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Anomalous Diffusion Limit of Kinetic Equations in Spatially Bounded Domains

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Abstract

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

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  1. DPMMS, Center for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK

    Ludovic Cesbron

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  1. Ludovic Cesbron
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Correspondence to Ludovic Cesbron.

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Communicated by W. Schlag

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Cesbron, L. Anomalous Diffusion Limit of Kinetic Equations in Spatially Bounded Domains. Commun. Math. Phys. 364, 233–286 (2018). https://doi.org/10.1007/s00220-018-3158-0

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