Contributions in Fractional Diffusive Limit and Wave Turbulence in Kinetic Theory Sara Merino Aceituno Lucy Cavendish College University of Cambridge Centre for Mathematical Sciences Cambridge Centre for Analysis A thesis submitted for the degree of Doctor of Philosophy (Mathematics) Supervised by: Cle´ment Mouhot and James Norris April 2015 2 A mis padres (To my parents): Paqui Aceituno Recio, Juan Antonio Merino Soler. Abstract This thesis is split in two different topics. Firstly, we study anomalous trans- port from kinetic models. Secondly, we consider the equations coming from weak wave turbulence theory and we study them via mean-field limits of fi- nite stochastic particle systems. Anomalous transport from kinetic models. The goal is to understand how fractional diffusion arises from kinetic equa- tions. We explain how fractional diffusion corresponds to anomalous trans- port and its relation to the classical diffusion equation. In previous works it has been seen that particles systems undergoing free transport and scattering with the media can give rise to fractional phenomena in two cases: firstly, if in the dynamics of the particles there is a heavy-tail equilibrium distribution; and secondly, if the scattering rate is degenerate for small velocities. We use these known results in the literature to study the emergence of frac- tional phenomena for some particular kinetic equations. Firstly, we study BGK-type equations conserving not only mass (as in previ- ous results), but also momentum and energy. In the hydrodynamic limit we obtain a fractional diffusion equation for the temperature and density mak- ing use of the Boussinesq relation and we also demonstrate that with the same rescaling fractional diffusion cannot be derived additionally for the momen- tum. But considering the case of conservation of mass and momentum only, we do obtain the incompressible Stokes equation with fractional diffusion in the hydrodynamic limit for heavy-tailed equilibria. Secondly, we will study diffusion phenomena arising from transport of en- ergy in an anharmonic chain. More precisely, we will consider the so-called FPU-β chain, which is a very simple model for a one-dimensional crystal in which atoms are coupled to their nearest neighbours by a harmonic potential, weakly perturbed by a nonlinear quartic potential. The starting point of our mathematical analysis is a kinetic equation; lattice vibrations, responsible for heat transport, are modelled by an interacting gas of phonons whose evolu- tion is described by the Boltzmann Phonon Equation. Our main result is the derivation of an anomalous diffusion equation for the temperature. Weak wave turbulence theory and mean-field limits for stochastic particle systems. The isotropic 4-wave kinetic equation is considered in its weak formulation using model homogeneous kernels. Existence and uniqueness of solutions is proven in a particular setting. We also consider finite stochastic particle sys- tems undergoing instantaneous coagulation-fragmentation phenomena and give conditions in which this system approximates the solution of the equa- tion (mean-field limit). 6 Acknowledgements Many thanks to my two supervisors, Cle´ment Mouhot and James Norris for their help, guidance and support these years. I would like to thank Cle´ment specially for his generosity, for sharing all his resources with me and making sure I was not in need of anything. Also thank you for making it possible for me to go to Maryland to work with Antoine and to Vienna to work with Sabine. I would like to thank James Norris specially for all his patience (I guess you did not have many students that did not study probability before); for all the interesting discussions that we had; for your advice and for the extra effort that you gave so that I could finish the project on Wave Turbulence on time. I would like to thank my two examiners Pierre-Emmanuel Jabin, from the University of Maryland, and Jose´ Antonio Carrillo de la Plata, from Imperial College London. A big thank you to my collaborator Sabine Hittmeir. It was great working with you and I had a great time in Vienna. You have been always very nice and supportive to me. Thanks to my collaborator Antoine Mellet for all those three great months of research at Maryland. I really enjoyed working in our project together and I learned a lot. It was a wonderful experience, probably the most important experience I had to mature as a researcher. Thanks to Arieh Iserles and James Norris for running and making possible the existence of the Cambridge Centre for Analysis. I cannot express enough gratitude to Amalio Ferna´ndez-Pacheco. He has become one of the fundamental pillars of my live. The reality surpassing the fiction. I love the way we help each other to grow as a human beings, please lets keep doing that forever. You are the greatest discovery I have done during the thesis... shame that it is not publishable! You have given me so much. My vision of life has changed for the better since meeting you. You make me a better, happier person. Thank you for sharing this life with me. Some books to remember from this last year: Good to great, Mindset, Search Inside Yourself, Miniaturas histo´ricas by Stephan Zweig, ’Marva Collins’ Way, Difficult conversations, Irvin Yalom’s books, Coach Wooden. Tu´ sigue da´ndome la brasa, no te canses. Another star that I found in the (PhD) path is Nayia Constantinou. You can never lose faith in humanity with people like her. A great friend, a great housemate, a great person, a great woman. A philosopher; I miss so much your questions. I miss your warmth. Pure generosity. I have so much to learn from you (and I have already learned so much from you). I really admire you. My dear Kostaki! It is hard to find words for you. You are such a dear and great friend to me. Never have a doubt about that. We may not talk feelings but it shows. I cannot believe how lucky I have been to share a flat with you in such random circumstances. Thank you for your friendship and company. Since you were learning Spanish here it goes: ‘Eres entran˜able’ (there is no English word close enough to the meaning of that one). The trip to Greece was one of the highlights from these years. I had such a great time there. I will go and visit you wherever you are! You cannot escape... We will watch IT crowd again. Special thanks to my mathematical brother, also known as “bro” or Marc Bri- ant. I am very glad that we had each other in the good and the bad periods, which there were many of both. You have meant a lot to me during these years. Thank you for your friendship, your generosity and your honesty. Thank you to the coolest officemates, Damon Civin and Edward Mottram. Yes, it was hard, but you made it. Thank you to all my friends from the CCA (and outskirts) that have made my years in Cambridge so colourful. Such a wonderful mixture of personalities and backgrounds! You will be what I remember the most from my years here. Thank you: Damon Civin, Meline Joaris, Edward Mottram, Marc Briant, Stephanie Le´fevre, Kolyan Ray, Anastasia Kisil, Kostas Papafitsoros, Bati Sengul, Marion Hesse, Spencer Hughes, Julio Brau, Nayia Constantinou, Luca Calatroni. 8 Thank you to the people that form the Kinetic Group in Cambridge. Many special thanks to Amit Einav for his constant kindness to me and his mentor- ship; you are a star as I said many times. Thanks to the Cambridge University Women Basketball Club and all my playmates there for all the great moments. Thanks to filmmaker Sameer Patel for embarking with me in this adventure of filming a small outreach video. It was a great experience thanks to you. Thanks to all the administrative staff from the Centre of Mathematical Sci- ences at the University of Cambridge, specially to our previous administra- tor, Emma Hacking. A big thank you to Sidney Sussex College for being so good and generous to Amalio and me. Thanks to my college, Lucy Cavendish College for its support. Thanks to CCSCAM, Eitan Tadmor and the University of Maryland for their hospitality during my three month stay in 2014. Thanks to the Ki-Net grant for providing a big part of the funding. Thanks also to Pierre-Emmanuel Jabin, Thomas Rey and their families for being so kind and welcoming during my stay. To reach this point in my life, not only the people that I have met during the PhD have been important but also all the people before that helped me to get here. I am very grateful to the Teide School in Viladecans, where I studied until I was sixteen. It was a great school. I am specially grateful to my dear teacher Jesu´s Garcı´a Herna´ndez who fundamentally taught me how to read and write; how to read critically, how to write like a writer would and to enjoy the reading. He also taught me English,... which somehow proved to be extremely useful. I really miss writing and him correcting my writings. I think everybody should have a teacher that leaves an imprint in them. I was lucky enough. I am very grateful too to my high school Sant Gabriel in Viladecans (Mod- olell school in my times). Thank you to my great teacher and mentor Carlos Gime´nez Estaban who made me enjoy so much my two years there; they are 9 full of good memories. It was great to find another lover of science. I cannot express how grateful I am for your mentoring, teaching and company. Thanks to the Facultat de Mate´matiques i Estadı´stica at Universitat Polite´cnica of Catalunya. Special thanks to Sebastian Xambo´-Descamps, who always took a great care of the students and help me in the start of my career. Thank you to all my friends there, I cannot believe that so many years have passed already. Special thanks to Sara Riera and Anna Ferrer for being great friends (with a lot of patience). Thanks to the ENSIMAG school in Grenoble where I spent two of the best years of my life and to all my friends there, the AST. I miss you all. Thanks to Emmanuel Maitre and Mae¨lle Nodet for their help. Thanks to my family for their support. Finally, I would like to thank the two people who have made everything pos- sible, who have accompanied me through every journey and to whom I owe who I am. Thank you to my parents Juan Antonio Merino Soler and Paqui Aceituno Recio. You are extraordinary parents. I would have never reached this far if it not were for you. What you have done in your life and for me has more merit than a thousand thesis put together. This thesis is yours. Even if you have not written a single letter, even if you may not understand a symbol, this work is as yours as it is mine. You are doctors; doctors in ’life’. You are the ones who have given me the most important lessons. Finalmente, mil gracias a las dos personas que han hecho que todo esto sea posible: a mi padre Juan Antonio Merino Soler y a mi madre Paqui Aceituno Recio. Gracias por acompan˜arme en cada etapa. Soy quien soy gracias a vosotros. Sois unos padres extraordinarios, jama´s hubiera llegado tan lejos sin vosotros. Lo que habe´is hecho por mı´ y en vuestra vida tiene ma´s me´rito que mil tesis juntas. Esta tesis os pertenece. Aunque no hayais escrito una sola letra, incluso aunque no entenda´is un so´lo sı´mbolo, esta obra es tanto mı´a como vuestra. Vosotros sois doctores. Doctores en ’vida’. Vosotros sois los que me habe´is ensen˜ado las lecciones ma´s importantes. 10 Statement of Originality I hereby declare that my dissertation entitled ‘Contributions in Fractional Dif- fusive Limit and Wave Turbulence in Kinetic Theory’ is not substantially the same as any that I have submitted for a degree or diploma or other qualifi- cation at any other University. I further state that no part of my dissertation has already been or is concurrently submitted for any such degree of diploma or other qualification. This dissertation is the result of my own work and in- cludes nothing which is the outcome of work done in collaboration except where specifically indicated in the text. Chapter 1 gives an overview of the mathematical methods and techniques needed for Part I. The information provided there comes from multiple refer- ences which are given. However Section 1.5.5 was developed by the author with the help of Professor James Norris. Even though no new results are given, as far as we know no other reference does the type of computations performed there. This literature review was done under the guidance, explanations and super- vision of Professor Cle´ment Mouhot. Chapter 2 is original research work produced in collaboration with Doctor Sabine Hittmeir from the Johann Radon Institute for Computational and Ap- plied Mathematics (RICAM), Linz, Austria. The original research problem was suggested by Professor Cle´ment Mouhot, who also established the collaboration and proofed-read the final work. Chapter 3 is original research work produced in collaboration with Professor Antoine Mellet from the University of Maryland, US. The original research problem was suggested by Professor Antoine Mellet and Professor Cle´ment Mouhot. Professor Cle´ment Mouhot was the one who sug- gested this collaboration. Chapter 4 is original research work produced under the supervision of Pro- fessor James Norris from the University of Cambridge, UK. Professor James Norris help me with numerous discussions and explanations, specially on his work on the Smoluchowski equation; and by reviewing the chapter presented here. Professor Cle´ment Mouhot was the one who suggested working on the 4- wave kinetic equation. 12 Contents I Fractional diffusion limit for some kinetic models 15 1 Overview: anomalous diffusion in kinetic theory 17 1.1 Preliminaries: Multiscale analysis . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2 Mathematical models and previous results . . . . . . . . . . . . . . . . . . 22 1.3 Contributions in this part of the dissertation . . . . . . . . . . . . . . . . . 31 1.4 Anomalous transport: super-diffusions . . . . . . . . . . . . . . . . . . . . 39 1.5 Methods in the diffusive limit . . . . . . . . . . . . . . . . . . . . . . . . . . 44 1.6 Summary and final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2 Kinetic derivation of fractional Stokes and Stokes-Fourier systems Joint work with Dr. Sabine Hittmeir 79 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.2 A priori estimates and the Cauchy problem . . . . . . . . . . . . . . . . . . 90 2.3 Weak formulation and auxiliary equation . . . . . . . . . . . . . . . . . . . 93 2.4 Derivation of the macroscopic dynamics . . . . . . . . . . . . . . . . . . . . 105 3 Anomalous energy transport in FPU-β chain Joint work with Dr. Antoine Mellet 109 3.1 Crystal vibrations: A kinetic description . . . . . . . . . . . . . . . . . . . . 110 3.2 FPU-β chain: The four phonon collision operator . . . . . . . . . . . . . . . 116 3.3 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.4 Properties of the operator L . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 3.5 Proof of Theorem 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 3.6 Proof of Proposition 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 3.7 Appendix: Origin of the collision frequency . . . . . . . . . . . . . . . . . . 147 II Wave turbulence theory and mean-field limits for stochastic particle sys- 13 CONTENTS tems 153 4 Isotropic Wave Turbulence with simplified kernels: existence, uniqueness and mean-field limit for a class of instantaneous coagulation-fragmentation pro- cesses 155 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 4.2 Existence of solutions for unbounded kernel . . . . . . . . . . . . . . . . . 168 4.3 Mean-field limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 4.5 Appendix: Some properties of the Skorokhod space . . . . . . . . . . . . . 198 4.6 Appendix: Formal derivation of the weak isotropic 4-wave kinetic equation 200 14 Part I Fractional diffusion limit for some kinetic models 15 Chapter 1 Overview: anomalous diffusion in kinetic theory In this chapter, we start by explaining the main idea of multiscale analysis, which is the derivation of macroscopic models from microscopic ones. We focus here on a particular type of multiscale analysis giving rise to the fractional diffusion equation. In Section 1.2 we define the diffusion and fractional diffusion equations followed by the linear Boltz- mann equation which is a kinetic model suitable for the study of the diffusive limit. In Section 1.2.3 we review the main results on this direction and explain the methods used to obtain them in Section 1.5. We will also spend some time explaining the type of phenomena that the fractional diffusion equation models (super-diffusions/anomalous transport) and relating it to the classical diffusion in Section 1.4. Contents 1.1 Preliminaries: Multiscale analysis . . . . . . . . . . . . . . . . . . . . . . 19 1.1.1 Multiscale analysis from kinetic models . . . . . . . . . . . . . . . 22 1.2 Mathematical models and previous results . . . . . . . . . . . . . . . . . 22 1.2.1 Classical and fractional diffusion equations . . . . . . . . . . . . . 22 1.2.2 Linear Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . 23 1.2.3 Diffusive limit in the literature . . . . . . . . . . . . . . . . . . . . 25 1.2.3.1 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.2.3.2 A formal computation . . . . . . . . . . . . . . . . . . . . 26 1.2.3.3 Fractional diffusion due to heavy-tail equilibria . . . . . 27 1.2.3.4 Fractional diffusion due to a degeneracy of the collision frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.2.3.5 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.2.4 From atomic models to diffusion phenomena . . . . . . . . . . . . 31 17 Overview: anomalous diffusion in kinetic theory 1.3 Contributions in this part of the dissertation . . . . . . . . . . . . . . . 31 1.3.1 Kinetic derivation of fractional Stokes and Stokes-Fourier sys- tems (joint work with Dr. Sabine Hittmeir) . . . . . . . . . . . . . 31 1.3.1.1 Classical hydrodynamic limit for the Stokes equation . . 32 1.3.1.2 Our contribution . . . . . . . . . . . . . . . . . . . . . . . 33 1.3.2 Anomalous transport in FPU-β chains (joint work with Professor Antoine Mellet) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.4 Anomalous transport: super-diffusions . . . . . . . . . . . . . . . . . . . 39 1.4.1 Rescaling invariance and self-similarity . . . . . . . . . . . . . . . 39 1.4.2 Diffusion vs fractional diffusion equation . . . . . . . . . . . . . . 40 1.4.2.1 Gaussian vs stable distributions . . . . . . . . . . . . . . 41 1.4.3 Fractional derivatives, heavy-tailed functions and non-locality . . 43 1.4.3.1 Anomalous Fourier law . . . . . . . . . . . . . . . . . . . 43 1.5 Methods in the diffusive limit . . . . . . . . . . . . . . . . . . . . . . . . 44 1.5.1 Toy example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 1.5.1.1 Diffusive limit . . . . . . . . . . . . . . . . . . . . . . . . 45 1.5.1.2 A priori estimates . . . . . . . . . . . . . . . . . . . . . . 46 1.5.2 Hilbert expansion (classical diffusion) . . . . . . . . . . . . . . . . 48 1.5.2.1 Classical diffusion limit . . . . . . . . . . . . . . . . . . . 48 1.5.2.2 Construction of the ansatz . . . . . . . . . . . . . . . . . 52 1.5.2.3 The fractional Hilbert expansion . . . . . . . . . . . . . . 56 1.5.3 Laplace-Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 57 1.5.3.1 The fractional symbol . . . . . . . . . . . . . . . . . . . . 59 1.5.4 Mellet’s moments methods . . . . . . . . . . . . . . . . . . . . . . 61 1.5.4.1 The idea behind the method: weak formulation . . . . . 61 1.5.4.2 The fractional symbol . . . . . . . . . . . . . . . . . . . . 64 1.5.5 Probabilistic approach . . . . . . . . . . . . . . . . . . . . . . . . . 65 1.5.5.1 Derivation of the linear Boltzmann equation . . . . . . . 66 1.5.5.2 Stable Le´vy processes . . . . . . . . . . . . . . . . . . . . 71 1.5.5.3 Fractional diffusive limit . . . . . . . . . . . . . . . . . . 72 1.6 Summary and final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 75 1.6.1 How does fractional phenomena arise . . . . . . . . . . . . . . . . 76 18 1.1. Preliminaries: Multiscale analysis 1.1 Preliminaries: Multiscale analysis In this first Section we give a general flavour of the idea of multiscale analysis. Much can be written about this subject. Here we present a very short introduction sketching some of the concepts in an informal way. One of the greatest applications of mathematics is the description and prediction of physical phenomena, for example, using Newton’s laws, we can predict with much pre- cision the movements of the planets; Newton’s laws are what is called a model in math- ematics. It is through models of physical phenomena, like the movement of the planets, that mathematics are used to get predictions. However, we can find two different models describing the same phenomenon. How is this be possible? An example of this are the model called Boltzmann equation and the model called diffusion equation. Both can be used to describe a gas. However, these models are ob- tained by studying the gas from different perspectives. On one hand, the model of the Boltzmann equation is obtained by studying the particles of the gas and their collisions; that is why is called microscopic model. On the other hand, the diffusion equation de- scribes what can be seen by the naked eye, i.e., how the flow of the gas behaves; this is a macroscopic model. The microscopic and macroscopic models are quantitatively and qualitatively different even if the physical phenomenon, the behaviour of a gas, is the same. This example evokes an ‘old’ idea; all the matter of the universe is formed by atoms, so if we know how atoms behave, we would expect to know how the world (the one that can be observed by the naked eye) behaves. Mathematically, this means that macroscopic models should be derived from microscopic ones. However, in the models of the gas, the diffusion equation was not derived from the Boltzmann equation; both models were obtained independently. Nevertheless, since both are good models, one would expect to find an a posteriori relation between them. In this document, it is shown how to derive a posteriori diffusion-type equations from Boltzmann-type equations. The set of methods for doing so are called multiscale analysis or scaling process or limiting process. This derivation gives insight on the relation between the two models. The origins of multiscale analysis can be found in Hilbert’s 6th problem. In the Inter- national Congress of Mathematics of 1900, Hilbert presented 23 main problems for the mathematics of the 20-th century. Problem number 6 is the axiomatization of physics. This means to find a set of axioms (describing how particles behave) from which to de- rive all the physical phenomena, in particular, the one that can be observed by the naked eye. Part of the problem states: 19 Overview: anomalous diffusion in kinetic theory 6-th Problem. Mathematical treatment of the axioms of physics1 “To treat [...], by means of axioms, those physical sciences in which mathematics plays an important part; in the first rank are the theory of probabilities and mechanics. As to the axioms of the theory of probabilities, it seems to me desirable that their logical investigation should be accompanied by a rigorous and satisfactory development of the method of mean values in mathematical physics, and in particular in the kinetic theory of gases. Important investigations by physicists on the foundations of mechan- ics are at hand; I refer to the writings of Mach, Hertz, Boltzmann and Volkmann. It is therefore very desirable that the discussion of the foun- dations of mechanics be taken up by mathematicians also. Thus Boltz- mann’s work on the principles of mechanics suggests the problem of de- veloping mathematically the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of con- tinua. [...] ” The mathematical derivation of macroscopic models from microscopic ones seemed an impossible task. What Hilbert suggested is to use an intermediate step: kinetic theory. We comment next on the meaning of this quote. Atomic systems are in general intractable at mathematical, computational and exper- imental level. Let us take for example Newton’s laws applied to the modelling of a gas. A gas can be described by giving the position and velocity of each particle at each instant of time. Therefore to each particle i, it corresponds the following dynamics: x˙i = vi v˙i = sum of forces, which depends on all the particles . This atomic model presents three problems: (i) mathematically, the problem of solving the system is related to the N -body prob- lem, which is a very hard problem to study; (ii) numerically, each particle has 6 degrees of freedom and the number of particles is too big to be computed (around 1020); (iii) experimentally, the equations tell us how the gas behaves, but we need to have an initial description of the gas, i.e., we need to know all the positions and velocities 1From http://aleph0.clarku.edu/ djoyce/hilbert/problems.html#prob6, David E. Joyce. 20 1.1. Preliminaries: Multiscale analysis Figure 1.1: In the Boltzmann equation, f(t, x, v)dxdv is the number of particles in an infinitesimal volume dxdv centred at the point (x, v) at a time t. of all the particles at a given time. However, technically, it is not known how to get those measurements. Summarizing, the microscopic model is very hard to deal with. Boltzmann proposed to model the same system but considering that the only known information is the distribution of the particles in the phase space (space of positions and velocities). Therefore, he assumes that we do not know exactly the position and velocity of each particle. The goal is then to study ft = ft(x, v) probability distribution of the particles in space and velocity for every given time. f(t, x, v)dxdv gives the number of particles in an infinitesimal volume dxdv around (x, v) at a time t (see figure 1.1). Originally Maxwell and afterwards Boltzmann avoided the intractability of the atomic model at the price of adding some ‘uncertainty’ into it: we do not have the whole infor- mation of the system, we will just look at how the positions and velocities of the particles evolve in average. This idea gave birth to Statistical Mechanics and the Boltzmann equation for rarefied gases is the most fundamental one in what is called kinetic theory. Kinetic theory studies mathematical models giving the evolution of a statistical or probability distribution for a given quantity. It started with the study of gas dynamics but it now extends to other areas like the study of plasmas. Hilbert’s suggestion is to derive macroscopic models from microscopic ones using an intermediate scale (mesoscale) which corresponds to kinetic models. 21 Overview: anomalous diffusion in kinetic theory 1.1.1 Multiscale analysis from kinetic models The idea of multiscale analysis is to derive mathematically a particular physical model from another one that contains more information than the original one. The two models or equations are at different scales and deriving one model from the other requires, in the cases that will be treated here, averaging and a limiting process. This means that a model at atomistic scale explains how particular physical phenomena may arise at observable scale. The kinetic equation has a solution that depends on space, time and velocity. The macroscopic equation depends only on space and time. The latter will be derived by averaging over the velocities the solution to the kinetic equation, and by performing a limiting process. In physical terms, rescaling space and time means the following. The micro time scale is the typical time a particle takes to change its velocities. For observable changes to happen in the bulk of the particles, we need to speed up time and consider macro time scales. In the same manner, we also make a zoom out in space, to focus on the bulk of particles instead of on the individual particles. This rescaling in time and space has to be done properly so that it stands out interest- ing phenomena: if we speed up time too much, the particles may escape to infinity and we will see nothing, i.e., in the limit we will get zero. If we do not speed up time fast enough, no changes will occur on the bulk of particles and no phenomena will arise. Summarising, in the limiting process information is lost and at the same time, the dynamics of the bulk of particles, that were only implicit in the kinetic equation, stand out. Notice that due to the loss of information during the limiting process, it is possible that different kinetic models lead to the same macroscopic equation. Example of limiting process from kinetic equations can be found in the classical references [SR09], [CIP94] and [Vil02]. In this part of the dissertation we will deal with a particular type of multiscale analysis called diffusive limit. We will present the diffusive limit from kinetic equations giving rise to the fractional diffusion equation. 1.2 Mathematical models and previous results 1.2.1 Classical and fractional diffusion equations In this section we will study the fractional diffusion equation and its classical counterpart, the diffusion equation. We give here the basic definitions and afterwards in Section 1.4 we will study their properties. 22 1.2. Mathematical models and previous results The classical diffusion equation is written as ∂tρ(t, x)−∇x · (D∇xρ(t, x)) = 0 in (0,∞)× RN (1.1) ρ(0, ·) = ρin in RN (1.2) where D is a diffusion matrix (positive definite). The Cauchy theory and the properties of this equation are well known and can be found in the classical reference [Eva98]. The fractional heat equation is a generalisation of the classical one: ∂tρ(t, x) + κ(−∆x)(α/2)ρ(t, x) = 0 in (0,∞)× RN , (1.3) ρ(0, ·) = ρin in RN (1.4) with κ ∈ R+. The fractional laplacian corresponds to the classical laplacian in the case α = 2 and it is defined as (−∆x)(α/2)ρ := F−1(|k|αF(ρ)(k)), α ∈ (0, 2) (1.5) where F stands for the Fourier transform in the space variable. This definition is the one considered in Laplace-Fourier methods (Section 1.5.3) but it is useful to have the following equivalent definition in terms of a principal value integral: (−∆)α/2f(x) := cN,αPV ˆ RN f(x)− f(y) ‖x− y‖N+α dy α ∈ (0, 2) (1.6) for cN,α a constant depending only on the dimension and the exponent α. This is the definition used in the moments methods (Section 1.5.4) and it already shows that the fractional laplacian is a non-local operator. Fractional and classical diffusion equations model transport phenomena. The case of the fractional diffusion is a particular instance of anomalous transport called super- diffusions. All these ideas are explained in Section 1.4. For the Cauchy problem on this equation the reader is referred to [dPQRV12, MLP01] and for regularity results to [VdPQR13] and the references therein. For some applications of fractional diffusive phenomena see [HBW+00] and [MS12]. 1.2.2 Linear Boltzmann equation We will use the linear Boltzmann equation to illustrate how fractional diffusion pheno- mena arises from this kinetic model and the different methods that exist to obtain this derivation. Notice that the equation presented here has some variants that appear in other contexts under different names, like radiative transport equation. The linear Boltzmann equation is a kinetic equation giving the distribution f = f(t, x, v) 23 Overview: anomalous diffusion in kinetic theory of particles undergoing free transport and collisions with the background (scattering), see figure 1.1 in page 21. The equation conserves the total mass and the scattering process makes the distribution of the particles relax to an equilibrium. In its general form, the linear Boltzmann equation is ∂tf(t, x, v) + v · ∇xf(t, x, v) = L(f)(t, x, v) in (0,∞)× RN × RN (1.7) f(0, ·) = f in(x, v) in RN × RN (1.8) where L is a linear Boltzmann operator L(f) := ˆ RN [ σ(v, v′)f(v′)− σ(v′, v)f(v)] dv′ (1.9) = K(f)− νf for K(f) = ˆ RN σ(v, v′)f(v′) dv′, ν(v) = ˆ RN σ(v′, v) dv′. The collision kernel or cross-section σ ≡ σ(v′, v) is non-negative and ν is called colli- sion frequency. The collision kernel indicates the proportion of particles whose velocity changes from v to v′. The operator L is linear, defined in L1(ν) and conservative, i.e., it preserves the total mass of the distribution: ˆ RN L(f) dv = 0 for f ∈ L1(ν) (2) . We say that a function M is an equilibria for L if L(M) = 0. We will consider in this document linear Boltzmann equations with two types of equilibria: either, Maxwellian distributions, whose normalise form is M(v) = 1 (2pi)N/2 exp(−‖v‖2/2) (1.10) or, we will consider heavy-tail functions under the following shape M(v) ≤ c0|v|−N−α for all v ∈ RN , M(v) = c0|v|−N−α if |v| ≥ 1, M ∈ C1(RN ) (1.11) with s0 > 0 and α > 0 . We will not study here the properties of this equation (existence of solutions, posi- tivity, dissipation of energy (entropy), maximum principle,...) but they can be found in [AG13] and [Mou13]. We will look for a priori estimates in Section 1.5.1. 