Archive for Rational Mechanics and Analysis

, Volume 231, Issue 2, pp 1217–1267 | Cite as

Dynamics and Flow Effects in the Beris-Edwards System Modeling Nematic Liquid Crystals

  • Hao Wu
  • Xiang XuEmail author
  • Arghir Zarnescu


We consider the Beris-Edwards system modelling incompressible liquid crystal flows of nematic type. This couples a Navier-Stokes system for the fluid velocity with a parabolic reaction-convection-diffusion system for the Q-tensors describing the average orientation of liquid crystal molecules. In this paper, we study the effect that the flow has on the dynamics of the Q-tensors by considering two fundamental aspects: the preservation of the eigenvalue-range and the dynamical emergence of defects in the limit of large Ericksen number.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



H. Wu is partially supported by NNSFC grant No. 11631011. X. Xu is supported by the start-up fund from the Department of Mathematics and Statistics at Old Dominion University. A. Zarnescu is partially supported by a Grant of the Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI, project number PN-II-RU-TE-2014-4-0657; by the Basque Government through the BERC 2018-2021 program; and by SpanishMinistry of Economy and CompetitivenessMINECO through BCAM Severo Ochoa excellence accreditation SEV-2013-0323 and through project MTM2017-82184-R funded by (AEI/FEDER, UE) and acronym “DESFLU”; and by Leverhulme grant RPG 2014-226. All authors would like to thank the anonymous referee for the careful reading of an initial version of this manuscript and for the very helpful comments that allowed us to improve the paper.


