Article Contents
Article Contents

# Global weak solutions to a general liquid crystals system

• 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:

•  [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.