Blow-up for the 3-dimensional axially symmetric harmonic map flow into S2
We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere $S^2$, \begin{align*} u_t & = Δu + |\nabla u|^2 u \quad \text{in } Ω\times(0,T) \\ u &= u_b \quad \text{on } \partial Ω\times(0,T) \\ u(\cdot,0) &= u_0 \quad \text{in } Ω, \end{align*} with $u(x,t): \bar Ω\times [0,T) \to S^2$. Here $Ω$ is a bounded, smooth axially symmetric domain in $\mathbb{R}^3$. We prove that for any circle $Γ\subset Ω$ 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 $Γ$, in fact $$ |\nabla u(\cdot ,t)|^2 \rightharpoonup |\nabla u_*|^2 + 8πδ_Γ\text{ as } t\to T . $$ for a regular function $u_*(x)$, where $δ_Γ$ 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.