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

We consider the Vlasov–Poisson system in a $C^3$ convex domain $D$ with a perfectly conducting wall. We introduce the asymptotic domain $D_{\mathrm{\infty }}$ 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 $\overline{D_{\mathrm{\infty }}}$ and the solutions exhibit the asymptotics of modified scattering.