Compactness and existence theory for a general class of stationary radiative transfer equations

In this paper, we study the steady-states of a large class of stationary radiative transfer equations in a $C^1$ convex bounded domain. Namely, we consider the case in which both absorption-emission and scattering coefficients depend on the local temperature $T$ and the radiation frequency $\nu.$ The radiative transfer equation determines the temperature of the material at each point. The main difficulty in proving existence of solutions is to obtain compactness of the sequence of integrals along lines that appear in several exponential terms. We prove a new compactness result suitable to deal with such a non-local operator containing integrals on a line segment. On the other hand, to obtain the existence theory of the full equation with both absorption and scattering terms we combine the compactness result with the construction of suitable Green functions for a class of non-local equations.