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

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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)345-466
Number of pages122
JournalInventiones Mathematicae
Volume219
Issue number2
Early online date27 Jul 2019
DOIs
Publication statusPublished - 29 Feb 2020

ASJC Scopus subject areas

  • Mathematics(all)

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