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 and Continuous Dynamical Systems, 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), 567-578. doi: 10.1007/BF01234431.

[2]

K. Chang, W. Ding and R. Ye, Finite time blow-up of the heat flow of harmonic maps from surfaces, JDG, 36 (1992), 507-515.

[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), 339-344.

[4]

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

[5]

J. L. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378.

[6]

J. Eells and J. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160. doi: 10.2307/2373037.

[7]

A. Freire, Uniqueness for the harmonic map flow from surfaces to general targets, Comm. Math. Helvetici., 70 (1995), 310-338. doi: 10.1007/BF02566010.

[8]

P. G. de Gennes and J. Prost, "The Physics of Liquid Crystals," New York, Oxford University Press, 1993.

[9]

J. Jost, Ein existenzbeiweis fiir harmonisch Abbildungen, die ein Dirichlet problem 16sen mittels der methode des warmeflusses, Manuscripta Math., 34 (1981), 17-25. doi: 10.1007/BF01168706.

[10]

H. Koch and D. Tataru, Well-posedness for theNavier-Stokes equations, Adv. Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937.

[11]

F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1968), 265-283. doi: 10.1007/BF00251810.

[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.

[13]

F. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537.

[14]

F. Lin and C. Liu, Partial regularities of nonlinear disspative systems modeling the flow of liquid crystals, DCDS, 2 (1996), 1-23.

[15]

F. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system, Arch. Rational Mech. Aanl., 154 (2000), 135-156.

[16]

F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flow in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x.

[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), 921-938. doi: 10.1007/s11401-010-0612-5.

[18]

H. Miura, Remark on uniqueness of mild solutions to the Navier-Stokes equations, J. Funt. Anal., 218 (2005), 110-129. doi: 10.1016/j.jfa.2004.07.007.

[19]

M. Struwe, On the evolution of harmonic maps in higher dimensions, J. Diff. Geom., 28 (1988), 485-502.

[20]

A. Soyeur, A global existence result for the heat flow of harmonic maps, Comm. PDE, 15 (1990), 237-244. doi: 10.1080/03605309908820685.

[21]

M. Struwe, On the evolution of harmonic maps of Riemannian surfaces, Comment. Math. Helv, 60 (1985), 558-581. doi: 10.1007/BF02567432.

[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), 1-19.

[23]

X. Xu and Z. Zhang, Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows, JDE, 252 (2012), 1169-1181. doi: 10.1016/j.jde.2011.08.028.

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), 567-578. doi: 10.1007/BF01234431.

[2]

K. Chang, W. Ding and R. Ye, Finite time blow-up of the heat flow of harmonic maps from surfaces, JDG, 36 (1992), 507-515.

[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), 339-344.

[4]

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

[5]

J. L. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378.

[6]

J. Eells and J. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160. doi: 10.2307/2373037.

[7]

A. Freire, Uniqueness for the harmonic map flow from surfaces to general targets, Comm. Math. Helvetici., 70 (1995), 310-338. doi: 10.1007/BF02566010.

[8]

P. G. de Gennes and J. Prost, "The Physics of Liquid Crystals," New York, Oxford University Press, 1993.

[9]

J. Jost, Ein existenzbeiweis fiir harmonisch Abbildungen, die ein Dirichlet problem 16sen mittels der methode des warmeflusses, Manuscripta Math., 34 (1981), 17-25. doi: 10.1007/BF01168706.

[10]

H. Koch and D. Tataru, Well-posedness for theNavier-Stokes equations, Adv. Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937.

[11]

F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1968), 265-283. doi: 10.1007/BF00251810.

[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.

[13]

F. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537.

[14]

F. Lin and C. Liu, Partial regularities of nonlinear disspative systems modeling the flow of liquid crystals, DCDS, 2 (1996), 1-23.

[15]

F. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system, Arch. Rational Mech. Aanl., 154 (2000), 135-156.

[16]

F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flow in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x.

[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), 921-938. doi: 10.1007/s11401-010-0612-5.

[18]

H. Miura, Remark on uniqueness of mild solutions to the Navier-Stokes equations, J. Funt. Anal., 218 (2005), 110-129. doi: 10.1016/j.jfa.2004.07.007.

[19]

M. Struwe, On the evolution of harmonic maps in higher dimensions, J. Diff. Geom., 28 (1988), 485-502.

[20]

A. Soyeur, A global existence result for the heat flow of harmonic maps, Comm. PDE, 15 (1990), 237-244. doi: 10.1080/03605309908820685.

[21]

M. Struwe, On the evolution of harmonic maps of Riemannian surfaces, Comment. Math. Helv, 60 (1985), 558-581. doi: 10.1007/BF02567432.

[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), 1-19.

[23]

X. Xu and Z. Zhang, Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows, JDE, 252 (2012), 1169-1181. doi: 10.1016/j.jde.2011.08.028.

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