March  2021, 26(3): 1243-1272. doi: 10.3934/dcdsb.2020161

Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows

1. 

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China

2. 

Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, China

3. 

The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Hong Kong

Received  February 2018 Revised  March 2020 Published  May 2020

In this paper, we consider the initial-boundary value problem to the non-isothermal incompressible liquid crystal system with both variable density and temperature. Global well-posedness of strong solutions is established for initial data being small perturbation around the equilibrium state. As the tools in the proof, we establish the maximal regularities of the linear Stokes equations and parabolic equations with variable coefficients and a rigid lemma for harmonic maps on bounded domains. This paper also generalizes the result in [5] to the inhomogeneous case.

Citation: Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161
References:
[1]

H. Abels, Nonstationary Stokes system with variable viscosity in bounded and unbounded domains, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 141-157.  doi: 10.3934/dcdss.2010.3.141.  Google Scholar

[2]

H. Abels and Y. Terasawa, On Stokes operators with variable viscosity in bounded and unbounded domains, Math. Ann., 344 (2009), 381-429.  doi: 10.1007/s00208-008-0311-7.  Google Scholar

[3]

R. A. Adams and J. F. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[4]

H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[5]

D. Bian and Y. Xiao, Global solution to the nematic liquid crystal flows with heat effect, J. Differential Equations, 263 (2017), 5298-5329.  doi: 10.1016/j.jde.2017.06.019.  Google Scholar

[6]

D. Bothe and J. Prüss, $L_p$-theory for a class of Non-Newtonian fluids, SIAM J. Math. Anal., 39 (2007), 379-421.  doi: 10.1137/060663635.  Google Scholar

[7]

R. Danchin, Density-dependent incompressible fluids in bounded domains, J. Math. Fluid Mech., 8 (2006), 333-381.  doi: 10.1007/s00021-004-0147-1.  Google Scholar

[8]

R. Denk, M. Hieber and J. Prüss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003). doi: 10.1090/memo/0788.  Google Scholar

[9]

W.-Y. Ding and F.-H. Lin, A generalization of Eells-Sampson's theorem, J. Partial Differential Equations, 5 (1992), 13-22.   Google Scholar

[10]

S. J. DingC. Y. Wang and H. Y. Wen, Weak solutions to compressible flows of nematic liquid crystals in dimension one, Discrete Contin. Dyn. Syst. B, 15 (2011), 357-371.  doi: 10.3934/dcdsb.2011.15.357.  Google Scholar

[11]

J. L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheology, 5 (1961), 23-34.  doi: 10.1122/1.548883.  Google Scholar

[12]

J. L. Ericksen, Continuum theory of nematic liquid crystals, Molecular Crystals, 7 (1969), 153-164.  doi: 10.1080/15421406908084869.  Google Scholar

[13]

E. FeireislM. FrémondE. Rocca and G. Schimperna, A new approach to non-isothermal models for nematic liquid crystals, Arch. Rational Mech. Anal., 205 (2012), 651-672.  doi: 10.1007/s00205-012-0517-4.  Google Scholar

[14]

E. FeireislE. Rocca and G. Schimperna, On a non-isothermal model for the nematic liquid crystals, Nonlinearity, 24 (2011), 243-257.  doi: 10.1088/0951-7715/24/1/012.  Google Scholar

[15]

G. P. Galdi, An Introduction to the Mathematical Theory of the Nevier-Stokes Equations. Vol. I. Linearized Steady Problems, Springer Tracts in Natural Philosophy, 38. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar

[16]

J. C. GaoQ. Tao and Z.-A. Yao, Long-time behavior of solution for the compressible nematic liquid crystal flows in $\mathbb{R}^3$, J. Differential Equations, 261 (2016), 2334-2383.  doi: 10.1016/j.jde.2016.04.033.  Google Scholar

[17]

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

[18]

B. L. GuoX. Y. Xi and B. Q. Xie, Global well-posedness and decay of smooth solutions to the non-isothermal model for compressible nematic liquid crystals, J. Differential Equations, 262 (2017), 1413-1460.  doi: 10.1016/j.jde.2016.10.015.  Google Scholar

[19]

