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 and Continuous Dynamical Systems, 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, 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.

show all references

References:
[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.

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