Existence of a stationary wave for the discrete Boltzmann equation in the half space
type: Item
instance of: scholarly article
title: Existence of a stationary wave for the discrete Boltzmann equation in the half space
zbMATH Open document ID: 0983.82015
author: Shuichi Kawashima
author: Shinya Nishibata
published in: Communications in Mathematical Physics
publication date: 2002-04-28T00:00:00Z
review text: 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.
reviewed by: Oleg A. Sinkevich
Mathematics Subject Classification ID: 82C40
Mathematics Subject Classification ID: 76P05
zbMATH DE Number: 1415646
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: stationary solutions
zbMATH Keywords: discrete model of Boltzmann equation
zbMATH Keywords: Dirichlet boundary conditions
zbMATH Keywords: algebraic decomposition
zbMATH Keywords: linearized Boltzmann equation
zbMATH Keywords: nonlinear Boltzmann equation
zbMATH Keywords: Maxwellian state
MaRDI profile type: Publication
full work available at URL: s002200050730
OpenAlex ID: W2062643684
DOI: 10.1007/S002200050730
description: scientific article; zbMATH DE number 1415646