Sharpening of decay rates in Fourier based hypocoercivity methods

TL;DR

This paper enhances Fourier-based hypocoercivity techniques by refining decay rate estimates for kinetic equations, comparing two methods involving nonlocal perturbations and Lyapunov-based scalar products, with applications to models like Fokker-Planck and Goldstein-Taylor.

Contribution

It introduces improved decay rate estimates in Fourier hypocoercivity methods using novel perturbations and scalar product modifications, advancing the analysis of kinetic equations.

Findings

Enhanced decay rate estimates for kinetic equations.

Comparison of two hypocoercivity methods on the Goldstein-Taylor model.

Improved understanding of convergence in Fokker-Planck and relaxation models.

Abstract

This paper is dealing with two $L^2$ hypocoercivity methods based on Fourier decomposition and mode-by-mode estimates, with applications to rates of convergence or decay in kinetic equations on the torus and on the whole Euclidean space. The main idea is to perturb the standard $L^2$ norm by a twist obtained either by a nonlocal perturbation build upon diffusive macroscopic dynamics, or by a change of the scalar product based on Lyapunov matrix inequalities. We explore various estimates for equations involving a Fokker-Planck and a linear relaxation operator. We review existing results in simple cases and focus on the accuracy of the estimates of the rates. The two methods are compared in the case of the Goldstein-Taylor model in one-dimension.