality. This inequality states that for all γ > 0, f ∈ Cb(XN) and any two
probability measures µ, ν ∈ P (XN), it holds that
(2.32) 〈µ, f〉 ≤
1
γ
(
HN(µ|ν) + log
〈
ν, exp(γf)
〉)
.
For example, if we set f(η) =
∑
x η(x), µ = µ
N
0 , and ν = ν
N
ρ in estimate (2.32), we obtain
that
〈µN0 , N
−d
∑
x η(x)〉 ≤
1
γ
(
N−dHN(µN0 |ν
N
ρ ) + log
〈
νNρ , exp(γη(0))
〉)
,
since νNρ is a product measure. Thus we obtain a bound on the average particle density
〈µN0 , N
−d
∑
x∈TdN
η(x)〉 ≤ C.
Note here that the infinite radius of convergence ρ∗ = +∞ of the partition function Z(·)
yields finite exponential moments
〈νNρ , exp(γη(0))〉 =
Z(ρeγ)
Z(ρ)
< +∞
for all γ ∈ R. Of course in our case, the bound on the average number of particles also
follows from fN0 ≤ ν
N
ρ and the monotonicity of N
−d
∑
x η(x) in η.
(3) While the size of κ is certainly not optimal, it is qualitatively correct since has the
expected polynomial dependence on N . As far as we are aware, it also constitutes the first
example of an explicit rate of convergence of the zero range process to the hydrodynamic
limit and the first example of a uniform-in-time rate of convergence. It has the added
advantage that the exponents do not depend on the function g. Thus, for example, it
should allow for perturbative arguments to be applied to prove the hydrodynamic limit
of small perturbations of the zero range process as given in the assumptions.
Proof of Corollary 2.4.9. First of all we can choose µN0 := ν
N
f0(·)
which was defined in
(2.11). Then in a rough first estimate it holds that
〈µN0 , ‖α
N,
η − f0‖L1〉 ≤ 〈µ
N
0 , ‖α
N,
η − f0‖L2〉
1
2 ≤
C
√
N
.
56
2.4. Hydrodynamic limit for zero range processes
We can deduce this as follows. First, it holds that
〈
µN0 , ‖α
N,
η − f0‖
2
L2
〉
=
∫
Td
〈
µN0 ,
1
N2d
∑
x,y
η(x)η(y)δ()x
N
(u)δ()y
N
(u)
−
2
Nd
∑
x
η(x)δ()x
N
f0(u) + f0(u)
2
〉
du.
Since η(x) and η(y) are independently distributed under µN0 as long as x 6= y, this equals
∫
Td
(
1
N2d
∑
x 6=y
f0( xN )f0(
y
N )δ
()
x
N
(u)δ()y
N
(u)
+
1
N2d
∑
x∈TdN
〈νNf0( xN ), η(0)
2〉δ()x
N
(u)2 −
2
Nd
∑
x∈TdN
f0( xN )δ
()
x
N
f0(u) + f0(u)
2
)
du.
Since f0 ∈ C1(Td) is bounded, it follows that the second moment 〈νNf0( xN ), η(0)
2〉 is bounded
uniformly in x and N . Furthermore it holds that ‖δ()x/N‖
2
L2 ≤ C
−d. Thus the above is
bounded from above by
∥
∥
∥
∥
1
Nd
∑
x∈TdN
f0( xN )δ
()
x
N
− f0
∥
∥
∥
∥
2
L2
+ O
( 1
(N)d
)
,
Note that
∫
Td χ(u) du = 1 and hence
(2.33)
∣
∣
∣
∣
1
(N)d
∑
x∈TdN
χ
( x
N
)
− 1
∣
∣
∣
∣ ≤ ‖∇χ‖∞
C
N
.
Since f0 ∈ C1(Td) is uniformly Lipschitz and the support of χ in (2.30) is compact, this
shows that ∣
∣
∣
∣
1
Nd
∑
x∈TdN
f0( xN )δ
()
x
N
(u)− f0(u)
∣
∣
∣
∣ ≤
C
N
.
Thus we have shown that
〈
µN0 , ‖α
N,
η − f0‖
2
L2
〉
≤
C
N
.
In order to estimate a rate of convergence for the hydrodynamic limit, we make the Ansatz
that  = (N), l = l(N), % = %(N), and T = T (N) are all monomials in N . If we optimize
over the set of possible powers, we find that indeed
〈
µNt , F
(
〈αNη , ϕ〉L2
)
− F
(
〈ft, ϕ〉L2
)〉
= O(N−κ)
for some κ > 0. Since d = 1, we obtain that, for example, we can achieve any rate κ with
κ < 1/6700. Note that this bound is much larger than one would expect in view of the
57
Quantitative hydrodynamic limits
law of large numbers and not optimal. On the other hand, it is independent of the rate
function g, as long as g satisfies Assumption 1.
As in Section 2.3 we have a collection of semigroups SNt and T
N
t describing the dynamics
of the particle process as well as S∞t and T
∞
t describing the evolution of the limit equation.
The semigroup S∞t is defined by
S∞t f0 := ft,
for all f0 ∈ H, where ft ∈ H is the solution (2.3) corresponding to the initial datum f0.
Uniqueness of the solutions shows that it is indeed a semigroup. Then T∞t is again given
by
T∞t Ψ(f) = Ψ(S
∞
t f)
for all Ψ ∈ Cb(H) and f ∈ H. The relationships of the semigroups can be summarized as
SNt : P (XN)→ P (XN) with dual T
N
t : Cb(XN)→ Cb(XN),
S∞t : H → H with pullback T
∞
t : Cb(H)→ Cb(H).
The semigroup TNt has generator G
N given in equation (2.2), whereas the time-derivative
G∞ of T∞t will be given in Lemma 2.4.18 below. Using the empirical measures, we also
define an embedding piN, : Cb(H)→ Cb(XN) via
(
piN,Ψ
)
(η) = Ψ(αN,η ),
where we shall usually drop the superscript  for ease of notation. This embedding will
allow us to compare the semigroups TNt and T
∞
t directly.
Remark 2.4.11. The heat equation admits measure-valued solutions and hence we could
give sense to ∆αNη . Even though there exist solutions to the nonlinear heat equation
∂tf = ∆σ(f) started from a measure, this approach is not feasible here, since we cannot
give sense to G∞Ψ(f) = DΨ(f)(∆σ(f)) if f is just a measure. This is the main reason
we need to use the mollified empirical measure
αN,η = α
N
η ∗ δ
(),
which can be seen as obtained through a convolution. The mollified empirical measure
induces the following local averages: For any function h : N→ R, we set
(2.34) (h ◦ η)()(u) :=
1
Nd
∑
x∈TdN
h(η(x))δ()x
N
(u)
where we recall that δ()(u) is given in (2.30). The mollification introduces another scale,
which might be called mesoscopic. It has microscopic size N and macroscopic size .
58
2.4. Hydrodynamic limit for zero range processes
Such an intermediate scale appears in virtually all works on the hydrodynamic limit,
see for example the replacement lemma. Here the intermediate scale is inherent in the
(mollified) empirical measure.
2.4.2 Regularity of the limit equation
We have seen in Section 2.3, that our proof of the hydrodynamic limit relies on a stability
and a consistency result. Since the limit equation in Section 2.3 is linear, its stability
is reduced to some bounds on its solutions. Here the question of stability is much more
difficult and relies on delicate estimates on the (uniform) propagation of higher regularity.
These estimates are crucial to obtaining stability by interpolating with weak contractivity
estimates. The weak contractivity estimates and some a-priori bounds on the solutions are
the content of the following Lemma 2.4.12, whereas the preservation of higher regularity
in d = 1 dimension will be proved in Lemma 2.4.13 below.
The filtration equation (2.3) satisfies the following basic estimates, which can all be shown
by simple differentiation in time, c.f. Vazquez [83].
Lemma 2.4.12 (Basic estimates). Let f0 ∈ H1(Td) ∩ L∞(Td) and denote by ft the
corresponding solution to (2.3) with initial datum f0. Then the following bounds hold
for all t ≥ 0. The Lp–norms do not grow, i.e.
(2.35) ‖ft‖Lp ≤ ‖f0‖Lp
for all 1 ≤ p ≤ ∞. The H1–norm of ft is bounded since
(2.36)
∫
Td
|∇σ(ft)|
2 du+ 2
∫ t
0
∫
Td
σ′(fs)|∆σ(fs)|
2 duds =
∫
Td
|∇σ(f0)|
2 du.
In particular, it follows ‖∇ft‖L2 ≤ C‖∇f0‖L2. If f˜t is another solution to (2.3) with
initial datum f˜0, the following stability estimates hold:
(2.37)
∫
Td
|f˜t(u)− ft(u)| du ≤
∫
Td
|f˜0(u)− f0(u)| du
for all f˜0, f0 ∈ H, t ≥ 0, as well as
(2.38) ‖f˜t − ft‖
2
H−1 + 2
∫ t
0
∫
Td
(
f˜s − fs
)(
σ(f˜s)− σ(fs)
)
duds = ‖f˜0 − f0‖
2
H−1
for all f˜0, f0 ∈ H such that
∫
Td f˜0(u) du =
∫
Td f0(u) du and t ≥ 0.
The above bound in H1(Td) does not suffice to prove stability. We will also need an
estimate on the conservation of higher regularity. This is harder for the nonlinear filtration
59
Quantitative hydrodynamic limits
equation (2.3) than it was for the heat equation. Given f0 ∈ H, consider the solution ft
to the filtration equation (2.3) with initial datum f0. It is a classical result that solutions
to the nonlinear, uniformly parabolic equation (2.3) are smooth, i.e. ft ∈ C∞(Td) for all
t > 0, if f0 ∈ H. On the other hand, in order to prove the hydrodynamic limit with an
explicit rate of convergence, we need to obtain explicit information on the size of ‖ft‖Hk
for some large enough k > 0 in terms of f0. In d = 1 dimensions, this can be achieved
by elementary calculations. This is the reason why in the hydrodynamic limit, Theorem
2.4.6, we restricted ourselves to the case d = 1. Recall that in order to emphasize the
generality of our approach, we still write out the parameter d, understanding d = 1 and
that we shall illustrate in Section 2.7 how we plan to remove the assumption in future
research.
For any integer k > 0 and f ∈ Hk(Td), let Dkf denote the k–th derivative of f , a tensor.
Furthermore we denote multi-indices in Nd by s, r and the corresponding scalar derivatives
by Dsft, Drft.
Lemma 2.4.13. Assume that d = 1. Then for every k > 0, it holds that
‖Dkft‖L2 ≤ C
(
1 + ‖Dkf0‖L2 + ‖f0‖
2k2
L∞‖∇f0‖
2k2+k
L2
)
for all t ≥ 0 and f0 ∈ Hk(Td).
Proof. Let s ∈ Nd be any multi-index such that |s| = k. The filtration equation yields
that
d
dt
∫
Td
|Dsft|
2 du = −2
∫
Td
∇DsftD
s
(
σ′(ft)∇ft
)
du
By Faa` di Bruno’s formula, this is bounded by
−2
∫
Td
σ′(ft)|∇D
sft|
2 du+
∑
m;ri
C(r)
∫
Td
σ(m+1)(ft) ∇D
sft · ∇D
s−
∑
i rift
m∏
j=1
Drjft
where the sum is over all integers m > 0 and multi-indices ri ∈ Nd, i = 1, . . . ,m, such
that
∑m
i=1 ri ≤ s and ri 6= 0 for all i. Here C(r) denotes a constant depending only on
r and we shall now bound each of the above summands of this sum. Thanks to Lemma
2.4.1, each summand is bounded by
C
∥
∥∇Dsft
∥
∥
L2
(1 + ‖ft‖
m
L∞)
∥
∥∇Ds−
∑
i rift
m∏
j=1
Drjft
∥
∥
L2
.
Now we choose any coefficients p, (pi)mi=1 such that 1/2 = 1/p+
∑
i 1/pi to obtain that
∥
∥∇Ds−
∑
i rift
m∏
j=1
Drjft
∥
∥
L2
≤
∥
∥∇Ds−
∑
i rift
∥
∥
Lp
m∏
j=1
∥
∥Drjft
∥
∥
Lpj
.
60
2.4. Hydrodynamic limit for zero range processes
Note that every order n of the derivatives appearing in this product satisfies 1 ≤ n ≤ k.
Recall that a generalized Gagliardo-Nirenberg-Sobolev inequality yields
‖Drf‖Lp ≤ C‖D
k+1f‖θL2‖∇f‖
1−θ
L2 ,
if
|r| −
d
p
= θ
(
k + 1−
d
2
)
+ (1− θ)
(
1−
d
2
)
,
as well as 0 ≤ θ ≤ 1, 0 ≤ |r| < k + 1, and θ ≥ (|r| − 1)/k, see [20]. Set
θi =
|ri| − 1 + d(12 −
1
pi
)
k
, θ =
|s−
∑
i ri|+ d(
1
2 −
1
p)
k
and note that θi ≥ (|ri| − 1)/k since pi ≥ 2 (likewise for θ). Summing over all θi yields
θ +
m∑
i=1
θi =
k −m+ d2(m+ 1)−
d
2
k
=
k − m2
k
≤ 1−
1
2k
,
since
∑
i 1/pi + 1/p = 1/2, 1 ≤ m ≤ k, and d = 1. We also calculate
(1− θ) +
m∑
i=1
(1− θi) = m+ 1−
k − m2
k
= m+
m
2k
≤ k +
1
2
Therefore the Gagliardo-Nirenberg-Sobolev inequality yields
∥
∥∇Ds−
∑
i rift
∥
∥
Lp
m∏
i=1
∥
∥Drift
∥
∥
Lpi
≤ C‖Dk+1ft‖
2k−m
2k
L2 ‖∇ft‖
m+m2k
L2
≤ C
(
‖Dk+1ft‖
2k−1
2k
L2 ‖∇ft‖
2k+1
2
L2 + 1
)
.
By Lemma 2.4.12, it holds that ‖∇ft‖L2 ≤ C‖∇f0‖L2 and ‖ft‖L∞ ≤ ‖f0‖L∞ . Therefore
we conclude that for any k > 0, there exist constants 0 < c and C <∞ such that
(2.39)
d
dt
‖Dkft‖
2
L2 ≤ −c‖D
k+1ft‖
2
L2 + C
(
‖Dk+1ft‖
2− 12k
L2 ‖f0‖
k
L∞‖∇f0‖
2k+1
2
L2 + 1
)
.
Since the integral of the derivative Dkft over the torus Td vanishes, Poincare´’s inequality
yields
‖Dkft‖
2
L2 ≤ C‖D
k+1ft‖
2
L2 .
Hence we can choose C ′ > 0 large enough, such that whenever ‖Dkft‖L2 ≥ C ′ holds, then
the right hand side of estimate (2.39) is negative. Therefore we deduce that
‖Dkft‖
1
2k
L2 ≤ C max
{
‖Dkf0‖
1
2k
L2 , ‖f0‖
k
L∞‖∇f0‖
2k+1
2
L2 + 1
}
,
61
Quantitative hydrodynamic limits
which concludes the proof of the proposition.
Remark. Lemma 2.4.12 yielded a-priori bounds in H1(Td), which is stronger in d = 1
dimension than the L∞(Td)–bound from the maximum principle. Furthermore L∞(Td) is
just the critical case, in which simple interpolation arguments do not yield a bound onDkft
of the form of Lemma 2.4.13. Thus, without any stronger bounds, we needed to restrict
ourselves to d = 1. This dichotomy in regularity between low and high dimensions is
natural for quasilinear parabolic equations like the filtration equation (2.3). The regularity
of the filtration equation in d = 1, 2 was known even before de Giorgi’s and Nash’s
famous results on the Ho¨lder-continuity of the solutions to parabolic equations. Indeed
our proposed approach in Section 2.7 depends on the regularity result of de Giorgi and
Nash.
2.4.3 Consistency and stability
Before we prove the hydrodynamic limit Theorem 2.4.6, we need to prove stability and
consistency estimates similar to Section 2.3. Since the limit PDE is nonlinear, this is
more involved and we shall mostly work with Hk–norms, not the Ck–norms of Section 2.3
(where the choice was convenient because of the use of Dirac distributions). Again, we first
show stability estimates on the filtration equation. These estimates allow us to control the
fluctuations around the hydrodynamic limit along the evolution of the filtration equation.
The consistency result concerns the closeness of the generators generators GN and G∞.
Lemma 2.4.14 (Stability). Assume d = 1 and let F ∈ C2b (R) and ϕ ∈ C
2(Td), and
define Ψ as in equation (2.19).
(i) For any t ≥ 0, the map S∞t : H → H is differentiable with respect to the H
−1–norm in
the sense given below. Its derivative DS∞t (f) : H → H can be given as the weak solution
vt := DS∞t (f)(f˜ − f) ∈ H to the linearized filtration equation
(2.40) ∂tvt = ∆ (σ
′(S∞t f)vt) ,
such that v0 = f˜ − f at time t = 0. Differentiability holds in the sense that for each
integer k > (d+ 2)/2 and
θ = θ(k) =
2k − d− 2
2k + 2
,
there exists a constant C < +∞ such that
‖S∞t f˜ − S
∞
t f‖H−1 ≤ ‖f˜ − f‖H−1 ,
‖S∞t f˜ − S
∞
t f −DS
∞
t (f)(f˜ − f)‖H−1 ≤ C max{Λ(f˜),Λ(f)}‖f˜ − f‖
1+θ
H−1 ,
62
2.4. Hydrodynamic limit for zero range processes
for all f˜ , f ∈ Hk(Td) such that
∫
Td f˜(u)du =
∫
Td f(u)du. The factor Λ(f) is given by
(2.41) Λ(f) = 1 + ‖f‖L∞
(
1 + ‖Dkf‖L2 + ‖f‖
2k2
L∞‖∇f‖
2k2+k
L2
) d+4
2k+2
for all f ∈ Hk(Td).
(ii) Furthermore S∞t : H → H satisfies the continuity estimates
‖S∞t f − f‖H−1 ≤ C‖f‖L2
√
t and ‖S∞t f − f‖H−1 ≤ C‖∇f‖L2t
for all f ∈ H and t ≥ 0.
(iii) For any t ≥ 0, the map T∞t Ψ : H → R is differentiable with respect to the H
−1–norm
in the sense detailed below and it holds that
(2.42) D (T∞t Ψ) (f) = F
′ (〈S∞t f, ϕ〉L2)DS
∞
t (f)
∗ϕ ∈ H1(Td)
for all f ∈ H. The function wt := DS∞t (f)
∗ϕ ∈ H is given as the weak solution of
(2.43) ∂twt = σ
′(S∞t f)∆wt,
such that w0 = ϕ. We have the estimates
|T∞t Ψ(f˜)− T
∞
t Ψ(f)| ≤ ‖F
′‖L∞‖ϕ‖H1‖f˜ − f‖H−1 and
|T∞t Ψ(f˜)− T
∞
t Ψ(f)−DT
∞
t Ψ(f)(f˜ − f)|
≤
1
2
‖F ′‖C1‖ϕ‖
2
H1‖f˜ − f‖
2
H−1 + C max{Λ(f˜),Λ(f)}‖F
′‖L∞‖ϕ‖H1‖f˜ − f‖
1+θ
H−1
for all f˜ , f ∈ Hk(Td) such that
∫
Td f˜(u)du =
∫
Td f(u)du, where Λ(·) denotes the same
function as in (i).
(iv) Furthermore, it holds that DS∞t (f)
∗ϕ ∈ L∞(0,∞;H1(Td)) ∩ L2(0,∞;H2(Td)) with
uniform bounds
‖∇DS∞t (f)
∗ϕ‖L2 ≤ ‖∇ϕ‖L2 , and
∫ t
0
‖∆DS∞s (f)
∗ϕ‖2L2 ds ≤ C‖∇ϕ‖
2
L2
for all ϕ ∈ H1(Td), f ∈ H, and t ≥ 0.
Proof. (i) The filtration equation (2.3) yields
d
dt
‖f˜t − ft‖
2
H−1 = −2
∫
(σ(f˜t)− σ(ft))(f˜t − ft) du ≤ 0
because σ is monotonous. This contraction property in weak measure distance is key to
most stability estimates.
63
Quantitative hydrodynamic limits
Let now additionally vt denote the solution to (2.40). The respective equations for ft, f˜t,
and vt yield
d
dt
‖f˜t − ft − vt‖
2
H−1 = −2
∫
(σ(f˜t)− σ(ft)− σ
′(ft)vt)(f˜t − ft − vt) du.
The mean value theorem implies that the right hand side is bounded by
−2
∫ (
σ′(ft)(f˜t − ft − vt)
2 +
1
2
σ′′(ξ)(f˜t − ft)
2(f˜t − ft − vt)
)
du
for some ξ(u) (measurable in u) in the interval between ft(u) and f˜t(u). Therefore the
bound (2.24) on σ′ yields that
d
dt
‖f˜t − ft − vt‖
2
H−1 ≤ −c‖f˜t − ft − vt‖
2
L2 −
∫
σ′′(ξ)(f˜t − ft)
2(f˜t − ft − vt) du
for a positive constant c > 0. Now, Lemma 2.4.1 yields |σ′′(ρ)| ≤ C(1+ρ). Hence Young’s
inequality yields a bound
(2.44)
d
dt
‖f˜t − ft − vt‖
2
H−1
≤ −
c
2
‖f˜t − ft − vt‖
2
L2 − C
(
1 + ‖f˜t‖
2
L∞ + ‖ft‖
2
L∞
)
∫
Td
(f˜t − ft)
4 du.
Now the Cauchy-Schwarz inequality in Fourier space and the standard Gagliardo-Nirenberg-
Sobolev inequality yield
‖f‖L2 ≤ C‖f‖
1
1+k
Hk ‖f‖
k
k+1
H−1
‖f‖L4 ≤ C‖f‖
d
4k
Hk‖f‖
1− d4k
L2
for all f ∈ Hk. Taken together these inequalities yield
‖f‖4L4 ≤ C‖f‖
d+4
k+1
Hk ‖f‖
4k−d
1+k
H−1 .
Since d = 1, Lemma 2.4.13 implies
d
dt
‖f˜t − ft − vt‖
2
H−1 ≤ C max{Λ(f˜),Λ(f)}‖f˜t − ft‖
4k−d
1+k
H−1 ,
where Λ(·) is given in (2.41). Thus the H−1–contractivity yields the desired result.
(ii) Recall that we set ft = S∞t f . Since
∫
Td ft(u)du =
∫
Td f(u)du, it holds that
d
dt
‖ft − f‖
2
H−1 = −2
∫
Td
σ(ft)(ft − f) du.
64
2.4. Hydrodynamic limit for zero range processes
Adding and subtracting σ(f) from σ(ft) yields
(2.45)
d
dt
‖ft − f‖
2
H−1 = −2
∫
Td
(σ(ft)− σ(f))(ft − f) du− 2
∫
Td
σ(f)(ft − f) du.
We apply ellipticity of σ on the first summand and the Cauchy-Schwarz inequality on the
second summand to obtain that
d
dt
‖ft − f‖
2
H−1 ≤ −c‖ft − f‖
2
L2 + C‖f‖L2‖ft − f‖L2 .
for some finite constants c, C > 0. Young’s inequality yields
d
dt
‖ft − f‖
2
H−1 ≤ C‖f‖
2
L2
which yields
‖ft − f‖
2
H−1 ≤ C‖f‖
2
L2t.
On the other hand, estimate (2.45) yields
d
dt
‖ft − f‖
2
H−1 ≤ −c‖ft − f‖
2
L2 + C‖∇f‖L2‖ft − f‖H−1 .
by interpolation between H1 and H−1, see (2.27). Here we have used again that
∫
Td
ft(u) du =
∫
Td
f(u) du.
Hence it holds that
d
dt
‖ft − f‖H−1 ≤ C‖∇f‖L2 ,
which yields the desired estimate by integration over t.
(iii) First we want to show that DS∞t (f)
∗ is the adjoint in L2(Td) of the operator DS∞t (f).
This is standard except for the time-dependence through ft of the coefficients of the
linearized equation (2.40). Let us denote for all 0 ≤ s < t the propagator from time
s to time t of the evolution equation (2.40) with time-dependent coefficients by S(t, s).
Likewise the propagator from s to t of equation (2.43) is denoted by S(t, s)∗. To be
precise, we denote by S(·, s)f the solution to equation (2.40) such that at time s the
solution equals f and similarly for S(t, s)∗. We show that S(t, 0)∗ is indeed the adjoint of
S(t, 0). For all 0 ≤ s < t, it holds that
d
ds
〈
S(t, s)(f˜ − f), S(s, 0)∗ϕ
〉
L2
=
〈
S(t, s)(f˜ − f), σ′(fs)∆[S(s, 0)
∗ϕ]
〉
L2
−
〈
∆
(
σ′(fs)[S(t, s)(f˜ − f)]
)
, S(s, 0)∗ϕ
〉
L2
= 0.
65
Quantitative hydrodynamic limits
Integrating with respect to s yields
(2.46)
〈
DS∞t (f)(f˜ − f), ϕ
〉
L2
=
〈
f˜ − f,DS∞t (f)
∗ϕ
〉
L2
.
The differentiability estimates are a consequence of the chain rule and (i) as follows.
Lipschitz continuity of F and (i) yield the first estimate. Using equation (2.46), the
second term to be estimated can be rewritten as
∣
∣
∣F
(
〈S∞t f˜ , ϕ〉
)
− F
(
〈S∞t f, ϕ〉
)
− F ′
(
〈S∞t f, ϕ〉
)
〈DS∞t (f)(f˜ − f), ϕ〉
∣
∣
∣.
We bound this term by the sum of the following two terms:
∣
∣
∣F ′
(
〈S∞t f, ϕ〉
)〈
S∞t f˜ − S
∞
t f −DS
∞
t (f)(f˜ − f), ϕ
〉∣
∣
∣
and ∣
∣
∣F
(
〈S∞t f˜ , ϕ〉
)
− F
(
〈S∞t f, ϕ〉
)
− F ′
(
〈S∞t f, ϕ〉
)
〈S∞t f˜ − S
∞
t f, ϕ〉
∣
∣
∣.
The results of part (i) yield a bound on the first term by
C‖F ′‖L∞‖ϕ‖H1 max{Λ(f˜),Λ(f)}‖f˜ − f‖
1+θ
H−1 ,
whereas the second term is bounded by
1
2
‖F ′‖C1
∣
∣〈S∞t f˜ − S
∞
t f, ϕ〉
∣
∣2 ≤
1
2
‖F ′‖C1‖ϕ‖
2
H1‖f˜ − f‖
2
H−1 .
(iv) Equation (2.43) yields
(2.47)
d
dt
∫
Td
|∇wt(u)|
2 du = −2
∫
Td
σ′(S∞t f(u))|∆wt(u)|
2 du ≤ 0,
since σ′ > 0. Since furthermore ∇w0(u) = ∇ϕ(u), integration of this equation yields
2
∫ ∞
0
∫
Td
σ′(S∞t f(u))|∆wt(u)|
2 dudt ≤ ‖∇ϕ‖2L2 ,
uniformly and f and ϕ. Thus we obtain that
∫ ∞
0
∫
Td
|∆wt(u)|
2 dudt ≤ C‖∇ϕ‖2L2
using the uniform ellipticity of σ′ again, see (2.24).
Next provide some large–time decay estimates which will enable us to provide uniform in
time bounds in the hydrodynamic limit.
66
2.4. Hydrodynamic limit for zero range processes
Lemma 2.4.15 (Spectral gap). Let F ∈ C2b (R) and ϕ ∈ C
3(Td) and define Ψ as in
(2.19). Using the notation of the stability lemma 2.4.14, let f˜ , f ∈ H. Furthermore let
f∞ =
∫
f(u)du denote the spatial average of f and set
(2.48) T1(f˜ , f ; Ψ) := Ψ(f˜)−Ψ(f)−DΨ(f)(f˜ − f).
(This is the difference of the function Ψ evaluated at f˜ ∈ H and its first Taylor polynomial
around f ∈ H.) Then there exist finite, positive constants c, C such that
‖S∞t f − f∞‖
p
Lp ≤ Ce
−ct‖f − f∞‖
p
Lp ,
‖S∞t f − f∞‖H−1 ≤ Ce
−ct‖f − f∞‖H−1 ,
‖T1(f˜ , f ;S∞t )‖
2
H−1 ≤ Ce
−ct
(
‖f − f∞‖
4
L4 + ‖f˜ − f˜∞‖
4
L4
)
, and
‖∇DS∞t (f)
∗ϕ‖L2 ≤ Ce
−ct‖∇ϕ‖L2
for all 2 ≤ p < +∞. Furthermore it holds that
|T1(f˜ , f ;T∞t Ψ)| ≤ Ce
−ct
(
‖f − f∞‖
2
L4 + ‖f˜ − f˜∞‖
2
L4
)
where Ψ is defined in (2.19) with F ∈ C2b (R) and ϕ ∈ C
2(Td).
Proof. We will use the notation of the stability lemma 2.4.14, letting
ft := S
∞
t f and f˜t := S
∞
t f˜
denote two solutions of the filtration equation and
vt := DS
∞
t (f)(f˜ − f) and wt := DS
∞
t (f)
∗ϕ
the linearization around ft as well as its L2-dual.
First of all, conservation of mass yields
∫
ft(u) du = f∞. Set f¯t := ft − f∞, which solves
the equation
∂tf¯t = ∇ ·
(
σ′(f¯t + f∞)∇f¯t
)
.
Note that in contrast to ft, the function f¯t is no longer non-negative everywhere. The
equation for f¯t yields
d
dt
∫
Td
|f¯t|
p du = p
∫
Td
|f¯t|
p−2f¯t∇ ·
(
σ′(f¯t + f∞)∇f¯t
)
du.
Now integration by parts yields
d
dt
∫
Td
|f¯t|
p du = −p(p− 1)
∫
Td
σ′(f¯t + f∞)|f¯t|
p−2|∇f¯t|
2 du.
67
Quantitative hydrodynamic limits
Since
|f¯t|
p−2|∇f¯t|
2 =
∣
∣|f¯t|
p/2−1∇f¯t
∣
∣2 =
4
p2
∣
∣∇|f¯t|
p/2
∣
∣2,
it holds that
d
dt
∫
Td
|f¯t|
p du = −
4(p− 1)
p
∫
Td
σ′(f¯t + f∞)
∣
∣∇|f¯t|
p/2
∣
∣2 du ≤ −c
∫
Td
∣
∣∇|f¯t|
p/2
∣
∣2 du.
Now Poincare´’s inequality in the L2–norm applied to |f¯t|p/2 yields
d
dt
∫
Td
|f¯t|
p du ≤ −c
∫
Td
|f¯t|
p du,
which yields the decay of the Lp–norms of f¯t = ft − f∞. Note that c, C can be taken to
be independent of the choice of p ≥ 2.
The decay of the H−1–norm follows from
d
dt
‖ft − f∞‖
2
H−1 = −2
∫
Td
(σ(ft(u))− σ(f∞))(ft(u)− f∞) du ≤ −c‖ft − f∞‖
2
H−1
by uniform ellipticity and (2.27).
The third statement follows directly from equation (2.44) and the exponential decay of
the L4–norm.
To prove the fourth statement, recall equation (2.47). The lower bound on σ′ yields
d
dt
∫
Td
|∇wt(u)|
2 du ≤ −c
∫
Td
|∆wt(u)|
2 du.
Applying once again Poincare´’s inequality yields
d
dt
∫
Td
|∇wt(u)|
2 du ≤ −c
∫
Td
|∇wt(u)|
2 du,
and hence exponential decay. Consequently, the proof of Lemma 2.4.14 yields
|T1(f˜ , f ;T∞t Ψ)| ≤ Ce
−ct
(
‖f − f∞‖
2
L4 + ‖f˜ − f˜∞‖
2
L4 + ‖f − f∞‖
2
H−1 + ‖f˜ − f˜∞‖
2
H−1
)
,
and the result follows since the H−1(Td)–norm is bounded by the L4(Td)–norm.
Many of the differentiability properties we have just shown fit in the framework of dif-
ferentiable functions on Cb(H) whose derivative is bounded uniformly with respect to a
weight function. The following definition formalizes this. It is an adaptation of Definition
2.10 in [67].
Definition 2.4.16. Let H˜1 and H˜2 be any two metric spaces and consider two Banach
spaces (H1, ‖ · ‖H1), (H2, ‖ · ‖H2) such that H˜i− H˜i ⊂ Hi, i = 1, 2. In general the metric of
68
2.4. Hydrodynamic limit for zero range processes
the subspace H˜i is stronger than the norm of Hi. We can understand Hi to be a tangent
space of H˜i. Let
Λ : H˜1 → [0,+∞)
be a weight function. Abusing notation, we also set
Λ(f˜ , f) := max{Λ(f˜),Λ(f)}.
Then we denote by C1,θΛ (H˜1, H1; H˜2, H2), the space of continuously differentiable function
from H˜1 to H˜2 whose derivative approximates the original function to the order 1 + θ and
is uniformly bounded with respect to the weight function Λ. Specifically, a function
Φ : H˜1 → H˜2 is in C
1,θ
Λ (H˜1, H1; H˜2, H2)
if and only if there exists a continuous function
DΦ : H˜1 → L(H1, H2)
and finite constants C1, C2, and C3, such that it holds
‖Φ(f˜)− Φ(f)‖H2 ≤ C1Λ(f˜ , f)‖f˜ − f‖H1(2.49)
‖DΦ(f)(f˜ − f)‖H2 ≤ C2Λ(f˜ , f)‖f˜ − f‖H1 and(2.50)
‖Φ(f˜)− Φ(f)−DΦ(f)(f˜ − f)‖H2 ≤ C3Λ(f˜ , f)‖f˜ − f‖
1+θ
H1(2.51)
for all f˜ , f ∈ H˜1. Let C
opt
i , i = 1, 2, 3, be the optimal constant in each of (2.49)-(2.51),
i.e. the infimum over all Ci, i = 1, 2, 3, such that each of the inequalities holds. Then we
set
[Φ]C0,1Λ := C
opt
1 , [Φ]C1,0Λ := C
opt
2 , and [Φ]C1,θΛ
:= Copt3 ,
which are seminorms associated to C1,θΛ (H˜1, H1; H˜2, H2).
Note that the proof of the hydrodynamic limit can be understood without using the
above notation, since in principle all we need to do is keep track of various differences.
We include it to provide context and simplify notation, especially in order to distinguish
clearly between stability and consistency estimates. Let us translate the stability result
(i) and (ii) of Lemma 2.4.14 to the language of Definition 2.4.16.
Corollary 2.4.17 (Stability in terms of differentiability). Let R ≥ 0, k ∈ N and denote
HR :=
{
f ∈ Hk(Td)|
∫
Td f(u) du = R
}
.
Furthermore, let F ∈ C2b (R) and ϕ ∈ C
2(Td), and define Ψ as in equation (2.19). Then
there exists a constant C = C(F, ϕ) independent of t and R such that the following holds.
69
Quantitative hydrodynamic limits
(i) For any t ≥ 0 and k > (d+ 2)/2, it holds that
S∞t ∈ C
1,θ
Λ (HR, H
−1(Td);HR, H−1(Td)) and [S∞t ]C1,θΛ ≤ C
in the sense of Definition 2.4.16, where
θ =
2k − d− 2
2k + 2
and Λ(f) = 1 + ‖f‖L∞
(
1 + ‖Dkf‖L2 + ‖f‖
2k2
L∞‖∇f‖
2k2+k
L2
) d+4
2k+2 .
(ii) For any t ≥ 0, it holds that
T∞t Ψ ∈ C
1,θ
Λ (HR, H
−1(Td);R,R) and [T∞t Ψ]C1,θΛ ≤ C
where θ and Λ are given as above.
Proof. Again, part (ii) is a direct consequence of the corresponding estimates on part (i)
and the regularity of F ∈ C2b (R) and ϕ ∈ C
2(Td). To show (i), just let H˜1 = H˜2 = HR
and H1 = H2 = H−1(Td). Note that since all functions in HR have the same integral over
Td, it holds that indeed
HR −HR ⊂ H
−1(Td).
The function S∞t maps HR to HR according to Lemma 2.4.13. Note that the (distribu-
tional) solution
vt = DS
∞
t (f)(f˜ − f)
to (2.40) also exists for initial data f˜ − f ∈ H−1(Td) and satisfies the bounds given in
Lemma 2.4.14. Hence [Ψ]C0,1Λ and [Ψ]C1,θΛ
of Definition 2.4.16 are indeed finite. We just
need to estimate [Ψ]C1,0Λ . Calculations similar to the proof of Lemma 2.4.14 yield
d
dt
‖vt‖
2
H−1 = −2
∫
Td
vtσ
′(ft)vt du ≤ 0,
if the initial datum satisfies f˜ − f ∈ L2(Td). This shows
‖DS∞t (f)(f˜ − f)‖H−1 ≤ ‖f˜ − f‖H−1 ,
if f˜ − f ∈ L2(Td), and by approximation in general. In particular, it holds that
DS∞t (f) ∈ L(H
−1(Td), H−1(Td)).
It remains to prove the continuity of DS∞t (f) ∈ L(H
−1(Td), H−1(Td)) with respect to
f ∈ HR (even though we shall not make use of this property in this thesis). This is easiest
to see in the dual setting. Let f˜ , f ∈ HR. As in the stability result, Lemma 2.4.14, we
70
2.4. Hydrodynamic limit for zero range processes
set wt := DS∞t (f)
∗ϕ and we also set w˜t := DS∞t (f˜)
∗ϕ. Then the continuity estimate
∥
∥DS∞t (f)−DS
∞
t (f˜)
∥
∥
L(H−1,H−1)
→ 0 as ‖f˜ − f‖Hk → 0
corresponds by duality to showing that
(2.52) ‖∇(w˜t − wt)‖L2 ≤ ω(‖f˜ − f‖Hk)‖∇ϕ‖L2
for some modulus of continuity ω, i.e. some function ω : [0,∞) → [0,∞) such that
ω(τ) → 0 as τ → 0. Note that we may allow ω to depend on ‖f˜‖Hk , ‖f‖Hk , and t. The
equations for w˜t and wt yield
d
dt
∫
Td
|∇(w˜t − wt)|
2 du = −2
∫
Td
∆(w˜t − wt)
(
σ′(f˜t)∆w˜t − σ
′(ft)∆wt
)
du.
Hence it holds that
d
dt
∫
Td
|∇(w˜t − wt)|
2 du = −2
∫
Td
|∆(w˜t − wt)|
2σ′(f˜t) du
− 2
∫
Td
∆(w˜t − wt)
(
σ′(f˜t)− σ
′(ft)
)
∆wt du.
Young’s inequality yields a bound on the right hand side by
(2.53) C
∥
∥(σ′(f˜t)− σ
′(ft))∆wt
∥
∥2
L2
.
The mean value theorem and Lemma 2.4.1 yield
∣
∣σ′(f˜t)− σ
′(ft)
∣
∣ ≤ C (‖f˜t‖L∞ + ‖ft‖L∞)‖f˜t − ft‖L∞ .
