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 $$u(t,r)=Q(λ(t)r)+ε(t,r)$$where $Q: \mathbb{R}^2 \to S^2$ is the ground state harmonic map and $λ(t) = t^{-1-ν}$ for any $ν> 0$. This extends the work [13], where such solutions were constructed under the assumption $ν> 1/2$. In light of a result of Struwe [22], our result is optimal for polynomial blow up rates.