Global existence for null-form wave equations with data in a Sobolev space of lower regularity and weight

TL;DR

This paper proves global existence of solutions for null-form wave equations in three dimensions with initial data in lower regularity weighted Sobolev spaces, using advanced energy estimates and a new approach to generator applications.

Contribution

It introduces a novel method to limit the use of hyperbolic generators in a priori estimates, enabling global solutions for radially symmetric data with weaker weighted norms.

Findings

Global existence for small data in lower weighted Sobolev spaces.

Reduction in the number of generator applications in estimates.

Applicability to radially symmetric initial data.

Abstract

Assuming initial data have small weighted $H^4\times H^3$ norm, we prove global existence of solutions to the Cauchy problem for systems of quasi-linear wave equations in three space dimensions satisfying the null condition of Klainerman. Compared with the work of Christodoulou, our result assumes smallness of data with respect to $H^4\times H^3$ norm having a lower weight. Our proof uses the space-time $L^2$ estimate due to Alinhac for some special derivatives of solutions to variable-coefficient wave equations. It also uses the conformal energy estimate for inhomogeneous wave equation $\Box u=F$. A new observation made in this paper is that, in comparison with the proofs of Klainerman and H\"ormander, we can limit the number of occurrences of the generators of hyperbolic rotations or dilations in the course of a priori estimates of solutions. This limitation allows us to obtain global solutions for radially symmetric data, when a certain norm with considerably low weight is small enough.