Stationary Waves for the Discrete Boltzmann Equations in the Half Space

Abstract

We study the unique existence and the asymptotic stability of stationary solutions for discrete velocity models of the Boltzmann equation with reflective boundary condition in the first half space. We obtain a sufficient condition that guarantees the existence and the uniqueness of stationary solutions satisfying the reflective boundary condition and the spatially asymptotic condition, which is a Maxwellian state. Also, it is shown that the stationary solution approaches the asymptotic Maxwellian state exponentially as the spatial variable tends to infinity. Moreover, we show the time asymptotic stability of the stationary solutions by employing the standard energy method.