Entropy production for ellipsoidal BGK model of the Boltzmann equation

TL;DR

This paper investigates the entropy production properties of the ellipsoidal BGK model for the Boltzmann equation, establishing key estimates and properties that enhance understanding of its thermodynamic behavior.

Contribution

It demonstrates the Cercignani type estimate for the ES-BGK model across a range of parameters and reveals a novel sign-definite property of the relaxation operator.

Findings

Cercignani type estimate holds for -1/2<ν<1

Ellipsoidal relaxation operator has a sign-definite property

Implications for entropy production in kinetic models

Abstract

The ellipsoidal BGK model (ES-BGK) is a generalized version of the BGK model of the Boltzmann equation designed to yield the correct Prandtl number in the Navier-Stokes limit. In this paper, we make two observations on the entropy production functional of the ES-BGK model. First, we show that the Cercignani type estimate holds for the ES-BGK model in the whole range of parameter $-1/2<\nu<1$. Secondly, we observe that the ellipsoidal relaxation operator satisfies an unexpected sign-definite property. Some implications of these observations are also discussed.