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

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