# American Institute of Mathematical Sciences

July  2013, 33(7): 2681-2710. doi: 10.3934/dcds.2013.33.2681

## Global weak solutions to a general liquid crystals system

 1 Department of Mathematics, Huzhou Teachers College, Zhejiang Huzhou, China 2 School of Mathematical Sciences, Fudan University, Shanghai, China, China

Received  January 2012 Revised  November 2012 Published  January 2013

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 $\mathbb{R}^3$.
Citation: Yuming Chu, Yihang Hao, Xiangao Liu. Global weak solutions to a general liquid crystals system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2681-2710. doi: 10.3934/dcds.2013.33.2681
##### References:
 [1] A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,", Birkhäuser, (1995).  doi: 10.1007/978-3-0348-9234-6.  Google Scholar [2] A. P. Calderon and A. Zygmund, On singular integrals,, Amer. J. Math., 78 (1956), 289.   Google Scholar [3] de Gennes, "The Physics of Liquid Crystals,", Claredon Press, (1993).   Google Scholar [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.  doi: 10.1007/s00205-011-0488-x.  Google Scholar [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.  doi: 10.1007/PL00000976.  Google Scholar [6] E. Feireisl, "Dynamics of Viscous Compressible Fluids,", Oxford University Press, (2004).   Google Scholar [7] E. G. Virga, "Variational Theories for Liquid Crystals,", Chapman & Hall press, (1994).   Google Scholar [8] F. H. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena,, Comm. Pure Appl. Math., 42 (1989), 789.  doi: 10.1002/cpa.3160420605.  Google Scholar [9] F. H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system,, Arch. Rational Mech. Anal., 154 (2000), 135.  doi: 10.1007/s002050000102.  Google Scholar [10] F. H. Lin and C. Liu, Static and dynamic theories of liquid crystals,, Journal of Partial Differential Equations, 14 (2001), 289.   Google Scholar [11] F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, Comm. Pure Appl. Math., 48 (1995), 501.  doi: 10.1002/cpa.3160480503.  Google Scholar [12] F. M. Leslie, Some constitutive equations for anisotropic fluids,, Quart. J. Mech. Appl. Math., 19 (1966), 357.   Google Scholar [13] F. M. Leslie, Some constitutive equations for liquid crystals,, Arch. Rational Mech. Anal., 28 (1968), 265.  doi: 10.1007/BF00251810.  Google Scholar [14] F. Jiang and Z. Tan, Global weak solution to the flow of liquid crystals system,, Math. Methods Appl. Sci., 32 (2009), 2243.  doi: 10.1002/mma.1132.  Google Scholar [15] G. P. Galdi, "An Introduction to the Mathematical Theory of the NavierStokes Equations I,", Springer-Verlag, (1994).   Google Scholar [16] J. L. Ericksen, Hydrostatic theory of liquid crystals,, Arch. Rational Mech. Anal., 9 (1962), 371.   Google Scholar [17] J. L. Ericksen, Some constitutive equations for liquid crystals,, Arch. Rational Mech. Anal., 28 (1968), 265.  doi: 10.1007/BF00251810.  Google Scholar [18] J. L. Ericksen, Anisotropic fluids,, Arch. Rational Mech. Anal., 4 (1960), 231.   Google Scholar [19] J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density and pressure,, SIAM J. Math. Anal., 21 (1990), 1093.  doi: 10.1137/0521061.  Google Scholar [20] L. C. Evans, "Partial Differential Equations,", Amer. Math. Soc. Providence, (1998).   Google Scholar [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.   Google Scholar [22] M. J. Stephen, Hydrodynamics of liquid crystals,, Phys. Rev. A, 2 (1970), 1558.   Google Scholar [23] O. Parodi, Stress tensor for a nematic liquid crystal,, J. Phys., 31 (1970), 581.   Google Scholar [24] P. L. lions, "Mathematical Topics in Fluid Dynamics, Vol.2. Compressible Models,", The Clarendon Press, (1998).   Google Scholar [25] R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,", North-Holland, (1977).   Google Scholar [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.  doi: 10.3934/dcdsb.2011.15.357.  Google Scholar [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.  doi: 10.3934/dcds.2012.32.539.  Google Scholar [28] X. G. Liu and Z. Y. Zhang, Existence of the flow of liquid crystals system,, Chinese Ann. Math. Ser. A, 30 (2009), 1.   Google Scholar [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.   Google Scholar [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.  doi: 10.1007/s00220-010-1017-8.  Google Scholar [31] Y. Z. Xie, "The Physics of Liquid Crystals,", Scientific Press, (1988).   Google Scholar

