### Abstract

We study the existence and the uniqueness of stationary solutions for discrete velocity models of the Boltzmann equation in the first half space. We obtain a sufficient condition that guarantees the existence and the uniqueness of solutions connecting the given boundary data and the Maxwellian state at a spatially asymptotic point. First, a sufficient condition is obtained for the linearized system. Then this result as well as the contraction mapping principle is applied to prove the existence theorem for the nonlinear equation. Also, we show that the stationary wave approaches the Maxwellian state exponentially at a spatially asymptotic point. We also discuss some concrete models of Boltzmann type as an application of our general theory. Here, it turns out that our sufficient condition is general enough to cover many concrete models.

Original language | English |
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Pages (from-to) | 385-409 |

Number of pages | 25 |

Journal | Communications in Mathematical Physics |

Volume | 207 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 1 1999 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*207*(2), 385-409. https://doi.org/10.1007/s002200050730

**Existence of a stationary wave for the discrete Boltzmann equation in the half space.** / Kawashima, Shuichi; Nishibata, Shinya.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 207, no. 2, pp. 385-409. https://doi.org/10.1007/s002200050730

}

TY - JOUR

T1 - Existence of a stationary wave for the discrete Boltzmann equation in the half space

AU - Kawashima, Shuichi

AU - Nishibata, Shinya

PY - 1999/1/1

Y1 - 1999/1/1

N2 - We study the existence and the uniqueness of stationary solutions for discrete velocity models of the Boltzmann equation in the first half space. We obtain a sufficient condition that guarantees the existence and the uniqueness of solutions connecting the given boundary data and the Maxwellian state at a spatially asymptotic point. First, a sufficient condition is obtained for the linearized system. Then this result as well as the contraction mapping principle is applied to prove the existence theorem for the nonlinear equation. Also, we show that the stationary wave approaches the Maxwellian state exponentially at a spatially asymptotic point. We also discuss some concrete models of Boltzmann type as an application of our general theory. Here, it turns out that our sufficient condition is general enough to cover many concrete models.

AB - We study the existence and the uniqueness of stationary solutions for discrete velocity models of the Boltzmann equation in the first half space. We obtain a sufficient condition that guarantees the existence and the uniqueness of solutions connecting the given boundary data and the Maxwellian state at a spatially asymptotic point. First, a sufficient condition is obtained for the linearized system. Then this result as well as the contraction mapping principle is applied to prove the existence theorem for the nonlinear equation. Also, we show that the stationary wave approaches the Maxwellian state exponentially at a spatially asymptotic point. We also discuss some concrete models of Boltzmann type as an application of our general theory. Here, it turns out that our sufficient condition is general enough to cover many concrete models.

UR - http://www.scopus.com/inward/record.url?scp=0033475357&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033475357&partnerID=8YFLogxK

U2 - 10.1007/s002200050730

DO - 10.1007/s002200050730

M3 - Article

AN - SCOPUS:0033475357

VL - 207

SP - 385

EP - 409

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -