# American Institute of Mathematical Sciences

January  2013, 12(1): 341-357. doi: 10.3934/cpaa.2013.12.341

## Global existence and stability for a hydrodynamic system in the nematic liquid crystal flows

 1 Institute of Applied Mathematics, College of Science, Northwest A\&F University, Yangling, Shaanxi 712100, China 2 Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China 3 Department of Mathematics, Sun Yat-Sen University, Guangzhou, Guangdong 510275

Received  June 2011 Revised  September 2011 Published  September 2012

In this paper we consider a coupled hydrodynamical system which involves the Navier-Stokes equations for the velocity field and kinematic transport equations for the molecular orientation field. By applying the Chemin-Lerner's time-space estimates for the heat equation and the Fourier localization technique, we prove that when initial data belongs to the critical Besov spaces with negative-order, there exists a unique local solution, and this solution is global when initial data is small enough. As a corollary, we obtain existence of global self-similar solution. In order to figure out the relation between the solution obtained here and weak solutions of standard sense, we establish a stability result, which yields in a direct way that all global weak solutions associated with the same initial data must coincide with the solution obtained here, namely, weak-strong uniqueness holds.
Citation: 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
##### References:
 [1] J.-Y. Chemin, "Perfect Incompressible Fluids,", Oxford Lecture Series in Mathematics and its Applications, 14 (1998). [2] R. Dachin, "Fourier Analysis Methods for PDE's,", 2005. Available from: http://perso-math.univ-mlv.fr/users/danchin.raphael/courschine.pdf., (). [3] J. L. Ericksen, Conservation laws for liquid crystals,, Trans. Soc. Rhe., 5 (1961), 23. doi: 10.1122/1.548883. [4] J. L. Ericksen, Continuum theory of nematic liquid crystals,, Res. Mechanica, 21 (1987), 381. [5] I. Gallagher and F. Planchon, On global infinite energy solutions to the Navier-Stokes equations in two dimensions,, Arch. Rational Mech. Anal., 161 (2002), 307. doi: 10.1007/s002050100175. [6] R. Hardt and D. Kinderlehrer, "Mathematical Questions of Liquid Crystal Theory,", The IMA Volumes in Mathematics andits Applications \textbf{5}, 5 (1987). [7] M. Hong, Global existence of solutions of the simplied Ericksen-Leslie system in dimension two,, Calc. Var., 40 (2011), 15. doi: 10.1007/s00526-010-0331-5. [8] X. Hu and D. Wang, Global solution to the three-dimensional incompressible flow of liquid crystals,, Commun. Math. Phys., 296 (2010), 861. doi: 10.1007/s00220-010-1017-8. [9] P.-G. Lemarié-Rieusset, "Recent Developments in the Navier-Stokes Problem,", Research Notes in Mathematics, (2002). [10] F. Leslie, Some constitutive equations for liquid crystals,, Arch. Rational Mech. Anal., 28 (1968), 265. doi: 10.1007/BF00251810. [11] F. Leslie, Theory of flow phenomenum in liquid crystals, In "The Theory of Liquid Crystals,", London-New York: Academic Press, 4 (1979), 1. [12] F. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena,, Comm. Pure Appl. Math., 42 (1989), 789. doi: 10.1002/cpa.3160420605. [13] F. Lin, J. Lin and C. Wang, Liquid crystal flows in two dimensions,, Arch. Rational Mech. Anal., 197 (2010), 297. doi: 10.1007/s00205-009-0278-x. [14] F. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, Comm. Pure Appl. Math., 48 (1995), 501. doi: 10.1002/cpa.3160480503. [15] F. Lin and C. Liu, Partial regularity of the dynamic system modeling the flow of liquid crystals,, Discrete. Contin. Dyn. Syst., 2 (1996), 1. doi: 10.3934/dcds.1996.2.1. [16] F. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals,, Chin. Ann. Math., 31B (2010), 921. doi: 10.1007/s11401-010-0612-5. [17] T. Runst and W. Sickel, "Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations,", de Gruyter Series in Nonlinear Analysis and Applications, 3 (1996). [18] H. Sun and C. Liu, On energetic variational approaches in modeling the nematic liquid crystal flows,, Discrete Contin. Dyn. Syst., 23 (2009), 455. doi: 10.3934/dcds.2009.23.455. [19] 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. doi: 10.1007/s00205-010-0343-5. [20] H. Wu, Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows,, Discrete Contin. Dyn. Syst., 26 (2010), 379. doi: 10.3934/dcds.2010.26.379. [21] H. Wu, X. Xu and C. Liu, Asymptotic behavior for a Nematic liquid crystal model with different kinematic transport properties,, DOI 10.1007/s00526-011-0460-5., (): 00526. doi: 10.1007/s00526-011-0460-5.