2 f ∈ L1(ν) if (definition) ˆ |f |ν dv <∞ 24 1.2. Mathematical models and previous results 1.2.3 Diffusive limit in the literature 1.2.3.1 Scaling Considering the conceptual meaning of the functions ρ = ρ(t, x) and f = f(t, x, v) as the distribution of particles in space and phase-space, respectively, we have: ρ(t, x) = ˆ RN f(t, x, v) dv . Given this relation one may think that solutions of the linear Boltzmann equation will give, by integration with respect to v, solutions of the diffusion equation. However, this first idea proves to be wrong; integrating the linear Boltzmann equation over the veloci- ties and using the mass conservation we have that ∂tρ+∇x · ˆ RN vf(t, x, v) dv = 0 which is not the diffusion equation. Therefore, solutions to the Boltzmann equation do not give directly solutions to the diffusion equation. We see next that we need to consider fε solution of the rescaled linear Boltzmann equation (for a well-chosen rescaling) to obtain in the limit ρ = limε→0 ´ RN fε dv a solution of the (fractional) diffusion equation. To perform the diffusive limit for the linear Boltzmann equation space and time are rescaled introducing the macroscopic variables x′ = εx, t′ = θ(ε)t (1.12) and the rescaled distribution function fε(t ′, x′, v) = f(t, x, v). (1.13) The rescaling means that a small variation in the macroscopic variables implies a big variation in the microscopic ones. Consider a point x′0 expressed in macroscopic variables and define x′1 as a small perturbation ∆ of x′0, x′1 := x′0 + ∆. Dividing by ε, we get x′1 ε = x′0 ε + ∆ ε where x1 := x′1/ε and x0 := x′0/ε are expressed in microscopic variables. Therefore a small perturbation ∆ in the macroscopic variables corresponds to a big perturbation ∆/ε in the microscopic variables. The rescaling corresponds to making the mean free path (distance between collisions) small and making the time scale very large. In other words, to observe macroscopic 25 Overview: anomalous diffusion in kinetic theory phenomena from the equation we make a zoom out in space and speed up time. In the limit the diffusion equation is obtained. The diffusion approximation to kinetic equations has been studied in various works, see for example [BSS84], [BLP79], [DGP00], [LK74], [PS00]. Now fε satisfies the following rescaled linear Boltzmann equation (we have skipped the primes): θ(ε)∂tfε(t, x, v) + ε v · ∇xfε(t, x, v) = K(fε)(t, x, v)− νfε(t, x, v) (1.14) fε(0, ·) = f in(x, v) (1.15) for (t, x, v) ∈ (0,∞)× RN × RN . Notice that the initial condition f in is independent of ε. 1.2.3.2 A formal computation A formal computation will help us understand how the diffusive limit comes out. Let us consider the classical diffusive limit. Formally, as ε → 0 we have that L(fε) → 0 so the limit f0 ∈ kerL = span{M}, therefore f ε → ρ(t, x)M(v). For the classical diffusion limit θ(ε) = ε2. Integrating (1.14) on the velocities we have: ∂t ˆ f ε dv︸ ︷︷ ︸ ρε + ε−1∇x · ˆ vf ε dv︸ ︷︷ ︸ =: I = 0 and using the self-adjointness of L we obtain formally I := ε−1∇x · ˆ RN L−1(v)L(f ε) dv = ∇x · ˆ RN L−1(v)v · ∇xf ε dv + ε∇x · ˆ RN L−1(v)∂tf ε dv → −∇x · (κ∇xρ(t, x)) where κ = − ˆ RN L−1(v)⊗ vM dv. Hence, we have obtained formally that fε → ρ(t, x)M(v) with ρ satisfying: ∂tρ = ∇x · (κ∇xρ(t, x)) as long as κ <∞. Since L is formed by a compact operator K and a multiplicative operator ν, κ can be 26 1.2. Mathematical models and previous results formally approximated as a constant matrix: κ ≈ ˆ RN ‖v‖2 ν M dv (1.16) Observe that we require κ < ∞ in order to have a finite diffusion coefficient. Frac- tional diffusion phenomena takes place when this integral is not finite. This can happen because there is a degeneracy in ν (Section 1.2.3.4) and/or not all the moments in M are finite, for example, because M is a heavy-tail function (1.11) (Section 1.2.3.3). Heavy-tail equilibria (also called power laws) are important distributions. We repeat here the account given in reference [MMM11] on different contexts where heavy-tail func- tions appear: on the velocity distribution in astrophysical plasmas [ST91], [MR94]; on dissipative collision mechanisms in granular gases [EB02], see also the review [Vil06]; on elastic collision mechanisms in mixture of gases with Maxwellian collision kernel [BG06]; in economy, through the Pareto distribution, see for example [New05], [Wri05]. In the case of a divergent diffusion coefficient κ, the multiscale analysis gives a frac- tional diffusion equation in the limit and has a fundamental change compared with the classical diffusion limit: we need a different time rescaling t′ := εγt for some γ > 0. 1.2.3.3 Fractional diffusion due to heavy-tail equilibria Next, we put together the main results in [MMM11]. The fractional diffusion is obtained as the limit of the linear Boltzmann equation and it is due to having a heavy-tail equilib- ria. We start with the assumptions on the equation Assumption (A1). The cross-section σ is locally integrable on R2N , non-negative, and the collision frequency ν is locally integrable on RN and satisfies ν(−v) = ν(v) > 0 for all v ∈ RN . Assumption (A2). There exists a function 0 ≤ M ∈ L1(ν) such that ‖v‖2ν(v)−1M is locally integrable and ν(v)M(v) = K(F )(v) = ˆ RN σ(v, v′)M(v′)dv′, and M(−v) = M(v) > 0 for all v ∈ RN and ˆ RN M(v) dv = 1. The existence of this function is a consequence of Krein-Rutman’s theorem (see [MMM11] and the references therein). 27 Overview: anomalous diffusion in kinetic theory Assumption (B1). There exists α > 0 and a slowly varying function l (explained below) such that M(v) = M0(v)l(‖v‖), where M0 is such that ‖v‖α+NM0(v)→ κ0 ∈ (0,∞) as ‖v‖ → ∞. A slowly varying function is a measurable function l : R+ → R such that l(λs) ∼ l(s) as s→∞ for all λ > 0. Assumption (B2). There exists β ∈ R and a positive constant ν0 such that ‖v‖−βν(v)→ ν0 as ‖v‖ → ∞. Assumption (B3). There exists a constant C such that ˆ RN M ′ ν b dv′ + (ˆ RN M ′ ν ′ b2 ν2 dv′ )1/2 ≤ C for all v ∈ RN , where b = b(v, v′) := σ(v, v′)M−1(v). Theorem 1.1 (Fractional diffusion limit, Theorem 3.2 in [MMM11]). Assume that Assump- tions (A1-A2) and (B1-B2-B3) hold with α > 0 and β < min{α; 2− α}. Define γ := α− β 1− β , and θ(ε) := l(ε − 1 1−β )εγ . Observe that this implies that β < 1 and γ < 2. Assume furthermore that f in ∈ L2(M−1) and let fε be the solution to (1.14), with that choice of θ and initial data f in. Then, fε converges in L∞(0, T ;L2(RN × RN ))-weak to a function ρ(t, x)F (v) where ρ(t, x) is the unique solution of the fractional diffusion equation of order γ: ∂tρ+ κ(−∆x)γ/2ρ = 0 in (0,∞)× RN ρ(0, ·) = ρin in RN , with κ given by κ = κ0ν0 1− β ˆ RN w21 ν20 + w 2 1 1 ‖w‖N+γ dw. Theorem 1.2 (Classical diffusion limit with anomalous time scale, Theorem 3.4 in [MMM11]). Assume that Assumptions (A1-A2) and (B1-B2-B3) hold with α > 1 and β = 2− α (i.e. γ = 2), 28 1.2. Mathematical models and previous results and l such that l(r) ln(r)→ +∞ as r → +∞ (in particular, the second moment of F is infinite). Then define θ(ε) = ε2l(ε − 1 1−β ln(ε−1). Assume furthermore that f in ∈ L2(M−1) and let fε be the solution of (1.14), with θ(ε) defined as above and initial data f in. Then, fε converges in L∞(0, T ;L2(RN × RN ))-weak to ρF where ρ = ρ(t, x) is the unique solution to the standard diffusion equation ∂tρ− κ∆xρ = 0 with κ given by κ = κ0ν0 (1− β limλ→0 1 ln(λ−1) ˆ ‖w‖≥λ w21 ν20 + w 2 1 1 ‖w‖N+2dw. Theorem 1.3 (Classical diffusion limit with classical time scale, Theorem 3.6 in [MMM11]). Assume that Assumptions (A1-A2) hold as well as the following bounds ˆ RN ( ν(v) b(v, v′) + ‖v′‖2 ν(v′) ) F ′ dv′ ≤ C for all v ∈ RN . Assume, furthermore, that f in ∈ L2(M−1) and let fε be the solution of (1.14), with θ(ε) = ε2 and initial data f in. Then fε converges in L∞(0, T ;L2(RN × RN ))-weak and in L2((0, T )× RN × RN ))-strong to a function ρF where ρ = ρ(t, x) is the unique solution of the standard diffusion equation with diffusion constant D = ˆ RN (v ⊗ χ) dv where L(χ) = −vM. 1.2.3.4 Fractional diffusion due to a degeneracy of the collision frequency This result in presented in [BAMP11]. Consider now that the Boltzmann operator is defined as: L(f) = ν(v)[ρνM(v)− f(v)] (1.17) where ρν = ρν(t, x) is defined such that there is conservation of the total mass: ˆ RN L(f) dv = 0 and M is a given probability density. 29 Overview: anomalous diffusion in kinetic theory Theorem 1.4 ([BAMP11]). Suppose that ν is bounded and that for δ > 0, β > 0, ν0 > 0 ν(v) = ν0‖v‖N+2+β, for ‖v‖ ≤ δ M(v) = M0 > 0 for ‖v‖ ≤ δ. where M is a probability distribution with ˆ RN vM(v) dv = 0 and ˆ ‖v‖≥δ ‖v‖2 ν(v) M(v) dv <∞, ˆ RN ν(v)2M(v) dv <∞. Define γ := 2− β β +N + 1 . Then the solution fε of εγ∂tfε + εv · ∇xfε = ν(v)[ρν,εM(v)− fε] f(0, ·) = f in converges weakly in L2νM−1(R N × RN × (0, T )) for all T > 0 to a function ρ(x, t)M(v) where ρ solves for some κ > 0  ∂tρ+ κ(−∆)γ/2ρ = 0 ρ(x, 0) = ´ RN f in dv . See Section 1.6.1 for an interpretation on why a degeneracy in the collision frequency gives fractional phenomena. 1.2.3.5 Methods In the literature, we have the following methods showing the fractional diffusion limit for the linear Boltzmann equation. • In [MMM11], explained in Section 1.2.3.3, the authors use a Laplace-Fourier trans- form method (Section 1.5.3). This method may not work for collision kernels de- pending on space or non linear operators. • In [JKO09] the limit is obtained using probabilistic methods. In Section 1.5.5 we will interpret the linear Boltzmann equation as giving the evolution of the prob- ability distribution of a particle undergoing a Markov process. We will show the fractional diffusive limit by proving the convergence to a stable Le´vy process. 30 1.3. Contributions in this part of the dissertation • In [Mel10] the limit is obtained via a moments method (Section 1.5.4). This allows to consider collision frequencies ν depending on the x variable (which was not possible with the Laplace-Fourier Transform method). • In [BAMP11] the fractional diffusion is obtained using also the moments method but for any given equilibria considering that the collision frequency ν has a singu- larity near v = 0, see the next section. • In [AMP10] a fractional Hilbert expansion is used (Section 1.5.2), this gives stronger convergence results than in the previous methods. 1.2.4 From atomic models to diffusion phenomena We conclude this section by mentioning that there exists results on the derivation of the linear Boltzmann equation with gaussian equilibrium starting from atomic (determinis- tic) models, [DP99], [vBLLS80]. Recently, in [BGSR13] the heat equation was derived directly from a deterministic particle system without using the kinetic scale as an intermediate step. No results of this type are known (to the best of our knowledge) for the fractional dif- fusive case or for linear Boltzmann-type equations with heavy-tail equilibria. The closest result in the literature can be found in [MT14], in this reference, the authors prove a ’su- perdiffusive’ central limit theorem for a periodic Lorentz gas. The limit in the end is classical but the rescaling in the central limit theorem is anomalous. 1.3 Contributions in this part of the dissertation 1.3.1 Kinetic derivation of fractional Stokes and Stokes-Fourier systems (joint work with Dr. Sabine Hittmeir) In Chapter 2 we extend the results presented in Section 1.2.3 to kinetic models preserving not only the total mass (0th moment) but also the first and second moment. We will consider a kinetic equation that resembles a linearised BGK equation. The BGK equation is the equation for a density distribution f = f(t, x, v). In dimension 3 it reads [SR09]: ∂tf + v · ∇xf =Mf − f where Mf := ρ(t, x) (2piT (t, x))3/2 exp ( −|v − U(t, x)| 2 2T (t, x) ) where ρ, U, T are the density, the momentum and the energy respectively, defined as ρ(t, x) = ˆ f(t, x, v) dv; ρU(t, x) = ˆ vf(t, x, v) dv; ρ(|U |2+3T )(t, x) = ˆ |v|2f(t, x, v) dv. 31 Overview: anomalous diffusion in kinetic theory Observe thatMf is a Maxwellian distribution. In our linearised equation sometimes we will consider that we have a Maxwellian equilibria and sometimes it will be substi- tuted by a heavy-tail distribution. In the limit we obtain what we call the fractional Stokes or Fourier-Stokes equation. Next we explain the classical Stokes limit, done in [GL02], as a background for our result. 1.3.1.1 Classical hydrodynamic limit for the Stokes equation The classical Stokes limit starts from the Boltzmann equation. Here we will give a sketch of this result. The Boltzmann equation has the following shape ∂th(t, x, v) + v · ∇xh(t, x, v) = Q(h, h) in (0,∞)× RN × RN for Q a particular bilinear operator. We will not describe here this equation, since it will not be necessary for the future but the reader is referred to [SR09], [CIP94] and [Vil02] for more information. The linearisation of the collision operatorQ around the equilibrium distribution (Maxwellian) M is written as: h(t, x, v) = M + δg(t, x, v) for some δ > 0. Then the linearised term g fulfils the equation ∂tg + v · ∇xg = −L(g) + δQ(g, g) (1.18) where L is the linearised Boltzmann operator and Q is a modified bilinear operator [SR09]. To perform the hydrodynamic limit, the kinetic equation (1.18) is rescaled in space and time (and in relation with the Knudsen number). Different rescaling produce in the limit different macroscopic equations like Euler, Navier-Stokes, Stokes or Acoustic equa- tions (see for example [Gol98], [SR09], [DMEL89], [Vil01]). In [GL02] (1.18) is rescaled as ε∂tgε + v · ∇xgε = −1 ε L(gε) + δε ε Q(gε, gε) (1.19) assuming that δε/ε→ 0 as ε→ 0. In this manner, it is proven in [GL02] that the nonlinear term in (1.19) vanishes in the limit and hence, the limiting behaviour is given by the linearised equation ε2∂tgε + εv · ∇xgε = −L(gε) (1.20) where we have abused notation writing again gε. 32 1.3. Contributions in this part of the dissertation Theorem 1.5 (Hydrodynamic limit for the Stokes equation, [GL02]). gε in (1.20) satisfies gε ⇀ g =  ˆ RN gdv︸ ︷︷ ︸ =:ρ + ˆ RN vg dv︸ ︷︷ ︸ =:−→m ·v + ˆ RN ( 1 2 ‖v‖2 − N 2 ) g dv︸ ︷︷ ︸ =:θ ( 1 2 ‖v‖2 − N 2 )M where the convergence is in some particular space that we do not make precise here (since it is not relevant for our work). The macroscopic quantities ρ,−→m, θ satisfy the (i) incompressibility condition: ∇x · −→m = 0; (ii) Boussinesq relation: ρ+ θ = 0; (iii) Stokes equation: ∂t −→m = ω∆x−→m +∇xp (1.21) N + 2 2 ∂tθ = κ∆xθ (1.22) for ω, κ > 0 and p = p(t, x) is a pressure term. The equations have some particular initial data (that we omit here, see [GL02]) fulfilling the incompressibility and Boussinesq relations. Remark 1.6. The following two properties of the linearised Boltzmann equation are fun- damental to perform the limit in Theorem (1.5) presented in [GL02]: (i) The conservation of the moments: for ψ(v) = 1, v, ‖v‖2, it holds ˆ RN ψ(v)L(g)dv = 0. (ii) The Kernel(L)=span{M, vM, ‖v‖2M}; g corresponds to the limit of the projection of gε onto the Kernel of L in the weighted L2(M−1dv) space. We will see all this in more detail in Section 2.1.1. 1.3.1.2 Our contribution Our goal is to find a result similar to the one in the preceding section (1.21)-(1.22) but having fractional diffusion equations for the 0th, 1st and 2nd moment. The structure of the equations and formulation is similar to the one for the classical Stokes limit but the proof is completely different, based on the moments methods [Mel10] explained in Section 1.5.4. The main results obtained are Theorems 2.1 (page 88) and 2.2 (page 89). 33 Overview: anomalous diffusion in kinetic theory 1.3.2 Anomalous transport in FPU-β chains (joint work with Professor An- toine Mellet) In Chapter 3, we investigate some aspects of the transport of energy in one dimensional chains of oscillators. The goal is to derive Fourier’s law, which is at the core of the heat equation and states that the heat flux ~j behaves as ~j = −κ∇xT (1.23) where T is the temperature and κ is a positive constant that may depend on the tem- perature itself. This law has been observed experimentally, but to this day there is not a complete and full mathematical justification describing how it arises from the atomic laws of the solid. Nevertheless, many mathematical works have been devoted in this direction and major progress has been achieved, see [BLRB00] for a review of this very challenging problem. At the microscopic level, solids can be modeled as lattices, were each node repre- sents an atom. For insulating crystals, where heat is transported by lattice vibrations (see [LLP03]), one possible approach to derive Fourier’s law relies on the introduction of the Boltzmann phonon equation, a kinetic equation that can play the role of an intermediate step between the microscopic atomic level and the macroscopic scale. It is this approach, first suggested by Peierls [Pei29], that we try to make rigorous in this paper in a very particular setting. The particular framework we are considering was made popular by a famous numeri- cal experiment performed by Fermi, Pasta and Ulam in the 1950’s at Los Alamos National Laboratories. The goal of their experiment was to investigate numerically the dynamic (and relaxation toward equilibrium) of the simplest model for a crystal: a chain of oscilla- tors coupled to their nearest neighbors by non-linear forces described by an Hamiltonian of the form H = 1 2 ∑ i∈Z [ p2i + V (qi+1 − qi) ] . When V is purely harmonic, the system has quasi-periodic solutions and does not relax to an equilibrium (see [BI05]). Fermi, Pasta and Ulam thus considered the next two simplest cases by adding a cubic potential V (r) = r2 + αr3 (this model is now referred to as the FPU-α chain) or a quartic potential V (r) = r2 + βr4 (the FPU-β chain). These models have been widely studied since that original experiment (see Lepri, Livi, Politi [LLP05] for a recent review of the work devoted to these models). Our goal in this paper is to derive Fourier’s law for the FPU-β chain (we will see later why we do not consider the FPU-α chain). To achieve this, we rely on an idea of Peierls [Pei29], who describes lattice vibrations, responsible for heat transport, as an interacting gas of phonons whose density distribution function (denoted W below) solves a Boltzmann 34 1.3. Contributions in this part of the dissertation phonon equation (also known as Peierls equation in this context). The mathematical derivation of this Boltzmann phonon equation starting from the microscopic equations for the motion of the atoms (Hamiltonian dynamic) has written formally by H. Spohn in [Spo06b]. We will thus not focus on this step, though we will spend some time in this paper discussing the results of [Spo06b]. Our focus instead will be on the rigorous deriva- tion of Fourier’s law from the Boltzmann phonon equation. The most remarkable aspect of our result is that we will not recover (1.23), but instead a non-local Fourier law corre- sponding to an anomalous diffusion equation (in place of the usual heat equation). This was not unexpected, since anomalous heat diffusion phenomena in the FPU-β frame- work have actually been observed numerically in dimension one and two (while normal diffusion is observed in the three dimensional case), see in particular [SMY+00], [LLP03], [LLP05], and also [AK01] for a study at the level of the kinetic equation. In fact, by using Peierls equation, it has previously been proved that the energy current correlation has a slow decay in time as t−3/5 indicating anomalous diffusive behavior (see [Per03, LS08]). Let us now describe our main result. As mentioned above, the starting point of our analysis is the Boltzmann phonon equation given by: ∂tW + ω ′(k)∂xW = C(W ) where the unknown W (t, x, k) is a function of the time t ≥ 0, the position x ∈ R and the wave vector k ∈ T := R/Z. This function is introduced in [Spo06b] as the Wigner transform of the displacement field of the atoms, but it can be interpreted as a density distribution function for a gas of interacting phonons (describing the chain vibrations). The function ω(k) is the dispersion relation for the lattice and the operator C describes the interactions between the phonons. We will discuss in Sections 3.1 and 3.2 the particular form of ω and C corresponding to our microscopic models. For the FPU-β chain, the operator C will be the so-called four phonon collision operator, which is an integral operator of Boltzmann type but cubic instead of quadratic (see (3.18)). As explained above, our goal is to derive a macroscopic equation for the tempera- ture. This is, at least in spirit, similar to the derivation of Navier-Stokes equations from the Boltzmann equation for diluted gas (see [BGL91] and references therein). We will consider a perturbation of a thermodynamical equilibrium W (k) = Tω(k) (note that the temperature is classically defined by the relation E = kBT where E = ´ T ω(k)W (k) dk and kB denotes Boltzmann’s constant - here, we choose temperature units so that kB = 1): W ε(t, x, k) = W (k)(1 + εf ε(t, x, k)). 35 Overview: anomalous diffusion in kinetic theory The function f ε then solves ∂tf ε + ω′(k)∂xf ε = L(f ε) +O(ε) where L is the linearized operator L(f) = 1 W DC(W )(Wf). As usual a macroscopic equation is derived after an appropriate rescaling of the time and space variable. More precisely, we will show (see Theorem 3.4) that the solution of ε 8 5∂tf ε + εω′(k)∂xf ε = L(f ε) converges to a function T (t, x) solution of ∂tT + κ T 6/5 (−∆) 45T = 0 thus giving the anomalous Fourier’s law (of order 3/5) ~j = −κ(T )∇(−∆)− 15T. The derivation of such a fractional diffusion equation from a kinetic equation is now classical (see Section 1.2.3 and references in Section 1.2.3.5). As in previous results (see in particular [BAMP11]), the order of the limiting diffusion process is determined by the de- generacy of the collision frequency of the operator L. Our work is thus greatly indebted to the work of J. Lukkarinen and H. Spohn [LS08] who carefully study the properties of the operator L and show in particular that the collision frequency behaves as |k|5/3 as k → 0. The main novelty here, compared with the results mentioned above, is the fact that the kernel of the collision operator L is 2 dimensional. The reason for that will be discussed in the next sections and it appears to be a mathematical artifact rather than being related to some physical phenomenon. It does, however, indicate some weakness in the mixing properties of the collision process (this will be even more obvious for the FPU-α chain, for which the collision operator vanishes altogether). And while the macroscopic behavior of f ε is completely determined by the function T (t, x), the other component of the projection of f ε onto the kernel of L will play a role in reducing the value of the diffusion coefficient κ. We point out that we will not attempt here to derive a nonlinear Fourier law by work- ing with the nonlinear operator C (rather than the linearized operator L). Such a deriva- tion is developed in [BK08] by Bricmont and Kupiainen, but under assumptions that en- sure that regular diffusion, rather than anomalous diffusion, takes place (non degeneracy 36 1.3. Contributions in this part of the dissertation of the collision frequency). To conclude this section, we mention that diffusive and superdiffusive heat transport has also been derived for FPU-type chains in a different mathematical setting using a probabilistic approach: in this setting the hamiltonian dynamics of the microscopic sys- tem are considered to have only an harmonic potential and the dynamics are perturbed by a stochastic noise conserving momentum and energy (see [BBO06], [BBO09], [BOS10] and the review paper [Oll09].) For a summary on the previous works, their relations and the place were our result takes place inside this area of research see scheme in next page. 37 MICRO FPU chain/lattice KINETIC MACRO (Anomalous) diffusion Harmonic potential No relaxation to eq. Cubic potential (FPU-alpha chain) Quartic potential (FPU-beta chain) Harmonic potential + noise conserving momentum and energy On site potential: - classic diffusion (Aoki, Lukkarinen, Spohn) Next neighbour pot.: - 1d, 2d anomalous dif. - 3d or higher, classic dif. (Lepri, Livi, Politi) Free transport eq. C(W)=0 3-phonon Boltz. eq. C(W): bilinear op. (we lose info in the kinetic limit; suggests weak mixing at micro level) next neighbour pot. 4-phonon Boltz. eq. C(W): trilinear op. Spurious conservation of nb. of phonons (Olla, Basile, et.al) Linear Boltzmann Equation Fractional heat equation Heat equationNonlinear heat equation (Mellet, Merino) (Bricmont - Kupianen) (Olla, et. al.) (Mellet, Mouhot, Mischler, et. al.) Weaker mixing; smaller coeff.; slows down convergence to eq. Linearised eq. (Lukkarinen, Spohn) Heat – lattice vibrations – phonons Spohn (formal) (Basile, Olla, Spohn) 1.4. Anomalous transport: super-diffusions 1.4 Anomalous transport: super-diffusions Fractional diffusion equations model super-diffusion phenomena. In super-diffusions, particles distributed according to ρ are transported spreading ‘faster than any diffusion’. Explaining the meaning of this will be the goal of this section as well as studying the properties of super-diffusions and compare them with the classical diffusion. One can compare the ‘speed of spreading’ through the Mean Square Displacement (MSD) of the particles which we explain in the next section. In classical diffusion the Mean Square Displacement is linear in time, in fractional phenomena this is not the case, that is why it is called anomalous transport. 1.4.1 Rescaling invariance and self-similarity The Mean Square Displacement (MSD) of a particle X(t) whose position over time is distributed according to the density probability ρ = ρ(t, x) is given by 〈X(t)2〉 := ˆ t 0 ˆ RN ‖X(s)‖2ρ(s, x)dsdx. In the case of the diffusion equation it holds 〈‖x‖2〉 = 2Dt. (1.24) where D > 0 is the diffusive constant. The relation between space and time established by the MSD is of paramount impor- tance since it implies a self-similarity of the trajectories of the particles. We explain this next. Observe that if we multiply this relation by ε2 we have that 〈‖εx‖2〉 = 2Dε2t and defining new variables x′ = εx and t′ = ε2t we get again the relation 〈‖x′‖2〉 = 2Dt′. Consequently, if space and time are rescaled by a factor of ε and ε2 respectively, we observe the same trajectories, i.e., the trajectories are self-similar. The MSD gives a good indication of the rescaling chosen in the diffusion limit since in most cases it is the one keeping the scaling invariance of the diffusion equation, i.e., (x, t) 7→ (εx, ε2t); which corresponds to the self-similarity of the trajectories and keeping the speed of dif- 39 Overview: anomalous diffusion in kinetic theory fusion, 2D, constant. For the fractional diffusion equation something analogous happens, the Mean Square Displacement in this case is not well defined since the variance is not finite (the density behaves asymptotically as a power law), however it holds that (see [MK00]) 〈‖x‖δ〉 ∼ t δα , 0 < δ < α < 2. Rescaling space and time as (x, t) 7→ (εx, εαt) the previous expression stays invariant. This is the chosen rescaling in fractional diffusive limits. As we said in the introduction one can compare the ‘speed of spreading’ through the Mean Square Displacement (MSD) of the particles. In classical diffusion it grows linearly in time, but in fractional diffusion phenomena it diverges because it scales faster, hence the spreading is faster. 1.4.2 Diffusion vs fractional diffusion equation In the diffusion or heat equation ∂tρ(t, x) = D∆xρ(t, x), D > 0 ρ(t, x) is a probability density giving the distribution of the particles in space at each time; it describes transport of particles. The diffusion equation is obtained by the combination of two rules: (i) Conservation of the total mass (number of particles); expressed as ∂tρ(t, x) = −∇x ·~j(t, x). where ~j is the flux of particles (rate at which particles cross an infinitesimal sur- face). Mathematically, the conservation of mass is a consequence of the divergence theorem. (ii) Fick’s law (or Fourier law for the temperature). This is a law observed experimen- tally: ~j(t, x) = −D∇xρ(t, x), meaning that particles move linearly from places of high concentration to places of low concentration following the gradient. D is a positive constant called ‘diffusivity constant’ and it is proportional to the speed at which particles spread (diffuse); recall the Mean Square Displacement (MSD) in (1.24). 40 1.4. Anomalous transport: super-diffusions A first intuition is that, since D dictates how fast the diffusion is taking place, if we want particles to spread faster than any diffusion, we need D to increase to infinity; meaning that the mean square displacement goes to infinity; hence, the variance will not be finite. This corresponds to having the constant κ in (1.16) equal to infinity. In super-diffusive phenomena for α ∈ (1, 2) the total number of particles is conserved but Fourier law is violated; it will need to be replaced as we explain in Section 1.4.3. 1.4.2.1 Gaussian vs stable distributions The fundamental solution of the diffusion equation is the gaussian (or Maxwellian) dis- tribution [Eva98], i.e., given initial data ρ(0, x) = δ(x− x0) its solution is ρ(t, x) = 1√ (2piDt)N exp ( −‖x− x0‖ 2 2Dt ) . Stable distribution, and not Gaussians, are the fundamental solution of the fractional diffusion equation (1.3) [MLP01]. Their density behaves asymptotically as a power law [FN99]. Stable distributions are defined in Section 1.5.5.2. In contrast with the gaussian distribution, stable distributions do not have all its mo- ments finite. Intuitively, this is coherent with the idea of super-diffusions: suppose that the variance of the fundamental solution is not finite, then particles are more likely to be further from their starting point x0 than they are to be with the Gaussian distribution. Consequently, particles spread faster than in a normal diffusion. Stable Le´vy processes. To the solutions of the diffusion and fractional diffusion equa- tions one can associate stochastic processes called, stable Le´vy processes (explained in Section 1.5.5.2). In the case of the classical diffusion, it corresponds to a 2-stable Le´vy process, which is Brownian motion. This is a gaussian process and its law is determined by the density solution of the diffusion equation. Likewise, for a fractional diffusion equation of order α, there is associated an α-stable Le´vy process, α ∈ (0, 2). This relation between diffusion equations and stable Le´vy processes is explained in more detail in [RW00, Ber] for the classical case and in [MS12, Section 4.5] for the frac- tional case. All these processes have in common the self-similarity of their trajectories, i.e., if (L (α) t )t≥0 is an α-stable Le´vy process then (L (α) t )t≥0 ∼ (εL(α)ε−αt)t≥0 41 Overview: anomalous diffusion in kinetic theory in law [Sat99]. The main difference between α = 2 and α ∈ (0, 2) is that Brownian motion is continuous almost everywhere, while the rest of stable processes are discontinuous; a particle makes a sequence of small jumps and from time to time it makes a large jump. This large jumps correspond to the idea that the particle is super-diffusing. (a) Brownian motion. (b) Le´vy process (jump points joined with a line). Figure 1.2: Source: by UserPAR, via Wikimedia Commons. Link with the Central Limit Theorem. At the very basis, the hydrodynamic limit is a manifestation of the Central Limit Theorem; given a sequence {X1, X2, . . .} of i.i.d ran- dom variables with expectation µ and variance σ2 <∞ then √ n ( 1 n n∑ k=0 Xk − µ ) d→ N (0, σ2) as n→∞ where d indicates convergence in distribution Note that: • the result is universal; it does not depend on the particular distribution of the ran- dom variables; they always converge to a Gaussian distribution; • it requires finite variance. Classical diffusive limits extend the idea of the Central Limit Theorem (or Generalised Central Limit Theorems in the case of fractional diffusions [MS12, Section 4.2]). The analogous idea in the case of stochastic processes is that, Brownian motion and Le´vy processes are obtained as limits of Random Walks (Donsker’s Theorem, [RW00]) and Continuous Time Random Walks [MS12, Section 4.4-4.5], respectively. 42 1.4. Anomalous transport: super-diffusions 1.4.3 Fractional derivatives, heavy-tailed functions and non-locality The explanation given here comes from reference [MS12]. Let us focus in the 1-dimensional case. A definition of fractional derivative in dimen- sion 1 is dα dxα f(x) = lim h→0 ∆αf(x) hα , α > 0 where ∆αf(x) := ∞∑ k=0 ( α k ) (−1)kf(x− kh) (1.25) and ( α k ) := Γ(α+ 1) k!