Equivariant wave maps exterior to a ball

TL;DR

This paper studies the long-term behavior of equivariant wave maps from 3+1D Minkowski space into the three-sphere, showing solutions converge to a static harmonic map, providing insights into dissipation and soliton resolution.

Contribution

It combines analytical and numerical methods to demonstrate convergence of solutions to a unique static harmonic map in a new geometric setting.

Findings

Solutions converge to a static harmonic map

Detailed description of the relaxation process

Model offers insights into dissipation-by-dispersion phenomena

Abstract

We consider the exterior Cauchy-Dirichlet problem for equivariant wave maps from 3+1 dimensional Minkowski spacetime into the three-sphere. Using mixed analytical and numerical methods we show that, for a given topological degree of the map, all solutions starting from smooth finite energy initial data converge to the unique static solution (harmonic map). The asymptotics of this relaxation process is described in detail. We hope that our model will provide an attractive mathematical setting for gaining insight into dissipation-by-dispersion phenomena, in particular the soliton resolution conjecture.