TY - JOUR

T1 - Milne problem for the linear and linearized Boltzmann equations relevant to a binary gas mixture

AU - Aoki, Kazuo

AU - Lin, Yu Chu

AU - Wu, Kung Chien

N1 - Funding Information:
The first author is supported by the Ministry of Science and Technology under the grant 107-2811-M-006-008- and National Center for Theoretical Sciences. The second author is supported by the Ministry of Science and Technology under the grant 108-2115-M-006-006-. The third author is supported by the Ministry of Science and Technology under the grant 108-2636-M-006-005- and National Center for Theoretical Sciences.

PY - 2020/6/15

Y1 - 2020/6/15

N2 - A stationary boundary-value problem for the Boltzmann equation in a half space is considered for a binary mixture of gases when the indata on the boundary are given for the both species. Under the assumption that one of the species is dominant and close to equilibrium but the density of the other is small, the problem is decomposed into two half-space problems: the so-called Milne problem for the linearized Boltzmann equation with a source term for the dominant species and that for the linear Boltzmann equation for the low-density species. The two problems are coupled in such a way that the source term in the former is specified by the solution of the latter. The existence and uniqueness of the solutions to the two problems are proved, and their accurate asymptotic behavior in the far field is obtained. In particular, the precise rate of approach of the solution to the state at infinity is expressed in terms of the decay rate in the molecular velocity of the boundary data for both species.

AB - A stationary boundary-value problem for the Boltzmann equation in a half space is considered for a binary mixture of gases when the indata on the boundary are given for the both species. Under the assumption that one of the species is dominant and close to equilibrium but the density of the other is small, the problem is decomposed into two half-space problems: the so-called Milne problem for the linearized Boltzmann equation with a source term for the dominant species and that for the linear Boltzmann equation for the low-density species. The two problems are coupled in such a way that the source term in the former is specified by the solution of the latter. The existence and uniqueness of the solutions to the two problems are proved, and their accurate asymptotic behavior in the far field is obtained. In particular, the precise rate of approach of the solution to the state at infinity is expressed in terms of the decay rate in the molecular velocity of the boundary data for both species.

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U2 - 10.1016/j.jde.2019.12.003

DO - 10.1016/j.jde.2019.12.003

M3 - Article

AN - SCOPUS:85076834881

VL - 269

SP - 257

EP - 287

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 1

ER -