Fractional Diffusion limit of a kinetic equation with Diffusive boundary conditions in the upper-half space

TL;DR

This paper derives a fractional diffusion equation with nonlocal boundary conditions from a kinetic model in the upper-half space, highlighting differences from previous models with specular reflection and emphasizing the impact of diffusive boundary conditions.

Contribution

It introduces a novel fractional diffusion limit for a kinetic equation with diffusive boundary conditions, expanding understanding of boundary effects in kinetic-to-diffusion transitions.

Findings

Derived a fractional diffusion equation with nonlocal Neumann boundary conditions.

Identified differences from previous models with specular reflection.

Provided a new perspective on boundary effects in kinetic equations.

Abstract

We investigate the fractional diffusion approximation of a kinetic equation in the upper-half plane with diffusive reflection conditions at the boundary. In an appropriate singular limit corresponding to small Knudsen number and long time asymptotic, we derive a fractional diffusion equation with a nonlocal Neumann boundary condition for the density of particles. Interestingly, this asymptotic equation is different from the one derived by L. Cesbron in [7] in the case of specular reflection conditions at the boundary and does not seem to have receive a lot of attention previously.