Abstract
Existence of solutions of weakly non-linear half-space problems for thegeneral discrete velocity (with arbitrarily finite number of velocities)model of the Boltzmann equation are studied. The solutions are assumed totend to an assigned Maxwellian at infinity, and the data for the outgoingparticles at the boundary are assigned, possibly linearly depending on thedata for the incoming particles. The conditions, on the data at theboundary, needed for the existence of a unique (in a neighborhood of theassigned Maxwellian) solution of the problem are investigated. In thenon-degenerate case (corresponding, in the continuous case, to the case whenthe Mach number at infinity is different of -1, 0 and 1) implicit conditionsare found. Furthermore, under certain assumptions explicit conditions arefound, both in the non-degenerate and degenerate cases. Applications toaxially symmetric models are studied in more detail.
Highlights
The planar stationary Boltzmann equation, see Ref. [16] and [17], with inflow boundary condition reads ξ1 ∂F = Q (F, F ), F = F (x, ξ)∂x F (0, ξ) = F0 (ξ) for ξ1 > 0 (1) F → M∞as x → ∞, where x ∈ R+, ξ = ξ1, ξ2, ξ3 ∈ R3, M∞
[38] Ukai, Yang and Yu studied the non-linear problem with inflow boundary conditions for a hard sphere gas, assuming that the solutions tend to an assigned Maxwellian at infinity
[18] Cercignani et al have shown that the solutions of the half-space problem for the general non-linear Discrete velocity models (DVMs) with inflow boundary conditions tend to Maxwellians at infinity
Summary
The planar stationary Boltzmann equation, see Ref. [16] and [17], with inflow boundary condition reads. [38] Ukai, Yang and Yu studied the non-linear problem with inflow boundary conditions for a hard sphere gas, assuming that the solutions tend to an assigned Maxwellian at infinity. [18] Cercignani et al have shown that the solutions of the half-space problem for the general non-linear DVM with inflow boundary conditions tend to Maxwellians at infinity (without specifying the Maxwellians). Implicit conditions for the existence of a unique (in a neighborhood of the assigned Maxwellian) solution in the non-degenerate case and for the degenerate case, but with some restrictions on the non-linear part of the collision operator, are obtained (Section 5). The results are in accordance with corresponding results for the continuous Boltzmann equation obtained in the non-degenerate case, with inflow boundary conditions in Ref.
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