Existence of a stationary wave for the discrete Boltzmann equation in the half space

We study the existence and the uniqueness of stationary solutions for discrete velocity models of the Boltzmann equation in the first half space. We obtain a sufficient condition that guarantees the existence and the uniqueness of solutions connecting the given boundary data and the Maxwellian state at a spatially asymptotic point. First, a sufficient condition is obtained for the linearized system. Then this result as well as the contraction mapping principle is applied to prove the existence theorem for the nonlinear equation. Also, we show that the stationary wave approaches the Maxwellian state exponentially at a spatially asymptotic point. We also discuss some concrete models of Boltzmann type as an application of our general theory. Here, it turns out that our sufficient condition is general enough to cover many concrete models.