American Institute of Mathematical Sciences

Advanced
October  1999, 5(4): 799-804. doi: 10.3934/dcds.1999.5.799

Harmonic maps on complete manifolds

Wenxiong Chen 1, and  Congming Li 2,

1. 

Department of Mathematics, Southwest Missouri State University
2. 

Department of Applied Mathematics, University of Colorado at Boulder

Received  November 1998 Revised  November 1998 Published  July 1999

In this article, we study harmonic maps between two complete noncompact manifolds M and N by a heat flow method. We find some new sufficient conditions for the uniform convergence of the heat flow, and hence the existence of harmonic maps.
Our condition are: The Ricci curvature of M is bounded from below by a negative constant, M admits a positive Green’s function and

$ \int_M G(x, y)|\tau(h(y))|dV_y $ is bounded on each compact subset. $\qquad$ (1)

Here $\tau(h(x))$ is the tension field of the initial data $h(x)$.
Condition (1) is somewhat sharp as is shown by examples in the paper.

Keywords: Harmonic maps between complete, heat flow method, noncompact manifolds, uniform convergence of heat flows..
Mathematics Subject Classification: 35J60, 53C9.
Citation: Wenxiong Chen, Congming Li. Harmonic maps on complete manifolds. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 799-804. doi: 10.3934/dcds.1999.5.799
[1]

Ze Li, Lifeng Zhao. Convergence to harmonic maps for the Landau-Lifshitz flows between two dimensional hyperbolic spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 607-638. doi: 10.3934/dcds.2019025
[2]

Junyu Lin. Uniqueness of harmonic map heat flows and liquid crystal flows. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 739-755. doi: 10.3934/dcds.2013.33.739
[3]

Y. Chen, S. Levine. The existence of the heat flow of H-systems. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 219-236. doi: 10.3934/dcds.2002.8.219
[4]

Andrey Shishkov, Laurent Véron. Propagation of singularities of nonlinear heat flow in fissured media. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1769-1782. doi: 10.3934/cpaa.2013.12.1769
[5]

Karl Kunisch, Markus Müller. Uniform convergence of the POD method and applications to optimal control. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4477-4501. doi: 10.3934/dcds.2015.35.4477
[6]

Zhong Tan, Qiuju Xu, Huaqiao Wang. Global existence and convergence rates for the compressible magnetohydrodynamic equations without heat conductivity. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 5083-5105. doi: 10.3934/dcds.2015.35.5083
[7]

Luis Caffarelli, Fanghua Lin. Nonlocal heat flows preserving the L2 energy. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 49-64. doi: 10.3934/dcds.2009.23.49
[8]

Eberhard Bänsch, Steffen Basting, Rolf Krahl. Numerical simulation of two-phase flows with heat and mass transfer. Discrete & Continuous Dynamical Systems, 2015, 35 (6) : 2325-2347. doi: 10.3934/dcds.2015.35.2325
[9]

Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327
[10]

Shouwen Fang, Peng Zhu. Differential Harnack estimates for backward heat equations with potentials under geometric flows. Communications on Pure & Applied Analysis, 2015, 14 (3) : 793-809. doi: 10.3934/cpaa.2015.14.793
[11]

J. R. L. Webb. Multiple positive solutions of some nonlinear heat flow problems. Conference Publications, 2005, 2005 (Special) : 895-903. doi: 10.3934/proc.2005.2005.895
[12]

Youcef Amirat, Kamel Hamdache. Strong solutions to the equations of flow and heat transfer in magnetic fluids with internal rotations. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3289-3320. doi: 10.3934/dcds.2013.33.3289
[13]

Mi-Ho Giga, Yoshikazu Giga, Takeshi Ohtsuka, Noriaki Umeda. On behavior of signs for the heat equation and a diffusion method for data separation. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2277-2296. doi: 10.3934/cpaa.2013.12.2277
[14]

Andrew M. Zimmer. Compact asymptotically harmonic manifolds. Journal of Modern Dynamics, 2012, 6 (3) : 377-403. doi: 10.3934/jmd.2012.6.377
[15]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268
[16]

Bernd Ammann, Robert Lauter and Victor Nistor. Algebras of pseudodifferential operators on complete manifolds. Electronic Research Announcements, 2003, 9: 80-87.

[17]

Yongfu Wang. Global strong solution to the two dimensional nonhomogeneous incompressible heat conducting Navier-Stokes flows with vacuum. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4317-4333. doi: 10.3934/dcdsb.2020099
[18]

Yong Hong Wu, B. Wiwatanapataphee. Modelling of turbulent flow and multi-phase heat transfer under electromagnetic force. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 695-706. doi: 10.3934/dcdsb.2007.8.695
[19]

Yasir Ali, Arshad Alam Khan. Exact solution of magnetohydrodynamic slip flow and heat transfer over an oscillating and translating porous plate. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 595-606. doi: 10.3934/dcdss.2018034
[20]

Yaguang Wang, Shiyong Zhu. Blowup of solutions to the thermal boundary layer problem in two-dimensional incompressible heat conducting flow. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3233-3244. doi: 10.3934/cpaa.2020141

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (71)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy