December  2019, 39(12): 6913-6943. doi: 10.3934/dcds.2019237

Blow-up for the 3-dimensional axially symmetric harmonic map flow into $ S^2 $

1. 

Instituto de Matemáticas, Universidad de Antioquia, Calle 67, No. 53–108, Medellín, Colombia

2. 

Departamento de Ingeniería Matemática-CMM, Universidad de Chile, Santiago 837-0456, Chile

3. 

Department of Mathematical Sciences University of Bath, Bath BA2 7AY, United Kingdom

4. 

Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada, V6T 1Z2

* Corresponding author: Juan Dávila

Dedicated to Luis Caffarelli on the occasion of his birthday

Received  August 2018 Revised  March 2019 Published  June 2019

We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere
$ S^2 $
,
$ \begin{align*} u_t & = \Delta u + |\nabla u|^2 u \quad \text{in } \Omega\times(0,T) \\ u & = u_b \quad \text{on } \partial \Omega\times(0,T) \\ u(\cdot,0) & = u_0 \quad \text{in } \Omega , \end{align*} $
with
$ u(x,t): \bar \Omega\times [0,T) \to S^2 $
. Here
$ \Omega $
is a bounded, smooth axially symmetric domain in
$ \mathbb{R}^3 $
. We prove that for any circle
$ \Gamma \subset \Omega $
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
$ \Gamma $
, in fact
$ | {\nabla} u(\cdot ,t)|^2 \rightharpoonup | {\nabla} u_*|^2 + 8\pi \delta_\Gamma \quad\mbox{as}\quad t\to T . $
for a regular function
$ u_*(x) $
, where
$ \delta_\Gamma $
denotes the Dirac measure supported on the curve. This the first example of a blow-up solution with a space-codimension 2 singular set, the maximal dimension predicted in the partial regularity theory by Chen-Struwe and Cheng [5,6].
Citation: Juan Dávila, Manuel Del Pino, Catalina Pesce, Juncheng Wei. Blow-up for the 3-dimensional axially symmetric harmonic map flow into $ S^2 $. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 6913-6943. doi: 10.3934/dcds.2019237
References:
[1]

J. B. van den BergJ. Hulshof and J. R. King, Formal asymptotics of bubbling in the harmonic map heat flow, SIAM Journal of Applied Mathematics, 63 (2003), 1682-1717.  doi: 10.1137/S0036139902408874.  Google Scholar

[2]

J. B. van den Berg and J. F. Williams, (In-)stability of singular equivariant solutions to the Landau-Lifshitz-Gilbert equation, European Journal of Applied Mathematics, 24 (2013), 921-948.  doi: 10.1017/S0956792513000247.  Google Scholar

[3]

K. C. Chang, Heat flow and boundary value problem for harmonic maps, Annales de l'Institut Henri Poincare C, Analyse non lineaire, 6 (1989), 363-395.  doi: 10.1016/S0294-1449(16)30316-X.  Google Scholar

[4]

K. C. ChangW. Y. Ding and R. Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, Journal of Differential Geometry, 36 (1992), 507-515.  doi: 10.4310/jdg/1214448751.  Google Scholar

[5]

Y. M. Chen and M. Struwe, Existence and partial regularity results for the heat flow for harmonic maps, Mathematische Zeitschrift, 201 (1989), 83-103.  doi: 10.1007/BF01161997.  Google Scholar

[6]

X. Cheng, Estimate of the singular set of the evolution problem for harmonic maps, Journal of Differential Geometry, 34 (1991), 169-174.  doi: 10.4310/jdg/1214446996.  Google Scholar

[7]

J. Dávila, M. Del Pino and J. Wei, Singularity formation for the two-dimensional harmonic map flow into the sphere, preprint, arXiv: 1702.05801. doi: 10.1007/BF02568328.  Google Scholar

[8]

W. Ding and G. Tian, Energy identity for a class of approximate harmonic maps from surfaces, Communications in Analysis and Geometry, 3 (1995), 543-554.  doi: 10.4310/CAG.1995.v3.n4.a1.  Google Scholar

[9]

J. Eells Jr and J. H. Sampson, Harmonic mappings of Riemannian manifolds, American Journal of Mathematics, 86 (1964), 109-160.  doi: 10.2307/2373037.  Google Scholar

[10]

J. Grotowski, Finite time blow-up for the harmonic map heat flow, Calculus of Variations and Partial Differential Equations, 1 (1993), 231-236.  doi: 10.1007/BF01191618.  Google Scholar

[11]

J. Grotowski, Harmonic map heat flow for axially symmetric data, Manuscripta Mathematica, 73 (1991), 207-228.  doi: 10.1007/BF02567639.  Google Scholar

[12]

F. H. Lin and C. Y. Wang, Energy identity of harmonic map flows from surfaces at finite singular time, Calculus of Variations and Partial Differential Equations, 6 (1998), 369-380.  doi: 10.1007/s005260050095.  Google Scholar

[13]

F. H. Lin and C. Y. Wang, The Analysis of Harmonic Maps and Their Heat Flows, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. doi: 10.1142/9789812779533.  Google Scholar

[14]

F. H. Lin and C. Y. Wang, Harmonic and quasi-harmonic spheres. Ⅲ. Rectifiability of the parabolic defect measure and generalized varifold flows, Annales de l'Institut Henri Poincaré C, Analyse non lineaire, 19 (2002), 209-259.  doi: 10.1016/S0294-1449(01)00090-7.  Google Scholar

[15]

J. Qing, On singularities of the heat flow for harmonic maps from surfaces into spheres, Communications in Analysis and Geometry, 3 (1995), 297-315.  doi: 10.4310/CAG.1995.v3.n2.a4.  Google Scholar

