Singularity formation for the two-dimensional harmonic map flow into $S^2$

TL;DR

This paper constructs solutions to the 2D harmonic map flow into S^2 that blow up at specified points, describing their profiles and a continuation method after blow-up that preserves topological class.

Contribution

It introduces a method to produce finite-time blow-up solutions with prescribed singularities and a continuation scheme post-blow-up that maintains the solution's homotopy class.

Findings

Solutions blow up at chosen points with specific profiles.

A continuation method after blow-up preserves homotopy class.

Explicit construction of initial data leading to prescribed blow-up points.

Abstract

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