Fractional Diffusion limit of a kinetic equation with Diffusive boundary conditions in a bounded interval

TL;DR

This paper proves that, under certain limits, the density of a kinetic equation in a bounded interval converges to the unique solution of a fractional diffusion equation with Neumann boundary conditions.

Contribution

It establishes the uniqueness of the fractional diffusion limit for a kinetic equation with diffusive boundary conditions in a bounded interval.

Findings

The asymptotic density solves a fractional diffusion equation with Neumann boundary conditions.

The analysis confirms the convergence of the kinetic model to the fractional diffusion limit.

Completes previous work by proving solution uniqueness in the bounded interval case.

Abstract

We investigate the fractional diffusion approximation of a kinetic equation set in a bounded interval with diffusive reflection conditions at the boundary. In an appropriate singular limit corresponding to small Knudsen number and long time asymptotic, we show that the asymptotic density function is the {\it unique solution} of a fractional diffusion equation with Neumann boundary condition. This analysis completes a previous work by the same authors in which a limiting fractional diffusion equation was identified on the half-space, but the uniqueness of the solution (which is necessary to prove the convergence of the whole sequence) could not be established.