M. HieberM. NesensohnJ. Prüss and K. Schade, Dynamics of nematic liquid crystal flows: The quasilinear approach, Ann. Ints. H. Poincaré, Analyse Nonlinéaire, 33 (2016), 397-408.  doi: 10.1016/j.anihpc.2014.11.001.  Google Scholar

[20]

M. Hieber and J. Prüss, Dynamics of the Ericksen-Leslie equations with general Leslie stress. I: The incompressible isotropic case, Math. Ann., 369 (2017), 977-996.  doi: 10.1007/s00208-016-1453-7.  Google Scholar

[21]

M. Hieber and J. W. Prüss, Modeling and analysis of the Ericksen-Leslie equations for nematic liquid crystal flows, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, Cham, (2016), 1075–1134.  Google Scholar

[22]

M.-C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc. Var. Partial Differential Equations, 40 (2011), 15-36.  doi: 10.1007/s00526-010-0331-5.  Google Scholar

[23]

M.-C. HongJ. K. Li and Z. P. Xin, Blow-up criteria of strong solutions to the Ericksen-Leslie system in $\mathbb{R}^3$, Comm. Partial Differential Equations, 39 (2014), 1284-1328.  doi: 10.1080/03605302.2013.871026.  Google Scholar

[24]

M.-C. Hong and Z. P. Xin, Global existence of solutions of the nematic liquid crystal flow for the Oseen-Frank model $\mathbb{R}^2$, Adv. Math., 231 (2012), 1364-1400.  doi: 10.1016/j.aim.2012.06.009.  Google Scholar

[25]

X. P. Hu and D. H. Wang, Global solution to the three-dimensional incompressible flow of liquid crystals, Commun. Math. Phys., 296 (2010), 861-880.  doi: 10.1007/s00220-010-1017-8.  Google Scholar

[26]

X. P. Hu and H. Wu, Global solution to the three-dimensional compressible flow of liquid crytals, SIAM J. Math. Anal., 45 (2013), 2678-2699.  doi: 10.1137/120898814.  Google Scholar

[27]

J. R. HuangF.-H. Lin and C. Y. Wang, Regularity and existence of global solutions to the Ericksen-Leslie system in $\mathbb{R}^2$, Commun. Math. Phys., 331 (2014), 805-850.  doi: 10.1007/s00220-014-2079-9.  Google Scholar

[28]

F. JiangS. Jiang and D. H. Wang, On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain, J. Funct. Anal., 265 (2013), 3369-3397.  doi: 10.1016/j.jfa.2013.07.026.  Google Scholar

[29]

Z. LeiD. Li and X. Y. Zhang, Remarks of global wellposedness of liquid crystal flows and heat flows of harmonic maps in tow dimensions, Proc. Amer. Math. Soc., 142 (2014), 3801-3810.  doi: 10.1090/S0002-9939-2014-12057-0.  Google Scholar

[30]

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

[31]

F. Leslie, Theory of Flow Phenomenum in Liquid Crystals, The Theory of Liquid Crystals, 4. London-New York, Academic Press, 1979, 1–81. Google Scholar

[32]

J. K. Li, Global strong and weak solutions to inhomogeneous nematic liquid crystal flows in two dimensions, Nonlinear Anal., 99 (2014), 80-94.  doi: 10.1016/j.na.2013.12.023.  Google Scholar

[33]

J. K. LiE. S. Titi and Z. P. Xin, On the uniqueness of weak solutions to the Ericksen-Leslie liquid crystal model in $\mathbb{R}^2$, Math. Models Meth. Appl. Sci., 26 (2016), 803-822.  doi: 10.1142/S0218202516500184.  Google Scholar

[34]

J. K. Li and Z. P. Xin, Global existence of weak solutions to the non-isothermal nematic liquid crystals in 2D, Acta Math. Sci. Ser. B Engl. Ed., 36 (2016), 973-1014.  doi: 10.1016/S0252-9602(16)30054-6.  Google Scholar

[35]

F.-H. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814.  doi: 10.1002/cpa.3160420605.  Google Scholar

[36]

F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure. Appl. Math., 48 (1995), 501-537.  doi: 10.1002/cpa.3160480503.  Google Scholar

