Existence of a stationary wave for the discrete Boltzmann equation in the half space
The existence and the uniqueness of stationary solutions of the discrete model of Boltzmann equation are studied analytically. The one-dimensional problem is considered in the first half space \((x>0)\). The Dirichlet boundary conditions that represent the pure diffuse reflection of rarefied gas are at fulfilled \(x=0\). The spatially asymptotic condition is posed at infinity. The Boltzmann equation is decomposed into linear algebraic equations and linear differential equations. Using the algebraic decomposition technique, the existence of a solution of the linearized Boltzmann equation is proved. The function space to, which solutions of the linearized Boltzmann equation belong, is constructed. Using the sufficient condition that is obtained for the linearized Boltzmann equation and the contraction mapping principle the existence theorem for the nonlinear Boltzmann equation is proved. It is shown that the stationary solutions of the discrete Boltzmann equation approaches the Maxwellian state exponentially at a spatial asymptotic point. Applications of the existence theorem to two kinds of Boltzmann type equations: a) the Cabannes' 14-velocity model; b) 6-velocity model with multiple collisions are given.
scientific article; zbMATH DE number 1415646