Existence and stability of stationary solutions to the discrete Boltzmann equation

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6 Citations (Scopus)

Abstract

The initial-boundary value problems and the corresponding stationary problems of the discrete Boltzmann equation are studied. It is shown that stationary solutions exist for any boundary data. These stationary solutions are unique in a neighborhood of a given constant Maxwellian. Furthermore, it is proved that if both initial and boundary data are close to a given constant Maxwellian, then unique solutions to the initial-boundary value problems exist globally in time and converge to the corresponding unique stationary solutions exponentially as time goes to infinity. The stability condition plays an essential role in proving the uniqueness and the time-asymptotic stability results.

Original languageEnglish
Pages (from-to)389-429
Number of pages41
JournalJapan Journal of Industrial and Applied Mathematics
Volume8
Issue number3
DOIs
Publication statusPublished - 1991 Oct 1
Externally publishedYes

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Boltzmann equation
Discrete Equations
Stationary Solutions
Boltzmann Equation
Boundary value problems
Initial-boundary-value Problem
Asymptotic stability
Stability Condition
Unique Solution
Asymptotic Stability
Uniqueness
Infinity
Converge

Keywords

  • discrete Boltzmann equation
  • stationary solution
  • time-asymptotic stability
  • time-global solution
  • uniqueness

ASJC Scopus subject areas

  • Engineering(all)
  • Applied Mathematics

Cite this

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author = "Shuichi Kawashima",
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AB - The initial-boundary value problems and the corresponding stationary problems of the discrete Boltzmann equation are studied. It is shown that stationary solutions exist for any boundary data. These stationary solutions are unique in a neighborhood of a given constant Maxwellian. Furthermore, it is proved that if both initial and boundary data are close to a given constant Maxwellian, then unique solutions to the initial-boundary value problems exist globally in time and converge to the corresponding unique stationary solutions exponentially as time goes to infinity. The stability condition plays an essential role in proving the uniqueness and the time-asymptotic stability results.

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KW - time-asymptotic stability

KW - time-global solution

KW - uniqueness

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