Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class

TL;DR

This paper constructs an energy class for wave maps from 2D Minkowski space to hyperbolic spaces and, assuming large data well-posedness, proves the non-existence of stationary, traveling, or self-similar solutions within this class, advancing the understanding of wave map regularity.

Contribution

It introduces a new energy class for wave maps and, under a conditional large data well-posedness assumption, rules out certain special solutions, supporting global regularity.

Findings

No stationary wave maps in the energy class

No self-similar wave maps in the energy class

Supports global regularity conjecture for wave maps

Abstract

Using the harmonic map heat flow, we construct an energy class for wave maps $\phi$ from two-dimensional Minkowski space $\R^{1+2}$ to hyperbolic spaces $\H^m$, and then show (conditionally on a large data well-posedness claim for such wave maps) that no stationary, travelling, self-similar, or degenerate wave maps exist in this energy class. These results form three of the five claims required in our earlier paper (arXiv:0805.4666) to prove global regularity for such wave maps. (The conditional claim of large data well-posedness is one of the remaining claims required in that paper.)