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

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.