### 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 language | English |
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Pages (from-to) | 183-206 |

Number of pages | 24 |

Journal | Communications in Mathematical Physics |

Volume | 211 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2000 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*211*(1), 183-206. https://doi.org/10.1007/s002200050808

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

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 211, no. 1, pp. 183-206. https://doi.org/10.1007/s002200050808

}

TY - JOUR

T1 - Stationary waves for the discrete Boltzmann equation in the half space with reflective boundaries

AU - Kawashima, Shuichi

AU - Nishibata, Shinya

PY - 2000/1/1

Y1 - 2000/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0034349434&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034349434&partnerID=8YFLogxK

U2 - 10.1007/s002200050808

DO - 10.1007/s002200050808

M3 - Article

AN - SCOPUS:0034349434

VL - 211

SP - 183

EP - 206

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -