# American Institute of Mathematical Sciences

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. 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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##### 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. 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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