[GS_M_APP] Stability of the Schwarzschild Spacetime in the Einstein-Vlasov System via Nonlinear Phase Mixing
ABSTRACT
We will discuss the proof of stability for the Schwarzschild black hole as a vacuum solution to the spherically symmetric Einstein-Vlasov system. We will focus on the regime where the characteristic curves of the Vlasov equation remain bounded, enabling the phase mixing mechanism to operate effectively within dynamical action-angle coordinates. This framework allows us to establish the polynomial decay of the time derivative of energy-momentum tensor and metric coefficients in time, up to a lifespan of $T_B = \epsilon^{-1}(\log(1/\epsilon))^2$. In this talk, we will review the foundational ideas of phase mixing for the gravitational Vlasov-Poisson system [1], and then discussing the new challenges introduced by the curved spacetime setting, including gauge choice and the monotonicity of the period function.
[1] Chaturvedi, S., and Luk, J. Linear and nonlinear phase mixing for the gravitational Vlasov-Poisson system under an external Kepler potential. to appear in Arch. Rat. Mech. Anal., 2026.
