Renormalization and blow up for wave maps from $S^2\times \RR$ to $S^2$
We construct a one parameter family of finite time blow ups to the co-rotational wave maps problem from $S^2\times \RR$ to $S^2,$ parameterized by $ν\in(1/2,1].$ The longitudinal function $u(t,α)$ which is the main object of study will be obtained as a perturbation of a rescaled harmonic map of rota...
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| Format: | Report |
| Language: | unknown |
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arXiv
2012
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| Online Access: | https://dx.doi.org/10.48550/arxiv.1203.4722 https://arxiv.org/abs/1203.4722 |
| Summary: | We construct a one parameter family of finite time blow ups to the co-rotational wave maps problem from $S^2\times \RR$ to $S^2,$ parameterized by $ν\in(1/2,1].$ The longitudinal function $u(t,α)$ which is the main object of study will be obtained as a perturbation of a rescaled harmonic map of rotation index one from $\RR^2$ 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 $(α,θ).$ In these coordinates $u(t,α)=Q(λ(t)α)+\mathcal{R}(t,α),$ where $Q(r)=2\arctan{r},$ is the standard co-rotational harmonic map to the sphere, $λ(t)=t^{-1-ν},$ and $\mathcal{R}(t,α)$ is the error with local energy going to zero as $t\rightarrow 0.$ Blow up will occur at $(t,α)=(0,0)$ due to energy concentration, and up to this point the solution will have regularity $H^{1+ν-}.$ : Modified argument in section 4 (with the corresponding sections of the introduction). arXiv admin note: text overlap with arXiv:math/0610248 by other authors |
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