2019/02/11 by Juan Dávila, Davila, Juan, Manuel del Pino +5 · 1 citation
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1902.03995
openalex publication_date 2019/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct finite time blow-up solutions to the 3-dimensional harmonic map\nflow into the sphere S2, \ut amp; =
Delta u + |
nabla u|2 u\n
quad
textin
Omega
times(0,T)
u amp;= ub
quad
texton
partial\n
Omega
times(0,T)
u(
cdot,0) amp;= u0
quad
textin
Omega , nwith u(x,t): \Ω\× [0,T) \→ S2. Here \Ω is a bounded,\nsmooth axially symmetric domain in \ℝ3. We prove that for any circle\n\Γ \⊂ \Ω with the same axial symmetry, and any sufficiently\nsmall T>0 there exist initial and boundary conditions such that u(x,t)\nblows-up exactly at time T and precisely on the curve \Γ, in fact \n|
nabla u(
cdot ,t)|2
rightharpoonup |
nabla u_*|2 + 8
pi
delta_
Gamma\n
text as t
to T . for a regular function u_*(x), where \δ_\Γ\ndenotes the Dirac measure supported on the curve. This the first example of a\nblow-up solution with a space-codimension 2 singular set, the maximal dimension\npredicted in the partial regularity theory by Chen-Struwe and Cheng.\n