On the existence and regularity of weakly nonlinear stationary Boltzmann equations : a Fredholm alternative approach

TL;DR

This paper develops a generalized Fredholm theory for identity power compact operators to establish the existence and regularity of solutions to the stationary Boltzmann equation in convex domains, advancing mathematical understanding of kinetic equations.

Contribution

It introduces a generalized Fredholm framework for identity power compact operators and applies it to prove existence and regularity of stationary Boltzmann solutions in bounded convex domains.

Findings

Established existence of stationary Boltzmann solutions in 3D convex domains.

Proved regularity properties of solutions using the generalized Fredholm theory.

Demonstrated the applicability of the theory to nonlinear kinetic equations.

Abstract

The celebrated Fredholm alternative theorem works for the setting of identity compact operators. This idea has been widely used to solve linear partial differential equations \cite{Evans}. In this article, we demonstrate a generalized Fredholm theory in the setting of identity power compact operators, which was suggested in Cercignani and Palczewski \cite{CP} to solve the existence of the stationary Boltzmann equation in a slab domain. We carry out the detailed analysis based on this generalized Fredholm theory to prove the existence theory of the stationary Boltzmann equation in bounded three-dimensional convex domains. To prove that the integral form of the linearized Boltzmann equation satisfies the identity power compact setting requires the regularizing effect of the solution operators. Once the existence and regularity theories for the linear case are established, with suitable bilinear estimates, the nonlinear existence theory is accomplished.