Abstract
The three-dimensional equations for the compressible flow of liquid crystals are considered. An initial-boundary value problem is studied in a bounded domain with large data. The existence and large-time behavior of a global weak solution are established through a three-level approximation, energy estimates, and weak convergence for the adiabatic exponent \({\gamma > \frac{3}{2}}\) .
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Caffarelli L., Kohn R., Nirenberg L.: Partial regularity of suitable weak solutions of Navier-Stokes equations. Commun. Pure Appl. Math. 35, 771–831 (1982)
Chandrasekhar S.: Liquid Crystals, 2nd edn. University Press, Cambridge (1992)
Ding, S., Lin, J., Wang, C., Wen, H.: Compressible hydrodynamic flow of liquid crystals in 1-D. Discrete Contin. Dyn. Syst. A (to appear)
DE Gennes, P. G.: The Physics of Liquid Crystals. Oxford, 1974
Ericksen J. L.: Hydrostatic theory of liquid crystal. Arch. Rational Mech. Anal. 9, 371–378 (1962)
Feireisl E.: Dynamics of Viscous Compressible Fluids Oxford Lecture Series in Mathematics and its Applications, vol 26 Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (2004)
Feireisl E., Novotný A., Petzeltová H.: On the existence of globally defined weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech. 3(4), 358–392 (2001)
Feireisl E., Petzeltová H.: Large-time behavior of solutions to the Navier-Stokes equations of compressible flow. Arch. Rational Mech. Anal. 150(1), 77–96 (1999)
Feireisl E., Rocca E., Schimperna G.: On a non-isothermal model for nematic liquid crystal.. Nonlinearity 24(1), 243–257 (2011)
Gilbarg D., Trudinger N. S.: Elliptic Partial Differential Equation of Second Order, 2nd edn. Grundlehren der Mathematischen Wissenschaften 224. Springer, Berlin (1983)
Hardt R., Kinderlehrer D.: Mathematical Questions of Liquid Crystal Theory The IMA Volumes in Mathematics and its Applications 5. Springer, New York (1987)
Hardt R., Kinderlehrer D., Lin F.-H.: Existence and partial regularity of static liquid crystal configurations. Commun. Math. Phys. 105(4), 547–570 (1986)
Hu X., Wang D.: Global Existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows. Arch. Rational Mech. Anal. 197(1), 203–238 (2010)
Hu X., Wang D.: Global solution to the three-dimensional incompressible flow of liquid crystals. Commun. Math. Phys. 296(3), 861–880 (2010)
Jiang F., Tan Z.: Global weak solution to the flow of liquid crystals system. Math. Methods Appl. Sci. 32(17), 2243–2266 (2009)
Leslie F.M.: Some constitutive equations for liquid crystals. Arch. Rational Mech. Anal. 28(4), 265–283 (1968)
Lin F.-H.: Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena. Commun. Pure Appl. Math. 42(6), 789–814 (1989)
Lin, F.-H.: Mathematics theory of liquid crystals. In: Applied Mathematics at the Turn of the Century. Lecture Notes of the 1993 Summer School. Universidat Complutense de Madrid, Madrid, 1995
Lin F.-H., Liu C.: Nonparabolic dissipative systems modeling the flow of liquid crystals. Commun. Pure Appl. Math. 48(5), 501–537 (1995)
Lin F.-H., Liu C.: Partial regularity of the dynamic system modeling the flow of liquid crystals. Discrete Contin. Dyn. Syst. 2(1), 1–22 (1996)
Lin F.-H., Lin J., Wang C.: Liquid crystal flows in two dimensions. Arch. Rational Mech. Anal. 197(1), 297–336 (2010)
Lions, P.-L.: Mathematical topics in fluid mechanics, vol. 2. Compressible models. Oxford Lecture Series in Mathematics and its Applications, vol. 10. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998
Lions J.-L.: Quelques méthodes de résolution des problèms aux limites nonlinéaires. Dunod, Gauthier-Villars, Paris (1960)
Liu C., Walkington N. J.: Approximation of liquid crystal flow. SIAM J. Numer. Anal. 37(3), 725–741 (2000)
Liu, X.-G., Qing, J.: Existence of globally weak solutions to the flow of compressible liquid crystals system. Preprint, 2011
Sun H., Liu C.: On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete Contin. Dyn. Syst. 23(1–2), 455–475 (2009)
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Wang, D., Yu, C. Global Weak Solution and Large-Time Behavior for the Compressible Flow of Liquid Crystals. Arch Rational Mech Anal 204, 881–915 (2012). https://doi.org/10.1007/s00205-011-0488-x
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DOI: https://doi.org/10.1007/s00205-011-0488-x