Remarks on a system of quasi-linear wave equations in $3$D satisfying the weak null condition

TL;DR

This paper provides an alternative proof for the global existence of solutions to a 3D quasi-linear wave system satisfying the weak null condition, avoiding Lorentz boosts and allowing less restrictive data assumptions.

Contribution

It introduces a new proof method that removes the need for Lorentz boosts and relaxes data support and regularity assumptions for the weak null condition system.

Findings

Eliminates the use of Lorentz boost operators in the proof.

Removes the compact support assumption of initial data.

Refines recent results for scalar wave equations.

Abstract

We give an alternative proof of the global existence result originally due to Hidano and Yokoyama for the Cauchy problem for a system of quasi-linear wave equations in three space dimensions satisfying the weak null condition. The feature of the new proof lies in that it never uses the Lorentz boost operator in the energy integral argument. The proof presented here has an advantage over the former one in that the assumption of compactness of the support of data can be eliminated and the amount of regularity of data can be lowered in a straightforward manner. A recent result of Zha for the scalar unknowns is also refined.