  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.

Analysis of PDEs