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

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Abstract

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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  3. Ball, J.: Differentiability properties of symmetric and isotropic functions. Duke Math. J. 51, 699–728 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ball, J.: Mathematics of Liquid Crystals. Cambridge Centre for Analysis, Short Course Slides, 13–17, 2012

  5. Ball, J.; Majumdar, A.: Nematic liquid crystals: from Maier-Saupe to a continuum theory. Mol. Cryst. Liq. Cryst. 525, 1–11 (2010)

    Article  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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, 1994

  9. Boyd, S.; Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  15. de Gennes, P.G.; Prost, J.: The Physics of Liquid Crystals. Oxford Science Publications, Oxford (1993)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  18. Guckenheimer, J., Holmes, P.J. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42, Springer Science & Business Media, 2013

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hartman, P.: Ordinary Differential Equations. Reprint of the second edition, Birkhäuser, Boston, MA (1982)

    MATH  Google Scholar 

  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, 2009

  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)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kato, T.; Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 41(7), 891–907 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu, C.; Calderer, M.C.: Liquid crystal flow: dynamic and static configurations. SIAM J. Appl. Math. 60(6), 1925–1949 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Mottram, N.J., Newton, J.P.: Introduction to Q-tensor theory. Preprint, arXiv:1409.3542, 2014

  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)

    Book  MATH  Google Scholar 

  29. Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow, vol. 27. Cambridge University Press, 2002

  30. Majumdar, A.: Equilibrium order parameters of nematic liquid crystals in the Landau-de Gennes theory. Eur. J. Appl. Math. 21, 181–203 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  32. Nomizu, K.: Characteristic roots and vectors of a diifferentiable family of symmetric matrices. Linear Multilinear Algebra 1(2), 159–162 (1973)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  35. Mkaddem, S.; Gartland Jr., E.C.: Fine structure of defects in radial nematic droplets. Phys. Rev. E 62(5), 6694 (2000)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44. Springer-Verlag, New York (1983)

    MATH  Google Scholar 

  38. Taylor, M.E.: Partial Differential Equations. III. Nonlinear Equations, Applied Mathematical Sciences, vol. 117, Springer-Verlag, New York, 1997

  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)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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.

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Authors and Affiliations

  1. School of Mathematical Sciences and Shanghai, Key Laboratory for Contemporary Applied Mathematics, Fudan University, Han Dan Road 220, Shanghai, 200433, China

    Hao Wu

  2. Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, Han Dan Road 220, Shanghai, 200433, China

    Hao Wu

  3. Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA, 23529, USA

    Xiang Xu

  4. IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013, Bilbao, Bizkaia, Spain

    Arghir Zarnescu

  5. BCAM, Basque Center for Applied Mathematics, Mazarredo 14, E48009, Bilbao, Bizkaia, Spain

    Arghir Zarnescu

  6. “Simion Stoilow” Institute of the Romanian Academy, 21 Calea Griviţei, 010702, Bucharest, Romania

    Arghir Zarnescu

Authors
  1. Hao Wu
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  2. Xiang Xu
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  3. Arghir Zarnescu
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Correspondence to Xiang Xu.

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Communicated by D. Kinderlehrer

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Wu, H., Xu, X. & Zarnescu, A. Dynamics and Flow Effects in the Beris-Edwards System Modeling Nematic Liquid Crystals. Arch Rational Mech Anal 231, 1217–1267 (2019). https://doi.org/10.1007/s00205-018-1297-2

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  • DOI: https://doi.org/10.1007/s00205-018-1297-2

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