Qualitative difference between solutions of stationary model Boltzmann equations in the linear and nonlinear cases

We consider problems for the nonlinear Boltzmann equation in the framework of two models{:} a new nonlinear model and the Bhatnagar–Gross–Krook model. The corresponding transformations reduce these problems to nonlinear systems of integral equations. In the framework of the new nonlinear model, we prove the existence of a positive bounded solution of the nonlinear system of integral equations and present examples of functions describing the nonlinearity in this model. The obtained form of the Boltzmann equation in the framework of the Bhatnagar–Gross–Krook model allows analyzing the problem and indicates a method for solving it. We show that there is a qualitative difference between the solutions in the linear and nonlinear cases{\rm:} the temperature is a bounded function in the nonlinear case, while it increases linearly at infinity in the linear approximation. We establish that in the framework of the new nonlinear model, equations describing the distributions of temperature, concentration, and mean-mass velocity are mutually consistent, which cannot be asserted in the case of the Bhatnagar–Gross–Krook model.