Exponential Entropy dissipation for weakly self-consistent Vlasov-Fokker-Planck equations

TL;DR

This paper investigates the long-term behavior of weakly self-consistent Vlasov-Fokker-Planck equations, establishing exponential convergence under specific Hessian matrix conditions related to a Lyapunov functional.

Contribution

It introduces Hessian matrix conditions on mean-field kernels that guarantee exponential convergence of solutions, a novel criterion for analyzing these equations.

Findings

Hessian matrix conditions ensure exponential convergence in $L^1$ distances.

The conditions are verified in specific examples.

A Lyapunov functional, auxiliary Fisher information, drives the analysis.

Abstract

We study long-time dynamical behaviors of weakly self-consistent Vlasov-Fokker-Planck equations. We introduce Hessian matrix conditions on mean-field kernel functions, which characterizes the exponential convergence of solutions in $L^1$ distances. The matrix condition is derived from the dissipation of a selected Lyapunov functional, namely auxiliary Fisher information functional. We verify proposed matrix conditions in examples.