Uniqueness of the critical mass blow up solution for the four dimensional gravitational Vlasov-Poisson system
Résumé
We study the gravitational Vlasov-Poisson system at f + v center dot del(x) f - E - del v f = 0, E (x) = del phi(x) (x), Delta phi (x) = integral f (x, v) dv, in dimension N = 4 where the problem is L-1 critical. We proved in [M. Lemou, F. Mehats, P. Raphael, On the orbital stability of the ground states and the singularity formation for the gravitational Vlasov Poisson system, preprint] a sharp criterion for the global existence of weak solutions based on the variational characterization of the polytropic steady states solutions. From the existence of a pseudo-conformal symmetry, this criterion is sharp and there exist critical mass blow up solutions. We prove in this paper the uniqueness of the critical mass blow up solution. This gives in particular a first dynamical classification of the polytropic stationary solutions. The proof is an adaptation of a similar result by Frank Merle [F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schrodinger equations with critical power, Duke Math. J. 69 (2) (1993) 427-454] for the L-2 critical nonlinear Schrodinger equation.