-
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
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 |
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
[Back to Top]