# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 2019, 39 (12) : 6913-6943. doi: 10.3934/dcds.2019237
##### 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.
 [1] Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021 [2] Xiaoliang Li, Baiyu Liu. Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3093-3112. doi: 10.3934/cpaa.2020134 [3] José M. Arrieta, Raúl Ferreira, Arturo de Pablo, Julio D. Rossi. Stability of the blow-up time and the blow-up set under perturbations. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 43-61. doi: 10.3934/dcds.2010.26.43 [4] Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318 [5] Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103 [6] Asma Azaiez. Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2397-2408. doi: 10.3934/cpaa.2019108 [7] Yohei Fujishima. On the effect of higher order derivatives of initial data on the blow-up set for a semilinear heat equation. Communications on Pure and Applied Analysis, 2018, 17 (2) : 449-475. doi: 10.3934/cpaa.2018025 [8] Yohei Fujishima. Blow-up set for a superlinear heat equation and pointedness of the initial data. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4617-4645. doi: 10.3934/dcds.2014.34.4617 [9] Sachiko Ishida, Tomomi Yokota. Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. Discrete and Continuous Dynamical Systems - B, 2013, 18 (10) : 2569-2596. doi: 10.3934/dcdsb.2013.18.2569 [10] Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388 [11] Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042 [12] Monica Marras, Stella Vernier Piro. Bounds for blow-up time in nonlinear parabolic systems. Conference Publications, 2011, 2011 (Special) : 1025-1031. doi: 10.3934/proc.2011.2011.1025 [13] Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225 [14] Yuya Tanaka, Tomomi Yokota. Finite-time blow-up in a quasilinear degenerate parabolic–elliptic chemotaxis system with logistic source and nonlinear production. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022075 [15] Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71 [16] Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399 [17] Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831 [18] Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006 [19] Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113 [20] Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1

2021 Impact Factor: 1.588