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Orbital stability of spherical galactic models
 Invent. math
, 2012
"... Dedicated to the memory of our friend Naoufel Ben Abdallah Abstract. We consider the three dimensional gravitational Vlasov Poisson system which is a canonical model in astrophysics to describe the dynamics of galactic clusters. A well known conjecture [6] is the stability of spherical models whic ..."
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Dedicated to the memory of our friend Naoufel Ben Abdallah Abstract. We consider the three dimensional gravitational Vlasov Poisson system which is a canonical model in astrophysics to describe the dynamics of galactic clusters. A well known conjecture [6] is the stability of spherical models which are nonincreasing radially symmetric steady states solutions. This conjecture was proved at the linear level by several authors in the continuation of the breakthrough work by Antonov [2] in 1961. In the previous work [29], we derived the stability of anisotropic models under spherically symmetric perturbations using fundamental monotonicity properties of the Hamiltonian under suitable generalized symmetric rearrangements first observed in the physics litterature [34, 12, 45, 1]. In this work, we show how this approach combined with a new generalized Antonov type coercivity property implies the orbital stability of spherical models under general perturbations. 1. Introduction and
STABLE GROUND STATES AND SELFSIMILAR BLOWUP SOLUTIONS FOR THE GRAVITATIONAL VLASOVMANEV SYSTEM
, 2012
"... Abstract. In this work, we study the orbital stability of steady states and the existence of blowup selfsimilar solutions to the socalled VlasovManev (VM) system. This system is a kinetic model which has a similar Vlasov structure as the classical VlasovPoisson system, but is coupled to a poten ..."
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Abstract. In this work, we study the orbital stability of steady states and the existence of blowup selfsimilar solutions to the socalled VlasovManev (VM) system. This system is a kinetic model which has a similar Vlasov structure as the classical VlasovPoisson 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 fractionalLaplacian 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 blowup of solutions to the VM 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 EulerLagrange equations (fractionalLaplacian 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 fractionalLaplacian EulerLagrange equations itself. In the second part of this work, we prove the existence of exact selfsimilar blowup solutions to the VlasovManev equation, with initial data arbitrarily close to ground states. This construction is based on a suitable variational problem with equimeasurability constraint. 1. Introduction and