Decay estimates of a tangential derivative to the light cone for the wave equation and their application

TL;DR

This paper develops new decay estimates for a tangential derivative near the light cone in 3D wave equations, providing a novel proof of global existence for nonlinear systems under the null condition without using scaling or pseudo-rotation operators.

Contribution

It introduces new weighted decay estimates for tangential derivatives and offers a simplified proof of global existence for nonlinear wave equations under the null condition.

Findings

Established new weighted $L^ ext{infty}$-$L^ ext{infty}$ estimates for tangential derivatives

Provided a proof of global existence without scaling or pseudo-rotation operators

Enhanced understanding of decay properties near the light cone in wave equations

Abstract

We consider wave equations in three space dimensions, and obtain new weighted $L^\infty$-$L^\infty$ estimates for a tangential derivative to the light cone. As an application, we give a new proof of the global existence theorem, which was originally proved by Klainerman and Christodoulou, for systems of nonlinear wave equations under the null condition. Our new proof has the advantage of using neither the scaling nor the pseudo-rotation operators.