A Large Data Regime for non-linear Wave Equations

TL;DR

This paper demonstrates that for certain semi-linear wave equations with null form nonlinearities in 3+1 dimensions, large initial data can still lead to global solutions, and it explores highly focused wave solutions along null geodesics.

Contribution

It introduces an open set of large-energy initial data that still produce global solutions and constructs localized solutions concentrated along specific null geodesics.

Findings

Existence of global solutions for large initial data.

Construction of highly focused wave solutions along null geodesics.

Energy confinement in tubular neighborhoods of geodesics.

Abstract

For semi-linear wave equations with null form non-linearities on $\mathbb{R}^{3+1}$, we exhibit an open set of initial data which are allowed to be large in energy spaces, yet we can still obtain global solutions in the future. We also exhibit a set of localized data for which the corresponding solutions are strongly focused, which in geometric terms means that a wave travels along an specific incoming null geodesic in such a way that almost all of the energy is confined in a tubular neighborhood of the geodesic and almost no energy radiating out of this tubular neighborhood.