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

2021/07/02 by Ludovic Cesbron, Cesbron, Ludovic, Antoine Mellet +3 · 1 citation
Mathematics · #35S15 26A33 82C40 #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #FOS: Physical sciences #Fractional Differential Equations Solutions #Mathematical Physics (math-ph) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2107.01011

openalex publication_date 2021/07/02 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

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.

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