Γ(α− k + 1) . Note that if α = n this corresponds to the classical derivatives where ∆nf(x) := n∑ k=0 ( n k ) (−1)kf(x− kh). The combinatorial number is generalised using the Gamma function Γ(n+ 1) = n!. Non-locality. To compute expression (1.25), the information over the entire space (x+δh for any δ ∈ N) is required. This implies that the fractional derivative (α 6= N) is non-local. Whereas the classical derivative is a local operator because the series defining ∆nf is actually just a finite sum. Heavy-tail functions. Expression (1.25) is a discrete convolution of f with the so called Grunwald weights, which have the following asymptotic property [MS12]: wj := (−1)k ( α k ) ∼ −α Γ(1− α)k −α−1 as k →∞. This is related to the appearance of the heavy-tail function. Moreover, for α ∈ (1, 2), wj > 0 for all j ≥ 2. For more details on this and other alternative definitions of the fractional derivative, check reference [MS12]. 1.4.3.1 Anomalous Fourier law We focus on the case when α ∈ (1, 2). Fourier law ~j = −D∇xρ(t, x) 43 Overview: anomalous diffusion in kinetic theory is replaced by the fractional Fourier law ~j = −D∇α−1ρ, α ∈ (1, 2) where∇α−1 = (∂α−11 , . . . ∂α−1N ) to give, combined with the conservation of the total mass (∂tρ = −∇ · ~j), the fractional diffusion equation ∂tρ = −D(−∆x)α/2ρ. What does this equation mean? Considering the Grunwald weights wj for α ∈ (1, 2) (wj > 0 for j ≥ 2), the discrete convolution in expression (1.25) means that particles are transported over the entire space in a heavy-tailed way (see figure 1.3). In contrast, in the normal diffusion particles are transported into a neighbourhood. Therefore, the fact that particles spread faster in fractional diffusion than in a normal one is a consequence of the non-locality of the operator (though, of course, not all non-local operators have this effect). Figure 1.3: With the fractional Fourier law, particles spread over the entire space in a heavy tail way (convolution). The figure is from reference [MS12]. For a discussion on the qualitative difference between α < 1 and α ∈ [1, 2), the reader is referred to [UZ99, Section 12.3]. 1.5 Methods in the diffusive limit 1.5.1 Toy example To explain the existing methods for the (fractional) diffusive limit, we will consider a simple case of the linear Boltzmann equation (1.7) in which σ(v, v′) = M(v), where M : RN → R has the following properties: M = M(v) > 0 a.e. in RN (1.26)ˆ RN M dv = 1 (1.27) L(M) = 0, (1.28) M(v) = M(−v) a.e. in RN . (1.29) 44 1.5. Methods in the diffusive limit M is the equilibrium distribution and we will consider either that it is Maxwellian (1.10) or a heavy-tail (1.11). Under these assumptions the linear Boltzmann equation (1.7) simplifies into: ∂tf(t, x, v) + v · ∇xf(t, x, v) + f(t, x, v) = M(v) ˆ RN f(t, x, v′) dv′ (1.30) f(0, ·) = f in (1.31) for (t, x, v) ∈ (0,∞)× RN × RN . 1.5.1.1 Diffusive limit We give next an example of fractional diffusion limit for this simpler case. Theorem 1.7 (Fractional diffusion limit for the linear Boltzmann equation, [MMM11]). Let M be a function fulfilling (1.11) with α ∈ (0, 2). Assume also that M satisfies (1.26), (1.27), (1.28), (1.29). Let f in ∈ L2(M−1dv) and let fε be the solution of the rescaled linear Boltzmann equation εα∂tfε + εv · ∇xfε = M ˆ RN fε dv − fε. (1.32) with initial data f(0, ·) = f in and for (t, x, v) ∈ (0,∞) × RN × RN . Then, when ε → 0, fε converges in L∞(0, T ;L2(RN × RN ))-weak to ρM with ρ = ρ(t, x) the unique solution to the fractional diffusion equation ∂tρ+ κ(−∆x)α/2ρ = 0 in (0,∞)× RN ρ(0, ·) = ˆ RN f in dv in RN for some constant κ > 0. In the following section we will present these methods on this toy example. We will start first by studying its classical counterpart: Theorem 1.8 (Classical diffusion limit). Let M be a function satisfying (1.26), (1.27), (1.28), (1.29) and also ˆ RN ‖v‖2M(v) dv <∞. Let f in ∈ L2(M−1dv) and let fε be the solution of the rescaled linear Boltzmann equation ε2∂tfε + εv · ∇xfε = M ˆ RN fε dv − fε. (1.33) with initial data f(0, ·) = f in for (t, x, v) ∈ (0,∞)×RN ×RN . Then, when ε→ 0, fε converges in L∞(0, T ;L2(RN × RN ))-weak to ρM with ρ = ρ(t, x) the unique solution to the diffusion 45 Overview: anomalous diffusion in kinetic theory equation ∂tρ−∇x · (D∇xρ) = 0 in (0,∞)× RN ρ(0, ·) = ˆ RN f(0, ·, v) dv in RN for D = ˆ RN M(v) v ⊗ v dv <∞. For some results on the classical diffusive limit for the linear Boltzmann equation see [AG13]. 1.5.1.2 A priori estimates To prove the diffusive limit we will need the following: Proposition 1.9 (A priori estimates). Consider the rescaled linear Boltzmann equation: εα∂tfε + εv · ∇xfε + fε −M ˆ RN fε dv = 0 in (0,∞)× RN × RN fε(0, ·) = f in in RN × RN . Then, we have the two following estimates sup t≥0 ˆ R2N (fε(t, ·))2 M dvdx ≤ ˆ R2N (f in)2 M dvdx = ‖f in‖2L2(M−1) (1.34) and ˆ ∞ 0 ˆ R2N [fε − ρεM ]2M−1 dvdxdt ≤ ε α 2 ‖f in‖2L2(M−1) . (1.35) Also, ρε(t, x), as well as L(fε), are well defined a.e., and sup t≥0 ˆ RN ρε(t, ·)2 dx ≤ ‖f in‖2L2(M−1) . (1.36) Proof. We start proving first the following Lemma 1.10. εα d dt ˆ R2N (fε) 2 2 M−1dvdx = − ˆ R2N [fε − ρεM ]2M−1dvdx . 46 1.5. Methods in the diffusive limit Proof of Lemma 1.10. εα d dt ˆ R2N (fε) 2 2 M−1dvdx (A.1) = ˆ R2N (ρεM − fε)fεM−1 dvdx (A.2) = ˆ R2N [(ρε) 2M − (fε)2M−1] dvdx (A.3) = − ˆ R2N [fε − ρεM ]2M−1dvdx . Equality (A.1) is justified by the following: εα d dt ˆ R2N (fε) 2 2 M−1dvdx = 2εα ˆ R2N fε (∂tfε)M −1 dvdx = −2 ˆ R2N ε(v · ∇xfε)fεM−1 dvdx + ˆ R2N (ρεM − fε)fεM−1 dvdx and 2 ˆ R2N (v · ∇xfε)fεM−1 dvdx = ˆ R2N v · ∇xf2ε M−1 dvdx = 0 where the last equality is due to the divergence theorem. Equality (A.2) is proven by rewriting (ρεM − fε)fεM−1 = ρεfε − (fε)2M−1 and computing ˆ R2N ρεfε dvdx = ˆ RN ρε (ˆ RN fε dv ) dx = ˆ RN (ρε) 2 dx = ˆ RN (ρε) 2 dx ˆ RN M dv = ˆ R2N (ρε) 2M dxdv . Finally, we prove equality (A.3). Note a := ρε, b := M , c := fε, then (ρε) 2M − (fε)2M−1 = a2b− c2b−1 (fε − ρεM)2M−1 = (c− ab)2b−1 = −c2b−1 + a2b− 2a2b+ 2ac . We just need to check that ˆ R2N (−2a2b+ 2ac) dxdv = 0, which is true: ˆ R2N (ρε) 2M dxdv − ˆ R2N ρεfε = ˆ RN (ρε) 2 (ˆ RN M dv ) ︸ ︷︷ ︸ =1 dx− ˆ RN ρε (ˆ RN fε dv ) ︸ ︷︷ ︸ =ρε dx = 0 . 47 Overview: anomalous diffusion in kinetic theory By Lemma 1.10 we know that ˆ R2N (fε) 2 2 M−1dvdx is a decreasing function on the time variable, hence we obtain the first estimate (1.34). The second estimate (1.35) is obtained by integrating in time the expression in Lemma 1.10: εα ˆ τ 0 d dt ˆ R2N (fε) 2 2 M−1dvdxdt = εα 2 (ˆ R2N fε(τ, ·)2M−1dvdx− ˆ R2N fε(0, ·)2M−1dvdx ) = εα 2 ( ‖fε(τ, ·)‖2L2(M−1) − ‖f inε ‖2L2(M−1) ) ≤ ε α 2 ‖fε(τ, ·)‖2L2(M−1) ≤ εα 2 ‖f inε ‖2L2(M−1) . Now, by the Cauchy-Schwarz inequality, we obtain estimate (1.36): ρε(t, x) = ˆ RN fε M1/2 M1/2 dv ≤ ( (fε) 2 M dv )1/2 , so that ρε(t, x), as well as L(fε), are well defined a.e., and sup t≥0 ˆ RN ρε(t, ·)2 dx ≤ ‖f in‖2L2(M−1) . 1.5.2 Hilbert expansion (classical diffusion) In this part we study the classical diffusion limit. This requires that the linear Boltzmann equation has an equilibrium distribution with finite second moment. We will use the Hilbert expansion, which is a common technique in multiscale analysis. This proof is an adaptation and combines the ones in [AG13] and [MMM11]. Examples of diffusion limits for some non linear collision operators can be found in [GM03], [MLT10], [Mel02]. 1.5.2.1 Classical diffusion limit In this section we will prove theorem 1.8. The idea of the proof is based on an ‘approximation’ (ansatz) of the solution fε of the form Fε = f0 + εf1 + ε 2f2 with ˆ RN Fε dv = ρ for all ε, where ρ will satisfy the diffusion equation (1.1). 48 1.5. Methods in the diffusive limit Specifically, we define the approximation to the solution fε as follows, Fε(t, x, v) = f0(t, x, v) + εf1(t, x, v) + ε 2f2(t, x, v) where f0 := M ρ (1.37) f1 := −v · ∇xf0 = −M v · ∇xρ (1.38) f2 := −∂tf0 − v · ∇xf1 = −M (∂tρ− v · ∇x(v · ρ)) (1.39) where ρ is the solution of the diffusion equation (1.1) with initial condition ρin = ´ RN f in dv. The construction of Fε will be explained in Section 1.5.2.2. Observe the following properties f0 −M ˆ RN f0 dv = 0 (1.40) ˆ RN f1 dv = 0 (1.41) ˆ RN f2 dv = 0 (1.42) (1.41)-(1.42) imply that ˆ RN Fε dv = ρ for any ε > 0 . (1.43) The properties (1.40)-(1.41) are readily proven, property (1.42) requires the diffusion ma- trix D := ˆ RN M(v) v ⊗ v dv to be well defined. For that, it is necessary and sufficient that the second moment of M to be bounded (as we will see). To prove Theorem 1.8 we are left to check the Proposition 1.11. It holds that lim ε→0 ˆ RN fε dv = lim ε→0 ˆ RN Fε dv = ρ. Proof of Proposition 1.11. We substitute the function Fε in the rescaled Boltzmann equation (1.33) and denote QFε := (ε2∂t + εv · ∇x + Id−K)Fε (1.44) where Id is the identity operator and KFε = M ´ RN Fε dv. Observe that the operator Q is 49 Overview: anomalous diffusion in kinetic theory linear. We compute Q f0 = ε2∂tf0 + εv · ∇xf0 + f0 −M ˆ RN f0 dv (1.40)-(1.38) = ε2∂tf0 − εf1 , Q εf1 = ε2∂tεf1 + εv · ∇xεf1 + εf1 − F ˆ RN εf1 dv (1.41)-(1.39) = ε3∂tf1 + ε 2(−f2 − ∂tf0) + εf1 , Q ε2f2 = ε2∂tε2f2 + εv · ε2∇xf2 + ε2f2 − F ˆ RN ε2f2 dv (1.42) = ε4∂tf2 + ε 3v · ∇xf2 + ε2f2 . Since Q is a linear operator, QFε = Q (f0 + εf1 + ε2f2) = ε3∂tf1 + ε4∂tf2 + ε3v · ∇xf2 . Therefore Rε := Fε − fε satisfies QRε = Q (Fε − fε) = ε3∂tf1 + ε4∂tf2 + ε3v · ∇xf2 . (1.45) At this stage, sometimes the convergence of solutions can be proven using estimate re- sults on the Boltzmann equation. However, we do not use this technique here, see [AG13] for more details. It holds that ˆ RN Rε dv = ˆ RN Fε − fε dv = ρε − ˆ RN fε(v)dv . Integrating the equation on Rε (1.44) w.r.t v, we obtain ˆ RN QRε dv = ε2∂t ˆ RN Rε dv + ε ˆ RN v · ∇xRε dv using that ´ RN M(v ′) ´ RN Rε dv dv ′ = ´ RN Rε dv ´ RN M(v ′) dv′ = ´ RN Rε dv. The factor ´ RN v ·∇xRε dv prevents to go further in the study of the limit of ´ RN Rε dv. In order to overcome this obstacle, we consider the Laplace-Fourier transform in t and x defined as follows, ĝ(p, k, v) := ˆ RN ˆ ∞ 0 e−pte−ik·xg(t, x, v)dtdx, g ∈ L∞(0,∞)× L1(RNx ), p > 0, k ∈ RN (1.46) and we apply this transformation to the equation on Rε (1.45). The function R̂ε satisfies ε2p R̂ε−ε2R̂inε +εi(v·k)R̂ε+R̂ε−M ˆ RN R̂ε dv = ε 3pf̂1 − ε3f̂ in1 + ε4pf̂2 − ε4f̂ in2 + ε3i(v · k)f2︸ ︷︷ ︸ =:bε(p,k,v) where gin = g(t = 0, ·) and the ̂ symbol means Fourier Transform (and not Laplace- 50 1.5. Methods in the diffusive limit Fourier Transform) in the functions with the in label. Isolate R̂ε R̂ε = M ´ RN R̂ε dv + ε 2R̂inε + b ε(t, x, v) ε2p+ εi(v · k) + 1 . Integrate w.r.t the variable v over RN , and using ´ RN M dv = 1( 1 ε2 ˆ RN ( 1 ε2p+ εi(v · k) + 1) − 1 ) M(v)dv ) ︸ ︷︷ ︸ =:aε(p,k,v) Ĝε + I ε(p, k, v) = 0 where Iε(p, k, v) := ˆ RN ( R̂inε + εpf̂1 − εf̂ in1 + ε2pf̂2 − ε2f̂ in2 + εi(v · k)f̂2 ε2p+ εi(v · k) + 1 ) dv . Finally, we compute the limit of Ĝε := ρ̂− ´ RN f̂ε(v)dv. From the previous step, we have aε(p, k, v)Ĝε = −Iε(p, k, v) . Lemma 1.12 (Laplace symbol). With the previous notations, aε(p, k, v) −→ ε→0 −p− s|k|2 for some s > 0. (1.47) (cf. proof in Section 1.5.3.1.) Lemma 1.13. With the previous notations, Iε(p, k, v) −→ ε→0 ρ̂in − ˆ RN f̂ in dv . Proof of Lemma 1.13. By definition, Rinε = F inε − f in = f in0 + εf in1 + ε2f in2 − f in. Hence, we can write Iε(p, k, v) = ˆ RN f̂ in0 − f̂ in + εpf̂1 + ε2pf̂2 + εi(v · k)f̂2 + εf in1 + ε2f in2 ε2p+ εi(v · k) + 1 dv . (1.48) By Lebesgue dominated convergence theorem: ˆ RN f̂ in0 ε2p+ εi(v · k) + 1 dv = ρ̂ in ˆ RN M(v) ε2p+ εi(v · k) + 1 dv −→ ρ̂ in ˆ RN M(v) dv = ρ̂in . 51 Overview: anomalous diffusion in kinetic theory Using that if a function g ∈ L2(RN ), then its Laplace-Fourier Transform ĝ ∈ L2(RN ) by Parseval equality, we can apply again Lebesgue dominated convergence theorem (since f in ∈ L2x(RN )× L2(M−1,RN ) by hypothesis). Therefore, we obtain ˆ RN f̂ in ε2p+ εi(v · k) + 1 dv −→ ˆ RN f̂ in dv . The rest of the terms in the integral (1.48) goes to zero as ε → 0 applying also the Lebesgue dominated convergence theorem. We can apply it with the next observation; f1, f2, (v · k)f2 ∈ L2(RN ), therefore f̂1, f̂2, (v · k)f̂2 ∈ L2(RN ). Considering Lemma (1.13) and that we chose the initial condition ρ(0, ·) to be ρin = ˆ RN f in(v) dv , we have that Iε(p, k, v) −→ ε→0 0 . Also, it will be proven that Lemma 1.14. The limit Ĝ0 := lim ε→0 Ĝε is well defined. (cf. proof of Lemma 1.18). Then, letting ε→ 0 in equation (1.47), we obtain (−p− s|k|2)Ĝ0 = 0 . This equation holds for all p > 0 and k ∈ RN , therefore 0 ≡ Ĝ0 = lim ε→0 Ĝε = ρ̂− lim ε→0 ˆ RN f̂ε(v) dv. Since the Laplace-Fourier Transform is a one-to-one map, we deduce that ρ = lim ε→0 ´ RN fε dv a.e.. 1.5.2.2 Construction of the ansatz In this section we explain how Fε is built. The key concept is that of Hilbert expansion, which is used to construct an ansatz, i.e., we suppose a priori that the solution of the equation has a particular shape, called the ansatz. Afterwards, the ansatz is substituted in the equation to study its properties. 52 1.5. Methods in the diffusive limit Consider a solution fε(t, x, v) to the rescaled linear Boltzmann equation (1.33) written as a Hilbert expansion of powers of ε fε(t, x, v) = ∑ n≥0 εnfn(t, x, v) (1.49) where the functions fn(t, x, v) are at least differentiable w.r.t t, twice differentiable w.r.t x and integrable w.r.t v. Observe that the Hilbert formal series do not converge in general for any value of ε > 0 and the functions fn are not necessarily non-negative (remember that the solution of the linear Boltzmann equation, to have physical coherence, must be non-negative). Formally, we substitute the Hilbert expansion (1.49) in the rescaled linear Boltzmann equation (1.33) and develop 0 = ε2∂tf0 + εv · ∇xf0 + f0 −M(v) ˆ RN f0 dv + ε3∂tf1 + ε 2v · ∇xf1 + εf1 − εM(v) ˆ RN f1 dv + ε4∂tf2 + ε 3v · ∇xf2 + ε2f2 − ε2M(v) ˆ RN f2 dv + ε5∂tf3 + ε 4v · ∇xf3 + ε3f3 − ε3M(v) ˆ RN f3 dv + . . . . For each degree on ε an equation is obtained. Degree 0 f0 −M(v)ρ0 = 0 =⇒ (Id−K)f0 = 0 . Degree 1 v · ∇xf0 + f1 −M(v)ρ1 = 0 =⇒ (Id−K)f1 = −v · ∇xf0 . Degree 2 ∂tf0 + v · ∇xf1 + f2 −M(v)ρ2 = 0 =⇒ (Id−K)f2 = −∂tf0 − v · ∇xf1 . Degree n > 1 In general, we will have the equation (Id−K)fn = −∂tfn−2 − v · ∇xfn−1 . We want to solve this system of equations: 53 Overview: anomalous diffusion in kinetic theory Solving the degree 0. (Id−K)f0 = 0 =⇒ f0 ∈ Ker(Id−K) . Therefore f0(t, x, v) = M(v)ρ0(t, x) . Solving the degree 1. (Id−K)f1 = −v · ∇xf0 . Now, we study the existence of solutions of this integral equation. Observe that the operator K is self-adjoint in L2(M−1): consider two functions f, g ∈ L2(M−1) and write ˆ RN g(v)K(f)(v)M−1(v) dv = ˆ RN g(v)M(v) (ˆ RN f(v′)dv′ ) M−1(v) dv = ˆ RN f dv ˆ RN g dv . In the same manner ˆ RN fK(g)M−1 dv = ˆ RN g dv ˆ RN f dv , which shows the self-adjointness. K is a compact operator in L2(M−1); consider a se- quence (ψn)n∈N bounded in L2(M−1), in particular, (ψn)n∈N is bounded in L1; therefore, ( ´ RN ψn dv)n∈N is a bounded sequence that does not depend on v; therefore, we can find a convergent subsequence in L2(M−1). We deduce that K is a compact operator since Kψn = M(v) ´ RN ψn dv. Since K is an operator self-adjoint and compact, we have that Ker(Id−K)⊥ = Im(Id−K) and Ker(Id−K) = { φ ∈ L2(M−1) s.t φ(t, x, v) = M(v) ˆ RN φ(v′)dv′ = M(v)ρφ(t, x) } . Therefore, Ker(Id−K)⊥ = { ψ ∈ L2(M−1) s.t ˆ RN ψ(v)φ(v)M−1(v) dv = 0 ∀φ ∈ Ker(Id−K) } = { ψ ∈ L2(M−1) s.t ρφ ˆ RN ψ(v) dv = 0 ∀φ ∈ Ker(Id−K) } = {ψ ∈ L2(M−1) s.t ˆ RN ψ dv = 0} . For the integral equation (Id−K)φ = ψ to have solution, it is required that ψ ∈ Im(Id−K) = Ker(Id−K)⊥, i.e., ˆ RN ψ dv = 0 . 54 1.5. Methods in the diffusive limit If this condition is fulfilled, then{ solutions φ ∈ L2(M−1) of (Id−K)φ = ψ, for ψ ∈ L2(M−1), ˆ RN ψ dv = 0 } = ψ+Ker(Id−K) (since we have (Id−K)ψ = ψ when ´RN ψ dv=0). Let us consider again, the equation on f1 (Id−K)f1 = −v · ∇xf0 , we have seen that, for this equation to have solution, we require ˆ RN −v · ∇xf0 dv = 0 which is true because ˆ RN −v · ∇xf0 dv = ˆ RN −M(v) v · ∇xρ0(t, x) dv and M is even. Therefore, the set of possible solutions f1 are f1 = −v · ∇xf0 + Ker(Id−K) = −M(v) v · ∇xρ0(t, x) + Ker(Id−K) . We consider f1 = −v · ∇xf0 = −M(v) v · ∇xρ0(t, x) , note that we already take the element in Ker(Id−K) to be zero, because this term will be killed in equation (1.50) by symmetry. Solving the degree 2 (Id−K)f2 = −∂tf0 − v · ∇xf1 . In the same way as before, for this integral equation to have solution, we require ˆ RN (−∂tf0 − v · ∇xf1) dv = 0 (1.50) this is equivalent to the condition ∂tρ0 − ˆ RN M(v) (v · ∇x(v · ∇xρ0)) dv = 0 . (1.51) This condition is a diffusion equation as we see next. If this condition (1.51) is satisfied, then f2(t, x, v) = M(v) (−∂tρ0 + v · ∇x(v · ∇xρ0)) + Ker(Id−K) . 55 Overview: anomalous diffusion in kinetic theory As in the previous case, to have ´ RN f2 dv = 0, we will need to choose the solution f2(t, x, v) = M(v) (−∂tρ0 + v · ∇x(v · ∇xρ0)) . Let us study the integral ˆ RN M(v) (v · ∇x(v · ∇xρ)) dv = ∇x · [(ˆ RN M(v) v ⊗ v dv ) ∇xρ ] . (1.52) Define the matrix D := ˆ RN M(v) v ⊗ v dv , then the condition ˆ RN f2 dv = 0 ⇐⇒ ∂tρ0 −∇x · (D∇xρ0)) = 0 is the diffusion equation. However, we need to check that the diffusion matrix D is well defined. Lemma 1.15. The diffusion matrix D is well defined if and only if the second moment order of M is finite. Proof. If the matrix D is well defined, then the second moment order of M is finite since it corresponds to the trace of D. If ˆ RN ‖v‖2M(v) dv <∞ then the matrix is well defined because, for each component of the matrix, vivj ≤ v 2 i 2 + v2j 2 ≤ ‖v‖ 2 2 + ‖v‖2 2 = ‖v‖2 (where the first inequality is Young’s inequality). Therefore, the second moment order of M bounds from above all the components of D and therefore it is well defined. 1.5.2.3 The fractional Hilbert expansion We conclude this section by mentioning that there exists a version of the Hilbert expan- sion method that allows to derive the fractional diffusion equation from the linear Boltz- mann equation. This has been done in reference [AMP10] and the reader is referred to it for more information. 56 1.5. Methods in the diffusive limit 1.5.3 Laplace-Fourier Transform Theorem 1.8 requires the second moment order of M to be finite. What happens when the second moment of M is not finite? As we saw in Section 1.2.3.2, the multiscale analysis gives a fractional diffusion equa- tion in the limit and has a fundamental change compared with the classical diffusion limit: we need a different time rescaling t′ := εαt, α ∈ (0, 2). Here we will proof Theorem 1.7 for α < 2 along with the classical diffusion (Theorem 1.8) for M having bounded second moment with the rescale in time t′ = ε2t. We will assume in the classical diffusion case thatM is rotationally invariant to simplify the proof. By performing the Laplace-Fourier Transform in space and time (1.46) on the linear Boltzmann equation, we will find the equation satisfied by ρ̂ε = ´ RN f̂ε dv (where the hat indicates the Laplace-Fourier Transform) and we will take the limit on this equation. We apply the Laplace-Fourier Transform to equation (1.30) εαpf̂ε − εαf̂ in + εi(v · k)f̂ε + f̂ε −M ˆ RN fε dv = 0 where f̂ in denotes the Fourier Transform in space (and not the Laplace-Fourier Transform in time and space as for the other terms). We isolate f̂ε f̂ε = M 1 + εαp+ εiv · k ρ̂ε + εαf̂ in 1 + εαp+ εiv · k and we integrate w.r.t v: ρ̂ε = (ˆ RN M 1 + εαp+ εiv · k dv ) ρ̂ε + (ˆ RN εαf̂ in 1 + εαp+ εiv · k dv ) . This last equation can be rewritten (because ´ RN M dv = 1) as aε(p, k)ρ̂ε + ˆ RN f̂ in 1 + εαp+ εiv · k dv = 0 (1.53) where aε(p, k) := 1 εα ˆ RN ( 1 1 + εαp+ εiv · k − 1 ) M(v) dv . We now compute the limit of each term in the previous equation when ε→ 0. Proposition 1.16. It holds that ˆ RN f̂ in 1 + εαp+ εiv · k dv → ˆ RN f̂ in dv 57 Overview: anomalous diffusion in kinetic theory for f in ∈ L2(M−1). Proof. The assumption f in ∈ L2(M−1) implies in particular that f in ∈ L2x(L1v). Hence, its Fourier Transform f̂ in also belongs to L2k(L 1 v) by Parseval equality, which means that f̂ in is integrable in v for almost all k. This allows to apply the Lebesgue dominated convergence theorem, which yields, for almost every k, ˆ RN f̂ in 1 + εαp+ εiv · k dv −→ ˆ RN f̂ in = ρ̂in . Proposition 1.17 (The fractional symbol). If α ∈ (0, 2), then aε(p, k)→ −p− s|k|α with s ∈ (0,∞) given by s = ˆ RN w21 1 + w21 s0 |w|N+α dw (w1 indicates the first coordinate of the vector w). Furthermore, aε(p, k) satisfies |aε(p, k)| ≤ |p|+ s|k|α . Proof. We postpone the proof of this crucial step to Section 1.5.3.1. Lemma 1.18. The limit ρ̂0 := lim ε→0 ρ̂ε is well defined. Before going into the proof of Lemma 1.18, let us explain how it allows to conclude that letting ε→ 0, we obtain ρ̂in + (−p− s|k|α)ρ̂0 = 0 for a.e p > 0, k ∈ RN . This equation is also satisfied by the unique solution of the fractional diffusion equation, so that ρ̂ = ρ̂0, and then ρ = ρ0 because the Laplace-Fourier Transform is a one-to-one mapping (say in S ′([0,∞) × RN ), the dual of the Schwartz space). Moreover, we have that fε → ρM weakly in L∞(0, T ;L2(RN × RN )) by the a priori estimates 1.9. We prove now the previous lemma: Proof of Lemma 1.18. From the bound sup t≥0 ˆ RN ρε(t, ·)2 dx ≤ ‖f in‖2L2(M−1) (1.54) established in 1.9, and up to the extraction of a subsequence, we know that there exists η ∈ L∞(0,∞;L2(RN )) such that ρε ⇀ η weakly in L∞(0,∞;L2(RN )). On one hand, from 58 1.5. Methods in the diffusive limit the precedent estimate, we have ρ̂ε is bounded in L∞(a,∞;L2(RN )) for any a > 0. On the other hand, consider the space of test functions D and the space of Schwartz S. Then, for any φ = φ(t) ∈ D(0,∞) its Laplace transform Lφ belongs to L1(0,∞) and for any ψ = ψ(x) ∈ S(RN ) its Fourier transform Fψ belongs to S(RN ) so that ˆ RN ρ̂ε (φ⊗ ψ) dv = ˆ RN ρε (Lφ⊗Fψ) dv → ˆ RN η (Lφ⊗Fψ) dv = ˆ RN η̂ (φ⊗ ψ) dv as ε→ 0. We deduce that ρ̂ε → η̂ weakly in L∞(a,∞;L2(RN )) for any a > 0. 1.5.3.1 The fractional symbol In this proof we see how the fractional symbol appears: Proof of Proposition 1.17. Remember that aε(p, k) := 1 εα ˆ RN ( 1 1 + εαp+ εiv · k − 1 ) M(v) dv where M(−v) = M(v) > 0, ˆ RN M dv = 1 α ∈ (0, 2] . We split aε into a sum of three terms: aε(p, k) = − 1 εα ˆ RN εαp+ εiv · k 1 + εαp+ εiv · kM(v) dv (1) = −p ˆ RN 1 + εαp (1 + εαp)2 + ε2(v · k)2M(v) dv︸ ︷︷ ︸ =:aε1(p,k) − 1 εα ˆ RN εiv · k (1 + εαp)2 + ε2(v · k)2M(v) dv︸ ︷︷ ︸ =:aε2(p,k) − 1 εα ˆ RN ε2(v · k)2 (1 + εαp)2 + ε2(v · k)2M(v) dv︸ ︷︷ ︸ =:aε3(p,k) . In (1), we multiply numerator and denominator by the conjugate of the denominator. Now, we look at each term: • the integrand of aε1(p, k) is bounded by pF uniformly on ε and the dominated con- vergence implies aε1(p, k) −→ ε→0 −p ˆ RN M(v) dv . • The integrand of aε2(p, k) is an odd function on v, therefore aε2(p, k) ≡ 0. 59 Overview: anomalous diffusion in kinetic theory • To study the limit of aε3(p, k) we will consider two cases: (i) α = 2 and the second moment order of M is bounded. aε3(p, k) = ˆ RN (v · k)2 (1 + ε2p)2 + ε2(v · k)2M(v) dv is uniformly bounded on ε by the maximum of |k|2F and |v|2M . By dominated convergence, aε3(p, k) −→ ε→0 ˆ RN (v · k)2M(v) dv . In particular, if we also assume that M is rotational invariant, we have that (v · k)2 = N∑ i v2i k 2 i + 2 N∑ i 6=j vikivjkj and then ˆ RN (v · k)2M(v) dv = N∑ i k2i ˆ RN v21M(v) dv + 2 N∑ i 6=j kikj ˆ RN v1v2M(v) dv . The last integral is zero applying Fubini and that F is an even function. There- fore the limit of aε3(p, k) is |k|2s with s = ˆ RN v21M(v) dv . (ii) α ∈ (0, 2) and the second moment order of M is unbounded. We consider that M is of the form indicated in (1.11). Lemma 1.19. Suppose α ∈ (0, 2), then dε := aε3 fulfills, for any p > 0, k ∈ RN |dε(p, k)| ≤ s|k|α, dε(p, k) −→ ε→0 s|k|α (1.55) with s ∈ (0,∞) given by s = ˆ RN w21 1 + w21 c0 |w|N+αdw . Proof of Lemma 1.19. The inequality (1.55) follows from estimate (1.34) in Proposition 1.9 0 ≤ dε(p, k) ≤ ˆ RN ε2−α(v · k)2 1 + ε2(v · k)2 c0 |v|N+αdv = s|k| α where the last equality is obtained by making the change of variables w := ε|k|v. We split 60 1.5. Methods in the diffusive limit dε between small and large velocities in the following way: dε(p, k) = dε1(p, k) + d ε 2(p, k) with dε1(p, k) = ˆ |v|≤1 ε2−α(v · k)2 (1 + εαp)2 + ε2(v · k)2M(v)dv ≤ ε2−α ˆ |v|≤1 (v · k)2M(v)dv ≤ ε2−α|k|2 ˆ |v|≤1 F (v)dv −→ 0 and dε2(p, k) = ˆ |v|≥1 ε2−α(v · k)2 (1 + εαp)2 + ε2(v · k)2 c0 |v|N+αdv = |k|α ˆ |w|≥ε|k| w21 (1 + εαp)2 + w21 c0 |w|N+αdw −→ s|k| α, where we use again the change of variables w := ε|k|v and the dominated convergence theorem. 1.5.4 Mellet’s moments methods In this Section we will prove theorem 1.7 using Mellet’s moments methods. We start by explaining how this method works. 1.5.4.1 The idea behind the method: weak formulation Consider equation (1.30) in distribution sense, i.e., for all distribution functions χ ∈ D((0,∞)× RN × RN ) we make the L2(M−1) product of the equation with χ: ˆ ∞ 0 ˆ R2N ∂tfεχM −1dxdvdt (1.56) = ε−α ˆ ∞ 0 ˆ R2N [ (ρεM − fε)χM−1 + fε εv · ∇xχM−1 ] dxdvdt. (1.57) Now the a priori estimates (1.34) imply that fε ⇀ ρ(t, x)M(v) in L∞((0,∞);L2(RN × RN ;M−1dv))−weak∗ 61 Overview: anomalous diffusion in kinetic theory and strong in L1((0, T ), L2(RN × RN ,M−1dv)). We can write fε as fε(t, x, v) = ρε(t, x)M(v) + gε(t, x, v) with ρε = ´ RN fε dv and where gε is the remainder. This remainder gives the behaviour of ρε in the hydrodynamic limit (fluctuations of fε). However, (i) the specific shape for gε is unknown as well as some properties like rotationally invariance; the velocities are not decoupled from time and space (like in the expres- sion ρ(t, x)M(v)). (ii) We do not have enough control on gε. We know that ‖gε‖L2((0,∞)×RN×RN ,M−1dv) ≤ Cεα/2 by estimate (1.35) which is insufficient to balance the factor ε−α in (1.57). Since the information that we have on gε is not enough, the idea of the moments method is to avoid computing directly on gε. For that, we consider a test function χ¯ε of the shape χ¯ε(t, x, v) = ϕ(t, x)M(v) + εψ(t, x, v) where the first term ϕ ∈ D((0,∞) × RN ) is a test function. Observe that ϕ(t, x)M(v) has the same shape as the limit ρ(t, x)M(v). ψ will help to ‘control’ the remainder term gε. Plugging-in χ¯ε in equation (1.56), we obtain ˆ ∞ 0 ˆ R2N ∂tfεχ¯εM −1dxdvdt (1.58) = ε−α ˆ ∞ 0 ˆ R2N [ (ρεM − fε) (ϕM + εψ)M−1 + fεεv · ∇xχ¯εM−1 ] dxdvdt (1.59) = ε−α ˆ ∞ 0 ˆ R2N [ (ρεM − fε) εψM−1 + fεεv · ∇xχ¯εM−1 ] dxdvdt. (1.60) We want to get rid of the term ˆ ∞ 0 ˆ R2N (−fε(εψ) + fεεv · ∇xχ¯ε)M−1dxdvdt (1.61) which can be achieved imposing −εψ + εv · ∇xχ¯ε = 0 which is equivalent to χ¯ε − εv · ∇xχ¯ε = ϕM(v) 62 1.5. Methods in the diffusive limit for which we know an explicit solution χ¯ε(t, x, v) = M(v) ˆ ∞ 0 e−zϕ(t, x+ εvz)dz. Hence, the equation becomes ˆ ∞ 0 ˆ R2N ∂tfεχ¯εM −1dxdvdt = ε−α ˆ ∞ 0 ˆ R2N (ρεM) εψM −1dxdvdt (1.62) = ε−α ˆ ∞ 0 ˆ R2N (ρεM) (χ¯εM −1 − ϕ)dxdvdt (1.63) This way we have gotten rid of gε. ρM is the limit of fε and, on the other hand, the balance between ε−α exploding in the limit and χ¯ε − ϕM going to zero in the limit will give the fractional symbol. Finally, we write χε = χ¯εM−1, χε satisfies the equation χε − εv · ∇xχε = ϕ whose solution is χε(t, x, v) = ˆ ∞ 0 e−zϕ(t, x+ εvz) dz. (1.64) By integration by parts twice we obtain χε − ϕ = ˆ ∞ 0 e−z (ϕ(t, x+ εvz)− ϕ(t, x)) dz = ε ˆ ∞ 0 e−zv · ∇xϕ(t, x+ εvz) dz = εv · ∇xϕ(t, x) + ε2 ˆ ∞ 0 e−zvTD2ϕ(t, x+ εvz) v dz (1.65) where T indicates transpose. We have that |χε − ϕ| ≤ ‖Dϕ‖L∞ε‖v‖ but we will need stronger convergence results: Lemma 1.20 (Convergence properties, [Mel10]). It holds that for any ϕ ∈ D(RN × [0,∞)) and χε defined as in (1.64) that ˆ (χε − ϕ) dv → 0 strongly in L2((0,∞)× RN ), ˆ (∂tχε − ∂tϕ) dv → 0 strongly in L2((0,∞)× RN ). 63 Overview: anomalous diffusion in kinetic theory The proof for this lemma can be found in [Mel10] and analogous ones will be proven in Chapter 2. 1.5.4.2 The fractional symbol We need to compute the limit in: ˆ ∞ 0 ˆ R2N ∂tfεχε dxdvdt = ε −α ˆ ∞ 0 ˆ R2N (ρεM) (χε − ϕ) dxdvdt (1.66) for χε − εv · ∇xχε = ϕ. On one hand we have that ˆ ∞ 0 ˆ R2N ∂tfεχεdxdvdt→ ˆ ∞ 0 ˆ R ∂tρϕ dxdt using Lemma 1.20 and the convergence properties of fε. On the other hand we have the following: Proposition 1.21 (Fractional symbol). It holds that ε−α ˆ ∞ 0 ˆ R2N (ρεM) (χε − ϕ)dxdvdt→ ˆ ∞ 0 ˆ RN ρ(−∆x)α/2ϕdxdt as ε→ 0. To prove this theorem we will use here the definition of the fractional Laplacian based on the singular integral (1.6). Proof of Proposition 1.21. We will focus on the integral in the velocities and split it into large and small velocities. We will need to prove the two following Lemmas: Lemma 1.22 (Large velocities). It holds that ε−α ˆ ‖v‖≥1 M(χε − ϕ)dv = c0 ˆ ∞ 0 e−zzα ˆ ‖w‖≥εz ϕ(t, x+ εvz)− ϕ(t, x) ‖w‖N+α dzdw. Lemma 1.23 (Small velocities). The following estimate holds:∣∣∣∣∣ε−α ˆ ‖v‖≤1 M(χε − ϕ)dv ∣∣∣∣∣ ≤ Cε2−α for some C > 0. To conclude the proof we will need to prove: 64 1.5. Methods in the diffusive limit Lemma 1.24 (Strong convergence of the fractional symbol). ˆ ∞ 0 e−zzα ˆ ‖w‖≥εz ϕ(t, x+ εvz)− ϕ(t, x) ‖w‖N+α dzdw → −κ(−∆x) α/2ϕ strongly in L2((0,∞)× RN ) for some κ > 0. Proof of Lemma 1.23. The statement is consequence of the expansion (1.65) (using the par- ity of v for the term of order ε1−α). Proof of Lemma 1.22. We compute on the integral to obtain (remember the shape of M in (1.11)) ε−α ˆ ‖v‖≥1 M(χε − ϕ) dv = ε−α ˆ ‖v‖≥1 M ˆ ∞ 0 e−z (ϕ(t, x+ εvz)− ϕ(t, x)) dzdv = c0ε −α ˆ ‖v‖≥1 ˆ ∞ 0 e−z ϕ(t, x+ εvz)− ϕ(t, x) ‖v‖N+α dzdv = c0ε −α ˆ ‖w‖≥εz ˆ ∞ 0 e−z|εz|N+αϕ(t, x+ w)− ϕ(t, x)‖w‖N+α dw |εz|N dz = c0 ˆ ∞ 0 e−zzα ˆ ‖w‖≥εz ϕ(t, x+ w)− ϕ(t, x) ‖w‖N+α dwdz doing the change of variables w = εvz. Proof of Lemma 1.24. The proof can be found in [Mel10] and also in this document in Chapter 2, Lemma 2.13. The combination of these three lemmas allows to conclude the statement using the weak convergence of ρε in L∞((0,∞);L2(RN )). 1.5.5 Probabilistic approach The solution to the linear Boltzmann equation is a probability density f = f(t, x, v), hence the probabilistic formulation of the equation in terms of the dynamics of a stochastic particle will provide insight on the equation. In this section we explain the dynamics of a stochastic particle (Xt, Vt)t≥0 under the law given by f . [AG13] gives a hint to what is explained here for a modified equation. Our goal is to model the dynamics of a single particle given by its position and ve- locity (X(t), V (t)) ∈ RN × RN over time t ∈ [0,∞). 65 Overview: anomalous diffusion in kinetic theory Suppose that we are given some random initial data (X0, V0) ∈ RN × RN whose dis- tribution has density function f(0, x, v) (X0, V0) ∼ f(0, x, v) . Consider now U1, U2, . . . i.i.d random variables with density function M and T1, T2, . . . also i.i.d with exponential distribution of parameter 1, E(1). We assume Ti, Ui to be pairwise independent. Modeling assumptions. In our model, the particle travels at a constant velocity and after an exponential time, the velocity ‘jumps’ to a new one with distribution given by the density functionM (note that, therefore, we are assuming that the velocity after the jump is independent from the velocity before the jump). With these considerations in mind, the exponential random variables Tn give the lapse of time between two consecutive jumps in the velocity and Un is the velocity of the particle after n jumps. 1.5.5.1 Derivation of the linear Boltzmann equation Consider the single particle model described in the previous section, our goal is to find an equation satisfied by the law of (Xt, Vt). Steps: • One finds the equations for (Xt, Vt). • One observes that (Xt, Vt)t≥0 is a Markov process in RN ×RN and finds its infinites- imal generator. • One identifies a martingale from the Markov process and using the martingale, finds an equation for the law of the process (Xt, Vt)t≥0. Characterization of the Markov process. Firstly, we find the equations for (Xt, Vt). De- fine, Jn = T1 + · · · + Tn, i.e., the lapse of time between time zero and the n-th jump. Then, Xt = X0 + ˆ t 0 Vs ds (1.67) Vt = V0 for t < J1 (1.68) Vt = Un for Jn ≤ t < Jn+1, n ≥ 1 (1.69) Proposition 1.25. (Vt)t≥0 is a pure jump Markov process in RN × RN with jump kernel pi(v, dv′) = M(v′)dv′ . 66 1.5. Methods in the diffusive limit (This measure is the probability of having post-jump values dv′ if the pre-jump value is v.) The holding times are given by the i.i.d random variables (Tn)n∈N, hence the rate of jumping is con- stant equal to 1. Before tackling the diffusive limit, our goal will be to prove the next theorem: Theorem 1.26. Consider the random process (Xt, Vt)t≥0 defined in Section 1.5.5.1 with initial distribution f(0, x, v) ∈ C1x×C0v . Then, the law ρ of the process has density function f ∈ C1x×C0v fulfilling the linear Boltzmann equation ∂tf(t, x, v) + v · ∇xf(t, x, v) = M(v) ˆ RN f(t, x, v′)dv′ − f(t, x, v) . Identification of a martingale. General results. This section is extracted and adapted from [DN08]. Consider a continuous- time Markov chain on a measurable state space, E, with holding times (Sn)n∈N and jump chain (Yn)n∈N. Define J0 = 0 and Jn = S0 + . . .