### Abstract

We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere S^{2}, ut = ∆u + |∇u|^{2}u in Ω × (0, T) u = ub on ∂Ω × (0, T) u(·, 0) = u_{0} in Ω, with u(x, t): Ω^{¯} × [0, T) → S^{2}. Here Ω is a bounded, smooth axially symmetric domain in R^{3}. 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} + 8πδ_{Γ} as t → T. for a regular function u∗(x), where δ_{Γ} 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 language | English |
---|---|

Pages (from-to) | 6913-6943 |

Number of pages | 31 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Volume | 39 |

Issue number | 12 |

DOIs | |

Publication status | Published - 31 Dec 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

*S*.

^{2}*Discrete and Continuous Dynamical Systems- Series A*,

*39*(12), 6913-6943. https://doi.org/10.3934/dcds.2019237

**Blow-up for the 3-dimensional axially symmetric harmonic map flow into S^{2}.** / Dávila, Juan; Pino, Manuel Del; Pesce, Catalina; Wei, Juncheng.

Research output: Contribution to journal › Article

*S*',

^{2}*Discrete and Continuous Dynamical Systems- Series A*, vol. 39, no. 12, pp. 6913-6943. https://doi.org/10.3934/dcds.2019237

*S*. Discrete and Continuous Dynamical Systems- Series A. 2019 Dec 31;39(12):6913-6943. https://doi.org/10.3934/dcds.2019237

^{2}}

TY - JOUR

T1 - Blow-up for the 3-dimensional axially symmetric harmonic map flow into S2

AU - Dávila, Juan

AU - Pino, Manuel Del

AU - Pesce, Catalina

AU - Wei, Juncheng

PY - 2019/12/31

Y1 - 2019/12/31

N2 - 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].

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].

KW - Blow-up

KW - Codimension 2 singular set

KW - Finite time blow-up

KW - Harmonic map flow

KW - Semilinear parabolic equation

UR - http://www.scopus.com/inward/record.url?scp=85072932156&partnerID=8YFLogxK

U2 - 10.3934/dcds.2019237

DO - 10.3934/dcds.2019237

M3 - Article

AN - SCOPUS:85072932156

VL - 39

SP - 6913

EP - 6943

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 12

ER -