In this paper we study the full system of incompressible liquid crystals, as modeled in the Q-tensor framework. Under certain conditions we prove the global existence of weak solutions in dimension two or three and the existence of global regular solutions in dimension two. We also prove the weak-strong uniqueness of the solutions, for sufficiently regular initial data.

MSC codes

  1. 35Q30
  2. 76A05
  3. 76A15


  1. nematic liquid crystal fluids
  2. Navier–Stokes equations
  3. global wellposedness

Get full access to this article

View all available purchase options and get full access to this article.


J. W. Barrett, C. Schwab, and E. Süli, Existence of global weak solutions for some polymeric flow models, Math. Models Methods Appl. Sci., 15 (2005), pp. 939–983.
A. N. Beris and B. J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure, Oxford Engrg. Sci. Ser. 36, Oxford University Press, Oxford, New York, 1994.
J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4), 14 (1981), pp. 209–246.
H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), pp. 677–681.
J. Y. Chemin, Perfect Incompressible Fluids, Oxford Lecture Ser. Math. Appl. 14, Clarendon Press, Oxford University Press, New York, 1998
J.-Y. Chemin and N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal., 33 (2001), pp. 84–112.
J. Y. Chemin and C. J. Xu, Inclusions de Sobolev en calcul de Weyl-Hörmander et champs de vecteurs sous-elliptiques, Ann. Sci. École Norm. Sup. (4), 30 (1997), pp. 719–751.
P. Constantin, C. Fefferman, E. S. Titi, and A. Zarnescu, Regularity of coupled two-dimensional nonlinear Fokker-Planck and Navier-Stokes systems, Comm. Math. Phys., 270 (2007), pp. 789–811.
P. Constantin and N. Masmoudi, Global well-posedness for a Smoluchowski equation coupled with Navier-Stokes equations in 2D, Comm. Math. Phys., 278 (2008), pp. 179–191.
P. Constantin and G. Seregin, Global Regularity of Solutions of Coupled Navier-Stokes Equations and Nonlinear Fokker-Planck Equations, preprint, 2009; available online from http://arxiv.org/abs/0901.4462.
P. G. de Gennes, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1974.
C. Denniston, E. Orlandini, and J. M. Yeomans, Lattice Boltzmann simulations of liquid crystals hydrodynamics, Phys. Rev. E., 63 (2001), 056702.
F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), pp. 265–283.
F. H. Lin, J. Y. Lin, and C. Wang, Liquid crystal flows in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), pp. 297–336.
F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), pp. 501–537.
F. H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system, Arch. Ration. Mech. Anal., 154 (2000), pp. 135–156.
F. H. Lin and C. Liu, Static and Dynamic Theories of Liquid Crystals, J. Partial Differential Equations, 14 (2001), pp. 289–330.
F. H. Lin, P. Zhang, and Z. Zhang, On the global existence of smooth solution to the 2-D FENE dumbbell model, Comm. Math. Phys., 277 (2008), pp. 531–553.
P. L. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chinese Ann. Math. Ser. B, 21 (2000), pp. 131–146.
A. Majumdar, Equilibrium order parameters of liquid crystals in the Landau-de Gennes theory, European J. Appl. Math., 21 (2010), pp. 181–203.
A. Majumdar and A. Zarnescu, Landau-de Gennes theory of nematic liquid crystals: The Oseen-Frank limit and beyond, Arch. Ration. Mech. Anal., 196 (2010), pp. 227–280.
N. Masmoudi, Well-posedness for the FENE dumbbell model of polymeric flows, Comm. Pure Appl. Math., 61 (2008), pp. 1685–1714.
M. Paicu, Équation periodique de Navier-Stokes sans viscosité dans une direction, Comm. Partial Differential Equations, 30 (2005), pp. 1107–1140.
M. Paicu and A. Zarnescu, Energy dissipation and regularity for a coupled Navier-Stokes and Q-tensor system, Arch. Ration. Mech. Anal., in press.
A. M. Sonnet, P. L. Maffettone, and E. G. Virga, Continuum theory for nematic liquid crystals with tensorial order, J. Non-Newtonian Fluid Mech., 119 (2004), pp. 51–59.
M. E. Schonbek, Existence and decay of polymeric flows, SIAM J. Math. Anal., 41 (2009), pp. 564–587.
G. Toth, C. Denniston, and J. M. Yeomans, Hydrodynamics of domain growth in nematic liquid crystals, Phys. Rev. E, 67 (2003), 051705.

Information & Authors


Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 2009 - 2049
ISSN (online): 1095-7154


Submitted: 14 April 2010
Accepted: 21 April 2011
Published online: 1 September 2011

MSC codes

  1. 35Q30
  2. 76A05
  3. 76A15


  1. nematic liquid crystal fluids
  2. Navier–Stokes equations
  3. global wellposedness



Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

View Options

View options


View PDF







Copy the content Link

Share with email

Email a colleague

Share on social media

On May 28, 2024, our site will enter Read Only mode for a limited time in order to complete a platform upgrade. As a result, the following functions will be temporarily unavailable: registering new user accounts, any updates to existing user accounts, access token activations, and shopping cart transactions. Contact [email protected] with any questions.