Concentration Compactness for Critical Radial Wave Maps

Abstract

We consider radially symmetric, energy critical wave maps from $$(1+2)$$ -dimensional Minkowski space into the unit sphere $$\mathbb {S}^m$$ , $$m \ge 1$$ , 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 and Tahvildar-Zadeh (Duke Math J 71(1):31–69, 1993; Pure Appl Math 46(7):1041–1091, 1993) and Struwe (Math Z 242(3):407–414, 2002; Calc Var Partial Differ Equ 16(4):431–437, 2003) as well as of Nahas (Calc Var Partial Differ Equ 46(1–2):427–437, 2013) 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 and Merle (Invent Math 166(3):645–675, 2006; Acta Math 201(2):147–212, 2008) and a “twisted” Bahouri–Gerard type profile decomposition (Am J Math 121(1):131–175, 1999), following the implementation of this strategy by the second author and Schlag (Concentration compactness for critical wave maps. EMS monographs in mathematics, European Mathematical Society (EMS), Zurich, 2012) for energy critical wave maps into the hyperbolic plane as well as by the last two authors (Ann PDE 1(1):1–208, 2015) for the energy critical Maxwell–Klein–Gordon equation.