EECT Home
Continuous maximal regularity on singular manifolds and its applications
June 2016, 5(2): 337-348. doi: 10.3934/eect.2016007
A remark on blow up criterion of three-dimensional nematic liquid crystal flows
Yinxia Wang 1,
School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011, China
Received January 2016 Revised March 2016 Published June 2016
In this paper, we study the initial value problem for the three-dimensional nematic liquid crystal flows. Blow up criterion of smooth solutions is established by the energy method, which refines the previous result.
Keywords: blow up criterion., smooth solutions, Nematic liquid crystal flows.
Mathematics Subject Classification: Primary: 76A15; Secondary: 76B0.
Citation: Yinxia Wang. A remark on blow up criterion of three-dimensional nematic liquid crystal flows. Evolution Equations & Control Theory, 2016, 5 (2) : 337-348. doi: 10.3934/eect.2016007
G. Brown and W. Shaw, The Mesomorphic State, Liquid Crystals,, Chem. Rev., 57 (1957), 1049. doi: 10.1021/cr50018a002. Google Scholar
Q. Chen, Z. Tan and G. Wu, LPS's criterion for incompressible nematics liquid crystal flows,, Acta Mathematica Scientia, 34 (2014), 1072. doi: 10.1016/S0252-9602(14)60070-9. Google Scholar
J. Chemin, perfect Incompressible Fluids,, Oxford Lecture Ser. Math. Appl. 14, 14 (1998). Google Scholar
J. Ericksen, Hydrostatic theory of liquid crystal,, Arch. Rational Mech. Anal., 9 (1962), 371. doi: 10.1007/BF00253358. Google Scholar
J. Ericksen and D. Kinderlehrer (eds.), Theory and Applications of Liquid Crystals,, The IMA Volumes in Mathematics and its Applications, 5 (1987), 21. doi: 10.1007/978-1-4613-8743-5. Google Scholar
J. Ericksen, Equilibrium theory of liquid crystals,, Advances in Liquid Crystals, 2 (1976), 233. doi: 10.1016/B978-0-12-025002-8.50012-9. Google Scholar
F. Frank, On the theory of liquid crystals,, Discussions Faraday Soc., 25 (1958), 19. Google Scholar
P. G. de Gennes, The Physics of Liquid Crystals,, Oxford, (1974). Google Scholar
Y. Hao and X. Liu, The existence and blow-up criterion of liquid crystals system in critical Besov space,, Commun. Pure Appl. Anal., 13 (2014), 225. Google Scholar
R. Hardt, D. Kinderlehrer and F. Lin, Existence and partial regularity of static liquid crystal configurations,, Comm. Math. Phys., 105 (1986), 547. doi: 10.1007/BF01238933. Google Scholar
M. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in $\mathbbR^2$,, Cal. Var. Part. Differ. Equ., 40 (2011), 15. doi: 10.1007/s00526-010-0331-5. Google Scholar
T. Huang and C. Wang, Blow up criterion for nematic liquid crystal flows,, Comm. Part. Differ.Equ., 37 (2012), 875. doi: 10.1080/03605302.2012.659366. Google Scholar
T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations,, Comm. Pure Appl. Math., 41 (1988), 891. doi: 10.1002/cpa.3160410704. Google Scholar
H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations,, Math. Z., 242 (2002), 251. doi: 10.1007/s002090100332. Google Scholar
F. Leslie, Some constitutive equations for liquid crystals,, Arch. Rational Mech. Anal., 28 (1968), 265. doi: 10.1007/BF00251810. Google Scholar
F. Leslie, Theory of flow phenomemum in liquid crystal,, In: Brown (ed.) Advances in Liquid Crystals, 4 (1979), 1. Google Scholar
F. Lin, Nonlinear theory of defects in nematic liquid crystal: Phase transition and flow phenomena,, Comm. Pure Appl. Math., 42 (1989), 789. doi: 10.1002/cpa.3160420605. Google Scholar
F. Lin and C. Liu, Nonparabolic Dissipative Systems Modeling the Flow of Liquid Crystals,, Comm. Pure Appl. Math., 48 (1995), 501. doi: 10.1002/cpa.3160480503. Google Scholar
F. Lin, J. Lin and C. Wang, Liquid crystal flows in two dimensions,, Arch. Rational Mech. Anal., 197 (2010), 297. doi: 10.1007/s00205-009-0278-x. Google Scholar
F. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals,, Chin. Ann. Math. Ser. B, 31 (2010), 921. doi: 10.1007/s11401-010-0612-5. Google Scholar
A. Majda and A. Bertozzi, Vorticity and Incompressible Flow,, Cambridge University Press: Cambridge, (2002). Google Scholar
C. Oseen, Die anisotropen Fliissigkeiten, Tatsachen und Theorien,, Forts. Chemie, 21 (1929), 25. Google Scholar
H. Triebel, Theory of Function Spaces,, Monograph in Mathematics, 78 (1983). doi: 10.1007/978-3-0346-0416-1. Google Scholar
Y. Wang and Y. Wang, Blow up criterion for three-dimensional nematic liquid crystal flows with partial viscosity,, Math. Meth. Appl. Sci. 36 (2013), 36 (2013), 60. doi: 10.1002/mma.2569. Google Scholar
Y. Wang, A logarithmically improved blow up criterion for three-dimensional nematic liquid crystal flows with partial viscosity,, Scienceasia, 39 (2013), 73. Google Scholar
Y. X. Wang, Blow-up criteria of smooth solutions to the three-dimensional magneto-micropolar fluid equations,, Boundary Value Problems, 2015 (2015). doi: 10.1186/s13661-015-0382-9. Google Scholar
H. Wen and S. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals,, Nonlinear Analysis: Real World Appl., 12 (2011), 1510. doi: 10.1016/j.nonrwa.2010.10.010. Google Scholar
Y. Zhang, Z. Tan and G. Wu, Blow up criterion for incompressible nematics liquid crystal flows,, preprint, (). Google Scholar
Z. Zhang, S. Liu, J. Pan and L. Ma, A refined blow up criterion for the nematics liquid crystals,, Int. J. Contemp. Math. Sciences, 9 (2014), 441. doi: 10.12988/ijcms.2014.4438. Google Scholar
Chun Liu, Huan Sun. On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 455-475. doi: 10.3934/dcds.2009.23.455
Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161
Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318
Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259
Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3093-3116. doi: 10.3934/dcds.2020039
Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439
Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388
Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391
Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264
Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216
Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020448
Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $ \mathbb{R}^4 $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052
Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304
Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268
Yi Guan, Michal Fečkan, Jinrong Wang. Periodic solutions and Hyers-Ulam stability of atmospheric Ekman flows. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1157-1176. doi: 10.3934/dcds.2020313
Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020385
Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366
Haili Yuan, Yijun Hu. Optimal investment for an insurer under liquid reserves. Journal of Industrial & Management Optimization, 2021, 17 (1) : 339-355. doi: 10.3934/jimo.2019114
Yahia Zare Mehrjerdi. A new methodology for solving bi-criterion fractional stochastic programming. Numerical Algebra, Control & Optimization, 2020 doi: 10.3934/naco.2020054
PDF downloads (57)
HTML views (0)
Yinxia Wang