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)
Cite this
Singularity formation for the two-dimensional harmonic map flow into S2. / Davila Bonczos, Juan; Del Pino, Manuel; Wei, Juncheng.
In: Inventiones Mathematicae, Vol. 219, No. 2, 29.02.2020, p. 345-466.Research output: Contribution to journal › Article
}
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 - 2020/2/29
Y1 - 2020/2/29
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
VL - 219
SP - 345
EP - 466
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
SN - 0020-9910
IS - 2
ER -