Kinetic and Related Models (KRM)

Kinetic derivation of fractional Stokes and Stokes-Fourier systems

Pages: 105 - 129, Volume 9, Issue 1, March 2016      doi:10.3934/krm.2016.9.105

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Sabine Hittmeir - RICAM Linz, Austrian Academy of Sciences, Altenberger Str. 69, 4040 Linz, Austria (email)
Sara Merino-Aceituno - Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom (email)

Abstract: In recent works it has been demonstrated that using an appropriate rescaling, linear Boltzmann-type equations give rise to a scalar fractional diffusion equation in the limit of a small mean free path. The equilibrium distributions are typically heavy-tailed distributions, but also classical Gaussian equilibrium distributions allow for this phenomena if combined with a degenerate collision frequency for small velocities. This work aims to an extension in the sense that a linear BGK-type equation conserving not only mass, but also momentum and energy, for both mentioned regimes of equilibrium distributions is considered. In the hydrodynamic limit we obtain a fractional diffusion equation for the temperature and density making use of the Boussinesq relation and we also demonstrate that with the same rescaling fractional diffusion cannot be derived additionally for the momentum. But considering the case of conservation of mass and momentum only, we do obtain the incompressible Stokes equation with fractional diffusion in the hydrodynamic limit for heavy-tailed equilibria.

Keywords:  Kinetic transport equation, linear BGK model, hydrodynamic limit, fractional diffusion, anomalous diffusive time scale, incompressible Stokes equation, Stokes-Fourier system.
Mathematics Subject Classification:  Primary: 35Q20, 35Q30, 35Q35, 35R11.

Received: September 2014;      Revised: July 2015;      Available Online: October 2015.