Localized minimizers of flat rotating gravitational systems

We study a two-dimensional system in solid rotation at constant angular velocity driven by a self-consistent three-dimensional gravitational field. We prove the existence of stationary solutions of such a flat system in the rotating frame as long as the angular velocity does not exceed some critical value which depends on the mass. The solutions can be seen as stationary solutions of a kinetic equation with a relaxation-time collision kernel forcing the convergence to the polytropic gas solutions, or as stationary solutions of an extremely simplified drift-diffusion model, which is derived from the kinetic equation by formally taking a diffusion limit. In both cases, the solutions are critical points of a free energy functional, and can be seen as localized minimizers in an appropriate sense.