Kinetic equations with Maxwell boundary conditions

TL;DR

This paper establishes global stability of renormalized solutions for kinetic equations with Maxwell boundary conditions, covering models like Boltzmann, Vlasov-Poisson, and Fokker-Planck, using advanced trace and convergence techniques.

Contribution

It proves the realization of Maxwell boundary conditions for kinetic equations, extending previous boundary inequality results to a boundary condition realization, with new convergence and trace theorems.

Findings

Proved stability of solutions with Maxwell boundary conditions.

Extended trace theorems for kinetic equations.

Developed new convergence results for weak solutions.

Abstract

We prove global stability results of {\sl DiPerna-Lions} renormalized solutions for the initial boundary value problem associated to some kinetic equations, from which existence results classically follow. The (possibly nonlinear) boundary conditions are completely or partially diffuse, which includes the so-called Maxwell boundary conditions, and we prove that it is realized (it is not only a boundary inequality condition as it has been established in previous works). We are able to deal with Boltzmann, Vlasov-Poisson and Fokker-Planck type models. The proofs use some trace theorems of the kind previously introduced by the author for the Vlasov equations, new results concerning weak-weak convergence (the renormalized convergence and the biting $L^1$-weak convergence), as well as the Darroz\`es-Guiraud information in a crucial way.