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##### References:
 [1] J.-Y. Chemin, "Perfect Incompressible Fluids,", Oxford Lecture Series in Mathematics and its Applications, 14 (1998). [2] R. Dachin, "Fourier Analysis Methods for PDE's,", 2005. Available from: http://perso-math.univ-mlv.fr/users/danchin.raphael/courschine.pdf., (). [3] J. L. Ericksen, Conservation laws for liquid crystals,, Trans. Soc. Rhe., 5 (1961), 23. doi: 10.1122/1.548883. [4] J. L. Ericksen, Continuum theory of nematic liquid crystals,, Res. Mechanica, 21 (1987), 381. [5] I. Gallagher and F. Planchon, On global infinite energy solutions to the Navier-Stokes equations in two dimensions,, Arch. Rational Mech. Anal., 161 (2002), 307. doi: 10.1007/s002050100175. [6] R. Hardt and D. Kinderlehrer, "Mathematical Questions of Liquid Crystal Theory,", The IMA Volumes in Mathematics andits Applications \textbf{5}, 5 (1987). [7] M. Hong, Global existence of solutions of the simplied Ericksen-Leslie system in dimension two,, Calc. Var., 40 (2011), 15. doi: 10.1007/s00526-010-0331-5. [8] X. Hu and D. Wang, Global solution to the three-dimensional incompressible flow of liquid crystals,, Commun. Math. Phys., 296 (2010), 861. doi: 10.1007/s00220-010-1017-8. [9] P.-G. Lemarié-Rieusset, "Recent Developments in the Navier-Stokes Problem,", Research Notes in Mathematics, (2002). [10] F. Leslie, Some constitutive equations for liquid crystals,, Arch. Rational Mech. Anal., 28 (1968), 265. doi: 10.1007/BF00251810. [11] F. Leslie, Theory of flow phenomenum in liquid crystals, In "The Theory of Liquid Crystals,", London-New York: Academic Press, 4 (1979), 1. [12] F. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena,, Comm. Pure Appl. Math., 42 (1989), 789. doi: 10.1002/cpa.3160420605. [13] F. Lin, J. Lin and C. Wang, Liquid crystal flows in two dimensions,, Arch. Rational Mech. Anal., 197 (2010), 297. doi: 10.1007/s00205-009-0278-x. [14] F. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, Comm. Pure Appl. Math., 48 (1995), 501. doi: 10.1002/cpa.3160480503. [15] F. Lin and C. Liu, Partial regularity of the dynamic system modeling the flow of liquid crystals,, Discrete. Contin. Dyn. Syst., 2 (1996), 1. doi: 10.3934/dcds.1996.2.1. [16] F. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals,, Chin. Ann. Math., 31B (2010), 921. doi: 10.1007/s11401-010-0612-5. [17] T. Runst and W. Sickel, "Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations,", de Gruyter Series in Nonlinear Analysis and Applications, 3 (1996). [18] H. Sun and C. Liu, On energetic variational approaches in modeling the nematic liquid crystal flows,, Discrete Contin. Dyn. Syst., 23 (2009), 455. doi: 10.3934/dcds.2009.23.455. [19] 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. doi: 10.1007/s00205-010-0343-5. [20] H. Wu, Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows,, Discrete Contin. Dyn. Syst., 26 (2010), 379. doi: 10.3934/dcds.2010.26.379. [21] H. Wu, X. Xu and C. Liu, Asymptotic behavior for a Nematic liquid crystal model with different kinematic transport properties,, DOI 10.1007/s00526-011-0460-5., (): 00526. doi: 10.1007/s00526-011-0460-5.
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