+ Sn. Then: Definition 1.27. The jump measure µ and the compensator ν of the Markov chain are random measures on (0,∞)× E given by µ = ∑ t:Xt 6=Xt− δ(t,Xt) = ∞∑ n=1 δ(Jn,Yn) and ν(dt,B) = q(Xt−)pi(Xt−, B)dt for all B ∈ E , the set of all sets of E. The term q(X) is the jump rate at X and pi is the jump kernel. Definition 1.28. The previsible σ-algebraP on Ω×(0,∞) is the σ-algebra generated by all the left-continuous adapted processes. A function defined on Ω×(0,∞)×E is previsible if it is P ⊗ E-measurable. Theorem 1.29 ([DN08], Appendix). Let H be previsible and assume that, for all t ≥ 0, E ˆ t 0 ˆ E |H(s, y)|ν(ds, dy) <∞. (1.70) Then the following process is a well-defined martingale M¯t = ˆ (0,t]×E H(s, y)(µ− ν)(ds, dy). 67 Overview: anomalous diffusion in kinetic theory The martingale. In this section we apply theorem 1.29 to construct a martingale. Propo- sition 1.25 provides the values of the jump rate, q(X) ≡ 1, and jump kernel, pi(v, dv′) = M(v′)dv′ of our Markov process; hence, the compensator defined in Definition 1.27 is in our case ν(ds, (Xs, dv)) = M(v) dvds. Now, for a function g ∈ C1x × L1v we have that g(Xt, Vt)−g(X0, V0) = ˆ t 0 Vs ·∇xg(Xs, Vs)Vs ds+ ˆ (0,t]×RN {g(Xs, v)−g(Xs, Vs−)}µ(ds, dv) where µ is the jump measure. Now, we have that g(Xt, Vt)−g(X0, V0) = ˆ t 0 Vs·∇xg(Xs, Vs) ds+M¯t+ ˆ (0,t]×RN {g(Xs, v)−g(Xs, Vs−)}M(v) dvds︸ ︷︷ ︸ ν(ds,(Xs,dv)) where M¯s = ˆ (0,t]×RN {g(Xs, v)− g(Xs, Vs−)}(µ− ν)(ds, dv) is a martingale by Theorem 1.29. Observe that condition (1.70) is fulfilled since we expect a finite number of jumps on a finite time. See the Appendix in reference [DN08] for more information on how to identify martingales out of Markov chains. An equation for the distribution. Lemma 1.30. There exists a density function for ρt, i.e., the measure ρt is absolutely continuous w.r.t the Lebesgue measure. Proof. We can write (Xt, Vt) = (Xt −XJ1 , Vt)1J1≤t (1.71) + (XJ1 , 0)1J1≤t (1.72) + (X0 + tV0, V0)1J1>t . (1.73) Remember that J1 ∼ E(1). Using the previous decomposition of (Xt, Vt), we have E(g(Xt, Vt)) = e −t ˆ R2N g(x+ tv, v)f(0, x, v) dxdv (1.74) + ˆ R2N×R2N ˆ t 0 e−sg(x+ sv + x′, v′)f(0, x, v)ρ∗t−s(dx ′, dv′) dxdvds, (1.75) where ρ∗t is the measure of the process starting from δ0(dx)M(v)dv (ρt is the measure of 68 1.5. Methods in the diffusive limit the process starting from f(0, x, v)dv). The first integral corresponds to the expected values taken by the function g when J1 > t, i.e., there is no jump in the velocity, and the second integral corresponds to the expected values taken by g when J1 < t. In integral (1.75) the parameter s represents the value of J1, i.e., the time of the first jump in the velocity, and x + sv is the position at time J1 = s. The values of x and v are given through the density function f(0, x, v) and the value of s is distributed as an exponential law of parameter 1, hence the presence of exp(−s). Now, the values of (x′, v′) correspond to the values of position and velocity taken by the particle after the first jump in velocity, i.e., for t ≥ J1 = s, and therefore their values are given by the distribution ρ∗t−s(dx′, dv′). This expectation can be written as: E(g(Xt, Vt)) = ˆ R2N g(x, v)f(t, x, v) dxdv where f(t, x, v) = e−tf(0, x− tv, v) +M(v) ˆ t 0 e−s ˆ R2N f(0, x− sv′ − x′, v′)ρ∗t−s(dx′, v)dv′ds . (1.76) To prove (1.76), we perform the change of variables y = x + sv + x′ and the distribution ρ∗t−s(dx′, dv′) is written as the product of the the distribution conditioning on {Vt−s = v′} times the probability of that being so, i.e.: ρ∗t−s(dx ′, dv′) = ρ∗t−s(dx ′, v′)M(v′)dv′ (1.77) where we have abused notation and written ρ∗t−s(dx′, v′) for the conditional distribution on {Vt−s = v′} (observe that it can be proven that ρ∗t−s can be written in this fashion). Then (1.75) gives: ˆ R2N×R2N ˆ t 0 e−sg(y, v′)f(0, y − sv − x′, v)M(v′)dv′ρ∗t−s(dx′, v′) dydvds = ˆ R2N g(y, v′) (ˆ R2N ˆ t 0 e−sf(0, y − sv − x′, v)M(v′)ρ∗t−s(dx′, v′)ds ) dydv′ and then by changing variables again y = x, v′ = v, v = v′. Remark 1.31. As a corollary of the previous result, if the initial distribution is bounded in the following sense f(0, x, v) ≤ f0(v) ∀x, f0 integrable, then f(t, x, v) ≤ e−tf0(v) + (1− e−t)‖f0‖L1M(v) . 69 Overview: anomalous diffusion in kinetic theory Write ρt for the distribution of (Xt, Vt), a probability measure on RN × RN . From the previous result we have that ρt(dx, dv) = f(t, x, v)dxdv . (1.78) Proof of Theorem 1.26. By Lemma 1.30 we know that ρ has a probability density f . We also know from the previous Section 1.5.5.1, we know that M¯(Xt, Vt) := g(Xt, Vt)− g(X0, V0)− ˆ t 0 Vt · ∇xg(Xs, Vs) ds − ˆ (0,t]×RN {g(Xs, v)− g(Xs, Vs−)}M(v) dvds is a martingale. Hence, E[M¯(Xt, Vt)] = E[M¯(X0, V0)] = 0 or, expressed in integral form, ˆ RN×RN g(x, v)ρt(dx, dv) = ˆ RN×RN f(0, x, v)g(x, v)dxdv (1.79) + ˆ t 0 ˆ RN×RN Gg(x, v)ρs(dx, dv)ds (1.80) where Gg(x, v) := v · ∇xg(x, v) + ˆ RN {g(x, v′)− g(x, v)}M(v′) dv′ . The integral (1.80) is equal to ˆ t 0 ˆ RN×RN g(x, v)G∗ρs(dx, dv) where G∗ is the adjoint of G. Since the equation holds for arbitrary g ∈ C1x × L1v and using (1.78) we have that (x, v)-a.e. f(t, x, v) = f(0, x, v) + ˆ t 0 G∗f(s, x, v)ds . (1.81) where, using integration by parts and Fubini’s theorem, G∗f(s, x, v) = −v · ∇xf(s, x, v) +M(v) ˆ RN f(s, x, v′) dv′ − f(s, x, v) . This expression is well defined because, thanks to (1.76), and assuming that f(0, x, v) ∈ C1x×C0v , f is continuous in the variables (x, v) for all t; it is defined as an integral on (x, v) plus a continuous function. Also, f is differentiable w.r.t x (using that expression (1.76) is differentiable). Moreover, we conclude that (1.81) is defined pointwise. 70 1.5. Methods in the diffusive limit Finally, observe that in (1.81) f is defined in terms of an integral w.r.t t, therefore it is continuous w.r.t t, hence, f is defined as an integral of a continuous function, so it is differentiable. Now, deriving w.r.t t in (1.81), one obtains the linear Boltzmann equation ∂tf(t, x, v) + v · ∇xf(t, x, v) = M(v) ˆ RN f(t, x, v′)dv′ − f(t, x, v) , which concludes the proof. 1.5.5.2 Stable Le´vy processes Here we expand what it was explained in Section 1.4.2.1. This will be needed to prove the fractional diffusive limit. Let us start with some introductory definitions and properties. Definition 1.32 (In [Ber96]). Let Y = (Yt, t ≥ 0) be a stochastic process taking values in RN . We say that Y has the scaling property of index α > 0 if, for every k > 0, the rescaled process (k−1/αYkt, t ≥ 0) has the same finite-dimensional distributions as Y . If Y is a Le´vy process with scaling property for α, then we say that Y is an α-stable Le´vy process. Definition 1.33 (In [RY99], Section III.4). A random variable Y is stable if, for every k, there are independent random variables Y1, . . . , Yk with the same law as Y and constants ak > 0, bk such that Y1 + . . .+ Yk (d) = akY + bk It can be proved that this equality forces ak = k1/α for some α ∈ [0, 2). As long as the Theorem 1.34 (In [RY99], Section III.4.). If Y is stable with index α ∈ (0, 2), then σ = 0 and the Le´vy measure has density (m11(x<0) +m21(x>0))|x|−(1+α) with m1 and m2 ≥ 0. Proposition 1.35 (In [Sat99], Representation of a non-trivial α-stable distribution µ with 0 < α < 2.). If 0 < α < 1, then µ has drift γ0 ∈ R and µ̂(z) = exp [ˆ S λ(dξ) ˆ ∞ 0 (ei〈z,rξ〉 − 1) dr r1+α + i〈γ0, z〉 ] . (1.82) If 1 < α < 2, then µ has center γ1 ∈ R and µ̂(z) = exp [ˆ S λ(dξ) ˆ ∞ 0 (ei〈z,rξ〉 − 1− i〈z, rξ〉) dr r1+α + i〈γ1, z〉 ] . (1.83) 71 Overview: anomalous diffusion in kinetic theory If α = 1, then for γ ∈ R we have µ̂(z) = exp [ˆ S λ(dξ) ˆ ∞ 0 (ei〈z,rξ〉 − 1− i〈z, rξ〉1(0,1](r)) dr r2 + i〈γ, z〉 ] . (1.84) In all cases λ is a finite measure on S, the unit circle in RN . (cf. the proof is in [Sat99], chapter 3, remark 14.6.) 1.5.5.3 Fractional diffusive limit Recall the single particle model defined in Section 1.5.5. In this section, we study the evolution of the spatial component Xt after rescaling space and time. Assume now that X0 = 0 and V0 ∼ M . Define Yt to be the random variable that describes the position of the particle at the jumps of velocity, i.e., Y0 = 0 Yt = XJn for Jn ≤ t < Jn+1 Y jumps at rate 1 with increment ∆Y = Yt − Yt− = UT (example, Y1 = XJ1 = J1 × V0 = T1U0). Hence, (Yt)t≥0 is a Le´vy process. Now, we know that a Le´vy process is characterised by a Le´vy triplet: • drift = 0 (because we are considering only the jumps), • diffusion = 0, • the Le´vy measure K(dy) is the distribution of UT (which corresponds to the dis- placement in position between two jumps in velocity), hence, for suitable function g ˆ g(y)K(dy) = E(g(UT )) = ˆ RN ˆ ∞ 0 e−tg(ut)M(u) dudt (1.85) since U ∼M and T ∼ E(1) are independent by hypothesis. We characterise the distribution of Y through its characteristic function. We know that for Le´vy processes E(eiθ TYt) = etψ(θ) (1.86) with characteristic exponent ψ(θ) = ˆ RN {eiθT y − 1− iθT y1‖y‖≤1}K(dy) . (1.87) 72 1.5. Methods in the diffusive limit Assume that M ∈ C1, it is even and F (v) = c0 ‖v‖N+α for ‖v‖ ≥ 1 , (1.88) for α ∈ (0, 2]. Theorem 1.36. The rescaled process Zε = εYε−αt converges weakly to a random variable Z which is an α-stable process with characteristic exponent ψα(θ) = C(α) ˆ S ˆ ∞ 0 eiθ T ξr − 1 r1+α drdξ where C(α) := c0 ˆ ∞ 0 sαe−s ds . (1.89) Proof. We prove the result by showing that the characteristic functions converge to the characteristic function of a particular α-stable process and then by applying Le´vy’s con- tinuity theorem for characteristic functions. So, firstly, we consider the characteristic function of the rescaled process logE[exp(iθT εYε−αt)] = ε −αtψ(εθ) by (1.86) = ε−αt ˆ RN ˆ ∞ 0 {eiθT εys − 1− iεθT ys1‖ys‖≤1}e−sM(y) dyds by (1.85) and (1.87) = ε−αt ˆ ‖y‖≤1 ˆ ∞ 0 {eiθT εys − 1}e−sM(y) dyds︸ ︷︷ ︸ =:I + ε−αt ˆ ‖y‖≥1 ˆ ∞ 0 {eiθT εys − 1}e−sM(y) dyds︸ ︷︷ ︸ =:II where the term iεθT ys1‖ys‖≤1 disappears in the last equality because it produces an odd integrand. Then, for the term I we have that eiθ T εys − 1 = cos(θT εys) + i sin(θT εys)− 1 . On one hand, observe that when we integrate the sin term, the integral gives 0 because 73 Overview: anomalous diffusion in kinetic theory the integrand is an odd function of y. On the other hand, we can bound using ‖y‖ ≤ 1 cos(θT εys)−1 = ∞∑ n=0 (−1)n (θ T εys)2n (2n)! −1 ≤ ∞∑ n>0 (−1)nε2n (‖θ‖s) 2n (2n)! = O(ε2) (≤ cos(ε‖θ‖s)) and we can exchange limit and integral with this bound using dominated convergence and the limit yields zero. Hence, I → 0, when ε→ 0. For the second term, we perform the change of variables εys = z, (εNsNdy = dz) II = tε−α−Ns−N ˆ ‖z‖≥εs ˆ ∞ 0 {eiθT z − 1}e−sM(z/εs) dzds = t ˆ ‖z‖≥εs ˆ ∞ 0 {eiθT z − 1}e−ssαM(z) dzds by (1.88) = t ˆ ∞ 0 sαe−s ˆ ‖z‖≥εs {eiθT z − 1}M(z) dzds . In the end lim ε→0 logE[exp(iθT εYε−αt)] = t ˆ ∞ 0 sαe−s ds ˆ RN {eiθT z − 1} c0‖z‖N+α dz . Define C(α) := c0 ˆ ∞ 0 sαe−s ds , then we rewrite lim ε→0 logE[exp(iθT εYε−αt)] = tC(α) ˆ RN eiθ T z − 1 ‖z‖N+α dz . Observe that C(α) is a gamma function C(α) = c0Γ(α+ 1), where Γ(α+ 1) = αΓ(α). In our case, we apply Proposition 1.35 with λ proportional to the identity, and there- fore ˆ RN ξλ(dξ) = 0 . Now, performing the change of variables z = ξr, (with ξ = ‖z‖) dz = rN−1drdξ, we have that C(α) ˆ RN eiθ T z − 1 ‖z‖N+α dz = C(α) ˆ S ˆ ∞ 0 eiθ T ξr − 1 r1+α drdξ Hence, in the notation of proposition 1.35, we have that λ = C(α)Id , (observe that C(α) > 0). 74 1.6. Summary and final remarks Summarizing, the limit of the characteristic functions of εYε−αt corresponds to the characteristic function ψα of an α-stable process. Now, we can apply Le´vy’s continu- ity theorem and since the characteristic function ψα is continuous at zero, there exists a random variable Z that has ψα as characteristic function and is the weak limit of εYε−αt. Using this probability approach we interpret how fractional phenomena arises in Sec- tion 1.6.1. 1.6 Summary and final remarks Here we summarise some of the concepts presented in this chapter and interpret why fractional phenomena arises in some particular cases using the probabilistic picture. • About the diffusion and fractional diffusion equations. – Both equations give the evolution of a probability density over time. – Both equations model transport phenomena, in the case of fractional diffusion it is called anomalous transport, in particular, it models superdiffusions. It is called anomalous diffusion because the Mean Square Displacement of the par- ticles is not linear in time, as it happens with the classical diffusion equation. It is called super-diffusion because particles spread faster than in the classical diffusion. – The solution of the diffusion equation is associated to a gaussian process, who is linked to Brownian motion. In the same way, fractional diffusion equations are linked to stable Le´vy processes, whose density behaves asymptotically as a power law. Brownian motion is just a particular case of stable Le´vy process. – The trajectories of the stochastic particles following stable processes are self- similar, this is linked to the scaling invariance of the equations. – Fractional Laplacians are non-local operators, classical Laplacian is a local op- erator. Brownian motion is continuous and the other stable Le´vy processes are jump processes and therefore discontinuous. • About the diffusive limit from the linear Boltzmann equation. – The rescaling in time t′ = εαt needed for the diffusive limit corresponds to the order of the fractional Laplacian obtained in the limit (−∆x)α/2. This corre- sponds to the scaling invariance of the equation. – The existing methods in Partial Differential Equations are the Hilbert expan- sion method, Laplace-Fourier Transform method and Moments method. There exist also some Probabilistic approaches. 75 Overview: anomalous diffusion in kinetic theory 1.6.1 How does fractional phenomena arise Remember that fractional phenomena occurs when in the formal computation presented in 1.2.3.2 we have that the diffusive constant obtained in the limit κ ∼ ˆ RN ‖v‖2 ν(v) M(v) dv = +∞ diverges. This can happen if the collision frequency ν and/or the equilibrium M take a particular shapes (degeneracy at zero/ heavy tail). We cannot conclude, however, that these are the only scenarios giving rise to fractional diffusion phenomena. Also, we have seen in Section 1.5.5 that the linear Boltzmann equation (1.30) gives the evolution of a single particle that undergoes free transport and scattering with the media. The scattering takes place at exponential rate of parameter ν, then the velocity of the particles changes and takes a new one with distribution M . With this information in mind, we interpret the causes for the emergence of fractional phenomena. Fractional phenomena due to a heavy-tail equilibria (large velocities) (Section1.2.3.3), [MMM11]. When the particle scatters and changes its velocity, in the presence of a heavy-tail equilibria, it is more probable to choose a large velocity since the variance is not finite. Hence, by choosing larger velocities, the particle makes larger displace- ments, therefore, it spreads faster than in the classical diffusive case. This gives rise to super-diffusion phenomena. Fractional phenomena due to a degeneracy in the collision frequency (small velocities) (Section 1.2.3.4), [BAMP11]. In the probabilistic model we have considered a collision frequency equal to 1. This can be generalised considering non-trivial collision frequen- cies, the change is in the rate at which the jumps happen; instead of being exponentials of rate 1, they will we exponential of rate ν(Vt−), i.e., the rate depends on the current velocity of the particle. Therefore, modifying this rate can contribute to having a faster spreading of the particles; making the rate of jumping very small for very small velocities. This is the case when the collision frequency presents a degeneracy at zero: ν(v) ∼ ν0‖v‖N+2+β, as ‖v‖ → 0 for some β > 0. This means that the smaller the velocity is, the more unlikely is to change. Then, a particle that moves at small speed but (almost) in a straight line ends up further from its starting point than a particle who takes higher velocities but changes the direction frequently, since most of the displacement is averaged out. This is why this degeneracy at zero gives fractional phenomena at macroscopic level (Theorem 1.4). 76 1.6. Summary and final remarks Remember that we have also seen in Section 1.2.3.3 that ν can contribute to the frac- tional phenomena by favouring high velocities (therefore large displacements), i.e., for a particular choice of ν, for higher velocities the rate of change is smaller. 77 Chapter 2 Kinetic derivation of fractional Stokes and Stokes-Fourier systems Joint work with Dr. Sabine Hittmeir Contents 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.1.1 The (classical) Stokes-Fourier Limit . . . . . . . . . . . . . . . . . 83 2.1.2 Rescaled equation for fractional Stokes-Fourier limit and func- tion spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 2.1.3 Summary of the assumptions and results . . . . . . . . . . . . . . 87 2.2 A priori estimates and the Cauchy problem . . . . . . . . . . . . . . . . 90 2.2.1 Integrability conditions on M . . . . . . . . . . . . . . . . . . . . . 90 2.2.2 A priori estimates and well-posedness . . . . . . . . . . . . . . . . 90 2.3 Weak formulation and auxiliary equation . . . . . . . . . . . . . . . . . 93 2.3.1 An auxiliary equation . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.3.2 The weak formulation . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.3.3 Convergence properties . . . . . . . . . . . . . . . . . . . . . . . . 95 2.4 Derivation of the macroscopic dynamics . . . . . . . . . . . . . . . . . . 105 2.4.1 Derivation of the fractional Stokes-Fourier system . . . . . . . . . 105 2.4.2 Derivation of the dynamics for fractional Stokes limit . . . . . . . 106 2.1 Introduction In this section we aim to extend the fractional diffusion limit presented in Section 1.2.3 to a kinetic transport equation conserving not only mass, but also momentum and en- 79 Kinetic derivation of fractional Stokes and Stokes-Fourier systems ergy. Many works have been investigating the incompressible fluid dynamical limit of the Boltzmann equation, see e.g. [BU91], [GL02], [SR09] and references therein. We shall review the basic formal derivation of the linear equations of the corresponding hydrody- namic limit below. On the other hand Navier-Stokes type of equations with a fractional Laplacian have gained also a lot of interest, and have been e.g. related to a model with modified dissipativity arising in turbulence in [BPFS79]. For an existence and unique- ness result in Besov spaces we refer to [Wu06]. A derivation of fractional fluid dynamical equations from kinetic transport equations would therefore be desirable to obtain. As a first step towards this direction we here analyse the linear case, i.e. we start from a linear kinetic transport equation of the form ∂tf + v · ∇xf = Lf , (2.1) where we assume the null space of L to be spanned by the equilibrium distribution M(v) satisfying M(v) = M(|v|) ≥ 0, M(v) <∞, ˆ RN M(v)dv = 1 , with the moment conditions ˆ RN M(v) dv = 1, ˆ RN |v|2M(v) dv = N, ˆ RN |v|4M(v) dv = N(N + 2) . (2.2) We assume in the following M(v) to be either the classical Gaussian M∗(v) = 1 (2pi)N/2 e− |v|2 2 , (2.3) or a heavy tailed distribution satisfying M˜(v) = c0 |v|α+N for |v| ≥ 1 (2.4) for some α > 4, that will be specified below and for some positive constant c0. For the Gaussian M∗(v) the moment conditions in (2.2) can be easily verified. For the second class of equilibrium distributions with heavy tails we only prescribe the behaviour for |v| ≥ 1 and assume M˜(v) to be smooth and bounded from above and below for small velocities. Hence for α > 4 the particularly chosen constants in (2.2) mean no loss of generality. If in the following we keep the general notation M(v), the statement holds for both M(v) = M∗(v) and M(v) = M˜(v). The macroscopic moments for density, momentum and temperature (actually, tem- 80 2.1. Introduction perature times density) of f are given by Uf =  ρf mf θf  = ˆ RN ζ(v)fdv, where ζ(v) =  1 v |v|2−N N  . We consider a linear collision operator of the form Lf = ν(v) (Kf − f) (2.5) with the operator K being defined as Kf = M(v)φ(v) · Uν,f = M(v) ( ρν,f + v ·mν,f + |v| 2 −N 2 θν,f ) , (2.6) where φ(v) =  1 v |v|2−N 2  (2.7) differs from ζ(v) only due to a normalising constant in the last component. The collision frequency is assumed to be velocity dependent in the sense that ν(v) = ν(|v|) ≥ 0 . For the Gaussian equilibrium distribution the corresponding collision frequency ν(v) = ν∗(v) is assumed to have a degeneracy as |v| → 0 of the following form ν∗(v) = |v|β∗ for |v| ≤ 1 , (2.8) for some β∗ > 0 specified below. Moreover ν∗(v) is assumed to be smooth and bounded from above and below by a positive constant for |v| ≥ 1. For the heavy-tailed equilibrium distribution the following far-field behaviour of the collision frequency ν(v) = ν˜(v) is assumed ν˜(v) = |v|β˜ for |v| ≥ 1 , (2.9) where β˜ < 1 will be coupled to the parameter α determining the tail of M˜(v). Here ν˜(v) is assumed to be smooth and bounded from above and below by a positive constant for 81 Kinetic derivation of fractional Stokes and Stokes-Fourier systems small velocities. The macroscopic quantity Uν,f = (ρν,f ,mν,f , θν,f )T is defined via ˆ RN νφfdv = AUν,f (2.10) in such a way that the collision operator satisfies the conservation laws ˆ RN φLfdv = 0 . (2.11) Using (2.6) this implies for the matrix A in (2.10) A = ˆ RN ν φ⊗ φMdv, where invertibility of A can be checked by direct calculation. Observe that for f of the form f = Mφ · U we have Uν,f = Uf = U . We can then express the linear operator K as Kf = M φ · Uν,f = M φ ·A−1 ˆ RN νφfdv . (2.12) Now the conservation properties can easily be checked ˆ RN φLf dv = ˆ RN νφfdv − ˆ RN νφMφ ·A−1dv ˆ RN νφ f dv = ( I − ˆ RN νφ⊗ φMdvA−1 )ˆ RN νφfdv = ( I −AA−1) ˆ RN νφfdv = 0 . Clearly the vector φ(v) in (2.11) can be replaced by the vector ζ(v), since their only dif- ference is a normalising constant factor in the last component. Integrating the kinetic transport equation against ζ(v), the conservation laws in terms of the macroscopic mo- ments read: ∂tρf +∇ ·mf = 0 , ∂tmf +∇x · ˆ RN v ⊗ v f dv = 0 , ∂tθf +∇x · ˆ RN v |v|2 −N N f dv = 0 . As mentioned above we will also investigate the limit to the fractional Stokes equation, hence in this case we shall only assume the conservation of mass and momentum. In this case we have φ¯(v) =  1 v  , U¯f =  ρf mf  = ˆ RN φ¯(v)fdv . (2.13) 82 2.1. Introduction Observe in particular that the corresponding A¯ is a diagonal matrix. In the remainder of introduction we are going to motivate the choice of the linear BGK model and recall the formal classical Stokes-Fourier limit as well as point out the difference to the regime with fractional rescaling. We then summarise the assumptions on the equilibrium distributions and the parameters involved and state the main results. Section 2 contains well-posedness and a priori estimates. We then introduce in a similar fashion to [Mel10] and [BAMP11] an auxiliary function on which the moments method is based upon in Section 3 and prove the necessary convergence properties for the in- dividual terms arising in the weak formulation. These are then unified for deriving the macroscopic dynamics in the fractional Stokes and Stokes-Fourier limit in Section 4. Before demonstrating the classical formal Stokes-Fourier limit we shall give a brief motivation for the choice of our collision operator. Struchtrup [SR09] and e.g. also [CDL05] used a power law form of ν in terms of |v − m/ρ| for large absolute values of the latter to obtain the correct Prandtl number out of a nonlinear BGK model of the following type: ∂tF + v · ∇xF = ν(|v −mF /ρF |) (M(ρν,F ,mν,F , θν,F )− F ) , (2.14) whereM denotes the Maxwellian M(U) = ρ (2piθ)N/2 e− |v−m/ρ|2 2θ . The macroscopic quantities Uν,F are again defined such that the conservation laws are guaranteed: ˆ RN ν(|v −mF /ρF |)φ(v)M(Uν,F )dv = ˆ RN ν(|v −mF /ρF |)φ(v)F dv . We assume to be close to the global equilibrium M(1, 0, 1) (which corresponds to M∗(v) from (2.3)). This means we can write for the remainder F −M(1, 0, 1) = δf for a small parameter δ. Then the linearised equation reads as follows ∂tf + v · ∇xf = ν(|v|) (∇UM(1, 0, 1) · Uν,f − f) , (2.15) where Uν,f is given by relation (2.10). Observing moreover ∇UM(1, 0, 1) = φ(v)M∗(v) we arrive at (2.1) with the operator given by (2.6). 2.1.1 The (classical) Stokes-Fourier Limit We shall briefly outline the formal derivation of the Stokes-Fourier system as the (classi- cal) diffusion limit from the linear kinetic transport equation with the diffusion scaling 83 Kinetic derivation of fractional Stokes and Stokes-Fourier systems γ = 2: ε2∂tf ε + εv · ∇xf ε = ν(Mφ · U εν − f ε) , (2.16) where here and in the following we denote the macroscopic moments of f ε by U ε := Ufε . For more details we refer e.g. to the work of [GL02], where the limit for the Boltzmann equation is carried out. Integration in v gives the macroscopic equation ε∂tρ ε +∇x ·mε = 0 , (2.17) which is closed in terms of the macroscopic moments. This equation formally provides the incompressibility condition for m in the limit ε → 0. Integrating (2.16) against v implies ∂tm ε + 1 ε ∇x · ˆ RN v ⊗ vf εdv = 0 . (2.18) We shall split the second moment as follows ∂tm ε + 1 ε ∇x ˆ RN |v|2 N f εdv + 1 ε ∇x · ˆ RN ( v ⊗ v − |v| 2 N I ) f εdv = 0 . (2.19) The second term can be expressed in terms of the macroscopic moments as follows: ˆ RN |v|2 N f εdv = ˆ RN ( |v|2 −N N ) f εdv + ˆ RN f εdv = θε + ρε , which provides the Boussinesq relation at leading order. The remaining terms of order 1 in the equation for m are of gradient type and therefore correspond to a pressure term, which vanishes when using divergence-free test functions. To analyse the behaviour of the third integral in (2.19) we employ the macro-micro decomposition f ε = Mφ · U εν + gεν , which inserted into the kinetic equation (2.16) formally gives gεν = −ε v ν M · ∇x(φ · U εν ) +O(ε2) = −ε v ν M · (φ · ∇xU ε) +O(ε2) , since knowing that gεν is O(ε), implies that U εν = U ε + O(ε). Now one can see that the macroscopic part of the antisymmetric integral term in (2.19) vanishes and we are left with ∂tm ε + 1 ε ∇x(ρε + θε) + 1 ε ∇x · ˆ RN ( v ⊗ v − |v| 2 N I ) gενdv = 0 . 84 2.1. Introduction The leading order term of gεν implies −1 ε ∇x ˆ RN ( v ⊗ v − |v| 2 N I ) gενdv = ∇x · ˆ RN ( v ⊗ v − |v| 2 N I ) M ν (v ⊗ v : ∇xmε)dv +O(ε) = µ0∇x · (∇xmε + (∇xmε)T ) +O(ε) = µ0∆xmε +O(ε) , for µ0 = ´ RN v 2 1v 2 2 M ν dv, where we have used the incompressiblity condition to leading order. Summarising we obtain from the equation for mε ∇x(ρε + θε) = O(ε) , (2.20) ∂tm ε = µ0∆xm ε +∇xpε +O(ε) . (2.21) We shall now turn to the equation for the temperature and therefore consider the follow- ing moment ∂t ˆ RN |v|2 − (N + 2) 2 f εdv + 1 ε ∇x · ˆ RN v |v|2 − (N + 2) 2 f εdv = 0 . (2.22) Note that due to the Boussinesq equation we have ˆ RN |v|2 − (N + 2) 2 f εdv = N 2 (θε − ρε) = Nθε +O(ε) . The choice of the moment is such that inserting the decomposition into the second inte- gral, the leading term vanishes: 1 ε ∇x · ˆ RN v |v|2 − (N + 2) 2 f εdv = 1 ε ∇x · ˆ RN v ⊗ v |v| 2 − (N + 2) 2 Mmεdv + 1 ε ∇x · ˆ RN v |v|2 − (N + 2) 2 gεdv = −∇x · ˆ RN v ⊗ v |v| 2 − (N + 2) 2 M ν ∇x(φ · U ε)dv +O(ε) = −∇x · ˆ RN v ⊗ v |v| 2 − (N + 2) 2 M ν ∇x ( ρε + |v|2 −N 2 θε ) dv +O(ε) = −Nκ0∆xθε +O(ε) , for κ0 = ´ RN |v|2(|v|2−(N+2))2 4N M ν dv > 0, where we used the Boussinesq relation to leading order. Hence formally we arrive in the limit ε → 0 at the incompressible Stokes-Fourier system: ρ+ θ = 0 , ∇x ·m = 0 ∂tm = µ0∆xm+∇xp ∂tθ = κ0∆xθ 85 Kinetic derivation of fractional Stokes and Stokes-Fourier systems Note that the momentum satisfies a heat equation up to a pressure gradient. This pres- sure term vanishes when using divergence-free testfunctions, which are typically used for incompressible fluid dynamical equations. 2.1.2 Rescaled equation for fractional Stokes-Fourier limit and function spaces As already mentioned in the introduction above it is our aim to analyse the Cauchy prob- lem for the kinetic equation with a rescaling in time of order γ ∈ (1, 2): εγ∂tf ε + εv · ∇xf ε = Lf ε (2.23) f ε(0, v, x) = f in(v, x) ∈ L2x,v(M−1) , satisfying ∇ · ˆ RN vf indv = 0 . Note that the latter condition guarantees that the initial data verifies the incompressibility condition∇x ·min = 0. Here and in the following we denote weighted L2-spaces as: ‖h‖2L2t,x,v(ω) = ˆ ∞ 0 ˆ R2N h2 ω dvdxdt . (2.24) The weight functions we are considering will only depend on v. To be more precise we will need the weight functions M−1, νM−1 and M . The spaces L2x,v(ω) and L2v(ω) are defined in a similar way, where integration in (2.24) is performed over x, v or v respec- tively. Also we shall use the abbrevations Lpt = L p(0,∞), Lpx,v = Lp(RN × RN ) and Lpt,x = L p((0,∞)× RN ). The conservation property of L implies for the zeroth moment of (2.23) after dividing by ε εγ−1∂tρε +∇x ·mε = 0 , (2.25) which provides again the incompressibility condition to leading order. Using the same macro-micro decomposition as above, we obtain for the first and second moment similar to before ∂tm ε + ε1−γ∇x(ρε + θε) = ε2−γ∇x · ˆ RN ( v ⊗ v − |v| 2 N I ) v ν M · ∇x(v ·mε)dv +O(ε) , ∂tθ ε = ε2−γ∇x · ˆ RN |v|2(|v|2 − (N + 2))2 4N M ν dv∇xθε +O(ε) . If we consider the fractional Stokes limit, then either the 2nd or the 6th moment of M/ν will be unbounded, but in such a way that it is balanced by the order ε2−γ in the limit ε → 0. Considering the fractional Stokes limit (i.e. there is no equation for θ) requires the 4th moment to be unbounded. This also explains why we cannot derive a fractional Stokes-Fourier system with a fractional Laplacian appearing in both equations for m and θ. 86 2.1. Introduction We shall also note that the scaling γ = 1 corresponds to the scaling for the acoustic limit. 2.1.3 Summary of the assumptions and results Assumption 1. [Assumptions on the parameters for the fractional Fourier-Stokes limit] (i) For the case of heavy-tailed equilibrium distributions M˜ we shall make the follow- ing assumptions on the parameters α, β˜ determining the behaviour of M˜ and the corresponding collision frequency ν˜ for large |v| (see (2.4) and (2.9)): Let α > 5 and β˜ < 1 satisfy 5 < α+ β˜ < 6 , β˜ < α− 4 2 . (2.26) The parameter γ˜ used for the rescaling in time then satisfies γ˜ = α− β˜ − 4 1− β˜ ∈ (1, 2) . Observe that this also includes a velocity independent collision frequency ν˜(v) ≡ 1. In this case the requirements on the parameters are β˜ = 0 , α = 5 + δ for δ ∈ (0, 1) , γ˜ = 1 + δ . (ii) For the Gaussian equilibrium distributions M∗ the collision frequency ν∗ is degen- erate as |v| → 0 with exponent β∗ > 1, see (2.8). For this exponent β∗ and the corresponding parameter γ∗ for the rescaling in time we assume N + 2 < β∗ < N + 3 , γ∗ = β∗ +N β∗ − 1 ∈ (1, 2) . These conditions stated in Assumption 1 imply for the heavy-tailed equilibrium dis- tribution the following integrability properties ˆ RN |v|k ν˜ M˜dv ≤ C (k ≤ 5), ˆ RN |v|6 ν˜ M˜dv =∞ , (2.27) whereas for the Gaussian equlibrium distribution the unboundedness occurs at the low- est order ˆ RN |v|2 ν∗ M∗dv =∞, ˆ RN |v|j ν∗ M∗dv ≤ C (j ≥ 3) . (2.28) If in the following the statements do hold for both cases of equilibrium distributions in Assumption 1 we write (M,γ), which can be either (M˜, γ˜) or (M∗, γ∗). 