By the maximum principle and since k > (d + 2)/2 yields an embedding Hk(Td) ↪→
L∞(Td), we can therefore bound (2.53) by
C
(
‖f˜‖Hk , ‖f‖Hk
)
‖f˜t − ft‖L∞‖∆wt‖
2
L2 ,
where C(‖f˜‖Hk , ‖f‖Hk) denotes some constant depending on its arguments, but not on t
or wt. Again since k > (d+2)/2, there exists a bounded embedding Hk−1(Td) ↪→ L∞(Td)
and therefore
‖f˜t − ft‖L∞ ≤ C‖f˜t − ft‖Hk−1 .
Hence interpolation yields
‖f˜t − ft‖Hk−1 ≤ ‖f˜t − ft‖
k
k+1
Hk ‖f˜t − ft‖
1
k+1
H−1
71
Quantitative hydrodynamic limits
Since Lemma 2.4.13 yields uniform-in-time bounds on the Hk–norms of f˜ and f , the
contractivity of S∞t with respect to the H
−1–norm yields
‖f˜t − ft‖
k
k+1
Hk ‖f˜t − ft‖
1
k+1
H−1 ≤ C
(
‖f˜‖Hk , ‖f‖Hk
)
‖f˜ − f‖
1
k+1
H−1 ,
for some (possibly different) constant C(‖f˜‖Hk , ‖f‖Hk). In summary, we have proved that
d
dt
‖∇(w˜t − wt)‖
2
L2 ≤ C
(
‖f˜‖Hk , ‖f‖Hk
)
‖f˜ − f‖
1
k+1
H−1‖∆wt‖
2
L2 .
Since w˜0 = ϕ = w0, integration with respect to time yields
‖∇(w˜t − wt)‖
2
L2 ≤ C
(
‖f˜‖Hk , ‖f‖Hk
)
‖f˜ − f‖
1
k+1
H−1
∫ t
0
‖∆ws‖
2
L2 ds.
Lemma 2.4.14 (iv) yields
‖∇(w˜t − wt)‖
2
L2 ≤ C
(
‖f˜‖Hk , ‖f‖Hk
)
‖f˜ − f‖
1
k+1
H−1‖∇ϕ‖
2
L2 ,
which shows the desired estimate (2.52).
Now let us identify an expression for the time-derivative dT∞t Ψ(f)/dt.
Lemma 2.4.18. Let F ∈ C1b (R) and ϕ ∈ C
2(Td), and define Ψ as in equation (2.19).
Furthermore let S∞t f solve the filtration equation with initial datum f ∈ H
k(Td) with
k > (d+ 2)/2. Then we obtain the following characterizations of the derivative in t of the
limit evolution.
(i) It holds that
d
dt
S∞t f = DS
∞
t (f)(∆σ(f)).
(ii) We can lift this result to the level of observables. Recalling the definition
G∞T∞t Ψ(f) :=
d
dt
T∞t Ψ(f),
the result (i) translates to T∞t as
G∞T∞t Ψ(f) = F
′ (〈S∞t f, ϕ〉L2) 〈∆σ(f), DS
∞
t (f)
∗ϕ〉L2 =: T
∞
t G
∞Ψ(f).
for all f ∈ Hk(Td) and all t ≥ 0, where Ψ is given in (2.19).
Proof. (i) Consider
I :=
1
s
(
S∞t+sf − S
∞
t f
)
−DS∞t (f)(∆σ(f))
72
2.4. Hydrodynamic limit for zero range processes
The semigroup property of S∞t (a direct consequence of uniqueness of solutions to the
filtration equation (2.3)) yields
I =
1
s
(
S∞t S
∞
s f − S
∞
t f
)
−DS∞t (f)(∆σ(f)).
Therefore, the stability result of Lemma 2.4.14 yields
(2.54) ‖I‖H−1 =
∥
∥
∥
∥
1
s
(
DS∞t (f)(S
∞
s f − f)
)
−DS∞t (f)(∆σ(f))
∥
∥
∥
∥
H−1
+ O
(
s−1Λ(f, S∞s f)‖S
∞
s f − f‖
1+θ
H−1
)
,
where θ = (2k−d−2)/(2k+2) > 0. The maximum principle and the improved regularity
of Lemma 2.4.13 yield that Λ(f, S∞s f) ≤ C(f) independent of s. Furthermore Lemma
2.4.14 (ii) yields
‖S∞s f − f‖H−1 ≤ C‖∇f‖L2s,
and hence
O
(
s−1Λ(f, S∞s f)‖S
∞
s f − f‖
1+θ
H−1
))
= O(sθ)
for fixed f ∈ Hk(Td). Since DS∞t (f) ∈ L(H
−1(Td), H−1(Td)) is a contraction, the other
summand on the right hand side of (2.54) equals
∥
∥
∥
∥DS
∞
t (f)
(
S∞s f − f
s
)
−DS∞t (f)(∆σ(f))
∥
∥
∥
∥
H−1
≤
∥
∥
∥
∥
S∞s f − f
s
−∆σ(f)
∥
∥
∥
∥
H−1
,
where we have used that
∫
Td
(
S∞s f(u)− f(u)
)
du = 0 =
∫
Td
∆σ(f(u)) du.
The right hand side vanishes since ft is a solution to equation (2.3).
(ii) This part is a direct consequence of (i) and the chain rule.
For the statement of the consistency result, we will need a replacement lemma, which in
the spirit of an ergodic theorem (or a law of large numbers) allows us to replace locally
the spatial average of a function of the number of particles over a small box with its
expectation value with respect to the local density in this box. Since we are interested
in obtaining qualitative results, we will prove a quantitative L2–version with an explicit
upper bound on the rate of convergence. The proof is deferred until Section 2.6.
Lemma 2.4.19 (A quantitative replacement lemma). Assuming that the initial data
possess bounded relative entropy and are bounded with respect to some Gibbs measure, i.e.
HN
(
µN0 |ν
N
ρ
)
≤ CNd, and µN0 ≤ ν
N
ρ
73
Quantitative hydrodynamic limits
for some ρ > 0. Then it holds that
(
1
T
∫ T
0
∫
Td
〈
µNt ,
∣
∣(g ◦ η)()(uN)− σ(η()(uN))
∣
∣2
〉
dudt
)1/2
≤ rRL(%, l, , N),
where we recall definition (2.34). The rate function rRL satisfies
rRL(%, l, , N) ≤ C
(
(N−
1
2 l
1
2 + 
1
2 l
1
4 )%l
d
4 + %l−
d
4 + %−
1
4 +
l
N
)
for all N ∈ N, 1/N <  < 1, l < N , and % > 0.
Now we are ready to state the consistency result.
Lemma 2.4.20 (Consistency). Let F ∈ C2b (R) and ϕ ∈ C
3(Td), and define Ψ as in
equation (2.19). Furthermore assume that the initial data µN0 of the ZRP have bounded
relative entropy and are bounded relative to some Gibbs measure
HN
(
µN0 |ν
N
ρ
)
≤ CNd and µN0 ≤ ν
N
ρ
for all N ∈ N. Then it holds that
∣
∣
∣
∣
∫ t
0
〈
µNs ,
(
GNpiN − piNG∞
)
T∞t−sΨ
〉
ds
∣
∣
∣
∣ ≤ rC(T, %, l, , N)
for all 0 ≤ t < +∞, T > 0, N ∈ N, and 1/N <  < 1. The consistency bound
rC(T, %, l, , N) is given explicitly by the function
(2.55) C
(
−
dθ
2 N1−θ(d+1)T sup
0≤s≤t
sup
x∼y
〈
µNt−s,
[
T∞s Ψ
]
C1+θ
Λ(αN,ηx,y , α
N,
η )N
−d
∑
z η(z)
〉
+ −(3+
d
2 ) sup
η∈XN
∫ ∞
T
‖∇(DS∞s (α
N,
η )
∗ϕ)‖L2(Td) ds
+N2+d sup
t≥T
∫ t
T
〈
µNt−s, sup
x∼y
∣
∣T1(α
N,
ηx,y , α
N,
η ;T
∞
s Ψ)
∣
∣
〉
ds
+
∥
∥∆(DS∞t (α
N,
η )
∗ϕ)
∥
∥
L∞η L
2
t,u
√
T
(
rRL(%, l, , N) + 
−(2+ d2 )N−2
))
.
for any θ = θ(k) and k > 0. Here θ, Λ, and [T∞s Ψ]C1+θΛ are given in Corollary 2.4.17 and
rRL(%, l, , N) stems from Lemma 2.4.19. The notation T1 is explained in Lemma 2.4.15
and the function DS∞s (α
N,
η )
∗ϕ is given in Lemma 2.4.14.
Note that in contrast to Section 2.3, the form (2.19) of Ψ is not conserved under the
application T∞t , hence we need to keep track of T
∞
t−s. This also explains why we needed
to derive the above fairly complex stability results.
74
2.4. Hydrodynamic limit for zero range processes
Proof of Lemma 2.4.20. Let us first assume that t ≤ T . Denote the quantity to be
estimated by
I :=
∣
∣
∣
∣
∣
∫ t
0
〈
µNt−s,
(
GNpiN − piNG∞
)
T∞s Ψ
〉
ds
∣
∣
∣
∣
∣
.
Inserting the expressions for the generators GN and G∞, cf. Lemma 2.4.18, we obtain
that
I =
∣
∣
∣
∣
∣
∫ t
0
〈
µNt−s, N
2
∑
x∼y
g(η(x))
[
T∞s Ψ(α
N,
ηx,y)− T
∞
s Ψ(α
N,
η )
]
− 〈DT∞s Ψ(α
N,
η ),∆σ(α
N,
η )〉L2
〉
ds
∣
∣
∣
∣
∣
,
where DT∞s Ψ is defined in equation (2.42). Linearizing T
∞
s Ψ around α
N,
η yields
I ≤ R1 +
∣
∣
∣
∣
∣
∫ t
0
〈
µNt−s, N
2
∑
x∼y
g(η(x))DT∞s Ψ(α
N,
η )(α
N,
ηx,y − α
N,
η )
− 〈DT∞s Ψ(α
N,
η ),∆σ(α
N,
η )〉L2
〉
ds
∣
∣
∣
∣
∣
with an error term
(2.56) R1 =
∫ t
0
〈
µNt−s, N
2
∑
x∼y
g(η(x))
∣
∣
∣T∞s Ψ(α
N,
ηx,y)− T
∞
s Ψ(α
N,
η )
−DT∞s Ψ(α
N,
η )(α
N,
ηx,y − α
N,
η )
∣
∣
∣
〉
ds.
Substituting the definition (2.29) of the empirical measure into the right hand side then
yields
I ≤ R1 +
∣
∣
∣
∣
∣
∫ t
0
〈
µNt−s, N
2−d
∑
x∼y
g(η(x))DT∞s Ψ(α
N,
η )(δ
()
y
N
− δ()x
N
)
− 〈DT∞s Ψ(α
N,
η ),∆σ(α
N,
η )〉L2
〉
ds
∣
∣
∣
∣
∣
.
75
Quantitative hydrodynamic limits
The explicit expression for DT∞s Ψ, see (2.42), then yields
I ≤ R1
+
∣
∣
∣
∣
∣
∫ t
0
〈
µNt−s, F
′
(〈
S∞s α
N,
η , ϕ
〉
L2
)(
N−d
∑
x
g(η(x))
〈
DS∞s (α
N,
η )
∗ϕ,∆Nδ
()
x
N
〉
L2
−
〈
DS∞s (α
N,
η )
∗ϕ,∆σ(αN,η )
〉
L2
)〉
ds
∣
∣
∣
∣
∣
.
Next, up to an error R2, we replace the discrete Laplacian ∆N by its continuous version
∆ and, after an integration by parts, obtain that
I ≤ R1 + R2 +
∣
∣
∣
∣
∣
∫ t
0
〈
µNt−s, F
′
(
〈S∞s α
N,
η , ϕ〉L2
)
∫
Td
(
N−d
∑
x
g(η(x))δ()x
N
(u)
− σ(αN,η (u))
)
∆
(
DS∞s (α
N,
η )
∗ϕ
)
(u) du
〉
ds
∣
∣
∣
∣
∣
.
The explicit expression for the error term is
(2.57) R2 =
∣
∣
∣
∣
∣
∫ t
0
〈
µNt−s, F
′
(
〈S∞s α
N,
η , ϕ〉L2
)
[
N−d
∑
x
g(η(x))〈DS∞s (α
N,
η )
∗ϕ,∆Nδ
()
x
N
−∆δ()x
N
〉L2
]〉
ds
∣
∣
∣
∣
∣
Recall that DS∞s (α
N,
η )
∗ϕ ∈ L2(0,∞;H2(Td)). Thus we apply Ho¨lder’s inequality to
obtain
I ≤ R1 + R2 + ‖F ′‖L∞×
(∫ t
0
∫
Td
〈
µNt−s,
( 1
Nd
∑
x
g(η(x))δ()x
N
(u)− σ(αN,η (u))
)2〉
duds
)1/2
×
∥
∥∆u(DS
∞
s (α
N,
η )
∗ϕ)
∥
∥
L∞η L2s,u
.
Since t ≤ T , Lemma 2.4.19 yields
I ≤ R1 + R2 +
√
T
∥
∥∆u(DS
∞
s (α
N,
η )
∗ϕ)
∥
∥
L∞η L2s,u
rRL(%, l, , N).
It remains to find a bound on R1 and R2 to finish the proof of the consistency result.
Since ∫
Td
αN,η (u) du =
∫
Td
αN,ηx,y(u) du,
76
2.4. Hydrodynamic limit for zero range processes
the first error term (2.56) is bounded from above by
R1 ≤ N2+dT sup
t∈[0,T ]
sup
s∈[0,t]
〈µNt−s,
[
T∞s Ψ
]
C1,θΛ
Λ(αN,ηx,y , α
N,
η )‖α
N,
ηx,y − α
N,
η ‖
1+θ
H−1〉,
where we have used the notation of Definition 2.4.16. To calculate the difference ‖αN,ηx,y −
αN,η ‖
1+θ
H−1 , we test with a function ω ∈ H
1(Td). Thus we obtain that
〈ω, αN,ηx,y − α
N,
η 〉L2 =
∫
ω(u)
1
Nd
(
δ()y
N
− δ()x
N
)
du
=
∫
1
Nd
(
ω(u+
y
N
)− ω(u+
x
N
)
)
δ()0 du
≤
C
Nd+1
‖∇ω‖L2‖δ
()
0 ‖L2 ,
and hence
‖αN,ηx,y − α
N,
η ‖H−1 ≤ C
−d/2N−(d+1).
The second error term (2.57) is bounded from above by
R2 ≤ g∗‖F ′‖L∞
(∫ t
0
〈
µNt−s,
∑
x∈TdN
η(x)
Nd
〉2
ds
) 1
2
× ‖∆DS∞s (α
N,
η )
∗ϕ‖L∞η L2s,u‖(−∆)
−1(∆N −∆)δ
()
0 ‖L2u .
Now it holds that
(2.58) ‖(−∆)−1(∆N −∆)δ
()
0 ‖L2 ≤ CN
−2‖D2δ()0 ‖L2 ≤
C
N22+d/2
.
Hence we obtain that
R2 ≤ C
√
T‖∆DS∞s (α
N,
η )
∗ϕ‖L∞η L2s,uN
−2−(2+d/2),
which completes the proof of Lemma 2.4.20 if t ≤ T . If t > T , all bounds remain valid
for ∣
∣
∣
∣
∣
∫ T
0
〈
µNt−s,
(
GNpiN − piNG∞
)
T∞s Ψ
〉
ds
∣
∣
∣
∣
∣
,
and it remains to estimate
II :=
∣
∣
∣
∣
∣
∫ t
T
〈
µNt−s,
(
GNpiN − piNG∞
)
T∞s Ψ
〉
ds
∣
∣
∣
∣
∣
.
77
Quantitative hydrodynamic limits
Again we will have three contributions to this quantity. The first one equals
R˜1 =
∫ t
T
〈
µNt−s, N
2
∑
x∼y
g(η(x))
∣
∣
∣T∞t Ψ(α
N,
ηx,y)− T
∞
t Ψ(α
N,
η )
−DT∞t Ψ(α
N,
η )(α
N,
ηx,y − α
N,
η )
∣
∣
∣
〉
ds.
It holds that
R˜1 ≤ CN2+d sup
t≥T
∫ t
T
〈
µNt−s, sup
x∼y
∣
∣T1(α
N,
ηx,y , α
N,
η ;T
∞
s Ψ)
∣
∣N−d
∑
z
η(z)
〉
ds,
where the second supremum is taken over all neighbours x and y in TdN . The second
contribution to II is given by
R˜2 ≤
∣
∣
∣
∣
∣
∫ t
T
〈
µNt−s, F
′(〈S∞s α
N,
η , ϕ〉L2)
(
N−d
∑
x
g(η(x))〈DS∞s (α
N,
η )
∗ϕ,∆Nδ
()
x
N
−∆δ()x
N
〉L2
)〉
ds
∣
∣
∣
∣
∣
.
Since the mass is conserved and bounded, it follows that
R˜2 ≤ C‖F ′‖L∞‖(∆−∆N)δ
()
0 ‖H−1 sup
η∈XN
∫ ∞
T
‖∇DS∞s (α
N,
η )
∗ϕ‖L2 ds
Similarly to (2.58), we obtain
‖(∆−∆N)δ
()
0 ‖H−1 ≤ CN
−2‖D3δ()0 ‖L2 ≤ C
−(3+d/2)N−2.
Thus we have shown
R˜2 ≤ C−(3+d/2)N−2 sup
η∈XN
∫ ∞
T
‖∇DS∞s (α
N,
η )
∗ϕ‖L2 ds
Finally, the remaining term is bounded by
∣
∣
∣
∣
∣
∫ t
T
〈
µNt−s, F
′(〈S∞s α
N,
η , ϕ〉L2)
∫
Td
∇
(
N−d
∑
x
g(η(x))δ()x
N
(u)
− σ(αN,η (u))
)
∇
(
DS∞s (α
N,
η )
∗ϕ
)
(u) du
〉
ds
∣
∣
∣
∣
∣
.
78
2.4. Hydrodynamic limit for zero range processes
Ho¨lder’s inequality yields a bound by
‖F ′‖L∞
∫ t
T
〈
µNt−s,
∥
∥
∥∇
(
N−d
∑
x
g(η(x))δ()x
N
(u)− σ(αN,η (u))
)∥∥
∥
L2
×
∥
∥
∥∇
(
DS∞s (α
N,
η )
∗ϕ
)∥∥
∥
L2
〉
ds.
Since g and σ are uniformly Lipschitz-continuous, the first term is bounded by
∥
∥
∥∇
(
N−d
∑
x
g(η(x))δ()x
N
− σ(αN,η )
)∥∥
∥
L2
≤ CN−d
∑
x
η(x)‖∇δ()x
N
‖L2 .
By the conservation of particles, this term is uniformly bounded by C−(1+d/2), which
completes the proof.
Remark 2.4.21. Similarly to the proof of Lemma 2.4.18, where we needed to take the
time-derivative at t = 0 and which can be understood as proving
piNG∞T∞t Ψ = pi
NT∞t G
∞Ψ = DT∞t Ψ(∆σ(α
N,)),
the term R1 can be seen as capturing the “commutation relation”
GNpiNT∞t Ψ(η) ≈ DT
∞
t Ψ(G
NαN,η ).
Closer inspection of the proof shows that in order to prove the former relationship between
G∞ and T∞t , we just need θ > 0 in the stability result, Lemma 2.4.14. On the other hand,
we need θ > 1/(d+1) in order to guarantee that the error term R1 coming from the latter
relationship between GN and T∞t vanishes in the limit.
2.4.4 Proof of the hydrodynamic limit
Using the stability and consistency results, we can prove the hydrodynamic limit.
Proof of Theorem 2.4.6. Let us split the left hand side of (2.31) into three separate con-
tributions which we shall call T1 to T3. It holds that
∣
∣
∣
〈
µNt , F
(
〈αNη , ϕ〉
)〉
− F
(
〈ft, ϕ〉L2
)∣∣
∣
≤
∣
∣
∣
〈
µNt , F
(
〈αNη , ϕ〉
)〉
− F
(
〈αN,η , ϕ〉L2
)∣∣
∣+
∣
∣
∣
〈
µNt , F
(
〈αN,η , ϕ〉L2
)〉
− F
(
〈ft, ϕ〉L2
)∣∣
∣.
As before, the definitions of Ψ, see (2.19), and the generators TNt and T
∞
t yield
∣
∣
∣
〈
µNt , F
(
〈αN,η , ϕ〉L2
)〉
− F
(
〈ft, ϕ〉L2
)∣∣
∣ =
∣
∣
∣
〈
µN0 , T
N
t pi
NΨ− T∞t Ψ(f0)
〉∣∣
∣.
79
Quantitative hydrodynamic limits
If we add and subtract 〈µN0 , pi
NT∞t Ψ〉 = 〈µ
N
0 , T
∞
t Ψ(α
N,
η )〉, we obtain that the right hand
side is bounded by
∣
∣
∣
〈
µN0 , T
N
t pi
NΨ− piNT∞t Ψ
〉∣∣
∣+
∣
∣
∣
〈
µN0 , T
∞
t Ψ(α
N,
η )− T
∞
t Ψ
(
f0
)〉∣∣
∣
+
∣
∣
∣
〈
µNt ,Ψ
(
αNη
)
−Ψ
(
αN,η
)〉∣∣
∣
=: T1 + T2 + T3,
where we have written Ψ(αNη ) = F (〈α
N
η , ϕ〉) in analogy with (2.19), even though α
N
η is
not in H. As in Section 2.3 it holds that
d
ds
(
TNs pi
NT∞t−s(η)
)
= TNs G
NpiNT∞t−sΨ(η)− T
N
s pi
NG∞T∞t−sΨ(η).
Hence Lemma 2.4.20 yields
T1 ≤
∫ t
0
∣
∣〈SNs µ
N
0 , (G
NpiN − piNG∞)(T∞t−sΨ)(η)〉
∣
∣ ds ≤ rC(T, %, l, , N),
where rC(T, %, l, , N) is given in the consistency lemma 2.4.20 as
C
(
−
dθ
2 N1−θ(d+1)T sup
0≤s≤t
sup
x∼y
〈
µNt−s,
[
T∞s Ψ
]
C1+θ
Λ(αN,ηx,y , α
N,
η )N
−d
∑
z η(z)
〉
+ −(3+
d
2 ) sup
η∈XN
∫ ∞
T
‖∇(DS∞s (α
N,
η )
∗ϕ)‖L2(Td) ds
+N2+d sup
t≥T
∫ t
T
〈
µNt−s, sup
x∼y
∣
∣T1(α
N,
ηx,y , α
N,
η ;T
∞
s Ψ)
∣
∣
〉
ds
+
∥
∥∆(DS∞t (α
N,
η )
∗ϕ)
∥
∥
L∞η L
2
t,u
√
T
(
rRL(%, l, , N) + 
−(2+ d2 )N−2
))
.
Next, using the stability result, we need to make the expression for rC(T, %, l, , N) explicit
in terms of l, , and N . Let us consider first the two middle terms, which describe the
large–time behaviour t ≥ T . Lemma 2.4.15 yields
∫ ∞
T
‖∇DS∞s (α
N,
η )
∗ϕ‖L2 ds ≤ Ce
−cT‖∇ϕ‖L2
as well as
∫ ∞
T
∣
∣T1(α
N,
ηx,y , α
N,
η ;T
∞
s Ψ)
∣
∣ ds ≤ Ce−cT
(
‖αN,ηx,y −N
−d
∑
z η(z)‖
2
L4
+ ‖αN,η −N
−d
∑
z η(z)‖
2
L4
)
,
80
2.4. Hydrodynamic limit for zero range processes
since ∫
Td
αN,ηx,y(u) du =
∫
Td
αN,η (u) du =
1
Nd
∑
z∈TdN
η(z).
The bound δ()0 (u) ≤ C
−d yields
‖αN,η ‖
4
L4u
=
∫
Td
∣
∣
∣
1
Nd
∑
x∈TdN
η(x)δ()x
N
(u)
∣
∣
∣
4
du ≤ C−4d
( 1
Nd
∑
z∈TdN
η(z)
)4
.
Collecting the three previous equations, we have shown that
sup
t≥T
∣
∣T1(α
N,
ηx,y , α
N,
η ;T
∞
t Ψ)
∣
∣ ≤ Ce−cT −2d
( 1
Nd
∑
z∈TdN
η(z)
)2
uniformly in x, y, and η. Conservation of the number of particles and the assumption
µN0 ≤ ν
N
ρ now yield uniform bounds
〈
µNt ,
( 1
Nd
∑
x∈TdN
η(x)
)m〉
=
〈
µN0 ,
( 1
Nd
∑
x∈TdN
η(x)
)m〉
≤
〈
νNρ ,
( 1
Nd
∑
x∈TdN
η(x)
)m〉
≤ Cm
(2.59)
for each m > 0.
The stability results of Lemma 2.4.14 yield
sup
s≥0
[
T∞s Ψ
]
C1,θΛ
≤ C.
Furthermore, it holds that
‖αN,η ‖L∞ ≤ C
1
dNd
∑
x∈TdN
η(x)
as well as
‖αN,η ‖Hk ≤ N
−d
∑
x∈TdN
η(x)‖δ()‖Hk ≤ C
−(k+ d2 )N−d
∑
x∈TdN
η(x).
Corollary 2.4.17 yields
Λ(αN,η ) ≤ C
(
1 + −(k+
d
2 )
d+4
2k+2−d
( 1
Nd
∑
x∈TdN
η(x)
) d+4
2k+2 +1
+ −(1+
d
2 )(d+4)
2k2+k
2k+2 −kd(d+4)−d
( 1
Nd
∑
x∈TdN
η(x)
)( 2k
2+k
2k+2 +k)(d+4)+1
)
81
Quantitative hydrodynamic limits
Since k > (d+ 2)/2 and  < 1 is small, conservation of the number of particles yields
〈
µNt ,Λ(α
N,
η )N
−d
∑
z η(z)
〉
≤ C−(1+
d
2 )(d+4)
2k2+k
2k+2 −kd(d+4)−d.
for some constant C depending on the quantities k, d, and ρ appearing in the statement
of the hydrodynamic limit. Replacing above η by ηx,y yields the identical bound
〈
µNt ,Λ(α
N,
η , α
N,
ηx,y)N
−d
∑
z η(z)
〉
≤ C−(1+
d
2 )(d+4)
2k2+k
2k+2 −kd(d+4)−d.
Finally we just have to recall that Lemma (2.4.14) (iv) yields the uniform bound
‖∆uDS
∞
s (α
N,
η )
∗ϕ‖L∞η L2s,u ≤ C,
in order to obtain that
T1 ≤ Ck
(
−(1+
d
2 )(d+4)
2k2+k
2k+2 −kd(d+4)−d−θ
d
2N−θ(d+1) + −(2+
d
2 )N−2
+ e−cTN2+d−2d + T
1
2 rRL(%, l, , N)
)
.
The stability estimate (2.37) yields
T2 =
∣
∣
〈
µN0 , T
∞
t Ψ
(
αN,η
)
− T∞t Ψ(f0)
〉∣
∣ ≤ ‖F ′‖L∞‖ϕ‖L∞〈µ
N
0 , ‖α
N,
η − f0‖L1〉.
The stability estimate (2.38) yields the validity of Remark 2.4.8, once we have substracted
∫
Td ϕ(y) du from ϕ. Finally, the last term equals
T3 =
∣
∣
〈
µNt , F (〈α
N
η , ϕ〉)− F (〈α
N,
η , ϕ〉L2)
〉∣
∣ ≤ ‖F ′‖L∞
∣
∣
〈
µNt , 〈α
N
η − α
N,
η , ϕ〉L2
〉∣
∣ .
Note that
〈αNη − α
N,
η , ϕ〉 =
1
Nd
∑
x∈TdN
η(x)
∫
Td
1
d
χ(u )(ϕ(
x
N )− ϕ(u+
x
N )) du
=
1
Nd
∑
x∈TdN
η(x)
∫
Td
χ(u)(ϕ( xN )− ϕ(u+
x
N ))) du,
and hence
∣
∣〈αNη − α
N,
η , ϕ〉
∣
∣ ≤ C‖∇ϕ‖L2
1
Nd
∑
x∈TdN
η(x)
Since the total mass is conserved and F ∈ C2b (T
d) is Lipschitz-continuous, it follows that
T3 ≤ C‖F ′‖L∞‖∇ϕ‖L2
〈
µNt ,
1
Nd
∑
x∈TdN
η(x)
〉
≤ C,
82
2.5. Convergence of the entropy
since the average number of particles is bounded. Collecting all available bounds, this
completes the proof of the hydrodynamic limit.
Remark. As in the proof of the hydrodynamic limit for independent random walks,
Theorem 2.3.1, the term T1 is a measure for the difference of the particle semigroup
and the limit semigroup on the level of observables and is (for fixed Ψ) bounded by a
consistency result on the two generators. Furthermore we have seen that this consistency
estimate is transported along the flow of the limit equation by the stability result. The
second term T2 measures the propagation of the difference between the initial data of
the particle system and the limit partial differential equation along the flow of the limit
equation. Hence it is bounded by a stability result on the limit equation. The last
term T3 did not appear in the proof of Theorem 2.3.1 and measures the error due to the
mollification of the empirical measure.
2.5 Convergence of the entropy
One important question in the field of statistical mechanics is the convergence of the
microscopic entropy N−dHN(µNt |ν
N
ρ ) towards the macroscopic entropy. In this section we
investigate this problem and its relation to entropic chaos. The problem can be thought
of independently from the results of the previous sections, once a hydrodynamic limit has
been established. In order to make this explicit, let us consider a zero range process with
generator (2.2) and with the filtration equation (2.3) as the limit equation. Suppose that
the function space for the limit equation is H ⊆ L∞(Td). The hydrodynamic limit will
be codified in Assumption 2. Throughout this section we suppose that any rate functions
r(N), which is only a function of N , vanishes in the limit as N →∞ and we absorb any
constants in the rate function, i.e. without loss of generality Cr(N) = r(N). Furthermore
suppose that all rate functions are polynomial in N .
Assumption 2. Let us fix solutions (ft)t≥0 and (µNt )t≥0 to the limit equation and the zero
range process, respectively, and assume that they satisfy a hydrodynamic limit, i.e. for all
F ∈ C2b (R), ϕ ∈ C
3(Td), and N ∈ N, it holds that
〈
µNt , F
(
〈αNη , ϕ〉L2
)
− F
(
〈ft, ϕ〉L2
)〉
≤ ‖F‖C2(1 + ‖ϕ‖
2
H1 + ‖ϕ‖C3)rHL(N)
for a rate function rHL(N) which vanishes as N →∞, cf. Corollary 2.4.9.
Furthermore assume that the limit solution ft ∈ H satisfies ft ∈ C3(Td) and that there
exists a constant c > 0 such that ft ≥ c for all t ≥ 0. Finally assume a replacement
83
Quantitative hydrodynamic limits
lemma with a rate rRL(,N), i.e.
1
TNd
∫ T
0
∑
x∈TdN
〈
µNt ,
∣
∣(g ◦ η)()(x)− σ(η()(x))
∣
∣
〉
dt ≤ rRL(,N)
for  > 0, N ∈ N such that N < 1 and all T > 0, where we suppose that
lim sup
→0
lim sup
N→∞
rRL(,N) = 0.
To keep notation simple, we allow all constants in this section (in contrast to the last) to
depend on (ft)t∈[0,∞) ⊂ C3(Td), although it is possible in principle to keep track of this
dependence as well. We set
f∞ =
∫
Td
ft(u) du,
which is independent of t since ft solves the limit equation (2.3). The notation is fur-
thermore justified on noting that we expect ft → f∞ as t → ∞, cf. Lemma 2.4.15.
Furthermore we denote the pressure by
(2.60) p(λ) = logZ(eλ),
where Z is the partition function given in equation (2.7). Then we define the macroscopic
entropy as
(2.61) H∞(ft) :=
∫
Td
h(ft(u)) du− h(f∞),
where the function h is given by
h(ρ) = ρ log σ(ρ)− p
(
log σ(ρ)
)
.
Let us find the corresponding macroscopic Fisher information by differentiating in time.
It holds that
d
dt
H∞(ft) =
∫
Td
(
∂tft log σ(ft) + ft
σ′(ft)
σ(ft)
∂tft − p
′(log σ(ft))
σ′(ft)
σ(ft)
∂tft
)
du.
Since σ is the inverse function of ρ∂ρ logZ(ρ), we find that p′(λ) = σ−1(eλ) and hence
(2.62)
d
dt
H∞(ft) = −
∫
Td
|∇σ(ft(u))|2
σ(ft(u))
du =: −D∞(ft),
where D∞(ft) is called the macroscopic Fisher information. Next we establish a micro-
scopic analogue of equation (2.62), relating the microscopic entropy HN(µNt |ν
N
ρ ) and its
84
2.5. Convergence of the entropy
Fisher information DN(µNt |ν
N
ρ ), to be defined presently. Let f
N
t ∈ Cb(XN) denote the
density of µNt ∈ P (XN) with respect to the grand-canonical measure ν
N
f∞ ∈ P (XN), i.e.
set
(2.63) fNt (η) :=
dµNt
dνNf∞
(η).
The microscopic Fisher information is then defined as
DN(µNt |ν
N
f∞) :=
∫
XN
√
fNt N
−2GN
√
fNt dν
N
f∞(2.64)
=
〈√
fNt , N
−2GN
√
fNt
〉
L2(νNf∞ )
.
Remark 2.5.1. Abusing notation, we shall sometimes refer to DN(µNt |ν
N
f∞) by D
N(fNt |ν
N
f∞),
where fNt is the density defined in (2.63). Also note that we have left out a factor of N
2 as
opposed to the natural (macroscopic) time-scaling, i.e. the time scale of the microscopic
Fisher information is the microscopic time scale.
As a first result we show the equivalence of the convergence of the entropy and entropic
chaos, by which we mean
lim
N→∞
1
Nd
HN(µNt |ν
N
ft(·)) = 0.
.
Lemma 2.5.2. Under assumption 2, it holds that
1
Nd
HN(µNt |ν
N
f∞) = H
∞(ft) +
1
Nd
HN(µNt |ν
N
ft(·)) + O
( 1
N
+ rHL(N)
)
.
In particular, the microscopic entropy N−dHN(µNt |ν
N
f∞) converges to the macroscopic en-
tropy H∞(ft) if and only if there is entropic chaos.
Proof. It holds that
1
Nd
HN(µNt |ν
N
f∞) =
1
Nd
∫
XN
log
(
dµNt
dνNf∞
)
dµNt
=
1
Nd
∫
XN
log
(
dµNt
dνNft(·)
)
dµNt +
∫
XN
log
(dνNft(·)
dνNf∞
)
dµNt .
By definition, it also holds that
(2.65)
dνNft(·)
dνNf∞
(η) =
∏
x∈TdN
Z(σ(f∞))
Z(σ(ft( xN )))
(
σ(ft( xN ))
σ(f∞)
)η(x)
.
85
Quantitative hydrodynamic limits
Consequently, the second term equals
1
Nd
∫
XN
log
(dνNft(·)
dνNf∞
)
dµNt
=
1
Nd
∑
x∈TdN
∫
XN
(
log
Z(σ(f∞))
Z(σ(ft( xN )))
+ η(x) log
σ(ft( xN ))
σ(f∞)
)
dµNt (η).
By Assumption 2, the macroscopic solution ft is differentiable, and the hydrodynamic
limit yields that the right hand side converges to
∫
Td
ft(u) log σ(ft(u)) du−
∫
Td
p
(
log σ(ft(u))
)
du− f∞ log σ(f∞) + p
(
log σ(f∞)
)
as N →∞. Thus, in view of (2.61), we have shown that
1
Nd
HN(µNt |ν
N
f∞) =
1
Nd
HN(µNt |ν
N
ft(·)) +H
∞(ft) + O
( 1
N
+ rHL(N)
)
.
which concludes the proof.
The main result of this section is the following theorem.
Theorem 2.5.3. Under Assumption 2, let the initial microscopic entropy converge, i.e.
let
(2.66)
∣
∣
∣
1
Nd
HN(µN0 |ν
N
f∞)−H
∞(f0)
∣
∣
∣ ≤ rH,0(N)
for some rate function rH,0(N). Then both the microscopic entropy and the time aver-
age of the Fisher information converge towards the corresponding macroscopic quantities.
Specifically, it holds that
∣
∣
∣
1
Nd
HN(µNt |ν
N
f∞)−H
∞(ft)
∣
∣
∣ ≤ rH,0(N) + C
(
+
1
N
+
1
3+d
rHL(N) + t rRL(,N)
)
,
and
∣
∣
∣
∫ t
0
4N2−dDN(µNs |ν
N
f∞) ds−
∫ t
0
D∞(fs) ds
∣
∣
∣
≤ rH,0(N) + C
(
+
1
N
+
1
3+d
rHL(N) + t rRL(,N)
)
for all  > 0, N ∈ N, and t ≥ 0 (note that this bound is not uniform in time). In
86
2.5. Convergence of the entropy
particular, it holds that
lim
N→∞
1
Nd
HN(µNt |ν
N
f∞) = H
∞(ft) and
lim
N→∞
∫ t
0
4N2−dDN(µNs |ν
N
f∞) ds =
∫ t
0
D∞(fs) ds
for all t ≥ 0.
The key to the proof of Theorem 2.5.3 is the following lemma.
Lemma 2.5.4. Under the assumptions of Theorem 2.4.6, it holds that
1
Nd
HN(µNt |ν
N
f∞) ≥ H
∞(ft)− C
( 1
N
+ rHL(N)
)
,
as well as
∫ t
0
4N2−dDN(µNs |ν
N
f∞) ds ≥
∫ t
0
D∞(ft) ds− C
( 1
N
+ +
1
3+d
rHL(N) + trRL(,N)
)
.
In particular, it holds that
lim inf
N→∞
1
Nd
HN(µNt |ν
N
f∞) ≥ H
∞(ft) and lim inf
N→∞
∫ t
0
4
N2
Nd
DN(µNs |ν
N
f∞) ds ≥ D
∞(ft)
for all t ≥ 0.
Proof. The bound on the relative entropy is a direct consequence of Lemma 2.5.2 and
the fact that all entropies HN are non-negative. As a warm-up for the bound on the
Fisher-information, let us give an alternative proof. The well-known variational formula
for the relative entropy, see [47], yields
HN(µNt |ν
N
f∞) = sup
f∈Cb(XN )
{
〈µNt , f〉 − log〈ν
N
f∞ , e
f〉
}
,
where the supremum is (formally) obtained by taking f = log dµNt /dν
N
f∞ . In view of the
hydrodynamic limit and Yau’s results using the relative entropy method, we expect that
µNt is close to the local Gibbs state ν
N
ft(·)
. Thus we choose
f(η) = log
dνNft(·)
dνNf∞
(η).
Then we obtain a lower bound
1
Nd
HN(µNt |ν
N
f∞) ≥
1
Nd
∫
XN
log
dνNft(·)
dνNf∞
(η)dµNt (η).
87
Quantitative hydrodynamic limits
Equation (2.65) thus yields a lower bound on the entropy of the form
1
Nd
∑
x∈TdN
∫
XN
(
η(x) log
σ
(
ft
(
x
N
))
σ(f∞)
+ log
Z(σ(f∞))
Z
(
σ
(
ft
(
x
N
)))
)
dµNt (η).
As in the proof of Lemma 2.5.2, since the macroscopic solution ft is differentiable, the
hydrodynamic limit yields a bound from below by
H∞(ft)− C
( 1
N
+ rHL(N)
)
.
To prove a similar estimate for the Fisher information, we need the following variational
formula, cf. [47]. It holds that
DN(µNt |ν
N
f∞) = sup
f
{
−
∫
XN
N−2GNf(η)
f(η)
dµNt (η)
}
,
where the supremum is taken over all positive f ∈ Cb(XN) such that f is strictly bounded
away from zero. The supremum is (formally) obtained at the function f =
√
dµNt /dν
N
f∞ ,
so here we choose
f(η) =
√
dνNft(·)
dνNf∞
(η).
After cancelling factors, this corresponds to taking
f(η) =
∏
x∈TdN
√
σ
(
ft(
x
N
))
η(x)
.
We obtain that
N−2GNf(η) =
∑
x∈TdN
∑
|e|=1
g(η(x))
(√
σ
(
ft
(
x+e
N
))
σ
(
ft
(
x
N
)) − 1
)
f(η),
and hence
GNf(η)
N2f(η)
=
∑
x∈TdN
∑
|e|=1
g(η(x))
√
σ
(
ft( xN
))
(√
σ
(
ft
(x+ e
N
))
−
√
σ
(
ft
( x
N
))
)
.
Hence it holds that
4N2−dDN(µNt |ν
N
f∞) ≥ −4
∫
XN
1
Nd
∑
x∈TdN
g(η(x))
√
σ
(
ft( xN
))∆N
√
σ
(
ft
( x
N
))
dµNt (η),
where we recall that ∆N denotes the discrete Laplacian (2.22). Even though f as chosen
88
2.5. Convergence of the entropy
above is not strictly bounded away from zero (uniformly in η), this can be made rigorous
by a standard approximation argument. Since ft ∈ C3(Td) is bounded away from zero,
we replace g(η(x)) by (g ◦ η)()(x), up to an error of O(). Now the replacement lemma
yields
∫ t
0
4N2−dDN(µNs |ν
N
f∞) ds ≥ −4
∫ t
0
∫
XN
(
1
Nd
∑
x∈TdN
σ(η()(x))
√
σ
(
fs( xN
))
×∆N
√
σ
(
fs
( x
N
))
)
dµNs (η)ds− C(+ trRL(,N)).
It holds that η()(x) = 〈αNη , δ
()
x
N
〉 and the following bounds are satisfied:
‖δ()x
N
‖C3 ≤ 
−3−d as well as ‖δ()x
N
‖2H1 ≤ 
−2−d.
Hence the hydrodynamic limit yields
4N2−dDN(µNt |ν
N
f∞) ≥ −4
∫
Td
√
σ(ft(u))∆N
√
σ(ft(u)) du
− C(+ trRL(,N) + 
−3−drHL(N)).
Finally we replace, up to an error O(N−1), the discrete Laplacian ∆N by its continuous
version ∆ and note that
−4
∫
Td
√
σ(ft(u))∆
√
σ(ft(u)) du =
∫
Td
|∇σ(ft(u))|2
σ(ft(u))
du.
This completes the proof of Lemma 2.5.4.
Proof of Theorem 2.5.3. Equation (2.62) yields
(2.67) H∞(ft) +
∫ t
0
D∞(fs) ds = H∞(f0).
Next, we establish a corresponding microscopic relation. First we note that fNt defined
in (2.63) satisfies the forward Kolmogorov equation
∂tf
N
t (η) = G
NfNt (η).
Since GN is self-adjoint in L2(νNf∞) and in particular 〈G
NfNt , 1〉L2(νNf∞ ) = 0, it holds that
(2.68)
d
dt
HN(µNt |ν
N
f∞) =
d
dt
∫
XN
fNt log f
N
t dν
N
f∞ =
∫
XN
(
GNfNt
)
log fNt dν
N
f∞ .
89
Quantitative hydrodynamic limits
Now we note that
DN(µNt |ν
N
f∞) =
〈
√
fNt , N
−2GN
√
fNt
〉
L2(νNf∞ )
=
1
2
∫
XN
∑
x∈TdN
∑
|e|=1
g(η(x))
(√
fNt (ηx,x+e)−
√
fNt (η)
)2
dνNf∞ ,
see for example [47, Appendix 1]. Polarization yields
∫
XN
(
N−2GNfNt
)
log fNt dν
N
f∞ = 〈N
−2GNfNt , log f
N
t 〉L2(νNf∞ )
=
1
2
∫
XN
∑
x∈TdN
|e|=1
g(η(x))
(
fNt (η
x,x+e)− fNt (η)
)(
log fNt (η
x,x+e)− log fNt (η)
)
dνNf∞ .
Therefore the elementary inequality (
√
a −
√
b)2 ≤ (a − b)(log a − log b)/4, which holds
for all a, b > 0, yields
∫
XN
(
N−2GNfNt
)
log fNt dν
N
f∞ ≥ 4
∫
XN
√
fNt N
−2GN
√
fNt dν
N
f∞ = 4D
N(µNt |ν
N
f∞).
Therefore time-integration of equation (2.68) yields
(2.69) HN(µNt |ν
N
f∞) + 4N
2
∫ t
0
DN(µNs |ν
N
f∞) ds ≤ H
N(µN0 |ν
N
f∞).
Note that this inequality holds for any ρ > 0 replacing f∞ > 0. Since the microscopic
entropy converges initially and Lemma 2.5.4 yields lower bounds on each term on the left
hand side, a simple technical lemma completes the proof, i.e. we set
a =
1
Nd
HN(µNt |ν
N
f∞)−H
∞(ft) and b =
∫ t
0
(
4N2−dDN(µNs |ν
N
f∞)−D
∞(fs)
)
ds
in Lemma 2.5.5.
Lemma 2.5.5. Let a, b ∈ R, and ri > 0, i = 1, 2, 3 be any reals such that
|a+ b| ≤ r1, a ≥ −r2, and b ≥ −r3.
Then it holds that
|a| ≤ r1 + r2 + r3 and |b| ≤ r1 + r2 + r3.
Proof. For any λ ∈ R, we set
λ+ = max{0, λ} and λ− = max{0,−λ},
90
2.6. The replacement lemma
i.e. λ = λ+ − λ−, |λ| = λ+ + λ−. The first inequality of the assumptions yields
a+ b = a+ − a− + b+ − b− ≤ |a+ b| ≤ r1.
On the other hand, the other two assumptions yield
a− ≤ r2 and b− ≤ r3
and hence
a+ + b+ ≤ r1 + r2 + r3.
Since both terms on the left hand side are positive and we already established bounds on
a− and b−, this shows the hypothesis.
As a consequence of Lemma 2.5.2 and Theorem 2.5.3, we recover Yau’s result on entropic
chaos. Of course, our proof is very different in spirit, since we assumed the hydrodynamic
limit in order to arrive at entropic chaos. In other words, we conclude the conservation
of the hydrodynamic limit in the strong form of Theorem 2.2.3 from the conservation of
the hydrodynamic limit in the weak form of Theorem 2.2.2.
Corollary 2.5.6. Under Assumption 2, suppose that entropic chaos holds initially, i.e.
lim
N→∞
1
Nd
HN
(
µN0 |ν
N
f0(·)
)
= 0.
Then this property is conserved along the evolution of the process, i.e.
lim
N→∞
1
Nd
HN
(
µNt |ν
N
ft(·)
)
= 0 for all t ≥ 0.
Note that so far we have not proved that the rate of convergence of the microscopic
entropy is uniform in time. This should be done in future work.
2.6 The replacement lemma
In this section we want to prove the replacement lemma, Lemma 2.4.19. This is a quan-
titative L2–version of the usual replacement lemma found in the literature, e.g. [47],with
an explicit estimate on its rate of convergence. The biggest difference from the classical
proof of Guo, Papanicolaou, and Varadhan [36] lies in the use of a logarithmic Sobolev
inequality (LSI) to obtain a rate of convergence. The classical reference for LSI is [33]; for
the zero range process, the LSI has been proven in [27] using only that the rate function
satisfies Assumption 1 (i)-(iii). The plan of this section is as follows. In Subsection 2.6.1,
91
Quantitative hydrodynamic limits
we use Assumption 1 (iv) to deduce uniform (in N) bounds on all moments. We state the
block estimates and use them to deduce the replacement lemma in Subsection 2.6.2. Sub-
section 2.6.3 deals with the equivalence of ensembles, a statement concerning the closeness
of grand-canonical and canonical measures and which appears in the proof of the block
estimates. In Subsection 2.6.4 we show how to restrict ourselves to bounded particle
configurations and consequently introduce the density bound %, which can be understood
as introducing a density scale. Finally, we prove the block estimates in Subsections 2.6.5
and 2.6.6.
The aim of this section is to prove Lemma 2.4.19, which we restate here for the reader’s
convenience.
Lemma (A quantitative replacement lemma). Assuming that the initial data possess
bounded relative entropy and are bounded with respect to some Gibbs measure, i.e.
HN
(
µN0 |ν
N
ρ
)
≤ CNd, and µN0 ≤ ν
N
ρ
for some ρ > 0. Then it holds that
(
1
T
∫ T
0
∫
Td
〈
µNt ,
∣
∣(g ◦ η)()(uN)− σ(η()(uN))
∣
∣2
〉
dudt
) 1
2
≤ rRL(%, l, , N),
where we recall definition (2.34). The rate function rRL satisfies
rRL(%, l, N) ≤ C
(
(N−
1
2 l
1
2 + 
1
2 l
1
4 )%l
d
4 + %l−
d
4 + %−
1
4 +
l
N
)
for all N ∈ N, 1/N <  < 1, l < N , and % > 0.
The proof will take up the rest of this section. First we note the following consequence
of the bound on the initial microscopic entropy. Equation (2.69) yields
(2.70) HN
(
µNt |ν
N
ρ
)
≤ CNd and
1
t
∫ t
0
DN(µNt |ν
N
ρ ) dt ≤ CN
d−2
for all t ≥ 0 and N ∈ N.
2.6.1 Attractivity and moment bounds
This section is the only place where we need to take advantage of attractivity, i.e. Assump-
tion 1 (iv). This assumption is useful when combined with a coupling of two processes
and allows us to provide uniform estimates on the particle moments. Since we do not
make use of attractivity anywhere else and the theory of attractivity is well-developed,
92
2.6. The replacement lemma
see [47, 54], we simply sketch the results here.
Consider two copies of the zero range process with initial configurations η, ζ ∈ XN , which
satisfy
η ≤ ζ, i.e. η(x) ≤ ζ(x) for all x ∈ TdN .
Assumption 1 (iv) simply states g(n + 1) ≥ g(n) for all n ∈ N and hence we can always
let particles of the process with more particles jump at a higher rate. Specifically, at an
arbitrary site x ∈ TdN , at time t = 0 where we have η(x) ≤ ζ(x), we let one particle at
x ∈ TdN of both processes η and ζ jump at the same time at a rate g(η(x)) and additionally
let just one particles of ζ jump at a rate g(ζ(x))−g(η(x)) ≥ 0. This coupling almost surely
preserves the property η(x) ≤ ζ(x) for all x ∈ TdN . Thus we have arrived at a random
particle process (ηt, ζt)t≥0 (understood as random variables) with state space XN × XN
and whose marginals ηt and ζt each are zero range processes with jump rate g, such that
the property ηt(x) ≤ ζt(x) for all x ∈ TdN is almost surely preserved by the evolution of
the process.
A consequence of this coupling is the preservation of stochastic ordering. Recall that in
Section 2.4, we defined a function fN ∈ Cb(XN) to be monotonous if fN(η) ≤ fN(ζ) for
all η ≤ ζ. Two probability measures µ, ν ∈ P (XN) were said to be ordered, µ ≤ ν, if
〈µ, fN〉 ≤ 〈ν, fN〉 for all monotonous fN ∈ Cb(XN).
Suppose now µN0 , µ˜
N
0 ∈ P (XN) are two initial measures of the zero range process such
that
µ˜N0 ≤ µ
N
0 .
It can be shown [54, Theorem II.2.4] that this property is equivalent to the existence of a
coupling measure on XN ×XN with marginals µ˜N0 and µ
N
0 that concentrates on {η ≤ ζ}.
As shown above, under the evolution of the coupled process, the support of the coupled
probability measure remains within {η ≤ ζ} and it follows, again by [54, Theorem II.2.4],
that µ˜Nt ≤ µ
N
t .
Let us now turn to the problem of bounding moments of the particle system. We define
the k-th order moment as
Mk
[
µNt
]
:=
〈
µNt ,
1
Nd
∑
x∈TdN
η(x)k
〉
.
Recall that in Remark 2.4.10 we have already obtained a bound on the average number
of particles M1[µNt ] (which is conserved by the evolution).
Lemma 2.6.1. Assume that the initial measure is bounded from above by νNρ for some
93
Quantitative hydrodynamic limits
ρ > 0, i.e. µN0 ≤ ν
N
ρ . Then for any k > 0, it holds that
Mk
[
µNt
]
≤Mk
[
νNρ
]
= Ck < +∞
for all N > 0 and t ≥ 0.
Proof. Attractivity yields
µNt ≤ ν
N
ρ ,
and hence
Mk
[
µNt
]
≤Mk
[
νNρ
]
by monotonicity of η(x)k. All moments of νNρ are finite and translation-invariance yields
Mk
[
νNρ
]
= 〈νNρ , η(0)
k〉 = Ck
uniformly in N .
2.6.2 Proof of the replacement lemma from the block estimates
In order to prove the replacement lemma with the correct rates, we shall separate scales
and introduce another scaling parameter l. Thus we shall first prove a one block estimate,
which corresponds to the replacement lemma on blocks of size l instead of N . Then we
will prove a two blocks estimate which allows us to estimate the difference in the scales l
and N and conclude the replacement lemma with an explicit rate of convergence. Here we
shall also use that the one block estimate is proved on boxes where δ() is approximately
constant and hence there is no added difficulty compared to the classical replacement
lemma in [36] by introducing χ different from a characteristic function. Similarly to the
definition (2.34), where we defined weighted averages (h ◦ η)()(u), we define the average
number of a function h : N→ R of the number of particles over the translation of a box
Λl by x ∈ TdN to be
h ◦ η
l
(x) =
1
(2l + 1)d
∑
|y|≤l
h(η(x+ y)).
Furthermore, we set ηl = id ◦ η
l
.
Lemma 2.6.2 (A quantitative one block estimate). Under the assumptions of the Lemma
2.4.19, it holds that
(
1
TNd
∫ T
0
∑
x∈TdN
〈
µNt ,
∣
∣
∣g ◦ η
l
(x)− σ(ηl(x))
∣
∣
∣
2 〉
dt
) 1
2
≤ rOBE(%, l, N)
94
2.6. The replacement lemma
for some rate function satisfying
rOBE(%, l, N) = C
(
N−
1
2 l
2+d
4 %+ %l−
d
4 + %−
1
4
)
for all l, N ∈ N, T > 0, 0 < l < N , and % > 0.
The two blocks estimate estimates the error made when averaging over the smaller box
of size l instead of the box of size N (with a weighted average given by χ).
Lemma 2.6.3 (A quantitative two blocks estimate). Under the assumptions of Lemma
2.4.19, it holds that
(
sup
|y|≤N
1
TNd
∫ T
0
∑
x∈TdN
〈
µNt ,
∣
∣ηl(x+ y)− η()(x)
∣
∣2
〉
dt
) 1
2
≤ rTBE(%, l, , N)
for some rate function satisfying
rTBE(%, l, , N) = C
((
N−
1
2 l
1
2 + 
1
2 l
1
4
)
l
d
4%+ %l−
d
4 + %−
1
4 +
l
N
)
for all l, N ∈ N, T > 0, 1/N <  < 1, 0 < l < N , and % > 0.
From these two estimate we shall presently deduce the Lemma 2.4.19.
Proof of Lemma 2.4.19. First we replace the integral over u ∈ Td by am discrete sum
over x ∈ TdN , thus allowing us to only bound the quantity
(2.71)
(
1
TNd
∫ T
0
∑
x∈TdN
〈
µNt ,
∣
∣(g ◦ η)()( xN )− σ(η
()( xN ))
∣
∣2
〉
dt
) 1
2
.
This will be done as follows. Let us define
V˜N,x(η) := (g ◦ η)
()( xN )− σ(η
()( xN )).
For notational convenience we leave out the integration over t and η in comparison to
(2.71) from now on. It holds that
∣
∣
∣
∣
∫
Td
V˜ 2N,uN du−
1
Nd
∑
x∈TdN
V˜ 2N,x
∣
∣
∣
∣
1
2
≤
(
1
Nd
∑
x∈TdN
sup
|e|≤
√
d
∣
∣V˜ 2N,x+e − V˜
2
N,x
∣
∣
) 1
2
,
95
Quantitative hydrodynamic limits
where the supremum is taken over all e ∈ Rd such that |e| ≤
√
d. It holds that
1
Nd
∑
x∈TdN
sup
|e|≤
√
d
∣
∣V˜ 2N,x+e − V˜
2
N,x
∣
∣
≤
(
1
Nd
∑
x∈TdN
sup
|e|≤
√
d
∣
∣V˜N,x+e − V˜N,x
∣
∣2
) 1
2
(
1
Nd
∑
x∈TdN
sup
|e|≤
√
d
∣
∣V˜N,x+e + V˜N,x
∣
∣2
) 1
2
by Cauchy-Schwarz. Next note that Lipschitz-continuity g∗ yields that
∣
∣(g ◦ η)()(x+eN )− (g ◦ η)
()( xN )
∣
∣ ≤
C
N
‖∇χ‖L∞
1
dNd
∑
|y|≤CN
η(y + x),
since x 7→ δ()0 (x/N) vanishes outside of a box of size CN , cf. the definition of the
smoothed empirical measure (2.29). Jensen’s inequality (convexity) yields
(2.72)
(
1
(2N + 1)d
∑
|y|≤N
η(x+ y)
)2
≤
1
(2N + 1)d
∑
|y|≤N
η(x+ y)2.
Therefore we obtain a bound
1
Nd
∑
x∈Td
sup
|e|≤
√
d
∣
∣(g ◦ η)()(x+eN )− (g ◦ η)
()( xN )
∣
∣2 ≤ C(N)−2
1
Nd
∑
x∈TdN
η(x)2.
Thus Lemma 2.6.1 allows us to replace, up to an error of O(1/N), the integral over
u ∈ Td by a discrete sum and we just need to estimate the term (2.71). Let us add and
subtract the expression
1
(N)d
∑
y∈TdN
χ
( y
N
)(
g ◦ η
l
(x+ y)− σ(ηl(x+ y))
)
96
2.6. The replacement lemma
inside the absolute value appearing in (2.71). Then we obtain
(
1
Nd
∑
x∈TdN
∣
∣(g ◦ η)()(x)− σ(η()(x))
∣
∣2
) 1
2
≤
(
1
Nd
∑
x∈TdN
∣
∣
∣
∣(g ◦ η)
()(x)−
1
(N)d
∑
y∈TdN
χ
( y
N
)
g ◦ η
l
(x+ y)
∣
∣
∣
∣
2) 12
+
(
1
Nd
∑
x∈TdN
∣
∣
∣
∣
1
(N)d
∑
y∈TdN
χ
( y
N
)(
g ◦ η
l
(x+ y)− σ(ηl(x+ y))
)∣∣
∣
∣
2) 12
+
(
1
Nd
∑
x∈TdN
∣
∣
∣
∣
1
(N)d
∑
y∈TdN
χ
( y
N
)
σ(ηl(x+ y))− σ(η()(x))
∣
∣
∣
∣
2) 12
.
We estimate the three terms separately. The first one equals
(
1
Nd
∑
x∈TdN
∣
∣
∣
∣
1
(N)d
∑
y∈TdN
χ
( y
N
)(
g(η(x+ y))− g ◦ η
l
(x+ y)
)∣∣
∣
∣
2) 12
.
Since
|χ( yN )− χ(
z
N )| ≤
l
N
‖∇χ‖L∞ if |z − y| ≤ l
and g(η(x)) ≤ g∗η(x), this term is bounded by
C
l
N
(
∑
x∈TdN
η(x)2
Nd
) 1
2
,
and hence by O(l/N) in view of the moment bound of Lemma 2.6.1. Recall that similarly
to the previous replacement of the integral over u ∈ Td by a discrete sum, the identity
∫
Td χ(u) du = 1 yields (2.33). A change of variables x → x + y yields a bound on the
second term by
C‖χ‖L∞
(
1
Nd
∑
x∈TdN
∣
∣
∣g ◦ η
l
(x+ y)− σ(ηl(x+ y))
∣
∣
∣
2
) 1
2
.
This in turn is bounded by the one block estimate. The third term is bounded from above
97
Quantitative hydrodynamic limits
as follows. It holds that
(2.73)
(
1
Nd
∑
x∈TdN
∣
∣
∣σ(η()(x))−
1
(N)d
∑
y∈TdN
χ
( y
N
)
σ(ηl(x+ y))
∣
∣
∣
2
) 1
2
≤
(
1
Nd
∑
x∈TdN
1
(N)d
∑
y∈TdN
χ
( y
N
)∣∣
∣σ(η()(x))− σ(ηl(x+ y))
∣
∣
∣
2
) 1
2
+ O
( 1
N
)
,
where we employed (2.33) and the moment bounds again. Recall that χ(y/(N)) = 0 if
|y| > N . Hence, up to a term O( 1N ), it holds that (2.73) is bounded from above by
C‖χ‖∞
(
sup
|y|≤N
1
Nd
∑
x∈TdN
∣
∣η()(x)− ηl(x+ y)
∣
∣2
) 1
2
which is bounded by the two blocks estimate. Note in view of the statement of the two
blocks estimate that if we include the the integration over t and η, we take the supremum
outside the integration.
Thus it remains to prove the one and two block estimates in order to deduce the replace-
ment lemma.
However, we are still missing one important ingredient in order to be able to present the
proof of the block estimates. This is the equivalence of ensembles which concerns the
closeness of the grand-canonical measures νLρ and the canonical measures ν
L,K for large
L under the condition that the densities are identical. Here L denotes the size of an
arbitrary lattice, not necessarily TdN .
2.6.3 Equivalence of ensembles
Equivalence of ensembles concerns the closeness of the canonical and the grand-canonical
measures in the limit of infinitely many sites. Consider a lattice ΛL of size |ΛL| = (2L+
1)d. For now, we shall not assume anything except Assumption 1 (i), (ii), (iii). These
assumptions guarantee the existence of the grand-canonical measures νLρ , defined in (2.6),
with finite exponential moments for all ρ ∈ R. Let us fix some notation: From now on,
for any integer l, we set l∗ := 2l + 1. Let Λl be a box of size |Λl| = ld∗ with l < L and
assume that Λl ⊂ ΛL.
We denote the standard deviation of the grand-canonical (product) measure νLρ by
s(ρ)2 = EνLρ
[
η(0)2
]
− EνLρ [η(0)]
2 .
98
2.6. The replacement lemma
Recall that, with σ given as the inverse function (2.23), the measure νLρ is the invariant
measure with particle density ρ, i.e.
EνLρ [η(0)] = ρ.
Let Hm denote the m-th Hermite polyomial
Hm(λ) = (−1)
me
λ2
2
dm
dλm
e−
λ2
2
for all λ ∈ R. Define two polynomials
q0(λ) =
1
√
2pi
e−
λ2
2 and
q1(λ) =
1
√
2pi
e−
λ2
2 H3(λ)
γ3(ρ)
6s(ρ)3
=
γ3(ρ)
6
√
2pis(ρ)3
e−
λ2
2 (λ3 − 3λ),
where
γ3(ρ) = EνLρ
[(
η(0)− ρ
)3]
.
The case of bounded densities
The contents of this section are mostly contained in [47, Appendix 2] and [52]. We include
them here for the reader’s convenience. Let us state, without proof, the following uniform
central limit theorem. The theorem, including higher order expansions, can be found in
[47], see also [73]. The results of this subsection are essentially valid without Assumption
1 (iii).
Lemma 2.6.4. For all 0 < ρ0 < +∞, there exist finite constants E1 = E1(ρ0) and
E2 = E2(ρ0) such that
sup
K≥0
∣
∣
∣
∣
√
Ld∗s(ρ)PνLρ
(∑
x∈ΛL
η(x) = K
)
− q0(λ)−
q1(λ)
√
Ld∗
∣
∣
∣
∣ ≤
E1
Ld∗s(ρ)2
for all ρ ≤ ρ0, such that s(ρ)2Ld∗ ≥ E2. Here we set λ = (K − L
d
∗ρ)/(L
d/2
∗ s(ρ)).
The following result is Corollary 1.6 in Appendix 2 of Kipnis and Landim [47].
Lemma 2.6.5. Fix 0 < ρ0 <∞, a positive integer l and a cylinder function f : NΛl → R
with finite second moment with respect to νLρ for all ρ ≤ ρ0. Then there exist finite
99
Quantitative hydrodynamic limits
constants E3 = E3(ρ0) and E4 = E4(ρ0), independent of the choice of f , such that
∣
∣
∣EνL,K [f ]− EνL
K/Ld∗
[f ]
∣
∣
∣(2.74)
≤ E3
ld∗
Ld∗
(
1
s(ρ)2
EνLρ
[∣
∣
∣f − EνLρ [f ]
∣
∣
∣
]
+
1
s(ρ)
√
EνLρ
[(
f − EνLρ [f ]
)2
])
for all L ≥ 2l and all K such that K/Ld∗ ≤ ρ0 and s(K/L
d
∗)
2Ld∗ ≥ E4. On the right hand
side of the inequality, we have set ρ = K/Ld∗.
Proof. First, we note that there exist constants 0 < c1(ρ0) and c2(ρ0) <∞ such that
c1 ≤
ρ
s(ρ)2
≤ c2 and c1 ≤
γ3(ρ)
s(ρ)2
≤ c2.
For ρ bounded away from zero, this is an obvious consequence of continuity in ρ, whereas
for small ρ, all moments of η grow linearly in ρ to first order, see equation (2.80) below.
It holds that
q1(λ)
√
Ld∗
≤ C
|λ|
√
Ld∗s(ρ)2
≤ C
( 1
Ld∗s(ρ)2
+ |λ|2
)
uniformly in ρ, λ, and L. Consequently, Lemma 2.6.4 yields
(2.75) sup
K≥0
∣
∣
∣
∣
√
Ld∗ − ld∗s(ρ)PνLρ
(∑
x∈ΛL\Λl
η(x) = K
)
−
1
√
2pi
e−
λ2
2
∣
∣
∣
∣
≤ C
( 1
Ld∗s(ρ)2
+ |λ|2
)
,
where
λ =
M(ξ)− ld∗ρ√
(Ld∗ − ld∗)s(ρ)2
and M(ξ) =
∑
x∈Λl
ξ(x). The difference |EνL,K [f ]− EνLρ [f ]| equals
(2.76)
∣
∣
∣
∣
∣
∑
ξ∈NΛl
νlρ(ξ)
(
f(ξ)− Eνlρ [f ]
){νLρ (
∑
x∈ΛL\Λl
η(x) = K −M(ξ))
νLρ (
∑
x∈ΛL
η(x) = K)
− 1
}∣∣
∣
∣
∣
.
Thus the central limit theorem, see Lemma 2.6.4, yields a bound on the term in braces
by
(2.77)
∣
∣
∣
∣
∣
√
Ld∗s(ρ)2√
(Ld∗ − ld∗)s(ρ)2
e−
λ2
2 − 1
∣
∣
∣
∣
∣
+ C
( 1
Ld∗s(ρ)2
+ |λ|2
)
,
100
2.6. The replacement lemma
where in the denominator, we have used that s(ρ)2Ld∗ ≥ E2. We now bound
|λ|2 =
1
Ld∗ − ld∗
(M(ξ)− ld∗ρ)
2
s(ρ)2
≤ C
ld∗
Ld∗
(
√
l−d∗ M(ξ)−
√
ld∗ρ)
2
s(ρ)2
and plug this into (2.77). Since s(ρ)2 ≤ C1(ρ0)ρ0, it follows that (2.77) is bounded by
E ′3(ρ0)
ld∗
Ld∗s(ρ)2
(
1 +
(√
l−d∗ M(ξ)−
√
ld∗ρ
)2
)
.
By a law of large numbers (a straightforward direct computation) together with bounds
similar to the bounds at the beginning of this proof, it holds that
Eνlρ
[∑
x∈Λl
(η(x)− ρ)4
]
≤ C
(
l2d∗ s(ρ)
4 + ld∗Eνlρ [(η(x)− ρ)
4]
)
≤ E ′3(ρ0)l
2d
∗ s(ρ)
2
after possibly changing the value of E ′3(ρ0). Therefore it holds that
Eνlρ
[(√
l−d∗ M(ξ)−
√
ld∗ρ
)4
]
≤ E ′3(ρ0)s(ρ)
2,
and therefore the Cauchy-Schwarz inequality yields a bound on (2.76) by
E3
ld∗
Ld∗
(
1
s(ρ)2
EνLρ
[∣
∣f − EνLρ [f ]
∣
∣
]
+
1
s(ρ)
√
EνLρ
[(
f − EνLρ [f ]
)2
]
)
for some constant E3 = E3(ρ0).
This lemma is used in [47] to prove the equivalence of ensembles without the lower bound
s(ρ)2Ld∗ ≥ E4. Since we are interested in obtaining explicit bounds on the rate of con-
vergence to the hydrodynamic limit, we need to be a bit more careful in our analysis to
identify the dependence on the size l and not just L. The good news so far is that E1 and
E2 do not depend on l and f , and that in our proof of the replacement lemma we shall
not need to consider any cylinder function f , but only the function
(2.78) f(ξ) :=
1
ld∗
∑
x∈Λl
g(ξ(x))2 ≤
(g∗)2
ld∗
∑
x∈Λl
ξ(x)2.
Carefully keeping track of the dependence on the integer l, we now prove equivalence of
ensembles.
Lemma 2.6.6 (Equivalence of ensembles for bounded densities). Fix 0 < ρ0 < ∞ and
let f as in (2.78). Then there exists a constant E5 = E5(ρ0) such that
∣
∣
∣
∣EνL,K [f ]− EνL
K/Ld∗
[f ]
∣
∣
∣
∣ ≤ E5
ld∗
Ld∗
101
Quantitative hydrodynamic limits
for all L large enough, all L ≥ 2l and all K such that 0 ≤ K/Ld∗ ≤ ρ0.
Proof. The proof follows Corollary 1.7 in Appendix II of [47]. Let E3, E4 as in Lemma
2.6.5 and consider first the case s(K/Ld∗)
2Ld∗ ≥ E4. In this case, if ρ = K/L
d
∗ is bounded
strictly away from zero, the bracket on the right hand side of the inequality in Lemma
2.6.5 is bounded by a constant since the variance s(ρ)2 and all the expectations on the
right hand side are continuous with respect to ρ ≤ ρ0. Hence let us consider ρ close to
zero and s(ρ)2Ld∗ ≥ E4. It holds
EνLρ [f ] =
∞∑
M=0
Eνl,M [f ] · ν
l
ρ
(∑
x∈Λl
ξ(x) = M
)
.
Since g grows at most linearly, the explicit form (2.78) yields that f(ξ) ≤ CM2 for particle
configurations ξ with M particles. For M ≥ 2 particles it holds that
(2.79) Eνl,M [f ] · ν
l
ρ
(∑
x∈Λl
ξ(x) = M
)
≤ CM2ρM ,
where we have used that νlρ(ξ(x) = n) = σ(ρ)
n/(Z(σ(ρ))g(n)!) ≤ Cρk since g ≥ δ˜k and
ρ ≤ ρ0. Furthermore, for M = 0, 1 we obtain
νlρ
(∑
x∈Λl
ξ(x) = M
)
≤



1
Z(ρ)l if M = 0,
l ρg(1)Z(ρ)l if M = 1.
Hence we can expand
Eνlρ [f ] = f(0) +
∑
x∈Λl
{f(dx)− f(0)}
ρ
g(1)
+ O(ρ2)
for ρ small enough, where 0 denotes the particle configuration in NΛl without any particles
and dx denotes the configuration containing only a single particle situated at the site
x ∈ Λl. Note that (2.79) yields that the term O(ρ2) is bounded independently of l and K
for small enough ρ. Likewise we obtain
Eνlρ
[(
f − Eνlρ [f ]
)2
]
=
∑
x∈Λl
{f(dx)− f(0)}
2 ρ
g(1)
+ O(ρ2) and
Eνlρ [f ] =
[∣
∣
∣
∣
∑
x∈Λl
{f(dx)− f(0)}
∣
∣
∣
∣+
∑
x∈Λl
|f(dx)− f(0)|
]
ρ
g(1)
+ O(ρ2).
Of course, in our case these expressions simplify due to f(0) = 0 but we shall not take
advantage of this just yet. Replacing f by ξ(0) we also see that
(2.80) s(ρ)2 =
ρ
g(1)
+ O(ρ2).
102
2.6. The replacement lemma
Putting all together we can bound the right hand side of inequality (2.74) by E3ld∗/L
d
∗.
Hence it remains only to consider the case of densities K/Ld∗ such that 0 ≤ s(K/L
d
∗)
2Ld∗ ≤
E4. Note that as before equation (2.80) implies that
0 < c1 ≤
s(ρ)2
ρ
≤ c2 < +∞
near ρ = 0 and hence for all 0 ≤ ρ ≤ ρ0. Hence the particle numbers K = ρLd∗ under
consideration are bounded by K ≤ c2E4. In the following we shall use the explicit form
of f . Translation invariance of the canonical measures yields that
EνL,K [f ] = EνL,K
[
1
ld∗
∑
x∈Λl
g(ξ(x))2
]
≤
(g∗)2
Ld∗
EνL,K
[
∑
x∈ΛL
ξ(x)2
]
≤
(g∗K)2
Ld∗
as well as
EνL
σ(K/Ld∗)
[f ] ≤ (g∗)2EνL
K/Ld∗
[η(0)2]
= (g∗)2
(
s( KLd∗ )
2 + ( KLd∗ )
2
)
≤
(g∗)2E2
Ld∗
+
(g∗K)2
L2d∗
.
Similar computations hold for the expectations of f 2. Again, this shows that the right
hand side of (2.74) is bounded by E5ld∗/L
d
∗. This concludes the proof of the equivalence
of ensembles in the case of bounded densities.
The case of large densities
Using Assumption 1 (i)-(iii), the following result has been shown in [52]. The proof is
similar to the proof of the equivalence of ensembles in the case of bounded densities, but
relies on Assumption 1 (iii) in order to obtain estimates on the growth of moments of η(x)
under νLρ .
Lemma 2.6.7. There exist 0 < ρ1 <∞ and constants E6, L0 such that
∣
∣
∣
∣EνL,K [f ]− EνL
K/Ld∗
[f ]
∣
∣
∣
∣ ≤ E6
ld∗
Ld∗
√
EνLρ
[(
f − EνLρ [f ]
)2
]
for all l > 0, cylinder functions f : NΛl → R with finite second moment with respect to
νLρ , L ≥ max{L0, 2l}, and K such that ρ = K/L
d
∗ ≥ ρ1.
Choosing f as in (2.78), we obtain that
EνLρ
[(
f − EνLρ [f ]
)2
]
≤ Cρ4
103
Quantitative hydrodynamic limits
for all ρ ≥ ρ1. Consequently Lemma (2.6.7) yields the equivalence of ensembles for large
densities.
Lemma 2.6.8 (Equivalence of ensembles for large densities). Let f as in (2.78). There
exist 0 < ρ1 <∞ and constants E7, L0 such that
∣
∣
∣
∣EνL,K [f ]− EνL
K/Ld∗
[f ]
∣
∣
∣
∣ ≤ E7
ld∗
Ld∗
ρ2
for all L ≥ L0 large enough, l < L/2, and K such that ρ = K/Ld∗ ≥ ρ1.
Equivalence of ensembles for arbitrary densities
Combining Lemmas 2.6.6 and 2.6.8, we arrive at the main result of this section.
Theorem 2.6.9. Let f as in (2.78). Then there exist constants E8 and L0 such that
∣
∣
∣
∣EνL,K [f ]− EνL
K/Ld∗
[f ]
∣
∣
∣
∣ ≤ E8
ld∗
Ld∗
(
1 +
K2
L2d∗
)
for all L ≥ L0 large enough, l < L/2 and K > 0.
2.6.4 Restriction to bounded particle configurations
As a first step we prove that it suffices to take bounded configurations.
Lemma 2.6.10. Under the assumptions of Lemma 2.4.19, there exists a constant C <
+∞ such that
1
Nd
∑
x∈TdN
EµNt
[
ηl(x)2χ{ηl(x)≥%}
]
≤
C
√
%
for all l > 0 and % > 0.
Proof. The proof follows Kipnis and Landim [47], Lemma V.4.2. A few modifications are
necessary in order to obtain an explicit, sufficiently fast rate of convergence.
First, by convexity it suffices to prove Lemma 2.6.10 with ηl(x)2 replaced by η(x)2 in the
hypothesis. Then Ho¨lder’s inequality yields
(2.81)
1
Nd
∑
x∈TdN
EµNt
[
η(x)2χ{ηl(x)≥%}
]
≤ EµNt
[
1
Nd
∑
x∈TdN
η(x)4
]1/2( 1
Nd
∑
x∈TdN
PµNt
(
ηl(x) ≥ %
)
)1/2
.
104
2.6. The replacement lemma
Lemma 2.6.1 yields a uniform bound on the fourth moment and we shall presently deduce
a vanishing bound on the probability P(ηl(x) ≥ %). The second term on the right hand
side of (2.81) can be estimated by the entropy inequality (2.32), which yields
1
Nd
∑
x∈TdN
PµNt
(
ηl(x) ≥ %
)
≤
1
γNd
(
HN(µNt |ν
N
ρ ) + log
∫
XN
exp
(
γ
∑
x∈TdN
χ{ηl(x)≥%}
)
dνNρ (η)
)
for any γ > 0. Since the relative entropy is bounded by assumption, we have that
1
γNd
HN
(
µNt |ν
N
ρ
)
≤
C0
γ
.
The second term on the right hand side can be split up as follows. Let the set Γx :=
{z ∈ TdN |z − x ∈ (2l + 1)Z
d} denote the sites in TdN equal to x modulo 2l + 1. With this
notation, we can write
∑
x∈TdN
χ{ηl(x)≥%} =
∑
|x|≤l
∑
y∈Γx
χ{ηl(y)≥%}.
Together with the independence of ηl(x) and ηl(y) for |x − y| > 2l, Ho¨lder’s inequality
yields
C0
γ
+
1
γNd
log
∫
XN
exp
[
γ
∑
x
χ{ηl(x)≥%}
]
dνNρ (η)
≤
C0
γ
+
1
γ(2l + 1)d
log
∫
XN
exp
[
γ(2l + 1)dχ{ηl(0)≥%}
]
dνlρ(η).
The presence of the indicator function means that the integral of the exponential on the
right hand side can be estimated by
1 + Eνlρ
[
χ{ηl(0)≥%}e
γ(2l+1)d
]
.
