Existence and stability of blowup for wave maps into negatively curved targets
A talk in the Keine Reihe series by
Irfan Glogic from Universität Wien
| Abstract: | Harmonic maps between Lorentzian and Riemannian manifolds go under the name of wave maps. We consider wave maps from $(1 + d)$-dimensional Minkowski space into negatively curved Riemannian manifolds. Furthermore, we are concerned with the question of blowup. Due to their resemblance to defocussing nonlinear wave equations, this kind of wave maps were believed to not exhibit generic (stable) blowup. Cazenave, Shatah and Tahvildar-Zadeh proved in 1998 that self-similar blowup exists for $d = 7$. We complement this result by constructing for each dimension $d \geq 8$ a negatively curved, $d$-dimensional
target manifold that allows for the existence of a self-similar wave map. What is more, we prove that our solutions provide a stable blowup mechanism for the corresponding Cauchy problem. This, in addition to being the first example of stable blowup for wave maps with negatively curved targets, also shows that intuition from above was in fact false. This is joint work with Roland Donninger, University of Vienna. |