Global existence for quasilinear wave equations satisfying the null condition

TL;DR

This paper proves the global existence of small-data solutions to quasilinear wave equations satisfying the null condition, using $r^p$-weighted energy estimates, and extends the approach to exterior domains without time-dependent vector fields.

Contribution

It introduces an adaptation of Dafermos and Rodnianski's $r^p$-weighted estimates to establish global existence for null condition wave equations, simplifying previous methods.

Findings

Global solutions exist for small initial data.

Method extends to wave equations outside star-shaped obstacles.

Avoids use of time-dependent vector fields.

Abstract

We explore the global existence of solutions to systems of quasilinear wave equations satisfying the null condition when the initial data are sufficiently small. We adapt an approach of Keel, Smith, and Sogge, which relies on integrated local energy estimates and a weighted Sobolev estimate that yields decay in $|x|$, by using the $r^p$-weighted local energy estimates of Dafermos and Rodnianski. One advantage of this approach is that all time-dependent vector fields can be avoided and the proof can be readily adapted to address wave equations exterior to star-shaped obstacles.