Global regularity of wave maps I. Small critical Sobolev norm in high dimension

TL;DR

This paper proves that wave maps from high-dimensional Minkowski space to a sphere are globally smooth for small initial data in the critical Sobolev space, overcoming technical challenges related to controlling the nonlinearity.

Contribution

It establishes global regularity for wave maps in high dimensions with small critical Sobolev norm, introducing techniques to handle borderline regularity issues.

Findings

Global smoothness for wave maps in high dimensions with small initial data.

Overcoming divergence issues using coordinate frames adapted to wave maps.

Framework applicable to the energy-critical two-dimensional case in future work.

Abstract

We show that wave maps from Minkowski space $R^{1+n}$ to a sphere are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space $\dot H^{n/2}$ in the high dimensional case $n \geq 5$. A major difficulty, not present in the earlier results, is that the $\dot H^{n/2}$ norm barely fails to control $L^\infty$, potentially causing a logarithmic divergence in the nonlinearity; however, this can be overcome by using co-ordinate frames adapted to the wave map by approximate parallel transport. In the sequel of this paper we address the more interesting two-dimensional case, which is energy-critical.