Calculus of Variations and Geometric Measure Theory

E. Chiodaroli - J. Krieger - J. Lührmann

Concentration compactness for critical radial wave maps

created by chiodaroli on 30 Nov 2016


Submitted Paper

Inserted: 30 nov 2016
Last Updated: 30 nov 2016

Year: 2016
Links: arxiv


We consider radially symmetric, energy critical wave maps from (1 + 2)-dimensional Minkowski space into the unit sphere Sm, m≥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 compactnessrigidity method of Kenig-Merle 6, 7 and a “twisted” Bahouri-Gérard 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.