Orbital stability and singularity formation for Vlasov–Poisson systems

Abstract

We study the gravitational Vlasov–Poisson system in dimension N = 3 and N = 4 and consider the problem of nonlinear stability of steady states solutions within the framework of concentration compactness techniques. In dimension N = 3 where the problem is subcritical, we prove the orbital stability in the energy space of the polytropes which are ground state type stationary solutions, which improves the already published results for this class. In dimension N = 4 where the problem is L 1 critical, polytropic steady states are obtained following Weinstein [M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys. 87 (1983) 567–576] by minimizing a suitable Gagliardo Nirenberg type inequality. Now a striking feature is the existence of a pseudo-conformal symmetry which allows us to derive explicit critical mass finite time blow up solutions. This is to our knowledge the first result of description of a singularity formation in a Vlasov setting. A general mass concentration phenomenon is eventually obtained for finite time blow up solutions. To cite this article: M. Lemou et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).