Uniqueness of harmonic map heat flows and liquid crystal flows

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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

Abstract
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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 $0< T ≤ +\infty.$

Keywords: liquid crystal flows, Heat flow, uniqueness, mild solution., harmonic maps.

Mathematics Subject Classification: 35Q35, 35K55, 76D03,76N1.

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

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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

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