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##### References:
 [1] A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,", Birkhäuser, (1995).  doi: 10.1007/978-3-0348-9234-6.  Google Scholar [2] A. P. Calderon and A. Zygmund, On singular integrals,, Amer. J. Math., 78 (1956), 289.   Google Scholar [3] de Gennes, "The Physics of Liquid Crystals,", Claredon Press, (1993).   Google Scholar [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.  doi: 10.1007/s00205-011-0488-x.  Google Scholar [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.  doi: 10.1007/PL00000976.  Google Scholar [6] E. Feireisl, "Dynamics of Viscous Compressible Fluids,", Oxford University Press, (2004).   Google Scholar [7] E. G. Virga, "Variational Theories for Liquid Crystals,", Chapman & Hall press, (1994).   Google Scholar [8] F. H. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena,, Comm. Pure Appl. Math., 42 (1989), 789.  doi: 10.1002/cpa.3160420605.  Google Scholar [9] F. H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system,, Arch. Rational Mech. Anal., 154 (2000), 135.  doi: 10.1007/s002050000102.  Google Scholar [10] F. H. Lin and C. Liu, Static and dynamic theories of liquid crystals,, Journal of Partial Differential Equations, 14 (2001), 289.   Google Scholar [11] F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, Comm. Pure Appl. Math., 48 (1995), 501.  doi: 10.1002/cpa.3160480503.  Google Scholar [12] F. M. Leslie, Some constitutive equations for anisotropic fluids,, Quart. J. Mech. Appl. Math., 19 (1966), 357.   Google Scholar [13] F. M. Leslie, Some constitutive equations for liquid crystals,, Arch. Rational Mech. Anal., 28 (1968), 265.  doi: 10.1007/BF00251810.  Google Scholar [14] F. Jiang and Z. Tan, Global weak solution to the flow of liquid crystals system,, Math. Methods Appl. Sci., 32 (2009), 2243.  doi: 10.1002/mma.1132.  Google Scholar [15] G. P. Galdi, "An Introduction to the Mathematical Theory of the NavierStokes Equations I,", Springer-Verlag, (1994).   Google Scholar [16] J. L. Ericksen, Hydrostatic theory of liquid crystals,, Arch. Rational Mech. Anal., 9 (1962), 371.   Google Scholar [17] J. L. Ericksen, Some constitutive equations for liquid crystals,, Arch. Rational Mech. Anal., 28 (1968), 265.  doi: 10.1007/BF00251810.  Google Scholar [18] J. L. Ericksen, Anisotropic fluids,, Arch. Rational Mech. Anal., 4 (1960), 231.   Google Scholar [19] J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density and pressure,, SIAM J. Math. Anal., 21 (1990), 1093.  doi: 10.1137/0521061.  Google Scholar [20] L. C. Evans, "Partial Differential Equations,", Amer. Math. Soc. Providence, (1998).   Google Scholar [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.   Google Scholar [22] M. J. Stephen, Hydrodynamics of liquid crystals,, Phys. Rev. A, 2 (1970), 1558.   Google Scholar [23] O. Parodi, Stress tensor for a nematic liquid crystal,, J. Phys., 31 (1970), 581.   Google Scholar [24] P. L. lions, "Mathematical Topics in Fluid Dynamics, Vol.2. Compressible Models,", The Clarendon Press, (1998).   Google Scholar [25] R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,", North-Holland, (1977).   Google Scholar [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.  doi: 10.3934/dcdsb.2011.15.357.  Google Scholar [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.  doi: 10.3934/dcds.2012.32.539.  Google Scholar [28] X. G. Liu and Z. Y. Zhang, Existence of the flow of liquid crystals system,, Chinese Ann. Math. Ser. A, 30 (2009), 1.   Google Scholar [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.   Google Scholar [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.  doi: 10.1007/s00220-010-1017-8.  Google Scholar [31] Y. Z. Xie, "The Physics of Liquid Crystals,", Scientific Press, (1988).   Google Scholar
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