Asymptotic Stability of Traveling Waves to a Certain Discrete Velocity Model of the Boltzmann Equation in the Half Space

Abstract

The present paper studies the asymptotic stability of a traveling wave for the Broadwell model in a half space. This model admits the traveling wave which connects two distinct Maxwellian states at the spatial asymptotic points. The traveling wave is shown to be time asymptotically stable if the fluid dynamical velocity is less than a certain positive value. This stability theorem is proved by applying the standard energy method. Here, the location of the traveling wave, which should be a time asymptotic state, is shifted by boundary effect. This shift is estimated by utilizing the property that the traveling wave converges to the Maxwellian states exponentially fast.