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

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.