Stable ground states and self-similar blow-up solutions for the gravitational Vlasov-Manev system

TL;DR

This paper investigates the stability of ground states and constructs self-similar blow-up solutions for the Vlasov-Manev system, a kinetic model with a fractional potential, advancing understanding of its dynamics and singularity formation.

Contribution

It establishes orbital stability of ground states and proves the existence of self-similar blow-up solutions for the Vlasov-Manev system, addressing mathematical challenges posed by fractional-Laplacian potentials.

Findings

Ground states are orbitally stable under certain conditions.

Existence of self-similar blow-up solutions near ground states.

Overcoming mathematical obstacles related to fractional-Laplacian equations.

Abstract

In this work, we study the orbital stability of steady states and the existence of blow-up self-similar solutions to the so-called Vlasov-Manev (VM) system. This system is a kinetic model which has a similar Vlasov structure as 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 states 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 VM system, which we overcome by imposing a sub-critical 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 equimeasurabilty 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.