Blow-up for the 3-dimensional axially symmetric harmonic map flow into S2

Abstract

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 $u(x,t):\bar{\Omega }\times [0,T)\to S^2$. 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 $$|\nabla u(\cdot ,t){|}^2\rightharpoonup |\nabla u_{\ast }{|}^2+8\pi {\delta }_{\Gamma }\text{ as }t\to T.$$ for a regular function $u_{\ast }(x)$, 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.

References in corpus (1)

• Stable blow up dynamics for the 1-corotational energy critical harmonic heat flow