Unconditional well-posedness for wave maps

TL;DR

This paper establishes the uniqueness of solutions to the wave map equation in a natural regularity class for dimensions four and higher, using gauge localization and curvature bounds.

Contribution

It proves unconditional well-posedness for wave maps in the critical regularity space in dimensions d≥4, advancing understanding of solution uniqueness.

Findings

Uniqueness of solutions in the critical class for d≥4

Use of gauge localization and curvature bounds

Establishment of solution difference estimates at lower regularity

Abstract

We prove uniqueness of solutions to the wave map equation in the natural class, namely $ (u, \partial_t u) \in C([0,T); \dot{H}^{d/2})\times C^1([0,T); \dot{H}^{d/2-1})$ in dimensions $d\geq 4$. This is achieved through estimating the difference of two solutions at a lower regularity level. In order to reduce to the Coulomb gauge, one has to localize the gauge change in suitable cones as well as estimate the difference between the frames and connections associated to each solutions and take advantage of the assumption that the target manifold has bounded curvature.