Fractional diffusion limit of a kinetic equation with diffusive boundary conditions in a bounded interval

Article type: Research Article

Authors: Cesbron, L.^{a; *} | Mellet, A.^b | Puel, M.^c

Affiliations: [a] Department of Mathematics, ETH Zürich, Switzerland | [b] Department of Mathematics and CSCAMM, University of Maryland, USA | [c] Département de Mathématiques, CY Cergy Paris Université, France

Correspondence: [*] Corresponding author. E-mail: ludovic.cesbron@univ-lille.fr.

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 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.

Keywords: Kinetic theory, fractional diffusion, boundary conditions

DOI: 10.3233/ASY-221755

Journal: Asymptotic Analysis, vol. 130, no. 3-4, pp. 367-386, 2022