Ellipsoidal BGK model near a global Maxwellian

TL;DR

This paper proves the existence of classical solutions for the ellipsoidal BGK model near a global Maxwellian, addressing its mathematical properties and degeneracy issues to improve its physical accuracy.

Contribution

It establishes the existence of solutions for the ES-BGK model with small perturbations, advancing the mathematical understanding of this improved kinetic model.

Findings

Existence of classical solutions near Maxwellian

Degeneracy comparable to original BGK and Boltzmann models

Valid for parameter range -1/2<ν<1

Abstract

The BGK model has been widely used in place of the Boltzmann equation because of the qualitatively satisfactory results it provides at relatively low computational cost. There is, however, a major drawback to the BGK model: The hydrodynamic limit at the Navier-Stokes level is not correct. One evidence is that the Prandtl number computed using the BGK model does not agree with what is derived from the Boltzmann equation. To overcome this problem, Holway \cite{Holway} introduced the ellipsoidal BGK model where the local Maxwellian is replaced by a non-isotropic Gaussian. In this paper, we prove the existence of classical solutions of the ES-BGK model when the initial data is a small perturbation of the global Maxwellian. The key observation is that the degeneracy of the ellipsoidal BGK model is comparable to that of the original BGK model or the Boltzmann equation in the range $-1/2<\nu<1$.