[37]

F.-H. Lin and C. Liu, Partial regularity of the nonlinear dissipative system modeling the flow of nematic liquid crystals, Discrete Comtin. Dyn. Syst., 2 (1996), 1-22.  doi: 10.3934/dcds.1996.2.1.  Google Scholar

[38]

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

[39]

F.-H. Lin and C. Y. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chin. Ann. Math. Ser. B, 31 (2010), 921-938.  doi: 10.1007/s11401-010-0612-5.  Google Scholar

[40]

F. H. Lin and C. Y. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimension three, Comm. Pure Appl. Math., 69 (2016), 1532-1571.  doi: 10.1002/cpa.21583.  Google Scholar

[41]

Q. LiuS. Q. LiuW. K. Tan and X. Zhong, Global well-posedness of the 2D nonhomogeneous incompressible nematic liquid crystal flows, J. Differential Equations, 261 (2016), 6521-6569.  doi: 10.1016/j.jde.2016.08.044.  Google Scholar

[42] P. Oswald and P. Pieranski, Nematic and Cholesteric Liquid Crystals: Concepts and Physical Properties Illustrated by Experiments, CRC Press, 2005.  doi: 10.1201/9780203023013.  Google Scholar
[43]

V. A. Solonnikov, $L_p$-estimates for solutions to the initial boundary-value problem for the generalized Stokes system in a bounded domain, J. Math. Sci. (New York), 105 (2001), 2448-2484.  doi: 10.1023/A:1011321430954.  Google Scholar

[44]

A. M. Sonnet and E. G. Virga, Dissipative Ordered Fluids: Theories for Liquid Crystals, Springer, New York, 2012. doi: 10.1007/978-0-387-87815-7.  Google Scholar

[45]

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

[46]

C. Y. 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.  doi: 10.1007/s00205-010-0343-5.  Google Scholar

[47]

M. Wang and W. D. Wang, Global existence of weak solution for the 2-D Ericksen-Leslie system, Calc. Var. Partial Differential Equations, 51 (2014), 915-962.  doi: 10.1007/s00526-013-0700-y.  Google Scholar

[48]

X. Xu and Z. F. Zhang, Global regularity and uniqueness of weak solutions for the 2-D liquid crystal flows, J. Differential Equations, 252 (2012), 1169-1181.  doi: 10.1016/j.jde.2011.08.028.  Google Scholar

show all references

References:
[1]

H. Abels, Nonstationary Stokes system with variable viscosity in bounded and unbounded domains, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 141-157.  doi: 10.3934/dcdss.2010.3.141.  Google Scholar

[2]

H. Abels and Y. Terasawa, On Stokes operators with variable viscosity in bounded and unbounded domains, Math. Ann., 344 (2009), 381-429.  doi: 10.1007/s00208-008-0311-7.  Google Scholar

[3]

R. A. Adams and J. F. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[4]

H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[5]

D. Bian and Y. Xiao, Global solution to the nematic liquid crystal flows with heat effect, J. Differential Equations, 263 (2017), 5298-5329.  doi: 10.1016/j.jde.2017.06.019.  Google Scholar

[6]

D. Bothe and J. Prüss, $L_p$-theory for a class of Non-Newtonian fluids, SIAM J. Math. Anal., 39 (2007), 379-421.  doi: 10.1137/060663635.  Google Scholar

[7]

R. Danchin, Density-dependent incompressible fluids in bounded domains, J. Math. Fluid Mech., 8 (2006), 333-381.  doi: 10.1007/s00021-004-0147-1.  Google Scholar

[8]

R. Denk, M. Hieber and J. Prüss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003). doi: 10.1090/memo/0788.  Google Scholar

[9]

W.-Y. Ding and F.-H. Lin, A generalization of Eells-Sampson's theorem, J. Partial Differential Equations, 5 (1992), 13-22.   Google Scholar

[10]

S. J. DingC. Y. Wang and H. Y. Wen, Weak solutions to compressible flows of nematic liquid crystals in dimension one, Discrete Contin. Dyn. Syst. B, 15 (2011), 357-371.  doi: 10.3934/dcdsb.2011.15.357.  Google Scholar

