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Archive for Rational Mechanics and Analysis

, Volume 197, Issue 1, pp 297–336 | Cite as

Liquid Crystal Flows in Two Dimensions

  • Fanghua LinEmail author
  • Junyu Lin
  • Changyou Wang
Article

Abstract

This paper is concerned with a simplified hydrodynamic equation, proposed by Ericksen and Leslie, modeling the flow of nematic liquid crystals. In dimension two, we establish both interior and boundary regularity theorems for such a flow under smallness conditions. As a consequence, we establish the existence of global (in time) weak solutions on a bounded smooth domain in \({\mathbb{R}^2}\) which are smooth everywhere with possible exceptions of finitely many singular times.

Keywords

Weak Solution Nematic Liquid Crystal Partial Regularity Global Weak Solution Parabolic Cylinder 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Chang K.C.: Heat flow and boundary value problem for harmonic maps. Annales de l’institut Henri Poincaré (C) Analyse non linéaire 6(5), 363–395 (1989)zbMATHGoogle Scholar
  2. 2.
    Chang K.C., Ding W.Y., Ye R.: Finite-time blow-up of the heat flow of harmonic maps from surfaces. J. Differ. Geom. 36, 507–515 (1992)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Constantin, P., Seregin, S.: Hölder continuity of solutions of 2D Navier-Stokes equations with singular forcing. Preprint, 2009Google Scholar
  4. 4.
    Chen Y.M., Lin F.H.: Evolution of harmonic maps with Dirichlet boundary conditions. Comm. Anal. Geom. 1(3–4), 327–346 (1993)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Caffarelli L., Kohn R., Nirenberg L.: Partial regularity of suitable weak solutions of Navier-Stokes equations. CPAM 35, 771–831 (1982)zbMATHMathSciNetADSGoogle Scholar
  6. 6.
    de Gennes, P.G.: The Physics of Liquid Crystals. Oxford, 1974Google Scholar
  7. 7.
    Ericksen J.L.: Hydrostatic theory of liquid crystal. Arch. Ration. Mech. Anal. 9, 371–378 (1962)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Hong, M.C.: Global existence of solutions of the simplified Ericksen-Leslie system in \({\mathbb{R}^2}\) . PreprintGoogle Scholar
  9. 9.
    Ladyzhenskaya, O.: The Mathematical Theory of Viscous Incompressible Fluid, Gordon and Breach, New York, 1969Google Scholar
  10. 10.
    Ladyzenskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and quasilinear equations of parabolic type. Am. Math. Soc., Providence, 1968Google Scholar
  11. 11.
    Leslie F.M.: Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal. 28, 265–283 (1962)MathSciNetGoogle Scholar
  12. 12.
    Lemaire L.: Applications harmoniques de surfaces riemanniennes. J. Differ. Geom. 13(1), 51–78 (1978)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Lin F.H.: A new proof of the Caffarelli-Kohn-Nirenberg Theorem. Comm. Pure. Appl. Math. LI, 0241–0257 (1998)CrossRefGoogle Scholar
  14. 14.
    Lin F.H., Liu C.: Nonparabolic dissipative systems modeling the flow of liquid crystals. CPAM XLVIII, 501–537 (1995)MathSciNetGoogle Scholar
  15. 15.
    Lin F.H., Liu C.: Partial regularity of the dynamic system modeling the flow of liquid crystals. DCDS 2(1), 1–22 (1998)Google Scholar
  16. 16.
    Lin F.H., Wang C.Y.: Harmonic and quasi-harmonic spheres. II. Comm. Anal. Geom. 10(2), 341–375 (2002)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Qing J.: On singularities of the heat flow for harmonic maps from surfaces into spheres. Comm. Anal. Geom. 3(1–2), 297–315 (1995)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Seregin G., Shilkin T., Solonnikov V.: Boundary partial regularity for the Navier-Stokes equations. J. Math. Sci. 132(3), 339–358 (2006)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Solonnikov, V.A.: On Schauder estimates for the evolution generalized stokes problem. Hyperbolic Problems and Regularity Questions, Trends in Mathematics, Birkhäuser, Basel, 197–205, 2007Google Scholar
  20. 20.
    Solonnikov V.A.: L p-Estimates for solutions to the initial boundary-value problem for the generalized Stokes system in a bounded domain. J. Math. Sci. 105(5), 2448–2484 (2001)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of 2-spheres. Ann. Math. (2) 113(1), 1--24 (1981)Google Scholar
  22. 22.
    Schoen, R., Uhlenbeck, K.: Approximation of Sobolev maps between Riemannian manifolds. Preprint (1984)Google Scholar
  23. 23.
    Struwe M.: On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helvetici 60, 558–581 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Temam, R.: Navier-Stokes equations. Studies in Mathematics and its Applications, Vol. 2, North Holland, Amsterdam, 1977Google Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Department of MathematicsSouth China University of TechnologyGuangzhouChina
  3. 3.Department of MathematicsUniversity of KentuckyLexingtonUSA

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