Exponential Mixing of Vlasov equations under the effect of Gravity and Boundary

TL;DR

This paper demonstrates that the Vlasov equations, modeling plasma dynamics under gravity and boundary effects, exhibit exponential mixing with solutions converging rapidly to equilibrium, supported by novel bounds on boundary fluxes.

Contribution

The study introduces a new method to prove exponential mixing for Vlasov equations with gravity and stochastic boundary conditions, including the construction of stationary solutions and uniform bounds on residual measures.

Findings

Moments of dynamical fluctuations decay exponentially in $L^ ablafty$.

Constructed stationary solutions and proved their stability.

Established uniform bounds on residual measures independent of bouncing number.

Abstract

In this paper, we study exponentially fast mixing induced/enhanced by gravity and stochastic boundary in the kinetic theory of Vlasov equations. We consider the Vlasov equations with and without a vertical magnetic field inside a horizontally-periodic 3D half-space equipped with a non-isothermal diffusive reflection boundary condition of bounded continuous boundary temperature at the bottom. We construct both stationary solutions and global-in-time dynamical solutions in $L^\infty$. We prove that moments of a dynamical fluctuation around the steady solutions decay exponentially fast in $L^\infty$. As a key of this proof, we establish a uniform bound of so-called residual measures independently of the bouncing number of stochastic characteristics, by constructing a continuous stationary outgoing boundary flux which is strictly positive almost everywhere.