Singularity formation for the two-dimensional harmonic map flow into $S^2$.

We construct finite time blow-up solutions to the 2-dimensional harmonic map flow into the sphere $S^2$, \begin{align*} u_t & = \Delta u + |\nabla u|^2 u \quad \text{in } \Omega\times(0,T) \\ u &= \varphi \quad \text{on } \partial \Omega\times(0,T) \\ u(\cdot,0) &= u_0 \quad \text{in } \Omega , \end{align*} where $\Omega$ is a bounded, smooth domain in $\mathbb{R}^2$, $u: \Omega\times(0,T)\to S^2$, $u_0:\bar\Omega \to S^2$ is smooth, and $\varphi = u_0\big|_{\partial\Omega}$. Given any points $q_1,\ldots, q_k$ in the domain, we find initial and boundary data so that the solution blows-up precisely at those points. The profile around each point is close to an asymptotically singular scaling of a 1-corrotational harmonic map. We build a continuation after blow-up as a $H^1$-weak solution with a finite number of discontinuities in space-time by "reverse bubbling", which preserves the homotopy class of the solution after blow-up.