The orbital stability of the ground states and the singularity formation for the gravitational vlasov poisson system

摘要

We study the gravitational Vlasov Poisson system f(t) + v . del(x)f - E . del(v)f = 0 where E( x) = del(x)phi f( x), Delta(x)phi =.rho(x), rho(x) = integral R-N f (x, v) dx dv, in dimension N = 3, 4. In dimension N = 3 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 in the energy space of the spherically symmetric polytropes and improves the nonlinear stability results obtained for this class in [ 11, 16, 19]. In dimension N = 4 where the problem is L-1 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 amass 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 of the one for the focusing nonlinear Schrodinger equation iu(t) = -Delta u-|u|(p-1)u in the energy space H-1(R-N).