### Abstract

We construct finite time blow-up solutions to the 2-dimensional harmonic map flow into the sphere S
^{2}, ut=Δu+|∇u|2uinΩ×(0,T)u=φon∂Ω×(0,T)u(·,0)=u0inΩ,where Ω is a bounded, smooth domain in R
^{2}, u: Ω × (0 , T) → S
^{2}, u: Ω ¯ → S
^{2} is smooth, and φ= u|
_{∂}
_{Ω}. Given any k points q
_{1}, … , 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-corotational 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. Furthermore, we prove the codimension one stability of the one point blow-up phenomenon.

Original language | English |
---|---|

Pages (from-to) | 345-466 |

Number of pages | 122 |

Journal | Inventiones Mathematicae |

Volume | 219 |

Issue number | 2 |

Early online date | 27 Jul 2019 |

DOIs | |

Publication status | Published - 29 Feb 2020 |

### ASJC Scopus subject areas

- Mathematics(all)