# American Institute of Mathematical Sciences

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

## 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

Received  July 2011 Revised  October 2011 Published  September 2012

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.
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.

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##### 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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