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

Analysis of PDEs