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
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 $\int_{0}^{T}(||\omega_{3}||_{L^{p}}^{\frac{2p}{2p-3}}+||u_{3}||_{L^{q}}^{\frac{2q}{q-3}} +||\nabla d||_{\dot{B}_{\infty, \infty}^{0}}^{2})d t < \infty, \frac{3}{2} < p\leq\infty, 3 < q\leq\infty.$
[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. |