February  2013, 33(2): 739-755. doi: 10.3934/dcds.2013.33.739

Uniqueness of harmonic map heat flows and liquid crystal flows

1. 

Department of Mathematics, South China University of Technology, Guangzhou, 510640

Received  July 2011 Revised  January 2012 Published  September 2012

In this paper, we prove a limiting uniqueness criterion to harmonic map heat flows and liquid crystal flows. We firstly establish the uniqueness of harmonic map heat flows from $R^n$ to a smooth, compact Riemannian manifold $N$ in the class $C([0,T),BMO_T(R^n,N))\cap L^\infty_{loc}((0,T);\dot{W}^{1,\infty}(R^n))$ for $0< T ≤ +\infty.$ For the nematic liquid crystal flows $(v,d)$, we show that the mild solution is unique under the class $C([0,T),BMO_T^{-1}(R^n))\cap L^\infty_{loc}((0,T);L^\infty(R^n))\times C([0,T),BMO_T(R^n,S^2))\cap L^\infty_{loc}((0,T);\dot{W}^{1,\infty}(R^n))$ for $0< T ≤ +\infty.$
Citation: Junyu Lin. Uniqueness of harmonic map heat flows and liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 739-755. doi: 10.3934/dcds.2013.33.739
References:
[1]

Y. Chen and W. Y. Ding, Blow up and global existence for heat flows of harmonic maps,, Invent. Math., 99 (1990), 567.  doi: 10.1007/BF01234431.  Google Scholar

[2]

K. Chang, W. Ding and R. Ye, Finite time blow-up of the heat flow of harmonic maps from surfaces,, JDG, 36 (1992), 507.   Google Scholar

[3]

J. M. Coron and J. M. Ghidaglia, Explosion en temps fini pour le flot des applications harmoniques,, C. R. Acad. Sci. Paris, 308 (1989), 339.   Google Scholar

[4]

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

[5]

J. L. Ericksen, Hydrostatic theory of liquid crystal,, Arch. Rational Mech. Anal., 9 (1962), 371.   Google Scholar

[6]

J. Eells and J. Sampson, Harmonic mappings of Riemannian manifolds,, Amer. J. Math., 86 (1964), 109.  doi: 10.2307/2373037.  Google Scholar

[7]

A. Freire, Uniqueness for the harmonic map flow from surfaces to general targets,, Comm. Math. Helvetici., 70 (1995), 310.  doi: 10.1007/BF02566010.  Google Scholar

[8]

P. G. de Gennes and J. Prost, "The Physics of Liquid Crystals,", New York, (1993).   Google Scholar

[9]

J. Jost, Ein existenzbeiweis fiir harmonisch Abbildungen, die ein Dirichlet problem 16sen mittels der methode des warmeflusses,, Manuscripta Math., 34 (1981), 17.  doi: 10.1007/BF01168706.  Google Scholar

[10]

H. Koch and D. Tataru, Well-posedness for theNavier-Stokes equations,, Adv. Math., 157 (2001), 22.  doi: 10.1006/aima.2000.1937.  Google Scholar

[11]

F. M. Leslie, Some constitutive equations for liquid crystals,, Arch. Rational Mech. Anal., 28 (1968), 265.  doi: 10.1007/BF00251810.  Google Scholar

[12]

J. Y. Lin and S. J. Ding, On the well-posedness for the heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals in critical spaces,, Math. Meth. Appl. Sciences, ().  doi: 10.1002/mma.1548.  Google Scholar

[13]

F. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, Comm. Pure Appl. Math., 48 (1995), 501.   Google Scholar

[14]

F. Lin and C. Liu, Partial regularities of nonlinear disspative systems modeling the flow of liquid crystals,, DCDS, 2 (1996), 1.   Google Scholar

[15]

F. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system,, Arch. Rational Mech. Aanl., 154 (2000), 135.   Google Scholar

[16]

F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flow in two dimensions,, Arch. Ration. Mech. Anal., 197 (2010), 297.  doi: 10.1007/s00205-009-0278-x.  Google Scholar

[17]

F. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals,, Chinese Annals of Mathematics, 31 (2010), 921.  doi: 10.1007/s11401-010-0612-5.  Google Scholar

[18]

H. Miura, Remark on uniqueness of mild solutions to the Navier-Stokes equations,, J. Funt. Anal., 218 (2005), 110.  doi: 10.1016/j.jfa.2004.07.007.  Google Scholar

