University of Connecticut

Analysis and Probability Seminar

 

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Title: Concentration Compactness for Critical Radial Wave Maps
Speaker: Jonas L├╝hrmann (Johns Hopkins University)
Time: Friday, March 24, 2017 at 1:30 pm
Place: MONT 226Abstract: 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 existence 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. The proof proceeds along the so-called concentration compactness/rigidity method of Kenig-Merle which in recent years has turned out to be a powerful strategy to study the long-time dynamics of solutions to critical dispersive or hyperbolic equations. I will explain the basic ideas of this method and the severe difficulties of implementing this method for energy critical wave maps due to the strong interactions in the wave maps nonlinearity. Then I will outline how these difficulties can be overcome by introducing a "twisted" profile decomposition. This is joint work with Elisabetta Chiodaroli and Joachim Krieger.


Organizer: Vasileios Chousionis and Oleksii Mostovyi