Then Chebyshev’s inequality yields a bound on the last expectation by
e(2l+1)
d(γ−%)Eνlρ
[
e
∑
|x|≤l η(x)
]
≤ exp
(
− (2l + 1)d(%− γ − logEρ(1))
)
,
where
Eρ(θ) := EνNρ
[
eθη(x)
]
=
Z(eθρ)
Z(ρ)
can be thought of as the Laplace transform of ν1ρ . Under Assumption 1, all exponential
105
Quantitative hydrodynamic limits
moments are finite. Now the elementary inequality log(1 + x) ≤ x yields a bound
1
Nd
∑
x∈TdN
PµNt
(
ηl(x) ≥ %
)
≤
C0
γ
+
1
γ(2l + 1)d
exp
[
− (2l + 1)d(%− γ − logEρ(1))
]
Hence we can choose γ = %− logEρ(1) and use that the second term vanishes at least as
fast as the first term if l > 0.
2.6.5 Proof of the one block estimate
The aim of this section is to prove Lemma 2.6.2. Up to an error of C/
√
%, Lemma 2.6.10
allows us to consider only configurations with bounded particle numbers. Therefore we
will assume that the support of the density function fNt , defined in equation (2.63), is
contained in
{
η ∈ XN | η
l(x) ≤ % for all x ∈ TdN
}
.
Specifically, recall that
fNt (η) =
dµNt
dνNf∞
(η).
Setting
(2.82) Vl,x(η) :=
∣
∣
∣g ◦ η
l
(x)− σ(ηl(x))
∣
∣
∣
2
and Vl := Vl,0, we need to control the term
1
TNd
∫ T
0
∑
x∈TdN
∫
XN
Vl,x(η)f
N
t (η)dν
N
ρ (η).
Next we reduce the problem to only a box of size l. Let us introduce the short-hand
(2.83) f
N
=
1
TNd
∫ T
0
∑
x
τxf
N
t dt,
were τx is the translation operator τxf(η) ≡ f(τxη) and (τxη)(y) = η(y − x). Then it
holds that
(2.84)
1
TNd
∫ T
0
∑
x∈TdN
∫
XN
Vl,x(η)f
N
t (η)dν
N
ρ (η) =
∫
XN
Vl(η)f
N
(η)dνNρ (η).
106
2.6. The replacement lemma
Due to convexity and translation invariance of the Fisher information, it also holds that
(2.85) DN
(
f
N
|νNρ
)
≤
1
T
∫ T
0
DN
(
fNt |ν
N
ρ
)
dt ≤ CNd−2,
where the last bound follows from (2.70). Now let f l(ξ) be the density with respect to ν
l
ρ
of the marginal of f
N
(η)dνNρ (η) on Λl, i.e. let
f l(ξ) =
1
νlρ(ξ)
∫
{η:η|Λl=ξ}
f
N
(η)dνNρ (η)
where
(2.86) Λl = {−l, . . . , l}
d,
is a box of size l and where νlρ is the Λl-marginal of the translation invariant measure ν
N
ρ .
Then it holds
(2.87)
∫
XN
Vl(η)f
N
(η)dνNρ (η) =
∫
NΛl
Vl(ξ)f l(ξ)dν
l
ρ(ξ).
The Fisher information on the box Λl is given by
(2.88) Dl
(
f |νlρ
)
=
∑
x,y∈Λl,x∼y
Ix,y(f |ν
l
ρ),
where
(2.89) Ix,y(f |ν
l
ρ) =
1
2
∫
g(ξ(x))
(√
f(ξx,y)−
√
f(ξ)
)2
dνlρ(ξ).
Thus we can formally write
Dl
(
f |νlρ
)
= −
∫
NΛl
√
fGN
√
f dνlρ(ξ),
if we neglect jumps that would take particles outside the box Λl. By convexity of the
“bond-Fisher-information” Ix,y we obtain
(2.90) Ix,y(f l|ν
l
ρ) ≤ Ix,y(f
N
|νNρ ).
The Fisher information and the density f
N
are translation invariant. Hence it holds
Ix,y(f
N
|νlρ) ≤
C
Nd
DN
(
f
N
|νlρ
)
.
107
Quantitative hydrodynamic limits
Summing (2.90) over all neighbours x ∼ y ∈ z + Λl yields
(2.91) NdDl
(
f l|ν
l
ρ
)
≤ (2l + 1)dDN
(
f
N
|νNρ
) (2.85)
≤ C(2l + 1)dNd−2.
Now we would like to apply the logarithmic Sobolev inequality. This means we have to
decompose along the canonical measures. Thus we shall only consider the problem on
hyperplanes of given particle numbers
ΩK :=
{
ξ ∈ NΛl
∣
∣
∣
∑
x∈Λl
ξ(x) = K
}
.
The canonical measures on Λl are given by
νl,K(ξ) = νlρ
(
ξ
∣
∣
∑
x∈Λl
ξ(x) = K
)
,
for all ξ ∈ ΩK , cf. (2.8). We decompose this problem along the canonical measures on
noting
νlρ =
∞∑
K=0
νlρ(ΩK)ν
l,K .
Thus we define
f l,K := Z
−1
K ν
l
ρ(ΩK)f l
∣
∣
ΩK
, where
ZK :=
∫
ΩK
f l
∣
∣
ΩK
(ξ)dνlρ(ξ).
This definition yields
∫
NΛl
f l,K(ξ) dν
l,K(ξ) = 1, ZK ≥ 0 and
∑
K
ZK = 1
Let us now consider the canonical Fisher information on Λl given by
Dl
(
f l,K |ν
l,K
)
:= −
∫
ΩK
√
f l,KG
N
√
f l,Kdν
l,K ,
where again we neglect possible jumps given by GN outside of ΛL, cf. (2.88). Denote the
relative entropy of measures on the smaller box by
Hl(µ|ν
l,K) :=
∫
NΛl
log
dµ
dνl,K
dµ
108
2.6. The replacement lemma
for all µ ∈ P (ΩK). Under Assumption 1 (i)-(iii), the canonical Fisher information satisfies
the logarithmic Sobolev inequality
(2.92) Hl(µ|ν
l,K) ≤ Cl2Dl(µ|νl,K)
for all square boxes Λl, all K > 0, and all µ ∈ P (ΩK), cf. (2.13) and [27]. The definitions
yield that
(2.93)
Dl
(
f l|ν
l
ρ
)
= −
∫
NΛl
√
f lG
N
√
f ldν
l
ρ
= −
∞∑
K=0
νlρ(ΩK)
∫
ΩK
√
f l
∣
∣
ΩK
GN
√
f l
∣
∣
ΩK
dνl,K
= −
∞∑
K=0
ZK
∫
ΩK
√
f l,KG
N
√
f l,Kdν
l,K
=
∞∑
K=0
ZKDl
(
f l,K |ν
l,K
)
.
Equation (2.93) and the logarithmic Sobolev inequality (2.92) then yield
∑
K
ZKH
l
(
f l,K |ν
l,K
) (2.92)
≤
∑
K
ZKCl
2Dl
(
f l,K |ν
l,K
) (2.93)
= Cl2Dl
(
f l|ν
l
ρ
)
(2.91)
≤ Cl2(2l + 1)dN−dDN
(
f
N
|νNρ
)
≤ C
l2
N2
(2l + 1)d.
Hence the Csisza´r-Kullback-Pinsker inequality, cf. [47], yields
(2.94)
∞∑
K=0
ZK
∥
∥
∥f
l,K
− 1
∥
∥
∥
L1(dνl,K)
≤ C(2l + 1)d/2
l
N
.
Remark 2.6.11. The two functional inequalities, the logarithmic Sobolev inequality and
the Csisza´r-Kullback-Pinsker inequality, were thus instrumental in replacing locally in
the infinitesimal volume element Λl the particle distribution by its local thermodynamic
equilibrium with an explicit error estimate.
Now we decompose the right hand side of (2.87) to obtain that
(2.95)
∫
NΛl
Vl(ξ)f l(ξ)dν
l
ρ(ξ) =
∞∑
K=0
ZK
∫
ΩK
Vl(ξ)f l,K(ξ)dν
l,K(ξ).
Recall that Lemma 2.6.10 allowed us to restrict ourselves to density functions with support
contained in {ηl(x) ≤ %}. This implies that ZK = 0 for all K > %(2l + 1)d. On ΩK , the
function Vl(ξ) is bounded by a multiple of %2. Thus estimate (2.94) implies that we can
109
Quantitative hydrodynamic limits
bound (2.95) by
(2.96) C(2l + 1)
d
2
l
N
%2 +
∑
K
ZK
∫
ΩK
Vl(ξ)dν
l,K(ξ).
Let us show that the equivalence of ensembles yields a bound on the latter term. To this
end, we introduce another parameter k < l, to be chosen later in terms of l, such that
(2k+ 1)d divides (2l+ 1)d, i.e. (2l+ 1)d/(2k+ 1)d = m ∈ N. Then split up the box Λl into
m disjoint smaller boxes Bi, such that for each i = 1, . . . ,m, the box Bi is a translation
of Λk, defined in (2.86). Furthermore we note that
σ
(
1
(2l + 1)d
∑
|x|≤l
ξ(x)
)
= Eνl
K/(2l+1)d
[
g(ξ(0))
]
under the law νl,K . Thus we split
∫
ΩK
Vl(ξ)dν
l,K(ξ)
=
∫
ΩK
∣
∣
∣
∣
1
(2l + 1)d
∑
|x|≤l
g(ξ(x))− σ
(
1
(2l + 1)d
∑
|x|≤l
ξ(x)
)∣
∣
∣
∣
2
dνl,K(ξ)
≤
m∑
i=1
|Bi|
|Λl|
∫
ΩK
∣
∣
∣
∣
1
|Bi|
∑
x∈Bi
g(ξ(x))− Eνl
K/(2l+1)d
[g(ξ(0))]
∣
∣
∣
∣
2
dνl,K(ξ),
by convexity of the function x 7→ x2. By translation invariance, the above sum can be
rewritten as
|Λk|
|Λl|
m∑
i=1
∫
ΩK
∣
∣
∣
∣
1
(2k + 1)d
∑
|x|≤k
g(ξ(x))− Eνl
K/(2l+1)d
[g(ξ(0))]
∣
∣
∣
∣
2
dνl,K(ξ).
Since by construction m(2k + 1)d/(2l + 1)d = 1, this in turn is bounded by
∫
ΩK
∣
∣
∣
∣
1
(2k + 1)d
∑
|x|≤k
g(ξ(x))− Eνl
K/(2l+1)d
[g(ξ(0))]
∣
∣
∣
∣
2
dνl,K(ξ)
Since K(2l + 1)−d ≤ %, the equivalence of ensembles, Lemma 2.6.9, yields a bound of
∫
NΛl
∣
∣
∣
∣
1
(2k + 1)d
∑
|x|≤k
g(ξ(x))− Eνl
K/(2l+1)d
[g(ξ(0))]
∣
∣
∣
∣
2
dνlK/(2l+1)d(ξ) + C
|Λk|
|Λl|
%2.
Finally due to the law of large numbers, this is bounded by
(2.97) C
(
1
(2k + 1)d
+
(2k + 1)d
(2l + 1)d
)
%2.
110
2.6. The replacement lemma
We can still choose k as a function of l. The asymptotically optimal choice is k =
√
l,
and in this case the last bound (2.97) vanishes as l−d/2%2. This error term together with
Lemma 2.6.10 and the error term appearing in (2.96) yield the one block estimate.
Remark 2.6.12. Note that the hydrodynamic limit is an asymptotic statement, in which
we choose l arbitrarily large. Therefore it was justified that k can be chosen in such a
way that (2k + 1)d divides (2l + 1)d and still k ∼
√
l.
2.6.6 Proof of the two blocks estimate
The aim of this section is to prove Lemma 2.6.3. First let us simplify the term under
investigation. Recall that
η()( xN ) =
1
(N)d
∑
z∈TdN
χ
( z
N
)
η(x+ z), and ηl(x) =
1
(2l + 1)d
∑
|z|≤l
η(x+ z).
Furthermore recall that the term to be estimated is
(
sup
|y|≤N
1
TNd
∫ T
0
∑
x∈TdN
〈
µNt ,
∣
∣ηl(x+ y)− η()(x)
∣
∣2
〉
dt
) 1
2
.
For ease of notation, let us temporarily drop the integral over t and η in the next line.
Since χ is differentiable, we can replace each η(x+z) appearing in the definition of η()(x)
by ηl(x+ z) to obtain
(
sup
|y|≤N
1
Nd
∑
x∈TdN
∣
∣ηl(x+ y)− η()( xN )
∣
∣2
) 1
2
≤
(
Cl2
(N)2
∑
x∈TdN
η(x)2
Nd
) 1
2
+
+
(
sup
|y|≤N
1
Nd
∑
x∈TdN
∣
∣
∣
∣η
l(x+ y)−
1
(N)d
∑
z∈TdN
χ
(
z
N
)
ηl(x+ z)
∣
∣
∣
∣
2
) 1
2
.
Hence upon inserting estimate (2.33) we deduce that, up to an error term O( lN ), the term
to be estimated in the two blocks estimate is bounded by
(
sup
cl<|y|≤2N
1
TNd
∫ T
0
∑
x∈TdN
EµNt
[∣
∣ηl(x)− ηl(x+ y)
∣
∣2
]
dt
) 1
2
for a suitable c > 0. Hence it suffices to estimate the rate of convergence of
sup
cl<|y|≤2N
1
TNd
∫ T
0
∑
x∈TdN
EµNt
[∣
∣ηl(x)− ηl(x+ y)
∣
∣2
]
dt.
111
Quantitative hydrodynamic limits
As before, let fNt denote the Radon-Nikodym density
fNt =
dµNt
dνNρ
.
Recalling definition (2.83) for the average f
N
over time T and lattice TdN of the density
fNt , the above expression is equal to
sup
cl<|y|≤2N
∫
XN
f
N
(η)
∣
∣ηl(0)− ηl(y)
∣
∣2 dνNρ (η).
Furthermore, we can bound the number of particles on the disjoint boxes x + Λl and
x + y + Λl as in Lemma 2.6.10. Again the resulting error is bounded by C/
√
% for some
constant C < +∞. Hence it is enough to consider
sup
cl<|y|≤2N
∫
XN
f
N
(η)V2,l(η)dν
N
ρ (η)
where
(2.98) V2,l(η) := χ{ηl(0)∨ηl(y)≤%}
∣
∣ηl(0)− ηl(y)
∣
∣2
with the notation a ∨ b = max{a, b}. From here the proof of the two blocks estimate is
similar to the proof of the one block estimate and we will only highlight the differences.
It would now suffice to restrict the problem to a union of boxes Λl ∪ (y + Λl), but, put
together, these do not form a square (hypercubic) box of equal side length and hence they
do not guarantee the validity of an LSI. Hence we take into account a larger hypercube
of side length 4l + 2, which contains (translations of) both boxes Λl and (y + Λl) glued
together. Since the number of sites in this larger box still scales as ld, this choice does not
change the scaling of the rate of convergence. Specifically, consider a square (hypercubic)
box in Zd with each side length being exactly equal to 4l + 2. Now we split up this box
along a plane into two equal halfs Λ1l and Λ
2
l and translate each part such that Λl ⊂ Λ
1
l ,
y + Λl ⊂ Λ2l . Since |y| > cl, we can choose c > 0 ( depending only on the dimension d)
such that Λ1l ∩ Λ
2
l = ∅. Then we set
Λy,l := Λ
1
l ∪ Λ
2
l
and we shall presently consider the process only on Λy,l. First we introduce some new
notation. Let X2,l := NΛ
1
l × NΛ
2
l be the configuration space on the two boxes, ν2,lρ be the
product measure νNρ restricted to X2,l, and ξ = (ξ1, ξ2) be a configuration in X2,l. Let f y,l
112
2.6. The replacement lemma
N
Λ2l
x∗
z∗
Λ1l
Figure 2.1: The two boxes Λ1l and Λ
2
l . Opposite faces have been chosen for Γ, defined
below, and it holds (x∗, z∗) ∈ Γ.
to be the density conditional on configurations ξ ∈ X2,l, i.e.
f y,l(ξ) =
1
ν2,lρ (ξ)
∫
X2,l
χ{η∈XN :η|Λy,l=ξ}f
N
(η)dν2,lρ (η).
In a next step, we need to obtain bounds on the Fisher information of f y,l. Since f y,l is a
density over two disjoint boxes, this is technically more involved than the corresponding
calculations in the proof of the one block estimate. First we recall that convexity yields
DN
(
f
N
|νNρ
)
≤
1
T
∫ T
0
DN
(
fNt |ν
N
ρ
)
dt ≤ CNd−2.
Now we compare with an appropriate Fisher information on X2,l. For all f ∈ Cb(X2,l),
we set
I1,lx,z(f |ν
2,l
ρ ) =
1
2
∫
X2,l
g(ξ(x))
(√
f(ξx,z1 , ξ2)−
√
f(ξ)
)2
dν2,lρ (ξ)
for all neighbours x, z ∈ Λ1l ,
I2,lx,z(f |ν
2,l
ρ ) =
1
2
∫
X2,l
g(ξ(x))
(√
f(ξ1, ξ
x,z
2 )−
√
f(ξ)
)2
dν2,lρ (ξ)
for all neighbours x, z ∈ Λ2l , and
I l,+x∗,z∗(f |ν
2,l
ρ ) =
1
2
∫
X2,l
g(ξ1(x∗))
(√
f(ξx∗,−1 , ξ
z∗,+
2 )−
√
f(ξ)
)2
dν2,lρ (ξ)
for all x∗ ∈ ∂Λ1l , z∗ ∈ ∂Λ
2
l on the boundaries of the boxes. Here ξ
x,z is the configuration ξ
after a particle jumped from site x to site z, cf. (2.2), and (ξx∗,−1 , ξ
z∗,+
2 ) is the configuration
ξ = (ξ1, ξ2) after a particle jumped from the boundary point x∗ of the first box to the
113
Quantitative hydrodynamic limits
boundary point z∗ of the second box, i.e. we set
ξx,±j (z) =
{
ξj(x)± 1 if z = x ∈ Λ
j
l ,
ξj(z) otherwise
for all z ∈ Λjl . Now consider one face each of the two rectangular boxes Λ
1
l ,Λ
2
l , which are
facing each other and along which we have split up the original box. Then we say that
(x∗, z∗) ∈ Γ if and only if x∗ is on the face belonging to one face of ∂Λ1l , and z∗ is the
corresponding site directly opposite x∗ on the face of ∂Λ2l belonging to the other box, see
Figure 2.1. In this fashion we join two faces of Λy,l back together. Note that it holds that
|Γ| = (4l+ 2)d−1. Using the above “bond-Fisher-information”, we define a corresponding
Fisher information on Λy,l as
(2.99) D2,l
(
f |ν2,lρ
)
:=
∑
(x∗,z∗)∈Γ
I l,+x∗,z∗(f |ν
2,l
ρ ) +
∑
|x−z|=1
(
I1,lx,z(f |ν
2,l
ρ ) + I
2,l
x,z(f |ν
2,l
ρ )
)
for all f ∈ Cb(X2,l). This Fisher information corresponds to a particle process where
particles move according to a zero range process on each box and where they can jump
from the boundary of one box to the other. As in inequality (2.90), convexity yields
I1,lx,z
(
f y,l|ν
2,l
ρ
)
≤ Ix,z
(
f
N)
and I2,lx,z
(
f y,l|ν
2,l
ρ
)
≤ Ix+y,z+y
(
f
N)
,
where Ix,z was defined in (2.89). Summing over all x, z ∈ TdN such that |x − z| = 1 we
obtain by translation invariance that
(2.100)
∑
|x−z|=1
(
I1,lx,z
(
f y,l|ν
2,l
ρ
)
+ I2,lx,z
(
f y,l|ν
2,l
ρ
))
≤ 2C(2l + 1)dN−2.
since DN(f
N
|νNρ ) ≤ CN
d−2. Recall from Subsection 2.6.3 that dx denotes the configura-
tion with a single particle at x ∈ TdN . Since
PνNρ
(
η(x) = n
)
=
σ(ρ)n
g(n)!Z(σ(ρ))
,
a change of variables ξ′ = ξ + dx∗ resp. η
′ = η + dx yields
I l,+x∗,z∗(f |ν
2,l
ρ ) =
σ(ρ)
2
∫
X2,l
(√
f(ξx∗,+1 , ξ2)−
√
f(ξ1, ξ
z∗,+
2 )
)2
dν2,lρ (ξ), and
Ix,z(f
N) =
σ(ρ)
2
∫
XN
(√
fN(ηx,+)−
√
fN(ηz,+)
)2
dνNρ (η),
114
2.6. The replacement lemma
where f ∈ Cb(X2,l) and fN ∈ Cb(XN), and where Ix,z(f) is the ordinary bond-Fisher
information defined in (2.89). Again we take advantage of convexity to see that
(2.101) I l,+x∗,z∗
(
f y,l|ν
2,l
ρ
)
≤
σ(ρ)
2
∫ (√
f
N
(ηx∗,+)−
√
f
N
(ηz∗,+)
)2
dνNρ (η)
for the average f
N
defined in (2.83). Of course the right hand side is not a summand of
DN(f
N
|νNρ ). Therefore we consider a path (xk)0≤k≤R from x∗ ∈ ∂Λ
1
l to z∗ ∈ ∂Λ
2
l . Here
R := ‖z∗ − x∗‖`1 =
∑
1≤j≤d
|z∗j − x∗j|
represents the `1–norm on TdN , and the path satisfies
x0 = x∗, xR = z∗, and |xk+1 − xk| = 1 for every 0 ≤ k ≤ R− 1.
The telescope identity
√
f
N
(ηx∗,+)−
√
f
N
(ηz∗,+) =
R−1∑
k=0
(√
f
N
(ηxk+1,+)−
√
f
N
(ηxk,+)
)
together with Cauchy–Schwarz inequality
(
R−1∑
k=0
ak
)2
≤ R
R−1∑
k=0
a2k
yields a bound on the right hand side of (2.101) by
Rσ(ρ)
2
R−1∑
k=0
∫
XN
(√
f
N
(ηxk,+)−
√
f
N
(ηxk+1,+)
)2
dνNρ (η) = R
R−1∑
k=0
Ixk,xk+1
(
f
N)
.
Since f
N
is translation-invariant and xk, xk+1 are neighbours, we obtain
Ixk,xk+1(f
N
) ≤ N−dDN(f
N
|νNρ )
and hence
I l,+x∗,z∗
(
f y,l|ν
2,l
ρ
)
≤ R2N−dDN
(
f
N
|νNρ
)
.
Without loss of generality we can assume that we joined the two boxes such that |x∗−z∗| ≤
|y| ≤ 2N and hence
R = ‖z∗ − x∗‖`1 ≤
√
d|y| ≤ 2
√
dN.
115
Quantitative hydrodynamic limits
The bound on the Fisher information DN(f
N
|νNρ ) ≤ CN
d−2 thus yields
I l,+x∗,z∗
(
f y,l|ν
2,l
ρ
)
≤ C2.
Summing over all pairs (x∗, z∗) ∈ Γ, yields another factor of Cld−1. In conjunction with
(2.100), we have shown that the Fisher information defined in (2.99) satisfies
(2.102) D2,l
(
f y,l|ν
2,l
ρ
)
≤ C
( ld
N2
+ 2ld−1
)
.
As before, we decompose the problem along hyperplanes of configurations on X2,l with
constant number of particles K and corresponding canonical measure
ν2,l,K(ξ) = ν2,lρ
(
ξ
∣
∣
∑
x∈Λ1l
ξ1(x) +
∑
x∈Λ2l
ξ2(x) = K
)
.
Similarly to the proof of the one block estimate we denote
Ω2K =
{
ξ ∈ Λy,l
∣
∣
∑
x∈Λ1l
ξ1(x) +
∑
x∈Λ2l
ξ2(x) = K
}
,
see Subsection 2.6.5. On Ω2K , we introduce the density
f 2,l,K := Z
−1
K ν
2,l
ρ (Ω
2
K)f y,l
∣
∣
Ω2K
,
ZK :=
∫
Ω2K
f y,l
∣
∣
Ω2K
dν2,lρ (η).
Then it holds that
(2.103)
∫
NΛl
V2,l(ξ)f y,l(ξ)dν
2,l
ρ (ξ) =
∞∑
K=0
ZK
∫
Ω2K
V2,l(ξ)f 2,l,K(ξ)dν
2,l,K(ξ).
By construction, the Fisher information (2.99) is equivalent to the Fisher information of
a ZRP on a box of side length 4l + 2. Thus it satisfies an LSI, if we replace ν2,lρ by its
canonical version ν2,l,K . The measure ν2,l,K is invariant with respect to this zero range
process and again we define a canonical Fisher information by
D2,l
(
f 2,l,K |ν
2,l,K
)
=
∑
(x∗,z∗)∈Γ
I l,+x∗,z∗(f |ν
2,l,K) +
∑
|x−z|=1
(
I1,lx,z(f |ν
2,l,K) + I2,lx,z(f |ν
2,l,K
)
.
We can formally write
D2,l
(
f 2,l,K |ν
2,l,K
)
=
∫
NΛy,l
√
f 2,l,KG
N
√
f 2,l,K dν
2,l,K
116
2.7. Perspective: the case of d ≥ 2 dimensions
if we neglect jumps outside of Λy,l. Of course, equation (2.93) holds for the equivalent
quantities on Λy,l as well. Also denote the relative entropy on X2,l by
H2,l(µ|ν2,l,K) :=
∫
X2,l
log
dµ
dν2,l,K
dµ
for all µ ∈ P (Ω2K). Furthermore, we constructed our canonical Fisher information in such
a way that it is the canonical Fisher information of a zero range process on the square
lattice Λy,l with the two faces glued together. Hence [27] yields the logarithmic Sobolev
inequality
H2,l(µ|ν2,l,K) ≤ Cl2D2,l(µ|ν2,l,K)
uniformly in l > 0, K > 0, and µ ∈ P (Ω2K), cf. (2.92). This logarithmic Sobolev inequality
yields
∑
K
ZKH
(
f 2,l,K |ν
2,l,K
)
≤
∑
K
ZKCl
2D2,l
(
f 2,l,K |ν
2,l,K
) (2.93)
= Cl2D2,l
(
f y,l|ν
2,l
ρ
)
as in Subsection 2.6.5. Now the Csisza´r-Kullback-Pinsker inequality and estimate (2.99)
yield
∞∑
K=0
ZK
∥
∥f 2,l,K − 1
∥
∥
L1(dν2,l,K)
≤ C
(
lN−1 + 
√
l
)
ld/2.
Therefore we can replace f 2,l,K by 1 in (2.103), up to an error bounded by
C
(
lN−1 + 
√
l
)
ld/2%2,
where % is the bound on the number of particles introduced in Lemma 2.6.10, see also
(2.98). Thus we are left to consider
∞∑
K=0
ZK
∫
Ω2K
V2,l(ξ)dν
2,l,K(ξ),
which is bounded by C%2l−d/2 analogously to Subsection 2.6.5 by the equivalence of en-
sembles and the law of large numbers.
2.7 Perspective: the case of d ≥ 2 dimensions
Disclaimer: This section is included for its mathematical interest, but does not constitute
a proof of any of its statements - all results are still work in progress.
So far in our proof of the hydrodynamic limit, we have assumed d = 1. Let us now give
an exposition of work in progress on how to remove this restriction. Regularity results
117
Quantitative hydrodynamic limits
for uniformly parabolic equations in higher dimensions usually rely on the famous results
of Nash, de Giorgi, and Moser. We shall see that this is the case here, too. First make
the following conjecture on the propagation of higher regularity in Hk(Td) for general
dimensions.
Conjecture 1 (Improved regularity). We conjecture that for each k > 0, there exist
constants C < +∞ and α, β > 0 such that
‖ft‖Hk ≤ C
(
1 + ‖f0‖Hk + ‖f0‖
α
L∞‖f0‖
β
Hk
)
for all f0 ∈ Hk(Td) and t ≥ 0.
Let us derive the consequences for the hydrodynamic limit if we assume this conjecture
to be true.
Stability estimates : Lemma 2.4.14 remains true if we change Λ to be
Λ(f) = ‖f0‖L∞
(
1 + ‖f0‖Hk + ‖f0‖
α
L∞‖f0‖
β
Hk
) d+4
2k+2 .
Abstract differential calculus of the semigroup: Lemma 2.4.17 remains true with Λ given
as in the previous equation.
Rate of convergence on the hydrodynamic limit : Theorem 2.4.6 remains true if we change
rHL to be
rHL(T, %, l, , N) ≤ C
(
+ −d−(dα+(k+
d
2 )β)
d+4
2k+2N1−θ(d+1) + −
4+d
2 N−2
+ e−cTN2+d−2d + T
1
2
(
N−
1
2 l
1
2 + 
1
2 l
1
4
)
%l
d
4 + %2l−
d
4 + %−
1
4 +
l
N
)
.
Optimal rate of convergence: Corollary 2.4.9 remains true with a different optimal rate
N−κ, with κ depending on the exact values of α and β.
The strategy for the proof is the following: We know that solutions to equation (2.3) are
smooth due to the ellipticity of σ. The difficulty lies in obtaining explicit bounds on the
propagation of the Sobolev norms in Hk(Td). First of all, let us go back to the proof of the
regularity estimate, Lemma 2.4.13. Instead of interpolating Drift between ‖Dk+1ft‖L2 and
‖∇ft‖L2 , we can interpolate between ‖Dk+1ft‖L2 and ‖ft‖L∞ . The generalized Gagliardo-
Nirenberg-inequality, see [20], yields
‖Drf‖Lp ≤ C‖D
k+1f‖θL2‖f‖
1−θ
L∞ ,
if
|r| −
d
p
= θ
(
k + 1−
d
2
)
118
2.7. Perspective: the case of d ≥ 2 dimensions
and 1 ≥ θ ≥ |r|/(k + 1). Interpolating between, say, ‖Dkft‖L2 , ‖Dk+1ft‖L2 , and ‖ft‖L∞ ,
we should be able to obtain a local in time bound of the form
‖Dkft‖L2 ≤ C‖D
kf0‖L2 for all t ≤ T (‖D
kf0‖L2 , ‖f0‖L∞)
with polynomial dependence of T (·, ·) in each argument. (Otherwise there are analytic
norms available to perform a similar job.) Thus we have reduced the problem to obtaining
“large” time bounds only. In the terms that appear in the proof of Lemma 2.4.13, we
choose pi = |ri|/(2(k + 1)), which corresponds to the critical case θi = |ri|/(k + 1). This
choice yields
d
dt
‖Dsft‖
2
L2 ≤ −c‖∇D
sft‖
2
L2 + C(1 + ‖f0‖
k+1
L∞ )‖∇D
sft‖
2
L2 ,
which is exactly the critical case. Therefore we need a slightly better space than L∞(Td) to
interpolate. A suitable function space is the space C0,s(Td) of s–Ho¨lder-continuous func-
tions. There are several ways to obtain an interpolation inequality between W |r|,p(Td),
Hk+1(Td) and C0,s(Td). A suitable approach could be Littlewood-Paley theory (microlocal
analysis). Assuming the correctness of this approach, we simply need to find a (polyno-
mial) bound on the C0,s–norm of the solutions for some s > 0. Now by Nash’s result [68]
on the Ho¨lder-continuity of solutions to uniformly parabolic equations there indeed exists
some s > 0 such that
[ft]C0,s ≤ C‖f0‖L∞t
s
2
for all t > 0. Applying this bound for t ≥ T (‖Dkft‖L2 , ‖ft‖L∞) completes the proof.
119
Quantitative hydrodynamic limits
120
Part II
Analysis of nonlinear Schro¨dinger
equations
121

Chapter 3
On the Cauchy-problem of nonlinear
Schro¨dinger equations with angular
momentum rotation term
The work in this chapter has been carried out in collaboration with Paolo Antonelli and
Christof Sparber, most of which has been published in [2].
3.1 Introduction
Ever since the realization of Bose-Einstein condensation (BEC) in dilute atomic gases,
much attention has been given to dynamical phenomena associated to its superfluid na-
ture. One remarkable feature of a superfluid is the appearance of quantized vortices, cf.
[1] for a broad introduction to these kind of phenomena. In physical experiments, the
BEC is thereby set into rotation by a stirring potential, which is usually induced by a laser
[61, 62, 64, 79] (see also [6] for numerical simulations). The corresponding mathematical
model is a nonlinear Schro¨dinger equation (NLS) of Gross-Pitaevskii type with angular
momentum rotation term, i.e.
(3.1) i~∂tψ = −
~2
2
∆ψ + λ|ψ|2ψ + U(x)ψ − Ω · Lψ, (t, x) ∈ R× R3,
where ψ = ψ(t, x) is the complex-valued wave function of the condensate and ~ is Planck’s
constant. In the physics literature, (3.1) is known as the Gross-Pitaevskii equation
for rotating Bose gases. The coupling constant λ ∈ R can be experimentally tuned to
account for both defocusing (λ > 0) and focusing (λ < 0) nonlinearities. The potential
123
On the NLS with rotation term
U(x) describes the magnetic trap and is usually assumed to be of the form
(3.2) U(x) =
1
2
3∑
j=1
γ2jx
2
j , γj ∈ R.
Finally, Ω · L denotes rotation term, where
(3.3) L := −ix ∧∇
is the quantum mechanical angular momentum operator and Ω ∈ R3 is a given angular
velocity vector. For a rigorous derivation of (3.1) in the stationary case, we refer to [53].
Furthermore, we remark that the appearance of quantized vortices has been rigorously
proved in [77], by means of a spontaneous symmetry breaking for the ground state of the
stationary equations (provided λ > 0 is sufficiently big). For further mathematical results
in this direction we refer to [1] and the references given therein.
The aforementioned works illustrate the fact that there is a considerable amount of math-
ematical studies devoted the stationary equation. On the other hand, the time-dependent
equation (3.1), has not been given as much attention, even though it is considered to pro-
vide the basis for a dynamical description of vortex creation. Indeed, except for numerical
simulations [6], we are only aware of [37, 38] providing rigorous results for (3.1). In [37]
global well-posedness of the Cauchy problem (in the energy space) is proved in the case
where λ > 0, U(x) = γ
2
2 |x|
2, i.e. an isotropic confinement, and |Ω| = γ. The analogous
result in d = 2 spatial dimensions is given in [38].
Remark 3.1.1. Let us mention that in [56, 57], the NLS model (3.1) is also rigorously
studied. The results, however, mainly concern an asymptotic regime, the so-called semi-
classical limit, and are thus very different from the present work.
In view of these results the main goal of our work is twofold: First, we shall prove global
well-posedness of (3.1) in the defocusing case, without any restriction on |Ω| or {γj}3j=1.
The latter is needed to describe actual physical experiments, which often require |Ω| 6= γ.
To this end, we shall show that by a suitable time-dependent change of coordinates,
we can transform equation (3.1) into a nonlinear Schro¨dinger equation without rotation
term but with a time-dependent trapping potential. In a second step, we shall analyze
the possibility of finite time blow-up of solutions, in the case of a focusing nonlinearity.
Recall that finite time blow-up means, that there is a T ∗ < +∞, such that
lim
t→T ∗
‖∇ψ(t)‖L2 = +∞.
As we shall see, the usual proof of finite time blow-up, based on the classical virial
argument of Glassey [32] (see also [19]), in general does not go through in a straight-
124
3.2. Mathematical setting and main result
forward way, due to the influence of the rotation term. Instead, it has to be slightly
modified, yielding blow-up conditions which depend on |Ω|, and which coincide with the
usual conditions in the limit |Ω| → 0.
3.2 Mathematical setting and main result
In the following we shall consider the Cauchy problem for the following, slightly more
general NLS type model
(3.4) i∂tψ = −
1
2
∆ψ + λ|ψ|2σψ + U(x)ψ − Ω · Lψ, ψ(0) = ψ0(x),
where x ∈ Rd, for d = 2 or d = 3, respectively, and σ < 2d−2 , i.e. the nonlinearity is
assumed to be energy-subcritical. In d = 2 the rotation term simply reads
(3.5) Ω · L = −iω(x1∂x2 − x2∂x1)
for some ω ∈ R.
Remark 3.2.1. In d = 3 we could, without restriction of generality, choose a reference
frame such that Ω = (0, 0, ω)>, ω ∈ R, yielding the same formula as in (3.5). For the
sake of generality we shall not do so but consider the term Ω · L ≡ −iΩ · (x ∧ ∇) in full
generality.
A potential U(x) ∈ Rd, satisfying (Ω · L)U(x) = 0, ∀x ∈ Rd, is said to be axially
symmetric (with respect to the rotation axis Ω ∈ R3). In particular, this holds in the case
of an isotropic trap potential, i.e. a potential of the form (3.2) with γ1 = γ2 = γ3.
Formally, (3.4) preserves the total mass
M :=
∫
Rd
|ψ(t, x)|2 dx,
and the energy
(3.6) EΩ :=
∫
Rd
1
2
|∇ψ|2 + U(x)|ψ|2 +
λ
σ + 1
|ψ|2σ+2 − ψ(Ω · L)ψ dx.
Note that, the last term is indeed real valued (as can be seen by a partial integration).
In order for these two quantities to be well defined, we shall study the Cauchy problem
corresponding to (3.4) in the space
Σ := {f ∈ H1(Rd) : |x|f ∈ L2(Rd)},
125
On the NLS with rotation term
equipped with the norm
‖f‖2Σ := ‖f‖
2
L2 + ‖∇f‖
2
L2 + ‖xf‖
2
L2 .
We remark that even if the potential U(x) is chosen to be identically zero, it would not
be enough to consider the Cauchy problem for ψ ∈ H1(Rd), since in this case we can no
longer guarantee that
(3.7) LΩ(t) :=
∫
Rd
ψ(t)(Ω · L)ψ(t) dx < +∞.
Physically speaking, this means that ψ has finite angular momentum. The choice of Σ is
therefore natural in our situation and not necessarily linked to the presence of a harmonic
trapping potential, in contrast to [16, 17, 18]. The usual definition of a solution ψ to the
nonlinear Schro¨dinger equation (3.4) is the following.
Definition 3.2.2. We say that ψ is a (mild) solution to (3.4) if ψ ∈ C([0, T ]; Σ) and it
holds that
(3.8) ψ(t) = S(t)ψ0 − i
∫ t
0
S(t− s)|ψ(s)|2σψ(s) ds,
where S(t) = eiHt denotes the unitary semigroup generated by the Hamiltonian
(3.9) H = −
1
2
∆ + U(x)− Ω · L,
which corresponds to solving the linear Schro¨dinger equation.
Equation (3.8) is usually called Duhamel’s formula. Existence of S(t) is established in
Yajima [85], cf. Section 3.3. We can now state the main result of this work.
Theorem 3.2.3. Let 0 < σ < 2/(d − 2), λ ∈ R, Ω ∈ Rd, for d = 2, 3 and denote the
smallest trap frequency by γ := min{γj}dj=1.
(1) Then, for any given initial data ψ0 ∈ Σ, there exists a unique global in-time solution
ψ ∈ C([0,∞); Σ) to (3.4), provided one of the following conditions is satisfied:
(i) the nonlinearity is L2-subcritical σ < 2/d, or
(ii) σ ≥ 2/d and λ ≥ 0, i.e. the nonlinearity is defocusing.