[19]

M. Struwe, On the evolution of harmonic maps in higher dimensions,, J. Diff. Geom., 28 (1988), 485.   Google Scholar

[20]

A. Soyeur, A global existence result for the heat flow of harmonic maps,, Comm. PDE, 15 (1990), 237.  doi: 10.1080/03605309908820685.  Google Scholar

[21]

M. Struwe, On the evolution of harmonic maps of Riemannian surfaces,, Comment. Math. Helv, 60 (1985), 558.  doi: 10.1007/BF02567432.  Google Scholar

[22]

C. Wang, Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data,, Arch. Rational Mech. Anal., 200 (2011), 1.   Google Scholar

[23]

X. Xu and Z. Zhang, Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows,, JDE, 252 (2012), 1169.  doi: 10.1016/j.jde.2011.08.028.  Google Scholar

show all references

References:
[1]

Y. Chen and W. Y. Ding, Blow up and global existence for heat flows of harmonic maps,, Invent. Math., 99 (1990), 567.  doi: 10.1007/BF01234431.  Google Scholar

[2]

K. Chang, W. Ding and R. Ye, Finite time blow-up of the heat flow of harmonic maps from surfaces,, JDG, 36 (1992), 507.   Google Scholar

[3]

J. M. Coron and J. M. Ghidaglia, Explosion en temps fini pour le flot des applications harmoniques,, C. R. Acad. Sci. Paris, 308 (1989), 339.   Google Scholar

[4]

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

[5]

J. L. Ericksen, Hydrostatic theory of liquid crystal,, Arch. Rational Mech. Anal., 9 (1962), 371.   Google Scholar

[6]

J. Eells and J. Sampson, Harmonic mappings of Riemannian manifolds,, Amer. J. Math., 86 (1964), 109.  doi: 10.2307/2373037.  Google Scholar

[7]

A. Freire, Uniqueness for the harmonic map flow from surfaces to general targets,, Comm. Math. Helvetici., 70 (1995), 310.  doi: 10.1007/BF02566010.  Google Scholar

[8]

P. G. de Gennes and J. Prost, "The Physics of Liquid Crystals,", New York, (1993).   Google Scholar

[9]

J. Jost, Ein existenzbeiweis fiir harmonisch Abbildungen, die ein Dirichlet problem 16sen mittels der methode des warmeflusses,, Manuscripta Math., 34 (1981), 17.  doi: 10.1007/BF01168706.  Google Scholar

[10]

H. Koch and D. Tataru, Well-posedness for theNavier-Stokes equations,, Adv. Math., 157 (2001), 22.  doi: 10.1006/aima.2000.1937.  Google Scholar

[11]

F. M. Leslie, Some constitutive equations for liquid crystals,, Arch. Rational Mech. Anal., 28 (1968), 265.  doi: 10.1007/BF00251810.  Google Scholar

[12]

J. Y. Lin and S. J. Ding, On the well-posedness for the heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals in critical spaces,, Math. Meth. Appl. Sciences, ().  doi: 10.1002/mma.1548.  Google Scholar

[13]

F. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, Comm. Pure Appl. Math., 48 (1995), 501.   Google Scholar

[14]

F. Lin and C. Liu, Partial regularities of nonlinear disspative systems modeling the flow of liquid crystals,, DCDS, 2 (1996), 1.   Google Scholar

[15]

F. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system,, Arch. Rational Mech. Aanl., 154 (2000), 135.   Google Scholar

[16]

F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flow in two dimensions,, Arch. Ration. Mech. Anal., 197 (2010), 297.  doi: 10.1007/s00205-009-0278-x.  Google Scholar

[17]

F. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals,, Chinese Annals of Mathematics, 31 (2010), 921.  doi: 10.1007/s11401-010-0612-5.  Google Scholar

[18]

H. Miura, Remark on uniqueness of mild solutions to the Navier-Stokes equations,, J. Funt. Anal., 218 (2005), 110.  doi: 10.1016/j.jfa.2004.07.007.  Google Scholar

[19]

M. Struwe, On the evolution of harmonic maps in higher dimensions,, J. Diff. Geom., 28 (1988), 485.   Google Scholar

[20]

A. Soyeur, A global existence result for the heat flow of harmonic maps,, Comm. PDE, 15 (1990), 237.  doi: 10.1080/03605309908820685.  Google Scholar

