Compactness property of the linearized Boltzmann operator for a polyatomic gas undergoing resonant collisions

In this paper, we investigate a compactness property of the linearized Boltzmann operator in the context of a polyatomic gas whose molecules undergo resonant collisions. The peculiar structure of resonant collision rules allows to tensorize the problem into a velocity-related one, neighbouring the monatomic case, and an internal energy-related one. Our analysis is based on a specific treatment of the contributions due to the internal energy of the molecules. We also propose a geometric variant of Grad's proof of the same compactness property in the monatomic case.