### 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 |
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Pages (from-to) | 1-122 |

Number of pages | 122 |

Journal | Inventiones Mathematicae |

Early online date | 27 Jul 2019 |

DOIs | |

Publication status | E-pub ahead of print - 27 Jul 2019 |

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Singularity formation for the two-dimensional harmonic map flow into S2.** / Davila Bonczos, Juan; Del Pino, Manuel; Wei, Juncheng.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Singularity formation for the two-dimensional harmonic map flow into S2

AU - Davila Bonczos, Juan

AU - Del Pino, Manuel

AU - Wei, Juncheng

PY - 2019/7/27

Y1 - 2019/7/27

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85069651630&partnerID=8YFLogxK

U2 - 10.1007/s00222-019-00908-y

DO - 10.1007/s00222-019-00908-y

M3 - Article

SP - 1

EP - 122

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

ER -