Global existence conditions for a nonlocal problem arising in statistical mechanics

Abstract

We consider the evolution of the density and temperature of a three-dimensional cloud of self-interacting particles. This phenomenon is modeled by a parabolic equation for the density distribution combining temperature-dependent diffusion and convection driven by the gradient of the gravitational potential. This equation is coupled with Poisson's equation for the potential generated by the density distribution. The system preserves mass by imposing a zero-flux boundary condition. Finally the temperature is fixed by energy conservation; that is, the sum of kinetic energy (temperature) and gravitational energy remains constant in time. This model is thermodynamically consistent, obeying the first and the second laws of thermodynamics. We prove local existence and uniqueness of weak solutions for the system, using a Schauder fixed-point theorem. In addition, we give sufficient conditions for global-in-time existence and blow-up for radially symmetric solutions. We do this using a comparison principle for an equation for the accumulated radial mass.