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Renormalization and blow-up for wave maps from $ S^2\times\mathbb{R}$ to $ S^2$


Author: Sohrab Shahshahani
Journal: Trans. Amer. Math. Soc. 368 (2016), 5621-5654
MSC (2010): Primary 74J30, 35A01, 35B44, 35C08
DOI: https://doi.org/10.1090/tran/6524
Published electronically: September 15, 2015
MathSciNet review: 3458393
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Abstract: We construct a one parameter family of finite time blow-ups to the co-rotational wave maps problem from $ S^2\times \mathbb{R}$ to $ S^2,$ parameterized by $ \nu \in (\frac {1}{2},1].$ The longitudinal function $ u(t,\alpha )$ which is the main object of study will be obtained as a perturbation of a rescaled harmonic map of rotation index one from $ \mathbb{R}^2$ to $ S^2.$ The domain of this harmonic map is identified with a neighborhood of the north pole in the domain $ S^2$ via the exponential coordinates $ (\alpha ,\theta ).$ In these coordinates $ u(t,\alpha )=Q(\lambda (t)\alpha )+\mathcal {R}(t,\alpha ),$ where $ Q(r)=2\arctan {r}$ is the standard co-rotational harmonic map to the sphere, $ \lambda (t)=t^{-1-\nu },$ and $ \mathcal {R}(t,\alpha )$ is the error with local energy going to zero as $ t\rightarrow 0.$ Blow-up will occur at $ (t,\alpha )=(0,0)$ due to energy concentration, and up to this point the solution will have regularity $ H^{1+\nu -}.$


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Additional Information

Sohrab Shahshahani
Affiliation: Section de Mathématiques, EPFL FSB SMA, Station 8 - Bâtiment MA, CH-1015 Lausanne, Switzerland
Address at time of publication: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109
Email: shahshah@umich.edu

DOI: https://doi.org/10.1090/tran/6524
Received by editor(s): May 25, 2014
Received by editor(s) in revised form: July 15, 2014
Published electronically: September 15, 2015
Article copyright: © Copyright 2015 American Mathematical Society