Global existence for quasilinear wave equations close to Schwarzschild

TL;DR

The paper proves global existence for small, smooth initial data of a quasilinear wave equation on metrics that are small perturbations of Schwarzschild. It develops a robust linear local energy estimate for perturbations of Schwarzschild, handling the photon sphere with slow decay, and couples this with a vector-field commutation framework to obtain high-order energy and pointwise decay. A careful bootstrap shows that the nonlinear terms remain controlled, yielding a global classical solution and precise decay/growth bounds. These results extend linear and semilinear stability techniques to a quasilinear setting near Schwarzschild, providing key local energy tools relevant to black hole stability analyses.

Abstract

In this article we study the quasilinear wave equation $\Box_{g(u, t, x)} u = 0$ where the metric $g(u, t, x)$ is close to the Schwarzschild metric. Under suitable assumptions of the metric coefficients, and assuming that the initial data for $u$ is small enough, we prove global existence of the solution. The main technical result of the paper is a local energy estimate for the linear wave equation on metrics with slow decay to the Schwarzschild metric.

Theorems & Definitions (31)

• Theorem 1
• Theorem 1
• Lemma 2
• Theorem 1
• Theorem 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4