2025 Theses Doctoral

Nonlinear Stability of sub-extremal Reissner-Norström Blackhole Spacetimes --- Hyperbolic Estimates for Nonlinear Positive-Spin Regge-Wheeler system

Wan, Jingbo

We initiate the study of the nonlinear stability of sub-extremal Reissner-Norström spacetimes within the Klainerman-Szfetel framework, extending methods developed for the Schwarzschild case {27} and the linear stability of Reissner-Norstrom {12, 13}. We aim to establish nonlinear stability without imposing additional symmetry assumptions, instead relying on a finite co-dimension constraint on perturbations, as introduced in {6} for the Schwarzschild case.

In this thesis, we focus on deriving the essential hyperbolic estimates for the nonlinear problem. A significant challenge arises from the interaction between gravitational and electromagnetic perturbations, which are governed by the nonlinear positive-spin Regge-Wheeler system.

To address this, we establish the energy, Morawetz, redshift, and 𝑟^𝑝-type estimates, in the spirit of Dafermos and Rodnianski’s linear analysis, while carefully managing the additional complexities introduced by the coupling and the intricate nonlinearities. As a result, we obtain decay estimates of the main variables under reasonable assumptions, providing the foundational hyperbolic estimates necessary to address the full nonlinear stability problem in sub-extremal Reissner-Norstrom spacetimes in the future.