Weakly turbulent dynamics on Schwarzschild-AdS black hole spacetimes

TL;DR

This paper demonstrates weakly turbulent dynamics characterized by energy transfer to higher frequencies in small-data solutions of a cubic wave equation on Schwarzschild-AdS black hole backgrounds, with implications for stability analysis.

Contribution

It establishes norm inflation and energy cascade phenomena for small solutions on Schwarzschild-AdS spacetimes, broadening understanding of nonlinear stability and trapping effects.

Findings

Energy transfer from low to high frequencies occurs for small-data solutions.

Norm inflation manifests as unbounded growth of Sobolev norms.

Results apply to a class of backgrounds with stable trapping of null geodesics.

Abstract

In the presence of confinement, small-data solutions to nonlinear dispersive equations can exhibit a gradual energy transfer from low to high frequencies, a mechanism driving the emergence of weakly turbulent dynamics. We show that such a forward energy transfer, manifested as arbitrary inflation of higher order Sobolev norms, occurs for small-data solutions of a quasilinear cubic wave equation on the Schwarzschild-AdS black hole exterior with Dirichlet conditions at infinity, for generic values of the mass parameter. This result is motivated by the question of nonlinear stability or instability of Schwarzschild-AdS as a solution to the Einstein vacuum equations, but the strategy of proof applies to a broader class of backgrounds exhibiting stable trapping of null geodesics. As an application, we obtain the analogous norm inflation statement on $\mathbb R \times \mathbb S^3_+$ for generic perturbations of the round metric on the hemisphere $\mathbb S^3_+$ preserving the trapping structure at the boundary.