On the Cauchy problem for a quantum kinetic equation linked to the Compton effect

Abstract

The Cauchy problem is studied for an homogeneous quantum kinetic equation describing the Compton effect. Since the collision kernel commonly used in physics is highly singular, numerical simulations are performed for related collision kernels to get a preliminary insight into the behavior of the solutions. Some of the numerical results are then given a theoretical explanation. Global existence of a solution to the Cauchy problem is proven when the L 1 initial data are a.e. smaller than the Planck distribution function, and non-existence of solutions to the Cauchy problem is proven when the L 1 initial data are a.e. bigger than the Planck distribution function.