The Vlasov-Poisson system with a perfectly conducting wall: Convex domains

TL;DR

This paper studies the behavior of solutions to the Vlasov-Poisson system in convex domains with conducting walls, showing particles' velocities become confined to an asymptotic domain and solutions exhibit modified scattering.

Contribution

It introduces the asymptotic domain for convex domains and demonstrates particle velocity confinement and modified scattering asymptotics.

Findings

Particles' velocities asymptotically supported in the domain closure

Solutions exhibit modified scattering behavior

Velocity support becomes confined to the asymptotic domain

Abstract

We consider the Vlasov--Poisson system in a $C^3$ convex domain $D$ with a perfectly conducting wall. We introduce the asymptotic domain $D_{\infty}$ for the domain $D$. Then under acceptable assumptions on $D$, we show that for localized initial data, the velocity of particles is asymptotically supported in the (closure of the) asymptotic domain $\overline{D_{\infty}}$ and the solutions exhibit the asymptotics of modified scattering.