Soliton resolution for equivariant wave maps to the sphere

TL;DR

This paper proves that finite energy equivariant wave maps to the sphere decompose into harmonic maps and either a smooth wave map or a scattering term over time, demonstrating soliton resolution in this setting.

Contribution

It establishes the soliton resolution conjecture for equivariant wave maps to the sphere, showing detailed decomposition of solutions over time.

Findings

Decomposition into harmonic maps and scattering components

Convergence of the error to zero in energy space

Validation of soliton resolution for this class of wave maps

Abstract

We consider finite energy corotationnal wave maps with target manifold $\m S^2$. We prove that for a sequence of times, they decompose as a sum of decoupled harmonic maps in the light cone, and a smooth wave map (in the blow case) or a linear scattering term (in the global case), up to an error which tends to 0 in the energy space.