Anomalous diffusion limit of kinetic equations in spatially bounded domains

TL;DR

This paper investigates the anomalous diffusion limit of kinetic equations with fractional Fokker-Planck operators in bounded domains, revealing boundary-dependent operators and establishing well-posedness of related equations.

Contribution

It introduces a new boundary-dependent non-local diffusion operator, the specular diffusion operator, and analyzes its properties in specific geometries.

Findings

The fractional diffusion limit is characterized by a Dirichlet problem in the absorption case.

The specular diffusion operator depends on domain geometry and boundary interactions.

Existence and uniqueness of solutions are proved for the associated equations.

Abstract

This paper is devoted to the anomalous diffusion limit of kinetic equations with a fractional Fokker-Planck collision operator in a spatially bounded domain. We consider two boundary conditions at the kinetic scale: absorption and specular reflection. In the absorption case, we show that the long time/small mean free path asymptotic dynamics are described by a fractional diffusion equation with homogeneous Dirichlet-type boundary conditions set on the whole complement of the spatial domain. On the other hand, specular reflections will give rise to a new operator which we call specular diffusion operator and write $(-\Delta)_{\text{SR}}^s$. This non-local diffusion operator strongly depends on the geometry of the domain and includes in its definition the interaction between the diffusion and the boundary. We consider two types of domains: half-spaces and balls in $\mathbb{R}^d$. In these domains, we prove properties of the specular diffusion operator and establish existence and uniqueness of weak solutions to the associated heat-type equation.