Some problems concerning the solvability of the nonlinear stationary Boltzmann equation in the framework of the BGK model

In the framework of the BGK (Bhatnagar–Gross–Krook) model, we derive a system of nonlinear integral equations for the macroscopic variables both in a finite plane channel $ \Pi _{r}$ of thickness $ r$ $ (r<+\infty )$ and in the subspace $ \Pi _\infty $ $ (r=+\infty )$ from the nonlinear integro-differential Boltzmann equation. Solvability problems are discussed and solution methods are suggested for these systems of nonlinear integral equations. Theorems on the existence of bounded positive solutions are proved and two-sided estimates of these solutions are obtained for the resulting nonlinear integral equations of the Urysohn type describing the temperature (Theorems 1 and 3). A theorem on the existence of a unique solution in the space $ L_1[0,r]$ is proved for the linear integral equations describing the velocity and density. Integral estimates for the solutions are obtained (see Theorem 2 and the Corollary).
The nonlinear system of integral equations in the subspace obtained for the macroscopic variables in the framework of the nonlinear BGK model of the Boltzmann equation is shown to have no bounded solutions with finite limit at infinity other than a constant solution.
The solution of the linear problem obtained by linearizing the corresponding nonlinear system is proved to be $ O(x)$ as $ x\rightarrow +\infty $ (Theorem 3).