(2) On the other hand, if λ < 0, and if either:
(i) (Ω · L)U = 0, i.e. U is axially symmetric, and σ ≥ 2/d,
126
3.2. Mathematical setting and main result
(ii) (Ω · L)U 6= 0, |Ω| ≤ γ, and σ ≥ κΩ/d, where
(3.10) κΩ :=
√
4γ2
γ2 − |Ω|2
, or
(iii) σ ≥ 2/d, without loss of generality Ω = (0, 0, ω)>, and there exists a T > 0 such
that
T <
2γ
|(γ21 − γ
2
2)ω|
and (EΩ − LΩ(0))T
2 + I˙(0)T + I(0) ≤ 0.(3.11)
Then there exist initial data ψ0 ∈ Σ such that finite time blow-up of the corresponding
solution ψ(t) occurs.
Note that assertions (2)(ii) coincides with (2)(i) in the limit Ω → 0 and that (2)(iii)
coincides with (2)(i) in the limit as γ1 − γ2 → 0.
In fact, we shall prove Assertions (1)(i) and (ii) under the more general assumptions on
U(x), see Assumption 3 below. This, together with the fact that no condition on the size
of |Ω| or {γj}dj=1 is required, generalizes the earlier results given in [37, 38].
Remark 3.2.4. The exact conditions on the initial data for Assertion (2) of the Theo-
rem 3.2.3 can be found in Lemma 3.4.1.
Concerning the possibility of finite time blow-up, we see that one has to distinguish
between the case of axially symmetric potential and the case where this symmetry is
broken. The reason will become clear in the proof given below. In the case of a non-
axially symmetric potential we can rigorously prove the occurrence of blow-up only under
the additional restrictions |Ω| ≤ γ, and σ ≥ κΩ/d, i.e. only for a limited range of
nonlinearities. It is easily seen that in d = 3, the set of σ’s satisfying our conditions
is non-empty, provided |Ω|2 < 89γ
2. Also note that in the case of vanishing rotation
lim|Ω|→0 κΩ = 2, yielding the usual range of L2-supercritical nonlinearities. At this point
it is not clear if these additional restrictions are only due to the strategy of our proof, or if
they indicate an actual difference in the behavior of solutions to (3.4). In particular, the
question whether or not finite time blow-up occurs in situations where Ω > γ is completely
open so far. In terms of physics, the latter would correspond to the case where the rotation
is stronger than the trap and thus one would expect a behavior which is similar (at least
qualitatively) to the “free” case, i.e. without any potential. We finally remark that the
question whether or not rotation can stabilize an attractive BEC is also debated from the
physics point of view, see [44] and [81].
This Chapter is now organized as follows: Section 3.3 is devoted to the proof of Assertion
(1) of Theorem 3.2.3. To this end, we shall first prove local in-time existence for solutions
127
On the NLS with rotation term
to (3.4). Also, we shall see that a naive use of the conservation laws for mass and energy
in general leads to restrictions on |Ω| or {γj}dj=1. We shall show in a second step how to
overcome this problems using a coordinate-change. Assertion (2) of our main Theorem is
then proved in Section 3.4 and we finally collect some concluding remarks in Chapter 5.
3.3 Local and global existence
In this section we shall allow for more general class of potentials U(x) satisfying the
following assumption.
Assumption 3. The potential U : Rd → R is assumed to be smooth and sub-quadratic,
i.e. for all multi-indices k ∈ Nd, with |k| ≥ 2, there exists a constant C = C(k) > 0 such
that
(3.12) |∂kU(x)| ≤ C for all x ∈ Rd.
Remark 3.3.1. Clearly, a harmonic trapping potential of the form (3.2) is sub-quadratic.
Assumption 3 allows us to take into account more general situations of physical interest,
such as a combined harmonic trap plus optical lattice potential, see e.g. [22]. Note how-
ever, that under Assumption 3, the potential is not necessarily bounded below (or con-
fining). In particular we can also allow for repulsive potentials such as U(x) = −γ2|x|2,
see [16].
As a first, preliminary step, we shall prove the following local well-posedness result.
Lemma 3.3.2. Let ψ0 ∈ Σ, ω ∈ R, and 0 < σ < 2/(d − 2). Moreover, assume that U
satisfies Assumption 3. Then there exists a time T = T (‖ψ0‖Σ) > 0 and a unique solution
ψ ∈ C([0, T ]; Σ) of equation (3.1) with ψ(0) = ψ0.
Thus we can construct a maximal solution ψ ∈ C([0, Tmax); Σ). The solution is maximal
in the sense that, if Tmax < +∞, then
lim
t→Tmax
‖∇ψ(t)‖L2 = +∞.
Moreover, the following conservation laws hold:
M(t) = M(0), and EΩ(t) = EΩ(0),(3.13)
whereas for the angular momentum we have
(3.14) LΩ(t) +
∫ t
0
∫
Rd
i|ψ|2(Ω · L)U(x)dx = LΩ(0).
128
3.3. Local and global existence
The proof is an adaptation of classical arguments, based on a contraction mapping (via
Duhamel’s formula (3.8)) and Strichartz estimates for the linear (unitary) group S(t) =
eitH generated by the Hamiltonian (3.9). In our case, Strichartz estimates can be obtained
by following the approach of Yajima [85]. As defined, the Hamiltonian is not of the form
necessary for the results in [85] to hold. However, the Hamiltonian can be rewritten as
H =
1
2
(i∇+ A(x))2 + U(x)−
1
2
A(x)2, where A(x) = Ω ∧ x.
Indeed it holds that
(i∇+ Ω ∧ x)2 = −∆ + (Ω ∧ x)2 + 2i(Ω ∧ x) · ∇+ i(∇ · A)
= −∆ + (Ω ∧ x)2 + 2iΩ · (x ∧∇).
Since the rotation B(x) = ∇ ∧ A(x) = 2Ω is constant and the potential U(x)− A(x)2/2
is subquadratic, the results in [85] imply that there exist finite, positive, constants C and
δ such that
‖S(t)ϕ‖L∞ ≤
C
|t|d/2
‖ϕ‖L1 , for |t| < δ.
In particular it follows that the Strichartz estimates for H are analogous to those found in
the well-known case of NLS with quadratic potentials [17], i.e. the rotation term does not
influence the dispersive behavior (locally in time). These Strichartz estimates have been
proven under general circumstances in [46]. Recall that p′ denotes the conjugate Ho¨lder
coefficient of p, i.e. 1/p+ 1/p′ = 1.
Lemma 3.3.3 (Strichartz estimates). The unitary group S(t) = eitH with Hamiltonian
H defined in (3.9) satisfies a local in time Strichartz estimate. There exist TS > 0 and
constants Cq, Cq,q˜ such that
‖S(t)ϕ‖Lp(0,TS ;Lq(Rd)) ≤ Cq‖ϕ‖L2(Rd)
for all ϕ ∈ L2(Rd) and
∥
∥
∥
∫ t
0
S(t− s)F (s) ds
∥
∥
∥
Lp(0,TS ;Lq(Rd))
≤ Cq,q˜‖F‖Lp˜′ (0,TS ;Lq˜′ (Rd))
if (p, q) and (p˜, q˜) are admissible coefficients, i.e. 2 ≤ q < 2d/(d − 2) and 2/p = δ(q) :=
d(1/2− 1/q) and likewise for (p˜, q˜).
Since the semigroup (S(t))t∈R ⊂ L(L2(Rd)) is continuous in time, the Strichartz estimates
also yield the following continuity property.
129
On the NLS with rotation term
Remark 3.3.4. Under the conditions of Lemma 3.3.3, it holds that
S(t)ϕ ∈ C([0, TS];L
2(Rd)) and
∫ t
0
S(t− s)F (s) ds ∈ C([0, TS];L
2(Rd)),
which will later allow us to prove continuity of time of fixed points of Duhamel’s for-
mula (3.8).
The existence of a local in-time solution is then standard and we repeat it here for the
reader’s convenience.
Proof of Lemma 3.3.2. The idea is to write the nonlinear Schro¨dinger equation as the
solution of a fixed point equation, using Duhamel’s formula:
ψ(t) = S(t)ψ0 − iλ
∫ t
0
S(t− s)
(
|ψ|2σ(s)ψ(s)
)
ds
=: Φ(ψ)(t).
Let us define parameters
q = 2σ + 2, p =
4σ + 4
dσ
, k =
2σ(2σ + 2)
2− (d− 2)σ
.
We will presently show that Φ as a maps the space
XT :=
{
ψ ∈ L∞(0, T ; Σ) : ψ, xψ,∇ψ ∈ Lp(0, T ;Lq(Rd))
}
onto itself. Indeed, setting R := ‖ψ0‖Σ, we shall establish that Φ is a contraction mapping
in
XT,R :=
{
ψ ∈ XT : ‖ψ‖L∞(0,T ;Σ) ≤ 2R,
‖xψ‖Lp(0,T ;Lq), ‖∇ψ‖Lp(0,T ;Lq) ≤ 2CqR
}
,
for all T small enough. We equip the space XT,R with a metric
d(ψ, ψ˜) = ‖ψ − ψ˜‖L∞t L2x + ‖ψ − ψ˜‖LptLqx .
Then (XT,R, d) forms a complete metric space, cf. [19].
130
3.3. Local and global existence
In order to proceed, we calculate the commutators [∇, S(t)] and [x, S(t)]. We find that
[∇, H] =−
1
2
[∇,∆] + [∇, U ] + i[∇,Ω · (x ∧∇)]
=∇U + i[∇, x · (∇∧ Ω)]
=∇U + i∇∧ Ω
by the well-known formula a ·(b∧c) = det(a, b, c) = (a∧b) ·c for three-dimensional vectors
a, b, c. Similar calculations yield
[x,H] = ∇− iΩ ∧ x.
Since
i∂t[∇, S(t)] = [∇, HS(t)] = H[∇, S(t)] + [∇, H]S(t),
we deduce that
(3.15) ∇Φ(ψ)(t) = S(t)∇ψ0 − iλ
∫ t
0
S(t− τ)∇
(
|ψ(τ)|2σψ(τ)
)
dτ
− i
∫ t
0
S(t− τ) (∇U − iΩ ∧∇) Φ(ψ)(τ) dτ,
and
xΦ(ψ)(t) = S(t)∇ψ0 − iλ
∫ t
0
S(t− τ)∇
(
|ψ(τ)|2σψ(τ)
)
dτ
− i
∫ t
0
S(t− τ) (∇− iΩ ∧ x) Φ(ψ)(τ) dτ.
Since T < TS, the Strichartz estimates of Lemma 3.3.3 yield
(3.16) ‖∇Φ(ψ)‖LptLqx ≤ Cq‖ψ0‖Σ + |λ|Cq,q
∥
∥∇
(
|ψ(τ)|2σψ(τ)
) ∥
∥
Lp
′
t L
q′
x
+ Cq,2
∥
∥∇UΦ(ψ)
∥
∥
L1tL
2
x
+ Cq,2
∥
∥Ω ∧∇Φ(ψ)
∥
∥
L1tL
2
x
.
Since
1
q′
=
2σ
q
+
1
q
and
1
p′
=
2σ
k
+
1
p
,
Ho¨lder’s inequality yields
∥
∥∇
(
|ψ|2σψ
)∥
∥
Lp
′
t L
q′
x
≤ (2σ + 1)
∥
∥|ψ|2σ∇ψ
∥
∥
Lp
′
t L
q′
x
≤ (2σ + 1)
∥
∥ψ
∥
∥2σ
LktL
q
x
∥
∥∇ψ
∥
∥
LptL
q
x
.
131
On the NLS with rotation term
Furthermore, it holds that
∥
∥ψ
∥
∥2σ
LktL
q
x
≤ T
2σ
k
∥
∥ψ
∥
∥
2σ
k
L∞t L
q
x
≤ T
2σ
k
∥
∥ψ
∥
∥2σ
L∞t H
1
x
where we used the Gagliardo-Nirenberg inequality in the last estimate. Recall that the
Gagliardo-Nirenberg inequality implies
‖ϕ‖Lqx ≤ C‖ϕ‖
1−δ(q)
L2x
‖∇ϕ‖δ(q)L2x
for all ϕ ∈ H1(Rd), where δ(q) is defined in the statement of Lemma 3.3.3. Furthermore
we can bound |∇U | ≤ Cx. Thus estimate (3.16) yields
‖∇Φ(ψ)‖LptLqx ≤ Cq‖ψ0‖Σ + CT
2σ
k
∥
∥ψ
∥
∥2σ
L∞t Σx
‖∇ψ‖LptLqx
+ CT
∥
∥xΦ(ψ)
∥
∥
L∞t L
2
x
+ CT
∥
∥∇Φ(ψ)
∥
∥
L∞t L
2
x
.
Similarly we obtain that
‖Φ(ψ)‖LptLqx ≤ Cq‖ψ0‖Σ + CT
2σ
k
∥
∥ψ
∥
∥2σ
L∞t Σx
‖ψ‖LptLqx
and
‖xΦ(ψ)‖LptLqx ≤ Cq‖ψ0‖Σ + CT
2σ
k
∥
∥ψ
∥
∥2σ
L∞t Σx
‖xψ‖LptLqx
+ CT
∥
∥∇UΦ(ψ)
∥
∥
L∞t L
2
x
+ CT
∥
∥∇Φ(ψ)
∥
∥
L∞t L
2
x
.
Since (∞, 2) is an admissible pair, the same estimates hold for the L∞t L
2
x-norm, with Cq
replaced by 1. If we choose T sufficiently small, this shows that Φ indeed constitutes a
mapping Φ : XT,R → XT,R. In order to show that Φ : XT,R → XT,R is also a contraction,
we take two functions ψ, ψ˜ ∈ XT,R and estimate the difference Φ(ψ)−Φ(ψ˜). In the same
way as above, we obtain that
∥
∥Φ(ψ)− Φ(ψ˜)
∥
∥
LptL
q
x
=
∥
∥
∥
∥
∫ t
0
S(t− s)
(
|ψ(s)|2σψ(s)− |ψ˜(s)|2σψ˜(s)
)
ds
∥
∥
∥
∥
LptL
q
x
≤
∥
∥|ψ|2σψ − |ψ˜|2σψ˜
∥
∥
Lp
′
t L
q′
x
,
The point-wise inequality
∣
∣|ψ|2σψ − |ψ˜|2σψ˜
∣
∣ ≤ 2σ
(
|ψ|2σ + |ψ˜)|2σ
)
|ψ − ψ˜|
132
3.3. Local and global existence
and the same Ho¨lder and Gagliardo-Nirenberg estimates used before then yield
∥
∥Φ(ψ)− Φ(ψ˜)
∥
∥
LptL
q
x
+
∥
∥Φ(ψ)− Φ(ψ˜)
∥
∥
L∞t L
2
x
≤ C
(
‖ψ‖2σLktLqx + ‖ψ˜‖
2σ
LktL
q
x
)
‖ψ − ψ˜‖LptLqx
≤ CT
2σ
k
(
‖ψ‖2σL∞t H1x + ‖ψ˜‖
2σ
L∞t H
1
x
)
‖ψ − ψ˜‖LptLqx .
Thus, choosing T ≤ TS small enough, there exists a fixed point ψ ∈ XT,R satisfying
Φ(ψ) = ψ. Remark 3.3.4 applied to the fixed point equations, e.g. (3.15), yields ψ ∈
C([0, T ]; Σ). Hence ψ is indeed a mild solution to the Schro¨dinger equation (3.4). The
conservation laws (3.13) follow from straightforward calculations in combination with a
standard density argument and reversibility (see e.g. [19]). Finally, in order to prove the
blow-up alternative we first notice that the above local existence argument can be iterated
as long as ‖ψ(t)‖Σ stays bounded. Thus we obtain a maximal solution ψ ∈ C([0, Tmax); Σ)
with Tmax > 0. Assume that Tmax < +∞. In view of mass conservation, it follows that
lim
t→Tmax
(
‖xψ(t)‖L2 + ‖∇ψ(t)‖L2
)
= +∞.
Furthermore we compute
(3.17)
d
dt
‖xψ(t)‖2L2 = 2Im
∫
Rd
xψ(t)∇ψ(t) dx ≤ ‖xψ(t)‖2L2 + ‖∇ψ(t)‖
2
L2 .
Thus, as long as ‖∇ψ(t)‖L2 is bounded, Gronwall’s inequality yields a bound on ‖xψ(t)‖L2
as well. The only obstruction to global existence is therefore given by the possible un-
boundedness of ‖∇ψ(t)‖L2 in [0, T ].
Remark 3.3.5. For quadratic potentials of the form (3.2), Strichartz estimates can be
obtained explicitly by invoking a generalization of Mehler’s formula for the kernel of S(t),
c.f. [18]. Indeed, by making the following ansatz
S(t)ψ0(x) =
d∏
j=1
(2piiµj(t))
−1/2
∫ d
R
e
i
2F (t,x,y)ψ0(y) dy,
where µj(t) ∈ R+ and F (t, x, y) is a general quadratic form in x and y with (yet to be
determined) time-dependent coefficients. Substituting this into the linear Schro¨dinger
equation yields a coupled system of differential equations for these coefficients. Solving
this system, however, is in general rather tedious. This approach is therefore only feasible
under some simplifying assumptions, such as Ω = 0 [16, 17], or U(x) = γ
2
2 |x|
2 with |Ω| = γ
as it is done in [37, 38].
In view, of (3.14), we immediately conclude the following important corollary.
133
On the NLS with rotation term
Corollary 3.3.6. If U(x) is such that (Ω · L)U = 0, then we also have conservation of
angular momentum, i.e. LΩ(t) = LΩ(0), and in addition it holds
(3.18) E0(t) ≡
∫
Rd
1
2
|∇ψ|2 + U(x)|ψ|2 +
λ
σ + 1
|ψ|2σ+2dx = E0(0).
Thus, in the case of axially symmetric potentials U(x), there are in fact two conserved
energy functionals corresponding to (3.4).
With a local existence result in hand, we can ask about global existence. In order to infer
T = +∞, one usually invokes the conservation of mass and energy (3.13). The problem
is, that due to the appearance of the angular momentum rotation term, the energy EΩ(t)
has no definite sign even if U ≥ 0 and λ ≥ 0 (defocusing nonlinearity). A possible strategy
to overcome this problem is to rewrite the linear Hamiltonian as
(3.19) H = −
1
2
∆− Ω · L+ U(x) =
1
2
(−i∇− A(x))2 + U(x)−
|Ω|2
2
r2,
where A(x) = Ω ∧ x and r = |x ∧ Ω|/|Ω| denotes the radial distance perpenticular
to Ω. Note that A(x) can be considered as the vector potential corresponding to a
constant magnetic field B = ∇ ∧ A = 2Ω. The corresponding “magnetic derivative”
DA := −i(∇+ A(x)) is known to satisfy, cf. [19, Chapter 7]:
‖∇|ψ|‖L2 ≤ ‖DAψ‖L2 ≤ ‖∇ψ‖L2 + ‖xψ‖L2 .
It can therefore be used to control the nonlinear potential energy∝ ‖ψ‖L2σ+2 via Gagliardo-
Nirenberg type inequalities. If in addition, U(x) is given by (3.2) with |Ω| ≤ γ we infer
that U(x) − |Ω|
2
2 r
2 ≥ 0. In this case, the linear part of the energy is seen to be a sum
of non-negative terms, and global existence can be concluded as in the case of NLS with
quadratic confinement [17]. However, it seems impossible to extend this approach to sit-
uations in which |Ω| > γ, even if λ > 0. In order to do so, we shall follow a different idea,
which invokes a time-dependent change of coordinates.
Proof of Assertion (1) of Theorem 3.2.3. We start with the L2-subcritical case, i.e. 0 <
σ < 2/d which follows by standard arguments. Recall that in the proof of Lemma 3.3.2
we showed that
‖ψ‖Lp(0,T ;Lq)∩L∞(0,T ;L2) ≤ C‖ψ0‖L2 + C‖ψ‖
2σ
Lk(0,T ;Lq) ,
where
q = 2σ + 2, p =
4σ + 4
dσ
, k =
2σ(2σ + 2)
2− (d− 2)σ
.
134
3.3. Local and global existence
Moreover, we showed that
‖xψ‖Lp(0,T ;Lq)∩L∞(0,T ;L2) + ‖∇ψ‖Lp(0,T ;Lq)∩L∞(0,T ;L2)
≤ C‖ψ0‖Σ + C‖ψ‖
2σ
Lk(0,T ;Lq)
(
‖xψ‖Lp(0,T ;Lq) + ‖∇ψ‖Lp(0,T ;Lq)
)
+
+ CT
(
‖xψ‖L∞(0,T ;L2) + ‖∇ψ‖L∞(0,T ;L2)
)
.
If σ < 2/d, it holds that 1/p < 1/k and thus
‖ψ‖Lk(0,T ;Lq) ≤ T
1/k−1/p‖ψ‖Lp(0,T ;Lq).
If we choose T = T ∗ > 0 small enough, we can absorb all terms on the right hand
side except the term involving ψ0 and obtain a bound on ‖ψ‖L∞(0,T ∗;Σ). Since we can
shift the time interval [0, T ∗] by an arbitrary amount of time, in the same way we
can get a uniform bound on ‖ψ‖L∞(0,T ∗;Σ) for every interval of length |I| ≤ T ∗. Thus,
by splitting any arbitrarily large time interval [0, T ] into sufficiently small sub-intervals
{In}Nn=1 such that |In| ≤ T
∗ and iterating the bound ‖ψ‖Lp(In;Lq)∩L∞(I;L2) ≤ C
∗, we infer
‖ψ‖Lp(0,T ;Lq)∩L∞(0,T ;L2) ≤ C where C < +∞ depends on the value of T . From here, we
proceed as before to obtain a uniform bound for the left hand side for small T ∗ and thus
by iteration for arbitrary time intervals [0, T ].
Next we consider the case of an L2-supercritical nonlinearity σ > 2/d. In this case,
the iterative argument given above breaks down. The basic idea is to use a change of
coordinates in order to bring equation (3.4) into a more suitable form. For the sake of
notation we shall only consider the case d = 3 in the following. We first note that by
using the skew-symmetric matrix
Θ :=



0 Ω3 −Ω2
−Ω3 0 Ω1
Ω2 −Ω1 0


 ,
the wedge product with the angular momentum can be written as
Ω ∧ x = −Θ · x.
Then the matrix-exponential
X(t, x) := eΘt · x
defines a rotation of the vector x ∈ R3 around the axis Ω by an angle of −|Ω|t. Its
time-derivative can be calculated as
(3.20) ∂tX(t, x) = Θ ·X(t, x) = −Ω ∧X(t, x).
135
On the NLS with rotation term
Denoting the wave function in rotated coordinates via
(3.21) ψ˜(t, x) = ψ(t,X(t, x)),
we conclude from (3.20) that
i∂tψ˜(t, x) = i∂tψ(t,X(t, x))− i (Ω ∧X(t, x)) · ∇ψ(t,X(t, x)).
Rewriting −i(Ω ∧X) · ∇ = −iΩ · (X ∧∇) = Ω · L, we arrive at
i∂tψ˜ = −
1
2
∆ψ˜ + λ|ψ˜|2σψ˜ + U(X(t, x))ψ˜,
where we have also used the fact that the Laplace operator is invariant with respect to
rotations, i.e.
∆Xψ(t,X(t, x)) = ∆xψ(t,X(t, x)).
Dropping all the tildes and denoting W (t, x) = U(X(t, x)), we conclude that up to
a change of coordinates, equation (3.4) is equivalent to the following NLS with time-
dependent potential
(3.22) i∂tψ = −
1
2
∆ψ + λ|ψ|2σψ +W (t, x)ψ.
Note that W (t, x) is smooth w.r.t. t ∈ R and sub-quadratic w.r.t. x ∈ R3 with the same
(uniform) constants C(k) as given in Assumption 3 for U(x). Moreover, if U(x) is axially
symmetric, i.e. (Ω · L)U(x) = 0, equation (3.20) implies that
∂tW (t, x) = −Ω ∧X(t, x) · ∇U(X(t, x)) = −i(Ω · L)U(X(t, x)) = 0,
and hence W (t, x) = W (0, x) ≡ U(x). The energy corresponding to the transformed NLS
(3.22) is given by
EW (t) :=
∫
1
2
|∇ψ(t, x)|2 + λ|ψ(t, x)|2σ+2 +W (t, x)|ψ(t, x)|2 dx.
However, since the potential W (t, x) in general is time-dependent, the EW (t) is no longer
a conserved quantity. Rather, we obtain that
(3.23)
d
dt
EW (t) =
∫
∂tW (t, x)|ψ(t, x)|
2 dx.
Nevertheless it is not hard to prove Assertion (1)(ii) of Theorem 3.2.3: In view of the
blow-up alternative, stated in Lemma 3.3.2, it suffices to show ‖∇ψ(t)‖L2 < +∞, for all
136
3.4. Finite time blow-up
T > 0. To this end, we first estimate
1
2
‖∇ψ(t)‖2L2 ≤ EW (t) +
∣
∣
∣
∣
∫
W (t, x)|ψ(t, x)|2 dx
∣
∣
∣
∣ ≤ EW (t) + C‖xψ(t)‖
2
L2 ,
under the assumption that λ > 0. Integrating equation (3.23) and having in mind that
W (t, x) is sub-quadratic in x, we obtain that
‖∇ψ(t)‖2L2 ≤EW (0) +
∫ t
0
d
ds
EW (s) ds+ C‖xψ(t)‖
2
L2(3.24)
≤C0
(
1 + ‖xψ(t)‖2L2 +
∫ t
0
‖xψ(s)‖2L2 ds
)
.
Recalling inequality (3.17), we infer
d
dt
‖xψ(t)‖2L2 + ‖xψ(t)‖
2
L2 ≤ C0
(
1 + ‖xψ(t)‖2L2 +
∫ t
0
‖xψ(s)‖2L2 ds
)
,
which by Gronwall’s inequality yields an uniform bound on ‖xψ(t)‖L2 for every time inter-
val [0, T ]. With this in hand, we can bound ‖∇ψ(t)‖L2 by simply using inequality (3.24)
once more.
Remark 3.3.7. In particular, for Ω = (0, 0, ω)> and U(x) given by (3.2), we explicitly
find
W (t, x) =
1
2
( (
γ21 cos
2(ωt) + γ22 sin
2(ωt)
)
x21
+
(
γ21 sin
2(ωt) + γ22 cos
2(ωt)
)
x22 + sin(2ωt)
(
γ21 − γ
2
2
)
x1x2 + γ
2
3x
2
3
)
.
Clearly, W = 12(γ
2
1x
2
1 + γ
2
2x
2
2 + γ
2
3x
2
3) in the axially symmetric case γ
2
1 = γ
2
2 .
3.4 Finite time blow-up
This section is devoted to the proof of assertion (2) of Theorem 3.2.3. The statements
(2)(i) and (ii) follow from the following lemma.
Lemma 3.4.1. Let λ < 0, σ < 2/(d− 2), Ω ∈ Rd, for d = 2, 3, and U(x) be a quadratic
potential of the form (3.2). Denote γ = min{γj}dj=1 and let κΩ be as in (3.10). If either
(i) (Ω · L)U = 0, σ ≥ 2/d, and E0(0) < 0, or
(ii) (Ω · L)U 6= 0, |Ω| ≤ γ, σ ≥ κΩ/d, and EΩ(0) < 0,
then the corresponding solution to equation (3.4) necessarily blows up in finite time.
137
On the NLS with rotation term
Note that the condition for the energy of the initial data ψ0 are not identical in both
cases. The reason will become clear in the proof given below.
Proof. To simplify the arguments later on, let us first compute the conservation laws
for the mass and momentum densities, i.e. ρ := |ψ|2 and J := Im(ψ∇ψ). Indeed a
straightforward calculation yields
(3.25) ∂tρ+ div J = iΩ · Lρ.
On the other hand, for the current density J we find
(3.26)
∂t
(
Im(ψ∇ψ)
)
= Im
((
−
i
2
∆ψ + iU(x)ψ + iλ|ψ|2σψ + iΩ · Lψ
)
∇ψ
)
+ Im
(
ψ∇
(
i
2
∆ψ − iU(x)ψ − iλ|ψ|2σψ + i(Ω · Lψ)
))
.
Next, we calculate
Im
(
ψ∇(iΩ · Lψ)
)
= Im
(
ψ(iΩ · L)∇ψ
)
− Ω ∧ J.
Thus we can combine the two terms in (3.26) which stem from the rotation via
Im
(
(iΩ · L)ψ∇ψ
)
+ Im
(
ψ(iΩ · L)∇ψ
)
− Ω ∧ J = (iΩ · L)J − Ω ∧ J
where we have used that iΩ · L is real-valued. The other terms in (3.26) are usual in
quantum hydrodynamics, see e.g. [3], yielding the following equation for J :
(3.27) ∂tJ + div
(
Re(∇ψ ⊗∇ψ)
)
+
λσ
σ + 1
∇|ψ|2σ+2 + ρ∇U =
1
4
∆∇ρ+ (iΩ ·L)J −Ω∧ J.
The proof of finite time blow-up now follows by the classical argument of Glassey [32].
To this end, we consider the time evolution of
I(t) :=
1
2
∫
Rd
|x|2|ψ(t, x)|2 dx.
Differentiating with respect to time and using (3.25), we obtain
d
dt
I(t) =
∫
Rd
x · J(t, x) dx+
∫
Rd
|x|2
2
(iΩ · L)ρ(t, x) dx.
Integrating by parts and using (Ω · L)|x|2 = 0 shows that the second integral in fact
vanishes, i.e. we have
d
dt
I(t) =
∫
Rd
x · J dx.
138
3.4. Finite time blow-up
Differentiating in time once more and using (3.27), we obtain
d
dt
∫
x · J dx =
∫
x ·
(
− div
(
Re(∇ψ ⊗∇ψ)
)
− λ
σ
σ + 1
∇|ψ|2σ+2 − ρ∇U
+
1
4
∆∇ρ+ (iΩ · L)J − Ω ∧ J
)
dx,
which we rewrite as
(3.28)
d
dt
∫
x · J dx =
∫ (
|∇ψ|2 + λ
dσ
σ + 1
|ψ|2σ+2 − ρx · ∇U
+ x · (iΩ · L)J − x · Ω ∧ J
)
dx.
Now we first note that for any potential U(x) of the form (3.2) we have x · ∇U = 2U .
Moreover, we compute
∫
Rd
x · (iΩ · LJ) dx =−
∫
Rd
(Ω · Lx) · J dx = −
∫
Rd
(Ω · (x ∧∇)x) · J dx
=−
∫
Rd
((Ω ∧ x) · ∇)x · J dx = −
∫
Rd
(Ω ∧ x) · J dx,
which shows that the last two terms in (3.28) cancel each other. In summary we arrive
at the following identity
(3.29)
d2
dt2
I(t) =
∫ (
|∇ψ|2 + λ
dσ
σ + 1
|ψ|2σ+2 − 2U |ψ|2
)
dx,
which is in fact exactly the same as in the case of NLS without rotation, c.f. [19].
We can now prove assertion (i): Recall from Corollary 3.3.6 that if the potential U(x) is
axially symmetric, then E0(t) = E0(0), with E0 defined in (3.18). Hence from (3.29) and
U ≥ 0 we can write
d2
dt2
I(t) ≤ 2E0 + λ
dσ − 2
σ + 1
∫
Rd
|ψ|2σ+2dx.
Assuming E0 < 0, λ < 0, and σ ≥ 2/d, we consequently obtain
d2
dt2
I(t) < −C,
for some constant C > 0. Integrating this relation twice, we obtain
I(t) < −
C
2
t2 + c1t+ c2
with some integration constants c1 and c2. Thus, if the solution ψ(t) ∈ Σ were to exist
for all times, there would be a time T ∗ < +∞, such that I(T ∗) < 0. This however is
139
On the NLS with rotation term
in contradiction with the fact that, by definition, I(t) ≥ 0 for all t ∈ R and hence the
assertion is proved.
In order to prove assertion (ii) we again consider (3.29): The problem is that in the case of
a non-axially symmetric potential (Ω · LU(x) 6= 0), the energy E0 is no longer conserved.
Rather we only have the conservation law for EΩ(t) = EΩ(0). In order to use this piece
of information, we first add and subtract to (3.29) a multiple of the angular momentum
LΩ(t), i.e.
d2
dt2
I(t) =
∫
Rd
(
|∇ψ|2 +
λσd
σ + 1
|ψ|2σ+2 − 2U |ψ|2 + κψΩ · Lψ
)
dx−
∫
κψΩ · Lψ dx,
where κ > 0 is a parameter to be chosen later on. Using Cauchy-Schwarz and Young’s
inequality, the last term on the right hand side can be bounded by
κ
∫
Rd
ψΩ · Lψdx ≤ κ|Ω|‖∇ψ‖L2‖xψ‖L2 ≤
κθ
2
‖∇ψ‖2L2 +
κ|Ω|2
2θ
‖xψ‖2L2 ,
where θ > 0 is another free parameter to be chosen later on. We consequently estimate
d2
dt2
I(t) ≤
∫
Rd
(
1 +
κθ
2
)
|∇ψ|2 +
λσd
σ + 1
|ψ|2σ+2 +
(
−2U +
κ|Ω|2
2θ
|x|2
)
|ψ|2 dx
−
∫
Rd
κψΩ · Lψ dx.
Now, we choose θ such that 2(1 + κθ2 ) = κ, that is θ =
κ−2
κ . In this way we have
d2
dt2
I(t) ≤
∫
Rd
κ
(
1
2
|∇ψ|2 + λ
1
σ + 1
|ψ|2σ+2 + U |ψ|2 − ψΩ · Lψ
)
dx
+
∫
Rd
λ
σd− κ
σ + 1
|ψ|2σ+2dx+
∫
Rd
(
−(κ+ 2)U +
κ2|Ω|2
2(κ− 2)
|x|2
)
|ψ|2dx.
Let γ := min(γ1, γ2, γ3), and choose κ such that
(κ+ 2)
2
γ2 = κ2
|Ω|2
2(κ− 2)
.
This yields κ = κΩ with
κΩ =
√
4γ2
γ2 − |Ω|2
.
By doing so, the last term in the previous inequality is seen to be non-positive and
furthermore we conclude that, for λ < 0 and σ ≥ κΩd , it holds:
(3.30)
d2
dt2
I(t) ≤ κΩEΩ(t) ≡ κΩEΩ(0).
140
3.4. Finite time blow-up
Thus, if the initial energy EΩ(0) < 0 the second derivative of I(t) is again negative and
we can argue (by contradiction) as before.
The next lemma gives a proof of assertion (2)(ii) in Theorem 3.2.3.
Lemma 3.4.2. Let λ < 0, σ < 2/(d− 2), d = 2, 3, and U(x) be a quadratic potential of
the form (3.2). Denote γ = min{γj}dj=1 and suppose that Ω = (0, 0, ω)
> if d = 3 or Ω · L
is of the form (3.5) if d = 2. If there exists a T > 0 such that
T <
2γ2
|(γ21 − γ
2
2)ω|
as well as
E0(0)T
2 + I˙(0)T + I(0) < 0,
then the corresponding solution to equation (3.4) necessarily blows up in finite time.
Note that if one does not care about the orientation of the rotation, one can always change
the sign of Ω such that LΩ(0) ≤ 0 in which case EΩ < 0 implies E0(0) < 0.
Proof. Again, we let
I(t) =
∫
x2|ψ(t, x)|2 dx
denote the second moment of |ψ|2. In equation (3.29), we already showed that
d2
dt2
I(t) =
∫
Rd
(
|∇ψ|2 +
λdσ
σ + 1
|ψ|2σ+2 − 2U |ψ|2
)
dx.
In the special case where Ω = (0, 0, ω)>, equation (3.14) yields
d
dt
LΩ(t) = (γ
2
1 − γ
2
2)ω
∫
Rd
x1x2|ψ|
2 dx.
Integrating twice and applying Ho¨lder’s inequality yields
∫ t
0
LΩ(s) ds = tLΩ(0) +
∫ t
0
∫ s
0
(γ21 − γ
2
2)ω
∫
Rd
x1x2|ψ(τ)|
2 dx dτds
≤ tLΩ(0) +
∫ t
0
∫ s
0
|(γ21 − γ
2
2)ω|
∫
Rd
1
2
(
x21 + x
2
2
)
|ψ(τ)|2 dx dτds.
The integrand on the right hand side is positive and we can estimate the integral over
{0 ≤ τ ≤ s} by the integral over the larger set {0 ≤ τ ≤ t}. Hence it holds that
∫ t
0
LΩ(s) ds ≤ tLΩ(0) + t
∫ t
0
|(γ21 − γ
2
2)ω|
∫
Rd
1
2
(
x21 + x
2
2
)
|ψ(s)|2 dx ds
≤ tLΩ(0) + t
∫ t
0
|(γ21 − γ
2
2)ω|
γ2
∫
Rd
U |ψ(s)|2 dx ds
(3.31)
141
On the NLS with rotation term
by the definition of γ. Integrating (3.29) yields
I˙(t) = I˙(0) +
∫ t
0
∫
Rd
(
|∇ψ|2 +
λdσ
σ + 1
|ψ|2σ+2 − 2U |ψ|2
)
dx ds
Adding and substracting the integral over 2LΩ yields
I˙(t) = I˙(0) +
∫ t
0
∫
Rd
(
|∇ψ|2 +
λdσ
σ + 1
|ψ|2σ+2 − 2U |ψ|2
)
dx ds
+ 2
∫ t
0
LΩ(s) ds− 2
∫ t
0
LΩ(s) ds.
The definition of the energy (3.6) thus yields
I˙(t) = I˙(0) +
∫ t
0
EΩ ds+
λ(dσ − 2)
σ + 1
∫ t
0
∫
Rd
|ψ|2σ+2 dx ds
− 4
∫ t
0
∫
Rd
U |ψ|2 dx ds− 2
∫ t
0
LΩ(s) ds.
The energy EΩ is constant, the term involving the nonlinearity is non-positive, and hence
estimate (3.31) yields
I˙(t) ≤ I˙(0) + 2t(EΩ − LΩ(0)) +
∫ t
0
(
2
(
t
|(γ21 − γ
2
2)ω|
γ2
− 2
)∫
Rd
U |ψ(s)|2 dx
)
ds
Of course, E0(0) = EΩ−LΩ(0). Under the conditions of the theorem, we obtain I(T ) ≤ 0
upon another integration with respect to time, thus yielding a contradiction!
3.5 Numerical simulations of a rotating Bose-Einstein
condensate
In this section, we will present some numerical results regarding the numerical simulation
of equation (3.4). Numerical simulations of the nonlinear Schro¨dinger equations will also
be necessary in Chapter 4, so that this section can be seen as a preparation for the next
chapter. It is well-known that the nonlinear Schro¨dinger equation without rotation term,
i.e. (3.4) with Ω = 0, can be efficiently simulated using operator splitting combined with
pseudo–spectral methods. For example, let us suppose we want to solve the rotation-less
NLS
(3.32) i∂tψ = −
1
2
∆ψ + U(x)ψ + λ|ψ|2σψ, ψ(t = 0) = ψ0(x),
142
3.5. Numerical simulations of a rotating Bose-Einstein condensate
with (t, x) ∈ [0, T ]×Rd. In order to perform numerical simulations, we restrict ourselves
to a bounded set, say, [−L,L]d for some L ∈ [0,∞) and periodic solutions ψ. This can
be justified by choosing a trapping potential U or a focusing nonlinearity λ < 0 which
is strong enough to ensure that the support of the solution effectively remains within
the box [−L,L]d. In practice, we stop simulations if the numerical support of the wave
function ψ reaches the boundary of our domain [−L,L]d. We discretize time and space
into the equidistant grid
tn = n∆t, n = 0, . . . , N such that tN = T and
xm = −L+m∆x,m = 0, . . . ,M such that ∆x =
2L
M
.
Our goal is to calculate approximations Ψn,m to the correct solution ψ(tn, xm) for all
n = 0, . . . , N and m = 0, . . . ,M . Let us first consider only the discretization in time. We
perform an operator-splitting method, i.e. at each time-step we split equation (3.32) into
two equations
(3.33) i∂tψ(t, x) = −
1
2
∆ψ(t, x), t ∈ [tn, tn+1]
and
(3.34) i∂tψ(t, x) = U(x)ψ(t, x) + λ|ψ(t, x)|
2σψ(t, x), t ∈ [tn, tn+1].
Thus, at each time step, given initial datum ψn(x) (which we take to be an approximation
of ψ(tn, x)), we solve the first equation to obtain a solution ψ˜n+1(x) at t = tn+1. Then we
use ψ˜n+1 as initial datum in the second equation to obtain ψn+1(x).
The first equation (3.33) is solved exactly in Fourier space
i∂tFψ(t, k) =
k2
2
Fψ(t, k), t ∈ [tn, tn+1], x ∈ Rd
where F : L2(Td)→ L2(Rd) is the Fourier transform. Thus we obtain that
(3.35) ψ˜n+1 = F−1
[
e−i
k2
2 ∆tFψ(tn, ·)
]
, t ∈ [tn, tn+1], x ∈ Rd,
where F−1 denotes the inverse Fourier transform. Since the probability density |ψ(t, x)|
is constant under the evolution of equation (3.34), the (exact) solution of equation (3.34)
is given by
ψn+1(x) = e
−i(U(x)+λ|ψ(t,x)|2)∆tψ˜n+1(x).
By iterating this procedure, we obtain an approximate solution ψ(tn, ·), n = 0, . . . , N .
When introducing the time-discretization equation, we just need to replace the Fourier
143
On the NLS with rotation term
transform in (3.35) by the discrete Fourier transform on the spatial lattice. Thus we
obtain an approximate function (ψ(tn, xm)), n = 0, . . . , N , m = 0, . . . ,M . The numerical
mass
∑
xm∈ΛM
|ψ(tn, xm)|
2(∆x)d
is conserved up to rounding errors.
In the linear case, convergence of the time-splitting scheme as ∆t → 0 is a direct conse-
quence of Trotter’s product formula
e(A+B)t = lim
N→∞
(
eA
t
N eB
t
N
)N
where A and B are generators of C0-continuous semigroups. Here we take A = i∆/2
to be the free Schro¨dinger operator and B = U to be the multiplaction operator with
the potential U . For examples of explicit error estimates, see the works [43, 80]. In the
nonlinear case, convergence has been investigated in, for example, [30, 60]. In practice
we use Fast Fourier Transform (FFT) and Strang-splitting which is second-order in time
and of spectral order in space. A Strang-splitting time-step consists of performing half a
time-step ∆t/2 solving (3.34), then a full time-step ∆t solving (3.33), followed by another
half-step ∆t/2 solving (3.34).
In the presence of the angular momentum rotation term −Ω · L with L given in (3.3),
we would need to include the rotation term in equation (3.33) or (3.34). Then it is in
general no longer possible to solve the separate equations exactly. In the literature sev-
eral discretizations for the nonlinear Schro¨dinger equations (3.4) have been suggested, see
[6, 8, 9, 88]. Here we point out a simple alternative approach based on the change of coor-
dinates (3.21) and present some numerical experiments. As detailed in section 3.3, after
the change of coordinates, the wave function ψ solves the NLS with a time-dependent
potential (3.22). Again we restrict ourselves to the finite box [−L,L]d for L large enough.
Then the time-splitting procedure given by (3.33) and (3.34) remains valid with U(x)
replaced by W (t, x).
This well-known pseudo-spectral time-splitting method is an efficient and easy-to-implement
alternative to numerical methods for equation (3.4) proposed in the literature [6, 8, 9, 88].
Note that many of the proposed methods are also valid for more general equations, e.g.
the NLS with a dissipation term. On the other hand, the pseudo-spectral method does
not introduce highly undesirable numerical dissipation to the solution of (3.4). Let us
compare the pseudo-spectral method based on the change of coordinates with an example
taken from the work [6]. We take d = 2, λ = 1000, and Ω = 0.9 in (3.4). We replace U(x)
by a time-dependent potential
W (t, x) =
1
2
(
(1 + )x˜2 + (1− )y˜2
)
,
144
3.5. Numerical simulations of a rotating Bose-Einstein condensate
where x˜ = x cos(ω˜t) + y sin(ω˜t), y˜ = y cos(ω˜t) − x sin(ω˜t) and we take  = 0.35. This a
rotating potential with frequency ω˜, which in rotating coordinates rotates with frequency
ω+ω˜. Note that our results also hold for time-dependent potentials, cf. [18]. The problem
is solved (in rotating coordinates) on the numerical domain [−14, 14] with N = 358 steps
and ∆t = 0.0001. The initial datum is taken to be a groundstate of (3.4) with  = 0 and
ω˜ = 0 which has been obtained by the normalized gradient flow for (3.4) with the backward
Euler finite difference method proposed in [10]. Our results are shown in Figure 3.1 and,
for comparison, the results of [6] are shown in Figure 3.2. Our results are transformed back
into the original (non-rotating) coordinates so that both figures show the wave functions
in the same coordinates. Note that the back-transformation from the rotated lattice to
the lattice in the original coordinates introduces small errors into some of the plots in
Figure 3.1, which are not present in Figure 3.2 and which could in principle be reduced
by using a more suitable (numerical) back-transformation. Comparison of the results
shows that the gradient-flow method produces slightly different vortex lattices as a ground
state. The lattice seems to be slightly less stable, but the results agree both in the number
of vortices as well as the orientation of the condensate, down to the position of the four
outer-most vortices (at t = 0, these are the two on the north-facing side of the condensate
and the two on the south-facing side).
145
On the NLS with rotation term
Figure 3.1: Contour plots of the density function |ψ(t, x)|2 for dynamics of a vortex lattice
at different times
146
3.5. Numerical simulations of a rotating Bose-Einstein condensate
Figure 3.2: Contour plots of the density function |ψ(t, x)|2 for dynamics of a vortex lattice
as found in [6]
147
On the NLS with rotation term
148
Chapter 4
Optimal bilinear control of
Gross–Pitaevskii equations
The work in this chapter has been carried out in collaboration with Michael Hintermu¨ller,
Peter A. Markowich, and Christof Sparber and is a slightly extended version of the pub-
lished work [39].
4.1 Introduction
4.1.1 Physics background
Ever since the first experimental realization of Bose–Einstein condensates (BECs) in 1995,
the possibility to store, manipulate, and measure a single quantum system with extremely
high precision has provided great stimulus in many fields of physical and mathematical
research, among them quantum control theory. In the regime of dilute gases, a BEC,
consisting of N particles, can be modeled by the Gross–Pitaevskii equation [74], i.e. a
cubically nonlinear Schro¨dinger equation (NLS) of the form
i~∂tψ = −
~2
2m
∆ψ + U(x)ψ +Ng|ψ|2ψ +W (t, x)ψ, x ∈ R3, t ∈ R,
with m denoting the mass of the particles, ~ Planck’s constant, g = 4pi~2asc/m, and
asc ∈ R their characteristic scattering length, describing the inter-particle collisions. The
function U(x) describes an external trapping potential which is necessary for the exper-
imental realization of a BEC. Typically, U(x) is assumed to be a harmonic confinement.
In situations where U(x) is strongly anisotropic, one experimentally obtains a quasi one-
dimensional (“cigar-shaped”), or quasi two-dimensional (“pancake shaped”) BEC, see for
instance [49]. In the following, we shall assume U(x) to be fixed. The condensate is
149
Optimal control of NLS
consequently manipulated via a time-dependent control potential W (t, x), which we shall
assume to be of the following form:
W (t, x) = α(t)V (x),
Here, α(t) denotes the control parameter (typically, a switching function acting within a
certain time-interval [0, T ]) and V (x) is a given potential. In our context, the potential
V (x) models the spatial profile of a laser field used to manipulate the BEC and α(t) its
intensity.
The problem of quantum control, i.e. the coherent manipulation of quantum systems
(in particular Bose–Einstein condensates) via external potentials W (t, x), has attracted
considerable interest in the physics literature, cf. [15, 23, 40, 41, 72, 75, 89]. From the
mathematical point of view, quantum control problems are a specific example of bilinear
control systems [24]. It is known that linear or nonlinear Schro¨dinger–type equations are
in general not exactly controllable in, say, L2(R3), cf. [82]. Similarly, approximate con-
trollability is known to hold for only some specific systems, such as [65]. More recently,
however, sufficient conditions for approximate controllability of linear Schro¨dinger equa-
tions with purely discrete spectrum have been derived in [21]. In [63] these conditions have
been shown to be generically satisfied, but, to the best of our knowledge, a generalization
to the case of nonlinear Schro¨dinger equations is still lacking.
The goal of the current paper is to consider quantum control systems within the framework
of optimal control, cf. [84] for a general introduction, from a partial differential equation
constrained point of view. The objective of the control process is thereby quantified
through an objective functional J = J(ψ, α), which is minimized subject to the condition
that the time-evolution of the quantum state is governed by the Gross–Pitaevskii equation
(GPE). Such objective functionals J(ψ, α) usually consist of two parts, one being the
desired physical quantity (observable) to be minimized, the other one describing the cost
it takes to obtain the desired outcome through the control process. In quantum mechanics,
the wave function ψ(t, ·) itself is not a physical observable. Rather, one considers self-
adjoint linear operators A acting on ψ(t, ·) and aims for a prescribed expectation value of
A at time t = T > 0, the final time of the control process. Such expectation values are
computed by taking the L2–inner product 〈ψ(T, ·), Aψ(T, ·)〉L2(Rd). Note that this implies
that the corresponding ψ(t, ·) is only determined up to a constant phase. This fact makes
quantum control less “rigid” when compared to classical control problems in which one
usually aims to optimize for a prescribed target state.
There are many possible ways of modeling the cost it takes to reach a certain prescribed
expectation value. The corresponding cost terms within J(ψ, α) are often given by the
norm of the control α(t) in some function space. Typical choices are L2(0, T ) or H1(0, T ).
However, these choices of function spaces for α(t) often lack a clear physical interpretation.
150
4.1. Introduction
In addition, cost terms based on, say, the L2–norm of α tend to yield highly oscillatory
optimal controls due to the oscillatory nature of the underlying (nonlinear) Schro¨dinger
equation. The same is true for quantum control via so-called Lyapunov tracking methods,
see, e.g., [25]. In the present work we shall present a novel choice for the cost term, which
is based on the corresponding physical work performed throughout the control process.
We continue this introductory section by describing the mathematical setting in more
detail.
4.1.2 Mathematical setting
We consider a quantum mechanical system described by a wave function ψ(t, ·) ∈ L2(Rd)
within d = 1, 2, 3 spatial dimensions. The case d = 1, 2 models the effective dynamics
within strongly anisotropic potentials (resulting in a quasi one or two-dimensional BEC).
The time-evolution of ψ(t, ·) is governed by the following generalized Gross–Pitaevskii
equation (rescaled into dimensionless form):
(4.1) i∂tψ = −
1
2
∆ψ + U(x)ψ + λ|ψ|2σψ + α(t)V (x)ψ, x ∈ Rd, t ∈ R,
with λ ≥ 0, σ < 2/(d− 2), and subject to initial data
ψ(0, ·) = ψ0 ∈ L
2(Rd), α(0) = α0 ∈ R.
For physical reasons we normalize ‖ψ0‖L2(Rd) = 1, which is henceforth preserved by the
time-evolution of (4.1). In addition, the control potential is assumed to be V ∈ W 1,∞(Rd),
whereas for U(x) we require
U ∈ C∞(Rd) such that ∂kU ∈ L∞(Rd) for all multi-indices k with |k| ≥ 2.
In other words, the external potential is assumed to be smooth and subquadratic. One
of the most important examples is the harmonic oscillator U(x) = 12 |x|
2. Due to the
presence of a subquadratic potential, we restrict ourselves to initial data ψ0 in the energy
space
(4.2) Σ :=
{
ψ ∈ H1(Rd) : xψ ∈ L2(Rd)
}
.
In particular, this definition guarantees that the quantum mechanical energy functional
(4.3) E(t) =
∫
Rd
1
2
|∇ψ(t, x)|2 +
λ
σ + 1
|ψ(t, x)|2σ+2 + (α(t)V (x) + U(x))|ψ(t, x)|2dx
associated to (4.1) is well defined.
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Optimal control of NLS
Remark 4.1.1. Note that σ < 2/(d − 2) allows for general power law nonlinearities in
dimensions d = 1, 2, whereas in d = 3 the nonlinearity is assumed to be less than quintic.
From the physics point of view a cubic nonlinearity σ = 1 is the most natural choice,
but higher order nonlinearities also arise in systems with more complicated inter-particle
interactions, in particular in lower dimensions; compare [49]. From the mathematical
point of view, it is well known that the restriction σ < 2/(d−2) guarantees well-posedness
of the initial value problem in the energy space Σ; see [18, 19]. In addition, the condition
λ ≥ 0 (defocusing nonlinearity) guarantees the existence of global in-time solutions to
(4.1); see [18]. Hence, we do not encounter the problem of finite-time blow-up in our
work.
Although (4.1) conserves mass, i.e. ‖ψ(t, ·)‖L2(Rd) = ‖ψ0‖L2(Rd) for all t ∈ R, the energy
E(t) is not conserved. This is in contrast to the case of time-independent potentials. In
our case, rather one finds that
(4.4)
d
dt
E(t) = α˙(t)
∫
Rd
V (x)|ψ(t, x)|2dx.
The physical work performed by the system within a given time-interval [0, T ] is therefore
equal to
(4.5) E(T )− E(0) =
∫ T
0
α˙(t)
∫
Rd
V (x)|ψ(t, x)|2dx dt.
Thus a control α(t) acting for t ∈ [0, T ] upon a system described by (4.1) requires a
certain amount of energy, which is given by (4.5). It, thus, seems natural to include such
a term in the cost functional of our problem in order to quantify the control action.
Indeed, for any given final control time T > 0, and parameters γ1 ≥ 0, γ2 > 0, we define
the following objective functional :
(4.6) J(ψ, α) := 〈ψ(T, ·), Aψ(T, ·)〉2L2(Rd) + γ1
∫ T
0
(E˙(t))2 dt+ γ2
∫ T
0
(α˙(t))2 dt,
where A : Σ → L2(Rd) is a bounded linear operator which is assumed to be essentially
self-adjoint on L2(Rd). In other words, A represents a physical observable with spectrum
spec (A) ⊆ R. A typical choice for A would be A = A′−a where a ∈ R is some prescribed
expectation value for the observable A′ in the state ψ(T, x). For example, if a ∈ spec (A′)
is chosen to be an eigenvalue of A′, the first term in J(ψ, α) is zero as soon as the target
state ψ(T, ·) is, up to a phase factor, given by an associated eigenfunction of A′. However,
one may consider choosing a ∈ R such that it “forces” the functional to equidistribute
between, say, two eigenfunctions.
Remark 4.1.2. We also remark that in the case A = Pϕ − 1, where Pϕ denotes the
152
4.1. Introduction
orthogonal projection onto a given target state ϕ ∈ L2(Rd), the first term on the right
hand side of (4.6) reads
(4.7) 〈ψ(T, ·), Aψ(T, ·)〉L2(Rd) =
∣
∣〈ψ(T, ·), ϕ(·)〉L2(Rd)
∣
∣2 − 1,
using the fact that ‖ψ(T, ·)‖L2(Rd) = 1. Expression (4.7) is the same as used in recent
works in the physics literature; see [40].
Using (4.4), we find that the objective functional J(ψ, α) explicitly reads
J(ψ, α) := 〈ψ(T, ·), Aψ(T, ·)〉2L2(Rd)
+ γ1
∫ T
0
(α˙(t))2
(∫
Rd
V (x)|ψ(t, x)|2 dx
)2
dt+ γ2
∫ T
0
(α˙(t))2 dt.
(4.8)
Here, the second line on the right hand side displays two cost (or penalization) terms for
the control: The first one, involving γ1 ≥ 0, is given by the square of the physical work,
i.e. the right hand side of (4.4). The second is a classical cost term as used in [40]. In our
case, the second term is required as a mathematical regularization of the optimal control
problem, since for general (sign changing) potentials V ∈ L∞(Rd) the weight factor
(4.9) ω(t) :=
∫
Rd
V (x)|ψ(t, x)|2dx
might vanish for some t ∈ R. In such a situation, the boundedness of variations of α(t)
is in jeopardy and the optimal control problem lacks well-posedness. Hence, we require
γ2 > 0 for our mathematical analysis, but typically take γ2  γ1 in our numerics in
Section 4.5 to keep its influence small. Note, however, that in the case where the control
potential satisfies the positivity condition
V (x) ≥ δ > 0 ∀x ∈ Rd,
we may choose γ2 = 0 and all of our results remain valid.
Remark 4.1.3. In situations where the above positivity condition on V (x) does not hold,
one might think of performing a time-dependent gauge transform of ψ, i.e.
ψ˜(t, x) = exp
(
−iκ
∫ t
0
α(s) ds
)
ψ(t, x),
with a constant κ > minx∈RdV (x), assuming that the minimum exists. This yields a
Gross–Pitaevskii equation for the wave function ψ˜ with modified control potential V˜ (x) =
(κ + V (x)) > 0 for all x ∈ Rd. Note, however, that this gauge transform leaves the
expression (4.8) unchanged and hence does not improve the stuation. Only if one also
153
Optimal control of NLS
changes the potential V (x) within J(ψ, α) into V˜ (x), the problem does not require any
regularization term (proportional to γ2). Note, however, that such a modification yields a
control system which is no longer (mathematically) equivalent to the original problem. In
fact, replacing V (x) by V˜ (x) in the objective functional J(ψ, α) corresponds to increasing
the parameter γ2 by κ.
4.1.3 Relation to other works and organization of the chapter
The mathematical research field of optimal bilinear control of systems governed by partial
differential equations is by now classical, cf. [31, 55] for a general overview. Surprisingly,
rigorous mathematical work on optimal (bilinear) control of quantum systems appears
very limited, despite the physical significance of the involved applications (cf. the refer-
ences given above). Results on simplified situations, as, e.g., for finite dimensional quan-
tum systems, can be found in [14] (see also the references therein). More recently, optimal
control problems for linear Schro¨dinger equations have been studied in [11, 13, 42]. In
addition, numerical questions related to quantum control are studied in [12, 87]. Among
these papers, the work in [42] appears closest to our effort. Indeed, in [42], the authors pro-
vide a framework for bilinear optimal control of abstract (linear) Schro¨dinger equations.
The considered objective functional involves a cost term proportional to the L2–norm of
the control parameter α(t). The present work goes beyond the results obtained in [42] in
several repects: First, we generalize the cost functional to account for oscillations in α(t)
and in particular for the physical work load performed throughout the control process. In
addition, we allow for observables A which are unbounded operators on L2. Second, we
consider nonlinear Schro¨dinger equations of Gross–Pitaevskii type, including unbounded
(subquadratic) potentials, which are highly significant in the quantum control of BECs.
This type of equation makes the study of the associated control problem considerably
more involved from a mathematical point of view.
The rest of this work is organized as follows. In section 4.2 we clarify existence of a
minimizer for our control problem. In particular, we prove that the corresponding optimal
solution ψ∗(t, x) is indeed a mild (and not only a weak) solution of (4.1), depending
continuously on the initial data ψ0. Then, in section 4.3 the adjoint equation is derived
and analyzed with respect to existence and uniqueness of a solution. It is our primary
tool for the description of the derivative of the objective function reduced onto the control
space through considering the solution of the Gross-Pitaevskii equation as a function of
the control variable α. The results of section 4.3 are paramount for the derivation of the
first order optimality system in section 4.4. In section 4.5 a gradient- and a Newton-type
descent method are defined, respectively, and then used for computing numerical solutions
for several illustrative quantum control problems. In particular, we consider the optimal
154
4.2. Existence of minimizers
shifting of a linear wave package, splitting of a linear wave package and splitting of a
BEC. The paper ends with conclusions on our findings in Chapter 5.
Throughout this chapter we shall denote strong convergence of a sequence (xn)n∈N by
xn → x and weak convergence by xn ⇀ x. For simplicity, we shall often write ψ(t) ≡
ψ(t, ·) and also use the shorthand notation LptL
q
x instead of L
p(0, T ;Lq(Rd)). Similarly,
H1t stands for H
1(0, T ), with dual (H1t )
∗ = (H1(0, T ))∗.
4.2 Existence of minimizers
We start by specifying the basic functional analytic framework. For any given T > 0, we
consider H1(0, T ) as the real vector space of control parameters α(t) ∈ R. It is known
[19] that for every α ∈ H1(0, T ), there exists a unique mild solution ψ ∈ C([0, T ]; Σ) of
the Gross-Pitaevskii equation, also see Chapter 3. More precisely, ψ solves
ψ(t, x) = S(t)ψ0(x)− i
∫ t
0
S(t− s)
(
λ|ψ(s, ·)|2σψ(s, ·) + α(s)V ψ(s, ·)
)
(x)ds,
where from now on we denote by
(4.10) S(t) = e−itH , H = −
1
2
∆ + U(x),
the group of unitary operators {S(t)}t∈R generated by the Hamiltonian H. In other words,
S(t) describes the time-evolution of the linear, uncontrolled system. Next, we define
(4.11) Υ(0, T ) := L2(0, T ; Σ) ∩H1(0, T ; Σ∗),
where Σ∗ is the dual of the energy space Σ. Then the appropriate space for our mini-
mization problem is
Λ(0, T ) := {(ψ, α) ∈ Υ(0, T )×H1(0, T ) : ψ is a mild solution of (4.1) }.
Since the control α is real-valued, it is natural to consider Λ(0, T ) as a real vector space
and we shall henceforth equip L2(Rd) with the scalar product
(4.12) 〈ξ, ψ〉L2(Rd) = Re
∫
Rd
ξ(x)ψ(x) dx,
which is subsequently inherited by all L2-based Sobolev spaces. (Note that this choice
is also used in [19].) From what is said above, we infer that the space Λ(0, T ) is indeed
nonempty.
155
Optimal control of NLS
With these definitions at hand, the optimal control problem under investigation is to find
(4.13) J∗ = inf
(ψ,α)∈Λ(0,T )
J(ψ, α).
We are now in the position to state the first main result of this work.
Theorem 4.2.1. Let λ ≥ 0, 0 < σ < 2/(d − 2), V ∈ W 1,∞(Rd), and U ∈ C∞(Rd) be
subquadratic. Then, for any T > 0, any initial data ψ0 ∈ Σ, α0 ∈ R and any choice of
parameters γ1 ≥ 0, γ2 > 0 the optimal control problem (4.13) has a minimizer (ψ∗, α∗) ∈
Λ(0, T ).
The proof of this theorem will be split into three steps: In subsection 4.2.1 we shall
first prove a convergence result for minimizing, or more precisely, infimizing sequences.
We consequently deduce in subsection 4.2.2 that the obtained limit ψ∗ is indeed a mild
solution of (4.1). Finally, we shall prove lower semicontinuity of J(ψ, α) with respect to
the convergence obtained before.
4.2.1 Convergence of infimizing sequences
First note that there exists at least one infimizing sequence with an infimum −∞ ≤ J∗ <
+∞, since Λ(0, T ) 6= ∅ and J : Λ(0, T ) → R. Then we have the following result for any
infimizing sequence.
Proposition 4.2.2. Let (ψn, αn)n∈N be an infimizing sequence of the optimal control prob-
lem given by (4.6). Then under the assumptions of Theorem 4.2.1 there exist a subse-
quence, still denoted by (ψn, αn)n∈N, and functions α∗ ∈ H1(0, T ), ψ∗ ∈ L∞(0, T ; Σ), such
that
αn ⇀ α∗ in H
1(0, T ), and αn → α∗ in L
2(0, T ),
ψn ⇀ ψ∗ in L
2(0, T ; Σ),
ψn → ψ∗ in L
2(0, T ;L2(Rd)) ∩ L2(0, T ;L2σ+2(Rd)),
as n→ +∞. Furthermore it holds that
(4.14) ψn(t)→ ψ∗(t) in L
2(Rd), and ψn(t) ⇀ ψ∗(t) in Σ
for almost all t ∈ [0, T ].
Proof. By definition, J ≥ 0 and thus it is bounded from below. For an infimizing sequence
(ψn, αn)n∈N the sequence of objective functional values (J(ψn, αn))n∈N converges and is
156
4.2. Existence of minimizers
bounded on R. Hence, it holds that J(ψn, αn) ≤ C < +∞ for all n ∈ N. Since γ2 > 0 it
follows that ∫ T
0
(α˙n(t))
2 dt ≤ C < +∞.
For smooth αn : [0, T ]→ R we compute
αn(t) = αn(0) +
∫ t
0
α˙n(s) ds ≤ αn(0) +
(
T
∫ T
0
(α˙n(s))
2 ds
)1/2
< +∞,
and thus αn is bounded in L∞(0, T ). By approximation (using the fact that αn(0) = α0
is fixed), the sequence (αn)n∈N is uniformly bounded in L∞(0, T ), which in turn implies a
uniform bound in L2(0, T ) and thus in H1(0, T ). Hence, there exists a subsequence, still
denoted (αn)n∈N, and α∗ ∈ H1(0, T ), such that
αn ⇀ α∗ ∈ H
1(0, T ).
Moreover, since H1(0, T ) is compactly embedded into L2(0, T ), we deduce that αn → α∗
in L2(0, T ). Next, we recall that
d
dt
En(t) = α˙n(t)
∫
Rd
V (x)|ψn(t, x)|
2 dx
and hence
‖E˙n‖L2t ≤ ‖α˙n‖L2t ‖V ‖L∞x ‖ψ0‖
2
L2x
,
in view of mass conservation ‖ψn(t)‖L2x = ‖ψ0‖L2x . Since En(0) = E0 depends only on
ψ0 and α0 (and is thus independent of n ∈ N), the same argument as before yields
‖En‖L∞t ≤ C. Recalling the definition of the energy (4.3) and the fact that λ ≥ 0, we
obtain
(4.15)
1
2
‖∇ψn(t)‖
2
L2x
≤ ‖En‖L∞t + c‖αn‖L∞t ‖ψ0‖
2
L2x
+ C‖xψn(t)‖
2
L2x
,
again using conservation of mass ‖ψn(t)‖L2x = ‖ψ0‖L2x . Furthermore, it holds that
d
dt
∫
Rd
|x|2|ψ|2 dx = 2 Re
∫
Rd
i|x|2ψ
(
1
2
∆ψ − λ|ψ|2σψ − λαV ψ − Uψ
)
dx
= 2 Im
∫
Rd
xψ∇ψ dx ≤ ‖xψ(t)‖2L2x + ‖∇ψ(t)‖
2
L2x
,
which, in view of the bound (4.15) and Gronwall’s inequality, yields
‖xψ(t)‖2L2x ≤ C
(
‖En‖L∞t + ‖αn‖L∞t ‖ψ0‖
2
L2x
)
,
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Optimal control of NLS
for all t ∈ [0, T ]. In summary we have shown
(4.16) ‖ψn(t)‖
2
Σ = ‖ψn(t)‖
2
H1x
+ ‖xψn(t)‖
2
L2x
≤ C,
where C > 0 is independent of n ∈ N and t ∈ [0, T ]. Hence, ψn is uniformly bounded in
L∞(0, T ; Σ) and in particular in L2(0, T ; Σ). By reflexivity of L2(0, T ; Σ), we consequently
infer the existence of a subsequence (denoted by the same symbol) such that
ψn ⇀ ψ∗ in L
2(0, T ; Σ) as n→ +∞.
To obtain the strong convergence announced above, we first note that (4.1) implies ∂tψn ∈
L∞(0, T ; Σ∗). Now we notice the following Lemma.
Lemma 4.2.3. The energy space Σ is compactly embedded in L2(Rd).
The proof of this statement can be found, for instance, in [26, Proposition 2.1]. We repeat
it here for the reader’s convenience. Due to the reflexivity of Σ it suffices to show that
ωn → ω in L2(Rd) whenever a sequence (ωn)n∈N ⊂ Σ satisfies ωn ⇀ ω in Σ as n → ∞.
Take any ball BR ⊂ Rd of radius R > 0 around the origin. Restricting to test functions
with support in BR, we see that ωn|BR ⇀ ω|BR in H
1(BR) as n → ∞. Since H1(BR) is
compactly embedded in L2(BR), it follows ωn|BR → ω|BR in L
2(BR). In order to show
convergence on the whole L2(Rd), we split up the L2–norm as follows. It holds that
∫
Rd
|ω(x)− ωn(x)|
2 dx =
∫
BR
|ω(x)− ωn(x)|
2 dx+
∫
Rd\BR
|ω(x)− ωn(x)|
2 dx
For fixed R > 0, the second term on the right hand side vanishes in the limit as n→∞.
On the other hand, the first term on the right hand side is bounded by C/R2, since
(ωn)n∈N is bounded in Σ by assumption. Given any  > 0, we thus choose first R > 0
large enough such that the first term is bounded by  and then n large enough such that
the second term is bounded by  in order to show that ωn → ω in L2(Rd) as n → ∞.
Hence Σ is indeed compactly embedded in L2(Rd).
Thus, we can apply the Aubin–Lions Lemma [78] to deduce
ψn
n→∞
−−−→ ψ∗ in L
2((0, T )× Rd).
In particular, there exists yet another subsequence (still denoted by the same symbol),
such that
ψn(t)
n→∞
−−−→ ψ∗(t) in L
2(Rd), for almost all t ∈ [0, T ].
In order to obtain weak convergence in the energy space, i.e. ψn(t) ⇀ ψ∗(t) in Σ, we fix
t ∈ [0, T ] such that ψn(t) → ψ∗(t) in L2(Rd). In view of (4.16), every subsequence of
ψn(t) has yet another subsequence such that ψn(t) converges weakly in Σ to some limit.
158
4.2. Existence of minimizers
On the other hand, this limit is necessarily given by ψ∗(t), since ψn(t)→ ψ∗(t) in L2(Rd).
Hence the whole sequence converges weakly in Σ to ψ∗(t). By lower-semicontinuity of the
Σ–norm we can deduce ‖ψn(t)‖Σ ≤ C and thus ψ∗ ∈ L∞(0, T ; Σ).
Finally, the announced convergence in L2(0, T ;L2σ+2(Rd)) is obtained by invoking the
Gagliardo–Nirenberg inequality, i.e.
(4.17) ‖ξ‖Lrx ≤ C‖ξ‖
1−δ(r)
L2x
‖∇ξ‖δ(r)L2x ,
where 2 ≤ r < 2dd−2 and δ(r) = d(
1
2−
1
r ). This concludes the proof of Proposition 4.2.2.
4.2.2 Minimizers as mild solutions
Next we prove that the limit ψ∗ obtained in the previous subsection is indeed a mild
solution of (4.1) with corresponding control α∗. From the physical point of view, this is
important since it implies continuous (in time) dependence of ψ∗ upon a given initial data
ψ0. To this end, one should also note that H1(0, T ) ↪→ C(0, T ) (using Sobolev imbed-
dings), and hence the obtained optimal control parameter α∗(t) is indeed a continuous
function on [0, T ].
Proposition 4.2.4. Let (ψ∗, α∗) ∈ Υ(0, T ) × H1(0, T ) be the limit obtained in Proposi-
tion 4.2.2. Then ψ∗ is a mild solution of (4.1) with control α∗ and
ψ∗ ∈ C([0, T ]; Σ) ∩ C
1([0, T ]; Σ∗).
In particular, this implies that the convergence result (4.14) holds for all t ∈ [0, T ].
Proof. First we note that, by construction, each ψn satisfies
(4.18) ψn(t) = S(t)ψ0 − i
∫ t
0
S(t− s)
(
λ|ψn(s)|
2σψn(s) + αn(s)V ψn(s)
)
ds
for all t ∈ [0, T ]. Here and in the following we shall suppress the x–dependence of ψ for
notational convenience. In order to prove that ψ∗ is a mild solution corresponding to
the control α∗, we take the L2–scalar product of the above equation with a test function
χ ∈ C∞0 (R
d). This yields
〈ψn(t), χ〉L2x = 〈S(t)ψ0, χ〉L2x − iλ
∫ t
0
〈
S(t− s)|ψn(s)|
2σψn(s), χ
〉
L2x
ds
− i
∫ t
0
〈
S(t− s)αn(s)V ψn(s), χ
〉
L2x
ds.
(4.19)
In view of Proposition 4.2.2, the term on the left hand side of this identity converges to
159
Optimal control of NLS
the desired expression for almost all t ∈ [0, T ], i.e.
lim
n→∞
〈ψn(t), χ〉L2x = 〈ψ∗(t), χ〉L2x .
In order to proceed further, we note that for any f ∈ D′(Rd) it holds that
(4.20) 〈S(t− s)f(s), χ〉L2x = 〈f(s), S(s− t)χ〉L2x ,
and we therefore define
(4.21) χ˜ : [0, t]× Rd → C, χ 7→ χ˜(·, x) := S(· − t)χ(x),
for which we can prove the following regularity properties.
Lemma 4.2.5. There exists a constant C = C(T ) > 0 such that for all t ∈ [0, T ] it holds
that
sup
s∈[0,t]
(
‖xχ˜(s)‖L2x + ‖∇χ˜(s)‖L2x
)
≤ C(T ) < +∞,
where the function χ˜ is defined in (4.21). In particular, the function χ˜ is bounded in
L∞(0, t;L2σ+2(Rd)).
Proof of Lemma 4.2.5. The norm ‖χ˜(s)‖L2x = ‖S(s − t)χ‖L2x is conserved since S(t) is a
unitary operator on L2(Rd). Furthermore, it holds that
i∂t[∇, S(t)] = H[∇, S(t)] + [∇, H]S(t) = H[∇, S(t)] +∇US(t),
and hence
[∇, S(t)] = −i
∫ t
0
S(t− s)∇U S(s) ds.
We can thus estimate
‖∇χ˜(s)‖L2x = ‖∇S(t− s)χ‖L2x
≤ ‖S(t− s)∇χ‖L2x +
∥
∥
∥
∫ t−s
0
S(t− s− τ)∇Uχ˜(τ) dτ
∥
∥
∥
L2x
≤ ‖∇χ‖L2x + C
∫ t−s
0
‖xχ˜(τ)‖L2x dτ,
(4.22)
since U is subquadratic, i.e. |∇U(x)| ≤ C|x|. Likewise, we deduce
[x, S(t)] = −i
∫ t
0
S(t− s)∇S(s) ds
160
4.2. Existence of minimizers
and hence
(4.23) ‖xχ˜(s)‖L2x ≤ ‖xχ‖L2x +
∫ t−s
0
‖∇χ˜(τ)‖L2x dτ.
Combining the estimates (4.22) and (4.23) and applying Gronwall’s inequality yields
‖xχ˜(s)‖L2x + ‖∇χ˜(s)‖L2x ≤ C
(
‖xχ‖L2x + ‖∇χ‖L2x
)
< +∞,
where C > 0. The bound in L∞(0, t;L2σ+2(Rd)) then follows from the uniform-in-time
bound in H1(Rd) and the Gagliardo–Nirenberg inequality (4.17).
With the result of Lemma 4.2.5 at hand, we consider the second term on the right hand
side of (4.19). Rewriting it using (4.20), we estimate
∣
∣
∣
∣
∫ t
0
〈
|ψn(s)|
2σψn(s)− |ψ∗(s)|
2σψ∗(s), χ˜(s)
〉
L2x
ds
∣
∣
∣
∣
≤
∫ t
0
∫
Rd
∣
∣|ψn(s, x)|
2σψn(s, x)− |ψ∗(s, x)|
2σψ∗(s, x)
∣
∣ |χ˜(s, x)| dx ds
≤ C
∫ t
0
∫
Rd
(
|ψn(s, x)|
2σ + |ψ∗(s, x)|
2σ
)
|ψn(s, x)− ψ∗(s, x)||χ˜(s, x)| dx ds.
By Ho¨lder’s inequality, it holds that
∫ t
0
∫
Rd
∣
∣|ψn(s, x)|
2σ + |ψ∗(s, x)|
2σ
∣
∣ |ψn(s, x)− ψ∗(s, x)||χ˜(s)| dx ds
≤
√
T
(
‖ψn‖
2σ
L∞t L
2σ+2
x
+ ‖ψ∗‖
2σ
L∞t L
2σ+2
x
)
‖ψn − ψ∗‖L2tL2σ+2x ‖χ˜‖L∞t L2σ+2x ,
where, in view of Lemma 4.2.5, we have ‖χ˜‖L∞t L2σ+2x < +∞. In addition, Proposition 4.2.2
implies that the factor inside the parentheses is bounded and that
lim
n→∞
‖ψn − ψ∗‖L2tL2σ+2x = 0.
Thus, we have shown that the second term on the right hand side of (4.19) vanishes in
the limit n→∞.
It remains to treat the last term on the right hand side of (4.19), rewritten via (4.20).
We first estimate
∣
∣
∣
∣
∫ t
0
〈
αn(s)V ψn(s)− α∗(s)V ψ∗(s), χ˜(s)
〉
L2x
ds
∣
∣
∣
∣
≤
∫ t
0
∫
Rd
|αn(s)| |V (x)| |ψn(s, x)− ψ∗(s, x)| |χ˜(s, x)| dx ds
+
∫ t
0
∫
Rd
|αn(s)− α∗(s)| |V (x)| |ψ∗(s, x)||χ˜(s, x)| dx ds.
161
Optimal control of NLS
Here, the last term on the right hand side can be bounded by
∫ t
0
∫
Rd
|αn(s)− α∗(s)||V (x)||ψ∗(s, x)||χ˜(s, x)| dx ds
≤ ‖αn − α∗‖L2t ‖V ‖L∞x ‖ψ‖L2tL2x‖χ˜‖L∞t L2x
n→∞
−−−→ 0,
in view of the convergence of αn → α∗ in L2(0, T ). For the remaining term we use the
fact that V ∈ L∞(Rd) and Ho¨lder’s inequality to obtain that
∫ t
0
∫
Rd
|αn(s)| |V (x)| |ψn(s, x)− ψ∗(s, x)||χ˜(s, x)| dx ds
≤ ‖αn‖L2t ‖V ‖L∞x ‖ψn − ψ∗‖L2tL2x‖χ˜‖L∞t L2x
n→∞
−−−→ 0,
due to the results of Proposition 4.2.2 and Lemma 4.2.5.
