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 $\mathrm{\Omega }$ is a bounded, smooth domain in ${\mathbb{R}}^2$, $u: \Omega\times(0,T)\to S^2$,$u_0:\bar\Omega \to S^2$issmooth,and$\varphi = u_0\big|_{\partial\Omega}$\mathrm{.}Givenanypoints$q_1,\ldots, q_k$ 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.