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Soap films with gravity and almost-minimal surfaces
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 |
$ 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*} $ |
$ u(x,t): \bar \Omega\times [0,T) \to S^2 $ |
$ \Omega $ |
$ \mathbb{R}^3 $ |
$ \Gamma \subset \Omega $ |
$ T>0 $ |
$ u(x,t) $ |
$ T $ |
$ \Gamma $ |
$ | {\nabla} u(\cdot ,t)|^2 \rightharpoonup | {\nabla} u_*|^2 + 8\pi \delta_\Gamma \quad\mbox{as}\quad t\to T . $ |
$ u_*(x) $ |
$ \delta_\Gamma $ |
References:
[1] |
J. B. van den Berg, J. 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. |
[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. |
[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. |
[4] |
K. C. Chang, W. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
J. Grotowski,
Harmonic map heat flow for axially symmetric data, Manuscripta Mathematica, 73 (1991), 207-228.
doi: 10.1007/BF02567639. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
M. Struwe,
On the evolution of harmonic mappings of Riemannian surfaces, Commentarii Mathematici Helvetici, 60 (1985), 558-581.
doi: 10.1007/BF02567432. |
[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. |
show all references
Dedicated to Luis Caffarelli on the occasion of his birthday
References:
[1] |
J. B. van den Berg, J. 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. |
[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. |
[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. |
[4] |
K. C. Chang, W. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
J. Grotowski,
Harmonic map heat flow for axially symmetric data, Manuscripta Mathematica, 73 (1991), 207-228.
doi: 10.1007/BF02567639. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
M. Struwe,
On the evolution of harmonic mappings of Riemannian surfaces, Commentarii Mathematici Helvetici, 60 (1985), 558-581.
doi: 10.1007/BF02567432. |
[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. |
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