Applied Mathematics Seminar

Title: Concentration Compactness for Critical Radial Wave Maps

Speaker: Jonas Lührmann (Johns Hopkins University)

Abstract: The wave maps equation is the natural generalization of the linear wave equation for scalar-valued fields to fields that take values in a Riemannian manifold. In this talk we consider radially symmetric, energy critical wave maps from (1+2)-dimensional Minkowski space into the unit sphere and prove global regularity and scattering for essentially arbitrary 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 and Struwe as well as of Nahas on radial wave maps in the case of the unit sphere as the target.

The proof proceeds along the concentration compactness/rigidity method of Kenig-Merle. I will explain the severe difficulties of implementing this strategy for energy critical wave maps due to strong low-high interactions in the wave maps nonlinearity and how these difficulties can be overcome by introducing a "twisted" profile decomposition. This is joint work with Elisabetta Chiodaroli and Joachim Krieger.