In summary this proves that ψ∗ ∈ Υ(0, T ) satisfies, for almost all t ∈ [0, T ],
ψ∗(t) = S(t)ψ0 − i
∫ t
0
S(t− s)
(
λ|ψ∗(s)|
2σψ∗(s) + α∗(s)V ψ∗(s)
)
ds,
i.e. ψ∗ is a weak Σ–solution in the terminology of [19, Definition 3.1.1] (where the analogous
notion of weak H1–solutions is introduced). In order to obtain that ψ∗ is indeed a mild
solution we note that
ψ∗ ∈ Υ(0, T ) ↪→ C([0, T ];L
2(Rd)) ∩ C([0, T ];L2σ+2(Rd))
by interpolation and the Gagliardo–Nirenberg inequality (4.17). Classical arguments
based on Strichartz estimates then yield uniqueness of the weak Σ–solution ψ∗. Arguing
as in the proof of [19, Theorem 3.3.9], we infer that ψ∗ is indeed a mild solution to (4.1),
satisfying ψ∗ ∈ C([0, T ]; Σ) ∩ C1([0, T ]; Σ∗).
4.2.3 Lower semicontinuity of objective functional
In order to conclude that the pair (ψ∗, α∗) ∈ Λ(0, T ) is indeed a minimizer of our optimal
control problem, it remains to show lower semicontinuity of the functional J(ψ, α) with
respect to the convergence results established in Proposition 4.2.2.
Lemma 4.2.6. For the sequence constructed in Proposition 4.2.2, it holds that
J∗ = lim inf
n→∞
J(ψn, αn) ≥ J(ψ∗, α∗).
Proof. Since A ∈ L(Σ, L2(Rd)) by assumption, the sequence (Aψn(T ))n∈N converges
weakly to Aψ(T ) in L2(Rd). In addition ψn(T ) → ψ∗(T ) in L2(Rd) as n → ∞ by
162
4.2. Existence of minimizers
Proposition 4.2.4, and hence the estimate
∣
∣〈ψn(T ), Aψn(T )〉L2x − 〈ψ∗(T ), Aψ∗(T )〉L2x
∣
∣
≤
∣
∣〈ψn(T )− ψ∗(T ), Aψn(T )〉L2x
∣
∣+
∣
∣〈ψ∗(T ), A(ψn(T )− ψ∗(T ))〉L2x
∣
∣ .
yields convergence of the corresponding term in the objective functional (4.8). Next, we
consider the cost term involving γ1. In view of (4.9), we define
ωn(t) :=
∫
Rd
V (x)|ψn(t, x)|
2dx, ω∗(t) :=
∫
Rd
V (x)|ψ∗(t, x)|
2dx,
and estimate
lim inf
n→∞
∫ T
0
(α˙n(t))
2ω2n(t)dt ≥
lim inf
n→∞
∫ T
0
(α˙n(t))
2ω2∗(t) dt+ lim infn→∞
∫ T
0
(α˙n(t))
2
(
ω2n(t)− ω
2
∗(t)
)
dt.
(4.24)
Note that 0 ≤ ωn(t) ≤ ‖V ‖L∞x ‖ψ0‖
2
L2x
independently of n ∈ N and t ∈ [0, T ] and that the
same holds for ω∗(t). The first term on the right hand side of (4.24) is convex in αn and
thus satisfies
(4.25) lim inf
n→∞
∫ T
0
(α˙n(t))
2ω2∗(t) dt ≥
∫ T
0
(α˙∗(t))
2ω2∗(t) dt,
since any convex and lower semicontinuous functional is weakly lower semicontinuous. On
the other hand, Proposition 4.2.2 implies
(4.26) lim inf
n→∞
ωn(t) ≥ ω∗(t) ≥ 0 for all t ∈ [0, T ].
Thus, using (4.25) and (4.26) together with Fatou’s Lemma yields
lim inf
n→∞
∫ T
0
(α˙n(t))
2ω2n(t) dt
≥
∫ T
0
(α˙∗(t))
2ω2∗(t) dt+
∫ T
0
lim inf
n→∞
(α˙n(t))
2 lim inf
n→∞
(
ω2n(t)− ω
2
∗(t)
)
dt
≥
∫ T
0
(α˙∗(t))
2ω2∗(t) dt.
Finally the cost term involving γ2 is lower semicontinuous by convexity and weak conver-
gence of αn in H1(0, T ).
In summary, we have shown that J∗ = lim infn→∞ J(ψn, αn) ≥ J(ψ∗, α∗) and thus indeed
J∗ = J(ψ∗, α∗). In other words, (ψ∗, α∗) ∈ Λ(0, T ) solves the optimization problem.
163
Optimal control of NLS
Remark 4.2.7. Note that the bound on xψn(t, ·) in L2(Rd), obtained in Proposition
4.2.2, is indeed crucial for proving the weak lower-semicontinuity of J(ψ, α). Without
such a bound on the second moment, we would only have
ψn(t)
n→∞
−−−→ ψ(t) in L2loc(R
d),
due to the lack of compactness of H1(Rd) ↪→ L2(Rd). In this case, the lower semi-
continuity of the term 〈ψ(T ), Aψ(T )〉L2x is not guaranteed. A possible way to circumvent
this problem would be to assume that A is positive definite, which, however, is not true
for general observables of the form A = A′ − a, with a ∈ R. A second possibility would
be to assume that A is localizing, i.e. for all ψ ∈ H1(Rd): suppx∈Rd(Aψ(x)) ⊆ B(R), for
some R < +∞.
4.3 Derivation and analysis of the adjoint equation
In order to give a characterization of a minimizer (ψ∗, α∗) ∈ Λ(0, T ), we need to derive
the first order optimality conditions for our optimal control problem (4.13). For this
purpose, we shall first formally compute the derivative of the objective functional J(ψ, α)
in the next subsection and consequently analyze the resulting adjoint problem. A rigorous
justification for the derivative will be given in Section 4.4.
4.3.1 Identification of the derivative of J(ψ, α)
The mild solution of the nonlinear Schro¨dinger equation (4.1), corresponding to the control
α ∈ H1(0, T ), induces a map
ψ : H1(0, T )→ Υ(0, T ) : α 7→ ψ(α).
Using this map we introduce the unconstrained or reduced functional
J : H1(0, T )→ R, α 7→ J(α) := J(ψ(α), α).
For the characterization of critical points, we need to compute the derivative of J. For this
calculation let δα ∈ H1(0, T ) with δα(0) = 0 be a feasible control perturbation. (Recall
that H1(0, T ) ↪→ C(0, T ) and hence it makes sense to evaluate δα(t) at t = 0.) Then the
chain rule yields
〈J′(α), δα〉(H1t )∗,H1t = 〈∂ψJ(ψ(α), α), ψ
′(α)δα〉Υ∗,Υ
+ 〈∂αJ(ψ(α), α), δα〉(H1t )∗,H1t
(4.27)
164
4.3. Derivation and analysis of the adjoint equation
where Υ∗ denotes the dual space of Υ ≡ Υ(0, T ) for any given T > 0. The main difficulty
lies in computing ψ′(α) since ψ is given only implicitly through the nonlinear Schro¨dinger
equation (4.1).
In the following, we shall write the (nonlinear) partial differential equation (4.1) in a more
abstract form, i.e.
(4.28) P (ψ, α) := i∂tψ −Hψ − α(t)V (x)ψ − λ|ψ|
2σψ = 0,
where H = −12∆ + U(x) denotes the linear, uncontrolled Hamiltonian operator. Setting
ψ = ψ(α) and differentiating with respect to α formally yields
d
dα
P (ψ(α), α) = ∂ψP (ψ(α), α)ψ
′(α) + ∂αP (ψ(α), α) = 0.
Next, assuming that ∂ψP is invertible, we solve for ψ′(α) via
ψ′(α) = −∂ψP (ψ, α)
−1∂αP (ψ(α), α).
Thus it holds that
〈∂ψJ(ψ(α), α), ψ
′(α)δα〉Υ∗,Υ
=
〈
−∂ψJ(ψ(α), α), ∂ψP (ψ(α), α)
−1∂αP (ψ(α), α)δα
〉
Υ∗,Υ
,
which can be rewritten as
〈∂ψJ(ψ(α), α), ψ
′(α)δα〉Υ∗,Υ
=
〈
−∂αP (ψ(α), α)
∗∂ψP (ψ(α), α)
−∗∂ψJ(ψ(α), α), δα
〉
(H1t )
∗,H1t
.
(4.29)
Here we abbreviate
∂ψP (ψ(α), α)
−∗ := (∂ψP (ψ(α), α)
∗)−1 = (∂ψP (ψ(α), α)
−1)∗.
Substituting (4.29) into equation (4.27), we see that critical points of (4.13) satisfy
0 = 〈J′(α), δα〉(H1t )∗,H1t = 〈∂αJ(ψ(α), α), δα〉(H1t )∗,H1t +
〈−∂αP (ψ(α), α)
∗∂ψP (ψ(α), α)
−∗∂ψJ(ψ(α), α), δα〉(H1t )∗,H1t
(4.30)
for all δα ∈ H1(0, T ) such that δα(0) = 0. In order to obtain (4.30) in a more explicit
form, we (formally) compute the derivative
∂ψP (ψ, α)ξ = i∂tξ −Hξ − α(t)V (x)ξ − λ(σ + 1)|ψ|
2σξ − λσ|ψ|2σ−2ψ2ξ,(4.31)
165
Optimal control of NLS
acting on ξ ∈ L2(Rd) ⊂ Σ∗. Analogously, we find
∂αP (ψ, α) = −V (x)ψ.
Next, we define
(4.32) ϕ := ∂ψP (ψ(α), α)
−∗∂ψJ(ψ(α), α),
which, in view of (4.30), allows us to express J′(α) ∈ (H1(0, T ))∗ in the following form:
(4.33) J′(α) = ∂αJ(ψ(α), α)− ∂αP (ψ(α), α)∗ϕ.
We consequently obtain J′(α) by explicitly calculating the right hand side of this equation
(given in (4.47) below), provided we can determine ϕ.
In order to perform this calculation, we recall that the duality pairing between ξ ∈
L2(Rd) ⊂ Σ∗ and ψ ∈ Σ can be expressed by the inner product defined in (4.12). Thus,
(4.32) implies
(4.34) 〈ϕ, ∂ψP (ψ(α), α)δψ〉L2tL2x = 〈∂ψJ(ψ(α), α), δψ〉L2tL2x ,
for all test functions δψ ∈ Υ(0, T ) such that δψ(0) = 0. This is the correct “tangent space”
for ψ in view of the Cauchy data
ψ(0) + δψ(0) = ψ0 and ψ(0) = ψ0.
By virtue of the symmetry of the linearized operator ∂ψP (ψ(α), α), equation (4.34) cor-
responds to the weak formulation of the following adjoint equation:
(4.35)



i∂tϕ−Hϕ− α(t)V (x)ϕ− λ(σ + 1)|ψ|
2σϕ− λσ|ψ|2σ−2ψ2ϕ =
δJ(ψ, α)
δψ(t)
,
for all t ∈ [0, T ] and with data: ϕ(T ) = i
δJ(ψ, α)
δψ(T )
.
Here, δJ(ψ,α)δψ(t) denotes the first variation of J(ψ, α) with respect to the value of ψ(t) ∈
H1(Rd), where ψ is the solution of (4.1) with control α. Likewise, δJ(ψ,α)δψ(T ) denotes the first
variation with respect to solutions of (4.1) evaluated at the final time t = T . Explicitly,
these derivatives are given by
δJ(ψ, α)
δψ(t)
= 4(α˙(t))2
(∫
Rd
V (x)|ψ(t, x)|2dx
)
V (x)ψ(t, x)
≡ 4(α˙(t))2ω(t)V (x)ψ(t, x),
(4.36)
166
4.3. Derivation and analysis of the adjoint equation
in view of the definition (4.9), and
δJ(ψ, α)
δψ(T )
= 4〈ψ(T, ·), Aψ(T, ·)〉L2xAψ(T, x).(4.37)
The system (4.35) consequently defines a Cauchy problem for ϕ with data given at t = T ,
the final time. Thus, one needs to solve (4.35) backwards in time, a common feature of
adjoint systems for time-dependent phenomena.
Remark 4.3.1. In fact, ϕ can also be seen as a Lagrange multiplier within the Lagrangian
formulation of the optimal control problem. In oder to see this, one defines the Lagrangian
L(ψ, α, ϕ) = J(ψ, α)− 〈ϕ, P (ψ, α)〉L2tL2x ,
where P (ψ, α) is the nonlinear Schro¨dinger equation given in (4.28). Formally, the Euler–
Lagrange equations associated to L(ψ, α, p) yield (4.33) and (4.35). In Section 4.5 we shall
use the Lagrangian formulation to formally compute the Hessian of the reduced objective
functional J(α).
4.3.2 Local and global existence theory for solutions of higher
regularity
In order to obtain existence of solutions to (4.35), we need sufficiently high regularity of
ψ, the solution of the Gross–Pitaevskii equation (4.1). For this purpose, for every m ∈ N
we define
Σm :=
{
ψ ∈ L2(Rd) : xj∂kψ ∈ L2(Rd) for all multi-indices j and k with
|j|+ |k| ≤ m} ,
equipped with the norm (note that Σ1 ≡ Σ):
‖ψ‖Σm :=
∑
|j|+|k|≤m
∥
∥xj∂kψ
∥
∥
L2x
.
Remark 4.3.2. If the external potential U(x) were in L∞(Rd), it would be enough to
work in the space Hm(Rd) instead of Σm. In the presence of an external subquadratic
potential, however, we also require control of higher moments of the wave function ψ with
respect to x.
The goal of this section is to show the following regularity result for solutions to (4.1).
167
Optimal control of NLS
Lemma 4.3.3. Let λ ≥ 0, σ ∈ N with σ < 2/(d− 2), and U ∈ C∞(Rd) be subquadratic.
For m > d/2, let ψ0 ∈ Σm, and V ∈ Wm,∞(Rd). Then the mild solution of (4.1) satisfies
ψ ∈ L∞(0, T ; Σm).
The proof of this lemma will require a local existence theory in Σm. This is the content of
Lemma 4.3.7. The proof of Lemma 4.3.3 then consists of showing that the local solutions
in Σm coincide with the (global) mild solutions in Σ. To proceed further, we need two
technical results. The first concerns estimates on the nonlinearity in Σm.
Lemma 4.3.4. Let σ ∈ N, and m > d/2. Then there exists a constant C > 0, such that
for all ψ, ψ˜ ∈ Σm it holds that
∥
∥|ψ|2σψ‖Σm ≤ C‖ψ‖
2σ
L∞‖ψ‖Σm
∥
∥|ψ|2σψ − |ψ˜|2σψ˜
∥
∥
Σm
≤ C
(
‖ψ‖2σL∞ + ‖ψ˜‖
2σ
L∞
)
‖ψ − ψ˜‖Σm .
In other words, ψ 7→ |ψ|2σψ is locally Lipschitz in Σm.
Proof. Consider two multi-indices j, k ∈ Nd such that n := |j|+ |k| ≤ m and any ψ ∈ Σm.
Then it holds that |xj∂kψ| is a sum of terms of the form
(4.38) |xj||ψ|2σ+1−r
r∏
l=1
∣
∣∂k
(l)
ψ
∣
∣
for some 1 ≤ r ≤ 2σ+ 1, and some multi-indices k(l), l = 1, . . . , r such that
∑r
l=1 k
(l) = k.
Using Ho¨lder’s inequality, we bound the L2–norm of each summand as follows:
∥
∥xj|ψ|2σ+1−r
r∏
l=1
∂k
(l)
ψ
∥
∥
L2
≤ ‖xj|ψ|2σ+1−r‖Lpr+1
r∏
l=1
∥
∥∂k
(l)
ψ
∥
∥
Lpl
,
where
pl =
2n
|k(l)|
, l = 1, . . . , r, and pr+1 =
2n
|j|
.
It holds that
∥
∥xj|ψ|2σ+1−r
∥
∥
Lpr+1
=
∥
∥|x|
|j|
2 pr+1|ψ|
2σ+1−r
2 pr+1
∥
∥
2
pr+1
L2
≤ ‖ψ‖
2σ+1−r− 2pr+1
L∞ ‖x
nψ‖
2
pr+1
L2 ≤ ‖ψ‖
2σ+1−r− |j|n
L∞ ‖ψ‖
|j|
n
Σm .
Furthermore the Gagliardo–Nirenberg inequality yields
∥
∥∂k
(l)
ψ
∥
∥
Lpl
≤ C‖ψ‖
1− |k
(l)|
n
L∞ ‖ψ‖
|k(l)|
n
Hn .
168
4.3. Derivation and analysis of the adjoint equation
Since |j|+
∑
l |k
(l)| = n ≤ m, it follows that
∥
∥xj|ψ|2σ+1−r
r∏
l=1
∂k
(l)
ψ
∥
∥
L2
≤ C‖ψ‖2σL∞‖ψ‖Σm ,
and hence
∥
∥xj∂k(|ψ|2σψ)
∥
∥
L2
≤ C‖ψ‖2σL∞‖ψ‖Σm .
Thus we conclude the proof of the first estimate of Lemma 4.3.4 by summing over all
multi-indices k and j such that |k|+ |j| ≤ m. The second estimate follows similarly once
we apply the expansion
r∏
l=1
al −
r∏
l=1
bl =
r∑
l=1
∏
l˜<l
al˜
∏
l˜>l
bl˜(al − bl)
to obtain
xj|ψ|2σ+1−r
r∏
l=1
∂k
(l)
ψ − xj|ψ˜|2σ+1−r
r∏
l=1
∂k
(l)
ψ˜
= xj
(
|ψ|2σ+1−r − |ψ˜|2σ+1−r
)
r∏
l=1
∂k
(l)
ψ
+ xj|ψ˜|2σ+1−r
r∑
l=1
∏
l˜<l
∂k
(l˜)
ψ
∏
l˜>l
∂k
(l˜)
ψ˜(∂k
(l)
ψ − ∂k
(l)
ψ˜).
Then we can estimate each term separately using the estimates above.
Remark 4.3.5. This technical lemma is ultimately the reason why we need to restrict
ourselves to σ ∈ N. If σ 6∈ N, we cannot guarantee that r ≤ 2σ + 1 in (4.38) and hence
‖xj|ψ|2σ+1−r‖Lpr+1 will not, in general, be bounded.
The second technical result yields a bound on the linear Schro¨dinger equation in Σm.
Lemma 4.3.6. Let S(t) be given by (4.10) with U ∈ C∞(Rd;R) and subquadratic. Then,
there exists a constant c > 0 such that
‖S(t)ψ0‖Σm ≤ e
ct‖ψ0‖Σm ,
for all t ∈ [0,∞) and ψ0 ∈ Σm.
The proof can be found in Kitada [48, Theorem 6.3]. Next we use this technical lemma
to prove local existence of mild solutions to equation (4.1) which lie in Σm.
Lemma 4.3.7. Let λ ≥ 0, σ ∈ N with σ < 2/(d− 2), and U ∈ C∞(Rd) be subquadratic.
For m > d/2, let ψ0 ∈ Σm, and V ∈ Wm,∞(Rd). Then there exists a τ > 0 and a unique
169
Optimal control of NLS
mild solution of (4.1) in ψ˜ ∈ L∞(0, τ ; Σm). If Tmax is the finite time of existence and
Tmax happens to be finite, it holds that
lim sup
t→Tmax
‖ψ(t)‖L∞ = +∞,
which is the blow-up alternative for solutions in L∞(0, τ ; Σm).
Proof. The proof is similar to the local existence proof of Lemma 3.3.2 and we only sketch
it here. The proof in Hm(Rd) in the absence of a subquadratic potential can be found
in [19, Theorem 4.10.1]. The idea is to find a mild solution as a fixed point of the map
Φ : Xτ,R → Xτ,R given by
Φ(ψ)(t) = S(t)ψ0 − i
∫ t
0
S(t− s)
(
λ|ψ(s)|2σψ(s) + α(s)V ψ(s)
)
ds,
in a suitable space Xτ,R. Here we let
Xτ,R :=
{
ψ ∈ L∞(0, τ ; Σm) : ‖ψ‖L∞(0,τ ;Σm) ≤ 2R
}
,
and set R := ‖ψ0‖Σm . Lemma 4.3.4 and 4.3.6 immediately yield
(4.39) ‖Φ(ψ)‖L∞(0,τ ;Σm) ≤ e
cτ‖ψ0‖Σm
+ C
∫ τ
0
eτ−s‖ψ‖2σL∞(0,τ ;L∞)‖ψ(s)‖L∞(0,τ ;Σm) ds ≤ 2R,
for all ψ ∈ Xτ,R, if τ > 0 is small enough. Furthermore, choosing τ possibly even smaller,
it holds that
‖Φ(ψ)− Φ(ψ˜)‖L∞(0,τ ;Σm)
≤ Cτ
(
‖ψ‖2σL∞(0,τ ;Σm) + ‖ψ˜‖
2σ
L∞(0,τ ;Σm)
)
‖ψ − ψ˜‖L∞(0,τ ;Σm) < 1.
Thus Φ : Xτ,R → Xτ,R is indeed a contraction mapping and has a unique fixed point.
This local solution can be extended as long as ‖ψ(t)‖Σm stays bounded. The blow-up
alternative follows from the fact that estimate (4.39) and Gronwall’s inequality yield a
bound on ‖ψ(t)‖Σm as long as ‖ψ(t)‖L∞ remains bounded.
Equipped with a local existence theory we now prove that the local solution ψ˜ ∈ L∞(0, τ ; Σm)
coincides with the solution ψ ∈ C([0, T ]; Σ), and hence the regularity in Σm propagates
for all times t ∈ [0, T ].
Proof of Lemma 4.3.3. The proof now closely follows the proof of [19, Theorem 5.5.1].
For the sake of the reader’s convenience, we state the proof in a self-contained form.
170
4.3. Derivation and analysis of the adjoint equation
The Σm–solution is global on [0, T ] and indeed coincides with the Σ–solution if we can
show that ψ ∈ L∞(0, T ; Σm). Let us first assume that ψ ∈ L2σ(0, T ;L∞(Rd)). Then
Lemma 4.3.4, Lemma 4.3.6, and the mild form of the NLS, cf. 4.18, yield
‖ψ(t)‖Σm ≤ C‖ψ0‖Σm + C
∫ t
0
‖ψ(s)‖2σL∞‖ψ(s)‖Σm ds
for all t ∈ [0, T ]. Thus Gronwall’s lemma yields
‖ψ(t)‖Σm ≤ C exp
(
C
∫ t
0
‖ψ(s)‖2σL∞ ds
)
,
which is bounded since ψ ∈ L2σ(0, T ;L∞(Rd)). Therefore the solution indeed exists in
L∞(0, T ; Σm).
It remains to prove ψ ∈ L2σ(0, T ;L∞(Rd)). If d = 1, this follows directly from the
embedding H1(R) ↪→ L∞(R), since ψ ∈ L∞(0, T ;H1(Rd)). If d ≥ 2, the local existence
theory in Σ, which we will not repeat here (instead we refer to the proof of Lemma 3.3.2),
yields that
ψ, xψ,∇ψ ∈ L
4σ+4
dσ (0, T ;L2σ+2(Rd)).
Now the same calculations, using Strichartz estimates, that yielded the existence of a
solution via a fixed point argument, yield
ψ, xψ,∇ψ ∈ Lp(0, T ;Lq(Rd))
for any admissible pair (p, q), cf. Lemma 3.3.3. Finally we choose some d < q < 2dσ/(dσ−
2) and let p such that (p, q) is an admissible pair. This is indeed possible since σ < 2/(d−2)
(σ < +∞, if d = 2), whence d < 2dσ/(dσ − 2). It follows p > 2σ and the embedding
W 1,r(Rd) ↪→ L∞(Rd) yields the desired property ψ inL2σ(0, T ;L∞(Rd)).
In the next subsection, we shall set up an existence theory for (4.35), which in turn will
be used to rigorously justify the above derivation in Section 4.4 below.
4.3.3 Existence of solutions to the adjoint equation
Having obtained ψ ∈ L∞(0, T ; Σm), we infer ψ ∈ L∞((0, T ) × Rd) by the Sobolev em-
bedding Hm(Rd) ↪→ L∞(Rd) whenever m > d/2. Thus, all the ψ–dependent coefficients
appearing in adjoint equation (4.35) are indeed in L∞.
Remark 4.3.8. Note that Lemma 4.3.3 requires us to impose σ ∈ N, which together
with the condition σ < 2/(d− 2) necessarily implies d ≤ 3. The reason is that for general
σ > 0 (not necessarily an integer) the nonlinearity |ψ|2σψ is not locally Lipschitz in Σm
171
Optimal control of NLS
(cf. Lemma 4.3.4) and the life-span of solution ψ(t, ·) ∈ Σm is in general not known, see
[19] for more details.
From now on, we shall always assume that V ∈ Wm,∞(Rd) for m > d/2 and U ∈ C∞(Rd)
subquadratic. With the above regularity result at hand, classical semigroup theory [71]
allows us to construct a solution to the adjoint problem.
Proposition 4.3.9. Let λ ≥ 0, σ ∈ N with σ < 2/(d − 2), and U ∈ C∞(Rd) be sub-
quadratic. For m > d/2, let ψ0 ∈ Σm, V ∈ Wm,∞(Rd). Then, (4.35) admits a unique
mild solution
ϕ ∈ C([0, T ];L2(Rd)).
Proof. First, we study the homogenous equation ∂ψP (ψ(α), α)ξ = 0, associated to (4.35).
It can be written as
∂tξ = −iHξ +B(t)ξ,
where
B(t)ξ := −i
(
λ(σ + 1)|ψ|2σξ + λσ|ψ|2σ−2ψ2ξ + α(t)V (x)ξ
)
.
The operator −iH : Σ2 → L2(Rd) is simply the generator of the Schro¨dinger group
S(t) = e−iHt. On the other hand, for any t ∈ [0, T ], B(t) is a linear operator on the
real vector space L2(Rd), equipped with the inner product (4.12) (the same would not be
true if we would consider L2(Rd) as a complex vector space). In addition, B(t)∗ = B(t)
is symmetric with respect to this inner product and the same is true for iB(t). Since
V ∈ Wm,∞(Rd), α ∈ L∞(0, T ) by assumption and ψ ∈ L∞((0, T )×Rd) in view of Lemma
4.3.3, we infer B ∈ L∞(0, T ;L(L2(Rd))). The operator B(t) may therefore be considered as
a (time-dependent) perturbation of the generator −iH. Following the construction given
in Proposition 1.2, Chapter 3 of [71], we obtain the existence of a propagator F (t, s), i.e.
a family of bounded operators
{F (t, s) : L2(Rd)→ L2(Rd)}s,t∈[0,T ]
which are strongly continuous in time and satisfy F (t, s) = F (t, r)F (r, s). This propagator
F (t, s) is implicitly given by
(4.40) F (t, s) = e−iH(t−s) +
∫ t
s
e−iH(t−τ)B(τ)F (τ, s) dτ.
It solves the homogeneous linearized equation in the sense that
(4.41)
d
dt
F (t, s)ξ = (−iH +B(t))F (t, s)ξ
172
4.3. Derivation and analysis of the adjoint equation
weakly in (Σ2)∗ for every ξ ∈ L2(Rd) and almost every t ∈ [0, T ]. Clearly, it provides
a unique mild solution ξ(t) = F (t, s)ϕ(s) of the homogenous equation. For the reader’s
convenience, we give here a detailed construction of F (t, s). We iteratively set
(4.42) F0(t, s) := e
−iH(t−s), Fn+1(t, s) :=
∫ t
s
e−iH(t−τ)B(τ)Fn(τ, s) dτ
for all s, t ∈ [0, T ]. As a first step, we show that
‖Fn(s, t)‖L(L2) ≤
‖B‖nL∞(0,T ;L(L2))(t− s)
n
n!
.
This is obviously true for n = 0. Assuming it is true for n ∈ N, it follows from the
definition of Fn+1 that
‖Fn+1(s, t)‖L(L2) ≤
‖B‖n+1L∞(0,T ;L(L2))
n!
∫ t
s
(τ − s)ndτ =
‖B‖n+1L∞(0,T ;L(L2))(t− s)
n+1
(n+ 1)!
.
Hence the series
(4.43) F (t, s) :=
∞∑
n=0
Fn(t, s)
converges absolutely in the operator topology of L(L2(Rd)) with uniform convergence
with respect to s and t on the (bounded) interval [0, T ]. Hence the definitions (4.43) and
(4.42) of F and Fn respectively, yield
F (t, s) =
∞∑
n=0
Fn(t, s) = e
−iH(t−s) +
∞∑
n=0
Fn+1(t, s)
= e−iH(t−s) +
∫ t
s
e−iH(t−τ)B(τ)
∞∑
n=0
Fn(τ, s) dτ,
due to the uniform in time convergence. Substituting the definition (4.43) of F , we obtain
the desired expression (4.40). Differentiation of equation (4.40) indeed yields (4.41).
Duhamel’s formula applied to the adjoint problem (4.35) consequently yields
(4.44) ϕ(t) = iF (t, T )
δJ(ψ, α)
δψ(T )
+ i
∫ T
t
F (t, s)
δJ(ψ, α)
δψ(s)
ds.
Under our assumptions on ψ and A we have that
δJ(ψ, α)
δψ(t)
∈ L1(0, T ;L2(Rd)),
δJ(ψ, α)
δψ(T )
∈ L2(Rd),
which in view of Duhamel’s formula (4.44) implies the existence of a mild solution
173
Optimal control of NLS
ϕ ∈ C([0, T ];L2(Rd)). Uniqueness follows from linearity and the uniqueness of the homo-
geneous equation.
4.4 Rigorous characterization of critical points
A classical approach for making the derivation of the adjoint system rigorous is based on
the implicit function theorem. The latter is used to show that ∂ψP (ψ(α), α) is indeed
invertible, but it requires the identification of a linear function space X such that
P : Υ(0, T )×H1(0, T )→ X ; (ψ, α) 7→ P (ψ, α),
and
∂ψP (ψ, α)
−1 : X → Υ(0, T ).
In other words, we require the solution of (4.35) with a right hand side in X to be in
Υ(0, T ). It seems, however, that the linearized operator ∂ψP (ψ, α)−1 is not sufficiently
regularizing to allow for an easy identification of X. Therefore we shall not invoke the
implicit function theorem but rather calculate the Gaˆteaux-derivative J′(α) directly. (We
do not prove Fre´chet-differentiability; see Remark 4.4.3 below.) To this end, we shall first
show that the solution ψ = ψ(α) to (4.1) depends Lipschitz-continuously on the control
parameter α. This will henceforth be used to estimate the error terms appearing in the
derivative of J(α).
4.4.1 Lipschitz continuity with respect to the control
As a first step towards full Lipschitz continuity, we prove local-in-time Lipschitz continuity
of ψ = ψ(α) with respect to the control parameter α.
Proposition 4.4.1. Let λ ≥ 0, σ ∈ N with σ < 2/(d − 2), and U ∈ C∞(Rd) be sub-
quadratic. For m > d/2, let V ∈ Wm,∞(Rd) and ψ˜, ψ ∈ L∞(0, T ; Σm) be two mild
solutions to (4.1), corresponding to initial data ψ˜0, ψ0 ∈ Σm and control parameters
α˜, α ∈ H1(0, T ), respectively. Assume that
‖α˜‖H1t , ‖α‖H1t , ‖ψ˜(t, ·)‖Σm , ‖ψ(t, ·)‖Σm ≤M
for some given M ≥ 0. Then there exist τ = τ(M) > 0 and a constant C = C(M) < +∞,
such that
(4.45) ‖ψ˜ − ψ‖L∞(It;Σm) ≤ C
(
‖ψ˜(t)− ψ(t)‖Σm + ‖α˜− α‖H1t
)
,
174
4.4. Rigorous characterization of critical points
where It := [t, t+τ ]∩ [0, T ]. In particular, the mapping α 7→ ψ(α) ∈ Υ(0, T ) is continuous
with respect to α ∈ H1(0, T ).
Proof. To simplify notation, let us assume t + τ ≤ T . By construction, there exists a
τ > 0 depending only on M , such that ψ|It is a fixed point of the mapping
ψ 7→ S( · )ψ0 − i
∫ ·
t
S(· − s)
(
λ|ψ(s)|2σψ(s) + α(s)V ψ(s)
)
ds,
which maps the set
Y = {ψ ∈ L∞(It; Σ
m) : ‖ψ‖L∞(It;Σm) ≤ 2M}
into itself. Of course, the same holds true for ψ˜ and α˜ in place of ψ and α, respectively.
In particular, the embedding Σm(Rd) ↪→ L∞(Rd), m > d/2, yields
‖ψ‖L∞(It×Rd) ≤ 2CM.
Subtracting the two fixed point expressions for ψ˜ and ψ gives
ψ˜(s)− ψ(s) = S(s− t)(ψ˜(t)− ψ(t))
− i
∫ s−t
0
S(s− r)
(
λ(|ψ˜|2σψ˜ − |ψ|2σψ) + V (x)(α˜ ψ˜ − αψ)
)
(τ) dτ
for all s ∈ [t, t + τ ]. Taking the L∞(It; Σm)-norm and recalling Lemma 4.3.6, together
with ‖ψ(s)‖Σm , ‖ψ˜(s)‖Σm ≤ 2M , for s ≤ t+ τ , yields
‖ψ˜ − ψ‖L∞(It;Σm) ≤ C‖ψ˜(t)− ψ(t)‖Σm + 2M‖α˜− α‖H1t ‖V ‖Wm,∞x
+ Cτ
(
C(2M) + ‖α˜‖H1t ‖V ‖Wm,∞x
)
‖ψ˜ − ψ‖L∞(t,t+τ ;Σm),
where C(2M) is the constant appearing in Lemma 4.3.4 with 2M replacing M . Since
‖α˜‖H1t ≤M , the estimate (4.45) follows from possibly choosing τ even smaller.
Finally, we show the continuity of the map H1(0, T )→ Υ(0, T ), α 7→ ψ(α). Set
t∗ := inf
{
0 ≤ t ≤ T : lim sup
α˜→α
‖ψ˜ − ψ‖L∞(0,t;Σm) > 0
}
,
with the convention inf ∅ := +∞. We have to show that t∗ = +∞. Assuming t∗ ≤ T <
+∞, fix M ′ ≥ M such that ‖ψ‖L∞(0,T ;Σm) ≤ M ′, let τ ′ = τ(M ′ + 1) > 0 be chosen as
above, with M ′+1 replacing M . Furthermore let ∆t = τ ′/2. The definition of of t∗ yields
lim sup
α˜→α
‖ψ˜ − ψ‖L∞(0,t∗−∆t;Σm) = 0.
175
Optimal control of NLS
In particular, it holds that ‖ψ˜‖L∞(0,t∗−∆t;Σm) ≤M
′ + 1 for all (α˜− α) small enough. But
now we see that the Lipschitz continuity (4.45) is satisfied by ψ˜ and ψ and such controls
α˜, α on the interval [t∗ −∆t, t∗ −∆t+ τ ′]. Hence
lim sup
α˜→α
‖ψ˜ − ψ‖L∞(0,t∗−∆t+τ ′;Σm) = 0,
a contradiction to the definition of t∗. Hence we must have t∗ = ∞, and continuity
holds.
As a direct consequence of this continuity result, we obtain uniform boundedness of the
solution ψ(α) on compact sets in α ∈ H1(0, T ). Of course, bounded sets in H1(0, T ) are in
general not compact and thus we have to restrict ourselves to finite-dimensional subsets.
Corollary 4.4.2. Under the assumptions of Proposition 4.4.1, let δα ∈ H1(0, T ) with
δα(0) = 0 be a direction of change for α and let ψ(α + εδα) be the solution to (4.1) with
control α + εδα and initial data ψ0 ∈ Σm, m > d/2. Then there exists M <∞ such that
‖ψ(α + εδα)‖L∞(0,T ;Σm) ≤M, ∀ ε ∈ [−1, 1].
Remark 4.4.3. This bound on finite dimensional subsets of H1(0, T ) is the reason why
we can only prove Gaˆteaux-differentiability. If we had a bound on ψ(α) in the Σm–norm
which was uniform in t ≤ T and ‖α‖H1t ≤ M , we could prove Fre´chet-differentiability.
For our further analysis, however, this will not be of any consequence.
Now we are ready to prove Lipschitz-continuity of the solution ψ(α) with respect to the
control parameter α ∈ H1(0, T ) on the whole control interval [0, T ].
Proposition 4.4.4. Let λ ≥ 0, σ ∈ N with σ < 2/(d − 2), and U ∈ C∞(Rd) be sub-
quadratic. For m > d/2, let V ∈ Wm,∞(Rd), ψ0 ∈ Σm, α ∈ H1(0, T ), and ψ ≡ ψ(α) ∈
L∞(0, T ; Σm) be the solution to (4.1). Set ψ˜ ≡ ψ(α˜) where for any ε ∈ [−1, 1], we let
α˜ := α+ εδα with δα ∈ H1(0, T ) such that δα(0) = 0. Then there exists a constant C > 0,
such that
(4.46) ‖ψ˜ − ψ‖L∞(0,T ;Σm) ≤ C‖α˜− α‖H1(0,T ) = C|ε|‖δα‖H1(0,T ).
In other words, the solution to (4.1) depends Lipschitz-continuously on the control α for
each fixed direction δα.
Proof. Since Corollary 4.4.2 provides a uniform (in ε) bound on ‖ψ˜‖L∞(0,T ;Σm), the quan-
tity τ in the local Lipschitz estimate (4.45) is now independent of  and t and the estimate
indeed holds on every interval [t, t+ τ ], i.e.
‖ψ˜ − ψ‖L∞(t,t+τ ;Σm) ≤ C
(
‖ψ˜(t)− ψ(t)‖Σm + ‖α˜− α‖H1(t,t+τ)
)
.
176
4.4. Rigorous characterization of critical points
Since both solutions ψ˜ and ψ coincide at t = 0, finite summation of this estimate over
intervals [nτ, (n+ 1)τ ] yields (4.46).
4.4.2 Proof of differentiability and characterization of critical
points
We are now in a position to state the second main result of this work.
Theorem 4.4.5. Let λ ≥ 0, σ ∈ N with σ < 2/(d−2), and U ∈ C∞(Rd) be subquadratic.
In addition, let ψ0 ∈ Σm, V ∈ Wm,∞(Rd) for some m ∈ N, m ≥ 2, and α ∈ H1(0, T ).
Then the solution of (4.1) satisfies ψ ∈ L∞(0, T ; Σm) and the functional J(α) is Gaˆteaux-
differentiable for all t ∈ [0, T ], with
(4.47) J′(α) = Re
∫
Rd
ϕ(t, x)V (x)ψ(t, x) dx− 2
d
dt
(
α˙(t)
(
γ2 + γ1ω
2(t)
))
,
in the sense of distributions, where ω(t) is the weight factor defined in (4.9) and ϕ ∈
C([0, T ];L2(Rd)) is the solution of the adjoint equation
(4.48)
i∂tϕ = −
1
2
∆ϕ+ U(x)ϕ+ α(t)V (x)ϕ+ λ(σ + 1)|ψ|2σϕ+ λσ|ψ|2σ−2ψ2ϕ
+ 4γ1(α˙(t))
2 ω(t)V (x)ψ,
subject to Cauchy data ϕ(T, x) = 4i〈ψ(T, ·), Aψ(T, ·)〉L2x Aψ(T, x).
