Existence and stability of steady states solutions of flat Vlasov-Poisson system with a central mass density

Abstract

We study a Newtonian model, that allows us to describe some phenomena in stellar dynamics. This model is described by a partial differential equation known as the Vlasov equation or the Liouville equation, whose solutions describe the temporal evolution of a collisionless particle system in the phase space, subject to a self-interacting gravitational potential. This work is divided in two parts. In the first part, we treat the Flat Vlasov-Poisson system with an external gravitational potential induced by a fixed mass density. This model describes some extremely flat objects in galactic dynamics. The aim of this part is the study of the existence and stability of steady states solutions of the Flat Vlasov-Poisson system in this case. We solve a variational problem to find minimizers for the Casimir-Energy functional in a suitable set of functions. The minimization problem is solved through a reduction of the original optimization problem (see [22]), but instead of a concentration-compactness argument, we use a symmetrization argument, taking the rearrangement of a minimizing sequence and we prove that it converges weakly to a minimizer in a suitable Lp space with p > 1. We prove that this minimizer induces a solution for the original minimization problem. The minimization problem give us a non-linear stability result for the steady state solution in the Lpspace. In the second part, we show the results published in [13]. We prove phase-space mixing for solutions of the Vlasov equation for integrable systems. Under a natural non-harmonicity condition, we obtain weak convergence of the distribution function with rate t^(−1). In one dimension, we also study the case where this condition fails at a certain energy, showing that mixing still holds but with a slower rate. When the condition holds and functions have higher regularity, the rate can be faster.