Global weak solutions to a general liquid crystals system

Previous Article
Weak attractor of the Klein-Gordon field in discrete space-time interacting with a nonlinear oscillator
DCDS Home
This Issue
Next Article
The period set of a map from the Cantor set to itself

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

Global weak solutions to a general liquid crystals system

Yuming Chu ^1, , Yihang Hao ^2, and Xiangao Liu ^2,

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

Abstract
Related Papers

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 $\mathbb{R}^3$.

Keywords: Bulk free energy, finite energy weak solutions, Navier-Stokes equations..

Mathematics Subject Classification: 76N10, 35Q35, 35Q3.

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.

[1]	Grzegorz Karch, Maria E. Schonbek, Tomas P. Schonbek. Singularities of certain finite energy solutions to the Navier-Stokes system. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 189-206. doi: 10.3934/dcds.2020008
[2]	Peter Anthony, Sergey Zelik. Infinite-energy solutions for the Navier-Stokes equations in a strip revisited. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1361-1393. doi: 10.3934/cpaa.2014.13.1361
[3]	Peter E. Kloeden, José Valero. The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 161-179. doi: 10.3934/dcds.2010.28.161
[4]	Bum Ja Jin, Kyungkeun Kang. Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4815-4834. doi: 10.3934/dcds.2017207
[5]	Xulong Qin, Zheng-An Yao. Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041
[6]	Tianwen Luo, Tao Tao, Liqun Zhang. Finite energy weak solutions of 2d Boussinesq equations with diffusive temperature. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3737-3765. doi: 10.3934/dcds.2019230
[7]	Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure and Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35
[8]	Daniel Pardo, José Valero, Ángel Giménez. Global attractors for weak solutions of the three-dimensional Navier-Stokes equations with damping. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3569-3590. doi: 10.3934/dcdsb.2018279
[9]	Fang Li, Bo You, Yao Xu. Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4267-4284. doi: 10.3934/dcdsb.2018137
[10]	Jian-Guo Liu, Zhaoyun Zhang. Existence of global weak solutions of $ p $-Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 469-486. doi: 10.3934/dcdsb.2021051
[11]	Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure and Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747
[12]	Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 769-801. doi: 10.3934/dcds.2011.29.769
[13]	Anthony Suen. Existence and a blow-up criterion of solution to the 3D compressible Navier-Stokes-Poisson equations with finite energy. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1775-1798. doi: 10.3934/dcds.2020093
[14]	Chérif Amrouche, María Ángeles Rodríguez-Bellido. On the very weak solution for the Oseen and Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 159-183. doi: 10.3934/dcdss.2010.3.159
[15]	Zhenhua Guo, Zilai Li. Global existence of weak solution to the free boundary problem for compressible Navier-Stokes. Kinetic and Related Models, 2016, 9 (1) : 75-103. doi: 10.3934/krm.2016.9.75
[16]	Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks and Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465
[17]	Fei Jiang, Song Jiang, Junpin Yin. Global weak solutions to the two-dimensional Navier-Stokes equations of compressible heat-conducting flows with symmetric data and forces. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 567-587. doi: 10.3934/dcds.2014.34.567
[18]	Flavia Smarrazzo, Alberto Tesei. Entropy solutions of forward-backward parabolic equations with Devonshire free energy. Networks and Heterogeneous Media, 2012, 7 (4) : 941-966. doi: 10.3934/nhm.2012.7.941
[19]	Vladislav Balashov, Alexander Zlotnik. An energy dissipative semi-discrete finite-difference method on staggered meshes for the 3D compressible isothermal Navier–Stokes–Cahn–Hilliard equations. Journal of Computational Dynamics, 2020, 7 (2) : 291-312. doi: 10.3934/jcd.2020012
[20]	Yutian Lei. On finite energy solutions of fractional order equations of the Choquard type. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1497-1515. doi: 10.3934/dcds.2019064

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Yuming Chu Yihang Hao Xiangao Liu
on Google Scholar
Yuming Chu Yihang Hao Xiangao Liu

[Back to Top]