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

Research output: Contribution to journalArticle

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 languageEnglish
Pages (from-to)1-122
Number of pages122
JournalInventiones Mathematicae
Early online date27 Jul 2019
DOIs
Publication statusE-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.

In: Inventiones Mathematicae, 27.07.2019, p. 1-122.

Research output: Contribution to journalArticle

@article{bb24fceef8844565945ac59414780982,
title = "Singularity formation for the two-dimensional harmonic map flow into S2",
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.",
author = "{Davila Bonczos}, Juan and {Del Pino}, Manuel and Juncheng Wei",
year = "2019",
month = "7",
day = "27",
doi = "10.1007/s00222-019-00908-y",
language = "English",
pages = "1--122",
journal = "Inventiones Mathematicae",
issn = "0020-9910",
publisher = "Springer New York",

}

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 -