\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Global weak solutions to a general liquid crystals system

Abstract Related Papers Cited by
  • We prove the global existence of finite energy weak solutions to the general liquid crystals system. The problem is studied in bounded domain of $\mathbb{R}^3$ with Dirichlet boundary conditions and the whole space $\mathbb{R}^3$.
    Mathematics Subject Classification: 76N10, 35Q35, 35Q30.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Birkhäuser, Berlin, 1995.doi: 10.1007/978-3-0348-9234-6.

    [2]

    A. P. Calderon and A. Zygmund, On singular integrals, Amer. J. Math., 78 (1956), 289-309.

    [3]

    de Gennes, "The Physics of Liquid Crystals," Claredon Press, 1993.

    [4]

    D. H. Wang and Y. Cheng, Global weak solution and large-time behavior for the compressible flow of liquid crystals, Arch. Rational Mech. Anal., 204 (2012), 881-915.doi: 10.1007/s00205-011-0488-x.

    [5]

    E. Feireisl, A. Novotny and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.doi: 10.1007/PL00000976.

    [6]

    E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford University Press, Oxford, 2004.

    [7]

    E. G. Virga, "Variational Theories for Liquid Crystals," Chapman & Hall press, 1994.

    [8]

    F. H. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814.doi: 10.1002/cpa.3160420605.

    [9]

    F. H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system, Arch. Rational Mech. Anal., 154 (2000), 135-156.doi: 10.1007/s002050000102.

    [10]

    F. H. Lin and C. Liu, Static and dynamic theories of liquid crystals, Journal of Partial Differential Equations, 14 (2001), 289-330.

    [11]

    F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537.doi: 10.1002/cpa.3160480503.

    [12]

    F. M. Leslie, Some constitutive equations for anisotropic fluids, Quart. J. Mech. Appl. Math., 19 (1966), 357-370.

    [13]

    F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1968), 265-283.doi: 10.1007/BF00251810.

    [14]

    F. Jiang and Z. Tan, Global weak solution to the flow of liquid crystals system, Math. Methods Appl. Sci., 32 (2009), 2243-2266.doi: 10.1002/mma.1132.

    [15]

    G. P. Galdi, "An Introduction to the Mathematical Theory of the NavierStokes Equations I," Springer-Verlag, New York, 1994.

    [16]

    J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Rational Mech. Anal., 9 (1962), 371-378.

    [17]

    J. L. Ericksen, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1968), 265-283.doi: 10.1007/BF00251810.

    [18]

    J. L. Ericksen, Anisotropic fluids, Arch. Rational Mech. Anal., 4 (1960), 231-237.

    [19]

    J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117.doi: 10.1137/0521061.

    [20]

    L. C. Evans, "Partial Differential Equations," Amer. Math. Soc. Providence, 1998.

    [21]

    M. E. Bogovskii, Solution of some problems of vector analysis, associated with the operators div and grad(in Russian), Trudy Sem. S. L. Sobolev, (1980), 5-40.

    [22]

    M. J. Stephen, Hydrodynamics of liquid crystals, Phys. Rev. A, 2 (1970), 1558-1562.

    [23]

    O. Parodi, Stress tensor for a nematic liquid crystal, J. Phys., 31 (1970), 581-584.

    [24]

    P. L. lions, "Mathematical Topics in Fluid Dynamics, Vol.2. Compressible Models," The Clarendon Press, Oxford University Press, New York, 1998.

    [25]

    R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," North-Holland, Amsterdam, 1977.

    [26]

    S. J. Ding, C. Y. Wang and H. Y. Wen, Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 357-371.doi: 10.3934/dcdsb.2011.15.357.

    [27]

    S. J. Ding, J. Y. Lin, C. Y. Wang and H. Y. Wen, Compressible hydrodynamic flow of liquid crystals in 1D, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 539-563.doi: 10.3934/dcds.2012.32.539.

    [28]

    X. G. Liu and Z. Y. Zhang, Existence of the flow of liquid crystals system, Chinese Ann. Math. Ser. A, 30 (2009), 1-20.

    [29]

    X. G. Liu and J. Qing, Globally weak solutions to the flow of compressible liquid crystals system, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013), 757-788.

    [30]

    X. P. Hu and D. H. Wang, Global solution to the three-dimensional incompressible flow of liquid crystals, Comm. Math. Phys., 296 (2010), 861-880.doi: 10.1007/s00220-010-1017-8.

    [31]

    Y. Z. Xie, "The Physics of Liquid Crystals," Scientific Press, Beijing, 1988.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(99) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return