zbMATH — the first resource for mathematics

Blow up criterion for compressible nematic liquid crystal flows in dimension three. (English) Zbl 1314.76010
Summary: 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.

76A15 Liquid crystals
35Q35 PDEs in connection with fluid mechanics
Full Text: DOI arXiv
[1] 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) · Zbl 0573.76029
[2] Bourguignon J., Brezis H.: Remarks on the Euler equation. J. Funct. Anal. 15, 341–363 (1974) · Zbl 0279.58005
[3] Choe H.J., Kim H.: Strong solutions of the Navier-Stokes equations for isentropic compressible fluids. J. Differ. Equ. 190, 504–523 (2003) · Zbl 1022.35037
[4] 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) · Zbl 1080.35066
[5] Chu, Y.M., Liu, X., Liu, X.G.: Strong solutions to the compressible liquid crystal system. Preprint, 2011 · Zbl 1451.76009
[6] 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) · Zbl 1212.35346
[7] 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) · Zbl 1206.35193
[8] 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)
[9] 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) · Zbl 1217.35099
[10] Ericksen J.L.: Hydrostatic theory of liquid crystal. Arch. Rational Mech. Anal. 9, 371–378 (1962) · Zbl 0105.23403
[11] Feireisl E.: Dynamics of Viscous Compressible Fluids. Oxford University Press, Oxford (2004) · Zbl 1080.76001
[12] de Gennes, P.G.: The Physics of Liquid Crystals. Oxford, 1974 · Zbl 0295.76005
[13] Hardt R., Kinderlehrer D., Lin F.: Existence and partial regularity of static liquid crystal configurations. Commun. Math. Phys. 105, 547–570 (1986) · Zbl 0611.35077
[14] Hong M.C.: Global existence of solutions of the simplified Ericksen-Leslie system in $${\(\backslash\)mathbb R\^2}$$ . Calc. Var. 40(1–2), 15–36 (2011) · Zbl 1213.35014
[15] 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 · Zbl 1241.35161
[16] Huang X., Li J., Xin Z.P.: Blowup criterion for viscous baratropic flows with vacuum states. Commun. Math. Phys. 301, 23–35 (2011) · Zbl 1213.35135
[17] Huang, T., Wang, C.Y.: Blow up criterion for nematic liquid crystal flows. Preprint (2011) · Zbl 1247.35103
[18] Huang, T., Wang, C.Y., Wen, H.Y.: Strong solutions of the compressible nematic liquid crystal flow. J. Diff. Equ. (2011, to appear) · Zbl 1233.35168
[19] Lin F.H.: Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena. CPAM 42, 789–814 (1989) · Zbl 0703.35173
[20] Lin F.H., Liu C.: Nonparabolic dissipative systems modeling the flow of liquid crystals. CPAM, XLVIII, 501–537 (1995) · Zbl 0842.35084
[21] Lin F.H., Liu C.: Partial regularity of the dynamic system modeling the flow of liquid cyrstals. DCDS 2(1), 1–22 (1998)
[22] 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
[23] 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) · Zbl 1208.35002
[24] Lin F.H., Wang C.Y.: The Analysis of Harmonic Maps and Their Heat Flows. World Scientific, Hackensack (2008) · Zbl 1203.58004
[25] Lin F.H., Lin J.Y., Wang C.Y.: Liquid crystal flows in two dimensions. Arch. Rational Mech. Anal. 197, 297–336 (2010) · Zbl 1346.76011
[26] 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) · Zbl 0866.76002
[27] 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) · Zbl 0908.76004
[28] 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) · Zbl 0589.76040
[29] Leslie F.M.: Some constitutive equations for liquid crystals. Arch. Rational Mech. Anal. 28, 265–283 (1968) · Zbl 0159.57101
[30] Morro A.: Modelling of nematic liquid crystals in electromagnetic fields. Adv. Theor. Appl. Mech. 2(1), 43–58 (2009) · Zbl 1193.78016
[31] Temam, R.: Navier-Stokes Equations. Theory and Numerical Analysis. Reprint of the 1984 edition. AMS Chelsea Publishing, Providence, 2001
[32] 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) · Zbl 1205.35212
[33] Wang C.Y.: Heat flow of harmonic maps whose gradients belong to $${L\^n_xL\^\(\backslash\)infty_t}$$ . Arch. Rational Mech. Anal. 188, 309–349 (2008) · Zbl 1156.35052
[34] Von Wahl W.: Estimating $${\(\backslash\)nabla u}$$ by div u and curl u. Math. Methods Appl. Sci. 15, 123–143 (1992) · Zbl 0747.31006
[35] Xu, X., Zhang, Z.F.: Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows. Preprint, 2011
[36] Yoshida Z., Giga Y.: Remarks on spectra of operator Rot. Math. Z. 204, 235–245 (1990) · Zbl 0676.47012
[37] 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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.