FREDHOLM PROPERTY OF THE LINEARIZED BOLTZMANN OPERATOR FOR A POLYATOMIC SINGLE GAS MODEL

. In the following work, we consider the Boltzmann equation that models a polyatomic gas by representing the microscopic internal energy by a continuous variable I. Under some convenient assumptions on the transition function B, we prove that the linearized Boltzmann operator L of this model is a Fredholm operator. For this, we write L as a perturbation of the collision frequency multiplication operator, and we prove that the perturbation operator /C is compact. The result is established after inspecting the kernel form of /C and proving it to be L2 integrable over its domain using elementary arguments.