Blow-up for the 3-dimensional axially symmetric harmonic map flow into S2

Juan Dávila, Manuel Del Pino, Catalina Pesce, Juncheng Wei

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Abstract

We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere S2, ut = ∆u + |∇u|2u in Ω × (0, T) u = ub on ∂Ω × (0, T) u(·, 0) = u0 in Ω, with u(x, t): Ω¯ × [0, T) → S2. Here Ω is a bounded, smooth axially symmetric domain in R3. We prove that for any circle Γ ⊂ Ω with the same axial symmetry, and any sufficiently small T > 0 there exist initial and boundary conditions such that u(x, t) blows-up exactly at time T and precisely on the curve Γ, in fact |∇u(·, t)|2 * |∇u∗|2 + 8πδΓ as t → T. for a regular function u∗(x), where δΓ denotes the Dirac measure supported on the curve. This the first example of a blow-up solution with a spacecodimension 2 singular set, the maximal dimension predicted in the partial regularity theory by Chen-Struwe and Cheng [5, 6].

Original languageEnglish
Pages (from-to)6913-6943
Number of pages31
JournalDiscrete and Continuous Dynamical Systems- Series A
Volume39
Issue number12
DOIs
Publication statusAccepted/In press - 31 Mar 2019

Keywords

  • Blow-up
  • Codimension 2 singular set
  • Finite time blow-up
  • Harmonic map flow
  • Semilinear parabolic equation

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Cite this

Blow-up for the 3-dimensional axially symmetric harmonic map flow into S2. / Dávila, Juan; Pino, Manuel Del; Pesce, Catalina; Wei, Juncheng.

In: Discrete and Continuous Dynamical Systems- Series A, Vol. 39, No. 12, 31.12.2019, p. 6913-6943.

Research output: Contribution to journalArticle

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AB - We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere S2, ut = ∆u + |∇u|2u in Ω × (0, T) u = ub on ∂Ω × (0, T) u(·, 0) = u0 in Ω, with u(x, t): Ω¯ × [0, T) → S2. Here Ω is a bounded, smooth axially symmetric domain in R3. We prove that for any circle Γ ⊂ Ω with the same axial symmetry, and any sufficiently small T > 0 there exist initial and boundary conditions such that u(x, t) blows-up exactly at time T and precisely on the curve Γ, in fact |∇u(·, t)|2 * |∇u∗|2 + 8πδΓ as t → T. for a regular function u∗(x), where δΓ denotes the Dirac measure supported on the curve. This the first example of a blow-up solution with a spacecodimension 2 singular set, the maximal dimension predicted in the partial regularity theory by Chen-Struwe and Cheng [5, 6].

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