# American Institute of Mathematical Sciences

December  2019, 39(12): 6913-6943. doi: 10.3934/dcds.2019237

## Blow-up for the 3-dimensional axially symmetric harmonic map flow into $S^2$

 1 Instituto de Matemáticas, Universidad de Antioquia, Calle 67, No. 53–108, Medellín, Colombia 2 Departamento de Ingeniería Matemática-CMM, Universidad de Chile, Santiago 837-0456, Chile 3 Department of Mathematical Sciences University of Bath, Bath BA2 7AY, United Kingdom 4 Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada, V6T 1Z2

* Corresponding author: Juan Dávila

Dedicated to Luis Caffarelli on the occasion of his birthday

Received  August 2018 Revised  March 2019 Published  June 2019

We construct finite time blow-up solutions to the 3-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 & = u_b \quad \text{on } \partial \Omega\times(0,T) \\ u(\cdot,0) & = u_0 \quad \text{in } \Omega , \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
 $| {\nabla} u(\cdot ,t)|^2 \rightharpoonup | {\nabla} u_*|^2 + 8\pi \delta_\Gamma \quad\mbox{as}\quad t\to T .$
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].
Citation: Juan Dávila, Manuel Del Pino, Catalina Pesce, Juncheng Wei. Blow-up for the 3-dimensional axially symmetric harmonic map flow into $S^2$. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6913-6943. doi: 10.3934/dcds.2019237
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