Localized minimizers of flat rotating gravitational systems

Abstract

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.

Boundary value problems for two dimensional steady incompressible fluids

“…A kinetic model, namely the Vlasov-Poisson equation, has been studied as well, see e.g. [13,23]. In fact, there is a relation between steady states of the Vlasov-Poisson equation and the compressible Euler equation, see [30] and references therein for an overview of the variational methods used in these problems.…”

Relative Equilibria in Continuous Stellar Dynamics

“…is also a serious cause of trouble, which clearly discards the possibility that the free energy functional can be bounded from below if ω = 0. This issue has been studied in [9], in the case of the so-called flat systems.…”

“…At this point, one should mention that the analysis is not exactly as simple as written above. Identity (9) indeed holds only component by component of the support of the solution, if this support has more than one connected component, and the Lagrange multipliers have to be defined for each component. The fact that…”

“…Also notice that, using the same function γ as in (9), to each distribution function f , we can associate a local equilibrium, or local Gibbs state,…”

“…Arnold (see [1,2,45]). The variational characterization of special stationary solutions and their orbital stability have been studied by Y. Guo and G. Rein in a series of papers [16,17,18,19,36,37,38,39] and by many other authors, see for instance [9,10,22,23,24,25,40,41,44].…”