The Shakhov model near a global Maxwellian

TL;DR

This paper investigates the mathematical properties of the Shakhov model, a kinetic equation approximation, focusing on existence and stability near equilibrium, and introduces a micro-macro system to address degeneracy issues related to the Prandtl number.

Contribution

It establishes the existence and asymptotic stability of solutions to the Shakhov model near equilibrium and analyzes the coercivity differences based on the Prandtl number.

Findings

Derived a dichotomy in coercive estimates for zero and non-zero Prandtl numbers.

Identified degeneracy in the linearized relaxation operator at zero Prandtl number.

Proposed a micro-macro system to recover full coercivity and address degeneracy.

Abstract

Shakhov model is a relaxation approximation of the Boltzmann equation proposed to overcome the deficiency of the original BGK model, namely, the incorrect production of the Prandtl number. In this paper, we address the existence and the asymptotic stability of the Shakhov model when the initial data is a small perturbation of global equilibrium. We derive a dichotomy in the coercive estimate of the linearized relaxation operator between zero and non-zero Prandtl number, and observe that the linearized relaxation operator is more degenerate in the former case. To remove such degeneracy and recover the full coercivity, we consider a micro-macro system that involves an additional non-conservative quantity related to the heat flux.