# Global existence for a system of quasi-linear wave equations in $3$D satisfying the weak null condition

## TL;DR

This paper proves the global existence of small solutions for a 3D quasi-linear wave system satisfying the weak null condition, extending Alinhac's ghost weight method to handle certain quadratic nonlinearities.

## Contribution

It extends Alinhac's ghost weight method to a system of quasi-linear wave equations with quadratic terms not satisfying the null condition, establishing global existence under the weak null condition.

## Key findings

- Global existence of small solutions in 3D for the system
- Extension of ghost weight method to quasi-linear systems
- Refinement of Alinhac's original theorem

## Abstract

We show global existence of small solutions to the Cauchy problem for a system of quasi-linear wave equations in three space dimensions. The feature of the system lies in that it satisfies the weak null condition, though we permit the presence of some quadratic nonlinear terms which do not satisfy the null condition. Due to the presence of such quadratic terms, the standard argument no longer works for the proof of global existence. To get over this difficulty, we extend the ghost weight method of Alinhac so that it works for the system under consideration. The original theorem of Alinhac for the scalar unknowns is also refined.