Anomalous diffusion limit of kinetic equations in spatially bounded\n domains

2016/11/19 by Ludovic Cesbron, Cesbron, Ludovic · 1 citation
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Mathematical Biology Tumor Growth

paper · pdf · doi:10.48550/arxiv.1611.06372

Abstract

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

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