Blow-up for the 3-dimensional axially symmetric harmonic map flow into <inline-formula><tex-math id="M1">$ S^2 $</tex-math></inline-formula>

We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere $ S^2 $, \begin{document}$ \begin{align*} u_t & = \Delta u + |\nabla u|^2 u \quad \text{in } \Omega\times(0,T) \\ u & = u_b \quad \text{on } \partial \Omega\times(0,T) \\ u(\cdot,0) & = u_0 \quad \text{in } \Omega , \end{align*} $\end{document} with $ u(x,t): \bar \Omega\times [0,T) \to S^2 $. Here $ \Omega $ is a bounded, smooth axially symmetric domain in $ \mathbb{R}^3 $. We prove that for any circle $ \Gamma \subset \Omega $ with the same axial symmetry, and any sufficiently small $ T>0 $ there exist initial and boundary conditions such that $ u(x,t) $ blows-up exactly at time $ T $ and precisely on the curve $ \Gamma $, in fact \begin{document}$ | {\nabla} u(\cdot ,t)|^2 \rightharpoonup | {\nabla} u_*|^2 + 8\pi \delta_\Gamma \quad\mbox{as}\quad t\to T . $\end{document} for a regular function $ u_*(x) $, where $ \delta_\Gamma $ denotes the Dirac measure supported on the curve. This the first example of a blow-up solution with a space-codimension 2 singular set, the maximal dimension predicted in the partial regularity theory by Chen-Struwe and Cheng [5,6].

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