87 Kinetic derivation of fractional Stokes and Stokes-Fourier systems Theorem 2.1. Let Assumption 1 hold. Then the solution f ε to (2.23) converges as ε→ 0 to f ε(t, x, v) ⇀∗ f(t, x, v) = M ( v ·m(x) + |v| 2 − (N + 2) 2 θ(t, x) ) in L∞(0, T ;L2x,v(νM −1)) , (2.29) where the macroscopic quantities are determined by m(x) = min(x), ∂tθ = −κ(−∆)γ/2θ, θ(0, x) = θin(x) , for a positive constant κ > 0, where the equations are understood in the weak sense. In particular ∂tm = 0 holds in distribution sense restricted to divergence-free testfunctions. The initial data U in = ˆ RN ζ(v)f in(x, v)dv is hereby assumed to satisfy ∇x ·min(x) = 0, ρin(x) + θin(x) = 0 . The derivation of this theorem shows that one cannot obtain a fractional derivative in all moments at the same time, since the chosen time scale is not the right one for the diffusive terms in the momentum equation. For the sake of completeness we shall recall here that the fractional Laplacian can be defined using the Fourier Transform F((−∆x)γ/2h)(k) = |k|γF(h)(k) . We will rather use the following alternative representation as a singular integral (−∆x)γ/2h = CN,γPV ˆ RN h(x)− h(y) |x− y|N+γ dy , see e.g. also [DNPV12]. Assumption 2. [Assumptions on the parameters for the fractional Stokes system without temperature] We shall here only consider the case of heavy-tailed equilibrium distribu- tions M˜ with corresponding collision frequency ν˜. For the parameters α and β˜ (see (2.4) and (2.9)) we make the following assumptions: Let α > 3 and β˜ < 1 satisfy 3 < α+ β˜ < 4 , β˜ < α− 2 2 . (2.30) 88 2.1. Introduction The parameter used for the rescaling in time then satisfies γ˜ = α− β˜ − 2 1− β˜ ∈ (1, 2) . Again this includes the case ν˜ ≡ 1 with the choice of parameters β˜ = 0 , α = 3 + δ for δ ∈ (0, 1) , γ˜ = 1 + δ . The corresponding conditions to (2.27) for these heavy-tailed equilibrium distribution read ˆ RN |v|k ν˜ M˜dv ≤ C (k ≤ 3), ˆ RN |v|4 ν˜ M˜dv =∞ . (2.31) Theorem 2.2. Let Assumption 2 hold. Then the solution f ε to (2.23) converges as ε→ 0 to f ε(t, x, v) ⇀∗ f(t, x, v) = M(ρ(x) + v ·m(t, x)) in L∞(0, T ;L2x,v(νM−1)) , (2.32) where the macroscopic quantities solve ρ(x) = ρin(x) , ∇ ·m = 0 , ∂tm = −κ(−∆)γ˜/2m+∇xp , m(0, x) = min(x) where the equation for the evolution of m holds in the weak sense. The pressure term p ∈ L2t,x vanishes when using divergence-free testfunctions. The initial data U¯ in = ´ RN φ¯f indv is assumed to satisfy∇ ·min = 0. In this regime the fractional diffusion only appears in the equation for the momen- tum, whereas the density does not change with time. This resembles well the Navier- Stokes equations, where the density (and temperature) are assumed to be constant and the continuity equation reduces to the incompressibility condition. Remark 2.3. The reason why the fractional Stokes limit cannot be carried out for the Gaus- sian equilibrium distribution is that in this case the fractional derivative arises from the unbounded second moment of M/ν and therefore appears for the density term. In the case of the Stokes-Fourier system the Boussinesq equation then relates the density to the temperature. In the Stokes limit however no such relation is available. 89 Kinetic derivation of fractional Stokes and Stokes-Fourier systems 2.2 A priori estimates and the Cauchy problem 2.2.1 Integrability conditions on M The above Assumptions 1 and 2 on the parameters determining the behaviour of M and ν guarantee the boundedness of the moments required for carrying out the macroscopic limit. We summarise these integrability conditions in the following Lemma: Lemma 2.4. Let (M,ν) be either given by (M˜, ν˜) or (M∗, ν∗). In both cases we assume that the corresponding conditions on the parameters stated in Assumption 1 are satisfied. Then the following integrability conditions hold ˆ |v|≥δ |v|2M(v) ν(v) dv ≤ C , ˆ RN |v|j+3M(v) ν(v) dv ≤ C for 0 ≤ j ≤ 2 , (2.33) ˆ RN |v|kν2(v)M(v)dv ≤ C , ˆ RN |v|kν(v)M(v)dv ≤ C for 0 ≤ k ≤ 4 , (2.34) where δ = 0 in the case of heavy-tailed equilibrium distributions, and 0 < δ = 1 (w.l.o.g.) in the case of the Gaussian equilibrium distributions. If only the conservation of mass and momentum hold, the order of integrable mo- ments reduces as follows: Lemma 2.5. For the heavy-tailed equilibrium distributions satisfying Assumption 2 the integra- bility conditions (2.33) hold for j = 0 and (2.34) is satisfied for 0 ≤ k ≤ 2. 2.2.2 A priori estimates and well-posedness Lemma 2.6. Let the equilibrium distribution M satisfy Assumption 1 or 2, then ‖νKf‖L2v(M−1) ≤ C‖f‖L2v(M−1) . Proof. The proof can be easily seen by first observing that ‖νKf‖L2v(M−1) = ˆ RN ν2M(φ · Uν)2dv ≤ C|Uν |2 , (2.35) where we have used the boundedness of M in (2.34), which can now be employed again together with the Cauchy-Schwarz inequality to conclude |Uν |2 = ∣∣∣∣A−1 ˆ RN νφfdv ∣∣∣∣2 ≤ C ˆ RN f2 M dv ˆ RN ν2|φ|2M dv ≤ C‖f‖2L2v(M−1) . (2.36) 90 2.2. A priori estimates and the Cauchy problem This continuity property of the linear collision operator allows to deduce well-posedness of the Cauchy-problem (2.1) with initial data f in ∈ L2x,v(M−1). The mild formulation reads f(t, x, v) = f in(x− vt, v)e−νt + ˆ t 0 e−ν(t−s)νKf(s, x− (t− s)v, v)ds . If the assumptions guaranteeing continuity of K as in Lemma 2.6 hold, then a standard contraction argument yields local well-posedness, which can be extended to a global result using the a priori estimate (2.41) below for ε = 1. Clearly also the Cauchy problem for the rescaled kinetic equation is well posed for any ε > 0: Corollary 2.7. Let Assumption 1 or Assumption 2 hold and let f in ∈ L2x,v(M−1). Then there exists a unique solution f ε ∈ L∞t (L2x,v(M−1)) to (2.23). Since we want to determine the convergence of f ε as ε → 0 we shall now investigate the a priori estimates for the rescaled problem. The basic L2-estimate for kinetic transport equations is obtained by integrating the equation against f ε/M . Similar to the formal derivation of the Fourier-Stokes limit in the introduction we shall introduce the micro- macro decompositions f ε = M φ · U ε + gε , (2.37) f ε = M φ · U εν + gεν = Kf ε + gεν , (2.38) whose remainder terms fulfill ˆ RN φgεdv = 0 , ˆ RN νφgενdv = 0 , (2.39) due to the definition of the macroscopic moments and the conservation properties respec- tively. In a similar fashion to [MMM11] and [BAMP11] we obtain the following lemma: Lemma 2.8. Let Assumption 1 or Assumption 2 hold. Then the operator 1νL is bounded in L2v(νM −1) and satisfies ˆ RN Lf f M dv = − ˆ RN ν M |f −Kf |2dv (2.40) for a positive constant C and for all f ∈ L2v(νM−1). Proof. To prove the boundedness of 1νL it remains to check the boundedness of K. In a similar fashion to (2.35) one can show that ‖Kf‖L2v(νM−1) ≤ C|Uν |2, and we conclude the boundedness with a slight modification of (2.36): |Uν |2 = ∣∣∣∣A−1 ˆ RN νφfdv ∣∣∣∣2 ≤ C ˆ RN ν M f2dv ˆ RN ν|φ|2Mdv ≤ C‖f‖2Lv(νM−1) . 91 Kinetic derivation of fractional Stokes and Stokes-Fourier systems To show (2.40) we first observe that due to the conservation properties ofL (2.11) we have ˆ RN LfKf M dv = ˆ RN φLf dv · Uν = 0 . Using this we can rewrite ˆ RN Lf f M dv = ˆ RN Lf f −Kf M dv = − ˆ RN ν M |f −Kf |2dv . This lemma now yields the basic ingredient for deriving the following a priori esti- mates: Proposition 2.9. Let Assumption 1 be satisfied. Then the solution f ε of (2.23) is bounded in L∞t (L2x,v(M−1)) uniformly with respect to ε. Moreover it satisfies the decomposition (2.38), where U εν and gεν are bounded by the initial data f in in the sense that sup t>0 ‖f ε‖L2x,v(M−1) ≤ ‖f in‖L2x,v(M−1) , (2.41) ‖gεν‖L2t,x,v(νM−1) ≤ ε γ/2‖f in‖L2x,v(M−1) , (2.42) sup t>0 ‖U εν (t, .)‖L2x ≤ C‖f in‖L2x,v(M−1) . (2.43) Proof. Using (2.40), the basic L2-estimate for the solution is obtained as follows εγ 2 d dt ‖f ε‖2L2x,v(M−1) = ˆ R2N Lf ε f ε M dvdx = − ˆ R2N ν M |f ε −Kf ε|2dvdx = − ˆ R2N ν M (gεν) 2dvdx . Integration in time implies (2.41) and (2.42). For the boundedness of the macroscopic moments U εν in (2.43) it only remains to integrate (2.36) over x and taking the supremum in time. Lemma 2.10. Let the assumptions of Proposition 2.9 hold. Then there exists aU ∈ L∞t (L2x), such that f ε ⇀∗ Mφ ·U in L∞((0, T );L2x,v(νM−1)) for any T > 0. In particular we have the conver- gence of the macroscopic moments U εν , U ε ⇀∗ U in L∞((0, T );L2x). In the case of heavy tailed equilibrium distributions M˜ moreover strong convergence of U εν − U ε → 0 in L∞((0, T );L2x) holds. Under Assumption 2 the same statements are valid for U¯ εν and U¯ ε respectively. Proof. To see the weak∗-convergence we first observe that the uniform bound of U εν in L∞t (L2x) given in (2.43) implies the existence of a U ∈ L∞t (L2x) such that U εν ⇀∗ U in L∞t (L2x). Moreover the bound (2.42) implies that f ε − Mφ · U εν → 0 in L2t,x,v(νM−1), which allows to deduce f ε ⇀∗ Mφ ·U in L∞((0, T );L2x,v(νM−1)) for any T > 0, implying also for the macroscopic moment U ε ⇀∗ U in L∞((0, T );L2x). 92 2.3. Weak formulation and auxiliary equation To show the strong convergence of U εν − U ε in the case of heavy-tailed equilibria we first note that integrating the difference of the decompositions (2.37)-(2.38) against φ gives A(U εν − U ε) = ˆ RN φ(gε − gεν)dv = − ˆ RN φgενdv . In the case of M(v) = M˜(v) the integrability of M in (2.33) holds for δ = 0 and we can thus employ the Cauchy-Schwarz inequality as follows ‖U εν (t, .)− U ε(t, .)‖2L2(RN ) ≤ C ˆ RN (ˆ RN φgενdv )2 dx ≤ C ˆ RN ˆ RN ν(gεν) 2 M dvdx ˆ RN |φ|2M ν dv ≤ Cεγ , where for the last inequality we applied (2.42). 2.3 Weak formulation and auxiliary equation 2.3.1 An auxiliary equation Analogously to Mellet [Mel10] and Ben-Abdallah et al. [BAMP11] we introduce an aux- iliary function χε(t, v, x) defined as the solution of ν(v)χε − εv · ∇xχε = ν(v)ϕ(t, x) , (2.44) whereϕ(t, x) is a test function inD([0,∞)×RN ) and hence χε ∈ L∞t,v((0,∞)×RN ;L2x(RN )). It is easy to verify that χε = ˆ ∞ 0 e−ν(v)zν(v)ϕ(t, x+ εvz)dz . Considering χε − ϕ = ˆ ∞ 0 νe−νz(ϕ(t, x+ εvz)− ϕ(t, x))dz , (2.45) it can easily be deduced that |χε − ϕ| ≤ ‖Dϕ‖∞ε|v|, which implies uniform convergence in space and time, but not with respect to v. The proof of Lemma 2.5 in [BAMP11] can easily be extended to give the following convergence results: φχε → φϕ strongly in L∞t (L2x,v(M)) , (2.46) φ∂tχ ε → φ∂tϕ strongly in L∞t (L2x,v(M)) , (2.47) 93 Kinetic derivation of fractional Stokes and Stokes-Fourier systems where the extension from φ ≡ 1 in [BAMP11] to φ given as in (2.7) is straightforward due to the weight M . The proof relies on a estimate of the form ‖φ(χε − ϕ)‖2L2x,v(M) = ˆ R2N M ∣∣∣∣ˆ ∞ 0 e−νzνφ(ϕ(x+ εvz)− ϕ(x))dz ∣∣∣∣2 dxdv ≤ ˆ RN ˆ ∞ 0 Me−νzν|φ|2‖ϕ(·+ εvz)− ϕ‖2L2xdzdv The fact that ‖ϕ(·+εvz)−ϕ‖L2x → 0 as ε→ 0 for all v and z, together with the integrability condition (2.34), allow to apply the Lebesgue dominated convergence theorem. A similar proof holds for the time derivative. 2.3.2 The weak formulation Since the macroscopic equation for ρε is closed in terms of the macroscopic moments U ε (see (2.25)), it is sufficient to consider test functions ϕ(t, x) ∈ D([0,∞)×RN ) independent of v. Note that this corresponds to building the inner product in L2t,x,v(M−1) of the kinetic equation with ϕ(t, x)M(v). − ˆ ∞ 0 ˆ RN ρε∂tϕdxdt− ˆ RN ρinϕ(t = 0)dx = ε1−γ ˆ ∞ 0 ˆ RN ∇xϕ ·mεdxdt (2.48) This equation will in the limit provide the incompressibility condition. In order to derive equations for the macroscopic momentum and temperature we consider the weak formulation of the rescaled kinetic equation (2.23) using testfunctions as introduced in the previous subsection. As for the classical Stokes-Fourier equations we shall consider the following moments corresponding to ψ(v) =  v |v|2−(N+2) 2  . We shall for each moment ψi consider a separated testfunction φi ∈ D([0,∞)× RN ) with its corresponding auxiliary function χεi . Integrating the kinetic equation against ψiχ ε i gives − ˆ ∞ 0 ˆ R2N ψif ε∂tχ ε idvdxdt− ˆ R2N ψif inχεi (t = 0)dvdx = ε−γ ˆ ∞ 0 ˆ R2N ψiLf ε χεidvdxdt+ ε1−γ ˆ ∞ 0 ˆ R2N ψiv f ε · ∇xχεidvdxdt = ε−γ ˆ ∞ 0 ˆ R2N ψiM φ · U εν χεidvdxdt+ ε−γ ˆ ∞ 0 ˆ R2N ψif ε(−νχεi + εv · ∇xχεi )dvdxdt = ε−γ ˆ ∞ 0 ˆ R2N νψiM φ · U εν χεidvdxdt− ε−γ ˆ ∞ 0 ˆ R2N νψif ε dv ϕi dxdt , 94 2.3. Weak formulation and auxiliary equation where we have used the auxiliary equation (2.44). Taking into account the conservation property of the collision operator (2.11) in the latter integral we finally obtain the weak formulation − ˆ ∞ 0 ˆ R2N ψif ε∂tχ ε idvdxdt− ˆ R2N ψif inχε(t = 0)dxdv = ε−γ ˆ ∞ 0 ˆ R2N ψiM φ · U εν ν(χεi − ϕi)dvdxdt . (2.49) In the following we will analyse the convergence properties of this weak form, in partic- ular the right hand side. In the next subsection we will analyse the limiting behaviours of the separate terms. These Lemmas will then be used in Section 2.4 to conclude the proofs of the Theorems 2.1 and 2.2. 2.3.3 Convergence properties We first derive the convergence results required for the macroscopic limit to the fractional Stokes-Fourier system. At the end of the subsection we will derive the corresponding convergence properties for the fractional Stokes limit for conservation of density and momentum only. In the following we will several times have to bound integrals of the form I(t, x) = ˆ RN f(v)g(t, x+ τv)dv in L2t,x for some τ ∈ R. This can be done by first applying the Cauchy-Schwarz inequality and then interchanging the order of integration: ‖I‖2L2t,x = ˆ ∞ 0 ˆ RN (ˆ RN f(v)g(t, x+ τv)dv )2 dxdt ≤ ˆ ∞ 0 ˆ RN |f(v)|dv ˆ RN |f(v)|g2(t, x+ τv)dvdxdt = ˆ ∞ 0 ˆ RN g2(t, x)dxdt (ˆ RN |f(v)|dv )2 = ‖g‖L2t,x‖f‖ 2 L1v . (2.50) We shall first consider the terms arising from the time derivative on the left hand side of the weak formulation in (2.49): Lemma 2.11. Let Assumption 1 hold and let χεi be auxiliary functions satisfying (2.44) for ϕi ∈ D([0,∞) × RN ) (i ∈ {1, . . . , N}). Let moreover fε be the weak solution as in Proposition 2.9. Then, as ε→ 0, the weak form of the time derivatives in (2.49) converges in the sense that ˆ ∞ 0 ˆ R2N ψif ε∂tχ ε idvdxdt+ ˆ R2N ψif inχεi (t = 0)dvdx 95 Kinetic derivation of fractional Stokes and Stokes-Fourier systems → ˆ RN ψiφM dv · (ˆ ∞ 0 ˆ RN U∂tϕidxdt+ ˆ RN U inϕi(t = 0)dx ) Proof. Due to the strong convergence of ψ∂tχεi → ψ∂tϕi in L∞((0,∞);L2x,v(M)) in (2.47) the weak convergence of f ε ⇀ Mφ · U in L∞((0, T );L2x,v(M−1)) and the fact that ϕi is a test function, the stated convergence can be deduced. For passing to the limit in the right hand side of the weak formulation in (2.49) we will make use of the following expansions of the auxiliary function obtained by integration by parts: ν(v)(χε(t, x, v)− ϕ(t, x)) = εv · ∇xϕ(t, x) (2.51) + ε2 ˆ ∞ 0 e−νzvT ·D2xϕ(t, x+ εvz) · vdz ν(v)(χε(t, x, v)− ϕ(t, x)) = ε ˆ ∞ 0 νe−νzv · ∇xϕ(t, x+ εvz)dz (2.52) We start with deriving the behaviour of the right hand side of (2.49) for ψi = vi (i ∈ {1, . . . , N}): Lemma 2.12. Let the assumptions of Lemma 2.11 hold, then ε−γ ˆ ∞ 0 ˆ R2N viM φ · U εν ν(χεi − ϕi)dvdxdt = ε1−γ ˆ ∞ 0 ˆ RN (ρεν + θ ε ν) ∂xiϕidxdt+R ε i ∈ {1, . . . , N} , where Rε → 0 as ε→ 0. Proof. We shall employ the expansion of ν(χεi − ϕi) according to (2.51): ε−γ ˆ ∞ 0 ˆ R2N viM φ · U εν ν(χεi − ϕi)dvdxdt = ε1−γ ˆ ∞ 0 ˆ RN (ˆ RN viM φ · U εν vdv ) · ∇xϕi dxdt +ε2−γ ˆ ∞ 0 ˆ R2N ˆ ∞ 0 vie −νzvTD2xϕi(t, x+ εvz)vdzM φ · U εν dvdxdt =: Iε1 + I ε 2 . We start with showing that Iε2 → 0 performing an estimation of the type (2.50): |Iε2 | ≤ Cε2−γ‖D2xϕi‖L2t,x‖U ε ν‖L2t,x ˆ RN |v|3 + |v|5 ν Mdv ≤ Cε2−γ → 0 . The integral Iε1 gives rise to the Boussinesq equation. The integrand of I ε 1 containing the 96 2.3. Weak formulation and auxiliary equation macroscopic momentum is odd and hence vanishes, such that Iε1 = ε 1−γ ˆ ∞ 0 ˆ RN (ρεν + θ ε ν)∂xiϕidxdt , which concludes the proof. Lemma 2.13. Let the assumptions of Lemma 2.11 hold. Then the fractional derivative arises from the following integrals as ε→ 0: (i) For the case of heavy-tailed equilibrium distributions, i.e. M = M˜ and ν = ν˜, we have ε−γ˜ ˆ RN ν˜M˜ |v|4 2N (χε − ϕ)dv → −κ˜(−∆x)γ˜/2ϕ strongly in L2t,x. (ii) For the case of Gaussian equilibrium distributions, i.e. M = M∗ and ν = ν∗, we have ε−γ ∗ ˆ RN ν∗M∗(χε − ϕ)dv → −κ∗(−∆x)γ∗/2ϕ strongly in L2t,x. Proof. We shall first demonstrate the convergence for the heavy-tailed equilibrium distri- butions stated in (i). We therefore split the domain of integration as follows: J˜ε1 = ε −γ˜ ˆ |v|≤1 |v|4ν˜M˜(χε − ϕ)dv, J˜ε2 = ε−γ˜ ˆ |v|≥1 |v|4ν˜M˜(χε − ϕ)dv . We expand the first integral using (2.51): J˜ε1 = ε 1−γ˜ ˆ |v|≤1 |v|4vM˜dv · ∇xϕ+ ε2−γ˜ ˆ |v|≤1 ˆ ∞ 0 |v|4e−ν˜zvTD2xϕ(t, x+ εvz)vM˜dzdv . The first integrand is odd, therefore the integral vanishes. The second integrand is uni- formly bounded in |v| ≤ 1, hence J˜ε1 → 0 as ε→ 0 uniformly in t, x and also L2t,x. For the integral J˜ε2 we use the behaviours of M˜ and v˜, as well as (2.45): J˜ε2 = ε −γ˜c0 ˆ |v|≥1 |v|4−N−α+β˜ ˆ ∞ 0 ν˜ε−ν˜z(ϕ(t, x+ εvz)− ϕ(t, x))dzdv = ε−γ˜c0 ˆ |v|≥1 |v|4−N−α+β˜ ˆ ∞ 0 e−s ( ϕ ( t, x+ ε v ν˜ s ) − ϕ(t, x) ) dsdv 97 Kinetic derivation of fractional Stokes and Stokes-Fourier systems where we substituted s = ν˜z. We recall that β˜ < 1 and perform the change of variables w = ε v |v|β˜ , dv = 1 1− β˜ ( |w|β˜ ε ) N 1−β˜ dw , (2.53) where for the calculation of the determinant of the Jacobian-matrix Silvester’s theorem can be applied. Recalling γ˜ = (α− β˜ − 4)/(1− β˜), we obtain J˜ε2 = c0 1− β˜ ˆ |w|≥ε ˆ ∞ 0 e−s ϕ(t, x+ ws)− ϕ(t, x) |w|N+γ˜ dsdw = c0 1− β˜ ˆ ∞ 0 ˆ |y|>εs ϕ(t, x+ y)− ϕ(t, x) |y|N+γ˜ dy e −ssγ˜ds where substituted y = ws. Due to the definition of the principle value we have the pointwise convergence in t, x of J˜ε2 → J˜0 with J0 being defined as J˜0 = c0 1− β˜ PV ˆ RN ˆ ∞ 0 e−s ϕ(t, x+ sw)− ϕ(t, x) |w|N+γ˜ dsdw = c0 1− β˜ PV ˆ RN ϕ(t, x+ y)− ϕ(t, x) |y|N+γ˜ dy ˆ ∞ 0 e−ssγ˜ds = Γ(1 + γ˜) κ˜ PV ˆ RN ϕ(t, x+ y)− ϕ(t, x) |y|N+γ˜ dy (2.54) = −κ˜(−∆)γ˜/2ϕ with κ˜ = c0Γ(γ˜+1) 1−β˜ . For proving convergence in L 2 t,x we proceed as in [BAMP11] and split J˜0 into 1 κ˜ J˜0 = ˆ |w|≥1 ˆ ∞ 0 e−s ϕ(t, x+ sw)− ϕ(t, x) |w|N+γ˜ dsdw + ˆ |w|≤1 ˆ ∞ 0 e−s ϕ(t, x+ sw)− ϕ(t, x)− sw · ∇xϕ(t, x) |w|N+γ˜ dsdw . (2.55) These integrals are defined in the classical sense. Splitting J˜ε2 into the integral over the domain {|w| ≥ 1} and {ε < |w| < 1} respectively, we obtain 1 κ˜ (J˜ε2 − J˜0) = − ˆ |w|≤ε ˆ ∞ 0 e−s ϕ(t, x+ sw)− ϕ(t, x)− sw · ∇xϕ(t, x) |w|N+γ˜ dsdw = − ˆ |w|≤ε ˆ ∞ 0 e−s wTD2xϕ(t, x+ sw) · w |w|N+γ˜ dsdw , (2.56) 98 2.3. Weak formulation and auxiliary equation where we have performed integration by parts twice. Due to the fact that ˆ ∞ 0 e−sds ˆ |w|≤ε 1 |w|N+γ˜−2dw ≤ Cε 2−γ˜ → 0 we deduce the (strong) L2t,x-convergence of J˜ε2 − J˜0 → 0, which concludes the proof for the heavy-tailed equilibrium distributions. We shall now derive the fractional Laplacian for the Gaussian equilibrium distribu- tions M∗(v) = 1 (2pi)N/2 e− |v|2 2 as stated in (ii). We proceed in a similar fashion to [BAMP11] and split the integral in (ii) as follows: Jε∗1 = ε −γ∗ ˆ |v|≤1 ν∗M∗(χε − ϕ)dv, Jε∗2 = ε−γ ∗ ˆ |v|≥1 ν∗M∗(χε − ϕ)dv . As we shall see below the degeneracy occurs in the first integral, whereas the second integral vanishes in the limit. Expanding Jε∗2 according to (2.51) we obtain Jε∗2 = ε 1−γ∗ ˆ |v|≥1 M∗vdv · ∇xϕ+ ε2−γ∗ ˆ |v|≥1 ˆ ∞ 0 e−ν˜ ∗zvTD2xϕ(t, x+ εvz)vM ∗dzdv . The first integral vanishes, since the integrand is odd. The second integrand is uniformly bounded in {|v| ≥ 1}, hence the second integral also converges to 0 uniformly and in L2t,x. We shall now turn to the integral Jε∗1 over the domain of small velocities. Observe that we cannot expand ν∗(χε − ϕ) according to (2.51) as above, since ´|v|≤1 |v|2M∗ν∗ dv is unbounded. Hence we expand ν∗(χε − ϕ) only up to first order as given in (2.52) and proceed as in [BAMP11]: Jε∗1 = ε 1−γ∗ ˆ |v|≤1 ˆ ∞ 0 e−ν ∗zν∗v · ∇xϕ(t, x+ εvz)dzM∗dv = ε1−γ ∗ ˆ |v|≤1 ˆ ∞ 0 e−sv · ∇xϕ ( t, x+ ε v ν∗ s ) dsM∗dv . We again perform a change of variables similar to (2.53), noting that here β∗ > 1, such that the domain of integration is inverted: w = ε v |v|β∗ , dv = 1 β∗ − 1 ( ε |w|β∗ ) N β∗−1 dw . Recalling γ∗ = (β∗ + d)/(β∗ − 1) we obtain Jε∗1 = 1 β∗ − 1 ˆ |w|≥ε ˆ ∞ 0 e−sw · ∇xϕ(t, x+ sw)ds|w|− β∗+N β∗−1 M∗ ( (ε/|w|) 1β∗−1 ) dw = 1 (2pi)N/2(β∗ − 1) ˆ |w|≥ε ˆ ∞ 0 e−s ϕ(t, x+ sw)− ϕ(t, x) |w|N+γ∗ ds e − 1 2 ( ε |w| ) 1 β∗−1 dw . 99 Kinetic derivation of fractional Stokes and Stokes-Fourier systems As above we introduce the integral J0∗ = 1 (2pi)N/2(β∗ − 1)PV ˆ RN ˆ ∞ 0 e−s ϕ(t, x+ sw)− ϕ(t, x) |w|N+γ∗ dsdw , satisfying the analogous relations given in (2.56). Moreover J0∗ can be split into two integrals according to (2.55), from which we can deduce the L2t,x convergence of Jε∗ → J0∗. From the Gaussian equilibrium distributions being non-constant for small velocities two more terms arise here compared to (2.56) and [BAMP11]: (β∗ − 1)(2pi)N/2(Jε∗1 − J0∗) = ˆ |w|≥1 ˆ ∞ 0 e−s ϕ(t, x+ sw)− ϕ(t, x) |w|N+γ∗ ds ( e − 1 2 ( ε |w| ) 2 β∗−1 − 1 ) dw + ˆ ε≤|w|≤1 ˆ ∞ 0 e−s ϕ(t, x+ sw)− ϕ(t, x)− sw · ∇xϕ(t, x) |w|N+γ∗ ds ( e − 1 2 ( ε |w| ) 2 β∗−1 − 1 ) dw + ˆ |w|≤ε ˆ ∞ 0 e−s ϕ(t, x+ sw)− ϕ(t, x)− sw · ∇xϕ(t, x) |w|N+γ∗ dsdw =: Lε∗1 + L ε∗ 2 + L ε∗ 3 . For the third integral Lε∗3 the convergence to 0 in L2t,x is obtained in the same fashion to (2.56) above. For Lε∗1 we employ an estimation as in (2.50): ‖Lε∗1 ‖L2t,x ≤ 2‖ϕ‖L2t,x ˆ s 0 e−sds ˆ |w|≥1 |w|−(N+γ∗) ( 1− e− 1 2 ( ε |w| ) 2 β∗−1 ) dw ≤ C ( 1− e− 12 ε 2 β∗−1 ) → 0 To see the convergence of the remaining term Lε∗2 we perform integration by parts twice and bound ‖Lε∗2 ‖L2t,x ≤ ‖D 2 xϕ‖L2t,x ˆ ε≤|w|≤1 ˆ s 0 e−s|w|−(N+γ∗−2) ( 1− e− 1 2 ( ε |w| ) 2 β∗−1 ) dsdw We now split the domain of integration in the latter integral once more. For any a ∈ (0, 1) ˆ ε≤|w|≤1 ˆ ∞ 0 e−s |w|N+γ∗−2 ( 1− e− 1 2 ( ε |w| ) 2 β∗−1 ) dsdw ≤ C ˆ 1 ε r1−γ ∗ ( 1− e− 12( εr ) 2 β∗−1 ) dr = C ˆ εa ε r1−γ ∗ ( 1− e− 12( εr ) 2 β∗−1 ) dr + ˆ 1 εa r1−γ ∗ ( 1− e− 12( εr ) 2 β∗−1 ) dr ≤ Cr2−γ∗∣∣εa ε + C 1− e− ε 2(1−a)β∗−12 → 0 . By dominated convergence, this implies the strong convergence of Jε∗2 to J0∗ in L2t,x, which concludes the proof of the Lemma. 100 2.3. Weak formulation and auxiliary equation Lemma 2.14. Let the assumptions of Lemma 2.11 hold and recall that ψN+1 = |v|2−(N+2) 2 . Then, as ε→ 0, we have ε−γ ˆ ∞ 0 ˆ R2N ψN+1M φ · U εν ν(χε − ϕ) dvdxdt → −κ ˆ ∞ 0 ˆ RN θ(−∆)γ/2ϕdxdt . Proof. We shall again employ the expansion of ν(χε − ϕ) according to (2.51): ε−γ ˆ ∞ 0 ˆ R2N ψN+1M φ · U εν ν(χε − ϕ) dvdxdt = ε1−γ ˆ ∞ 0 ˆ RN ˆ RN ψN+1M φ · U ενv dv · ∇xϕdxdt +ε2−γ ˆ ∞ 0 ˆ R2N ˆ ∞ 0 ψN+1e −νzvTD2xϕ(x+ εvz, t)v dzM φ · U εν dvdxdt =: Iε1 + I ε 2 . The part in the integrand of Iε1 containing the macroscopic density and temperature is odd and hence vanishes, therefore we are left with computing only the part containing the momentum: 2Iε1 = ε 1−γ ˆ ∞ 0 ˆ RN (ˆ RN (|v|2 − (N + 2)) v ⊗ vM dv ·mεν) · ∇xϕdxdt = 0 , which holds due to the moment conditions in (2.2). We now turn to the second integral term Iε2 , which gives rise to the fractional Laplacian. We first order the moments accord- ingly 2 Iε2 = ε 2−γ ˆ ∞ 0 ˆ R2N ˆ ∞ 0 (|v|2 − (N + 2))e−νzvTD2xϕ(t, x+ εvz) vdzM φ · U εν dvdxdt = ε2−γ(N + 2) ˆ ∞ 0 ˆ R2N ˆ ∞ 0 e−νzvTD2xϕ(x+ εvz, t)vdzMdv ( −ρεν + N 2 θεν ) dxdt +ε2−γ ˆ ∞ 0 ˆ R2N ˆ ∞ 0 ( (|v|2 − (N + 2))v ·mεν + |v|2 (ρεν − (N + 1)θεν) ) · ·e−νzvTD2xϕ(x+ εvz, t)vdzM dvdxdt + ε2−γ 2 ˆ ∞ 0 ˆ R2N ˆ ∞ 0 |v|4θενe−νzvTD2xϕ(x+ εvz, t)vdzM dvdxdt =: Lε1 + L ε 2 + L ε 3 . We start with showing that Lε2 → 0 for both cases of equilibrium distributions due to (2.27) and (2.28) |Lε2| ≤ Cε2−γ‖D2xϕ‖L2t,x‖U ε ν‖L2t,x ˆ |v|3 + |v|5 ν M dv ≤ Cε2−γ → 0 . Moreover for the heavy-tailed equilibrium distributions the integral term Lε1 also van- 101 Kinetic derivation of fractional Stokes and Stokes-Fourier systems ishes in the limit due to (2.27) using the same argumentation. The third integral term Lε3 corresponds, after integration by parts twice and inserting the definition of ν(χε − ϕ), to the integral in Lemma 2.13 (i) and hence converges towards the fractional Laplacian. For the case of Gaussian equilibrium the roles of the integrals Lε1 and L ε 3 are interchanged, namely Lε3 vanishes and from L ε 1 we obtain the fractional Laplacian according to Lemma 2.13 (ii). We shall now state the corresponding convergence properties for the fractional Stokes limit without temperature. In fact, in the weak form (2.49) we only need to consider the moment ψ¯(v) = v. Since in this case we only treat the case of heavy-tailed equilibrium distributions as stated in Assumption 2, no distinction between the types of equilibrium distributions has to made here. Hence for the fractional Stokes limit we skip the tildes for M and ν in the following. Lemma 2.15. Let Assumption 2 hold and let χεi be the auxiliary functions as defined above (2.44) for corresponding ϕi ∈ D((0,∞)× RN ) and let f ε be the weak solution as in Proposition 2.9. (i) The weak form of the time derivatives in (2.49) for ψ¯ = v converges in the sense of Lemma 2.11 with the macroscopic moments U being replaced by U¯ as ε→ 0. (ii) For ψ¯i = vi we have for the right hand side in the weak formulation of (2.49): ε−γ ˆ ∞ 0 ˆ R2N viM φ¯ · U¯ εν ν(χεi − ϕi) dvdxdt = −ε1−γ ˆ ∞ 0 ˆ RN ϕi∂xiρ ε νdxdt− κ ˆ ∞ 0 ˆ RN mi(−∆)γ/2ϕi dxdt+ R¯εi , where R¯εi → 0 for all i ∈ {1, . . . , N}. Proof. The convergence of the terms involving time derivatives in (i) is similar to the proof of Lemma 2.11. To derive the integral identity in (ii) we first split the integral into the terms containing ρεν and mεν respectively: ε−γ ˆ ∞ 0 ˆ R2N viM φ¯ · U¯ εν ν(χεi − ϕi) dvdxdt = ε−γ ˆ ∞ 0 ˆ R2N viMρ ε ν ν(χ ε i − ϕi) dvdxdt+ ε−γ ˆ ∞ 0 ˆ R2N viM v ·mενν(χεi − ϕi) dvdxdt =: I¯ε1 + I¯ ε 2 . 102 2.3. Weak formulation and auxiliary equation We expand ν(χεi − ϕi) according to (2.51) in I¯ε1 : I¯ε1 = ε 1−γ ˆ ∞ 0 ˆ R2N vivMdv · ∇xϕi ρενdxdt + ε2−γ ˆ ∞ 0 ˆ R2N ˆ ∞ 0 e−νzvivD2xϕ(t, x+ εvz)vM dzdvρ ε νdxdt = ε1−γ ˆ ∞ 0 ˆ RN ρεν∂xiϕidxdt+ Rˆ ε i where the latter integral vanishes in the limit ε→ 0: |Rˆεi | ≤ Cε2−γ‖D2xϕi‖L2t,x‖ρ ε ν‖L2t,x ˆ RN |v|3 ν M dv ≤ Cε2−γ → 0 . We shall now derive the fractional Laplacian from the integral I¯ε2 and therefore, similar to above, split the integral into I¯ε2 = ε −γ ˆ ∞ 0 ˆ RN ˆ |v|≤1 viM v ·mενν(χεi − ϕi) dvdxdt +ε−γ ˆ ∞ 0 ˆ RN ˆ |v|≥1 viMρ ε ν v ·mενν(χεi − ϕi) dvdxdt =: J¯ε1 + J¯ ε 2 . Inserting (2.51) it is easy to see that J¯ε1 vanishes in the limit ε→ 0. We insert (2.45) in the integrand of J¯ε2 to obtain after substituting s = νz J¯ε2 = ε −γ ˆ ∞ 0 ˆ RN ˆ |v|≥1 ˆ ∞ 0 νe−sviv ·mενM ( ϕi ( t, x+ ε v ν s ) − ϕi(t, x) ) dsdvdxdt Recalling the definition of γ = (α−β−2)/(1−β) and using the same change of variables as in (2.53) we obtain (1− β)(γ +N)I¯ε2 = (γ +N) ˆ ∞ 0 ˆ RN (ˆ |w|≥ε ˆ ∞ 0 e−s wiw |w|2 1 |w|γ+N (ϕi(t, x+ sw)− ϕi(t, x))dsdw ) ·mενdxdt = − ˆ ∞ 0 ˆ RN (ˆ |w|≥ε ˆ ∞ 0 e−s∇w ( 1 |w|γ+N ) wi(ϕi(t, x+ sw)− ϕi(t, x))dsdw ) ·mενdxdt = ˆ ∞ 0 ˆ RN (ˆ |w|≥ε ˆ ∞ 0 e−s 1 |w|γ+N (ϕi(t, x+ sw)− ϕi(t, x))dsdw ) mενidxdt + ˆ ∞ 0 ˆ RN (ˆ |w|≥ε ˆ ∞ 0 e−s wi |w|γ+N s∇xϕi(t, x+ sw)dsdw ) ·mενdxdt 103 Kinetic derivation of fractional Stokes and Stokes-Fourier systems + ˆ ∞ 0 ˆ RN (ˆ |w|=ε ˆ ∞ 0 e−s wi |w|γ+N sϕi(t, x+ sw) w |w|dsdσ ) ·mενdxdt =: L¯ε1 + L¯ ε 2 + b¯ ε , where we performed integration by parts and used the fact that the outer unit normal on the sphere is w/|w|. The convergence of L¯ε1 towards the integral involving the fractional Laplacian L¯ε1 → κ ˆ ∞ 0 ˆ RN mi(−∆) γ 2ϕi dxdt is deduced as in the proof of Lemma 2.14. Hence to conclude the proof it remains to show that L¯ε2 and b¯ ε vanish in the limit. Therefore we first observe (1− β)(γ +N)L¯ε2 = ˆ ∞ 0 ˆ RN (ˆ |w|≥ε ˆ ∞ 0 e−s wi |w|γ+N s∇xϕi(t, x+ sw)dsdw ) ·mενdxdt = − ˆ ∞ 0 ˆ RN ˆ |w|≥ε ˆ ∞ 0 e−s wi |w|γ+N s(∇x ·m ε ν)ϕi(t, x+ sw)dsdwdxdt = − ˆ ∞ 0 ˆ R2N ˆ ∞ 0 e−s wi |w|γ+N s(∇x ·m ε ν)ϕi(t, x+ sw)dsdwdxdt + ˆ ∞ 0 ˆ RN ˆ |w|≤ε ˆ ∞ 0 e−s wi |w|γ+N s(∇x ·m ε ν)ϕi(t, x+ sw)dsdwdxdt =: K¯ε1 + K¯ ε 2 . For the first integral K¯ε1 we shall use the fact that∇ ·mεν ⇀ 0 in L2t,x. Hence, if ˆ RN wi |w|γ+N ˆ ∞ 0 se−sϕi(t, x+ sw)dsdw (2.57) is bounded in L2t,x, then K¯ε1 → 0. Proceeding as in (2.50) we can bound the L2t,x-norm of the integral (2.57) over the domain {|w| ≥ 1} directly by C‖ϕi‖L2t,x ˆ ∞ 1 |w|−γ−N+1dw ≤ C . For the integral (2.57) over the domain {|w| ≤ 1} we observe that se−s = ∂s((s + 1)e−s). Integrating by parts in swe can then bound the L2t,x-norm using an estimation of the type (2.50) by C‖∇xϕ‖L2t,x ˆ |w|≤1 |w|−γ−N+2dw ≤ C from which we can now deduce K¯ε1 → 0 (note that the boundary term is odd in w and 104 2.4. Derivation of the macroscopic dynamics hence vanishes). To see K¯ε2 → 0 we integrate by parts additionally in x |Kε2 | ≤ C‖mεν‖L2t,x‖D 2 xϕ‖L2t,x ˆ |w|≤ε |w|−γ−N+2dw ≤ Cε2−γ → 0 . It now remains to show that the boundary terms vanish. We employ integration in parts twice |b¯ε| = ∣∣∣∣∣ 1γ +N ˆ ∞ 0 ˆ RN ˆ |w|=ε ˆ ∞ 0 e−swD2xϕ(t, x+ sw)w wi |w|γ+Nm ε ν · w |w|dσdwdxdt ∣∣∣∣∣ ≤ C‖D2xϕ‖L2t,x‖m ε ν‖L2t,x ˆ |w|=ε |w|4 |w|γ+N+1dσ ≤ Cε 2−γ → 0 . 2.4 Derivation of the macroscopic dynamics 2.4.1 Derivation of the fractional Stokes-Fourier system The convergence of the solution f ε of the Cauchy problem in (2.23) was already shown. We will now derive the macroscopic equations determining the limiting solution stated in Theorem 2.1. Proof of Theorem 2.1. We start by deriving the incompressibility condition from equation (2.48). Since ∂tϕ and ρε are both uniformly bounded in L2t,x, multiplying (2.48) with εγ−1 and using the fact that mε ⇀ m in L2t,x we obtain the incompressibility condition in the limit ε→ 0. We shall now turn to the weak form of the first moments. Due to Lemma 2.12 we know that − ˆ ∞ 0 ˆ R2N vif ε∂tχ ε idvdxdt− ˆ R2N vif inχεi (t = 0)dxdv = ε1−γ N ˆ ∞ 0 ˆ RN (ρεν + θ ε ν) ∂xiϕidxdt+R ε i , i ∈ {1, . . . , N} . (2.58) Again, due to the boundedness of the terms on the left hand side and the remainder Rε, which vanishes in the limit ε→ 0, we obtain after multiplying by εγ−1:∣∣∣∣ˆ ∞ 0 ˆ RN (ρεν + θ ε ν) ∂xiϕidxdt ∣∣∣∣ ≤ Cεγ−1 for all i ∈ {1, . . . , N} . (2.59) Hence using the fact that U εν ⇀ U in L2t,x, we obtain the Boussinesq relation. Moreover, 105 Kinetic derivation of fractional Stokes and Stokes-Fourier systems carrying out the limit in the equation for mε we obtain ∂tm = ∇xp in the weak sense, where p(t, x) is the remainder of the Boussinesq relation: p(t, x) = lim ε→0 ε1−γ (ρεν + θ ε ν) = lim ε→0 ε1−γ ( ρεν − ρ+ (θεν − θ) √ 2/N ) which is bounded inL2t,x due to (2.59). Using divergence-free testfunctions, i.e. ∑ i ∂xiϕi = 0, we obtain ∂tm = 0. We shall now turn to the equation for θ. Herefore we use the weak form of the mo- ment corresponding to ψN+1 = |v|2−(N+2) 2 . Lemma 2.11 and the Boussinesq relation im- ply − ˆ ∞ 0 ˆ R2N ψN+1f ε∂tχ εdvdxdt− ˆ R2N ψN+1f inχε(t = 0)dxdv → ( 1 + N 2 ) ˆ ∞ 0 ˆ R2N θ∂tϕdxdt− ( 1 + N 2 ) ˆ R2N θinϕ(t = 0)dxdv where we have used the Boussinesq equation for the limiting solution and the assump- tion on the initial data ρin+θin = 0. Lemma 2.14 completes the derivation of the dynamics for the limiting function f = Mφ · U . 2.4.2 Derivation of the dynamics for fractional Stokes limit We finally give the proof for the limiting solution stated in Theorem 2.2. Proof of Theorem 2.2. The incompressibility condition from equation (2.48) can be deduced as in the proof of Theorem 2.1 above. Lemma (2.15) implies − ˆ ∞ 0 ˆ R2N vif ε∂tχ ε idvdxdt− ˆ R2N vif inχεi (t = 0)dvdx = −ε1−γ ˆ ∞ 0 ˆ R2N ϕi∂xiρ ε νdxdt− κ ˆ ∞ 0 ˆ RN mi(−∆)γ/2ϕi dxdt+ R¯εi , where R¯εi → 0 as ε → 0. Using divergence-free testfunctions, i.e. considering Φ = (ϕ1, . . . , ϕN ) T with∇ · Φ = 0, we obtain in the limit − ˆ ∞ 0 ˆ RN m · ∂tΦdxdt− ˆ RN min · Φ(t = 0)dx = κ ˆ ∞ 0 ˆ RN m · (−∆)γ/2Φ dxdt , 106 2.4. Derivation of the macroscopic dynamics which gives ∂tm = −κ(−∆x)− γ 2m m(0, x) = min(x) in the distribution sense for divergence-free testfunctions. 107 Chapter 3 Anomalous energy transport in FPU-β chain Joint work with Dr. Antoine Mellet We recall the reader that we have motivated this chapter in Section 1.3.2 where an intro- duction and previous results are given. This chapter is organized as follows: In the next section, we describe the original prob- lem (chains of coupled harmonic oscillators) and its relation to the Boltzmann phonon equation. We then introduce the collision operators C that appears in the context of FPU chains. In that section, we will see in particular that this kinetic description cannot be used to study the FPU-α chain because the collision operator C vanishes in that case. This section is mostly based on the paper of H. Spohn [Spo06b]. In Section 3.2, we investigate the properties of the four phonon collision operators, appearing in the context of the FPU-β chain as well as its linearization around an equilib- rium (this section is largely based on the work of J. Lukkarinen and H. Spohn [LS08]). The main result of our paper is finally stated in Section 3.3 and its proof is divided between Sections 3.4 and 3.5. Contents 3.1 Crystal vibrations: A kinetic description . . . . . . . . . . . . . . . . . . 110 3.1.1 The FPU framework . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.1.2 The dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . 112 3.1.3 The interaction operator C . . . . . . . . . . . . . . . . . . . . . . 112 3.2 FPU-β chain: The four phonon collision operator . . . . . . . . . . . . . 116 3.2.1 Conserved quantities . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3.2.2 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 109 Anomalous energy transport in FPU-β chain 3.2.3 Stationary solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.2.4 The linearized operator . . . . . . . . . . . . . . . . . . . . . . . . 118 3.2.5 Formal asymptotic limit . . . . . . . . . . . . . . . . . . . . . . . . 119 3.3 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.4 Properties of the operator L . . . . . . . . . . . . . . . . . . . . . . . . . . 124 3.5 Proof of Theorem 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 3.5.1 A priori estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 3.5.2 Laplace Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 131 3.5.3 Proofs of the asymptotic results . . . . . . . . . . . . . . . . . . . . 135 3.6 Proof of Proposition 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 3.7 Appendix: Origin of the collision frequency . . . . . . . . . . . . . . . . 