  1. 1.
    Abels, H.; Dolzmann, G.; Liu, Y.N.: Well-posedness of a fully coupled Navier-Stokes/Q-tensor system with inhomogeneous boundary data. SIAM J. Math. Anal. 46, 3050–3077 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abels, H.; Dolzmann, G.; Liu, Y.N.: Strong solutions for the Beris-Edwards model for nematic liquid crystals with homogeneous Dirichlet boundary conditions. Adv. Differ. Equ. 21, 109–152 (2016)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Ball, J.: Differentiability properties of symmetric and isotropic functions. Duke Math. J. 51, 699–728 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ball, J.: Mathematics of Liquid Crystals. Cambridge Centre for Analysis, Short Course Slides, 13–17, 2012Google Scholar
  5. 5.
    Ball, J.; Majumdar, A.: Nematic liquid crystals: from Maier-Saupe to a continuum theory. Mol. Cryst. Liq. Cryst. 525, 1–11 (2010)CrossRefGoogle Scholar
  6. 6.
    Barberi, R., Ciuchi, F., Durand, G.E., Iovane, M., Sikharulidze, D., Sonnet, A.M., Virga, E.G.: Electric field induced order reconstruction in a nematic cell. Eur. Phys. J. E Soft Matter Biol. Phys. 13(1), 61–71 (2004)Google Scholar
  7. 7.
    Bauman, P.; Phillips, D.: Regularity and the behavior of eigenvalues for minimizers of a constrained Q-tensor energy for liquid crystals. Calc. Var. Partial Differ. Equ. 55, 55–81 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Beris, A.N., Edwards, B.J.: Thermodynamics of Flowing Systems with Internal Microstructure. Oxford Engineerin Science Series, vol. 36. Oxford university Press, Oxford, New York, 1994Google Scholar
  9. 9.
    Boyd, S.; Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  10. 10.
    Cavaterra, C.; Rocca, E.; Wu, H.; Xu, X.: Global strong solutions of the full Navier-Stokes and Q-tensor system for nematic liquid crystal flows in two dimensions. SIAM J. Math. Anal. 48(2), 1368–1399 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dai, M.M.; Feireisl, E.; Rocca, E.; Schimperna, G.; Schonbek, M.: On asymptotic isotropy for a hydrodynamic model of liquid crystals. Asymptot. Anal. 97, 189–210 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    De Anna, F.: A global 2D well-posedness result on the order tensor liquid crystal theory. J. Differ. Equ. 262(7), 3932–3979 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    De Anna, F.; Zarnescu, A.: Uniqueness of weak solutions of the full coupled Navier-Stokes and Q-tensor system in 2D. Commun. Math. Sci. 14, 2127–2178 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fefferman, C.L.; McCormick, D.S.; Robinson, J.C.; Rodrigo, J.L.: Higher order commutator estimates and local existence for the non-resistive MHD equations and related models. J. Funct. Anal. 267(4), 1035–1056 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    de Gennes, P.G.; Prost, J.: The Physics of Liquid Crystals. Oxford Science Publications, Oxford (1993)Google Scholar
  16. 16.
    Evans, L.C.; Kneuss, O.; Tran, H.: Partial regularity for minimizers of singular energy functionals, with application to liquid crystal models. Trans. Am. Math. Soc. 368, 3389–3413 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Feireisl, E.; Rocca, E.: Schimperna, G., arnescu, A.: Evolution of non-isothermal Landau-de Gennes nematic liquid crystals flows with singular potential. Commun. Math. Sci. 12, 317–343 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Guckenheimer, J., Holmes, P.J. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42, Springer Science & Business Media, 2013Google Scholar
  19. 19.
    Guillén-González, F.; Rodríguez-Bellido, M.A.: Weak time regularity and uniqueness for a Q-tensor model. SIAM J. Math. Anal. 46, 3540–3567 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Guillén-González, F.; Rodríguez-Bellido, M.A.: Weak solutions for an initial-boundary Q-tensor problem related to liquid crystals. Nonlinear Anal. 112, 84–104 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Feireisl, E.; Rocca, E.; Schimperna, G.; Zarnescu, A.: Nonisothermal nematic liquid crystal flows with the Ball-Majumdar free energy. Annali di Mat. Pura ed App. 194(5), 1269–1299 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hartman, P.: Ordinary Differential Equations. Reprint of the second edition, Birkhäuser, Boston, MA (1982)zbMATHGoogle Scholar
  23. 23.
    Ionescu, A.D., Kenig, C.E.: Local and global well-posedness of periodic KP-I equations. Mathematical Aspects of Nonlinear Dispersive Equations. Annals of Mathematics Studies, Vol. 163, Princeton University Press, 181–212, 2009Google Scholar
  24. 24.
    Iyer, G.; Xu, X.; Zarnescu, A.: Dynamic cubic instability in a 2D Q-tensor model for liquid crystals. Math. Models Methods Appl. Sci. 25(8), 1477–1517 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kato, T.; Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 41(7), 891–907 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Liu, C.; Calderer, M.C.: Liquid crystal flow: dynamic and static configurations. SIAM J. Appl. Math. 60(6), 1925–1949 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mottram, N.J., Newton, J.P.: Introduction to Q-tensor theory. Preprint, arXiv:1409.3542, 2014
  28. 28.
    Majda, A.J.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, vol. 53. Springer-Verlag, New York (1984)CrossRefzbMATHGoogle Scholar
  29. 29.
    Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow, vol. 27. Cambridge University Press, 2002Google Scholar
  30. 30.
    Majumdar, A.: Equilibrium order parameters of nematic liquid crystals in the Landau-de Gennes theory. Eur. J. Appl. Math. 21, 181–203 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Majumdar, A.; Zarnescu, A.: Landau-De Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond. Arch. Ration. Mech. Anal. 196, 227–280 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Nomizu, K.: Characteristic roots and vectors of a diifferentiable family of symmetric matrices. Linear Multilinear Algebra 1(2), 159–162 (1973)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Paicu, M.; Zarnescu, A.: Global existence and regularity for the full coupled Navier-Stokes and Q-tensor system. SIAM J. Math. Anal. 43, 2009–2049 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Paicu, M.; Zarnescu, A.: Energy dissipation and regularity for a coupled Navier-Stokes and Q-tensor system. Arch. Ration. Mech. Anal. 203, 45–67 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Mkaddem, S.; Gartland Jr., E.C.: Fine structure of defects in radial nematic droplets. Phys. Rev. E 62(5), 6694 (2000)ADSCrossRefGoogle Scholar
  36. 36.
    Murza, A.C.; Teruel, A.E.; Zarnescu, A.: Shear flow dynamics in the Beris-Edwards model of nematic liquid crystals. Proc. R. Soc. A 474(2210), 20170673 (2018)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44. Springer-Verlag, New York (1983)zbMATHGoogle Scholar
  38. 38.
    Taylor, M.E.: Partial Differential Equations. III. Nonlinear Equations, Applied Mathematical Sciences, vol. 117, Springer-Verlag, New York, 1997Google Scholar
  39. 39.
    Wilkinson, M.: Strict physicality of global weak solutions of a Navier-Stokes Q-tensor system with singular potential. Arch. Ration. Mech. Anal. 218, 487–526 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical Sciences and Shanghai, Key Laboratory for Contemporary Applied MathematicsFudan UniversityShanghaiChina
  2. 2.Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University)Ministry of EducationShanghaiChina
  3. 3.Department of Mathematics and StatisticsOld Dominion UniversityNorfolkUSA
  4. 4.IKERBASQUE, Basque Foundation for ScienceBilbaoSpain
  5. 5.BCAM, Basque Center for Applied MathematicsBilbaoSpain
  6. 6.“Simion Stoilow” Institute of the Romanian AcademyBucharestRomania

Personalised recommendations