Mathematical study of long-range interactions in kinetic equations

Abstract

On the one hand, we are concerned with the dynamics of gas particles, captured by a partial differential equation named after Ludwig Boltzmann. On the other hand, we study the dynamics of charged particles, described through an equation derived by Lev Landau. A solution of these equations encodes the density of gas or charged particles, respectively, which at a given time are located at a certain point in space and travel at a certain velocity. Both equations are written with a transport operator on the left hand side, which accounts for the trajectories travelled by a particle in absence of any external forces. The right hand side describes the fluctuations in velocity resulting from particle collisions. We investigate when these equations derived from physical principles are rigorously justifiable. We focus on long-range interactions, where there is a high frequency of interactions at small deviation angle. These particle interactions induce fluctuations in velocity that generate a Lévy process, which explains the regularisation effect that takes place. We capture this regularisation effect on the behaviour of solutions in form of a priori estimates. Boltzmann: In the case of the Boltzmann equation, we assume the underlying potential to be moderately soft, and we work in a conditional regime, where we assume that solutions admit bounded mass, energy and entropy. In this regime, the Boltzmann equation can be written as a transport operator and a non-local diffusion operator with rough coefficients that are elliptic on average. In the first part, we explain why solutions are Hölder continuous. We derive the Harnack inequalities for solutions to the Boltzmann equation, and we discuss Schauder theory: the gain of regularity in Hölder spaces from the coefficients onto the solution. In particular, a bootstrapping argument permits to deduce that solutions to the Boltzmann equation are smooth, provided that the associated macroscopic quantities such as mass, energy and entropy are bounded. Therefore, if a singularity forms and the equation was to blow up, we would observe this phenomenon on the level of physically relevant quantities. Finally, we discuss how to obtain bounds on the asymptotic behaviour of the fundamental solution associated to the equation. In short: we evolve around the study of local regularity properties of solutions to non-local hypoelliptic equations. In a second part, we discuss the behaviour of solutions on the boundary. To this end, we introduce a suitable notion of solutions that is weak enough to be constructed in the space-homogeneous case, and moreover, is able to treat physically reasonable boundary assumptions. We show that these suitable weak solutions to the inhomogeneous non-cutoff Boltzmann equation with moderately soft potentials generate any pointwise polynomial velocity decay. Landau: Finally in the last part, we move away from hypoelliptic structures, but we consider the most singular of all long-range interactions: we study the space-homogeneous Landau equation with Coulomb interactions. We construct smooth solutions with initial data in sub-critical Lebesgue spaces, first global in time but in a perturbative regime, then local in time for large data. Then, building on the work of Guillen-Silvestre, we construct global smooth solutions for rough, slowly decaying initial data in critical Lebesgue spaces, and finally, we discuss how to treat the super-critical regime.