抄録
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.
| 元の言語 | English |
|---|---|
| ページ（範囲） | 183-206 |
| ページ数 | 24 |
| ジャーナル | Communications in Mathematical Physics |
| 巻 | 211 |
| 発行部数 | 1 |
| DOI | |
| 出版物ステータス | Published - 2000 1 1 |
| 外部発表 | Yes |
Fingerprint
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
これを引用
Stationary waves for the discrete Boltzmann equation in the half space with reflective boundaries. / Kawashima, Shuichi; Nishibata, Shinya.
：: Communications in Mathematical Physics, 巻 211, 番号 1, 01.01.2000, p. 183-206.研究成果: Article
}
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 -