147 3.7.1 Four phonons collision operator. . . . . . . . . . . . . . . . . . . . 148 3.1 Crystal vibrations: A kinetic description In this section, we recall the results from the paper of H. Spohn [Spo06b] that are relevant to our present study. Our goal is to detail the relation between the Boltzmann phonon equation that we are considering in this paper and the microscopic models. At the micro- scopic level, we consider an infinite lattice Zn describing the equilibrium positions of the atoms of a crystal (we briefly introduce the model in general dimension, though starting in the next section, we will focus solely on the one-dimensional case). The deviation of the atom i ∈ Zn from its equilibrium position is denoted by qi, and the conjugate momen- tum variable is denoted by pi. We consider the dynamics associated to the Hamiltonian H(q, p) = 1 2 ∑ i∈Z p2i + Vh(q) + √ λV (q) where Vh is a harmonic potential (quadratic) and √ λV is a small anharmonic pertur- bation (the kinetic equation is obtained in the limit λ → 0). The general form of the harmonic potential is Vh(q) = 1 2 ∑ i,j∈Zn α(i− j)qiqj + ω 2 0 2 ∑ i∈Zn q2i , (3.1) while V is typically a cubic or quartic potential of the form V (q) = ∑ i∈Zn γ(qi) or V (q) = ∑ i, j ∈ Zn |i− j| = 1 γ(qj − qi). 110 3.1. Crystal vibrations: A kinetic description In order to understand how energy is being transported by the vibration of the atoms in the lattice, we will replace this very large system of ODE by a kinetic equation (the so-called Botzmann phonon equation) whose unknown W (x, k, t) will be interpreted as a density distribution function for a gas of interacting phonons. The idea of describing the lattice vibrations by interacting phonons, whose evolution would be described by a Boltzmann type equation first appeared in a paper of Peierls [Pei29]. This derivation was made more rigorous by H. Spohn [Spo06b] using Wigner transforms and asymptotic analysis. We will not give any details concerning this derivation (we refer the interested reader to the work of H. Spohn [Spo06b]). We just claim that (formally at least) an appropri- ately rescaled Wigner transform of the displacement field q converges when λ → 0 to a function W (t, x, k) solution of the Boltzmann phonon equation ∂tW +∇kω(k) · ∇xW = C(W ). (3.2) The function W depends on the time t ≥ 0, the position x ∈ Rn and a wave vector k which lies in the Torus Tn = Rn/Zn. The function ω(k) is the dispersion relation of the lattice. It is determined by the harmonic part of the potential. For general potential given by (3.1), we have: ω(k) = (ω20 + α̂(k)) 1/2 (3.3) where α̂(k) is the Fourier transform of α, defined by α̂(k) = ∑ j∈Zn e−i2pik·jα(j). The operator C in the right hand side of (3.2) is an integral collision operator which depends on the anharmonic potential V (q). Of course this operator C is crucial in deter- mining the long time behavior of the solutions of this equation, so we will spend a bit of time discussing its properties in this introduction. Note that while the relation between W (t, x, k) and the microscopic variable qi and pi is rather complicated, the total energy of the system is given by ˆ Rn ˆ Tn ω(k)W (t, x, k) dk dx = 1 2 ˆ |p̂(k)|2 + ω(k)2|q̂(k)|2 dk = ∑ i∈Zn 1 2 p2i + Vh(q). (3.4) 3.1.1 The FPU framework As explained in the introduction, we now focus on the FPU chain model. For this model, we have N = 1 (we denote by T the torus T = R/Z) and the potential describes only 111 Anomalous energy transport in FPU-β chain nearest neighbors interactions. The harmonic potential is thus given by: Vh(q) = 1 8 ∑ i∈Z (qi+1 − qi)2, and the anharmonic potential V is either cubic (FPU-α chain) or quartic (FPU-β chain): V (q) = ∑ i∈Zn γ(qi+1 − qi), γ(q) = 1 3 q3 or γ(q) = 1 4 q4. The corresponding microscopic dynamics is given by d dt qi(t) = pi(t) (3.5) d dt pi(t) = 1 4 qi+1(t)− 1 2 qi(t) + 1 4 qi−1(t)− √ λ[γ′(qi − qi−1)− γ′(qi+1 − qi)]. 3.1.2 The dispersion relation When Vh is given by Vh(q) = 1 2 ω20 ∑ i∈Z q2i + 1 8 ∑ i∈Z (qi+1 − qi)2, (3.6) equation (3.3) gives the following formula for the dispersion relation: ω(k)2 = ω20 + 1 2 − 1 4 ( ei2pik + e−i2pik ) and so ω(k) = ( ω20 + 1 2 (1− cos(2pik)) )1/2 , k ∈ T. (3.7) For the FPU model, we have ω0 = 0, and so the dispersion relation is given by ω(k) = √ 1 2 (1− cos(2pik)) = | sin(pik)|. 3.1.3 The interaction operator C The operator C in the right hand side of (3.2) is determined by the non-harmonic pertur- bation of the potential V . Cubic potentials: Three phonons operator When the anharmonic potential is cubic, that is V = 1 3 ∑ i∈Z q3i , (3.8) 112 3.1. Crystal vibrations: A kinetic description k1 k k1 k2 k k2 Figure 1: Three phonons interactions we find F (k, k1, k2) 2 = 8 sin2(πk) sin2(πk1) sin 2(πk2) ωω1ω2 . Going back to (10), we note that the first term can be interpreted as describing a wave vector k merging with a wave vector k1 and leading to a new wave vector k2 (k + k1 → k2), while the second term describes the splitting of wave vector k into k1 and k2 (k → k1 + k2). See Figure 1. These interactions conserve the energy (ω+ω1 = ω2), but the momentum is conserved only modulo integers: the δ-function in the first term yields the constraint k+ k1 = k2 + n, n ∈ , k, k1, k2 ∈ (one talks of normal process when n = 0, and umklapp process when n ￿= 0). This quadratic operator is reminiscent of the Boltzmann operator for the theory of dilute gas. There is however an essential difference: The kinetic energy 12v 2 is replaced here by the dispersion relation ω(k). In order to further study this integral operator, it is thus essential to characterize the set of (k, k1, k2) such that the δ-functions are not zero, that is:￿ k + k1 = k2 ω(k) + ω(k1) = ω(k2) or ω(k) + ω(k1) = ω(k + k1), (k, k1) ∈ , (11) This is much more delicate than for the usual Bolzmann operator and for general dispersion relation ω, it is not obvious that (11) has any solutions. In our framework, that is when ω is given by (7) (nearest neighbor har- monic coupling) we actually can prove that ω(k) + ω(k1)− ω(k + k1) ≥ ω0 2 so (11) has no solutions when ω0 > 0 and only the trivial solution k1 = 0 when ω0 = 0. 9 Figure 3.1: Three phonons interactions or V = 1 3 ∑ i∈Z (qi+1 − qi)3 (3.9) (the latter one corresponds to the FPU-α chain), the collision operator is give by C(W ) = 4pi ˆ ˆ F (k, k1, k2) 2 × [ 2δ(k + k1 − k2)δ(ω + ω1 − ω2)(W1W2 +WW2 −WW1) + δ(k − k1 − 2)δ(ω − ω1 − ω2)(W1W2 −WW1 −WW2) ] dk1dk2 (3.10) where we used the notation ωi = ω(ki) and Wi = W (ki). The formula for the collision rate F (k, k1, k2) can be found in [Spo06b]. In particular, when V is given by (3.8) (on-site potential) then F (k, k1, 2) 2 = (8ωω1ω2) −1 When V is the nearest neighbor interaction potential (3.9) and ω0 = 0 (that is for the FPU-α chain), the collision rate becomes F (k, k1, k2) 2 = (8ωω1ω2) −1|[exp(i2pik)− 1][exp(i2pik1)− 1][exp(i2pik2)− 1]|2. Using the fa t that | exp(i2pik)− 1|2 = 4 sin2(pik), we find F (k, k1, k2) 2 = 8 sin2(pik) sin2(pik1) sin 2(pik2) ωω1ω2 . Going back to (3.10), we note that the first term can be interpreted as describing a wave vector k merging with a wave vector k1 and leading to a new wave vector k2 (k + k1 → k2), while the second term describes the splitting of wave vector k into k1 and k2 (k → k1 + k2). See Figure 3.1. These interactions conserve the energy (ω + ω1 = ω2), but the momentum is conserved only modulo integers: the δ-function in the first term yields 113 Anomalous energy transport in FPU-β chain the constraint k + k1 = k2 + n, n ∈ Z, k, k1, k2 ∈ T (one talks of normal process when n = 0, and umklapp process when n 6= 0). This quadratic operator is reminiscent of the Boltzmann operator for the theory of dilute gas. There is however an essential difference: The kinetic energy 12v 2 is replaced here by the dispersion relation ω(k). In order to further study this integral operator, it is thus essential to characterize the set of (k, k1, k2) such that the δ-functions are not zero, that is:  k + k1 = k2 ω(k) + ω(k1) = ω(k2) or ω(k) + ω(k1) = ω(k + k1), (k, k1) ∈ T, (3.11) This is much more delicate than for the usual Bolzmann operator and for general disper- sion relation ω, it is not obvious that (3.11) has any solutions. In our framework, that is when ω is given by (3.7) (nearest neighbor harmonic cou- pling) we actually can prove that ω(k) + ω(k1)− ω(k + k1) ≥ ω0 2 so (3.11) has no solutions when ω0 > 0 and only the trivial solution k1 = 0 when ω0 = 0. It follows [Spo06b]: Theorem 3.1. When ω is given by (3.7) with ω0 ≥ 0, then the three phonon collision operator (3.10) satisfies C(W ) = 0 for all W . In particular, this implies that for the FPU-α chain, the collision operator vanishes, and the corresponding Boltzmann phonon equation reduces to pure transport. This sug- gests poor relaxation to equilibrium for the microscopic model, and it means that this kinetic approach is of no use in studying the long time behavior of the hamiltonian sys- tem. This is of course the reason why we focus in this paper on the FPU-β chain. Remark 3.2. As noted in [Spo06b], equation (3.11) might have non trivial solutions for other dispersion relations (for instance ω(k) = ω0 + 2(1− cos(2pik))), so this three phonon operator is of interest in other framework (different harmonic potential Vh). Quartic potentials: Four phonons operator. We now consider the quartic potential given by V (q) = 1 4 ∑ i∈Z q4i (3.12) 114 3.1. Crystal vibrations: A kinetic description k1 k2 k3 k k1 k k2 k3 Figure 2: Four phonons interactions The term proportional to W is the loss term, while the gain term is W1W2W3 (which is always positive). Again, we can interpret the differ- ent terms as pair collisions or merging/splitting of phonons (see Figure 2). In order to understand the collision rule, we note that for pair collisions (k, k1) → (k2, k3) (which correspond to the terms such that ￿3 j=1 σj = −1 in the integral), we need to solve ω(k) + ω(k1) = ω(k2) + ω(k + k1 − k2) (16) while for three phonons mergers (or splitting) (k, k1, k2)→ k3 we have ω(k) + ω(k1) + ω(k2) = ω(k + k1 + k2). (17) Again, in general, it is not possible to solve these equations explicitly, and it is not obvious that either of these equations should be satisfied on a set of positive measure In fact, when ω is given by (7) (nearest neighbor couplings), it can be shown (see [23]) that (17) has no solution (so collision processes in which three phonons are merged into one, or one phonon splits into three are impossible). As a consequence, the only interactions that are allowed are pair collisions, which, in particular, preserve the total number of phonons. This preservation of the number of phonons, reminiscent of the preservation of the number of particles in gas dynamics, does not follow here from a fundamental physical principle, but is instead a mathematical artifact. This property is however stable under small perturbation of ω, and it also holds for the nonlinear wave equation for which ω(k) = |k| (k ∈ 3). As a consequence, the operator C can be rewritten as C(W ) = 36π ￿ ￿ ￿ F (k, k1, k2, k3) 2δ(k + k1 − k2 − k3)δ(ω + ω1 − ω2 − ω3) [W1W2W3 +WW2W3 −WW1W3 −WW1W2] dk1 dk2 dk3. (18) 11 Figure 3.2: Four phonons interactions or V = 1 4 ∑ i∈Z (qi+1 − qi)4. (3.13) The corresponding collision operator then reads C(W ) = 12pi ∑ σ1,σ2,σ3=±1 ˆ ˆ ˆ F (k, k1, k2, k3) 2 × δ(k + σ1k1 + σ2k2 + σ3k3)δ(ω + σ1ω1 + σ2ω2 + σ3ω ) × (W1W2W3 +W (σ1W2W3 +W1σ2W3 +W1W2σ3)) dk1 dk2 dk3 (3.14) with F (k, k1, k2, k3) 2 = (16ωω1ω2ω3) −1 for on-site poten ial (3.12) and F (k, k1, k2, k3) 2 = 3∏ i=0 2 sin2(piki) ω(ki) . (3.15) for nearest neighbor coupling (3.13). The term proportional to W is the loss term, while the gain term is W1W2W3 (which is always positive). Again, we can interpret the different terms as pair collisions or merg- ing/splitting of phonons (se Figure 3.2). In order to understand the collision rule, we note that for pair collisions (k, k1) → (k2, k3) (which correspond to the terms such that∑3 j=1 σj = −1 in the integral), we need to solve ω(k) + ω(k1) = ω(k2) + ω(k + k1 − k2) (3.16) while for three phonons mergers (or splitting) (k, k1, k2)→ k3 we have ω(k) + ω(k1) + ω(k2) = ω(k + k1 + k2). (3.17) In general, it is not possible to solve these equations explicitly, and it is not obvious that either of these equations should be satisfied on a set of positive measure. In fact, 115 Anomalous energy transport in FPU-β chain when ω is given by (3.7) (nearest neighbor couplings), it can be shown (see [Spo08]) that (3.17) has no solution (so collision processes in which three phonons are merged into one, or one phonon splits into three are impossible). As a consequence, the only interactions that are allowed are pair collisions, which, in particular, preserve the total number of phonons. This preservation of the number of phonons, reminiscent of the preservation of the number of particles in gas dynamics, does not follow here from a fundamental physical principle, but is instead a mathematical artifact. This property is however stable under small perturbation of ω, and it also holds for the nonlinear wave equation for which ω(k) = |k| (k ∈ R3). As a consequence, the operator C can be rewritten as C(W ) = 36pi ˆ ˆ ˆ F (k, k1, k2, k3) 2δ(k + k1 − k2 − k3)δ(ω + ω1 − ω2 − ω3) [W1W2W3 +WW2W3 −WW1W3 −WW1W2] dk1 dk2 dk3. (3.18) When ω is given by (3.7), we will see later on that (3.16) has non trivial solutions on a set of full measure, that is ˆ T ˆ T δ(ω(k) + ω(k1)− ω(k2)− ω(k + k1 − k2)) dk1 dk2 6= 0. In particular this operator C is non trivial. 3.2 FPU-β chain: The four phonon collision operator In this section, we briefly summarize the properties of the four phonon collision operator (3.18) which arises in the modeling of the FPU-β chain. 3.2.1 Conserved quantities All the collision operatorsC mentioned above conserve the energy. This can be expressed by the following condition: ˆ T ω(k)C(W )(k) dk = 0 for all functions W . The four phonon collision operator (3.18), corresponding to the quartic potential, also satisfies ˆ T C(W )(k) dk = 0 which can be interpreted as the conservation of the number of phonons ´ TW dk. How- ever, this quantity has no microscopic equivalent, and does not correspond to any phys- ical principle. Rather it is a consequence of the symmetry of the operator, which follows 116 3.2. FPU-β chain: The four phonon collision operator from the fact that 3 phonon merger cannot take place ((3.17) has no solutions). In partic- ular, this equality does not hold for the three phonon operator. Note that the first moment k is preserved in the wave kinetic equation case (where k ∈ RN ). However, this conservation is broken here by umklapp processes. 3.2.2 Entropy The Boltzmann phonon operators satisfy an entropy inequality, similar to Boltzmann H- Theorem in gas dynamic. In particular, for the four phonon operator we can rewrite (3.18) as follows: C(W ) = 36pi ˆ ˆ ˆ F (k)2δ(k + k1 − k2 − k3)δ(ω + ω1 − ω2 − ω3) WW1W2W3[W −1 +W−11 −W−12 −W−13 ]dk1 dk2 dk3 and we then see that (assuming all integrals are well defined): ˆ T1 W−1(k)C(W )(k) dk (3.19) = 9pi ˆ ˆ ˆ ˆ F (k)2δ(k + k1 − k2 − k3)δ(ω + ω1 − ω2 − ω3) ·WW1W2W3[W−1 +W−11 −W−12 −W−13 ]2dk1 dk2 dk3 ≥ 0. 3.2.3 Stationary solutions It is easy to check that the distributions Wb(k) = 1 bω(k) for any b > 0 satisfy C(Wb) = 0 for all the operators C considered above. This fact is in accordance with equilibrium statistical mechanics (see [Spo06b]). It is more delicate to check that these are the only solutions. In fact it is not always true. For the four phonon collision operator (3.18), we can check that Wa,b(k) = 1 a+ bω(k) (3.20) is an equilibrium for all a, b > 0. Conversely, the entropy inequality (3.19) implies that if C(W ) = 0 then ψ(k) = W (k)−1 is a collision invariant, that is ψ(k) + ψ(k1) = ψ(k2) + ψ(k3) 117 Anomalous energy transport in FPU-β chain for all k, k1, k2, k3 such that k + k1 = k2 + k3, and ω(k) + ω(k1) = ω(k2) + ω(k3). An obvious candidate is ψ(k) = a + bω(k). Under general conditions on ω, Spohn proved that these are indeed the only collision invariants in dimension N ≥ 2 [Spo06a]. The same result is proved by Lukkarinen and Spohn [LS08] in our framework (dimension 1). As a conclusion, (3.20) are the only solutions of C(W ) = 0 for the four phonon col- lision operator (3.18). Note that the fact that we can take a 6= 0 is a consequence of the conservation of the number of phonons for the four phonon collision operator (which, as explained above, follows from the fact that equation (3.17) describing merging and splitting of phonons has no solutions). 3.2.4 The linearized operator As mentioned in the introduction, we will be interested in the behavior of the solutions of the Boltzmann phonon equation in the neighborhood of a thermodynamical equilibrium. Given W (k) = Tω(k) , we thus introduce the linearized operator L(f) = 1 W DC(W )(Wf) where DC denotes the derivative of the operator C. By differentiating the equation C(Wa,b) = 0 with respect to a and b, we get: L(1) = 0 and L(ω−1) = 0, which suggests (as will be proved later) that the kernel of L is two dimensional and spanned by 1 and ω−1. In our framework, the later mode, ω−1 is singular (not integrable) for k = 0. Because of natural a priori bounds on the solutions of the Boltzmann Phonon equation, it will be easy to see that this mode is not present in the macroscopic limit. It will however play an important role in the derivation of a macroscopic model. Note that it comes from the derivation with respect to the spurious coefficient a. Similarly, differentiating the conservation equations ˆ ωC(W + tWf) dk = 0 and ˆ C(W + tWf) dk = 0 with respect to t, we deduce that ˆ L(f) dk = 0, and ˆ ω−1L(f) dk = 0. 118 3.2. FPU-β chain: The four phonon collision operator The properties of L will be further investigated in Section 3.4. For now, we just state the following proposition without proof, since it is all we need to formally derive a macroscopic equation. Proposition 3.3. The operator L : L2(T1, V (k) dk) −→ L2(T1, V (k)−1 dk) (where V is defined by (3.31)) is a bounded self-adjoint operator which satisfies (i) ker(L) = Span {1, ω(k)−1} (ii) R(L) = {h ∈ L2(T1, V (k)−1 dk) ; ´T h(k) dk = ´ T ω −1(k)h(k) dk = 0 } We end this section by deriving the explicit formula for the operator L: A direct com- putation gives (when W (k) = Tω(k) ): DC(W )(Wf) = 36pi ˆ ˆ ˆ F (k, k1, k2, k3) 2δ(k + k1 − k2 − k3)δ(ω + ω1 − ω2 − ω3) ×WW1W2W3 [ f3W −1 3 + f2W −1 2 − f1W−11 − fW−1 ] dk1 dk2 dk3 = 36piT 3 ˆ ˆ ˆ F (k, k1, k2, k3) 2 ωω1ω2ω3 δ(k + k1 − k2 − k3)δ(ω + ω1 − ω2 − ω3) × [ ω3f3 + ω2f2 − ω1f1 − ωf ] dk1 dk2 dk3 Using (3.15), we see that F (k, k1, k2, k3) 2 ωω1ω2ω3 = 16 and we deduce: L(f) = 576piT 2 ω ˆ ˆ ˆ δ(k + k1 − k2 − k3)δ(ω + ω1 − ω2 − ω3) × [ ω3f3 + ω2f2 − ω1f1 − ωf ] dk1 dk2 dk3. (3.21) 3.2.5 Formal asymptotic limit We now have all the ingredient to perform the usual asymptotic analysis and attempt to derive (formally) a diffusion equation from the Boltzmann phonon equation (we will see however that it fails in our framework). The starting point is the following rescaled equation in the FPU-β chain framework detailed above: ε2∂tW + εω ′(k)∂xW = C(W ), (3.22) where C is the four phonon collision operator (3.18) with collision frequency given by (3.15), and we consider a solution which is a perturbation of a thermodynamical equilib- rium: W ε(t, x, k) = W (k)(1 + εf ε(t, x, k)) 119 Anomalous energy transport in FPU-β chain where W = Tω(k) for some constant T > 0. We introduce the operators Q(f, f) = 1 W D2C(W )(Wf,Wf), and R(f, f, f) = 1 W D3C(W )(Wf,Wf,Wf) so that (we recall that C is a cubic operator): 1 W C(W ε) = εL(f) + ε2 1 2 Q(f, f) + ε3 1 6 R(f, f, f) where L is given by (3.21). The function f ε solves ε2∂tf ε + εw′(k)∂xf ε = L(f ε) + ε 1 2 Q(f ε, fε) + ε2 1 6 R(f ε, fε, fε). (3.23) Taking the limit ε→ 0 in (3.23), we get L(f0) = 0 and so Proposition 3.3 (i) implies f0(t, x, k) = T (t, x) + S(t, x)ω(k)−1. Since equation (3.23) preserves the L1 norm, it is natural to assume that f0(t, x, k) ∈ L1(R× T). We note however that ω(k) ∼ |k| as |k| → 0, and so we must have S(t, x) = 0. Next, integrating (3.23) with respect to k yields ∂tT ε + ∂xJ ε = 0 with T ε = 〈f ε〉, Jε(t, x) = 1 ε 〈ω′f ε〉 where we use the notation 〈·〉 = ´T · dk. We now need to compute J = limε→0 Jε. Recalling that L is a self adjoint operator, we write ε−1〈ω′f ε〉 = 〈L−1(ω′)L(f ε)〉 120 3.2. FPU-β chain: The four phonon collision operator and using (3.23), we replace L(f ε) in the right hand side: ε−1〈ω′f ε〉 = 〈L−1(ω′)ω′∂xf ε〉 − 〈L−1(ω′)Q(f ε, fε)〉+O(ε). Formally, we thus get lim ε→0 ε−1〈ω′f ε〉 = 〈L−1(ω′)ω′〉∂xT − 〈L−1(ω′)Q(T, T )〉. Finally, a direct computation gives Q(f, f) = 576piT 2 ω ˆ ˆ ˆ δ(k + k1 − k2 − k3)δ(ω + ω1 − ω2 − ω3)[ 2(ω − ω3)[f1f2 − ff3] + (ω + ω1)[f2f3 − ff1] ] dk1 dk2 dk3, and it is readily seen that Q(T, T ) = 0. We thus get the following relation J = 〈L−1(ω′)ω′〉∂xT which is Fourier’s law with diffusion coefficient κ = −〈L−1(ω′)ω′〉 > 0. We conclude this section with the following remarks: (i) The non linear termQ(T, T ) = 0 does not contribute to the limiting equation. In the next section, we will drop this term altogether. (ii) The fact that S = 0 will need to be addressed very carefully in the rigorous proof. In particular, we will see that while we do indeed have f0 = T , the term S plays a significant role in the rigorous derivation of the diffusion equation (see next sec- tion). (iii) Perhaps the most important remark is that one need to check that κ is well defined. In fact, it can be proved that the integrand in the definition of the diffusion coeffi- cient behaves like |k|−5/3 for small k. It follows that κ = +∞ so the limit presented above does not give any equation for the evolution of T . Such a phenomenon is not uncommon, and based on previous work (see [MMM11]), we expect that by taking a different time scale in (3.23) we can derive an anomalous diffusion equation for the evolution of the temperature T . This is of course the goal of this paper as explained in the next section. 121 Anomalous energy transport in FPU-β chain 3.3 Main result In view of the formal asymptotic limit detailed in the previous section, we now consider the following linear equation: εα∂tf ε + εω′(k)∂xf ε = T 2 L(f ε), x ∈ R, k ∈ T (3.24) where ω(k) = | sin(pik)| and L is defined by L(f) = ω ˆ ˆ ˆ δ(k + k1 − k2 − k3)δ(ω + ω1 − ω2 − ω3)[ ω3f3 + ω2f2 − ω1f1 − ωf ] dk1 dk2 dk3. (3.25) Note also that we have made L independent of the equilibrium temperature T and set all other constant in L equal to 1 for the sake of clarity. The existence of a solution to this equation is fairly classical. We recall it for the sake of completeness in Proposition 3.15. Our main result is then the following: Theorem 3.4 (Fractional diffusion limit for the linearised equation). Let f ε be a solution of equation (3.24) and take α = 85 , with initial data f0 ∈ L2(R× T). Then f ε(t, x, k) ⇀ T (t, x) L∞((0,∞);L2(R× T))-weak ∗ where T solves the fractional diffusion equation ∂tT + κ T 6/5 (−∆x)4/5T = 0 in (0,∞)× R (3.26) with initial condition T (0, x) = T0(x) := ˆ 1 0 f0(x, k) dk. (3.27) The diffusion coefficient κ ∈ (0,∞) is given by κ = κ1 − κ 2 2 κ3 ∈ (0,∞) where κ1, κ2, κ3 are defined in Proposition 3.19. First, we note that it is enough to consider the case T = 1 122 3.3. Main result since we can recover the general case by a simple rescaling t 7→ T 2t, x 7→ T 2x. The main difficulty here, compared with previous work devoted to fractional diffu- sion limit of kinetic equations, is the fact that the kernel of L is spanned by 1 and ω(k)−1. This last mode should not appear in the limit since it is not square integrable, but it will nevertheless play an important role. In fact, we will prove that f ε can be expanded as follows: f ε(t, x, k) = T ε(t, x) + ε 3 5Sε(t, x)ω(k)−1 + ε 4 5hε(t, x, k) where T ε is bounded in L∞(0,∞;L2(R)), hε is bounded in L2V (T × R) and Sε converges in some weak sense to a non trivial function. More precisely we will prove in Section 3.6: Proposition 3.5. The function Sε(t, x) converges in distribution sense to S(t, x) = −κ2 κ3 (−∆)3/10T (t, x). In particular, as mentioned above, this means that the mode ω(k)−1 vanishes in the limit and the macroscopic behavior of the phonon distribution is completely described by T = limε→0 T ε. However, projecting equation (3.24) onto the constant mode of the kernel of L, we will find the following equation of the evolution of T : ∂tT + κ1(−∆)4/5T + κ2(−∆)1/2S = 0. (3.28) We see that S = limε→0 Sε plays a role in the evolution of T . To understand this, we note (anticipating a bit on the result of the next section) that the reason we are observing anomalous diffusion phenomena here (as opposed to standard diffusion as described in the previous section), is the fact that phonons with frequency k close to zero encounter very few collisions (degenerate collision frequency). And the term ε 3 5Sε(t, x)ω(k)−1, while small, is heavily concentrated around k = 0 (non integrable singularity at k = 0). The competition between the smallness and the singularity gives rise to a term of order 1 in the equation. In order to describe the evolution of T , we now need to obtain an equation for S. By projecting equation (3.24) onto the ω(k)−1 mode of the kernel of L, we will prove that: κ2(−∆)1/2T + κ3(−∆)1/5S = 0. (3.29) We note that there is no ∂tS in (3.29) (unlike the corresponding equation for T ). The reason is that due to the singularity of ω(k)−1 for k = 0, the quantity S diffuses faster than T (so we would have to take a smaller α in (3.24) in order to observe the diffusion of S). At our time scale (given by α = 85 ), S has thus already reached equilibrium, and can 123 Anomalous energy transport in FPU-β chain be expressed (in view of (3.29)) as S = −κ2 κ3 (−∆)3/10T. Inserting this expression into (3.28), we find ∂tT + κ(−∆)4/5T = 0 where κ = κ1 − κ 2 2 κ3 . Of course, we will show that κ > 0 (once the explicit expressions for the κi are given, it will be a very simple consequence of Cauchy-Schwarz inequality - see Lemma 3.21). It is interesting to note that the effect of the mode ω−1 on the macroscopic equation is to reduce the diffusion coefficient (and thus to slow down the diffusion). This can be understood by noting that the fact that the kernel of L does not contain only the natural constant mode, is due to the lack of merging k + k1 + k2 → k3 and splitting k → k1 + k2 + k3 interactions for phonons in the non linear collision operator C (fewer interactions⇒ slower relaxation). 3.4 Properties of the operator L The asymptotic behavior of the solution of (3.24) depends very strongly on the properties of the operatorL. This operator is studied in great detail in [LS08], and we will recall their main results in this section. The operator L can be written as L(f) = ˆ K(k, k′)f(k′) dk′ − V (k)f(k) where K(k, k′) = ω(k)ω(k′) ˆ T 2 δ(ω(k) + ω(k1)− ω(k′)− ω(k + k1 − k′)) − δ(ω(k) + ω(k′)− ω(k1)− ω(k + k′ − k1) dk1 (3.30) and V (k) = ω(k)2 ˆ T δ(ω(k) + ω(k1)− ω(k′)− ω(k + k1 − k′)) dk1dk′. (3.31) The fact that ´ T L(f) dk = 0 for all f implies V (k) = ˆ T K(k′, k) dk′ (this equality can be checked also from the formula forK and V , but it is much easier this 124 3.4. Properties of the operator L way) and a short computation shows that K(k, k′) = K(k′, k). In particular, L is a self adjoint operator in L2(T). It is also positive since we have − ˆ T L(f)f dk = 1 4 ˆ ˆ ˆ ˆ δ(k + k1 − k2 − k3)δ(ω + ω1 − ω2 − ω3) [ω3f3 + ω2f2 − ω1f1 − ωf ]2 dk dk1 dk2 dk3 (3.32) ≥ 0 for all f . One of our goals will be to improve this inequality and show thatL has a spectral gap property in the appropriate functional spaces. For that, we will need to show that the integral operator K(f) = ˆ K(k, k′)f(k′) dk′ (3.33) is a compact operator (in an appropriate functional spaces) The first step, in view of (3.30) is to study the solution set of the equation of conserva- tion of energy: ω(k) + ω(k1) = ω(k ′) + ω(k + k1 − k′). (3.34) We recall the following result: Proposition 3.6 ([LS08]). The equation (3.34) has the trivial solutions k′ = k and k′ = k1, and the (non trivial) solution k1 = h(k, k ′) where h(k, k′) = k′ − k 2 + 2 arcsin ( tan |k′ − k| 4 cos k + k′ 4 ) (and there are no other solutions of (3.34)). With this proposition in hand, one can now compute the kernel K(k, k′) and the mul- tiplicative function V (k). We recall here the main result of [LS08]. The first one states that the function V (k) is degenerate for k → 0 (note that W in [LS08] corresponds to our V ): Proposition 3.7 ([LS08, Lemma 4.1]). The function V : R → R+ is symmetric (V (1 − k) = V (k)), continuous and satisfies c1| sin(pik)|5/3 ≤ V (k) ≤ c2| sin(pik)|5/3 (3.35) for all k ∈ R, for some c1, c2 > 0. Moreover, lim k→0 ( |sinpik|−5/3 V (k) ) = v0 > 0. 125 Anomalous energy transport in FPU-β chain Because of the degeneracy of V for k = 0, we do not expect the operator L to have a spectral gap in L2. We thus introduce the operator L0(f) := V −1/2L(V −1/2f) We note that this operator has the form L0(f) = K0(f)− f with K0(f) = V −1/2K(V −1/2f). To prove that L0 has good properties in L2(T), we need to study the operator K0. Again, it is proved in [LS08] thatK0 : L2(T1)→ L2(T1) is a compact, self-adjoint operator, which implies that K : L2(T1, V dk)→ L2(T1, V −1dk) is a compact, self-adjoint operator. To be more precise, in [LS08], the kernel K is first written as K(k, k′) = 2ω(k)K2(k, k′)ω(k′)− ω(k)K1(k, k′)ω(k′) where K1(k, k ′) := 4 1 (F−(k, k′) > 0)√ F−(k, k′) and K2(k, k′) := 2√ F+(k, k′) (3.36) for k, k′ ∈ [0, 1] and F±(k, k′) = ( cos(pik) + cos(pik′) )2 ± 4 sin(pik) sin(pik′). and the main result of [LS08] is the following: Proposition 3.8 ([LS08, Propositions 4.3 and 4.4.]). Let ψ : [0, 1] → R be given, and assume that there are C, p > 0 such that |ψ(k)| ≤ C (sinpik)p for all k ∈ [0, 1]. Then the kernels ψ(k)∗K2(k, k′)ψ(k′) and ψ(k)∗K1(k, k′)ψ(k′) define compact, self-adjoint integral operators in L2(T). We immediately conclude: Corollary 3.9. The kernel K0(k, k ′) = V −1/2(k)ω(k) ( 2K2(k, k ′)−K1(k, k′) ) ω(k′)V −1/2(k′) (3.37) 126 3.4. Properties of the operator L defines a compact self-adjoint operator in L2((0, 1)). As a consequence, the kernel K(k, k′) = V 1/2(k)K0(k, k′)V 1/2(k′) defines a compact self-adjoint operator from L2(T1, V (k) dk) onto L2(T1, V (k)−1 dk). In partic- ular, ˆ T |K(f)(k)|2V (k)−1 dk ≤ C ˆ T |f(k)|2V (k) dk. (3.38) Proof. Indeed, by Proposition 3.7 we have that V −1/2(k)ω(k) ≤ c2 (sinpik)1/6 and the claim follows from Proposition 3.8. Furthermore, we note that we have not used the full potential of Proposition (3.8). We can thus improve (3.38) as follows: Corollary 3.10. The kernel K˜0(k, k ′) := (sin(pik))−1/6+ηK0(x, k′) ( sin(pik′) )−1/6+η η > 0 defines a compact self-adjoint operator in L2((0, 1)). In particular, for all η > 0, there exists C(η) such that ˆ T |K(f)(k)|2(sin(pik))− 13+ηV (k)−1 dk ≤ C ˆ T |f(k)|2(sin(pik)) 13−ηV (k) dk (3.39) Proof. Using Proposition 3.7 we have that V −1/2(k)ω(k) (sinpik)−1/6+η ≤ c2 (sinpik)1/6 (sinpik)−1/6+η = c2 (sinpik) η the claim follows from Proposition 3.8. We have thus showed that L : L2(T1, V (k) dk) −→ L2(T1, V (k)−1 dk) was a bounded operator. Next, we characterize the kernel ofL: First, we note that given f ∈ L2(T1, V (k) dk), inequality (3.32) implies that if L(f) = 0 then ˆ ˆ ˆ ˆ δ(k + k1 − k2 − k3)δ(ω + ω1 − ω2 − ω3) × [ω3f3 + ω2f2 − ω1f1 − ωf ]2 dk dk1 dk2 dk3 = 0. So f must satisfy ω(k)f(k) + ω(k1)f(k1) = ω(k2)f(k2) + ω(k + k1 − k2)f(k + k1 − k2) 127 Anomalous energy transport in FPU-β chain whenever ω(k) + ω(k1) = ω(k2) + ω(k + k1 − k2). We also say that ω(k)f(k) must be a collision invariant. Such invariants have been char- acterized in [LS08]: Theorem 3.11 ([LS08]). A function ψ ∈ L1(T) is a collisional invariant if and only if there exists c1 and c2 such that ψ(k) = c1 + c2ω(k). As a consequence, we deduce: Corollary 3.12. The kernel of L is the two dimensional subspace of L2(T1, V (k) dk) spanned by the functions 1 and ω(k)−1 (note that both of those functions belongs to L2(T1, V (k) dk) thanks to (3.35)) Finally, the compactness of K and inequality (3.32) implies Lemma 3.13. There exists c0 > 0 such that − ˆ T1 L(f)f dk ≥ c0 ˆ V (k)|f −Π(f)|2 dk for all f ∈ L2(T1, V (k) dk), where Π(f) denotes the orthogonal projection of f onto ker(L). To summarize, we have thus showed: Proposition 3.14. The operator L : L2(T1, V (k) dk) −→ L2(T1, V (k)−1 dk) is bounded and satisfies: (i) The kernel of L has dimension 2 and is spanned by 1 and 1ω(k) . (ii) For all f ∈ L2(T1, V (k) dk), we have ˆ T1 L(f) dk = 0 and ˆ T1 1 ω(k) L(f) dk = 0. (3.40) (iii) There exists c0 > 0 such that − ˆ T1 L(f)f dk ≥ c0 ˆ V (k)|f −Π(f)|2 dk for all f ∈ L2(T1, V (k) dk), where Π(f) denotes the orthogonal projection of f onto ker(L). Note that the projection of f onto ker(L) can be written as Π(f) = T + S [〈V 〉ω(k)−1 − 〈V ω−1〉] 128 3.5. Proof of Theorem 3.4 with T = 1 〈V 〉 ˆ V (k)f(k) dk and S = 1 m0 ˆ [ 〈V 〉V (k) ω(k) − 〈V ω−1〉V (k) ] f(k) dk where m0 = 〈V 〉2〈V ω−2〉 − 〈V ω−1〉2〈V 〉 is a normalization constant. The operator Π is a continuous operator in L2(V (k) dk). We finish this section commenting on the existence of solutions for the equation for the sake of completeness: Proposition 3.15 (Cauchy Problem). There exists a unique solution in L∞((0,∞);L2(R×T)) for equation (3.24) with initial data f0 ∈ L2(R× T). Proof. A traditional method for solving the Cauchy problem for this type of equations uses an iterative scheme based on the mild formulation: f(t, x, k) = f0(x− ω′(k)t, k) + ˆ t 0 Lf(x− (t− s)ω′(k), s)ds together with the estimate ‖L(f)‖L2(R×T) ≤ C‖f‖L2(R×T). This last estimate is consequence of (3.38) and the boundedness of the function V . We refer to [AG13] and [Mou13] for further details on this method. 