Remark 4.4.6. When compared to the assumptions of Theorem 4.2.1, the result of
Theorem 4.4.5 requires additional regularity (and stronger decay) of the initial data ψ0 and
the potential V (plus, we need to restrict ourselves to σ ∈ N). Note that the requirement
m ∈ N and m ≥ 2 implies m > d/2 for d = 1, 2, 3 spatial dimensions.
Proof. We need to prove that J′(α) is of the form (4.4.2). For this purpose, let ψ = ψ(α),
ψ˜ = ψ(α˜) with α˜ = α + εδα, satisfy the assumptions of Lemma 4.4.4 and consider the
difference of the corresponding objective functionals J(α), J(α˜). This difference can be
written as the sum of three terms
J(α˜)− J(α) = I + II + III,
where we define
I := 〈ψ˜(T ), Aψ˜(T )〉2L2x − 〈ψ(T ), Aψ(T )〉
2
L2x
, II := γ2
∫ T
0
( ˙˜α(t))2 − (α˙(t))2 dt,
177
Optimal control of NLS
and
III := γ1
∫ T
0
( ˙˜α(t))2
(∫
Rd
V (x)|ψ˜(t, x)|2 dx
)2
dt
− γ1
∫ T
0
(α˙(t))2
(∫
Rd
V (x)|ψ(t, x)|2 dx
)2
dt.
The general strategy will be to use the Lipschitz property established in Lemma 4.4.4 and
rewrite the terms I, II, and III in such a way that
J(α˜)− J(α) = linear terms in (α˜− α) + O(‖α˜− α‖2H1t ).
Since α˜ = α+εδα and thus O(‖α˜−α‖2H1t ) = O(ε
2), the limit ε→ 0 then yields the desired
functional derivative.
We start by considering the term I. It can be rewritten in the form
I = 〈ψ˜(T ), Aψ˜(T )〉2L2x − 〈ψ(T ), Aψ(T )〉
2
L2x
= 2〈ψ(T ), Aψ(T )〉L2x
(
〈ψ˜(T ), Aψ˜(T )〉L2x − 〈ψ(T ), Aψ(T )〉L2x
)
+
(
〈ψ˜(T ), Aψ˜(T )〉L2x − 〈ψ(T ), Aψ(T )〉L2x
)2
.
Using the essential self-adjointness of A, the terms within the parentheses yield
〈ψ˜(T ), Aψ˜(T )〉L2x − 〈ψ(T ), Aψ(T )〉L2x
= 2〈ψ˜(T )− ψ(T ), Aψ(T )〉L2x + 〈ψ˜(T )− ψ(T ), A(ψ˜(T )− ψ(T ))〉L2x .
Using the Lipschitz-estimate (4.46), we obtain
∣
∣
∣〈ψ˜(T )− ψ(T ), A(ψ˜(T )− ψ(T ))〉L2x
∣
∣
∣ ≤ ‖A‖L(Σ,L2x)‖ψ˜(T )− ψ(T )‖
2
Σ ≤ Cε
2‖δα‖
2
H1t
,
and hence
〈ψ˜(T ), Aψ˜(T )〉L2x − 〈ψ(T ), Aψ(T )〉L2x = 2〈ψ˜(T )− ψ(T ), Aψ(T )〉L2x + O(‖α˜− α‖
2
H1t
).
Squaring the above result and plugging it into our expression for I consequently yields
(4.49) I = 4〈ψ(T ), Aψ(T )〉L2x 〈ψ˜(T )− ψ(T ), Aψ(T )〉L2x + O(‖α˜− α‖
2
H1t
).
178
4.4. Rigorous characterization of critical points
Next we consider II, which can be written as
II = 2γ2
∫ T
0
α˙(t)
(
˙˜α(t)− α˙(t)
)
dt+ γ2
∫ T
0
(
˙˜α(t)− α˙(t)
)2
dt
= 2γ2
∫ T
0
α˙(t)
(
˙˜α(t)− α˙(t)
)
dt+ O(‖α˜− α‖2H1t ).
The first term in the second line is thereby seen to be of the form given in (4.4.2). Finally
we consider III, which in view of definition (4.9) can be written as
III = γ1
∫ T
0
(
( ˙˜α(t))2 − (α˙(t))2
)
ω2(t) dt
+ γ1
∫ T
0
( ˙˜α(t))2
((∫
Rd
V (x)|ψ˜(t, x)|2dx
)2
− ω2(t)
)
dt.
As before, we can expand these terms using quadratic expansions in both ψ˜ and α˜. In
view of the Lipschitz estimate (4.46), any quadratic error ‖ψ˜ − ψ‖2L∞t L2x is bounded by
O(‖α˜− α‖2H1t ) and hence we obtain
III = 4γ1
∫ T
0
(α˙(t))2 ω(t)
(
Re
∫
Rd
(
(ψ˜ − ψ)V ψ
)
(t, x) dx
)
dt
+ 2γ1
∫ T
0
(
˙˜α(t)− α˙(t)
)
α˙(t)ω2(t) dt+ O(‖α˜− α‖2H1t ).
(4.50)
Here the second term on the right hand side is linear in (α˜− α) and hence of the desired
form. In order to treat the first term, we note that the expression
4γ1
(
(α˙(t))2
∫
Rd
V (x)|ψ(t, x)|2 dx
)
V (x)ψ(t, x)
appears as a source term in the adjoint equation (4.48). Thus we obtain
4γ1
∫ T
0
(α˙(t))2 ω(t)
(
Re
∫
Rd
(
(ψ˜ − ψ)V ψ
)
(t, x) dx
)
dt
= Re
∫ T
0
∫
Rd
ϕ(t, x)
(
∂ψP (ψ, α)(ψ˜ − ψ)
)
(t, x) dx
− Re
∫
Rd
i ϕ(T, x)
(
ψ˜(T, x)− ψ(T, x)
)
dx,
(4.51)
where we recall that ∂ψP (ψ, α) denotes the linearized Schro¨dinger operator obtained in
(4.31). The last term on the right hand side of (4.51) stems from the boundary condition
at t = T . Note that the boundary term at t = 0 vanishes since ψ˜(0) = ψ0 = ψ(0) by
179
Optimal control of NLS
assumption. We recall that ϕ ∈ C([0, T ];L2(Rd)) and
ψ˜, ψ ∈ L∞(0, T ; Σm) ∩W 1,∞(0, T ; Σm−2), with m ≥ 2,
and hence the right hand side of (4.51) is well-defined. In addition, since both ψ˜ and ψ
solve the nonlinear Schro¨dinger equation (4.1), we can write
∂ψP (ψ, α)(ψ˜ − ψ) = i∂t(ψ˜ − ψ)−H(ψ˜ − ψ)− V (x)(α˜(t)ψ˜ − α(t)ψ)
+ λ|ψ|2σψ − λ|ψ˜|2σψ˜ + (α˜(t)− α(t))V (x)ψ˜ + %(ψ˜, ψ)
= (α˜(t)− α(t))V (x)ψ˜ + %(ψ˜, ψ),
(4.52)
where the remainder %(ψ˜, ψ) is given by
1
λ
%(ψ˜, ψ) = |ψ˜|2σψ˜ − |ψ|2σψ − (σ + 1)|ψ|2σ(ψ˜ − ψ)− σ|ψ|2σ−2ψ2
(
ψ˜ − ψ
)
.
Since 2σ ≥ 2 by assumption, ‖ψ˜‖L∞t L∞x , ‖ψ‖L∞t L∞x ≤ C in view of Corollary 4.4.2, and
Σm ↪→ L∞(Rd), the remainder can be bounded by
|%(ψ˜, ψ)| ≤ C
(
|ψ˜|2σ−1 + |ψ|2σ−1
)
|ψ˜ − ψ|2 ≤ C|ψ˜ − ψ|2.
In addition, since ϕ ∈ C([0, T ];L2(Rd)) and Σm ⊂ Hm(Rd) ↪→ L4(Rd), we find that
∫
Rd
|ϕ(t, x)||ψ˜(t, x)− ψ(t, x)|2 dx ≤ ‖ϕ‖L∞t L2x‖ψ˜ − ψ‖
2
L∞t L
4
x
= O(‖α˜− α‖2H1t ).
Furthermore, the contribution of (α˜(t)− α(t))V (x)ψ˜ in (4.52) equals
(α˜(t)− α(t))V (x)ψ + (α˜(t)− α(t))V (x)(ψ˜ − ψ),
where the latter term can be estimated by O(‖α˜ − α‖2H1t ) as before. In summary, this
shows that
(4.51) =
∫ T
0
(α˜(t)− α(t))Re
∫
Rd
ϕ(t, x)V (x)ψ(t, x)dxdt+O(‖α˜− α‖2H1t )
− 4〈ψ(T ), Aψ(T )〉L2x 〈ψ˜(T )− ψ(T ), Aψ(T )〉L2x ,
(4.53)
where we have used the fact that the data of the adjoint problem at t = T is given by
ϕ(T, x) = 4i〈ψ(T, ·), Aψ(T, ·)〉L2x Aψ(T, x).
Thus, we infer that, up to quadratic errors, the second line in (4.53) cancels with the
terms obtained in (4.49). Collecting all the expressions obtained for I, II, III and taking
180
4.4. Rigorous characterization of critical points
the limit ε→ 0, we have shown that J(α) is Gaˆteaux-differentiable with derivative J′(α)
given by (4.4.2). This concludes proof of Theorem 4.4.5.
Equation (4.47) yields the following characterization of the critical points α∗ ∈ H1(0, T ),
i.e. points where J′(α∗) = 0.
Corollary 4.4.7. Let ψ∗ be the solution of (4.1) with control α∗. Also, let ϕ∗ be the
corresponding solution of the adjoint equation (4.48), and denote by ω∗ the function defined
in (4.9) with ψ replaced by ψ∗. Then α∗ ∈ C2(0, T ) is a classical solution of the following
ordinary differential equation
(4.54)
d
dt
(
α˙∗(t)
(
γ2 + γ1ω
2
∗(t)
))
=
1
2
Re
∫
Rd
ϕ∗(t, x)V (x)ψ∗(t, x) dx,
subject to α∗(0) = α0, α˙∗(T ) = 0.
Remark 4.4.8. In the case γ1 = 0 this simplifies to the expression used in the physics
literature; cf. [40].
Proof. Let µ ∈ C∞0 (0, T ) be a test function with compact support in (0, T ). Then,
Theorem 4.2.1 and Theorem 4.4.5 imply that there exists α∗ ∈ H1(0, T ) such that J′(α∗) =
0, satisfying (4.47) in the sense of distributions, i.e.
∫ T
0
α˙∗(t)µ˙(t)
(
γ2 + γ1ω
2
∗(t)
)
dt =
1
2
Re
∫ T
0
∫
Rd
µ(t)ϕ∗(t, x)V (x)ψ∗(t, x)dxdt,
where we have used the fact that the boundary terms at t = 0 and t = T vanish due to
the compact support of µ(t). We shall show that the weak solution α∗ is in fact unique.
This can be seen by considering two different α1∗(t), α
2
∗(t), satisfying α
1
∗(0) = α
2
∗(0) = α0.
Denoting their difference by β∗ = α1∗ − α
2
∗, we have that β∗(t) solves
∫ T
0
β˙∗(t)µ˙(t)
(
γ2 + γ1ω
2
∗(t)
)
dt = 0, for all µ ∈ C∞0 (0, T ).
Since γ2 > 0 and γ1 ≥ 0, this implies that β˙∗(t) = 0 in the sense of distributions. However,
since α1∗, α
2
∗ ∈ H
1(0, T ) ↪→ C(0, T ), we conclude that β∗ ∈ C(0, T ) and thus β∗(t) = const
for all t ∈ [0, T ]. Since β∗(0) = 0 by assumption, we infer uniqueness of the weak solution
α∗(t). On the other hand, standard arguments imply that (4.54) admits a unique classical
solution α∗ ∈ C2(0, T ), provided ω∗ ∈ C1(0, T ) and the (source term on the) right hand
side is continuous in time. The latter is obviously true in view of Proposition 4.2.4 and
Proposition 4.3.9. In addition, since V ∈ W 1,∞(Rd), we infer that for all ψ(t) ∈ Σ it holds
that χ(t) := (V (x)ψ(t)) ∈ Σ. From Proposition 4.2.4 it follows that
ω˙∗(t) = 2Re
∫
Rd
V (x)∂tψ∗(t, x)ψ∗(t, x) dx = 2〈χ(t), ψ˙∗(t)〉Σ,Σ∗ < +∞.
181
Optimal control of NLS
Thus, ω(t) ∈ C1(0, T ), yielding the existence of a unique classical solution α∗ ∈ C2(0, T ).
We therefore conclude that the unique weak solution α∗ obtained above is in fact a classical
solution, satisfying (4.54) subject to α∗(0) = α0, α˙∗(T ) = 0.
We call α∗ ∈ H1(0, T ) a critical or stationary point of the problem
(4.55) minimize J(α) over α ∈ H1(0, T ),
if J′(α∗) = 0, where J′ is given in Theorem 4.4.5. In order the check computationally
whether α∗ is critical, one needs to solve (4.1) for α = α∗ to obtain ψ∗ and then the adjoint
equation (4.48) with ψ = ψ∗ and α = α∗ to compute ϕ∗. Inserting (α, ψ, ϕ) = (α∗, ψ∗, ϕ∗)
in (4.47) yields J′(α∗) which has to vanish for α∗ to be critical, i.e., (4.54) is satisfied. We
therefore call (4.1), (4.48) and (4.54) the first order optimality conditions associated with
(4.55).
4.5 Numerical simulation of the optimal control prob-
lem
For our numerical treatment we simplify to the case d = σ = 1. In this case, the first
order optimality conditions for our optimal control problem are given by:



d
dt
(
α˙(t)
(
γ2 + γ1ω
2(t)
))
=
1
2
Re
∫
Rd
ϕ(t, x)V (x)ψ(t, x) dx,
i∂tψ +
1
2
∂2xψ = (U(x) + α(t)V (x))ψ + λ|ψ|
2ψ,
i∂tϕ+
1
2
∂2xϕ = (U(x) + α(t)V (x))ϕ+ 2λ|ψ|
2ϕ+ λψ2ϕ+ 4γ1(α˙)
2 ω(t)V (x)ψ,
subject to the following conditions: α(0) = α0, α˙(T ) = 0, and
ψ(0, x) = ψ0(x), ϕ(T, x) = 4i〈ψ(T, ·), Aψ(T, ·)〉L2xAψ(T, x).
In our numerical simulations, we solve the minimization problem (4.55) iteratively, con-
structing a minimizing sequence (αk)k ⊂ H1(0, T ). We determine a sequence of descent
directions (δkα)k ⊂ H
1(0, T ), i.e., for every k ∈ N
(4.56) J(αk + δkα) < J(αk) = J(ψ(αk), αk)
is satisfied, where the iteration is given by αk+1 = αk + δkα. If αk is such that there is
no descent direction, αk is indeed a (local) minimum of J and ψ(αk) is the corresponding
(locally) optimal solution to the NLS (4.1).
182
4.5. Numerical simulation of the optimal control problem
We solve the resulting Cauchy problems for Schro¨dinger–type equations by a time-splitting
spectral method of second order (Strang-splitting), as can be found in Section 3.5 or, for
instance, in [7]. This computational approach is unconditionally stable and allows for
spectral accuracy in the resolution of the wave function ψ(t, x). This is needed due
to the highly oscillatory nature of solutions to (nonlinear) Schro¨dinger–type equations.
We consequently perform our simulations on a numerical domain Ω ⊂ R, equipped with
periodic boundary conditions. The trapping potential U(x) is thereby chosen such that the
“effective” (i.e. the numerically relevant) support of the wave function ψ(t, x) stays away
from the boundary. In doing so, the boundary conditions do not significantly influence
our results. A good test of the accuracy of our numerical code is given by the fact that the
Gross-Pitaevskii equation conserves the physical mass (i.e. the L2-norm of ψ(t)). Indeed,
in all our numerical examples presented in Section 4.5.3 below, we find that the L2-norm
is numerically preserved up to relative errors of the order 10−13.
4.5.1 Gradient-related descent method
Once a suitable solver for the state and the adjoint equations is at hand, our gradient-
related descent scheme operates as follows. Given αk ∈ H1(0, T ), determine δkα such that
the condition (4.56) is satisfied. A simple Taylor expansion of J around αk shows that
〈J′(αk), δkα〉 < 0
is sufficient for δkα to be a descent direction for J at αk. We are in particular interested in
gradient-related descent directions which satisfy
Mδkα = −J
′(αk), where M : H
1(0, T )→ H1(0, T )∗
is a suitably chosen positive definite operator.
A rather straightforward choice of M is given by M = ∂2αJ(ψ, αk). In this case δ
k
α is
obtained as the solution of the following ordinary differential equation (of second order):
J′(αk) = 2
d
dt
(
δ˙kα(t)
(
γ2 + γ1
(∫
R
V (x)|ψk(t, x)|
2dx
)2
))
,
with δkα(0) = 0 and δ˙kα(T ) = 0. Here ψk(t, x) denotes the solution of the Gross–Pitaevskii
equation with α(t) = αk(t). With this choice of a descent direction, we then perform a
line search in order to decide on the length of the step taken along δkα. In fact, we seek
for νk > 0 such that
(4.57) J(αk + νkδkα) ≤ J(αk) + µνk〈J
′(αk), δ
k
α〉
183
Optimal control of NLS
with some fixed µ ∈ (0, 1). Within each line search, we determine νk iteratively by a
backtracking strategy. Starting from ν(0)k > 0, we iteratively test the condition (4.57); if
it holds for ν(`)k , then we accept αk+1 := αk + ν
(`)
k δ
k
α, and if not, we choose ν
(`+1)
k < ν
(`)
k
and repeat. Thus, the whole procedure amounts to an Armijo line search method with
backtracking. Of course, more elaborate strategies based on interpolation or alternative
line search criteria are possible; see for instance [69] for more details.
We stop the gradient descent method whenever
(4.58) ‖J′(αk)‖H−1t ≤ TOL · ‖J
′(α1)‖H−1t
is satisfied for the first time. Here, TOL ∈ (0, 1) is a given stopping tolerance and α1 ∈
H1(0, T ) is the initial guess satisfying the boundary conditions α1(0) = α0 and α˙1(T ) = 0.
As a safeguard, also an upper bound on the number of iterations is implemented.
In our tests, we observe the usual behavior of steepest descent type algorithms, i.e., the
method exhibits rather fast progress towards a stationary point in early iterations, but
then suffers from scaling effects reducing the convergence speed. Therefore, often the
maximum number of iterations is reached. Thus, we connect the first-order, gradient
method to a Newton-type method which relies on second derivatives or approximations
thereof.
4.5.2 Newton method
The majority of iterations within our simulations are performed via a second order
method, Newton’s method, for which we use the full Hessian
M := D2αJ(αk) : H
1(0, T )×H1(0, T )→ R,
or a sufficiently close positive definite approximation thereof. Note that we can also
consider the Hessian as a map D2αJ : H
1(0, T ) → H1(0, T )∗. Recall that the gradient-
related method above simply uses M = ∂2αJ(ψ, αk).
We derive D2αJ formally form the Lagrangian formulation; see Remark 4.3.1. The La-
grangian is given by
L(ψ, α, ϕ) = J(ψ, α)− 〈ϕ, P (ψ, α)〉L2t,x ,
where ϕ is the solution to the adjoint equation (4.48) and P (ψ, α) is the Gross–Pitaevksii
operator written in abstract form. Proceeding formally, we find
〈(D2αJ)δα, δ˜α〉L2t = 〈(∂
2
ψL)δψ, δ˜ψ〉L2t,x + 〈(∂ψαL)δα, δ˜ψ〉L2t,x
+ 〈(∂αψL)δ˜α, δψ〉L2t + 〈(∂
2
αL)δα, δ˜α〉L2t ,
184
4.5. Numerical simulation of the optimal control problem
where δψ and δ˜ψ solve the linearized Gross–Pitaevksii equation with controls δα, δ˜α, re-
spectively. In view of the derivation given in Section 4.3.1 we have
δψ = ψ
′(α)δα = −∂ψP (ψ(α), α)
−1∂αP (ψ(α), α)δα,
and analogously for δ˜ψ. Hence we conclude that
(4.59)
〈(D2αJ)δα, δ˜α〉L2t = 〈(∂
2
ψJ)ψ
′(α)δα, ψ
′(α)δ˜α〉L2t,x + 〈(∂αψJ)δα, ψ
′(α)δ˜α〉L2t,x
+ 〈(∂ψαJ)ψ
′(α)δα, δ˜α〉L2t + 〈(∂
2
αJ)δα, δ˜α〉L2t − 〈ϕ,
(
D2αP (ψ, α)δα
)
δ˜α〉L2t,x ,
where
(D2αP (ψ, α)δα)δ˜α =
(
∂2ψP (ψ, α)(ψ
′(α)δα)
)
(ψ′(α)δ˜α)
+
(
∂αψP (ψ, α)δα
)
(ψ′(α)δ˜α) +
(
∂ψαP (ψ, α)(ψ
′(α)δα)
)
δ˜α,
since ∂2αP (ψ, α) = 0. All of the terms appearing on the right hand side of (4.59) can
be evaluated by replacing ψ′(α)δα by −∂ψP (ψ, α)−1∂αP (ψ, α)δα. Consequently for calcu-
lating the action of the Hessian this requires to solve several linearized Schro¨dinger-type
equations with different source terms and boundary data. For example, the term involving
(∂αψJ) can be evaluated by using
χ := ∂ψP (ψ, α)
−∗((∂αψJ)δα),
which solves the following Cauchy problem
i∂tχ+
1
2
∂2xχ = U(x)χ+ α(t)V (x)χ+ 2λ|ψ|
2χ+ λψ2χ+ 8γ1h(t, x)ψ,
where h(t, x) := ω(t)α˙(t)δ˙α(t)V (x) and
χ(T, x) =
δ2J(ψ, α)
δψ(T, x) δα(T )
= 0.
Bearing this in mind, we have to solve the following equation for δkα ∈ H
1(0, T ):
(4.60) Mδkα = D
2
αJδ
k
α = −J
′(αk) ∈ H
1(0, T )∗.
Hence, we need to invert D2αJ, which, in view of (4.59) is not directly possible. Rather we
resort to an iterative method, the preconditioned MINRES algorithm, see [70], with the
preconditioner (∂2αJ(ψ, α))
−1 : H1(0, T )∗ → H1(0, T ).
Remark 4.5.1. (1) Note that D2αJ maps H
1(0, T ) to H1(0, T )∗ and thus makes it neces-
sary to precondition the iterative MINRES algorithm with a map H1(0, T )→ H1(0, T )∗.
185
Optimal control of NLS
The choice ∂2αJ(ψ(αk), αk) is easy to implement and can be expected to contain some of
the features of the full Hessian D2αJ, thus making the inversion problem for the MINRES
algorithm better-conditioned.
(2) We choose here the MINRES algorithm over alternatives like the conjugated gradient
(CG) method, because the Hessian D2αJ is symmetric but not necessarily positive defi-
nite. In this setting the MINRES algorithm is superior to the CG-method. Note that
the partial derivative ∂2αJ(ψ(αk), αk) is indeed positive definite and therefore is a valid
preconditioner.
(3) The map ∂2αJ(ψ(αk), αk) can also be understood as inducing a scalar product in the
dual space H1(0, T )∗. The correspondence between the scalar product in the dual space
and the preconditioner of the MINRES algorithm has been investigated in [35].
The description of the MINRES-algorithm is given in Algorithm 1. Setting A := D2αJ(αk),
R := ∂2αJ(ψ(αk), αk)
−1, and r0 := −J′(αk), it seeks at the `-th step to minimize the
residual r` of Aδα` + J′(αk) with respect to the norm 〈r`,Rr`〉 over all δα ∈ r0 +
span{Ar0, . . . ,A`r0} using the result of the previous step. Here and in the description
of Algorithm 1 the brackets 〈·, ·〉 denote the dual product between H1(0, T ) and H1(0, T )∗.
We emphasize that here we aim to study the behavior of solutions of our control problem
rather than at optimizing the respective solution algorithm or its implementation.
4.5.3 Numerical examples
In all our examples, we choose the numerical domain Ω = [−L,L] with L = 20 and periodic
boundary conditions. The number of spatial grid points is N = 256. In addition, we set
the final control time to be T = 10, and we use M = 1024 equidistant time steps. In order
to avoid the influence of the boundary, we choose a trapping potential U(x) = 30
(
x
L
)2
.
The initial guess for the control is taken to be just α1 ≡ 0 in the linear case (λ = 0),
whereas each algorithm in the nonlinear case (λ 6= 0) is started from the control obtained
by solving the linear problem. In our tests of the first-order gradient method, we choose
TOL = 10−8 in the terminating condition (4.58) for the whole algorithm, µ = 10−3,
and a maximum number of 20000 iterations. For the Newton method, we likewise set
TOL = 10−8 and we stop the algorithm after at most 45 Newton steps.
Example: shifting a linear wave packet
For validation purposes, we consider the time-evolution of a linear wave packet, i.e. λ = 0,
whose center of mass we aim to shift towards a prescribed point y1 ∈ [−L,L]. For this
186
4.5. Numerical simulation of the optimal control problem
Algorithm 1: Preconditioned MINRES algorithm
Given δkα0, set r0 := −Aδ
k
α0 − J
′(αk), z1 = Rr0, β1 = 〈r0, z1〉
1
2 , z1 = z1β1 , v1 =
r0
β1
,
γ0 = γ1 = 1, σ0 = σ1 = 0, η = β1 foreach ` = 1, 2, . . . do
if η < TOLMINRES then
Stop, δkα := δ
k
α` is the solution to (4.60).
else
Set µ` = 〈z`,Rz`〉,
v`+1 = Az` − β`v` − µ`v`,
z`+1 = Rv`+1,
β`+1 = 〈z`+1,Rz`+1〉
1
2 ,
z`+1 =
z`+1
β`+1
,
v`+1 =
v`+1
β`+1
,
ρ0 = γ`µ` − σ`γ`−1β`,
ρ1 =
√
ρ20 + β
2
`+1,
ρ2 = σ`µ` + γ`−1γ`β`,
ρ3 = σ`−1β`,
γ`+1 =
ρ0
ρ1
,
σ`+1 =
β`+1
ρ1
,
w`+1 = 1ρ1
(
z`+1 − ρ3w`−1 − ρ2w`
)
,
x`+1 = x` + γ`+1ηw`+1,
η = −σ`+1η
end
end
purpose consider a control potential
V (x) =
3
10
+
3x
200
≥ 0, ∀x ∈ [−L,L],
and the observable
A(x) = 1− e−(κ(x−y1))
2/L2 .
In this case, we find that the algorithm converges well even if we only invoke the first order
gradient method. Indeed, as we decrease the regularization parameters γ1, γ2  1, we
approach an optimal solution which, as it seems, cannot be improved upon. This optimal
solution, or, more precisely, its spatial density ρ = |ψ|2, is depicted in Figure 4.1 (right
plot), where we denote by “target” the function proportional to 1 − A(x) with κ = 0.07
and y1 = −2L/8, such that it has the same L2–norm as ψ0. The left plot shows the
associated control.
Since this solution seems optimal, the choice of γ1, γ2 becomes negligible below a certain
threshold. Thus, it suffices to consider γ1 = 0 and only include the cost term proportional
to γ2. Similar results hold for any other given point y1 ∈ Ω, provided y1 stays sufficiently
187
Optimal control of NLS
Figure 4.1: Shifting a linear wave paket
far away from the boundary.
Example: splitting a linear wave paket
We still consider the linear case, i.e., λ = 0, and aim to split a given initial wave packet
into two separate packets centered around y1 and y2, respectively. The control potential
is chosen as
V (x) = e−8x
2/L2 ≥ 0,
and the observable
A(x) = 1−
(
e−(κ(x−y1))
2/L2 + e−(κ(x−y2))
2/L2
)
.
In the following we fix κ = 0.07, y1 = −2L/8, and y2 = 2L/8. In this case we find that
the residual of the first order gradient method does not drop below the tolerance given
in (4.58) before the maximum number of iterations is reached. With the Newton method,
however, we find a (local) minimum of the objective functional J(ψ, α) in less than 20
Newton iterations. Of course there is no guarantee that this is a global minimum.
In order to illustrate our results, we consider the case where γ1 = 0, γ2 = 1.5× 10−6. At
the final control time T = 10 we then obtain:
〈Aψ(T ), ψ(T )〉2L2x ≈ 2.261× 10
−3.
The spatial density ρ = |ψ|2 of the corresponding solution is shown in the right plot of
Figure 4.2. The associated control is depicted in the left plot. If, instead, we choose
γ1 = 4× 10−5, γ2 = 1× 10−9, we find
〈Aψ(T ), ψ(T )〉2L2x ≈ 2.269× 10
−3,
188
4.5. Numerical simulation of the optimal control problem
Figure 4.2: Splitting a linear wave paket with γ1 = 0
and the corresponding solution is given in Figure 4.3. Here the intermediate state is a
plot of ρ(t) at t = 4 = 0.4× T .
Figure 4.3: Splitting a linear wave paket with γ1 > 0
A direct comparison of the (spatial densities of the) resulting wave functions and the
respective controls is given in Figure 4.4. We see that the spatial densities are nearly
identical, but the variability of the respective control parameters is not the same. This
is, of course, related to time–evolution of the weight factor ω(t), defined in (4.9), which
is shown in Figure 4.5 for the case of γ1 = 4× 10−5 and γ2 = 1× 10−9.
By construction, the time–integral of ω(t) can be interpreted as the physical work
performed during the control process. We find that compared to the case γ1 > 0, the
term ‖E(·)‖2L2t is around 30% larger (64.5 versus 49.1) and ‖E˙(·)‖
2
L2t
is around twice as
large (95.0 versus 43.4) in the case where γ1 = 0, yielding a significant advantage of our
control cost over terms considering the H1-norm only; see [40] for the latter.
Finally, Figure 4.6 shows an example of the evolution of the objective functional J(ψ, α)
189
Optimal control of NLS
Figure 4.4: Direct comparison between results
Figure 4.5: The weight factor ω =
∫
V |ψ|2dx over time
over the number of iterations of the Newton method, here for the case where γ1 = 0.
Example: splitting a Bose–Einstein condensate
We consider the same situation as in the previous example, but with an additional (cubic)
nonlinearity. More precisely, we choose λ = 8 > 0. It turns out that the conclusions are
similar to the ones found in the linear case (λ = 0). Qualitatively, the main difference
is that during the time–evolution, the wave function spreads out more because of the
additionally repulsive (defocusing) nonlinearity. In the linear case, the widest extension
of the wave packet is always comparable to its final value. Choosing as before γ1 = 4×10−5
and γ2 = 1× 10−9, we obtain the solution depicted in the right plot of Figure 4.7, where
we show the spatial density at the times t = 0, t = T = 10 and at the intermediate time
t = 4. The control is shown in the left plot. In comparison to the linear case (λ = 0), the
190
4.5. Numerical simulation of the optimal control problem
Figure 4.6: Value of J(ψ, α) over number of iterations
observable term in the objective functional J(ψ, α) is found to be slightly larger. Indeed,
we obtain
〈Aψ(T ), ψ(T )〉2L2x ≈ 3.720× 10
−3.
This seems to indicate that nonlinear effects counteract the influence of the control po-
tential.
Figure 4.7: Splitting a condensate with γ1 > 0
We again compare the present case with the one where γ1 = 0 (i.e. no cost term propor-
tional to the physical work) and γ2 = 1.5× 10−6. First, we find that
〈Aψ(T ), ψ(T )〉2L2x ≈ 3.382× 10
−3.
Moreover, ‖E˙‖2L2t is about 150% larger (172.1 versus 68.5) than in the case where γ1 6= 0.
Similarly, the total energy ‖E‖2L2t is around 15% larger (91.8 versus 79.5).
191
Optimal control of NLS
Example: splitting an attractive Bose–Einstein condensate
Our numerical method allows us to go beyond the rigorous mathematical theory developed
in the early chapters. In particular we may try to control the behavior of attractive
condensates, which are modeled by (4.1) with λ < 0, i.e. a focusing nonlinearity. Here
we choose λ = −1, whereas the parameters γ1 = 4× 10−5, γ2 = 1× 10−9 are the same as
before. The results are shown in Figure 4.8 (control in the left plot and the state at times
t = 0, 10, 4 in the right plot). The observable part of the objective functional satisfies
〈Aψ(T ), ψ(T )〉2L2x ≈ 2.143× 10
−3.
Figure 4.8: Splitting a focusing condensate with γ1 > 0
In comparison to the case of a repulsive (defocusing) nonlinearity the final value for the
observable term 〈Aψ(T ), ψ(T )〉2L2x is much smaller, confirming the basic intuition that an
attractive condensate does not tend to spread out as much as in the repulsive case.
192
Chapter 5
Concluding remarks and future work
5.1 On the Cauchy-problem of nonlinear Schro¨dinger
equations with angular momentum rotation term
In Chapter 3, we have investigated local and global existence for the nonlinear Schro¨dinger
equation with angular momentum rotation term, generalizing earlier results in the liter-
ature [37, 38]. As we have seen there, equation (3.4) can be considered (upon a change
of coordinates) as a special case of NLS with time-dependent potentials (sub-quadratic
in x). This class of models has recently been studied in [18]. Following the arguments
given therein, one could infer global in-time existence of (3.4) for sufficiently small initial
data ψ0 ∈ Σ, regardless of the sign of the nonlinearity. Moreover, growth rates for higher
order (weighted) Sobolev norms can also be obtained as in [18]. In addition, we note
that for a repulsive, isotropic quadratic potential V (x) = −γ
2
2 |x|
2, the time-dependent
change of coordinates is trivial and we could henceforth conclude global in-time existence
for sufficiently large γ > 0 by following the arguments given in [16].
We also want to point out that for the usual NLS with σ = 2/d there is an extra symmetry
which has been successfully deployed in the study of blow-up (yielding explicit blow-up
solutions and blow-up rates), see e.g. [76]. Using the so-called Lens transform [45] one
can transfer (most of) these results to the case of NLS with isotropic time dependent
quadratic potential W (t, x) = γ(t)|x|2, see [18]. However, it is argued in [18] that such
an approach is only feasible in the case of isotropic potentials and thus, we cannot expect
from it any further insight on the possibility of blow-up in our case, when (L ·Ω)V (x) 6= 0
and |Ω| > γ.
Finally, it is worth noting that the effect of the angular momentum rotation term in our
model is very different from other situations. For example, it has been shown for the
Euler equations with Coriolis force that blow-up can be delayed through a sufficiently
193
Concluding remarks and future work
strong rotation term [58] (see also as [4] for a related result). Clearly, the situation in our
model is much more involved, and we can not expect an analogous result to be true (the
counterexample being the case where V (x) is axially symmetric).
In future work, we would like to numerically test the blow-up conditions of Theorem 3.2.3.
In particular, we are intested in establishing whether the conditions are due to technical
difficulties or present real obstacles to blow-up.
5.2 Optimal bilinear control of Gross-Pitaevskii equa-
tions
In Chapter 4, we have introduced a rigorous mathematical framework for optimal quantum
control of linear and nonlinear Schro¨dinger equations of Gross-Pitaevskii type. We remark
that in the physics literature, L2(Rd) is usually considered as a complex Hilbert space,
equipped with the inner product 〈ϕ, ξ〉 =
∫
Rd ϕ(x)ξ(x)dx, whereas we consider L
2(Rd) as
a real Hilbert space (of complex functions), equipped with (4.12). Note, however, that
the expectation value of any physical observable A and thus also J(ψ, α) is the same for
both choices.
Let us briefly discuss possible generalizations for which our results remain valid. First,
we point out that in our analysis above, we did not take advantage of the fact that γ1 > 0
and hence all of our results remain true in the case γ1 = 0. However, Example 4.5.3
shows a significant quantitative difference in the behavior of the cost functionals with and
without the term proportional to γ1.
Second, it is straightforward to extend our analysis to the case of several control param-
eters, i.e.
V (t, x) =
K∑
k=1
αk(t)Vk(x), K ≥ 2.
Clearly, for Vk ∈ Wm,∞(Rd), m ≥ 2 > d/2, all of our results remain valid. In addition,
it is not difficult to extend our framework to cases of more general control potentials
V (α(t), x), not necessarily given in the form of a product. Such potentials are of physical
significance; see cf. [40]. From the mathematical point of view, all of our results still apply
provided that
‖V (α, ·)‖Wm,∞x ≤ C1, ‖∂
s
αV (α, ·)‖L∞x ≤ C2, ∀|s| ≤ 2.
Note that in this case, the cost term in J(ψ, α), which is proportional to the physical
194
5.3. Uniform quantitative hydrodynamic limits
work performed throughout the control process, reads
∫ T
0
(E˙(t))2dt =
∫ T
0
(α˙(t))2
(∫
Rd
∂αV (α(t), x)|ψ(t, x)|
2 dx
)2
dt.
It is more problematic to provide a rigorous mathematical framework for control potentials
V (α, x) which are unbounded with respect to x ∈ Rd. Only in the case where V (α, x) is
subquadratic with respect to x and in L∞(Rd) with respect to α, existence of a minimizer
can be proved along the lines of the proof of Theorem 4.2.1. More general unbounded
control potentials V (α, x) definitely require new mathematical techniques. Note that in
this case, even the existence of solutions to the nonlinear Schro¨dinger equation is not
obvious.
Finally, we want to mention that it is possible to extend our results (with some technical
effort) to the case of focusing nonlinearities, λ < 0, provided σ < 2/d. The latter prohibits
the appearance of finite-time blow-up in the dynamics of the Gross–Pitaevskii equation.
Clearly, the optimal control problem ceases to make sense if the solution to the underlying
partial differential equation no longer exists.
5.3 Uniform quantitative hydrodynamic limits
First and foremost, it remains to pursue the strategy outline in Section 2.7 to extend our
results to general dimensions.
In Section 2.5, we showed, assuming initial convergence of the microscopic entropy towards
the macroscopic entropy, this convergence is propagated along the evolution of the system.
It would be interesting to prove uniformity in time of this convergence. The difficulty
here lies in the fact that in general, the microscopic relative entropy HN(µNt |ν
N
f∞) does
not decay in t, since the weight of PµNt (
∑
x η(x) = K) on the hyperplanes of constant
particles is invariant under the evolution of the particle system. Thus as long as µN0 is
not chosen exactly such that
PµNt
(∑
x
η(x) = K
)
= PνNf∞
(∑
x
η(x) = K
)
,
we cannot take advantage of decay of the microscopic entropy. On the other hand, by the
equivalence of ensembles, we expect that
PµNt
(∑
x
η(x) = K
)
≈ PνNf∞
(∑
x
η(x) = K
)
195
Concluding remarks and future work
and that we can still deduce a uniform-in-time convergence of the entropy. This remains
to be clarified in future work. Another question, answered in Kosygina [50] for simple
exclusion processes, is whether the entropies converge for all positive times even if we
only assume a hydrodynamic limit (macroscopic profile) initially and no convergence of
the initial entropies.
It should be possible to prove a strong conservation of local equilibrium using our tech-
niques, see Remark 2.3.3 (4). In the case of attractive processes, this is a known result
- however, our method has the advantage of yielding explicit uniform-in-time bounds on
the rate of convergence.
Finally, it remains to be seen if this method can be extended to problems where the
hydrodynamic limit is not yet known, especially limit equations that can exhibit shocks.
196
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