Entropy production estimate for the ES-BGK model with the correct Prandtl number

TL;DR

This paper derives an improved entropy-entropy production estimate for the ES-BGK model, covering the full Prandtl range and providing sharper bounds, enhancing understanding of the model's thermodynamic properties.

Contribution

It extends previous entropy estimates for the ES-BGK model to the entire Prandtl parameter range and offers a more precise bound with explicit coefficients.

Findings

Full Prandtl range covered, including the critical case.

Sharper entropy production bounds obtained.

Explicit coefficient characterization provided.

Abstract

In this paper, we establish the entropy-entropy production estimate for the ES-BGK model, a generalized version of the BGK model of the Boltzmann equation introduced for better approximation in the Navier-Stokes limit. Our result improves the previous entropy production estimate [39] in that (1) the full range of Prandtl parameters $-1/2\leq\nu <1$ including the critical case $\nu=-1/2$ is covered, and (2) a sharper entropy production bound is obtained. An explicit characterization of the coefficient of the entropy-entropy production estimate is also presented.