[16]

J. Qing and G. Tian, Bubbling of the heat flows for harmonic maps from surfaces, Communications on Pure and Applied Mathematics, 50 (1997), 295-310.  doi: 10.1002/(SICI)1097-0312(199704)50:4<295::AID-CPA1>3.0.CO;2-5.  Google Scholar

[17]

P. Raphaël and R. Schweyer, Stable blowup dynamics for the 1-corotational energy critical harmonic heat flow, Communications on Pure and Applied Mathematics, 66 (2013), 414-480.  doi: 10.1002/cpa.21435.  Google Scholar

[18]

M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces, Commentarii Mathematici Helvetici, 60 (1985), 558-581.  doi: 10.1007/BF02567432.  Google Scholar

[19]

P. M. Topping, Repulsion and quantization in almost-harmonic maps, and asymptotics of the harmonic map flow, Annals of Mathematics, 159 (2004), 465-534.  doi: 10.4007/annals.2004.159.465.  Google Scholar

show all references

Dedicated to Luis Caffarelli on the occasion of his birthday

References:
[1]

J. B. van den BergJ. Hulshof and J. R. King, Formal asymptotics of bubbling in the harmonic map heat flow, SIAM Journal of Applied Mathematics, 63 (2003), 1682-1717.  doi: 10.1137/S0036139902408874.  Google Scholar

[2]

J. B. van den Berg and J. F. Williams, (In-)stability of singular equivariant solutions to the Landau-Lifshitz-Gilbert equation, European Journal of Applied Mathematics, 24 (2013), 921-948.  doi: 10.1017/S0956792513000247.  Google Scholar

[3]

K. C. Chang, Heat flow and boundary value problem for harmonic maps, Annales de l'Institut Henri Poincare C, Analyse non lineaire, 6 (1989), 363-395.  doi: 10.1016/S0294-1449(16)30316-X.  Google Scholar

[4]

K. C. ChangW. Y. Ding and R. Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, Journal of Differential Geometry, 36 (1992), 507-515.  doi: 10.4310/jdg/1214448751.  Google Scholar

[5]

Y. M. Chen and M. Struwe, Existence and partial regularity results for the heat flow for harmonic maps, Mathematische Zeitschrift, 201 (1989), 83-103.  doi: 10.1007/BF01161997.  Google Scholar

[6]

X. Cheng, Estimate of the singular set of the evolution problem for harmonic maps, Journal of Differential Geometry, 34 (1991), 169-174.  doi: 10.4310/jdg/1214446996.  Google Scholar

[7]

J. Dávila, M. Del Pino and J. Wei, Singularity formation for the two-dimensional harmonic map flow into the sphere, preprint, arXiv: 1702.05801. doi: 10.1007/BF02568328.  Google Scholar

[8]

W. Ding and G. Tian, Energy identity for a class of approximate harmonic maps from surfaces, Communications in Analysis and Geometry, 3 (1995), 543-554.  doi: 10.4310/CAG.1995.v3.n4.a1.  Google Scholar

[9]

J. Eells Jr and J. H. Sampson, Harmonic mappings of Riemannian manifolds, American Journal of Mathematics, 86 (1964), 109-160.  doi: 10.2307/2373037.  Google Scholar

[10]

J. Grotowski, Finite time blow-up for the harmonic map heat flow, Calculus of Variations and Partial Differential Equations, 1 (1993), 231-236.  doi: 10.1007/BF01191618.  Google Scholar

[11]

J. Grotowski, Harmonic map heat flow for axially symmetric data, Manuscripta Mathematica, 73 (1991), 207-228.  doi: 10.1007/BF02567639.  Google Scholar

[12]

F. H. Lin and C. Y. Wang, Energy identity of harmonic map flows from surfaces at finite singular time, Calculus of Variations and Partial Differential Equations, 6 (1998), 369-380.  doi: 10.1007/s005260050095.  Google Scholar

[13]

F. H. Lin and C. Y. Wang, The Analysis of Harmonic Maps and Their Heat Flows, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. doi: 10.1142/9789812779533.  Google Scholar

[14]

F. H. Lin and C. Y. Wang, Harmonic and quasi-harmonic spheres. Ⅲ. Rectifiability of the parabolic defect measure and generalized varifold flows, Annales de l'Institut Henri Poincaré C, Analyse non lineaire, 19 (2002), 209-259.  doi: 10.1016/S0294-1449(01)00090-7.  Google Scholar

[15]

J. Qing, On singularities of the heat flow for harmonic maps from surfaces into spheres, Communications in Analysis and Geometry, 3 (1995), 297-315.  doi: 10.4310/CAG.1995.v3.n2.a4.  Google Scholar

[16]

J. Qing and G. Tian, Bubbling of the heat flows for harmonic maps from surfaces, Communications on Pure and Applied Mathematics, 50 (1997), 295-310.  doi: 10.1002/(SICI)1097-0312(199704)50:4<295::AID-CPA1>3.0.CO;2-5.  Google Scholar

[17]

P. Raphaël and R. Schweyer, Stable blowup dynamics for the 1-corotational energy critical harmonic heat flow, Communications on Pure and Applied Mathematics, 66 (2013), 414-480.  doi: 10.1002/cpa.21435.  Google Scholar

[18]

M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces, Commentarii Mathematici Helvetici, 60 (1985), 558-581.  doi: 10.1007/BF02567432.  Google Scholar

[19]

P. M. Topping, Repulsion and quantization in almost-harmonic maps, and asymptotics of the harmonic map flow, Annals of Mathematics, 159 (2004), 465-534.  doi: 10.4007/annals.2004.159.465.  Google Scholar

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