[11]

J. L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheology, 5 (1961), 23-34.  doi: 10.1122/1.548883.  Google Scholar

[12]

J. L. Ericksen, Continuum theory of nematic liquid crystals, Molecular Crystals, 7 (1969), 153-164.  doi: 10.1080/15421406908084869.  Google Scholar

[13]

E. FeireislM. FrémondE. Rocca and G. Schimperna, A new approach to non-isothermal models for nematic liquid crystals, Arch. Rational Mech. Anal., 205 (2012), 651-672.  doi: 10.1007/s00205-012-0517-4.  Google Scholar

[14]

E. FeireislE. Rocca and G. Schimperna, On a non-isothermal model for the nematic liquid crystals, Nonlinearity, 24 (2011), 243-257.  doi: 10.1088/0951-7715/24/1/012.  Google Scholar

[15]

G. P. Galdi, An Introduction to the Mathematical Theory of the Nevier-Stokes Equations. Vol. I. Linearized Steady Problems, Springer Tracts in Natural Philosophy, 38. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar

[16]

J. C. GaoQ. Tao and Z.-A. Yao, Long-time behavior of solution for the compressible nematic liquid crystal flows in $\mathbb{R}^3$, J. Differential Equations, 261 (2016), 2334-2383.  doi: 10.1016/j.jde.2016.04.033.  Google Scholar

[17]

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

[18]

B. L. GuoX. Y. Xi and B. Q. Xie, Global well-posedness and decay of smooth solutions to the non-isothermal model for compressible nematic liquid crystals, J. Differential Equations, 262 (2017), 1413-1460.  doi: 10.1016/j.jde.2016.10.015.  Google Scholar

[19]

M. HieberM. NesensohnJ. Prüss and K. Schade, Dynamics of nematic liquid crystal flows: The quasilinear approach, Ann. Ints. H. Poincaré, Analyse Nonlinéaire, 33 (2016), 397-408.  doi: 10.1016/j.anihpc.2014.11.001.  Google Scholar

[20]

M. Hieber and J. Prüss, Dynamics of the Ericksen-Leslie equations with general Leslie stress. I: The incompressible isotropic case, Math. Ann., 369 (2017), 977-996.  doi: 10.1007/s00208-016-1453-7.  Google Scholar

[21]

M. Hieber and J. W. Prüss, Modeling and analysis of the Ericksen-Leslie equations for nematic liquid crystal flows, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, Cham, (2016), 1075–1134.  Google Scholar

[22]

M.-C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc. Var. Partial Differential Equations, 40 (2011), 15-36.  doi: 10.1007/s00526-010-0331-5.  Google Scholar

[23]

M.-C. HongJ. K. Li and Z. P. Xin, Blow-up criteria of strong solutions to the Ericksen-Leslie system in $\mathbb{R}^3$, Comm. Partial Differential Equations, 39 (2014), 1284-1328.  doi: 10.1080/03605302.2013.871026.  Google Scholar

[24]

M.-C. Hong and Z. P. Xin, Global existence of solutions of the nematic liquid crystal flow for the Oseen-Frank model $\mathbb{R}^2$, Adv. Math., 231 (2012), 1364-1400.  doi: 10.1016/j.aim.2012.06.009.  Google Scholar

[25]

X. P. Hu and D. H. Wang, Global solution to the three-dimensional incompressible flow of liquid crystals, Commun. Math. Phys., 296 (2010), 861-880.  doi: 10.1007/s00220-010-1017-8.  Google Scholar

[26]

X. P. Hu and H. Wu, Global solution to the three-dimensional compressible flow of liquid crytals, SIAM J. Math. Anal., 45 (2013), 2678-2699.  doi: 10.1137/120898814.  Google Scholar

[27]

J. R. HuangF.-H. Lin and C. Y. Wang, Regularity and existence of global solutions to the Ericksen-Leslie system in $\mathbb{R}^2$, Commun. Math. Phys., 331 (2014), 805-850.  doi: 10.1007/s00220-014-2079-9.  Google Scholar

[28]

