Research article

Blow-up criterion for the 3D nematic liquid crystal flows via one velocity and vorticity components and molecular orientations

  • Received: 05 October 2019 Accepted: 02 December 2019 Published: 16 December 2019
  • MSC : 35B65, 35Q35, 76A15

  • In this paper, we are devoted to investigating the blow-up criteria for the three dimensional nematic liquid crystal flows. More precisely, we proved that the smooth solution $(u, d)$ can be extended beyond T, provided that $\int_{0}^{T}(||\omega_{3}||_{L^{p}}^{\frac{2p}{2p-3}}+||u_{3}||_{L^{q}}^{\frac{2q}{q-3}} +||\nabla d||_{\dot{B}_{\infty, \infty}^{0}}^{2})d t < \infty, \frac{3}{2} < p\leq\infty, 3 < q\leq\infty.$

    Citation: Qiang Li, Baoquan Yuan. Blow-up criterion for the 3D nematic liquid crystal flows via one velocity and vorticity components and molecular orientations[J]. AIMS Mathematics, 2020, 5(1): 619-628. doi: 10.3934/math.2020041

    Related Papers:

  • In this paper, we are devoted to investigating the blow-up criteria for the three dimensional nematic liquid crystal flows. More precisely, we proved that the smooth solution $(u, d)$ can be extended beyond T, provided that $\int_{0}^{T}(||\omega_{3}||_{L^{p}}^{\frac{2p}{2p-3}}+||u_{3}||_{L^{q}}^{\frac{2q}{q-3}} +||\nabla d||_{\dot{B}_{\infty, \infty}^{0}}^{2})d t < \infty, \frac{3}{2} < p\leq\infty, 3 < q\leq\infty.$



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    [1] H. Bahouri, J. Y. Chemin, R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Heidelberg: Springer, 2011.
    [2] J. T. Beale, T. Kato, A. Majda, Remarks on breakdown of smooth solutions for the 3D Euler equations, Commun. Math. Phys., 94 (1984), 61-66.
    [3] C. S. Cao, Sufficient conditions for the regularity to the 3D Navier-Stokes equations, Discrete Contin. Dyn. Syst., 26 (2010), 1141-1151.
    [4] C. S. Cao, E. S. Titi, Regularity criteria for the three-dimensional Navier-Stokes equations, Indiana Univ. Math. J., 57 (2008), 2643-2661.
    [5] P. Constantin, C. Fefferman, Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana Univ. Math. J., 42 (1993), 775-789.
    [6] L. Escauriaza, G. Seregin, V. Šverák, Backward uniqueness for parabolic equations, Arch. Ration. Mech. Anal., 169 (2003), 147-157.
    [7] B. Q. Dong, Z. F. Zhang, The BKM criterion for the 3D Navier-Stokes equations via two velocity components, Nonlinear Anal. Real, 11 (2010), 2415-2421.
    [8] J. L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961) 23-34.
    [9] H. Kozono, T. Ogawa, Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z., 242 (2002), 251-278.
    [10] T. Huang, C. Y. Wang, Blow up criterion for nematic liquid crystal flows, Commun. Part. Diff. Eq., 37 (2012), 875-884.
    [11] T. Kato, G. Ponce, Commutator estimates and the Euler and the Navier-Stokes equations, Commun. Pur. Appl. Math., 41 (1988), 891-907.
    [12] F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.
    [13] F. M. Leslie, Theory of Flow Phenomenon in Liquid Crystals, In: Advances in Liquid Crystals, New York: Academic Press, 4 (1979), 1-81.
    [14] F. H. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena, Commun. Pur. Appl. Math., 42 (1989), 789-814.
    [15] F. H. Lin, C. Liu, Nonparabolic dissipative systems modelling the flow of liquid crystals, Commun. Pur. Appl. Math., 48 (1995), 501-537.
    [16] F. H. Lin, C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals, Discrete Contin. Dynam. Syst., 2 (1996), 1-22.
    [17] G. Prodi, Un teorema di unicit per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48 (1959), 173-182.
    [18] P. Penel, M. Pokorný, Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity, Appl. Math., 49 (2004), 483-493.
    [19] C. Y. Qian, Remarks on the regularity criterion for the nematic liquid crystal flows in $ \mathbb{R}^3$, Appl. Math. Comput., 274 (2016), 679-689.
    [20] C. Y. Qian, A further note on the regularity criterion for the 3D nematic liquid crystal flows, Appl. Math. Comput., 290 (2016), 258-266.
    [21] J. Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Probl. Proc. Symp., 1963 (1963), 69-98.
    [22] H. Beirão da Veiga, A new regularity class for the Navier-Stokes equations in $\mathbb{R}^n$, Chinese Ann. Math. Ser. B, 16 (1995), 407-412.
    [23] H. Beirão da Veiga, L. Berselli, On the regularzing effect of the vorticity direction in incompressible viscous flows, Differ. Integral Equ., 15 (2002), 345-356.
    [24] C. Y. Wang, Heat flow of harmonic maps whose gradients belong to $L_{x}^{n} L_{t}^{\infty}$, Arch. Ration. Mech. Anal., 188 (2008), 309-349.
    [25] H. Y. Wen, S. J. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals, Nonlinear Anal. Real, 12 (2011), 1510-1531.
    [26] R. Y. Wei, Z. A. Yao, Y. Li, Regularity criterion for the nematic liquid crystal flows in terms of velocity, Abst. Appl. Anal., 2014 (2014), 234809.
    [27] B. Q. Yuan, C. Z. Wei, BKM's criterion for the 3D nematic liquid crystal flows in Besov spaces of negative regular index, J. Nonlinear Sci. Appl., 10 (2017), 3030-3037.
    [28] B. Q. Yuan, C. Z. Wei, Global regularity of the generalized liquid crystal model with fractional diffusion, J. Math. Anal. Appl., 467 (2018), 948-958.
    [29] J. H. Zhao, BKM's criterion for the 3D nematic liquid crystal flows via two velocity components and molecular orientations, Math. Method. Appl. Sci., 40 (2016), 871-882.
    [30] Z. F. Zhang, Q. L. Chen, Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equationsin $\mathbb{R}^3$, J. Differ. Equations, 216 (2005), 470-481.
    [31] Z. J. Zhang, G. Zhou, Serrin-type regularity criterion for the Navier-Stokes equations involving one velocity and one vorticity component, Czech. Math. J., 68 (2018), 219-225.
    [32] Y. Zhou, A new regularity criterion for weak solutions to the Navier-Stokes equations, J. Math. Pure. Appl., 84 (2005), 1496-1514.
    [33] Y. Zhou, M. Pokorný, On the regularity to the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity, 23 (2010), 1097-1107.
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