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 + | abla u|^2 u quad text{in } Omegatimes(0,T) u &= varphi quad text{on } partial Omegatimes(0,T) u(cdot,0) &= u_0 quad text{in } Omega , end{align*} where $Omega$ is a bounded, smooth domain in $mathbb{R}^2$, $u: Omegatimes(0,T)to S^2$, $u_0:barOmega to S^2$ is smooth, and $varphi = u_0big|_{partialOmega}$. 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.