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

We construct a one parameter family of finite time blow-ups to the co-rotational wave maps problem from S 2 × R S^2\times \mathbb {R} to S 2 , S^2, parameterized by ν ∈ ( 1 2 , 1 ] . \nu \in (\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 R 2 \mathbb {R}^2 to S 2 . S^2. The domain of this harmonic map is identified with a neighborhood of the north pole in the domain S 2 S^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 )+\mathcal {R}(t,\alpha ), where Q ( r ) = 2 arctan ⁡ r Q(r)=2\arctan {r} is the standard co-rotational harmonic map to the sphere, λ ( t ) = t − 1 − ν , \lambda (t)=t^{-1-\nu }, and R ( t , α ) \mathcal {R}(t,\alpha ) is the error with local energy going to zero as t → 0. t\rightarrow 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 H 1 + ν − . H^{1+\nu -}.