[21]

M. Struwe, On the evolution of harmonic maps of Riemannian surfaces,, Comment. Math. Helv, 60 (1985), 558.  doi: 10.1007/BF02567432.  Google Scholar

[22]

C. Wang, Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data,, Arch. Rational Mech. Anal., 200 (2011), 1.   Google Scholar

[23]

X. Xu and Z. Zhang, Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows,, JDE, 252 (2012), 1169.  doi: 10.1016/j.jde.2011.08.028.  Google Scholar

[1]

T. Tachim Medjo. On the existence and uniqueness of solution to a stochastic simplified liquid crystal model. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2243-2264. doi: 10.3934/cpaa.2019101

[2]

Domenico Mucci. Maps into projective spaces: Liquid crystal and conformal energies. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 597-635. doi: 10.3934/dcdsb.2012.17.597

[3]

Zhenlu Cui, M. Carme Calderer, Qi Wang. Mesoscale structures in flows of weakly sheared cholesteric liquid crystal polymers. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 291-310. doi: 10.3934/dcdsb.2006.6.291

[4]

Jihong Zhao, Qiao Liu, Shangbin Cui. Global existence and stability for a hydrodynamic system in the nematic liquid crystal flows. Communications on Pure & Applied Analysis, 2013, 12 (1) : 341-357. doi: 10.3934/cpaa.2013.12.341

[5]

Shanshan Guo, Zhong Tan. Energy dissipation for weak solutions of incompressible liquid crystal flows. Kinetic & Related Models, 2015, 8 (4) : 691-706. doi: 10.3934/krm.2015.8.691

[6]

Chun Liu, Huan Sun. On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 455-475. doi: 10.3934/dcds.2009.23.455

[7]

Chun Liu, Jie Shen. On liquid crystal flows with free-slip boundary conditions. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 307-318. doi: 10.3934/dcds.2001.7.307

[8]

Xiaoli Li, Boling Guo. Well-posedness for the three-dimensional compressible liquid crystal flows. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1913-1937. doi: 10.3934/dcdss.2016078

[9]

Zhenlu Cui, Qi Wang. Permeation flows in cholesteric liquid crystal polymers under oscillatory shear. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 45-60. doi: 10.3934/dcdsb.2011.15.45

[10]

Qiang Tao, Ying Yang. Exponential stability for the compressible nematic liquid crystal flow with large initial data. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1661-1669. doi: 10.3934/cpaa.2016007

[11]

Sili Liu, Xinhua Zhao, Yingshan Chen. A new blowup criterion for strong solutions of the compressible nematic liquid crystal flow. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4515-4533. doi: 10.3934/dcdsb.2020110

[12]

Francisco Guillén-González, Mouhamadou Samsidy Goudiaby. Stability and convergence at infinite time of several fully discrete schemes for a Ginzburg-Landau model for nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4229-4246. doi: 10.3934/dcds.2012.32.4229

[13]

Jishan Fan, Fei Jiang. Large-time behavior of liquid crystal flows with a trigonometric condition in two dimensions. Communications on Pure & Applied Analysis, 2016, 15 (1) : 73-90. doi: 10.3934/cpaa.2016.15.73

[14]

Yang Liu, Sining Zheng, Huapeng Li, Shengquan Liu. Strong solutions to Cauchy problem of 2D compressible nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3921-3938. doi: 10.3934/dcds.2017165

[15]

Yinxia Wang. A remark on blow up criterion of three-dimensional nematic liquid crystal flows. Evolution Equations & Control Theory, 2016, 5 (2) : 337-348. doi: 10.3934/eect.2016007

[16]

Shengquan Liu, Jianwen Zhang. Global well-posedness for the two-dimensional equations of nonhomogeneous incompressible liquid crystal flows with nonnegative density. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2631-2648. doi: 10.3934/dcdsb.2016065

[17]

Hao Wu. Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 379-396. doi: 10.3934/dcds.2010.26.379

[18]

Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020161

[19]

Yang Liu, Xin Zhong. On the Cauchy problem of 3D nonhomogeneous incompressible nematic liquid crystal flows with vacuum. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5219-5238. doi: 10.3934/cpaa.2020234

[20]

Blanca Climent-Ezquerra, Francisco Guillén-González. Global in time solution and time-periodicity for a smectic-A liquid crystal model. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1473-1493. doi: 10.3934/cpaa.2010.9.1473

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (36)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]