Characterization of large energy solutions of the equivariant wave map problem: II

TL;DR

This paper classifies all degree 1 global solutions of the equivariant wave map problem with energy less than three times the harmonic map energy, showing they asymptotically resemble a harmonic map plus radiation.

Contribution

It provides a complete classification of degree 1 wave maps with sub-threshold energy, detailing their asymptotic behavior and decoupling into harmonic maps and radiation.

Findings

All degree 1 solutions with energy < 3E(Q) asymptotically resemble a rescaled harmonic map plus radiation.

The classification complements previous work on finite-time blow-up, covering the entire energy regime [E(Q), 3E(Q)).

Solutions with energy below 3E(Q) do not develop singularities and exhibit predictable asymptotic behavior.

Abstract

We consider 1-equivariant wave maps from 1+2 dimensions to the 2-sphere of finite energy. We establish a classification of all degree 1 global solutions whose energies are less than three times the energy of the harmonic map Q. In particular, for each global energy solution of topological degree 1, we show that the solution asymptotically decouples into a rescaled harmonic map plus a radiation term. Together with our companion article, where we consider the case of finite-time blow up, this gives a characterization of all 1-equivariant, degree 1 wave maps in the energy regime [E(Q), 3E(Q)).