Nonlinear asymptotic stability of inhomogeneous steady solutions to boundary problems of Vlasov-Poisson equation

TL;DR

This paper proves the nonlinear asymptotic stability of certain inhomogeneous steady solutions to the Vlasov-Poisson equations in a half-space, showing exponential decay of small perturbations under specific conditions.

Contribution

It constructs Lipschitz continuous inhomogeneous steady states and demonstrates their exponential stability, advancing understanding of kinetic PDE stability in gravitational and electrostatic contexts.

Findings

Exponential stability of steady states under small perturbations.

Construction of Lipschitz continuous steady solutions.

Well-posedness and regularity results for the Vlasov-Poisson system.

Abstract

We consider an ensemble of mass collisionless particles, which interact mutually either by an attraction of Newton's law of gravitation or by an electrostatic repulsion of Coulomb's law, under a background downward gravity in a horizontally-periodic 3D half-space, whose inflow distribution at the boundary is prescribed. We investigate a nonlinear asymptotic stability of its generic steady states in the dynamical kinetic PDE theory of the Vlasov-Poisson equations. We construct Lipschitz continuous space-inhomogeneous steady states and establish exponentially fast asymptotic stability of these steady states with respect to small perturbation in a weighted Sobolev topology. In this proof, we crucially use the Lipschitz continuity in the velocity of the steady states. Moreover, we establish well-posedness and regularity estimates for both steady and dynamic problems.