Boundary Layers and Shock Profiles for the Broadwell Model

We consider the existence of nonlinear boundary layers and the typically nonlinear problem of existence of shock profiles for the Broadwell model, which is a simplified discrete velocity model for the Boltzmann equation. We find explicit expressions for the nonlinear boundary layers and the shock profiles. In spite of the few velocities used for the Broadwell model, the solutions are (at least partly) in qualitatively good agreement with the results for the discrete Boltzmann equation, that is the general discrete velocity model, and the full Boltzmann equation.


Introduction
The Boltzmann equation (BE) is a fundamental equation in kinetic theory.Half-space problems for the BE are of great importance in the study of the asymptotic behavior of the solutions of boundary value problems of the BE for small Knudsen numbers [1,2] and have been extensively studied both for the full BE [3,4] and for the discrete Boltzmann equation (DBE) [5][6][7][8].The half-space problems provide the boundary conditions for the fluid-dynamic-type equations and Knudsen-layer corrections to the solution of the fluiddynamic-type equations in a neighborhood of the boundary.In [8] nonlinear boundary layers for the DBE, the general discrete velocity model (DVM) was considered.Existence of weakly nonlinear boundary layers was proved.Here we exemplify the theory in [8] for a simplified model, the Broadwell model [9], where the whole machinery is actually not really needed, even if it helps out.For the nonlinear Broadwell model, we obtain explicit expressions for boundary layers near a wall moving with a constant speed.The number of conditions, on the assigned data for the outgoing particles at the boundary, needed for the existence of a unique (in a neighborhood of the assigned Maxwellian at infinity) solution of the problem is in complete agreement with the results in [8] for the DBE and [3] for the full BE.Here we also want to mention a series of papers studying initial boundary value problems for the Broadwell model using Green's functions [10][11][12][13][14][15][16].
We also consider the question of existence of shock profiles [17,18] for the same model [9,19].The shock profiles can then be seen as heteroclinic orbits connecting two singular points (Maxwellians) [20].In [20] existence of shock profiles for the DBE in the case of weak shocks was proved.We exemplify the theory in [20] for the Broadwell model, where again the whole machinery is not really needed, even if it helps out.In this way we, in a new way, obtain similar explicit solutions, not only for weak shocks, as the ones obtained in [19] for the same problem.
The paper is organized as follows.In Section 2 we introduce the Broadwell model and find explicit expressions for the nonlinear boundary layers near a wall moving with a constant speed, and in Section 3 we find explicit expressions for the shock profiles for the Broadwell model.

Nonlinear Boundary Layers for the Broadwell Model Near a Moving Wall
In this section we study boundary layers for the nonlinear Broadwell model near a wall moving with a constant speed .
Here we consider the stationary nonlinear system The linearized collision operator is symmetric and semipositive and have the null-space for some  1 and  2 , such that Note also that ⟨ (ℎ, ℎ) , ⟩ = 0, ∀ ∈  () .
Below we consider the remaining different cases.

Shock Profiles
In this section we are concerned with the existence of shock profiles [17,18] for the Boltzmann equation Here x = ( 1 , . . .,   ) ∈ R  ,  = ( 1 , . . .,   ) ∈ R  , and  ∈ R + denote position, velocity, and time, respectively.Furthermore,  denotes the speed of the wave.The profiles are assumed to approach two given Maxwellians (, u, and  denote density, bulk velocity, and temperature, resp.) as  → ±∞, which are related through the Rankine-Hugoniot conditions.The (shock wave) problem is to find a solution  = (, ) ( =  1 − ) of the equation such that  →  ± as  → ±∞. (45) In [20] the shock wave problem (44), (45) for the DBE was considered.Existence of shock profiles in the case of weak shocks was proved.Here we exemplify the theory in [20] in the case of a simplified model, where the whole machinery is actually not really needed, even if it helps out.In this way we, in a different way, obtain similar results as is obtained in [19] for the same problem.
We study the reduced system (2) of the classical Broadwell model in (1) [9] in space.The collision invariants are given by (3) and the Maxwellians (equilibrium distributions) by ( 4).
Remark 3. We can instead of system (50) consider with and in a similar way as above, we obtain Example 4. If  = 1 then we have Furthermore, and the other Maxwellian is International Journal of Differential Equations 7 Example 5. Similar results can be obtained for the (reduced) plane Broadwell model (75) The shock strength (cf.[19]) is given by the density ratio