3.5 Proof of Theorem 3.4 3.5.1 A priori estimates As a first step in the proof of Theorem 3.4, we establish some a priori estimates. The coercivity property of L (Lemma 3.13) gives the following proposition: Proposition 3.16. Assume that f0 ∈ L2(R × T). Then, the function f ε(t, x, k), solution of (3.24) satisfies ||f ε(t)||L2(R×T) ≤ ||f0||L2(R×T) for all t ≥ 0. (3.41) Furthermore, f ε can be expanded as follows: f ε = Π(f ε) + ε4/5hε, (3.42) where ‖hε‖L2V ((0,∞)×R×T) ≤ C||f0||L2(R×T) (3.43) 129 Anomalous energy transport in FPU-β chain and Π(f ε) is the projection of f ε onto ker(L), given by Π(f ε)(t, x, k) = T˜ ε(t, x) + S˜ε(t, x) [〈V 〉ω(k)−1 − 〈V ω−1〉] with T˜ ε(t, x) = 1 〈V 〉 ˆ V (k)f ε(t, x, k) dk , S˜ε(t, x) = 1 m0 ˆ [ 〈V 〉V (k) ω(k) − 〈V ω−1〉V (k) ] f ε(t, x, k) dk (3.44) where T˜ ε, S˜ε are bounded in L∞((0,∞);L2(R)). Proof. Multiplying (3.24) by f ε and integrating with respect to x and k, we get 1 2 d dt ‖f ε(t)‖2L2(R×T1) − 1 εα ˆ R ˆ T1 L(f ε)f ε dk dx = 0. Integrating with respect to t and using (Lemma 3.13), we deduce 1 2 ‖f ε(t)‖2L2(R×T1) + c0 εα ˆ t 0 ˆ R ˆ T1 V (k)|f ε −Π(f ε)|2 dk dx ds ≤ 1 2 ‖f ε0 ||2L2(R×T1). which implies the proposition. The fact that T˜ ε, S˜ε ∈ L∞((0,∞);L2(R)) is a direct conse- quence of this estimate and Cauchy-Schwartz. Because the singular terms in Π(f ε) (those involving ω(k)−1) play a particular role in the sequel, we will prefer to write Π(f ε) as follows: Πf ε = T ε + 〈V 〉 ω S˜ε(x, t) with T ε(t, x) = T˜ ε(t, x)− S˜ε(t, x)〈V ω−1〉 Finally, we set Sε(t, x) = ε−3/5〈V 〉S˜ε(t, x), (3.45) leading to the following expansion of f ε: f ε(t, x, k) = T ε(t, x) + ε 3 5Sε(t, x)ω(k)−1 + ε 4 5hε(t, x, k). (3.46) Note that while T ε and hε are clearly bounded (in appropriate functional spaces) in view of Proposition 3.16, the scaling of Sε may seem arbitrary at this point. However, we will see later on that Sε defined as in (3.45) indeed converges to a non trivial function (in some weak sense). 130 3.5. Proof of Theorem 3.4 3.5.2 Laplace Fourier Transform As in [MMM11], the main tool in deriving the macroscopic equation for T is the use of the Laplace-Fourier transform. More precisely, we define f̂ ε(p, ξ, k) = ˆ R ˆ ∞ 0 e−pte−iξxf ε(t, x, k) dt dx. We also denote by f̂0(ξ, k) the Fourier transform of f0(x, k). Remark 3.17. We recall that the Fourier transform preserves the L2(R) norm (Parseval’s theorem). It is also easy to see that the Laplace transform of an L1 function is in L∞. However our functions are not L1 with respect to t. Instead, we will make use of the simple fact that for a given function g(t), its Laplace transform ĝ(p) satisfies |ĝ(p)| ≤ 1 p ‖g‖L∞(0,∞) and |ĝ(p)| ≤ 1√ 2p ‖g‖L2(0,∞) (3.47) for all p > 0. Taking the Laplace Fourier transform of the equation, we obtain: εαpf̂ ε − εαf̂0 + iεω′(k)ξf̂ ε = K(f̂ ε)− V f̂ ε which easily yields f̂ ε(p, ξ, k) = εα εαp+ V (k) + iεω′(k)ξ f̂0 + 1 εαp+ V (k) + iεω′(k)ξ K(f̂ ε). (3.48) We recall thatL(f) = K(f)−V f withK(f) = ´ K(k, k′)f(k′) dk′. The fact that ´ L(f) dk = 0 and ´ 1 ω(k)L(f) dk = 0 for all f implies V (k) = ˆ K(k′, k)dk′, V (k) ω(k) = ˆ K(k′, k) 1 ω(k′) dk′ Multiplying (3.48) by K(k′, k) and integrating with respect to k and k′, we get ˆ T K(f̂ ε)(k′)dk′ = ˆ T ˆ T εαK(k′, k) εαp+ V (k) + iεω′(k)ξ f̂0(ξ, k) dk dk ′ + ˆ T ˆ T K(k′, k) εαp+ V (k) + iεω′(k)ξ K(f̂ ε)(k) dk dk′ = ˆ T εαV (k) εαp+ V (k) + iεω′(k)ξ f̂0(ξ, k) dk + ˆ T V (k) εαp+ V (k) + iεω′(k)ξ K(f̂ ε)(k) dk. 131 Anomalous energy transport in FPU-β chain We deduce 0 = ˆ T V (k) εαp+ V (k) + iεω′(k)ξ f̂0(ξ, k) dk + ε−α ˆ T ( V (k) εαp+ V (k) + iεω′(k)ξ − 1 ) K(f̂ ε)(k) dk. (3.49) Similarly, multiplying (3.48) by K(k′, k) ε 3 5 ω(k′) , and we get: 0 = ε 3 5 ˆ T V (k) εαp+ V (k) + iεω′(k)ξ f̂0(ξ, k) ω(k) dk + ε−αε 3 5 ˆ T ( V (k) εαp+ V (k) + iεω′(k)ξ − 1 ) K(f̂ ε)(k) ω(k) dk. (3.50) Next, we write K(f̂ ε) = K(Π(f̂ ε)) +K(f̂ ε −Π(f̂ ε)) = VΠ(f̂ ε) +K(f̂ ε −Π(f̂ ε)) where we rewrite Π(f̂ ε) = T̂ ε + ε3/5 1 ω(k) Ŝε. We can thus rewrite (3.49) as follows: Fε1(f̂0) + aε1(p, ξ)T̂ ε(p, ξ) + aε2(p, ξ)Ŝε(p, ξ) +Rε1(p, ξ) = 0 (3.51) and (3.50) as follows: Fε2(f̂0) + aε2(p, ξ)T̂ ε(p, ξ) + aε3(p, ξ)Ŝε(p, ξ) +Rε2(p, ξ) = 0, (3.52) where for α = 8/5, we have: Fε1(f̂0) = ˆ T V (k) ε 8 5 p+ V (k) + iεω′(k)ξ f̂0(ξ, k) dk Fε2(f̂0) = ε 3 5 ˆ T V (k) ε 8 5 p+ V (k) + iεω′(k)ξ f̂0(ξ, k) ω(k) dk, aε1(p, ξ) := ε − 8 5 ˆ T ( V (k) ε 8 5 p+ V (k) + iεω′(k)ξ − 1 ) V (k) dk aε2(p, ξ) := ε − 8 5 ˆ T ( V (k) ε 8 5 p+ V (k) + iεω′(k)ξ − 1 ) ε 3 5V (k) ω(k) dk = ε−1 ˆ T ( V (k) ε 8 5 p+ V (k) + iεω′(k)ξ − 1 ) V (k) ω(k) dk 132 3.5. Proof of Theorem 3.4 aε3(p, ξ) := ε −1 ˆ T ( V (k) ε 8 5 p+ V (k) + iεω′(k)ξ − 1 ) ε 3 5V (k) ω(k)2 dk and Rε1(ξ, p) := ε − 8 5 ˆ T ( V (k) ε 8 5 p+ V (k) + iεω′(k)ξ − 1 ) K(f̂ ε −Π(f̂ ε))(k) dk Rε2(ξ, p) := ε −1 ˆ T ( V (k) ε 8 5 p+ V (k) + iεω′(k)ξ − 1 ) 1 ω(k) K(f̂ ε −Π(f̂ ε))(k) dk In order to prove the main theorem, we now need to pass to the limit in (3.51) and (3.52). The following three propositions give the necessary results for that. First, we have the following limits for the terms involving the initial data: Proposition 3.18. The following limits hold for all P ≥ 0: Fε1(f̂0)(ξ, p) −→ ˆ T f̂0(ξ, k) dk = T̂0(ξ) in L2((0, P )× R) Fε2(f̂0)(ξ, p) −→ 0 in L1((0, P )× R) when ε→ 0. Next, we pass to the limit in the symbol aεi (p, ξ): Proposition 3.19. The following limits hold pointwise (p, ξ) ∈ (0,∞) × R and strongly in Lploc((0,∞)× R) for all p ∈ (1,∞): aε1(p, ξ) −→ −p− κ1|ξ| 8 5 with κ1 = 6 5 ( pi v0 )3/5 ˆ ∞ 0 z3/5 z2 + 1 dz (3.53) aε2(p, ξ) −→ −κ2|ξ| with κ2 = 6 5 ˆ ∞ 0 1 z2 + 1 dz (3.54) aε3(p, ξ) −→ −κ3|ξ| 2 5 with κ3 = 6 5 (v0 pi )3/5 ˆ ∞ 0 z−3/5 z2 + 1 dz (3.55) Furthermore, aε1, a ε 2, a ε 3 ∈ L∞loc((0,∞)× R) uniformly with respect to ε. Finally, we need to show that the remainder terms, involving f ε −Π(f ε), go to zero: Proposition 3.20. For all 0 < a < P and K > 0, we have Rεi → 0 in L2((a, P )× (−K,K)) as ε→ 0 for i = 1, 2. The proofs of these three propositions are given in Section 3.5.3. 133 Anomalous energy transport in FPU-β chain Proof of Theorem 3.4. We are now ready to prove Theorem 3.4. First, using Proposition 3.16, we see that up to a subsequence, T ε(t, x) converges weakly to T (t, x) inL2((0, τ)×R) for all τ (the uniqueness of the limit will give the convergence of the whole sequence). Next, for a given test function ϕ(p, ξ) in D((0,∞)× R), we then have ˆ ∞ 0 ˆ R T̂ ε(p, ξ)ϕ(p, ξ) dξ dp = ˆ ∞ 0 ˆ R T ε(t, x)ϕ̂(t, x) dx dt (3.56) where ϕ̂ ∈ L2((0,∞)×R). This last fact is the classical Parseval inequality for the Fourier transform, while for the Laplace transform, it follows from Minkowski’s integral inequal- ity: (ˆ ∞ 0 (ˆ ∞ 0 e−ptϕ(p) dp )2 dt )1/2 ≤ ˆ ∞ 0 (ˆ ∞ 0 e−2pt dt )1/2 ϕ(p) dp ≤ ˆ ∞ 0 1√ 2p ϕ(p) dp <∞. Thus T̂ ε converges to T̂ in D′((0,∞) × R). Since T̂ ε is also bounded in L2loc((0,∞) × R) (using (3.47)), we deduce that (up to another subsequence) it converges weakly in L2loc((0,∞)× R) to T̂ . In order to derive the equation satisfied by T̂ , we need to pass to the limit in (3.51) and (3.52). However, we do not know that Sε (defined in (3.45)) is bounded in some functional space. So we multiply equation (3.51) by aε3 and (3.52) by a ε 2 and consider their difference, in order to get rid of the terms in Ŝε: 0 = aε3(p, ξ)Fε1(f̂0) + ( aε3(p, ξ)a ε 1(p, ξ)− (aε2(p, ξ))2 ) T̂ ε(p, ξ) + aε3(p, ξ)R ε 1(p, ξ)− aε2(p, ξ)Fε2(f̂0)− aε2(p, ξ)Rε2(p, ξ). Using Proposition 3.19, Proposition 3.20 and Proposition 3.18, we can now pass to the limit in this equation in D′((0,∞)× R) and deduce: −κ3|ξ|2/5T̂0 + ( −κ3|ξ|2/5(−p− κ1|ξ|8/5)− κ22|ξ|2 ) T̂ = 0 in D′((0,∞)× R). Furthermore, factorizing −κ3|ξ|2/5 in this last equation we get −κ3|ξ|2/5 ( T̂0 − pT̂ − (κ1 + κ 2 2 κ3 )|ξ|8/5T̂ ) = 0 in D′((0,∞)× R). This implies that the function g(p, ξ) := T̂0 − pT̂ − ( κ1 − κ 2 2 κ3 ) |ξ|8/5T̂ , (3.57) 134 3.5. Proof of Theorem 3.4 which belongs to L2loc((0,∞)× R), satisfies g(p, ξ) = 0 a.e. in (0,∞)× R which gives (3.26)-(3.27). To complete the proof of Theorem 3.4, it remains to show that f ε converges to T (t, x) (weakly in L∞((0,∞), L2(R × T))). Since f ε is bounded in L∞(0,∞;L2(R × T), and in view of the expansion (3.46), it is enough to show that ε3/5Sε converges to zero in some weak sense. This follows from Proposition 3.5, the proof of which uses equation (3.52) and some bounds from below on aε3(p, ξ) and will be detailed in Section 3.6. We end this section by proving that the diffusion coefficient κ is indeed positive: Lemma 3.21. The coefficients κ1, κ2 and κ3 are such that κ1 − κ 2 2 κ3 > 0. Proof. Indeed, this is equivalent to κ22 < κ1κ3 and using the explicit formula for κ1, κ2 and κ3, we see that this is equivalent to(ˆ ∞ 0 1 1 + z2 dz )2 < ˆ ∞ 0 z3/5 1 + z2 dz ˆ ∞ 0 z−3/5 1 + z2 dz which is an immediate consequence of Ho¨lder inequality. 3.5.3 Proofs of the asymptotic results We recall here that T denotes the torus R/Z and that ω(k) = | sin(pik)|. Since the dispersion relation ω is degenerate at k = 0 ± n, it will be easier in the computation below to work with k in the symmetric interval (−12 , 12) (when working with the interval (0, 1), we have to deal with both endpoints 0 and 1). Note that the function ω is even in that interval and that ω′(k) = sgn (k)pi cos(pik). Finally, Proposition 3.7 implies: Proposition 3.22. The function k 7→ V (k) is even and non-negative on the interval (−12 , 12). Furthermore the function W (k) := V (k)|k|−5/3 for k ∈ (−12 , 12) satisfies lim k→0 W (k) = w0 := v0pi 5/3 135 Anomalous energy transport in FPU-β chain and C−10 ≤W (k) ≤ C0 for some C0 > 0. Proof of Proposition 3.18. The first part of the proposition follows immediately from Lebesgue dominated convergence theorem, since∣∣∣∣∣ V (k)ε 85 p+ V (k) + iεω′(k)ξ ∣∣∣∣∣ = V (k)√( ε 8 5 p+ V (k) )2 + (εω′(k)ξ)2 ≤ 1 and V (k) ε 8 5 p+ V (k) + iεω′(k)ξ −→ 1 as ε→ 0. For the second part, we note that∣∣∣∣∣ V (k)ε 85 p+ V (k) + iεω′(k)ξ ∣∣∣∣∣ ≤ V (k)ε 85 p+ V (k) and so |Fε2(f̂0)|(ξ, p) ≤ ε 3 5 ˆ T V (k) ε 8 5 p+ V (k) f̂0(ξ, k) ω(k) dk ≤ Cε 35 ||f̂0(ξ, ·)||L∞(T) ˆ 1/2 0 |k|2/3 ε 8 5 p+ |k|5/3 dk ≤ Cε 35 ||f̂0(ξ, ·)||L∞(T)(1 + | ln(ε 8 5 p)|) (3.58) hence the result, since this last inequality implies (integrating with respect to ξ and p) ||Fε2(f̂0)||L1((0,P )×R) ≤ Cε 3 5 ||f0||L∞(R×T)P (1 + | ln(ε 8 5P )|). Proof of Proposition 3.19. First, we write 1− V (k) ε 8 5 p+ V (k) + iεω′(k)ξ = ε 8 5 p+ iεω′(k)ξ ε 8 5 p+ V (k) + iεω′(k)ξ = ε 8 5 p+ V (k) (ε 8 5 p+ V (k))2 + (εω′(k)ξ)2 ε 8 5 p + V iεω′(k)ξ (ε 8 5 p+ V (k))2 + (εω′(k)ξ)2 136 3.5. Proof of Theorem 3.4 + (εω′(k)ξ)2 (ε 8 5 p+ V (k))2 + (εω′(k)ξ)2 (3.59) Using the fact that V (k) = V (−k), ω′(−k) = −ω′(k), we deduce that aε1(p, ξ) :=− p ˆ 1 2 − 1 2 ε 8 5 p+ V (k) (ε 8 5 p+ V (k))2 + (εω′(k)ξ)2 V (k) dk − ε− 85 ˆ 1 2 − 1 2 (εω′(k)ξ)2 (ε 8 5 p+ V (k))2 + (εω′(k)ξ)2 V (k) dk. Dominated convergence immediately implies that the first term converges to −p, so we only have to consider the term dε(p, ξ) = ε− 8 5 ˆ 1 2 − 1 2 (εω′(k)ξ)2 (ε 8 5 p+ V (k))2 + (εω′(k)ξ)2 V (k) dk. For some δ ∈ (0, 12), we write dε(p, ξ) = dε1(p, ξ) + d ε 2(p, ξ) where dε1(p, ξ) = ε − 8 5 ˆ k ∈ (− 1 2 , 1 2 ) |k| ≥ δ (εω′(k)ξ)2 (ε 8 5 p+ V (k))2 + (εω′(k)ξ)2 V (k) dk ≤ Cε− 85 ˆ k ∈ (− 1 2 , 1 2 ) |k| ≥ δ (εξ)2 V (k) dk ≤ C(δ)|ξ|2ε 25 and dε2(p, ξ) = ε − 8 5 ˆ |k|≤δ (εω′(k)ξ)2 (ε 8 5 p+ V (k))2 + (εω′(k)ξ)2 V (k) dk = 2ε− 8 5 ˆ δ 0 (εpi cos(pik)ξ)2 (ε 8 5 p+W (k)|k|5/3)2 + (εpi cos(pik)ξ)2 W (k)|k|5/3 dk = 2ε− 8 5 ˆ δ 0 (pi cos(pik))2 (ε 3 5 p |ξ| +W (k) |k|5/3 ε|ξ| ) 2 + (pi cos(pik))2 W (k)|k|5/3 dk. We now do the change of variable w = |k|5/3 ε|ξ| , dk = 3 5 (ε|ξ|)3/5w−2/5dw, (3.60) 137 Anomalous energy transport in FPU-β chain which yields dε2(p, ξ) = 2ε − 8 5 ˆ δ5/3 ε|ξ| 0 zε(w)( ε 3 5 p |ξ| +W ε(w)w )2 + zε(w) W ε(w)ε|ξ|w3 5 (ε|ξ|)3/5w−2/5 dw = |ξ|8/5 6 5 ˆ δ5/3 ε|ξ| 0 zε(w)( ε 3 5 p |ξ| +W ε(w)w )2 + zε(w) W ε(w)w3/5 dw where zε(w) = (pi cos(pi(ε|ξ|w)3/5))2 W ε(w) = W ((ε|ξ|w)3/5). In particular, the integrand converges pointwise (for all w and ξ), as ε goes to zero, to pi2 (w0w) 2 + pi2 w0w 3/5 and it is bounded by pi2( C−10 w )2 + (pi cos(piδ))2 C0w 3/5. We deduce that |dε2(p, ξ)| ≤ C|ξ|8/5 for some constant C and that dε2(p, ξ) −→ |ξ|8/5 6 5 ˆ ∞ 0 pi2 (w0w) 2 + pi2 w0w 3/5 dw = κ1|ξ| 85 (recall that w0 = v0pi5/3) which concludes the proof of the first part. Note that we have also proved that |aε1(p, ξ)| ≤ p+ Cε 2 5 |ξ|2 + C|ξ|8/5. In particular, aε1(p, ξ) is bounded in L ∞ loc((0,∞) × R). Since it converges pointwise, a classical argument shows that it also converges strongly in Lploc((0,∞) × R) for all 0 < p <∞. The convergence of aε2 is proved similarly: Using (3.59), we find aε2(p, ξ) :=− ε 3 5 p ˆ 1 2 − 1 2 ε 8 5 p+ V (k) (ε 8 5 p+ V (k))2 + (εω′(k)ξ)2 V (k) ω(k) dk − ε−1 ˆ 1 2 − 1 2 (εω′(k)ξ)2 (ε 8 5 p+ V (k))2 + (εω′(k)ξ)2 V (k) ω(k) dk. 138 3.5. Proof of Theorem 3.4 The first term is bounded by ε 3 5 p ˆ 1 2 − 1 2 1 ε 8 5 p+ V (k) V (k) ω(k) dk ≤ Cε 35 p ˆ 1 2 0 1 ε 8 5 p+ C−10 k 5 3 k 2 3 dk ≤ Cε−1 ˆ 1 2 0 1 1 + ε− 8 5 p−1k 5 3 k 2 3 dk ≤ Cε−1p ε 85 ˆ Cε− 85 p−1 0 1 1 + w dw ≤ Cp ε 35 ln(1 + Cε− 85 p−1) and thus converges to zero as ε→ 0 (here we used the change of variablew = ε− 85 p−1k 53 ). For the second term the same decomposition of the integral in the interval |k| ≤ δ and |k| ≥ δ. The integral in |k| ≥ δ is bounded by C(δ)ε|ξ|2. For the integral in |k| ≤ δ, the change of variable (3.60) gives that it is bounded by C|ξ| and converges to |ξ|6 5 ˆ ∞ 0 pi2 (w0w) 2 + pi2 w0 pi dw = κ2|ξ|. Note that |aε2(p, ξ)| ≤ Cp ε 3 5 ln(1 + Cε− 8 5 p−1) + C(δ)ε|ξ|2 + C|ξ| (3.61) so aε2 ∈ L∞loc((0,∞)×R) implying, next to the pointwise convergence, the Lploc((0,∞)×R) strong convergence for 0 < p <∞. Finally, using (3.59), we find aε3(p, ξ) :=− ε 6 5 p ˆ 1 2 − 1 2 ε 8 5 p+ V (k) (ε 8 5 p+ V (k))2 + (εω′(k)ξ)2 V (k) ω(k)2 dk − ε− 25 ˆ 1 2 − 1 2 (εω′(k)ξ)2 (ε 8 5 p+ V (k))2 + (εω′(k)ξ)2 V (k) ω(k)2 dk. (3.62) The first term is bounded by ε 6 5 p ˆ 1 2 − 1 2 1 ε 8 5 p+ V (k) V (k) (ω(k))2 dk ≤ Cε 65 p ˆ 1 2 0 1 ε 8 5 p+ C−10 k 5 3 k− 1 3 dk ≤ Cε− 25 ˆ 1 2 0 1 1 + ε− 8 5 p−1k 5 3 k− 1 3 dk 139 Anomalous energy transport in FPU-β chain ≤ Cε 2425 p 45 ˆ Cε− 85 p−1 0 w− 3 5 1 + w dw ≤ Cε 2425 p 45 ˆ ∞ 0 w− 3 5 1 + w dw and thus converges to zero as ε → 0. For the second term, the same decomposition of the integral in the interval |k| ≤ δ and |k| ≥ δ. The integral in |k| ≥ δ is bounded by C(δ)ε8/5|ξ|. The integral in |k| ≤ δ, the change of variable (3.60) gives that it is bounded by C|ξ|2/5 and converges to |ξ| 25 6 5 ˆ ∞ 0 pi2 (w0w) 2 + pi2 w0 pi2 w− 3 5 dw = κ3|ξ| 25 . Analogously as in the previous cases, we have that |aε3(p, ξ)| ≤ Cε 24 25 p 4 5 + C(δ)ε8/5|ξ|+ C|ξ|2/5 so aε3 ∈ L∞loc((0,∞)×R) which, next to the pointwise convergence, implies theLploc((0,∞)× R) strong convergence p ∈ (0,∞). It only remain to prove Proposition 3.20. For that we will require the following lemma: Lemma 3.23. For all η ∈ (0, 13 ], we have ˆ T ∣∣∣∣∣ V (k)ε 85 p+ V (k) + iεω′(k)ξ − 1 ∣∣∣∣∣ 2 V (k)(sin(pik)) 1 3 −η dk ≤ C [ (ε 8 5 p) 8 5 + (ε|ξ|) 95− 3η5 + (ε|ξ|)2 ] (3.63) and ˆ T ∣∣∣∣∣ V (k)ε 85 p+ V (k) + iεω′(k)ξ − 1 ∣∣∣∣∣ 2 V (k) ω(k)2 (sin(pik)) 1 3 −η dk ≤ C [ (ε 8 5 p) 3 5 (1−η) + (εξ) 3 5 (1−η) + (ε|ξ|)2 ] (3.64) We note that when η = 13 (that is when we do not have the term (sin(pik)) 1 3 −η in the integral), then the integral behaves like ε 8 5 . As we will see below, this would be just enough to show that the remainder termRε1 is bounded, but not to show that it converges to zero. The improvement of the norm of K given by (3.39) is thus essential here. We first prove Proposition 3.20 (using Lemma 3.23), before giving the proof of Lemma 3.23: 140 3.5. Proof of Theorem 3.4 Proof of Proposition 3.20. Using (3.39), we get: |Rε1(p, ξ)| = ε− 8 5 ∣∣∣∣∣ ˆ T ( V (k) ε 8 5 p+ V (k) + iεω′(k)ξ − 1 ) K(f̂ ε −Π(f̂ ε))(k) dk ∣∣∣∣∣ ≤ ε− 85 ˆ T ∣∣∣∣∣ V (k)ε 85 p+ V (k) + iεω′(k)ξ − 1 ∣∣∣∣∣ 2 V (k)(sin(pik)) 1 3 −η dk 1/2 × (ˆ T K(f̂ ε −Π(f̂ ε))2(sin(pik))− 13+ηV −1(k) dk )1/2 ≤ Cε− 85 ˆ T ∣∣∣∣∣ V (k)ε 85 p+ V (k) + iεω′(k)ξ − 1 ∣∣∣∣∣ 2 V (k)(sin(pik)) 1 3 −η dk 1/2 × (ˆ T (f̂ ε −Π(f̂ ε))2(sin(pik)) 13−ηV (k) dk )1/2 and using (3.63), we deduce that for p < P and |ξ| ≤ K, we have |Rε1(p, ξ)| ≤ C(P,K)ε− 8 5 ε 9−3η 10 (ˆ T (f̂ ε −Π(f̂ ε))2V (k) dk )1/2 . For all 0 < a < P and K > 0, we deduce ˆ P a ˆ K −K |Rε1(p, ξ)|2 dξ dp ≤ C(P,K)ε− 16 5 ε 9−3η 5 ‖f̂ ε −Π(f̂ ε)‖2L∞((a,∞);L2V (T×R)) ≤ C(P,K) 1 2a ε −7−3η 5 ‖f ε −Π(f ε)‖2L2((0,∞);L2V (T×R)) ≤ C(a, P,K)ε−7−3η5 ε 85 ≤ C(a, P,K)ε 1−3η5 where we have used (3.47). Clearly, this implies Proposition 3.20 for i = 1. Proceeding similarly, we have that |Rε2(p, ξ)| ≤ Cε−1 ( (ε 8 5 p) 3 5 (1−η) + (εξ) 3 5 (1−η) + (ε|ξ|)2 )1/2 × (ˆ T (f̂ ε −Π(f̂ ε))2V (k)dk )1/2 (3.65) and therefore ˆ P a ˆ K −K |Rε2(p, ξ)|2 dp dξ ≤ C(P,K)ε−2ε 3(1−η) 5 ‖(f̂ ε −Π(f̂ ε))‖2L∞((a,∞);L2V (T×R)) 141 Anomalous energy transport in FPU-β chain ≤ C(a, P,K)ε−7−3η5 ‖f ε −Π(f ε)‖2L2((0,∞);L2V (T×R)) ≤ C(a, P,K)ε 1−3η5 (3.66) which converges to zero for any η ∈ (0, 13). Proof of Lemma 3.23. We write: ˆ T ∣∣∣∣∣ V (k)ε 85 p+ V (k) + iεω′(k)ξ − 1 ∣∣∣∣∣ 2 V (k)(sin(pik)) 1 3 −η dk = ˆ T ∣∣∣∣∣ ε 8 5 p+ iεω′(k)ξ ε 8 5 p+ V (k) + iεω′(k)ξ ∣∣∣∣∣ 2 V (k)(sin(pik)) 1 3 −η dk = ˆ T (ε 8 5 p)2 + (εω′(k)ξ)2 (ε 8 5 p+ V (k))2 + (εω′(k)ξ)2 V (k)(sin(pik)) 1 3 −η dk = I1 + I2 where I1 := ˆ T (ε 8 5 p)2 (ε 8 5 p+ V (k))2 + (εω′(k)ξ)2 V (k)(sin(pik)) 1 3 −η dk ≤ 2 ˆ 1/2 0 (ε 8 5 p)2 (ε 8 5 p+ V (k))2 V (k)(sin(pik)) 1 3 −η dk ≤ 2 ˆ 1/2 0 (ε 8 5 p)2 (ε 8 5 p+ k5/3)2 k5/3 dk (note we do not need to use the (sin(pik)) 1 3 −η to control this term) and I2 := ˆ T (εω′(k)ξ)2 (ε 8 5 p+ V (k))2 + (εω′(k)ξ)2 V (k)(sin(pik)) 1 3 −η dk ≤ 2 ˆ 1/4 0 (εω′(k)ξ)2 V (k)2 + (εω′(k)ξ)2 V (k) (sin(pik)) 1 3 −η dk + 2 ˆ 1/2 1/4 (εpiξ)2 V (k) dk ≤ C ˆ 1/4 0 (εξ)2 k10/3 + (εξ)2 k2−η dk + Cε2|ξ|2 here the (sin(pik)) 1 3 −η is essential. Using the change of variable w = k 5/3 ε8/5p in I1 and w = k 5/3 εξ in I2, we find I1 ≤ C(ε8/5p)8/5 ˆ ∞ 0 w3/5 (1 + w)2 dw I2 ≤ C(εξ)9/5−3η/5 ˆ ∞ 0 w4/5−3η/5 1 + w2 dw + Cε2|ξ|2 142 3.5. Proof of Theorem 3.4 where the integral in the right hand side are clearly finite (recall that η ∈ (0, 13). Inequality (3.63) follows. We now proceed similarly to prove (3.64): First, we write ˆ T ∣∣∣∣∣ V (k)ε 85 p+ V (k) + iεω′(k)ξ − 1 ∣∣∣∣∣ 2 V (k) ω(k)2 (sin(pik)) 1 3 −η dk = ˆ T (ε 8 5 p)2 + (εω′(k)ξ)2 (ε 8 5 p+ V (k))2 + (εω′(k)ξ)2 V (k) ω(k)2 (sin(pik)) 1 3 −η dk = I˜1 + I˜2 where I˜1 := ˆ T (ε 8 5 p)2 (ε 8 5 p+ V (k))2 + (εω′(k)ξ)2 V (k) ω(k)2 (sin(pik)) 1 3 −η dk ≤ 2 ˆ 1/2 0 (ε 8 5 p)2 (ε 8 5 p+ V (k))2 V (k) ω(k)2 (sin(pik)) 1 3 −ηdk ≤ 2 ˆ 1/2 0 (ε 8 5 p)2 (ε 8 5 p+ k5/3)2 k−η dk and I˜2 := ˆ T (εω′(k)ξ)2 (ε 8 5 p+ V (k))2 + (εω′(k)ξ)2 V (k) ω(k)2 (sin(pik)) 1 3 −η dk ≤ 2 ˆ 1/2 0 (εω′(k)ξ)2 V (k)2 + (εω′(k)ξ)2 V (k) ω(k)2 (sin(pik)) 1 3 −η dk ≤ 2 ˆ 1/4 0 (εξ)2 k10/3 + (εξ)2 k−η dk + C(ε|ξ|)2 Using the change of variable w = k 5/3 ε8/5p in I˜1 and w = k 5/3 εξ in I˜2, we find I˜1 ≤ C(ε 85 p) 35 (1−η) ˆ ∞ 0 w−3/5η (1 + w)2 dw I˜2 ≤ C(εξ) 35 (1−η) ˆ ∞ 0 w−2/5−3η/5 1 + w2 dw + C(ε|ξ|)2 which yields (3.64). 143 Anomalous energy transport in FPU-β chain 3.6 Proof of Proposition 3.5 The proof of Proposition 3.5 relies on the following crucial bound: Lemma 3.24. There exists a constant c such that for all K and for all ε such that εK ≤ 1, the following lower bound holds |aε3(p, ξ)| ≥ c ε 6 25 p 2 5 + c|ξ| 25 for 0 ≤ p ≤ K, |ξ| ≤ K. (3.67) Proof of Lemma 3.24. We recall that aε3(p, ξ) is given by (3.62). In particular, we note that for all (p, ξ) 6= (0, 0), we have aε3(p, ξ) < 0. Furthermore, we can write (using the fact that all the terms in (3.62) have the same sign): −aε3(p, k) ≥ ε 6 5 p ˆ 1 4 0 ε 8 5 p+ V (k) (ε 8 5 p+ V (k))2 + (εω′(k)ξ)2 V (k) ω(k)2 dk + ε− 2 5 ˆ 1 4 0 (εω′(k)ξ)2 (ε 8 5 p+ V (k))2 + (εω′(k)ξ)2 V (k) ω(k)2 dk. Using the fact that for k ∈ (0, 1/4) we have C−10 |k|5/3 ≤ V (k) ≤ C0|k|5/3, pi√2 ≤ ω′(k) ≤ pi and pi2k ≤ ω(k) ≤ pik, we obtain the following lower bound (for some constant c > 0): −aε3(p, k) ≥ c ε 6 5 p ˆ 1 4 0 |k|4/3 (ε 8 5 p+ C0|k|5/3)2 + (εpiξ)2 dk + c ε− 2 5 ˆ 1 4 0 (εpiξ)2k−1/3 (ε 8 5 p+ C0|k|5/3)2 + (εpiξ)2 dk. (3.68) From now on, we fix K and assume that 0 < p ≤ K and that |ξ| ≤ K. We also assume that ε is such that εK ≤ 1. In order to establish (3.67), we consider two cases, and in each case we use only one of the integrals in (3.68): (i) First, assume that p and ξ are such that |ξ| ≤ ε 35 p. (3.69) Then, using only the first integral in (3.68), we get (using (3.69)): −aε3(p, k) ≥ c ε 6 5 p ˆ 1 4 0 |k|4/3 (ε 8 5 p+ C0|k|5/3)2 + (piε 85 p)2 dk and the change of variable w = (ε 8 5 p)− 3 5k yields −aε3(p, k) ≥ c ε 6 5 p (ε 8 5 p) 7 5 (ε 8 5 p)2 ˆ 1 4(ε 8 5 p) 3 5 0 |w|4/3 (1 + C0|w|5/3)2 + pi2 dw 144 3.6. Proof of Proposition 3.5 and using the fact that ε 8 5 p ≤ 1, we deduce (for a different constant c): −aε3(p, k) ≥ c ε 6 25 p 2 5 . Finally, using (3.69), we also get −aε3(p, k) ≥ c|ξ| 2 5 and so (3.67) holds in this case. (ii) Next, we assume that p and ξ are such that ε 3 5 p ≤ |ξ| (3.70) and using only the second integral in (3.68), we get (using (3.70)): −aε3(p, k) ≥ c ε− 2 5 ˆ 1 4 0 (εpiξ)2k−1/3 (ε|ξ|+ C0|k|5/3)2 + (εpiξ)2 dk and the change of variable w = (ε|ξ|)− 35k, yields: −aε3(p, k) ≥ c ε− 2 5pi2 ˆ 1 4(εξ) 3 5 0 (εξ) 2 5w−1/3 (1 + C0|w|5/3)2 + pi2 dw ≥ c|ξ| 25 (using the fact that ε|ξ| ≤ 1). Finally, using (3.70), we also get −aε3(p, k) ≥ c ε 6 25 p 2 5 and so (3.67) holds also in this case. Proof of Proposition 3.5. We use equation (3.52) to determine Ŝε: Ŝε = 1 aε3 ( −Fε2(f̂0)−Rε2 − aε2T̂ ε ) . (3.71) Note that we can do this since aε3(p, ξ) < 0 as long as p and ξ are not simultaneously zero. We now need to show that we can pass to the limit in all the terms in the right hand side. First, using Lemma 3.24 and the estimate (3.61), we deduce that for a given K and 145 Anomalous energy transport in FPU-β chain for all ε ≤ K−1, we have aε2(p, ξ) aε3(p, ξ) = ∣∣∣∣aε2(p, ξ)aε3(p, ξ) ∣∣∣∣ ≤ Cpε3/5 ln(1 + Cε−8/5p−1) cε 6 25 p 2 5 + Cε|ξ|2 c|ξ| 25 + C|ξ| c|ξ| 25 ≤ Cp 35 ε 925 ln(1 + Cε−8/5p−1) + Cε|ξ| 85 + C|ξ| 35 ≤ C(K) for all 0 ≤ p ≤ K and |ξ| ≤ K. Furthermore, this uniform bound, together with Proposi- tion 3.19 implies that aε2(p, ξ) aε3(p, ξ) −→ κ2 κ3 |ξ|3/5 pointwise and in Lploc((0,∞)× R) strong. Next, for ε sufficiently small we can use Lemma 3.24 along with the estimates on Fε2(fˆ0) in (3.58) to conclude that 1 aε3 Fε2(fˆ0)→ 0 in D′((0,∞)× R). Finally, we need to bound the quantity∣∣∣∣Rε2(p, ξ)aε3(p, ξ) ∣∣∣∣ . For that, we fix 0 < a < P and for p ∈ (a, P ) and |ξ| ≤ K, estimate (3.65) then implies |Rε2(p, ξ)| ≤ Cε−1 ( ε 24 25 (1−η) + (εξ) 3 5 (1−η) )1/2(ˆ T (f̂ ε −Π(f̂ ε))2V (k)dk )1/2 ≤  Cε−1ε 12 25 (1−η) (´ T(f̂ ε −Π(f̂ ε))2V (k)dk )1/2 if |ξ| ≤ ε3/5 Cε−1(εξ) 3 10 (1−η) (´ T(f̂ ε −Π(f̂ ε))2V (k)dk )1/2 if |ξ| ≥ ε3/5 and we are going to use the following consequence of Lemma 3.24: −aε3(p, ξ) ≥  c(a)ε6/25 if |ξ| ≤ ε3/5 c|ξ|2/5 if |ξ| ≥ ε3/5 We deduce ∣∣∣∣Rε2(p, ξ)aε3(p, ξ) ∣∣∣∣ ≤  Cε −19−12η 25 (´ T(f̂ ε −Π(f̂ ε))2V (k)dk )1/2 if |ξ| ≤ ε3/5 Cε−1ε 3 10 (1−η)|ξ|−1−3η10 (´ T(f̂ ε −Π(f̂ ε))2V (k)dk )1/2 if |ξ| ≥ ε3/5 . 146 3.7. Appendix: Origin of the collision frequency Finally, using the condition |ξ| ≥ ε3/5 in the second case, we deduce that∣∣∣∣Rε2(p, ξ)aε3(p, ξ) ∣∣∣∣ ≤ C(a, p,K)ε−19−12η25 (ˆ T (f̂ ε −Π(f̂ ε))2V (k)dk )1/2 for all p ∈ (a, P ) and |ξ| ≤ K. We deduce(ˆ P a ˆ K −K ∣∣∣∣Rε2(p, ξ)aε3(p, ξ) ∣∣∣∣2 dξ dp )1/2 ≤ C(a, P,K)ε−19−12η25 ‖(f̂ ε −Π(f̂ ε))‖L∞((a,∞);L2V (T×R)) ≤ C(a, P,K)ε−19−12η25 ‖f ε −Π(f ε)‖L2((0,∞);L2V (T×R)) ≤ C(a, P,K)ε 1−12η25 which goes to zero as ε→ 0. We can now pass to the limit in (3.71) to conclude that Ŝε −→ Ŝ = κ2 κ3 |ξ|3/5T̂ in D′((0,∞)× R) which completes the proof of Proposition 3.5. 3.7 Appendix: Origin of the collision frequency Microscopic dynamics. Consider the microscopic level with the Hamiltonian H(q, p) = 1 2 ∑ i∈Z p2i + 1 8 ∑ i∈Z (qi+1 − qi)2 + 1 4 β ∑ i∈Z (qi+1 − qi)4 which gives the following dynamics d dt qi(t) = pi(t) (3.72) d dt pi(t) = 1 4 qi+1 − 1 2 qi + 1 4 qi−1 +β(qi+1 − qi)3 − β(qi − qi−1)3. Dispersion relation. The dispersion relation is defined in [Spo06b] expression (2.14). The dispersion relation comes only from the harmonic part of the potential. In our case, from Vharm = 1 8 ∑ i∈Z (qi+1 − qi)2. 147 Anomalous energy transport in FPU-β chain Consider the discrete Fourier Transform defined as f̂(k) = ∑ x∈Z e−i2pikxfx. The wave vector k lies in the torus T1 = R/Z. Alternatively, we can take k ∈ I = [0, 1] and assume that all functions are 1-periodic with respect to k. We compute the discrete Fourier Transform of ∂ ∂qi Vharm = −1 4 qi+1 + 1 2 qi − 1 4 qi−1 to obtain ∂̂ ∂qi Vharm(k) = 1 2 − 1 4 ( ei2pik + e−i2pik ) q̂(k) := √ ω(k)q̂(k). The dispersion relation is given by ω(k) = √ 1 2 (1− cos(2pik)) = sin(pik). Note that ω(k) ≥ 0 for k ∈ I , and one can indeed look at ω as a 1-periodic function defined on R by ω(k) = | sin(pik)| rather than a function defined on T1. However, ω is not differentiable at k = 0. 3.7.1 Four phonons collision operator. The kinetic limit of the previous hamiltonian will give the 4-phonon Boltzmann equation ∂tW + ω ′(k)∂xW = C(W ) (3.73) as we already saw in Section 3.1, with C(W ) = 36pi ˆ ˆ ˆ F (k, k1, k2, k3) 2δ(k + k1 − k2 − k3)δ(ω + ω1 − ω2 − ω3) [W1W2W3 +WW2W3 −WW1W3 −WW1W2] dk1 dk2 dk3. (3.74) The dispersion relation ω and the collision frequency F depend on the shape of the origi- nal Hamiltonian. In most frameworks an on-site potential for the Hamiltonian is consid- ered. This gives rise to a collision kernel F of the form F (k, k1, k2, k3) 2 = (ω(k)ω(k1)ω(k2)ω(k3)) −1. However, for FPU-β case the potential depends on the nearest neighbours (3.72)), 148 3.7. Appendix: Origin of the collision frequency giving F (k, k1, k2, k3) 2 = 3∏ i=0 2 sin2(piki) ω(ki) . We explain here how this expression is obtained. Origin of the collision frequency. From the Hamiltonian dynamics (3.72), we define as in [Spo06b] a(k) = 1√ 2 (√ ωqˆ(k) + i 1√ ω pˆ(k) ) . The collision frequency appears in the dynamics of a: d dt a(k, σ) = −iσω(k)a(k, σ) − iσβ ∑ σ′∈{±1}3 ˆ [0,1]3 d3k′ δ k − 3∑ j=1 k′j F (k′) 3∏ j=1 a(k′j , σ ′ j) Therefore we compute d dt a(k) = 1√ 2 √ ω d dt qˆ(k) + i 1√ 2 √ ω pˆ(k) = 1√ 2 √ ωpˆ(k) + i 1√ 2 √ ω 1 4 ( e2piikqˆ(k)− 2qˆ(k) + e−2piikqˆ(k) ) + iβ 1√ 2 √ ω ( ̂(qi+1 − qi)3 − ̂(qi − qi−1)3 ) ︸ ︷︷ ︸ =:Qˆ = 1√ 2 √ ωpˆ(k) + i 1√ 2 √ ω ω2 ˆq(k) + iβQˆ = −iωa(k, t) + iβQˆ. The first term will give the transport part in equation (3.73) and the β part gives the collision term. Now we study Qˆ. We have that ̂(qi+1 − qi)3 = ( (e2piik − 1)qˆ(k) )∗3 = ˆ ˆ ˆ (e1 − 1)(e2 − 1)(e3 − 1)qˆ1qˆ2qˆ3δ(k − k1 − k2 − k3)dk1dk2dk3 denoting ei = e 2piiki and qˆi = qˆ(ki). Therefore ̂(qi+1 − qi)3 − ̂(qi − qi−1)3 = ˆ ( (e1 − 1)(e2 − 1)(e3 − 1)− (1− e1)(1− e2)(1− e3) ) 149 Anomalous energy transport in FPU-β chain × qˆ1qˆ2qˆ3δ(k − k1 − k2 − k3)dk1dk2dk3 Now we use that qˆi = 1√ 2 √ ω(ki) ( a(ki) + a(−ki) ) ︸ ︷︷ ︸ =:Ai so that Qˆ = ˆ ( (e1 − 1)(e2 − 1)(e3 − 1)− (1− e1)(1− e2)(1− e3) ) ( √ 2)4 √ ω(k)ω(k1)ω(k2)ω(k3)︸ ︷︷ ︸ =:Φ A1A2A3δ(k−k1−k2−k3)dk1dk2dk3 (3.75) Φ is what will give the collision frequency which is the square of Φ. Now by observing that e2piiki − 1 = e2piiki ( 1− e−2piiki ) i.e. (ei − 1) = ei(1− ei) we can rewrite the numerator of Φ as N(k, k1, k2, k3) = e1e2e3(1− e1)(1− e2)(1− e3)− (1− e1)(1− e2)(1− e3) since k − k1 − k2 − k3 = 0 in (3.75) we have that e1e2e3 = e 2pii(k1+k2+k3) = e2piik so N(k, k1, k2, k3) = (ek − 1)(1− e1)(1− e2)(1− e3). Using analogously that 1− ei = ei(ei − 1) we have that also N(k, k1, k2, k3) = (1− ek)(e1 − 1)(e2 − 1)(e3 − 1). So summing the two previous expressions we check that N is a real number since one is the conjugate of the other, i.e., N(k, k1, k2, k3) = 1 2 ( (ek − 1)(1− e1)(1− e2)(1− e3) + (1− ek)(e1 − 1)(e2 − 1)(e3 − 1) ) = Re ( (ek − 1)(1− e1)(1− e2)(1− e3) ) . To compute the collision frequency, we square expression Φ. However since expression N is a real number, squaring it corresponds to taking the modulus square that we will 150 3.7. Appendix: Origin of the collision frequency denote by | · |. We obtain F (k, k1, k2, k3) = |Φ|2 = |(ek − 1)(1− e1)(1− e2)(1− e3)| 2( ( √ 2)4 √ ω(k)ω(k1)ω(k2)ω(k3) )2 = 3∏ j=0 |ei − 1|2 2ω(ki) = 3∏ j=0 4 sin2(piki) 2ω(ki) where we have used that |ei − 1|2 = (cos(2piki)− 1)2 + sin2(2piki) = cos2(2piki) + 1− 2 cos(2piki) + sin2(2piki) = 2− 2 cos(2piki) = 4 sin2(piki). 151 Part II Wave turbulence theory and mean-field limits for stochastic particle systems 153 Chapter 4 Isotropic Wave Turbulence with simplified kernels: existence, uniqueness and mean-field limit for a class of instantaneous coagulation-fragmentation processes This work has been done under the supervision of Professor James Norris. The isotropic 4-wave kinetic equation is considered in its weak formulation using model homogeneous kernels. Existence and uniqueness of solutions is proven in a par- ticular setting. We also consider finite stochastic particle systems undergoing instanta- neous coagulation-fragmentation phenomena and give conditions in which this system approximates the solution of the equation (mean-field limit). Contents 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 4.1.1 The 4-wave kinetic equation. . . . . . . . . . . . . . . . . . . . . . 157 4.1.2 The simplified weak isotropic 4-wave kinetic equation. . . . . . . . 158 4.1.2.1 Summary of results . . . . . . . . . . . . . . . . . . . . . 161 4.1.3 Some notes on the physical theory of Wave Turbulence . . . . . . 164 4.2 Existence of solutions for unbounded kernel . . . . . . . . . . . . . . . 168 4.2.1 Proof of Theorem 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . 169 4.3 Mean-field limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 4.3.1 The instantaneous coagulation-fragmentation stochastic process 179 4.3.2 First result on mean-field limit . . . . . . . . . . . . . . . . . . . . 181 155 Isotropic Wave Turbulence and mean-field limits 4.3.2.1 Mean-field limit for bounded jump kernel . . . . . . . . 181 4.3.2.2 Proof of Theorem 4.18 . . . . . . . . . . . . . . . . . . . . 183 4.3.2.3 Proof of Theorem 4.7 (unbounded kernel) . . . . . . . . 189 4.3.3 Second result on mean-field limit . . . . . . . . . . . . . . . . . . . 190 4.3.3.1 A coupling auxiliary process . . . . . . . . . . . . . . . . 190 4.3.3.2 Proof of Theorem 4.8 . . . . . . . . . . . . . . . . . . . . 193 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 4.5 Appendix: Some properties of the Skorokhod space . . . . . . . . . . . 198 4.6 Appendix: Formal derivation of the weak isotropic 4-wave kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 Notation R+ = [0,∞); B = space of bounded measurable functions with bounded support; D = {(ω1, ω2, ω3) ∈ R3+ |ω1 + ω2 ≥ ω3}; k wavevector, it belongs to RN ; ω(k) dispersion relation; T = T (k1,k2,k3,k) interaction coefficient; P(R+) space of probability measures in R+ M(R+) set of finite measures on R+. 4.1 Introduction Wave turbulence ([ZDP04, ZLF92, Naz11], [S+06, Entry turbulence]) describes weakly non-linear systems of dispersive waves. The present work focuses in the case of 4 inter- acting waves. We start with a brief presentation of the general 4-wave kinetic equation and move quickly to consider the isotropic case with simplified kernels, which is the object of study of the present work, and present the main results. We give a brief account on the theory of wave turbulence in Section 4.1.3. The rest of the text consists on the proofs of the main theorems. 