IUMJ

Title: Global existence and nonlinear stability for the relativistic Vlasov-Poisson system in the gravitational case

Authors: Mahir Hadzic and Gerhard Rein

Issue: Volume 56 (2007), Issue 5, 2453-2488

Abstract:

As is well known from the work of R. Glassey and J. Schaeffer [R.T. Glassey and J. Schaeffer, \textit{On symmetric solutions of the relativistic Vlasov-Poisson system}, Comm. Math. Phys. \textbf{101} (1985), 459--473], the main energy estimates which are used in global existence results for the gravitational Vlasov-Poisson system do not apply to the relativistic version of this system, and smooth solutions to the initial value problem with spherically symmetric initial data of negative energy blow up in finite time. For similar reasons the variational techniques by which Y. Guo and G. Rein obtained nonlinear stability results for the Vlasov-Poisson system [see Y. Guo, \textit{Variational method in polytropic galaxies}, Arch. Ration. Mech. Anal. \textbf{150} (1999), 209--224; Y. Guo, \textit{On the generalized Antonov's stability criterion}, Contemp. Math. \textbf{263} (2000), 85--107; Y. Guo and G. Rein, \textit{Stable steady states in stellar dynamics}, Arch. Ration. Mech. Anal. \textbf{147} (1999), 225--243; Y. Guo and G. Rein, \textit{Isotropic steady states in galactic dynamics}, Comm. Math. Phys. \textbf{219} (2001), 607--629; G. Rein, \textit{Reduction and a concentration-compactness principle for energy-Casimir functionals}, SIAM J. Math. Anal. \textbf{33} (2002), 896--912; G. Rein, \textit{Nonlinear stability of Newtonian galaxies and stars from a mathematical perspective}, in: "Nonlinear Dynamics in Astronomy and Physics," Annals of the New York Academy of Sciences. The New York Academy of Sciences \textbf{1045} (2005), 103--119] do not apply in the relativistic situation. In the present paper a direct, non-variational approach is used to prove nonlinear stability of certain steady states of the relativistic Vlasov-Poisson system against spherically symmetric, dynamically accessible perturbations. The resulting stability estimates imply that smooth solutions with spherically symmetric initial data which are sufficiently close to the stable steady states exist globally in time.