Fractional hypocoercivity in bounded domains in the anomalous diffusion limit

TL;DR

This paper establishes exponential stability for dissipative linear kinetic equations with heavy-tailed equilibria, using an $L^2$-hypocoercivity approach that remains effective in the anomalous diffusion limit and in bounded domains with various boundary conditions.

Contribution

It introduces a robust hypocoercivity framework that handles heavy-tailed equilibria, bounded domains, and complex boundary conditions in kinetic equations.

Findings

Proves exponential stability in the anomalous diffusion limit.

Provides uniform estimates across different boundary conditions.

Handles linear collisional operators acting on both velocity and space.

Abstract

In this paper, we provide a result of exponential stability for several dissipative linear kinetic equations with heavy-tailed equilibria. The approach, inspired by the so-called $L^2$-hypocoercivity method, is robust enough to provide estimates that are uniform in the anomalous diffusion limit. Moreover, it is able to deal with bounded domains with periodic boundary condition or general Maxwell boundary condition (from the pure specular to the pure diffusive case). In addition, our framework accommodates linear collisional operators that act simultaneously on the velocity and spatial variables.