LPS´S CRITERION FOR INCOMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOWS

doi:10.1016/S0252-9602(14)60070-9

Abstract

Abstract:

In this paper we derive LPS´s criterion for the breakdown of classical solutions to the incompressible nematic liquid crystal flow, a simplified version of Ericksen-Leslie system modeling the hydrodynamic evolution of nematic liquid crystals in R3. We show that if 0 < T < +∞ is the maximal time interval for the unique smooth solution u ∈ C^∞([0, T),R³),
then |u| + |∇d| /∈ L^q([0, T], L^p(R³)), where p and q safisfy the Ladyzhenskaya-Prodi-Serrin´s condition: 3/p+2/q= 1 and p ∈ (3,+∞].

Key words: incompressible nematic liquid crystal flow, Ladyzhenskaya-Prodi-Serrin´s cri-terion

CLC Number:

35Q35

CHEN Qing, TAN Zhong, WU Guo-Chun. LPS´S CRITERION FOR INCOMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOWS[J].Acta mathematica scientia,Series B, 2014, 34(4): 1072-1080.

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References

[1] Beale J T, Kato T, Majda A. Remarks on the breakdown of smooth solutions for the 3-D Euler equation. Commun Math Phys, 1984, 94: 61–66

[2] DE Gennes P G. The Physics of Liquid Crystals. Oxford, 1974

[3] Ericksen J L. Hydrostatic theory of liquid crystal. Arch Ration Mech Anal, 1962, 9: 371–378

[4] Escauriaza L, Seregin G, Sver´ak V. L3,1 solutions of the Navier-Stokes equations and backward uniqueness. Russ Math Surv, 2003, 58: 211–250

[5] Hong M C. Global existence of solutions of the simplified Ericksen-Leslie system in R2. Calc Var Partial
Differential Equations, 2011, 40: 15–36

[6] Hong M C, Xin Z P. Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in R2. Adv Math, 2012, 231: 1364–1400

[7] Huang T, Wang C Y. Blow up criterion for nematic liquid crystal flows. Comm Partial Differ Equ, 2012, 37: 875–884

[8] Jiang F, Jiang S, Wang D H. Global weak solutions to the equations of compressible flow of nematic liquid crystals in Two Dimensions. Preprinted

[9] Jiang F, Jiang S, Wang D H. On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain. J Funct Anal, 2013, 265(12): 3369–3397

[10] Jiang F, Tan Z. Global weak solution to the flow of liquid crystals system. Math Methods Appl Sci, 2009, 32(17): 2243–2266

[11] Ladyzhenskaya O. Uniqueness and smoothness of generalized solutions of Navier-Stokes equations. Zap Nauv cn Sem Leningrad Otdel Mat Inst Steklov (LOMI), 1967, 5: 169–185

[12] Leray J. Sur le mouvement d´un liquide visqueux emplissant l´espace. Acta Math, 1934, 63: 193–248

[13] Leslie F M. Some constitutive equations for liquid crystals. Arch Ration Mech Anal, 1962, 28: 265–283

[14] Li J K. Global strong and weak solutions to nematic liquid crystal flow in two dimensions. Nonlinear Anal: TMA, 2014, 99: 80–94

[15] Li J K, Xin Z P. Global Weak Solutions to Non-isothermal Nematic Liquid Crystals in 2D.

[16] Lin F H. Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena. Commun Pure Appl Math, 1989, 42: 789–814

[17] Lin F H, Liu C. Nonparabolic dissipative systems modeling the flow of liquid crystals. Commun Pure Appl Math, 1995, 48: 501–537

[18] Lin F H, Liu C. Partial regularity of the dynamic system modeling the flow of liquid crystals. Discrete Contin Dyn Syst, 1996, 2: 1–22

[19] Lin F H, Lin J, Wang C Y. Liquid crystal flows in two dimensions. Arch Rational Mech Anal, 2010, 197: 297–336

[20] Lin F H, Lin J, Wang C Y. On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals. Chinese Ann Math, 2010, 31B: 921–928

[21] Liu Q, Zhao J H. Logarithmical blow-up criteria for the nematic liquid crystal flows. Nonlinear Anal: RWA, 2014, 16: 178–190

[22] Majda A. Compressible Fluid Flow and System of Conservation Laws in Several Space Variables. Applied Mathematical Sciences, 53. New York: Springer-Verlag, 1984

[23] Prodi G. Un teorema di unicit per le equationi di Navier-Stokes. Ann Mat Pura Appl, 1959, 48(4): 173–182

[24] Serrin J. The initial value problem for the Navier-Stokes equations. Nonlinear Problems (Proc Sympos, Madison, Wis). Madison: University of Wisconsis Press, 1963: 69–98

[25] Xu X, Zhang Z F. Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows. J Differ Equ, 2012, 252: 1169–1181

[26] Zhou Y, Fan J S. A regularity criterion for the nematic liquid crystal flows. J Inequal Appl 2010, Art. ID 589697, 9 pp.