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KRM

Existence of solutions of weakly non-linear half-space problems for the
general discrete velocity (with arbitrarily finite number of velocities)
model of the Boltzmann equation are studied. The solutions are assumed to
tend to an assigned Maxwellian at infinity, and the data for the outgoing
particles at the boundary are assigned, possibly linearly depending on the
data for the incoming particles. The conditions, on the data at the
boundary, needed for the existence of a unique (in a neighborhood of the
assigned Maxwellian) solution of the problem are investigated. In the
non-degenerate case (corresponding, in the continuous case, to the case when
the Mach number at infinity is different of -1, 0 and 1) implicit conditions
are found. Furthermore, under certain assumptions explicit conditions are
found, both in the non-degenerate and degenerate cases. Applications to
axially symmetric models are studied in more detail.

KRM

We consider the discrete Boltzmann equation for binary gas mixtures. Some known results for half-space problems and shock profile solutions of the discrete Boltzmann for single-component gases are extended to the case of two-component gases. These results include well-posedness results for
half-space problems for the linearized discrete Boltzmann equation, existence
results for half-space problems for the weakly non-linear discrete Boltzmann equation, and existence results for shock profile solutions of the discrete Boltzmann equation.
A characteristic number, corresponding to the speed of sound in the continuous case, is calculated for a class of symmetric models. Some explicit calculations are also made for a simplified 6 + 4 -velocity model.

keywords:
Boltzmann equation
,
discrete velocity models
,
half-space problems
,
boundary layers
,
shock profiles.
,
mixtures

KRM

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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