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Globally weak solutions to the flow of compressible liquid crystals system
1. | Institute of Mathematics, Fudan University, Shanghai, China |
2. | Mathematics Department, UC at Santa Cruz, United States |
References:
[1] |
M. C. Calderer and C. Liu, Liquid crystal flow: Dynamic and staic configurations, SIAM J. Appl. Math., 60 (2000), 1925-1949. |
[2] |
D. Coutand and S. Shkoller, Well posedness of the full Ericksen-Leslye model of nematic liquid crystals, Note C. R. A. S, Paris, Math., 333 (2001), 919-924. |
[3] |
J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Rational Mech. Anal., 9 (1962), 371-378. |
[4] |
J. L. Ericksen and D. Kinderlehrer, "Theory and Applications of Liquid Crystals," IMA Vol. 5, Springer-Verlag, New York, 1986. |
[5] |
E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford University Press, Oxford, 2004. |
[6] |
D. Forster, T. Lubensky, P. Martin, J. Swift and P. Pershan, Hydrodynamics of liquid crystals, Phys. Rev. Lett., 26 (1971), 1016-1019.
doi: 10.1103/PhysRevLett.26.1016. |
[7] |
E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392. |
[8] |
de Gennes, "The Physics of Liquid Crystals," Claredon Press, 1993. |
[9] |
D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Diff. equns, 120 (1995), 215-254.
doi: 10.1006/jdeq.1995.1111. |
[10] |
D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Rational Mech. Anal., 132 (1995), 1-14. |
[11] |
S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Comm. Math. Phys., 215 (2001), 559-589. |
[12] |
O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," Gordon and Breach, New York, 1969. |
[13] |
O. A. Ladyzhenskaya, N. A. Solonnikov and N. N. Uraltseva, "Linear and Quasilinear Equations Of Parabolic Type," Transl. Math. Monographs, 23, American Mathematical Society, 1968. |
[14] |
F. M. Leslie, Some constitutive equations for anisotropic fluids, Quart. J. Mech. Appl. Math., 19 (1966), 357-370.
doi: 10.1093/qjmam/19.3.357. |
[15] |
F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1968), 265-283.
doi: 10.1007/BF00251810. |
[16] |
F. H. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena, Comm. Pure. Appl. Math., 42 (1989), 789-814. |
[17] |
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. |
[18] |
F. H. Lin and C. Liu, Static ang dynamic theories of liquid crystals, Journal of Partial Differential Equations, 14 (2001), 289-330. |
[19] |
F. H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie System, Arch. Rational Mech. Aanl., 154 (2000), 135-156. |
[20] |
F. H. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals, Discrete and Continuous Dynamic Systems, 2 (1996), 1-23. |
[21] |
J. L. Lions, "Quelques Méthodes Derésolution des Problèms aux Limites Nonlinéaires," Dunod, Gauthier-Villars, Paris, 1960. |
[22] |
P. L. Lions, "Mathematical Topics in Fluid Dynamics," Vol.1. Compressible models. Oxford Science Publication, Oxford, 1998. |
[23] |
P. L. Lions, "Mathematical Topics in Fluid Dynamics," Vol.2. Compressible models. Oxford Science Publication, Oxford, 1998. |
[24] |
X. Liu and Z. Zhang, Global existence of weak solutions for the incompressible liquid crystals, Chinese Anna. Math., 30 (2009), 1-20. |
[25] |
A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Birkhäuser, Berlin 1995. |
[26] |
A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible and heat-conductives fluids, Proc. Japan Acad., A, 55 (1979), 337-342.
