Boundary layers for discrete kinetic models: Multicomponent mixtures, polyatomic molecules, bimolecular reactions, and quantum kinetic equations

Niclas Bernhoff

doi:10.3934/krm.2017037

Article Contents

2017, Volume 10, Issue 4: 925-955. Doi: 10.3934/krm.2017037

This issue Previous Article Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction Next Article Grossly determined solutions for a Boltzmann-like equation

Boundary layers for discrete kinetic models: Multicomponent mixtures, polyatomic molecules, bimolecular reactions, and quantum kinetic equations

Niclas Bernhoff

Department of Mathematics and Computer Science, Karlstad University, 65188 Karlstad, Sweden

Received: March 2016

Revised: December 2016

Published: December 2017

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Abstract

We consider some extensions of the classical discrete Boltzmann equation to the cases of multicomponent mixtures, polyatomic molecules (with a finite number of different internal energies), and chemical reactions, but also general discrete quantum kinetic Boltzmann-like equations; discrete versions of the Nordheim-Boltzmann (or Uehling-Uhlenbeck) equation for bosons and fermions and a kinetic equation for excitations in a Bose gas interacting with a Bose-Einstein condensate. In each case we have an H-theorem and so for the planar stationary half-space problem, we have convergence to an equilibrium distribution at infinity (or at least a manifold of equilibrium distributions). In particular, we consider the nonlinear half-space problem of condensation and evaporation for these discrete Boltzmann-like equations. We assume that the flow tends to a stationary point at infinity and that the outgoing flow is known at the wall, maybe also partly linearly depending on the incoming flow. We find that the systems we obtain are of similar structures as for the classical discrete Boltzmann equation (for single species), and that previously obtained results for the discrete Boltzmann equation can be applied after being generalized. Then the number of conditions on the assigned data at the wall needed for existence of a unique solution is found. The number of parameters to be specified in the boundary conditions depends on if we have subsonic or supersonic condensation or evaporation. All our results are valid for any finite number of velocities.

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Mathematics Subject Classification: Primary: 82C40; Secondary: 35Q20, 76P05.

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Figure 1. 27-velocity model for a mixture of three species with mass ratios 2, 3/2, and 3

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Figure 2. 24-velocity supernormal DVM for internal energies E, 2E, and 3E

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