Transport Equations with Second-Order Differential Collision Operators

Abstract

This paper discusses existence, uniqueness, and a priori estimates for time-dependent and time-independent transport equations with unbounded collision operators. These collision operators are described by second-order differential operators resulting from diffusion in the velocity space. The transport equations are degenerate parabolic-elliptic partial differential equations, that are treated by modifications of the Fichera–Oleinik–Radkevic Theory of second-order equations with nonnegative characteristic form. We consider weak solutions in spaces that are extensions of $L^P $ to include traces on certain parts of the boundary. This extension is necessary due to the nonclassical boundary conditions imposed by the transport problem, which requires a specific analysis of the behavior of our weak solutions.