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


We construct finite time blow-up solutions to the 2-dimensional harmonic map flow into the sphere $S^2$, $$\begin{array}{c} u_t = \Delta u + |\nabla u|^2 u \quad &\text{in } \Omega\times(0,T)\\ u = \vp \quad &\text{on } \pp\Omega\times(0,T)\\ u(\cdot,0) = u_0 \quad & \text{in } \Omega \end{array} $$ where $\Omega$ is a bounded, smooth domain in $\R^2$ and $u: \Omega\times(0,T)\to S^2$, $u_0:\bar\Omega \to S^2$, smooth, $\vp= u_0\big|_{\pp\Omega}$. 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 analyze stability of this phenomenon if $k=1$. This is joint work with Juan D\'avila and Juncheng Wei.

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