On global existence and trend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system with exterior confining potential

TL;DR

This paper establishes global existence and convergence to equilibrium for the Vlasov-Poisson-Fokker-Planck system with an exterior confining potential, even with low regularity initial data and small nonlinearities.

Contribution

It introduces a fixed point approach combined with sharp semi-group estimates to handle low regularity data and large confining potentials in 2D and 3D.

Findings

Proves global existence for low regularity initial data.

Shows trend to equilibrium under small nonlinear terms.

Utilizes sharp short and long time semi-group estimates.

Abstract

We prove a global existence result with initial data of low regularity, and prove the trend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system with small non linear term but with a possibly large exterior confining potential in dimension $d=2$ and $d=3$. The proof relies on a fixed point argument using sharp estimates (at short and long time scales) of the semi-group associated to the Fokker-Planck operator, which were obtained by the first author.