Blow Up Construction and Stability of Stationary Maps

This thesis is a study of wave maps from a curved background to the standard two dimensional sphere S2. The target is always assumed to be embedded in R3 in the standard way. The do- main manifold (the "curved background") will be diffeomorphic to S2 × R, but the Lorentzian metric will not necessarily be the standard one. The current work is based on the following two results by Shatah, Tahvildar-Zadeh [47] and Krieger, Schlag, Tataru [25]. In [47] Shatah and Tahvildar-Zadeh established the existence and stability in the energy norm of a compact family of stationary (satisfying a certain periodicity condition in time) and equiv- ariant (satisfying a certain spatial symmetry) wave maps from S2 × R to S2 equipped with the standard metrics. Stability was proved under perturbations in the same spatial equivariance class. In [25], the authors construct a one parameter family of blow up solutions to the co-rotational wave maps equation from R2+1 to S2 with the standard metrics. Co-rotational means that the wave maps are assumed to have rotation number equal to one. These solutions are obtained as perturbations of the rescaled standard rotational harmonic map 2arctanr from R2 to S2. Here, we prove the conclusions of these papers for different metrics on the domains of wave maps. More specifically, our results can be grouped into two parts: stationary maps and blow ups. In the first part following the method of [25], we construct a one parameter family of co-rotational blow up wave maps from S2 ×R to S2. These solutions can be parameterized by ν ∈ ( 1 , 1]. The metric on the domain sphere is taken to be the standard one. 2 In the second part, we consider wave maps from S2 × R to S2 where the metric on the do- main S2 is now taken to be a general SO(1)−symmetric metric of the form dr2 + f 2(r)dθ2. The special case f (r ) = sin r corresponds to the standard metric considered by Shatah and Tahvildar-Zadeh. We prove the same stability result as in [47] in this general setting.