Optimal polynomial blow up range for critical wave maps

Abstract

We prove that the critical Wave Maps equation with target $S^2$ and origin ${\mathbb{R}}^{2+1}$ admits energy class blow up solutions of the form $$u(t,r)=Q(\hat{I}»(t)r)+\hat{I}\mu (t,r)$$where $Q:{\mathbb{R}}^2\to S^2$ is the ground state harmonic map and $\hat{I}»(t)=t^{-1-\hat{I}½}$ for any $\hat{I}½>0$. This extends the work [13], where such solutions were constructed under the assumption $\hat{I}½>1/2$. In light of a result of Struwe [22], our result is optimal for polynomial blow up rates.