Stable Ground States and Self-Similar Blow-Up Solutions for the Gravitational Vlasov--Manev System

Abstract

In this work, we study the orbital stability of steady states and the existence of self-similar blow-up solutions to the so-called Vlasov--Manev system. This system is a kinetic model which has a similar Vlasov structure to the classical Vlasov--Poisson system but is coupled to a potential in $-1/r- 1/r^2$ (Manev potential) instead of the usual gravitational potential in $-1/r$, and in particular the potential field does not satisfy a Poisson equation but a fractional Laplacian equation. We first prove the orbital stability of the ground state--type solutions which are constructed as minimizers of the Hamiltonian, following the classical strategy: compactness of the minimizing sequences and the rigidity of the flow. However, in driving this analysis, there are two mathematical obstacles: the first one is related to the possible blow-up of solutions to the Vlasov--Manev system, which we overcome by imposing a subcritical condition on the constraints of the variational problem. The second difficulty (and the most important) is related to the nature of the Euler--Lagrange equations (fractional Laplacian equations) to which classical results for the Poisson equation do not extend. We overcome this difficulty by proving the uniqueness of the minimizer under equimeasurability constraints, using only the regularity of the potential and not the fractional Laplacian Euler--Lagrange equations itself. In the second part of this work, we prove the existence of exact self-similar blow-up solutions to the Vlasov--Manev equation, with initial data arbitrarily close to ground states. This construction is based on a suitable variational problem with equimeasurability constraint.