Renormalization and blow-up for wave maps from 𝑆²×ℝ to 𝕊²

Shahshahani, Sohrab

doi:10.1090/tran/6524

Renormalization and blow-up for wave maps from $S^2\times \mathbb {R}$ to $S^2$
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Trans. Amer. Math. Soc. 368 (2016), 5621-5654 Request permission

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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