Stationary Flows of the ES-BGK Model with the Correct Prandtl Number

SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6361-6397, October 2024.
Abstract. The 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 parameter [math], so that the correct Prandtl number can be computed in the Navier–Stokes limit. In this work, we consider steady rarefied flows arising from the evaporation and condensation process between two parallel condensed phases, which is formulated in this paper as the existence problem of stationary solutions to the ES-BGK model in a bounded interval with the mixed boundary conditions. One of the key difficulties arises in the uniform control of the temperature tensor from below. In the noncritical case [math], we utilize the property that the temperature tensor is equivalent to the temperature. In the critical case, [math], where such equivalence relation breaks down, we observe that the size of bulk velocity in the [math] direction can be controlled by the discrepancy of boundary flux, which enables one to bound the temperature tensor from below.