Boundedness of the Stationary Solution to the Boltzmann Equation with Spatial Smearing, Diffusive Boundary Conditions, and Lions' Collision Kernel

TL;DR

This paper proves that stationary solutions to a bounded Boltzmann equation with specific boundary conditions and collision kernel are essentially bounded, extending previous existence results to include bounds.

Contribution

It establishes bounds for stationary solutions of the Boltzmann equation with spatial smearing and diffusive boundary conditions, under Lions' collision kernel, beyond mere existence.

Findings

Stationary solutions are almost everywhere positively bounded from below and above.

Existence and uniqueness of stationary solutions are confirmed under certain conditions.

The results apply to bounded physical and velocity spaces.

Abstract

We investigate the Boltzmann equation with spatial smearing, diffusive boundary conditions, and Lions' collision kernel. Both, the physical as well as the velocity space, are assumed to be bounded. Existence and uniqueness of a stationary solution, which is a probability density, has been demonstrated in [3] under a certain smallness assumption on the collision term. We prove that whenever there is a stationary solution then it is a.e. positively bounded from below and above.