The Non-cutoff Vlasov-Poisson-Boltzmann and Vlasov-Poisson-Landau Systems in Union of Cubes

TL;DR

This paper proves the global stability and exponential decay of solutions to the non-cutoff Vlasov-Poisson-Boltzmann and Vlasov-Poisson-Landau systems in a union of cubes, using energy estimates and boundary conditions.

Contribution

It establishes the first global stability results for these systems with Coulomb interaction in a bounded union of cubes domain.

Findings

Global stability and exponential decay proven

Compatible boundary conditions for derivatives developed

Energy estimates with velocity weights obtained

Abstract

This work concerns the Vlasov-Poisson-Boltzmann system without angular cutoff and Vlasov-Poisson-Landau system including Coulomb interaction in bounded domain, namely union of cubes. We establish the global stability, exponential large-time decay with specular-reflection boundary condition when an initial datum is near Maxwellian equilibrium. We provide the compatible specular boundary condition for high-order derivatives and a velocity weighted energy estimate.