American Institute of Mathematical Sciences

Advanced
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

Jihong Zhao 1, , Qiao Liu 2, and  Shangbin Cui 3,

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.
Keywords: Liquid crystal flow, stability, global existence, blow up, weak-strong uniqueness..
Mathematics Subject Classification: Primary: 35A01, 35B35; Secondary: 76A1.
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, vol. 14. The Clarendon Press, Oxford University Press: New York, 1998.  Google Scholar

[2]

R. Dachin, "Fourier Analysis Methods for PDE's,", 2005. Available from: http://perso-math.univ-mlv.fr/users/danchin.raphael/courschine.pdf., ().   Google Scholar

[3]

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

[4]

J. L. Ericksen, Continuum theory of nematic liquid crystals, Res. Mechanica, 21 (1987), 381-392. Google Scholar

[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), 307-337. doi: 10.1007/s002050100175.  Google Scholar

[6]

R. Hardt and D. Kinderlehrer, "Mathematical Questions of Liquid Crystal Theory," The IMA Volumes in Mathematics andits Applications 5, New York: Springer-Verlag, 1987.  Google Scholar

[7]

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

[8]

X. Hu and D. 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

[9]

P.-G. Lemarié-Rieusset, "Recent Developments in the Navier-Stokes Problem," Research Notes in Mathematics, Chapman & Hall/CRC, 2002.  Google Scholar

[10]

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

[11]

F. Leslie, Theory of flow phenomenum in liquid crystals, In "The Theory of Liquid Crystals," London-New York: Academic Press, 4 (1979), 1-81. Google Scholar

[12]

F. 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

[13]

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

[14]

F. 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

[15]

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

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

[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, vol. 3. Walter de Gruyter & Co.: Berlin, 1996.  Google Scholar

[18]

H. Sun and C. Liu, On energetic variational approaches in modeling the nematic liquid crystal flows, Discrete Contin. Dyn. Syst., 23 (2009), 455-475. doi: 10.3934/dcds.2009.23.455.  Google Scholar

[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), 1-19. doi: 10.1007/s00205-010-0343-5.  Google Scholar

[20]

H. Wu, Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows, Discrete Contin. Dyn. Syst., 26 (2010), 379-396. doi: 10.3934/dcds.2010.26.379.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

J.-Y. Chemin, "Perfect Incompressible Fluids," Oxford Lecture Series in Mathematics and its Applications, vol. 14. The Clarendon Press, Oxford University Press: New York, 1998.  Google Scholar

[2]

R. Dachin, "Fourier Analysis Methods for PDE's,", 2005. Available from: http://perso-math.univ-mlv.fr/users/danchin.raphael/courschine.pdf., ().   Google Scholar

[3]

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

[4]

J. L. Ericksen, Continuum theory of nematic liquid crystals, Res. Mechanica, 21 (1987), 381-392. Google Scholar

[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), 307-337. doi: 10.1007/s002050100175.  Google Scholar

[6]

R. Hardt and D. Kinderlehrer, "Mathematical Questions of Liquid Crystal Theory," The IMA Volumes in Mathematics andits Applications 5, New York: Springer-Verlag, 1987.  Google Scholar

[7]

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

[8]

X. Hu and D. 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

[9]

P.-G. Lemarié-Rieusset, "Recent Developments in the Navier-Stokes Problem," Research Notes in Mathematics, Chapman & Hall/CRC, 2002.  Google Scholar

[10]

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

[11]

F. Leslie, Theory of flow phenomenum in liquid crystals, In "The Theory of Liquid Crystals," London-New York: Academic Press, 4 (1979), 1-81. Google Scholar

[12]

F. 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

[13]

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

[14]

F. 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

[15]

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

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

[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, vol. 3. Walter de Gruyter & Co.: Berlin, 1996.  Google Scholar

[18]

H. Sun and C. Liu, On energetic variational approaches in modeling the nematic liquid crystal flows, Discrete Contin. Dyn. Syst., 23 (2009), 455-475. doi: 10.3934/dcds.2009.23.455.  Google Scholar

[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), 1-19. doi: 10.1007/s00205-010-0343-5.  Google Scholar

[20]

H. Wu, Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows, Discrete Contin. Dyn. Syst., 26 (2010), 379-396. doi: 10.3934/dcds.2010.26.379.  Google Scholar

[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.  Google Scholar

[1]

Hongjun Gao, Šárka Nečasová, Tong Tang. On weak-strong uniqueness and singular limit for the compressible Primitive Equations. Discrete & Continuous Dynamical Systems, 2020, 40 (7) : 4287-4305. doi: 10.3934/dcds.2020181
[2]

Wenjing Zhao. Weak-strong uniqueness of incompressible magneto-viscoelastic flows. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2907-2917. doi: 10.3934/cpaa.2020127
[3]

Etienne Emmrich, Robert Lasarzik. Weak-strong uniqueness for the general Ericksen—Leslie system in three dimensions. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4617-4635. doi: 10.3934/dcds.2018202
[4]

Sili Liu, Xinhua Zhao, Yingshan Chen. A new blowup criterion for strong solutions of the compressible nematic liquid crystal flow. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4515-4533. doi: 10.3934/dcdsb.2020110
[5]

Xianpeng Hu, Hao Wu. Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3437-3461. doi: 10.3934/dcds.2015.35.3437
[6]

T. Tachim Medjo. On the existence and uniqueness of solution to a stochastic simplified liquid crystal model. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2243-2264. doi: 10.3934/cpaa.2019101
[7]

Qiang Tao, Ying Yang. Exponential stability for the compressible nematic liquid crystal flow with large initial data. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1661-1669. doi: 10.3934/cpaa.2016007
[8]

Yinxia Wang. A remark on blow up criterion of three-dimensional nematic liquid crystal flows. Evolution Equations & Control Theory, 2016, 5 (2) : 337-348. doi: 10.3934/eect.2016007
[9]

Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211
[10]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843
[11]

Bagisa Mukherjee, Chun Liu. On the stability of two nematic liquid crystal configurations. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 561-574. doi: 10.3934/dcdsb.2002.2.561
[12]

Yi-hang Hao, Xian-Gao Liu. The existence and blow-up criterion of liquid crystals system in critical Besov space. Communications on Pure & Applied Analysis, 2014, 13 (1) : 225-236. doi: 10.3934/cpaa.2014.13.225
[13]

Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072
[14]

Guochun Wu, Yinghui Zhang. Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1411-1429. doi: 10.3934/dcdsb.2018157
[15]

Shihui Zhu. Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 5201-5221. doi: 10.3934/dcds.2016026
[16]

Junyu Lin. Uniqueness of harmonic map heat flows and liquid crystal flows. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 739-755. doi: 10.3934/dcds.2013.33.739
[17]

Shanshan Guo, Zhong Tan. Energy dissipation for weak solutions of incompressible liquid crystal flows. Kinetic & Related Models, 2015, 8 (4) : 691-706. doi: 10.3934/krm.2015.8.691
[18]

Rafael O. Ruggiero. Shadowing of geodesics, weak stability of the geodesic flow and global hyperbolic geometry. Discrete & Continuous Dynamical Systems, 2006, 14 (2) : 365-383. doi: 10.3934/dcds.2006.14.365
[19]

François James, Nicolas Vauchelet. One-dimensional aggregation equation after blow up: Existence, uniqueness and numerical simulation. Networks & Heterogeneous Media, 2016, 11 (1) : 163-180. doi: 10.3934/nhm.2016.11.163
[20]

Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3093-3116. doi: 10.3934/dcds.2020039

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences