Stable soliton resolution for exterior wave maps in all equivariance classes

TL;DR

This paper proves that finite energy -equivariant wave maps from 1+3D Minkowski space outside a ball into the 3-sphere asymptotically approach a harmonic map, extending previous results to all equivariance classes.

Contribution

It establishes soliton resolution for all -equivariant exterior wave maps, confirming a conjecture and generalizing prior 1-equivariant results.

Findings

Wave maps scatter to harmonic maps in all equivariance classes.

Extension of soliton resolution results to higher equivariance classes.

Numerical conjecture of Bizon, Chmaj, and Maliborski fully resolved.

Abstract

In this paper we consider finite energy, \ell-equivariant wave maps from 1+3-dimensional Minkowski space exterior to the unit ball at the origin into the 3-sphere. We impose a Dirichlet boundary condition at r=1, which in this context means that the boundary of the unit ball in the domain gets mapped to the north pole. Each such \ell-equivariant wave map has a fixed integer-valued topological degree, and in each degree class there is a unique harmonic map, which minimizes the energy for maps of the same degree. We prove that an arbitrary \ell-equivariant exterior wave map with finite energy scatters to the unique harmonic map in its degree class, i.e., soliton resolution. This extends the recent results of the first, second, and fourth authors on the 1-equivariant equation to higher equivariance classes, and thus completely resolves a conjecture of Bizon, Chmaj and Maliborski, who observed this asymptotic behavior numerically. The proof relies crucially on exterior energy estimates for the free radial wave equation in dimension d = 2 \ell +3, which are established in a companion paper.