Linear and non-linear collisionless many-particle systems

Abstract

The present thesis focuses on linear and non-linear stability results for collisionless many-particle systems described by distribution functions satisfying classical kinetic models arising in astrophysics. On the one hand, we study the large time behavior of solutions to the Vlasov--Poisson system, which describes a self-gravitating system interacting through the laws of gravity set by Newton. On the other hand, we study the large time behavior of solutions to the Einstein--Vlasov system, which describes a self-gravitating system interacting through gravity, according to the theory of general relativity set by Einstein. Moreover, we obtain dispersion estimates for solutions to non-relativistic and relativistic Vlasov equations on geometric backgrounds. Chapter 1 contains a summary of the results presented in this thesis. The first half of this thesis concerns non-relativistic collisionless many-particle systems. In Chapter 2, we obtain propagation in time of Gevrey regularity for solutions to the Vlasov--Poisson system, in the torus and the whole space. The main result is based on quantitative a priori estimates for the Gevrey norm and the radius of Gevrey regularity of the solution, in terms of the force field and the volume of the momentum support of the distribution function. Consequently, we obtain global existence of Gevrey solutions for the three dimensional Vlasov--Poisson system, in the torus and the whole space. In Chapter 3, we obtain linear and non-linear stability results for dispersive collisionless many-particle systems, in which the Hamiltonian dynamics described by its particles are hyperbolic. Firstly, we study small data solutions for the Vlasov--Poisson system under the simplest external potential, for which unstable trapping holds for the associated Hamiltonian flow. More precisely, we prove optimal decay estimates in space and time for small data solutions to the Vlasov--Poisson system with the unstable trapping potential $-\frac{|x|^2}{2}$, in dimension three or higher, through the use of a commuting vector field approach. Secondly, we study dispersion estimates for solutions to the Vlasov equation on the hyperbolic space. Specifically, we obtain optimal decay estimates in time for the induced spatial density, by making use of commuting vector fields lying in the stable and unstable invariant distributions of phase space. The two sections in this chapter are joint works with Anibal Velozo Ruiz. The second half of this thesis concerns relativistic collisionless many-particle systems in the exterior of black hole spacetimes. In Chapter 4, we prove a non-linear asymptotic stability result of the Schwarzschild family of black holes as solutions to the spherically symmetric Einstein--massless Vlasov system in the exterior of the black hole region. The proof exploits the normal hyperbolicity of the null geodesic flow in a neighborhood of the trapped set to show quantitative decay estimates for the stress energy momentum tensor of matter. In particular, we prove exponential decay for the stress energy momentum tensor in a bounded region of spacetime. The main result requires precise estimates for the derivatives of the Vlasov field, which we obtain by studying the Jacobi fields on the null-shell of spacetime, in terms of the Sasaki metric. In Chapter 5, we obtain decay estimates for massive collisionless many-particle systems on the exterior of a Schwarzschild black hole. More precisely, we prove slow decay estimates for the stress energy momentum tensor associated with the massive Vlasov equation on Schwarzschild spacetime, for initial data compactly supported away from the region of phase space where bounded timelike geodesics do not cross the event horizon. As a result, we prove that the stress energy momentum tensor decays like $v^{-\frac{1}{3}}$ in a bounded region and like $u^{-\frac{1}{3}}r^{-2}$ in an unbounded region, where $u$ and $v$ denote the standard Eddington--Finkelstein retarded and advanced time coordinates, respectively. The dispersion estimates for the stress energy momentum tensor follow by proving decay in time for the volume of the momentum support of the distribution function.