Kinetic derivation of fractional Stokes and Stokes-Fourier systems

TL;DR

This paper extends the kinetic derivation of fractional hydrodynamic equations from linear Boltzmann-type models, demonstrating fractional diffusion for temperature and density, and deriving fractional Stokes equations under certain conservation laws.

Contribution

It introduces a BGK-type kinetic model conserving mass, momentum, and energy, and derives fractional diffusion and Stokes equations in the hydrodynamic limit for different equilibrium distributions.

Findings

Fractional diffusion for temperature and density derived from kinetic models.

Fractional Stokes equations obtained for heavy-tailed equilibria.

Fractional diffusion for momentum not derived under full conservation laws.

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.