Publication

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

Sohrab Mirshams Shahshahani
2016
Journal paper
Abstract

We construct a one parameter family of nite time blow ups to the co-rotational wave maps problem from S2×RS^2\times R to S2S^2 parameterized by νϵ(12,1]\nu \epsilon (\frac{1}{2},1]. The longitudinal function u(t,α)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 R2\mathbb{R}^2 to S2S^2. The domain of this harmonic map is identied with a neighborhood of the north pole in the domain S2S^2 via the exponential coordinates (α,θ)(\alpha ,\theta). In these coordinates u(t,α)=Q(λ(t)α)+R(t,α)u(t,\alpha) = Q(\lambda(t)\alpha) + R(t,\alpha), where Q(r)=2arctanrQ(r) = 2\,arctan\,r is the standard co-rotational harmonic map to the sphere, λ(t)=t1ν\lambda (t) = t^{-1-\nu}, and R(t,α)R(t,\alpha) is the error with local energy going to zero as t ! 0: Blow up will occur at (t,α)=(0,0)(t,\alpha) = (0,0) due to energy concentration, and up to this point the solution will have regularity H1+νH^{1+\nu-}.

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