Global stability for nonlinear wave equations satisfying a generalized null condition

TL;DR

This paper establishes the global stability of nonlinear wave equations under a generalized null condition, utilizing advanced energy estimates and geometric analysis of null hypersurfaces.

Contribution

It introduces a generalized null condition allowing bounded coefficient null forms and combines bilinear energy estimates with $r^p$ techniques for stability proof.

Findings

Proves global stability for a broad class of nonlinear wave systems.

Demonstrates decay rates and stability using geometric and energy methods.

Extends previous null condition results to more general coefficient settings.

Abstract

We prove global stability for a system of nonlinear wave equations satisfying a generalized null condition. The generalized null condition allows for null forms whose coefficients have bounded $C^k$ norms. We prove both pointwise decay and improved decay of good derivatives using bilinear energy estimates and duality arguments. Combining this strategy with the $r^p$ estimates of Dafermos--Rodnianski then allows us to prove global stability. The proof requires analyzing the geometry of intersecting null hypersurfaces adapted to solutions of wave equations.