doi: 10.3792/pjaa.55.337. |
[27] | |
[28] |
S. V. Pasechnik, V. G. Chigrinov, D. V. Shmeliova, "Liquid Crystals: Viscous and Elastic Properties," Wiley-VCH, 2009. |
[29] |
D. Serre and J.L. Lions(Rapporteur), Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible, C. R. Acad. Sci. Paris, 303 (1986), 639-642. |
[30] |
M. J. Stephen, Hydrodynamics of liquid crystals, Phys. Rev. A, 2 (1970), 1558-1562. |
[31] |
R. Temam, "Navier-Stokes Equations," rev. ed., Studies in Mathematics and its Applications 2, North-Holland, Amsterdam, 1977. |
[32] |
W. Stewart, "The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction," Liquid crystals bookseries, Taylor & Francis, London, 2004. |
[33] |
Y. Z. Xie, "The Physics of Liquid Crystals," Scientific Press, Beijing, 1988. |
[34] |
D. Wang and C. Yu, Global weak solution and large-time behavior for the compressible flow of liquid crystals, arXiv:1108.4939. |
show all references
References:
[1] |
M. C. Calderer and C. Liu, Liquid crystal flow: Dynamic and staic configurations, SIAM J. Appl. Math., 60 (2000), 1925-1949. |
[2] |
D. Coutand and S. Shkoller, Well posedness of the full Ericksen-Leslye model of nematic liquid crystals, Note C. R. A. S, Paris, Math., 333 (2001), 919-924. |
[3] |
J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Rational Mech. Anal., 9 (1962), 371-378. |
[4] |
J. L. Ericksen and D. Kinderlehrer, "Theory and Applications of Liquid Crystals," IMA Vol. 5, Springer-Verlag, New York, 1986. |
[5] |
E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford University Press, Oxford, 2004. |
[6] |
D. Forster, T. Lubensky, P. Martin, J. Swift and P. Pershan, Hydrodynamics of liquid crystals, Phys. Rev. Lett., 26 (1971), 1016-1019.
doi: 10.1103/PhysRevLett.26.1016. |
[7] |
E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392. |
[8] |
de Gennes, "The Physics of Liquid Crystals," Claredon Press, 1993. |
[9] |
D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Diff. equns, 120 (1995), 215-254.
doi: 10.1006/jdeq.1995.1111. |
[10] |
D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Rational Mech. Anal., 132 (1995), 1-14. |
[11] |
S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Comm. Math. Phys., 215 (2001), 559-589. |
[12] |
O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," Gordon and Breach, New York, 1969. |
[13] |
O. A. Ladyzhenskaya, N. A. Solonnikov and N. N. Uraltseva, "Linear and Quasilinear Equations Of Parabolic Type," Transl. Math. Monographs, 23, American Mathematical Society, 1968. |
[14] |
F. M. Leslie, Some constitutive equations for anisotropic fluids, Quart. J. Mech. Appl. Math., 19 (1966), 357-370.
doi: 10.1093/qjmam/19.3.357. |
[15] |
F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1968), 265-283.
doi: 10.1007/BF00251810. |
[16] |
F. H. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena, Comm. Pure. Appl. Math., 42 (1989), 789-814. |
[17] |
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. |
[18] |
F. H. Lin and C. Liu, Static ang dynamic theories of liquid crystals, Journal of Partial Differential Equations, 14 (2001), 289-330. |
[19] |
F. H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie System, Arch. Rational Mech. Aanl., 154 (2000), 135-156. |
[20] |
F. H. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals, Discrete and Continuous Dynamic Systems, 2 (1996), 1-23. |
[21] |
J. L. Lions, "Quelques Méthodes Derésolution des Problèms aux Limites Nonlinéaires," Dunod, Gauthier-Villars, Paris, 1960. |
[22] |
P. L. Lions, "Mathematical Topics in Fluid Dynamics," Vol.1. Compressible models. Oxford Science Publication, Oxford, 1998. |
[23] |
P. L. Lions, "Mathematical Topics in Fluid Dynamics," Vol.2. Compressible models. Oxford Science Publication, Oxford, 1998. |
[24] |
X. Liu and Z. Zhang, Global existence of weak solutions for the incompressible liquid crystals, Chinese Anna. Math., 30 (2009), 1-20. |
[25] |
A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Birkhäuser, Berlin 1995. |
[26] |
A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible and heat-conductives fluids, Proc. Japan Acad., A, 55 (1979), 337-342.
doi: 10.3792/pjaa.55.337. |
[27] | |
[28] |
S. V. Pasechnik, V. G. Chigrinov, D. V. Shmeliova, "Liquid Crystals: Viscous and Elastic Properties," Wiley-VCH, 2009. |
[29] |
D. Serre and J.L. Lions(Rapporteur), Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible, C. R. Acad. Sci. Paris, 303 (1986), 639-642. |
[30] |
M. J. Stephen, Hydrodynamics of liquid crystals, Phys. Rev. A, 2 (1970), 1558-1562. |
[31] |
R. Temam, "Navier-Stokes Equations," rev. ed., Studies in Mathematics and its Applications 2, North-Holland, Amsterdam, 1977. |
[32] |
W. Stewart, "The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction," Liquid crystals bookseries, Taylor & Francis, London, 2004. |
[33] |
Y. Z. Xie, "The Physics of Liquid Crystals," Scientific Press, Beijing, 1988. |
[34] |
D. Wang and C. Yu, Global weak solution and large-time behavior for the compressible flow of liquid crystals, arXiv:1108.4939. |
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