Concentration Compactness for Critical Radial Wave Maps

TL;DR

This paper proves global regularity, scattering, and concentration compactness properties for radially symmetric, energy-critical wave maps from 3D Minkowski space into spheres, extending classical results with modern concentration compactness techniques.

Contribution

It establishes new global regularity and scattering results for radial wave maps and develops concentration compactness properties using advanced profile decomposition methods.

Findings

Proved global regularity and scattering for radial wave maps.

Established concentration compactness properties for bounded energy sequences.

Extended classical results with modern analytical techniques.

Abstract

We consider radially symmetric, energy critical wave maps from (1 + 2)-dimensional Minkowski space into the unit sphere $\mathbb{S}^m$, $m \geq 1$, and prove global regularity and scattering for classical smooth data of finite energy. In addition, we establish a priori bounds on a suitable scattering norm of the radial wave maps and exhibit concentration compactness properties of sequences of radial wave maps with uniformly bounded energies. This extends and complements the beautiful classical work of Christodoulou-Tahvildar-Zadeh [3, 4] and Struwe [31, 33] as well as of Nahas [22] on radial wave maps in the case of the unit sphere as the target. The proof is based upon the concentration compactness/rigidity method of Kenig-Merle [6, 7] and a "twisted" Bahouri-Gerard type profile decomposition [1], following the implementation of this strategy by the second author and Schlag [17] for energy critical wave maps into the hyperbolic plane as well as by the last two authors [16] for the energy critical Maxwell-Klein-Gordon equation.