Long time behavior of discrete velocity kinetic equations

TL;DR

This paper investigates the long-term decay to equilibrium of nonlinear discrete velocity kinetic equations in one and three dimensions, providing explicit convergence rates using Lyapunov functionals.

Contribution

It introduces a method to prove exponential decay for a broad class of kinetic equations, including Goldstein-Taylor and Carleman models, with explicit convergence estimates.

Findings

Solutions decay exponentially to equilibrium in L^2 space

Applicable to various interaction rates including Goldstein-Taylor and Carleman equations

Constructive estimates on convergence rates

Abstract

We study long time behavior of some nonlinear discrete velocity kinetic equations in the one and three dimensions with periodic boundary conditions. We prove the exponential time decay of solutions towards the global equilibrium in the $L^2$ space. Our result holds for a wide class of interaction rates including the Goldstein-Taylor and Carleman equations, and the estimates on the rate of convergence are explicit and constructive. The technique is based on the construction of suitable Lyapunov functionals by modifying Boltzmann's entropy.