Nonlinear Phase Mixing in Einstein-Vlasov System near Schwarzschild Spacetime

RESUMO

We study the Einstein-Vlasov system near the Schwarzschild spacetime in spherical symmetry. In particular, we consider the case when the matter is supported on bounded geodesics. In this regime, the decay of the matter is driven by phase mixing rather than dispersion. For the linearized problem, we obtain quantitative phase mixing; the key new ingredient proved here is a monotonicity property of the period function. For the nonlinear coupled problem, we prove phase mixing up to the timescale expected from the nonlinear echo mechanism; this is achieved by analysis on dynamical action angle variables and the integral estimates with the vector field method. This construction is made possible by our gauge choice, an isotropic coordinate system together with a foliation of prescribed mean curvature, in which the Einstein equations gain derivatives over the matter due to its elliptic nature.