Dispersive estimates for a system of tensorial quasilinear wave equations satisfying the weak-null condition

TL;DR

This paper proves global existence and decay for small data solutions to a broad class of tensorial quasilinear wave systems in three dimensions, including Einstein equations with non-null matter sources, using novel energy estimate techniques.

Contribution

It introduces a new decoupling method for higher order energy estimates applicable to systems satisfying the weak-null condition, extending previous results to new non-linearities.

Findings

Established global existence for small data solutions.

Proved decay properties for solutions of complex tensorial wave systems.

Extended techniques to include non-null non-linearities in Einstein-like systems.

Abstract

We establish both global existence and decay properties for solutions with small data for a general class of coupled system of tensorial quasilinear hyperbolic wave equations in three space dimensions, that covers the dynamical Einstein equations coupled to a class of non-linear matter sources that do not satisfy the null condition of Christodoulou and Klainerman, and have new different non-linearities than the one treated by Lindblad-Rodnianski, for which their celebrated seminal $L^\infty$-estimate does not work, to the best of our knowledge. Global existence of solutions for a general class of quasilinear wave equations satisfying the weak-null condition, with small initial data, is largely an open problem at present. There is no known theory to prove decay for the class of non-linear hyperbolic partial differential equations that we treat in this paper. We establish a technique based on novel decoupling of the higher order energy estimates, at the level of the $L^2$-norm of the Lie derivatives of the tangential components, without involving all the other components, up to some good factor. This generalizes our previous results to include new non-linearities that are not present in the Einstein-Yang-Mills system in the Lorenz gauge.