Stationary waves for the discrete Boltzmann equation in the half space with reflective boundaries

Shuichi Kawashima, Shinya Nishibata

Research output: Contribution to journalArticle

22 Citations (Scopus)

Abstract

The present paper is concerned with stationary solutions for discrete velocity models of the Boltzmann equation with reflective boundary condition in the first half space. We obtain a sufficient condition that guarantees the existence and the uniqueness of stationary solutions satisfying the reflective boundary condition as well as the spatially asymptotic condition given by a Maxwellian state. First, the sufficient condition is obtained for the linearized system. Then, this result is applied to prove the existence theorem for the nonlinear equation through the contraction mapping principle. Also, it is shown that the stationary solution approaches the asymptotic Maxwellian state exponentially as the spatial variable tends to infinity. Moreover, we show the time asymptotic stability of the stationary solutions. In the proof, we employ the standard energy method to obtain a priori estimates for nonstationary solutions. The exponential convergence at the spatial asymptotic state of the stationary solutions gives essential information to handle some error terms. Then we discuss some concrete models of the Boltzmann type as an application of our general theory.

Original languageEnglish
Pages (from-to)183-206
Number of pages24
JournalCommunications in Mathematical Physics
Volume211
Issue number1
DOIs
Publication statusPublished - Jan 1 2000

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Discrete Equations
half spaces
Stationary Solutions
Boltzmann Equation
Half-space
existence theorems
boundary conditions
energy methods
uniqueness
infinity
nonlinear equations
contraction
Discrete Velocity Models
Contraction Mapping Principle
Boundary conditions
Exponential Convergence
Sufficient Conditions
Energy Method
Error term
A Priori Estimates

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Stationary waves for the discrete Boltzmann equation in the half space with reflective boundaries. / Kawashima, Shuichi; Nishibata, Shinya.

In: Communications in Mathematical Physics, Vol. 211, No. 1, 01.01.2000, p. 183-206.

Research output: Contribution to journalArticle

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