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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, (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.
|
[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. |
[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. |
[6] |
E. Feireisl, "Dynamics of Viscous Compressible Fluids,", Oxford University Press, (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.
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.
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.
|
[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. |
[12] |
F. M. Leslie, Some constitutive equations for anisotropic fluids,, Quart. J. Mech. Appl. Math., 19 (1966), 357.
|
[13] |
F. M. Leslie, Some constitutive equations for liquid crystals,, Arch. Rational Mech. Anal., 28 (1968), 265.
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.
doi: 10.1002/mma.1132. |
[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.
|
[17] |
J. L. Ericksen, Some constitutive equations for liquid crystals,, Arch. Rational Mech. Anal., 28 (1968), 265.
doi: 10.1007/BF00251810. |
[18] |
J. L. Ericksen, Anisotropic fluids,, Arch. Rational Mech. Anal., 4 (1960), 231.
|
[19] |
J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density and pressure,, SIAM J. Math. Anal., 21 (1990), 1093.
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.
|
[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).
|
[25] |
R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,", North-Holland, (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.
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.
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.
|
[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. |
[31] |
Y. Z. Xie, "The Physics of Liquid Crystals,", Scientific Press, (1988). Google Scholar |
show all references
References:
[1] |
A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,", Birkhäuser, (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.
|
[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. |
[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. |
[6] |
E. Feireisl, "Dynamics of Viscous Compressible Fluids,", Oxford University Press, (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.
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.
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.
|
[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. |
[12] |
F. M. Leslie, Some constitutive equations for anisotropic fluids,, Quart. J. Mech. Appl. Math., 19 (1966), 357.
|
[13] |
F. M. Leslie, Some constitutive equations for liquid crystals,, Arch. Rational Mech. Anal., 28 (1968), 265.
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.
doi: 10.1002/mma.1132. |
[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.
|
[17] |
J. L. Ericksen, Some constitutive equations for liquid crystals,, Arch. Rational Mech. Anal., 28 (1968), 265.
doi: 10.1007/BF00251810. |
[18] |
J. L. Ericksen, Anisotropic fluids,, Arch. Rational Mech. Anal., 4 (1960), 231.
|
[19] |
J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density and pressure,, SIAM J. Math. Anal., 21 (1990), 1093.
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.
|
[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).
|
[25] |
R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,", North-Holland, (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.
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.
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.
|
[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. |
[31] |
Y. Z. Xie, "The Physics of Liquid Crystals,", Scientific Press, (1988). Google Scholar |
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