Globally weak solutions to the flow of compressible liquid crystals system

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February 2013, 33(2): 757-788. doi: 10.3934/dcds.2013.33.757

Globally weak solutions to the flow of compressible liquid crystals system

Xian-Gao Liu ^1, and Jie Qing ^2,

1.	Institute of Mathematics, Fudan University, Shanghai, China
2.	Mathematics Department, UC at Santa Cruz, United States

Received July 2011 Revised October 2011 Published September 2012

Abstract
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We study a simplified system for the compressible fluid of Nematic Liquid Crystals in a bounded domain in three Euclidean space and prove the global existence of the finite energy weak solutions.

Keywords: global existence., compressible, Nematic liquid crystals.

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

Citation: Xian-Gao Liu, Jie Qing. Globally weak solutions to the flow of compressible liquid crystals system. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 757-788. doi: 10.3934/dcds.2013.33.757

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]	J. Phillips, "Liquid Crystals," mcgill. ca, April 2005.
[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]	J. Phillips, "Liquid Crystals," mcgill. ca, April 2005.
[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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