Evaporation law in kinetic gravitational systems described by simplified Landau models

This paper is devoted to a mathematical and numerical study of a simplified kinetic model for evaporation phenomena in gravitational systems. This is a first step towards a mathematical understanding of more realistic kinetic models in this area. It is well known in the astrophysics literature that the appropriate kinetic model to describe escape (evaporation) from stars clusters is the so-called Vlasov-Landau-Poisson system with vanishing boundary condition at positive microscopic energies. Since collisions between stars and their self-consistent interactions are both taken into account in this model, its mathematical analysis is difficult, and so far not achieved. Here, as a first step, we focus on a simplified framework of this model and make the following assumptions: i) Only homogenous (space-independent) distributions functions are considered, leading to a collisional kinetic model with a vanishing boundary condition in velocity. ii) The interaction potential involved in the Landau collision operator is of Maxwellian type. iii) The escape velocity (or energy) is supposed to be constant. Using these assumptions, we first establish the well-posedness of the associated Cauchy problem. Then, we focus on the long time behavior of the solution and prove that the energy of the system decreases in time as $O(1/\log(t))$ (logarithmic evaporation), with convergence to a Dirac distribution in velocity when time goes to infinity. Finally, a suitable numerical scheme is constructed for this model and some simulations are performed to illustrate the theoretical study.