On shock wave solutions for discrete velocity models of the Boltzmann equation

Abstract

The existence of weak shock wave solutions for discrete velocity models of the Boltzmann equation is proved. Specifically, consider triples (M− M+,c) of two Maxwellians M− M+ and shock speeds c ∈ ℛ satisfying the Rankine-Hugoniot conditions. The triple (M−, M− , c) satisfies these conditions trivially for any c. If c0 is chosen to be such that the manifold defined by the Rankine-Hugoniot conditions exhibits a transcritical bifurcation at C0, then we prove that there is a one-sided, one-parameter family of triples M−, M+(ε) and c(ε), with M+(0) =M− and c(0) = c0, such that there is a rarefied shock wave solution for the discrete velocity model connecting M− with M+(ε), moving with shock speed c(ε), and satisfying the entropy condition.