Global existence for systems of nonlinear wave equations with bounded, stable asymptotic systems

TL;DR

This paper establishes global existence results for certain nonlinear wave systems by demonstrating that bounded, stable asymptotic systems ensure solutions exist for small initial data, extending beyond classical null condition requirements.

Contribution

It introduces a new criterion based on boundedness and stability of the asymptotic system, avoiding structural assumptions on nonlinear terms, and provides examples related to Lie group Euler equations.

Findings

Bounded, stable asymptotic systems guarantee global solutions.

Examples include systems generalizing Euler equations on Lie groups.

Avoids previous structural conditions on nonlinear terms.

Abstract

Some systems of nonlinear wave equations admit global solutions for all sufficiently small initial data, while others do not. The (classical) null condition guarantees that such a result holds, but it is too strong to capture certain systems -- most famously the Einstein equations -- which nevertheless admit global solutions for small initial data. The weak null condition has been proposed as a sufficient condition for such a result to hold; it takes the form of a condition on a related set of nonlinear ODEs known as the "asymptotic system". Previous results in this direction have required certain structural conditions on the asymptotic system in addition to the weak null condition. In this work we show that, if the solutions to the asymptotic system are bounded (given small initial data), and, in addition, if these solutions are stable against rapidly decaying perturbations, then the corresponding system of nonlinear wave equations admits global solutions for all sufficiently small initial data. This avoids any direct assumptions on the structure of the nonlinear terms. We also give an example of a class of systems obeying this condition but not obeying previously identified structural conditions. For this class the asymptotic system arises as a generalisation of the "Euler equations" for rigid body motion, associated with a left-invariant Hamiltonian flow on a finite dimensional Lie group. This work relies heavily on the author's previous work, "The weak null condition and global existence using the p-weighted energy method".