2 research outputs found
Dynamical stability criterion for inhomogeneous quasi-stationary states in long-range systems
We derive a necessary and sufficient condition of linear dynamical stability
for inhomogeneous Vlasov stationary states of the Hamiltonian Mean Field (HMF)
model. The condition is expressed by an explicit disequality that has to be
satisfied by the stationary state, and it generalizes the known disequality for
homogeneous stationary states. In addition, we derive analogous disequalities
that express necessary and sufficient conditions of formal stability for the
stationary states. Their usefulness, from the point of view of linear dynamical
stability, is that they are simpler, although they provide only sufficient
criteria of linear stability. We show that for homogeneous stationary states
the relations become equal, and therefore linear dynamical stability and formal
stability become equivalent.Comment: Submitted to Journal of Statistical Mechanics: Theory and Experimen
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