F. JiangS. Jiang and D. H. Wang, On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain, J. Funct. Anal., 265 (2013), 3369-3397.  doi: 10.1016/j.jfa.2013.07.026.  Google Scholar

[29]

Z. LeiD. Li and X. Y. Zhang, Remarks of global wellposedness of liquid crystal flows and heat flows of harmonic maps in tow dimensions, Proc. Amer. Math. Soc., 142 (2014), 3801-3810.  doi: 10.1090/S0002-9939-2014-12057-0.  Google Scholar

[30]

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

[31]

F. Leslie, Theory of Flow Phenomenum in Liquid Crystals, The Theory of Liquid Crystals, 4. London-New York, Academic Press, 1979, 1–81. Google Scholar

[32]

J. K. Li, Global strong and weak solutions to inhomogeneous nematic liquid crystal flows in two dimensions, Nonlinear Anal., 99 (2014), 80-94.  doi: 10.1016/j.na.2013.12.023.  Google Scholar

[33]

J. K. LiE. S. Titi and Z. P. Xin, On the uniqueness of weak solutions to the Ericksen-Leslie liquid crystal model in $\mathbb{R}^2$, Math. Models Meth. Appl. Sci., 26 (2016), 803-822.  doi: 10.1142/S0218202516500184.  Google Scholar

[34]

J. K. Li and Z. P. Xin, Global existence of weak solutions to the non-isothermal nematic liquid crystals in 2D, Acta Math. Sci. Ser. B Engl. Ed., 36 (2016), 973-1014.  doi: 10.1016/S0252-9602(16)30054-6.  Google Scholar

[35]

F.-H. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814.  doi: 10.1002/cpa.3160420605.  Google Scholar

[36]

F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure. Appl. Math., 48 (1995), 501-537.  doi: 10.1002/cpa.3160480503.  Google Scholar

[37]

F.-H. Lin and C. Liu, Partial regularity of the nonlinear dissipative system modeling the flow of nematic liquid crystals, Discrete Comtin. Dyn. Syst., 2 (1996), 1-22.  doi: 10.3934/dcds.1996.2.1.  Google Scholar

[38]

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

[39]

F.-H. Lin and C. Y. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chin. Ann. Math. Ser. B, 31 (2010), 921-938.  doi: 10.1007/s11401-010-0612-5.  Google Scholar

[40]

F. H. Lin and C. Y. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimension three, Comm. Pure Appl. Math., 69 (2016), 1532-1571.  doi: 10.1002/cpa.21583.  Google Scholar

[41]

Q. LiuS. Q. LiuW. K. Tan and X. Zhong, Global well-posedness of the 2D nonhomogeneous incompressible nematic liquid crystal flows, J. Differential Equations, 261 (2016), 6521-6569.  doi: 10.1016/j.jde.2016.08.044.  Google Scholar

[42] P. Oswald and P. Pieranski, Nematic and Cholesteric Liquid Crystals: Concepts and Physical Properties Illustrated by Experiments, CRC Press, 2005.  doi: 10.1201/9780203023013.  Google Scholar
[43]

V. A. Solonnikov, $L_p$-estimates for solutions to the initial boundary-value problem for the generalized Stokes system in a bounded domain, J. Math. Sci. (New York), 105 (2001), 2448-2484.  doi: 10.1023/A:1011321430954.  Google Scholar

[44]

A. M. Sonnet and E. G. Virga, Dissipative Ordered Fluids: Theories for Liquid Crystals, Springer, New York, 2012. doi: 10.1007/978-0-387-87815-7.  Google Scholar

[45]

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

[46]

C. Y. 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.  doi: 10.1007/s00205-010-0343-5.  Google Scholar

[47]

M. Wang and W. D. Wang, Global existence of weak solution for the 2-D Ericksen-Leslie system, Calc. Var. Partial Differential Equations, 51 (2014), 915-962.  doi: 10.1007/s00526-013-0700-y.  Google Scholar

[48]

X. Xu and Z. F. Zhang, Global regularity and uniqueness of weak solutions for the 2-D liquid crystal flows, J. Differential Equations, 252 (2012), 1169-1181.  doi: 10.1016/j.jde.2011.08.028.  Google Scholar

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