Long time solutions for wave maps with large data

TL;DR

This paper proves the existence of long-time solutions for 2+1 dimensional wave maps with large initial energy, without symmetry assumptions, ensuring solutions persist for any prescribed time interval.

Contribution

It establishes the existence of long-time solutions for large data wave maps in 2+1 dimensions without symmetry or harmonic map proximity assumptions.

Findings

Solutions exist for any positive time interval with large initial energy.

No symmetry or harmonic map closeness required for long-time existence.

Solutions can have arbitrarily large initial energy.

Abstract

For 2 + 1 dimensional wave maps with $\mathbb{S}^2$ as the target, we show that for all positive numbers $T_0 > 0$ and $E_0 > 0$, there exist Cauchy initial data with energy at least $E_0$, so that the solution's life-span is at least $[0,T_0]$. We assume neither symmetry nor closeness to harmonic maps.