Hilbert expansion for kinetic equations with non-relativistic Coulomb collision

TL;DR

This paper rigorously derives the compressible Euler equations from the Landau equation with Coulomb interactions and proves the convergence of solutions of the Vlasov-Maxwell-Landau system to the Euler-Maxwell system in the hydrodynamic limit, using novel energy estimates.

Contribution

It provides the first rigorous derivation of hydrodynamic limits for Landau equations with Coulomb potential via Hilbert expansion and establishes convergence results for the Vlasov-Maxwell-Landau system.

Findings

Derivation of compressible Euler equations from Landau equation.

Proof of convergence of Vlasov-Maxwell-Landau solutions to Euler-Maxwell solutions.

Introduction of new energy estimates for kinetic equations.

Abstract

In this paper, we study the hydrodynamic limits of both the Landau equation and the Vlasov-Maxwell-Landau system in the whole space. Our main purpose is two-fold: the first one is to give a rigorous derivation of the compressible Euler equations from the Landau equation via the Hilbert expansion; while the second one is to prove, still in the setting of Hilbert expansion, that the unique classical solution of the Vlasov-Maxwell-Landau system converges, which is shown to be globally in time, to the resulting global smooth solution of the Euler-Maxwell system, as the Knudsen number goes to zero. The main ingredient of our analysis is to derive some novel interplay energy estimates on the solutions of the Landau equation and the Vlasov-Maxwell-Landau system which are small perturbations of both a local Maxwellian and a global Maxwellian, respectively. Our result solves an open problem in the hydrodynamic limit for the Landau-type equations with Coulomb potential and the approach developed in this paper can seamlessly be used to deal with the problem on the validity of the Hilbert expansion for other types of kinetic equations.