Soliton resolution for energy-critical wave maps in the equivariant case

TL;DR

This paper proves that all finite energy solutions to the equivariant wave maps equation in two dimensions decompose over time into harmonic maps and free radiation, confirming the soliton resolution conjecture in this setting.

Contribution

It establishes the soliton resolution for all equivariance classes of the energy-critical wave maps in two dimensions, a significant advance in understanding their long-term behavior.

Findings

Finite energy solutions resolve into harmonic maps and radiation

The decomposition occurs continuously in time

The result applies to all equivariance classes

Abstract

We consider the equivariant wave maps equation $\mathbb{R}^{1+2} \to \mathbb{S}^2$, in all equivariance classes $k \in \mathbb{N}$. We prove that every finite energy solution resolves, continuously in time, into a superposition of asymptotically decoupling harmonic maps and free radiation.