Stationary Flows of the ES-BGK model with the correct Prandtl number

TL;DR

This paper establishes the existence and uniqueness of stationary solutions for the ES-BGK model with correct Prandtl number in a slab, addressing challenges in controlling the temperature tensor under various conditions.

Contribution

It provides the first rigorous analysis of stationary solutions for the ES-BGK model with mixed boundary conditions, including the critical case where the temperature tensor control is complex.

Findings

Proved existence and uniqueness of stationary solutions.

Developed methods to control the temperature tensor from below.

Analyzed both non-critical and critical cases for the Prandtl parameter.

Abstract

Ellipsoidal BGK model (ES-BGK) is a generalized version of the BGK model where the local Maxwellian in the relaxation operator of the BGK model is extended to an ellipsoidal Gaussian with a Prandtl parameter $\nu$, so that the correct transport coefficients can be computed in the Navier-Stokes limit. In this work, we consider the existence and uniqueness of stationary solutions for the ES-BGK model in a slab imposed with the mixed boundary conditions. One of the key difficulties arise in the uniform control of the temperature tensor from below. In the non-critical case $(-1/2<\nu<1)$, we utilize the property that the temperature tensor is equivalent to the temperature in this range. In the critical case, $(\nu=-1/2)$, where such equivalence relation breaks down, we observe that the size of bulk velocity in $x$ direction can be controlled by the discrepancy of boundary flux, which enables one to bound the temperature tensor from below.