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Strictly Physical Global Weak Solutions of a Navier–Stokes Q-tensor System with Singular Potential

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Abstract

We study the existence, regularity and so-called ‘strict physicality’ of global weak solutions of a Beris–Edwards system which is proposed as a model for the incompressible flow of nematic liquid crystal materials. An important contribution to the dynamics comes from a singular potential introduced by John Ball and Apala Majumdar which replaces the commonly employed Landau-de Gennes bulk potential. This is built into our model to ensure that a natural physical constraint on the eigenvalues of the Q-tensor order parameter is respected by the dynamics of this system. Moreover, by a maximum principle argument, we are able to construct global strong solutions in dimension two.

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References

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

    Article  Google Scholar 

  2. Beris A.N., Edwards B.J.: Thermodynamics of Flowing Systems: With Internal Microstructure, vol. 36. Oxford University Press, Oxford, USA (1994)

    Google Scholar 

  3. Bollobás B.: Linear Analysis. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

  4. Chandrasekhar S.: Liquid Crystals. Cambridge University Press, Cambridge (1992)

    Book  Google Scholar 

  5. Constantin, P., Kiselev, A., Ryzhik, L., Zlatoš, A.: Diffusion and mixing in fluid flow. Ann. Math. 168(2) (2008)

  6. de Gennes P.-G., Prost J.: The Physics of Liquid Crystals, International Series of Monographs on Physics, vol. 83. Oxford University Press, Oxford (1993)

    Google Scholar 

  7. Denniston C., Orlandini E., Yeomans J.M.: Lattice Boltzmann simulation of liquid crystal hydrodynamics. Phys. Rev. E 63, 056702 (2001)

    Article  ADS  Google Scholar 

  8. Doi M., Edwards S.F.: The Theory of Polymer Dynamics, vol. 73. Oxford University Press, Oxford (1988)

    Google Scholar 

  9. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems, vol. 1. North Holland, 1976

  10. Evans, L.C.: Partial Differential Equations, Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence, RI, 1998

  11. Fatkullin I., Slastikov V.: On spatial variations of nematic ordering. Phys. D 237, 2577–2586 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Frank F.C.: I. Liquid crystals. On the theory of liquid crystals. Discuss. Faraday Soc. 25, 19–28 (1958)

    Article  Google Scholar 

  13. Iyer G.: A stochastic Lagrangian proof of global existence of the Navier–Stokes equations for flows with small Reynolds number. Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 181–189 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Katriel J., Kventsel G.F., Luckhurst G.R., Sluckin T.J.: Free energies in the Landau and molecular field approaches. Liq. Cryst. 1(4), 337–355 (1986)

    Article  Google Scholar 

  15. Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasi-Linear Equations of Parabolic Type. American Mathematical Society, Providence, RI, 1968

  16. Leslie F.M.: Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal. 28(4), 265–283 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lieb, E.H., Loss, M.: Analysis, Second, Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence, RI, 2001

  18. Lin F.H., Liu C.: Nonparabolic dissipative systems modeling the flow of liquid crystals. Commun. Pure Appl. Math. 48(5), 501–537 (1995)

    Article  MATH  Google Scholar 

  19. Majumdar A.: Equilibrium order parameters of nematic liquid crystals in the Landau-de Gennes theory. Euro. J. Appl. Math. 21(2), 181–203 (2010)

    Article  MATH  Google Scholar 

  20. Mottram, N.J., Newton, C.: Introduction to Q-tensor theory. University of Strathclyde, Department of Mathematics, Research Report, 10, 91, (2004)

  21. Paicu M., Zarnescu A.: Global existence and regularity for the full coupled Navier–Stokes and Q-tensor system. SIAM J. Math. Anal. 43(5), 2009–2049 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  22. Paicu, M., Zarnescu, A.: Energy dissipation and regularity for a coupled Navier–Stokes and Q-tensor system. Arch. Ration. Mech. Anal. 203, (2012)

  23. Pucci, P., Serrin, J.: The Maximum Principle, vol. 73. Birkhäuser, 2007

  24. Rockafellar, R.T.: Convex Analysis, vol. 28. Princeton University Press, Princeton 1997

  25. Temam, R.: Navier–Stokes Equations and Nonlinear Functional Analysis, vol. 66. Society for Industrial Mathematics, 1995

  26. Tóth, G., Denniston, C., Yeomans, J.M.: Hydrodynamics of domain growth in nematic liquid crystals. Phys. Rev. E 67, 051705, (2003)

  27. Wilkinson, M.: Some problems on the dynamics of nematic liquid crystals. University of Oxford, D.Phil thesis, 2013

  28. Yosida K.: Functional Analysis, Classics in Mathematics. Springer, Berlin (1995)

    Google Scholar 

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Authors and Affiliations

  1. École Normale Supérieure, Paris, France

    Mark Wilkinson

  2. Courant Institute of Mathematical Sciences, New York, NY, USA

    Mark Wilkinson

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  1. Mark Wilkinson
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Correspondence to Mark Wilkinson.

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Communicated by F. Lin

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Wilkinson, M. Strictly Physical Global Weak Solutions of a Navier–Stokes Q-tensor System with Singular Potential. Arch Rational Mech Anal 218, 487–526 (2015). https://doi.org/10.1007/s00205-015-0864-z

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