156 4.1. Introduction 4.1.1 The 4-wave kinetic equation. Using in shorthand ni = n(ki, t), nk = n(k, t), ωi = ω(ki) and ω = ω(k), the 4-wave kinetic equation is given by d dt n(k, t) = 4pi ˆ R3N T 2 (k1,k2,k3,k)(n1n2n3 + n1n2nk − n1n3nk − n2n3nk) (4.1) ×δ(ω1 + ω2 − ω3 − ω)δ(k1 + k2 − k3 − k)dk1dk2dk3. where k ∈ RN is called wavevector; the function n = n(k, t) can be interpreted as the spectral density (in k-space) of a wave field and it is called energy spectrum; ω(k) is the dispersion relation; and T 123k := T (k1,k2,k3,k) is the interaction coefficient. E = ˆ RN ω(k)n(k)dk, W = ˆ RN n(k)dk correspond to the total energy and the waveaction (total number of waves), respectively. These two quantities are conserved formally. Properties of the dispersion relation and the interaction coefficient. ω(k) and T123k are homogeneous, i.e., for some α > 0 and β ∈ R ω(ξk) = ξαω(k), T (ξk1, ξk2, ξk3, ξk) = ξ βT (k1,k2,k3,k) ξ > 0. Moreover the interaction coefficient possesses the following symmetries T 123k = T 213k = T 12k3 = T 3k12. Example: shallow water. In the case of shallow water we deal with weakly-nonlinear waves on the surface of an ideal fluid in an infinite basin of constant depth h small. In this case ([Zak99]) we have that α = 1, β = 2, dimension is 2 and T (k1,k2,k3,k) = − 1 16pi2h 1 (k1k2k3k)1/2 [(k1 · k2)(k3 · k) + (k1 · k3)(k2 · k) + (k1 · k)(k2 · k3)] . (4.2) In general T will be given by very complex expressions, see for example [ZLF92]. Resonant conditions and the δ distributions. The delta distributions appearing in equa- tion (4.1) correspond to the so-called resonant conditions: k1 + k2 = k3 + k 157 Isotropic Wave Turbulence and mean-field limits ω(k1) + ω(k2) = ω(k3) + ω(k). This imposes the conservation of energy and momentum in the wave interactions. 4.1.2 The simplified weak isotropic 4-wave kinetic equation. We focus our study on the weak formulation of the isotropic 4-wave kinetic equation de- fined against functions in B(RN ); the set of bounded measurable functions with bounded support in RN . More specifically, we assume that n(k) = n(k) is a radial function (isotropic). Then, using the relation ω(k) = kα, we study the evolution of the angle-averaged frequency spectrum µ = µ(dω) which corresponds to µ(dω) := |SN−1| α ω N−α α n(ω1/α)dω, where SN−1 is the N dimensional sphere. The total number of waves (waveaction) and the total energy are now expressed respectively as W = ˆ ∞ 0 µ(dω) (4.3) E = ˆ ∞ 0 ωµ(dω). (4.4) The weak form of the isotropic equation is given formally by µt = µ0 + ˆ t 0 Q(µs, µs, µs) ds (4.5) where Q is defined against functions f ∈ B(R+) as 〈f,Q(µ, µ, µ)〉 = 1 2 ˆ D µ(dω1)µ(dω2)µ(dω3)K(ω1, ω2, ω3) ×[f(ω1 + ω2 − ω3) + f(ω3)− f(ω2)− f(ω1)] where D := {R3+ ∩ (ω1 + ω2 ≥ ω3)}. See appendix 4.6 for the formal derivation of this equation. Formally K = K(ω1, ω2, ω3) is written as K(ω1, ω2, ω3) = 8pi α|SN−1|4 (ω1 + ω2 − ω3) N−α α (4.6) ˆ (SN−1)4 ds1ds2ds3dsT 2 (ω 1/α 1 s1, ω 1/α 2 s2, ω 1/α 3 s3, (ω1 + ω2 − ω3)1/αs) ×δ(ω1/α1 s1 + ω1/α2 s2 − ω1/α3 s3 − (ω1 + ω2 − ω3)1/αs). 158 4.1. Introduction Notice that formally K is homogeneous of degree λ := 2β − α α . (4.7) Our starting point is equation (4.5) considering simplified kernels K. In this work we do not study the relation between the interaction coefficient T and K. Specifically, we will consider the following type of kernels: Definition 4.1. We say that K is a model kernel if • K : R3+ → R+; • K is continuous in R3+ = [0,∞)3; • K is homogeneous of degree λ; • K(ω1, ω2, ω3) = K(ω2, ω1, ω3) for all (ω1, ω2, ω3) ∈ R3+. Some examples of model kernels are: K(ω1, ω2, ω3) = 1 2 (ωp1ω q 2ω r 3 + ω q 1ω p 2ω r 3) with p+ q + r = λ, K(ω1, ω2, ω3) = (ω1ω2ω3) λ/3, (4.8) K(ω1, ω2, ω3) = 1 3 (ωλ1 + ω λ 2 + ω λ 3 ). The main question we want to address is: FOR WHICH TYPES OF KERNELS K THERE IS EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR EQUATION (4.5) AND, MOREOVER, CAN THIS SOLUTION(S) BE TAKEN AS THE MEAN-FIELD LIMIT OF A SPECIFIC STOCHASTIC PARTICLE SYSTEM? The present work gives a positive answer for a particular class of kernels as explained in the next section, but first, for the motivation of the problem, we need to answer the two following questions: a) Why is it relevant to study the weak isotropic 4-wave kinetic equation with simpli- fied kernels? The present work is inspired on the article [Con09] from the physics literature on wave turbulence. In [Con09] the author works with the 3-wave kinetic equation and considers its isotropic version also assuming simplified kernels. The idea is that the 3-wave kinetic equation can be interpreted as a process where particles coagulate and fragment. This interpretation allows to use numerical methods coming from the theory 159 Isotropic Wave Turbulence and mean-field limits of coagulation-fragmentation processes, which can be applied to this type of simplified kernels. As in [Con09], ignoring the specific shape of the interaction coefficient T is not un- common in the wave turbulence literature; in general the shape of T is too complex, too messy to extract information. Moreover, the most important feature in wave turbulence, the steady states called KZ-spectrum, depend only on the parameters α, β and N . That is why in the physics literature T plays a secondary role, sometimes no role at all. It is believed that only the asymptotic scaling properties of the kernel will affect the asymptotic behaviour of the solution. This is similar to what happens in the case of the Smoluchowski’s coagulation equation, where homogeneous kernels give rise to self- similar solutions (scaling solutions) in some cases. The hypothesis that solutions become self-similar in the long run under the presence of an homogeneous kernel is called dy- namical scaling hypothesis, see [MC11] for more on this. In the case of wave turbulence we expect this self-similar solutions to correspond to the steady states given by the KZ- spectrum. Proving the dynamical scaling hypothesis for the simplified isotropic 4-wave kinetic equation under the assumptions of Theorem 4.6 (existence of solutions) will imply prov- ing the validity of the KZ-spectrum for this simplified kernels (if there is correspondence between the two). This would provide a great indication of the mathematical validity of the theory of wave turbulence. b) Why consider the isotropic case? There are examples in the physics literature where the phenomena are considered to behave isotropically (like in Langmuir waves for isotropic plasmas and shallow water with flat bottom). The main reason though to consider the isotropic case is that it makes easier to get a mean-field limit from discrete stochastic particle systems. Suppose that we want to find a discrete particle system that approximates the dynamics of (4.1). For given waves with wavenumbers k1,k2,k3, we want to see if they interact. On one hand, due to the resonance conditions k defined as k = k1 + k2 − k3 is uniquely determined. On the other hand, on top we must add the constraint ω = ω1 + ω2 − ω3 and this in general will not be satisfied. Therefore, if we consider systems with a finite number of particles, in general, interactions will not occur and the dynamics will be con- stant. We go around this problem by considering the isotropic case. By assuming that n = 160 4.1. Introduction n(k) is a rotationally invariant function, we add the degree of freedom that we need. 4.1.2.1 Summary of results Next we summarise the main results in the present work. These results are the analogous ones presented in the papers [Nor99, Nor00] for the Smoluchowski equation (coagulation model). Remark 4.2 (Strategy). We will adapt the proofs by Norris in [Nor99] and [Nor00] for coagulation phenomena. In the proof by Norris in [Nor99] sublinear functions ϕ : R+ → R+ are used, i.e., ϕ(λx) ≤ λϕ(x), λ ≥ 1 ϕ(x+ y) ≤ ϕ(x) + ϕ(y). These functions are the key to get bounds because of the following property: let (µnt )t≥0 be a stochastic coagulation process with n particles, if initially 〈ϕ, µn0 〉 ≤ Λ for some Λ <∞, for all n ∈ N, then 〈ϕ, µnt 〉 ≤ Λ for all n, t. Actually, what we obtain is that 〈ϕ, µnt 〉 ≤ 〈ϕ, µn0 〉 thanks to the sublinearity of ϕ; say that two particles of masses x, y ∈ R+ coagulate creating a particle of mass x+ y, then ϕ(x+ y) ≤ ϕ(x) + ϕ(y) (4.9) by sublinearity. In general, this idea to get bounds cannot be applied to the type of stochastic particle processes that we are going to consider because they also include fragmentation pheno- mena; we will have that in an interaction two particles of masses ω1, ω2 ∈ R+ disappear and two particles of masses ω1 + ω2 − ω3, ω3 ∈ R+ are created. To get bounds on this stochastic process using the method above we need an expres- sion analogous to (4.9), i.e., ϕ(ω1 + ω2 − ω3) + ϕ(ω3) ≤ ϕ(ω1) + ϕ(ω2). 161 Isotropic Wave Turbulence and mean-field limits Therefore we can use Norris method with the appropriate adaptations for the particular case where ϕ(ω) = ω + c for a constant c, which we will take to be one. Notice that this works as a consequence of the conservation of the energy (given by the ω’s, see (4.4)) and the conservation of the total number of particles at each interaction. Definition 4.3. Consider ϕ(ω) = ω + 1. We say that a kernel K is sub-multiplicative if K(ω1, ω2, ω3) ≤ ϕ(ω1)ϕ(ω2)ϕ(ω3). (4.10) A. Existence and uniqueness of solutions. Definition 4.4 (Solution and types of solutions). We will say that (µt)t 0 where k = ‖k‖ and T123k = (k1k2k3k) β/4 (4.12) for some β ∈ R. The particular case α = 12 corresponds to the Majda-McLaughlin-Tabak (MMT) model. We have a four-wave interaction process with resonant conditions: k1 + k2 = k3 + k |k1|1/2 + |k2|1/2 = |k3|1/2 + |k|1/2. In this case wave numbers that are non-trivial solutions to these conditions cannot have all the same sign. Moreover, non-trivial solutions can be parametrized by a two parameter family A and ξ > 0: k1 = −A2ξ2, k2 = A2(1 + ξ + ξ2)2, k3 = A2(1 + ξ)2, k = A2ξ2(1 + ξ)2. (4.13) When β = 0 the collision rate is bounded. In [ZDP04] the authors obtain the following Kolmogorov-type solutions for α = 1/2 and β = 0: n ∼ |k|−5/6 (4.14) n ∼ |k|−1 . (4.15) 167 Isotropic Wave Turbulence and mean-field limits The derivation of the kinetic wave equation is done from the equation i ∂ψ ∂t = ∣∣∣∣ ∂∂x ∣∣∣∣α ψ︸ ︷︷ ︸ dispersive +λ ∣∣∣∣ ∂∂x ∣∣∣∣β/4 ∣∣∣∣∣ ∣∣∣∣ ∂∂x ∣∣∣∣β/4 ψ ∣∣∣∣∣ 2 ∣∣∣∣ ∂∂x ∣∣∣∣β/4 ψ  ︸ ︷︷ ︸ non-linearity λ = ±1 where ψ(x, t) denotes a complex wave field. Other examples in wave turbulence are (taken from [Naz11]): • 4-wave examples – surface gravity waves; N = 2, α = 1/2, β = 3; – langmuir waves in isotropic plasmas, spin waves; N = 3, α = 2, β = 2; – waves on elastic plates: N = 2, α = 2, β = 3; – Bose-Einstein condensates and non-linear optics: α = 2, β = 0; – Gravity waves on deep water: N = 2, α = 1/2, β = 3. • 3-wave examples – capillary waves: N = 2, α = 3/2; – acoustic turbulence, waves in isotropic elastic media; N = 3, α = 1; – interval waves in stratified fluids: N = 1, α = −1; • other examples – Kelvin waves on vortex filaments: N = 1, 6-wave interaction, α = 2. 4.2 Existence of solutions for unbounded kernel In this section we will follow the steps in [Nor99, Theorem 2.1] (see Remark 4.2). Remark 4.9. We make some comments about Theorem 4.6: (i) The statement is correct even if K(ω1, ω2, ω3) ≤ Cϕ(ω1)ϕ(ω2)ϕ(ω3) for some positive constant C <∞. This only changes the ζ(µ0) into ζ(µ0) = 〈ϕ2, µ0〉−1C−1〈ϕ, µ0〉−1. Also notice that by scaling time, we can eliminate the multiplicative constant. 168 4.2. Existence of solutions for unbounded kernel (ii) Notice that in the coagulation case, existence of strong solutions is assured for times T ′ = 〈ϕ2, µ0〉−1. We expect that in the 4-wave equations we can assure existence of strong solutions for larger times. The reason that we do not get that is because when bounding (4.30), we ignore some negative factors. (iii) We will need to use that for ϕ(ω) = ω+1, it holds that for any local solution (µt)t 0 depending only on ϕ andB, and a unique local solution (µt, λt)t≤T starting from (µ0, λ0). Then we will see that µt ≥ 0 for all t ∈ [0, T ]. This will be enough to prove the proposition: if we put f = 0 and a = 1 in (4.17) we get d dt λt = 1 2 ˆ D ϕ(ω1 + ω2 − ω3)1ω1+ω2−ω3 /∈BK(ω1, ω2, ω3)µ(dω1)µ(dω2)µ(dω3) +(λ2 + 2λ〈ϕ, µ〉) ˆ ∞ 0 ϕ(ω)2µ(dω). So, since µt ≥ 0, we deduce that λt ≥ 0 for all t ∈ [0, T ]. Next, we put f = ϕ and a = 1 170 4.2. Existence of solutions for unbounded kernel to see that d dt 〈ϕ, µt〉+ λt = 1 2 ˆ D (ϕ(ω1 + ω2 − ω3) + ϕ(ω3)− ϕ(ω1)− ϕ(ω2)) (4.18) ×K(ω1, ω2, ω3)µ(dω1)µ(dω2)µ(dω3) = 0 which is zero given that ϕ(ω) = ω + 1. Therefore, ‖µT ‖+ |λT | ≤ 〈ϕ, µT 〉+ λT = 〈ϕ, µ0〉+ λ0 ≤ 1. We can now start again from (µT , λT ) at time T to extend the solution to [0, 2T ], and so on, to prove the proposition. We use the following norm onMB × R: ‖(µ, λ)‖ = ‖µ‖+ |λ|. Note the following estimates: there is a constant C = C(ϕ,B) < ∞ such that for all µ, µ′ ∈MB and all λ, λ′ ∈ R ‖LB(µ, λ)‖ ≤ C‖(µ, λ)‖3 (4.19) ‖LB(µ, λ)− LB(µ′, λ′)‖ ≤ C ( ‖µ− µ′‖ (‖µ‖2 + ‖µ‖‖µ′‖+ ‖µ′‖2) (4.20) +(|λ|+ |λ′|)|λ− λ′|‖µ‖+ |λ′|2‖µ− µ′‖ +|λ− λ′|‖µ‖2 + |λ′| (‖µ‖‖µ− µ′‖+ ‖µ′‖‖µ′ − µ‖)) Observe that we get these estimates because we are working on a bounded set B. We turn to the iterative scheme. Set (µ0t , λ0t ) = (µ0, λ0) for all t and define inductively a sequence of continuous maps t 7→ (µnt , λnt ) : [0,∞)→MB × R by (µn+1t , λ n+1 t ) = (µ0, λ0) + ˆ t 0 LB(µns , λ n s ) ds. Set fn(t) = ‖(µnt , λnt )‖ then f0(t) = fn(0) = ‖(µ0, λ0)‖ ≤ 1 and by the estimate (4.19) we have that fn+1(t) ≤ 1 + C ˆ t 0 fn(s) 3 ds. 171 Isotropic Wave Turbulence and mean-field limits Hence fn(t) ≤ (1− 2Ct)−1/2 for t < (2C)−1. This last assertion is checked by induction. Suppose that it holds for n then fn+1(t) ≤ 1 + C ˆ t 0 (1− 2Cs)−3/2 ds = 1 + (1− 2Cs)−1/2|s=ts=0. Therefore, for all n setting T = (4C)−1, we have ‖(µnt , λnt )‖ ≤ √ 2 t ≤ T. (4.21) Next set g0(t) = f0(t) and for n ≥ 1 gn(t) = ‖(µnt , λnt )− (µn−1t , λn−1t )‖. By estimates (4.20) and (4.21), there is a constant C = C(B,ϕ) <∞ such that gn+1(t) ≤ C ˆ t 0 gn(s) ds t ≤ T. Hence by the usual arguments (Gronwall, Cauchy sequence), (µnt , λnt ) converges inMB× R uniformly in t ≤ T , to the desired local solution, which is also unique. Moreover, for some constant C <∞ depending only on ϕ and B we have ‖(µt, λt)‖ ≤ C t ≤ T. Finally, we are left to check that µt ≥ 0. For this, we need the following result: Proposition 4.11. Let (t, ω) 7→ ft(ω) : [0, T ]×B → R be a bounded measurable function, having a bounded partial derivative ∂f/∂t. Then, for all t ≤ T , d dt 〈ft, µt〉 = 〈∂f/∂t, µt〉+ 〈(ft, 0), LB(µt, λt)〉. The proof is a straightforward adaptation of the same Proposition (with different LB) in [Nor99, Proposition 2.3]. For t ≤ T , set θt(ω1) = exp ˆ t 0 (ˆ R2+∩(ω1+ω2≥ω3) K(ω1, ω2, ω3)µs(dω2)µs(dω3) + ( λ2s + 2λs〈ϕ, µs〉 ) ϕ(ω1) ) ds 172 4.2. Existence of solutions for unbounded kernel and define Gt :MB →MB by 〈f,Gt(µ)〉 = 1 2 ˆ D ((fθt)(ω1 + ω2 − ω3)1ω1+ω2−ω3∈B + (fθt)(ω3)) ×K(ω1, ω2, ω3)θt(ω1)−1θt(ω2)−1θt(ω3)−1 ×µ(dω1)µ(dω2)µ(dω3) Note that Gt(µ) ≥ 0 whenever µ ≥ 0 and for some C = C(ϕ,B) <∞we have ‖Gt(µ)‖ ≤ C‖µ‖3 (4.22) ‖Gt(µ)−Gt(µ′)‖ ≤ C‖µ− µ′‖ (‖µ‖2 + ‖µ′‖‖µ‖+ ‖µ′‖2) . (4.23) Set µ˜t = θtµt. By Proposition 4.11, for all bounded measurable function f we have d dt 〈f, µ˜t〉 = 〈f ∂θ ∂t , µt〉+ 〈(fθt, 0), LB(µt, λt)〉 so, using the symmetry of ω1 and ω2 in LB we get d dt 〈f, µ˜t〉 = 〈f,Gt(µ˜t)〉. (4.24) Thus, the function θt is simply designed as an integrating factor, which removes the neg- ative terms appearing in LB . Define inductively a new sequence of measures µ˜nt by setting µ˜0t = µ0 and for n ≥ 0 µ˜n+1 = µ0 + ˆ t 0 Gs(µ˜ n s ) ds. By an argument similar to that used for the original iterative scheme, the proof is com- pleted: we can show, first, and possibly for a smaller value of T > 0, but with the same dependence, that ‖µ˜nt ‖ is bounded, uniformly in n, for t ≤ T , and then that ‖µ˜nt − µ˜t‖ → 0 as n→∞. Since µ˜nt ≥ 0 for all n, we deduce µ˜t ≥ 0 and hence µt ≥ 0 for all t ≤ T . We fix now µ0 ∈ M with µ0 ≥ 0 and 〈ϕ, µ0〉 < ∞. For each bounded set B ⊂ [0,∞), let µB0 = 1Bµ0, λ B 0 = ˆ [0,∞)\B ϕ(ω)µ0(dω) (4.25) and denote by (µBt , λBt )t≥0 the unique solution to (4.17), starting from (µB0 , λB0 ), provided by Proposition 4.10. We have that for B ⊂ B′, µBt ≤ µB ′ t , 〈ϕ, µBt 〉+ λBt = 〈ϕ, µB ′ t 〉+ λB ′ t . The inequality will be proven in Proposition 4.12 and the equality is consequence of ex- 173 Isotropic Wave Turbulence and mean-field limits pression (4.18) and the fact that 〈ϕ, µB0 〉+ λB0 = 〈ϕ, µB ′ 0 〉+ λB ′ 0 by expression (4.25). Moreover, it holds that for any local solution (νt)t b then it must hold that ω1 + ω2 − ω3 < b. Proposition 4.13. Suppose that (νt)t 0 depending only on ‖f‖∞ and Λ. First we use Markov’s and Jensen’s inequalities to get P(w′(〈f,Xn〉, δ, T ) ≥ η) ≤ E[w ′(〈f,Xn〉, δ, T )] η ≤ ( E[w′(〈f,Xn〉, δ, T )2])1/2 η . (w′ is defined in Theorem 4.40). Now, for a given partition {ti}ni=1, sup r1,r2∈[ti−1,ti) |〈f,Xnr1 −Xnr2〉| ≤ 2 sup r∈[ti−1,ti) |〈f,Xnr −Xnti−1〉|. Denote by i∗ the point where the maximum on the right hand side is attained (the number of points in each partition is always finite). Now we want to consider a partition such that maxi|ti − ti−1| = δ + ε for some ε > 0 so w′(〈f,Xn〉, δ, T ) ≤ 2 sup r∈[ti∗−1,ti∗−1+δ+ε) |〈f,Xnr −Xnti∗−1〉| a.s.. Therefore we are just left to check that E [ sup r∈[s,s+δ+ε) |〈f,Xnr −Xns 〉|2 ] ≤ η 4 2 which is fulfilled thanks to the bound (4.40) by taking, for example, δ = √ 1 + η4 2A − 1− ε for ε small enough. Proof of Lemma 4.23. We will use Theorem 4.39 to prove this. To check condition (i), we consider the compact set W ∈ P(R+) (compact with respect to the topology induced by the weak convergence of measures) defined as W := { τ ∈ P(R+) : ˆ R+ ω τ(dω) ≤ C } . Consider (Ln)n∈N the family of probability measures in P(D([0,∞);W )) which are the laws of (Xn)n∈N. We have that Ln(D([0,∞);W ) = 1 for all n ∈ N 186 4.3. Mean-field limit by the conservation of the total energy and its boundedness (assumption (B1)): ˆ R+ ωXnt (dω) = 1 n n∑ i=0 ωn,it = 1 n n∑ i=0 ωn,i0 = ˆ R+ ωµn0 (dω) ≤ C a.s.. Now, to check condition (ii) we will use the family of continuous functions in P(R+) defined as F = {F : P(R+)→ R : F (τ) = 〈f, τ〉 for some f ∈ Cb(R+)}. This family is closed under addition since Cb(R+) is, it is continuous in P(R+), and sepa- rates points in P(R+): if F (τ) = F (τ¯) for all F ∈ F then ˆ R+ f(k)d(τ − τ¯)(k) = 0 ∀f ∈ Cb(R+) hence τ ≡ τ¯ . So we are left with proving that for every f ∈ Cb(R+) the sequence {〈f,Xn〉}n∈N is tight. This was proven in Lemma 4.22. Step 3: convergence for the trilinear term Proposition 4.24 (Convergence for the trilinear term). It holds that ˆ t 0 〈f,Q(n)(Xnks )〉 ds→ ˆ t 0 〈f,Q(µs, µs, µs)〉 ds weakly. Lemma 4.25 (Continuity of the limit). The weak limit of (Xnkt )t≥0 as k →∞ is continuous in time a.e.. Lemma 4.26 (Uniform convergence). For all f ∈ Cb(R+), it holds sup s≤t |〈f,Xnks − µs〉| → 0 weakly as k →∞. Lemma 4.27. It holds that sup s≤t |〈f,Q(n)(Xnks )−Q(µs)〉| → 0 weakly as k →∞. Proof of Proposition 4.24. By Lemma 4.27 we can pass the limit inside the integral in time. 187 Isotropic Wave Turbulence and mean-field limits Proof of Lemma 4.25. We have that for any f ∈ Cb(R+) |〈f,Xnkt 〉 − 〈f,Xnkt− 〉| ≤ 4 nk ‖f‖∞ applying Theorem 4.41 we get that 〈f, µt〉 is continuous for any f ∈ Cb(R+) and this implies the continuity of (µt)t≥0. Proof of Lemma 4.26. We know by Lemma 4.25 that the limit of (Xnk)k∈N is continuous. The statement is consequence of the continuity mapping theorem in the Skorokhod space (proven using the Skorokhod representation theorem 4.38) and the fact that g(X)(t) = sups≤t |X| is a continuous function in this space. Proof of Lemma 4.27. We abuse notation and denote by (Xnt )n∈N the convergent subse- quence. We split the proof in two parts, we will prove for all f ∈ Cb(R+): (i) sups≤t |〈f, ( Q−Q(n)) (Xns )〉| → 0 as n→∞, (ii) sups≤t |〈f,Q (Xns )−Q (µs)〉| → 0 as n→∞. (i) is consequence of |〈f, ( Q−Q(n) ) (Xns )〉| = 1 2 1 n ˆ 2ω2≥ω3 (f(2ω2 − ω3) + f(ω3)− 2f(ω2)) ×K(ω2, ω2, ω3)Xns (dω2)Xns (dω3) ≤ 2 n ‖f‖∞Λ. (4.41) Now, for (ii) we compute we have that sup s≤t |〈f,Q(Xns )−Q(µs)〉| ≤ 1 2 ˆ D K(ω1, ω2, ω3) |f(ω1 + ω2 − ω3) + f(ω3)− f(ω1)− f(ω2)| × sup s≤t |Xns (dω1)Xns (dω2)Xns (dω3)− µs(dω1)µs(dω2)µs(dω3)| (4.42) We conclude (ii) with an argument analogous to Lemma 4.26 and the fact that Xnt ⊗Xnt ⊗Xnt → µt ⊗ µt ⊗ µt weakly (consequence of Le´vy’s continuity theorem). 188 4.3. Mean-field limit 4.3.2.3 Proof of Theorem 4.7 (unbounded kernel) Remark 4.28. The proof that we already wrote in the case of bounded kernels works here in most parts substituting Λ by M where ˆ R+ ωXn(dω) ≤M = 〈ω, µ0〉. The only places where we need to be careful are Lemmas 4.26 and 4.27. Lemma 4.29 (Convergence of a subsequence). There exists a subsequence (Xnkt )k∈N that converges weakly in D([0,∞)× P(R+)) as k →∞. Proof. The proof is exactly the one as in Section 4.3.2.2 and Proposition 4.21 using the bound on the jump kernel K, for example in the proof of Lemma 4.22, in the bounds of expressions (4.38) and (4.39), the value of Λ will be substituted by M3. Lemma 4.30. For any f ∈ Cb(R+), t ≥ 0 it holds that E [ sup s≤t |Mn,fs |2 ] ≤ 1 n 32‖f‖2∞M6t. Proof. The proof is the same one as in Proposition 4.20 using the bound on the jump kernel K. Lemma 4.31. It holds that for any t ≥ 0 sup s≤t |〈f,Q(n)(Xns )−Q(µs)〉| → 0 weakly for f continuous and of compact support. Proof. Here everything works as in Section 4.3.2.2, but we need to find the bounds (4.41) and (4.42). We use a similar approach as in [Nor99]. Firstly, we will prove an analogous bound to (4.42). Fix ε > 0 and define p(ε) = ε−1/γ . Then for ω ≥ p(ε) it holds ϕ˜(ω) ω ≤ ε. Now choose κ ∈ (0, γ/[2(1− γ)]). We split the domain into F p1 := {(ω1, ω2, ω3) : ω1 ≤ pκ(ε), ω2 ≤ pκ(ε), ω3 ≤ pκ(ε)} and F p2 its complementary. In F p1 the kernel is bounded and we have, with obvious notations, sup s≤t |〈f,Q1(Xns )−Q1(µs)〉| → 0 weakly. 189 Isotropic Wave Turbulence and mean-field limits On the other hand, in F p2 , at least one of the components is greater than p κ(ε). Assume, without loss of generality that ω3 ≥ pκ(ε). Then |〈f,Q2(Xnt )〉| = ∣∣∣∣ ˆ D {f(ω1 + ω2 − ω3) + f(ω3)− f(ω1)− f(ω2)}K(ω1, ω2, ω3) ×Xnt (dω1)Xnt (dω2)Xnt (dω3) ∣∣∣∣ ≤ 4‖f‖∞ ˆ D ϕ˜(ω1)ϕ˜(ω2)ϕ˜(ω3)X n t (dω1)X n t (dω2)X n t (dω3) ≤ 4‖f‖∞max { (pκ(ε))2(1−γ) ε〈ω, µ0〉, (pκ(ε))1−γ ε2〈ω, µ0〉2, ε3〈ω, µ0〉3 } ≤ cεη for η = 1− 2κ(1− γ)/γ > 0. and analogously |〈f,Q2(µt)〉| ≤ cεη. This implies that lim sup n→∞ sup s≤t |〈f,Q2(Xns )−Q2(µs)〉| ≤ 2cεη but ε is arbitrary so the limit is proved. We are left with proving an analogous estimate to (4.41), which is obtained straight- forwardly since we restrict ourselves to continuous functions of compact support. Proof of Theorem 4.7. Thanks to the previous Lemmas we know that there exists conver- gent subsequence Xnkt → µt weakly as k →∞ such that 〈f, µt〉 = 〈f, µ0〉+ ˆ t 0 〈f,Q(µs)〉ds for any f is continuous of compact support. Now using the bounds on the jump ker- nel and that 〈ω, µt〉 ≤ 〈ω, µ0〉 and a limit argument, we can extend this equation to all bounded measurable functions f . 4.3.3 Second result on mean-field limit 4.3.3.1 A coupling auxiliary process Write Xn0 = 1 n n∑ i=1 δωi , for ωi ∈ R+. Define for B ⊂ R+ bounded XB,n0 = 1 n n∑ i :ωi∈B δωi . 190 4.3. Mean-field limit Consider ΛB,n0 such that for each B ′ ⊂ R+ bounded such that B ⊂ B′ it holds XB,n0 ≤ XB ′,n 0 , 〈ϕ,XB,n0 〉+ ΛB,n0 = 〈ϕ,XB ′,n 0 〉+ ΛB ′,n 0 . (4.43) Set νB = (ΛB,n0 ) 2 + 2ΛB,n0 〈ϕ,XB,n0 〉 − 1 n2 ∑ k,j :ωj /∈B or ωk /∈B ϕ(ωj)ϕ(ωk). Note that νB decreases as B increases and ν[0,∞) = (ΛB,n0 ) 2 + 2ΛB,n0 〈ϕ,XB,n0 〉 ≥ 0. For i < j take independent exponential random variables Tijk of parameterK(ωi, ωj , ωk)/n2. Set Tijk = Tjik. Also, for i 6= j, take independent exponential random variables Sijk of parameter (ϕ(ωi)ϕ(ωj)ϕ(ωk)−K(ωi, ωj , ωk)) /n2 (in all these cases we assume that ωi + ωj ≥ ωk). We can construct, independently for each i, a family of independent exponential random variables SBi , increasing in B, with S B i having parameter ϕ(ωi)ν B . Set TBi = min k,j :ωj /∈B or ωk /∈B (Tijk ∧ Sijk) ∧ SBi , TBi is an exponential random variable of parameter 1 n2 ∑ k,j :ωj /∈B or ωk /∈B ϕ(ωi)ϕ(ωj)ϕ(ωk) + ϕ(ωi)ν B = ϕ(ωi) ( (ΛB,n0 ) 2 + 2ΛB,n0 〈ϕ,XB,n0 〉 ) . For each B, the random variables (Tijk, T B i : i, j, k such that ωi, ωj , ωk ∈ B, i < j) form an independent family. Suppose that i is such that ωi ∈ B and that j is such that ωj /∈ B or k is such that ωk /∈ B, then we have TBi ≤ Tijk and for B ⊂ B′ and all i, we have (as a consequence of (4.43)) TBi ≤ TB ′ i . Now set T = ( min i 0 and t < T . Since (µt)t δ ) → 0 as n→∞, as required. 4.4 Conclusions In this work we have dealt with the weak isotropic 4-wave kinetic equation with simpli- fied kernels. When the kernels are at most linear we have given conditions for the local existence and uniqueness of solutions. We have also derived the equation as a mean-field limit of interacting particle system given by a simultaneous coagulation-fragmentation: three particles interact with a coagulation-fragmentation phenomenon where one of the particles seem to act as a catalyst. As we saw in the introduction, this theory can be applied to physical scenarios that include Langmuir waves, shallow water and waves on elastic plates. Moreover, using the interacting particle system, numerical methods can be devised to simulate the solution of the equation (as done by [Con09] for the 3-wave kinetic equation), by adapting the methods in [EW00]. Finally, these numerical simulations would allow the study of steady state solutions 3Remember the definition of the metric d given in (4.11). Since d(XB,ns , µBs ) → 0, we have that for all f bounded continuous function on R+ˆ fϕ(XB,ns − µBs ) = ˆ fϕ1B(X B,n s − µBs )→ 0 since ϕ restricted to B is also bounded and continuous. 197 Isotropic Wave Turbulence and mean-field limits and to check if they match the Kolmogorov-Zakharov spectra. 4.5 Appendix: Some properties of the Skorokhod space Theorem 4.34 (Prohorov’s theorem ([EK09]), Chapter 3). Let (S, d) be complete and separa- ble, and letM∈ P(S). Then the following are equivalent: (i) M is tight. (ii) For each ε > 0, there exists a compact K ∈ S such that inf P∈M P (Kε) ≥ 1− ε where Kε := {x ∈ S : infy∈K d(x, y) < ε}. (iii) M is relatively compact. Let (E, r) be a metric space. The space D([0,∞);E) of cadlag functions taking values in E is widely used in stochastic processes. In general we would like to study the con- vergence of measures on this space, however, most of the tools known for convergence of measures are for measures in P(S) for S a complete separable metric space. Therefore, it would be very useful to find a topology in D([0,∞)×E) such that it is a complete and separable metric space. This can be done when E is also complete and separable; and the metric considered is the Skorokhod one. This is why in this case the space of ca`dla`g functions is called Skorohod space. Some important properties of this space are the following: Proposition 4.35 ([EK09], Chapter 3). If x ∈ D([0,∞);E), then x has at most countably many points of discontinuity. Theorem 4.36 ([EK09], Chapter 3). If E is separable, then D([0,∞);E) is separable. If (E, r) is complete, then (D([0,∞);E), d) is complete, where d is the Skorokhod metric. Theorem 4.37. The Skorokhod space is a complete separable metric space. Theorem 4.38 (The almost sure Skorokhod representation theorem, [EK09], Theorem 1.8, Chapter 3). Let (S, d) be a separable metric space. Suppose Pn, n = 1, 2, . . . and P in P(S) satisfy limn→∞ ρ(Pn, P ) = 0 where ρ is the metric in P(S). Then there exists a probability space (Ω,F , ν) on which are defined S- valued random variable Xn, n = 1, 2, . . . and X with distributions Pn, n = 1, 2, . . . and P , respectively such that limn→∞Xn = X almost surely. Theorem 4.39 (Tightness criteria for measures on the Skorokhod space, [Jak86] Theorem 3.1). Let (S, T ) be a completely regular topological space with metrisable compact sets. Let G be a family of continuous functions on S. Suppose that G separates points in S and that it is closed 198 4.5. Appendix: Some properties of the Skorokhod space under addition. Then a family {Ln}n∈N of probability measures in P(D([0,∞);S) is tight iff the two following conditions hold: (i) For each ε > 0 there is a compact set Kε ⊂ S such that Ln(D([0,∞);Kε)) > 1− ε, n ∈ N. (ii) The family {Ln}n∈N is G-weakly tight. Theorem 4.40 (Criteria for tightness in Skorokhod spaces ([EK09], Corollary 7.4, Chapter 3)). Let (E, r) be a complete and separable metric space, and let {Xn} be a family of processes with sample paths in D([0,∞);E). Then {Xn} is relatively compact iff the two following conditions hold: (i) For every η > 0 and rational t ≥ 0, there exists a compact set Λη,t ⊂ E such that lim inf n→∞ P{Xn(t) ∈ Λ η η,t} ≥ 1− η. (ii) For every η > 0 and T > 0, there exits δ > 0 such that lim sup n→∞ P{w′(Xn, δ, T ) ≥ η} ≤ η. where we have used the modulus of continuity w′ defined as follows: for x ∈ D([0,∞) × E), δ > 0, and T > 0: w′(x, δ, T ) = inf {ti} max i sup s,t∈[ti−1,ti) r(x(s), x(t)), where {ti} ranges over all partitions of the form 0 = t0 < t1 < . . . < tn−1 < T ≤ tn with min1≤i≤n(ti − ti−1) > δ and n ≥ 1 Theorem 4.41 (Continuity criteria for the limit in Skorokhod spaces ([EK09], Theorem 10.2, Chapter 3)). Let (E, r) be a metric space. Let Xn, n = 1, 2, . . . , and X be processes with sample paths in D([0,∞);E) and suppose that Xn converges in distribution to X . Then X is a.s. continuous if and only if J(Xn) converges to zero in distribution, where J(x) = ˆ ∞ 0 e−u[J(x, u) ∧ 1] du for J(x, u) = sup 0≤t≤u r(x(t), x(t−)). 199 Isotropic Wave Turbulence and mean-field limits 4.6 Appendix: Formal derivation of the weak isotropic 4-wave kinetic equation Suppose that n(k) = n(k) is a radial function (isotropic). The waveaction in the isotropic case can be written as W = ˆ RN n(k)dk = ˆ R+×SN−1 n(k)kN−1dkds = |SN−1| α ˆ ∞ 0 n(ω)ω N−α α dω, where SN−1 is the N − 1 dimensional sphere. From this expression, one can denote the angle-averaged frequency spectrum µ = µ(dω) as µ(dω) := |SN−1| α ω N−α α n(ω)dω. The total number of waves (waveaction) and the total energy are respectively W = ˆ ∞ 0 µ(dω) E = ˆ ∞ 0 ωµ(dω). The isotropic version of the weak 4-wave kinetic equation can be written as µt = µ0 + ˆ t 0 Q(µs, µs, µs) ds (4.49) where Q is defined against test functions g ∈ S(R+) as 〈g,Q(µ, µ, µ)〉 = 1 2 ˆ D µ(dω1)µ(dω2)µ(dω3)K(ω1, ω2, ω3) (4.50) ×[g(ω1 + ω2 − ω3) + g(ω3)− g(ω2)− g(ω1)] where D := {R3+ ∩ (ω1 + ω2 ≥ ω3)} and K(ω1, ω2, ω3) = 8pi α|SN−1|4 (ω1 + ω2 − ω3) N−α α (4.51) ˆ (SN−1)4 ds1ds2ds3dsT 2 (ω 1/α 1 s1, ω 1/α 2 s2, ω 1/α 3 s3, (ω1 + ω2 − ω3)1/αs) ×δ(ω1/α1 s1 + ω1/α2 s2 − ω1/α3 s3 − (ω1 + ω2 − ω3)1/αs) Next we explain the formal derivation of the weak isotropic 4-wave kinetic equation 200 4.6. Appendix: Formal derivation of the weak isotropic 4-wave kinetic equation (4.5). We have that ˆ (0,∞) ∂tµ(ω)dω = ˆ RN ∂tn(k)dk = 4pi ˆ Ω4×S4 T 2 (k1s1, k2s2, k3s3, ks) ×δ(k1s1 + k2s2 − k3s3 − ks)δ(ω1 + ω2 − ω3 − ω) ×(n1n2n3 + n1n2n− n1n3n− n2n3n)(kk1k2k3)N−1dkds = 4pi α|SN−1|4 ˆ R4+×S4 dω0123ds0123T 2(ω 1/α 1 s1, ω 1/α 2 s2, ω 1/α 3 s3, ω 1/αs) ×δ(ω1/α1 s1 + ω1/α2 s2 − ω1/α3 s3 − ω1/αs)δ(ω1 + ω2 − ω3 − ω) ×(µ(ω1)µ(ω2)µ(ω3)ω N−α α + µ(ω1)µ(ω2)µ(ω)ω N−α α 3 −µ(ω1)µ(ω3)µ(ω)ω N−α α 2 − µ(ω2)µ(ω3)µ(ω)ω N−α α 1 ) = ˆ R4+ dω0123F (ω1, ω2, ω3, ω)δ(ω1 + ω2 − ω3 − ω) ×(µ(ω1)µ(ω2)µ(ω3)ω N−α α + µ(ω1)µ(ω2)µ(ω)ω N−α α 3 −µ(ω1)µ(ω3)µ(ω)ω N−α α 2 − µ(ω2)µ(ω3)µ(ω)ω N−α α 1 ) for Si = (SN−1)i, dω0123 = dωdω1dω2dω3, ds0123 = ds1ds2ds3ds, and F (ω1, ω2, ω3, ω) = 4pi α|SN−1|4 ˆ S4 ds0123T 2 (ω 1/α 1 s1, ω 1/α 2 s2, ω 1/α 3 s3, ω 1/αs) ×δ(ω1/α1 s1 + ω1/α2 s2 − ω1/α3 s3 − ω1/αs). Hence, µω satisfies ∂tµ(ω) = ˆ R3+ dω123F (ω1, ω2, ω3, ω)δ(ω1 + ω2 − ω3 − ω) (4.52) ×(µ(ω1)µ(ω2)µ(ω3)ω N−α α + µ(ω1)µ(ω2)µ(ω)ω N−α α 3 (4.53) −µ(ω1)µ(ω3)µ(ω)ω N−α α 2 − µ(ω2)µ(ω3)µ(ω)ω N−α α 1 ) Its weak formulation µt = µ in + ˆ Ω3 Q(µs, µs, µs) ds is defined against functions g ∈ S(R+) as 〈g,Q(µ, µ, µ)〉 = ˆ R4+ dω0123µ(ω1)µ(ω2)µ(ω3)ω N−α α ×[F1230δ(ω1230)g(ω) + F1203δ(ω1203)g(ω3) 201 Isotropic Wave Turbulence and mean-field limits −F1032δ(ω1032)g(ω2)− F0231δ(ω0231)g(ω1)] = ˆ R4+ dω0123µ(ω1)µ(ω2)µ(ω3)ω N−α α F1230δ(ω 12 30) ×[g(ω) + g(ω3)− g(ω2)− g(ω1)] = 1 2 ˆ D dω123µ(ω1)µ(ω2)µ(ω3)K(ω1, ω2, ω3) ×[g(ω1 + ω2 − ω3) + g(ω3)− g(ω2)− g(ω1)] (4.54) To conclude we assumed that T is symmetric in all its variables. We used that changing labels we get that dω123F1230δ(ω 12 30)g(ω) + F1203δ(ω 12 03)g(ω3)− F1032δ(ω1032)g(ω2)− F0231δ(ω0231)g(ω1) = dω123F1230δ(ω 12 30)g(ω) + F1203δ(ω 12 03)g(ω3)− F3012δ(ω3012)g(ω2)− F0321δ(ω0321)g(ω1) and the properties of the function F to factorise it. We used the notation δ(ωijlp) = δ(ωi + ωj − ωl − ωp) and K(ω1, ω2, ω3) := 2(ω1 + ω2 − ω3) N−α α F (ω1, ω2, ω3, ω1 + ω2 − ω3). For the last line we used the sifting property of the delta distribution i.e. ˆ b a f(t)δ(t− d) dt =  f(d) for d ∈ [a, b] 0 otherwise . (4.55) Remark 4.42. In reference [ZLF92, Section 3.1.3], the authors state that even in isotropic medium, the interaction coefficient T in the 4-wave case cannot be considered to be isotropic too. In the 3-wave case it is possible, but not for the 4-wave. We can rewrite |T (k1,k2,k3,k)|2 = T 20k2βf2 ( k1 k , k2 k , k3 k ) (4.56) for some dimensionless constant T 0 and some dimensionless function f2. 202 Bibliography [AG13] G Allaire and F Golse, Transport et diffusion, Lecture notes, Ecole polytech- nique (in French) (2013). (Cited on pages 24, 46, 48, 50, 65, and 129). [AK01] Kenichiro Aoki and Dimitri Kusnezov, Fermi-pasta-ulam β model: Boundary jumps, fourier’s law, and scaling, Phys. Rev. Lett. 86 (2001), 4029–4032. 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