# The Weak Null Condition and Kaluza-Klein Spacetimes

## TL;DR

This paper proves the non-linear stability of certain wave equations satisfying the weak null condition, including Minkowski spacetime times a torus, extending stability results to these Kaluza-Klein spacetimes.

## Contribution

It establishes the non-linear stability of Kaluza-Klein spacetimes with perturbations depending only on non-compact coordinates, using methods similar to Lindblad-Rodnianski 2010.

## Key findings

- Proved non-linear stability of Minkowski times a torus.
- Extended stability results to Kaluza-Klein spacetimes.
- Validated the weak null condition for these systems.

## Abstract

In this paper we prove the non-linear stability of a system of non-linear wave equations satisfying the weak null condition. In particular, this includes the case of the non-linear stability of Minkowski spacetime times a $d$-torus subject to perturbations depending only on the non-compact coordinates. Our argument very closely follows the proof of the non-linear stability of Minkowski spacetime in [Lindblad-Rodnianski-2010].