Abstract
In this paper, we consider the short time strong solution to a simplified hydrodynamic flow modeling compressible, nematic liquid crystal materials in dimension three. We establish a criterion for possible breakdown of such solutions at a finite time in terms of the temporal integral of both the maximum norm of the deformation tensor of the velocity gradient and the square of the maximum norm of the gradient of a liquid crystal director field.
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Beale J.T., Kato T., Majda A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equation. Commun. Math. Phys. 94, 61–66 (1984)
Bourguignon J., Brezis H.: Remarks on the Euler equation. J. Funct. Anal. 15, 341–363 (1974)
Choe H.J., Kim H.: Strong solutions of the Navier-Stokes equations for isentropic compressible fluids. J. Differ. Equ. 190, 504–523 (2003)
Cho Y., Choe H.J., Kim H.: Unique solvability of the initial boundary value problems for compressible viscous fluids. J. Math. Pures Appl. 83, 243–275 (2004)
Chu, Y.M., Liu, X., Liu, X.G.: Strong solutions to the compressible liquid crystal system. Preprint, 2011
Chen G.Q., Osborne D., Qian Z.: The Navier-Stokes equations with the kinematic and vorticity boundary conditions on non-flat boundaries. Acta Math. Sci. Ser. B Engl. Ed. 29(4), 919–948 (2009)
Chen G.Q., Qian Z.: A study of the Navier-Stokes equations with the kinematic and Navier boundary conditions. Indiana Univ. Math. J. 59(2), 721–760 (2010)
Ding, S.J., Lin, J.Y., Wang, C.Y., Wen, H.Y.: Compressible hydrodynamic flow of liquid crystals in 1D. DCDS Series A (to appear)
Ding S.J., Wang C.Y., Wen H.Y.: Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one. DCDS Series B 2, 357–371 (2011)
Ericksen J.L.: Hydrostatic theory of liquid crystal. Arch. Rational Mech. Anal. 9, 371–378 (1962)
Feireisl E.: Dynamics of Viscous Compressible Fluids. Oxford University Press, Oxford (2004)
de Gennes, P.G.: The Physics of Liquid Crystals. Oxford, 1974
Hardt R., Kinderlehrer D., Lin F.: Existence and partial regularity of static liquid crystal configurations. Commun. Math. Phys. 105, 547–570 (1986)
Hong M.C.: Global existence of solutions of the simplified Ericksen-Leslie system in \({\mathbb R^2}\) . Calc. Var. 40(1–2), 15–36 (2011)
Huang, X., Li, J., Xin, Z.P.: Serrin Type Criterion for the Three-Dimensional Viscous Compressible Flows. Preprint. http://arxiv.org/list/math.AP/1004.4748, 2010
Huang X., Li J., Xin Z.P.: Blowup criterion for viscous baratropic flows with vacuum states. Commun. Math. Phys. 301, 23–35 (2011)
Huang, T., Wang, C.Y.: Blow up criterion for nematic liquid crystal flows. Preprint (2011)
Huang, T., Wang, C.Y., Wen, H.Y.: Strong solutions of the compressible nematic liquid crystal flow. J. Diff. Equ. (2011, to appear)
Lin F.H.: Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena. CPAM 42, 789–814 (1989)
Lin F.H., Liu C.: Nonparabolic dissipative systems modeling the flow of liquid crystals. CPAM, XLVIII, 501–537 (1995)
Lin F.H., Liu C.: Partial regularity of the dynamic system modeling the flow of liquid cyrstals. DCDS 2(1), 1–22 (1998)
Liu, X.G., Liu, L.M.: A blow-up criterion for the compressible liquid crystals system. Preprint. http://arxiv.org/list/math.AP/1011.4399
Lin F.H., Wang C.Y.: On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals. Chinese Ann. Math. B 31(6), 921–938 (2010)
Lin F.H., Wang C.Y.: The Analysis of Harmonic Maps and Their Heat Flows. World Scientific, Hackensack (2008)
Lin F.H., Lin J.Y., Wang C.Y.: Liquid crystal flows in two dimensions. Arch. Rational Mech. Anal. 197, 297–336 (2010)
Lions P.L.: Mathematical topics in fluid mechanics, vol 1 Incompressible models Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications, The Clarendon Press Oxford University Press, New York (1996)
Lions P.L.: Mathematical topics in fluid mechanics, vol. 2. Compressible models Oxford. Lecture Series in Mathematics and its Applications, 10 Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1998)
Ponce G.: Remarks on a paper: “Remarks on the breakdown of smooth solutions for the 3-D Euler equations”. Commun. Math. Phys. 98(3), 349–353 (1985)
Leslie F.M.: Some constitutive equations for liquid crystals. Arch. Rational Mech. Anal. 28, 265–283 (1968)
Morro A.: Modelling of nematic liquid crystals in electromagnetic fields. Adv. Theor. Appl. Mech. 2(1), 43–58 (2009)
Temam, R.: Navier-Stokes Equations. Theory and Numerical Analysis. Reprint of the 1984 edition. AMS Chelsea Publishing, Providence, 2001
Sun Y., Wang C., Zhang Z.: A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations. J. Math. Pures. Appl. 95, 36–47 (2011)
Wang C.Y.: Heat flow of harmonic maps whose gradients belong to \({L^n_xL^\infty_t}\) . Arch. Rational Mech. Anal. 188, 309–349 (2008)
Von Wahl W.: Estimating \({\nabla u}\) by div u and curl u. Math. Methods Appl. Sci. 15, 123–143 (1992)
Xu, X., Zhang, Z.F.: Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows. Preprint, 2011
Yoshida Z., Giga Y.: Remarks on spectra of operator Rot. Math. Z. 204, 235–245 (1990)
Zakharov A.V., Vakulenko A.A.: Orientational dynamics of the compressible nematic liquid crystals induced by a temperature gradient. Phys. Rev. E 79, 011708 (2009)
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Huang, T., Wang, C. & Wen, H. Blow Up Criterion for Compressible Nematic Liquid Crystal Flows in Dimension Three. Arch Rational Mech Anal 204, 285–311 (2012). https://doi.org/10.1007/s00205-011-0476-1
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DOI: https://doi.org/10.1007/s00205-011-0476-1