5 research outputs found
A new variational approach to the stability of gravitational systems
We consider the three dimensional gravitational Vlasov Poisson system which
describes the mechanical state of a stellar system subject to its own gravity.
A well-known conjecture in astrophysics is that the steady state solutions
which are nonincreasing functions of their microscopic energy are nonlinearly
stable by the flow. This was proved at the linear level by several authors
based on the pioneering work by Antonov in 1961. Since then, standard
variational techniques based on concentration compactness methods as introduced
by P.-L. Lions in 1983 have led to the nonlinear stability of subclasses of
stationary solutions of ground state type.
In this paper, inspired by pioneering works from the physics litterature
(Lynden-Bell 94, Wiechen-Ziegler-Schindler MNRAS 88, Aly MNRAS 89), we use the
monotonicity of the Hamiltonian under generalized symmetric rearrangement
transformations to prove that non increasing steady solutions are local
minimizer of the Hamiltonian under equimeasurable constraints, and extract
compactness from suitable minimizing sequences. This implies the nonlinear
stability of nonincreasing anisotropic steady states under radially symmetric
perturbations
The Orbital Stability of the Ground States and the Singularity Formation for the Gravitational Vlasov Poisson System
International audienceWe study the gravitational Vlasov Poisson system where , , \rho(x)=\int_{\RR^N} f(x,v)dxdv, in dimension . In dimension where the problem is subcritical, we prove using concentration compactness techniques that every minimizing sequence to a large class of minimization problems attained on steady states solutions are up to a translation shift relatively compact in the energy space. This implies in particular the orbital stability {\it in the energy space} of the spherically symmetric polytropes what improves the nonlinear stability results obtained for this class in \cite{Guo,GuoRein,Dol}. In dimension where the problem is critical, we obtain the polytropic steady states as best constant minimizers of a suitable Sobolev type inequality relating the kinetic and the potential energy. We then derive using an explicit pseudo-conformal symmetry the existence of critical mass finite time blow up solutions, and prove more generally a mass concentration phenomenon for finite time blow up solutions. This is the first result of description of a singularity formation in a Vlasov setting. The global structure of the problem is reminiscent to the one for the focusing non linear Schrödinger equation in the energy space
Orbital stability of spherical galactic models
International audienceWe 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 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 in 1961. In a previous work, we derived the stability of anisotropic models under {\it spherically symmetric perturbations} using fundamental monotonicity properties of the Hamiltonian under suitable generalized symmetric rearrangements first observed in the physics litterature. In this work, we show how this approach combined with a {\it new generalized} Antonov type coercivity property implies the orbital stability of spherical models under general perturbations
Evolution of single-particle structure of silicon isotopes
New data on proton and neutron single-particle energies of Si isotopes with neutron number N from 12 to 28 as well as occupation probabilities of single-particle states of stable isotopes 28, 30Si near the Fermi energy were obtained by the joint evaluation of the stripping and pick-up reaction data and excited state decay schemes of neighboring nuclei. The evaluated data indicate the following features of single-particle structure evolution: persistence of Z = 14 subshell closure with N increase, the new magicity of the number N = 16, and the conservation of the magic properties of the number N = 20 in Si isotopic chain. The features were described by the dispersive optical model. The calculation also predicts the weakening of N = 28 shell closure and demonstrates evolution of a bubble-like structure of the proton density distributions in neutron-rich Si isotopes