Solutions of quasi-linear wave equations polyhomogeneous at null infinity in high dimensions

TL;DR

This paper establishes the propagation of regularity and polyhomogeneity for solutions to certain quasi-linear hyperbolic systems, including Einstein-Maxwell equations, in high-dimensional space-times, particularly at null infinity.

Contribution

It proves propagation of weighted Sobolev regularity and polyhomogeneity for solutions of quasi-linear hyperbolic systems in high dimensions, extending results to Einstein-Maxwell equations.

Findings

Propagation of Sobolev regularity for solutions in dimensions ≥7

Polyhomogeneity at null infinity in dimensions ≥9

Small data solutions of Einstein equations are polyhomogeneous at null infinity

Abstract

We prove propagation of weighted Sobolev regularity for solutions of the hyperboloidal Cauchy problem for a class of quasi-linear symmetric hyperbolic systems, under structure conditions compatible with the Einstein-Maxwell equations in space-time dimensions $n+1\ge 7$. Similarly we prove propagation of polyhomogeneity in dimensions $n+1\ge 9$. As a byproduct we obtain, in those last dimensions, polyhomogeneity at null infinity of small data solutions of vacuum Einstein, or Einstein-Maxwell equations evolving out of initial data which are stationary outside of a ball.