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

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.