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arXiv · 2511.13323

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

Marianne Bessemoulin-Chatard·Tino Laidin·Thomas Rey
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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.

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Marianne Bessemoulin-Chatard, Tino Laidin, Thomas Rey. 2025-11-17. Discrete $H$-theorem for a finite volume discretization of a nonlinear kinetic system: application to hypocoercivity. https://arxiv.org/abs/2511.13323

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