Discrete $H$-theorem for a finite volume discretization of a nonlinear kinetic system: application to hypocoercivity

TL;DR

This paper proves a discrete $H$-theorem for a finite-volume scheme of a nonlinear kinetic system, demonstrating exponential convergence to equilibrium for a broad class of initial data, and highlighting the scheme's stabilizing dissipation effects.

Contribution

It extends previous exponential convergence results to more general initial data by leveraging entropy properties and the scheme's dissipation, without requiring proximity to equilibrium.

Findings

Proves a discrete $H$-theorem for the scheme.

Establishes exponential decay to equilibrium for broad initial data.

Highlights the stabilizing role of numerical dissipation.

Abstract

In this article, we study the long-time behavior of a finite-volume discretization for a nonlinear kinetic reaction model involving two interacting species. Building upon the seminal work of [Favre, Pirner, Schmeiser, ARMA, 2023], we extend the discrete exponential convergence to equilibrium result established in [Bessemoulin-Chatard, Laidin, Rey, IMAJNA, 2025], which was obtained in a perturbative framework using weighted $L^2$ estimates. The analysis applies to a broader class of exponentially decaying initial data, without requiring proximity to equilibrium, by exploiting the properties of the Boltzmann entropy. The proof relies on the propagation of the initial $L^\infty$ bounds, derived from monotonicity properties of the scheme, allowing controlled linearizations within the nonlinear entropy estimates. Moreover, we show that the time-discrete dissipation inherent to the numerical scheme plays a crucial stabilizing role, providing control over the nonlinear terms.