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Solutions to the discrete Boltzmann equation with general boundary conditions. (English) Zbl 0939.35156

This paper contains results of two different types.
First, existence and uniqueness results for the initial-boundary value problems for discrete velocity models of the Boltzmann equation in an interval (“slab”) are proved for mixed-type boundary conditions combining partial stochastic reflection and inflow. Under reasonable assumptions on the boundary condition, it follows that no particles are created in the diffuse reflection process. This is used for the global existence proof.
Secondly, existence of stationary solutions is proved by a Leray-Schauder type fixed point argument under the stricter assumption that the reflection on one part of the boundary is really strictly partial.

MSC:

35Q35 PDEs in connection with fluid mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
35L45 Initial value problems for first-order hyperbolic systems
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