In this paper, we obtain a regularity criterion via horizontal components of velocity and molecular orientations for the 3D nematic liquid crystal flows. That is the smooth solution $ (u, d) $ can be extended beyond T, provided that $ \int_{0}^{T}(||u_{h}||_{\dot{B}_{\infty, \infty}^{0}}^{2}+||\nabla d||_{\dot{B}_{\infty, \infty}^{0}}^{2}) \mbox{d} t < \infty $.
Citation: Qiang Li, Mianlu Zou. A regularity criterion via horizontal components of velocity and molecular orientations for the 3D nematic liquid crystal flows[J]. AIMS Mathematics, 2022, 7(5): 9278-9287. doi: 10.3934/math.2022514
In this paper, we obtain a regularity criterion via horizontal components of velocity and molecular orientations for the 3D nematic liquid crystal flows. That is the smooth solution $ (u, d) $ can be extended beyond T, provided that $ \int_{0}^{T}(||u_{h}||_{\dot{B}_{\infty, \infty}^{0}}^{2}+||\nabla d||_{\dot{B}_{\infty, \infty}^{0}}^{2}) \mbox{d} t < \infty $.
[1] | H. Bahouri, J. Y. Chemin, R. Danchin, Fourier analysis and nonlinear partial differential equations, Berlin, Heidelberg: Springer, 2011. https://doi.org/10.1007/978-3-642-16830-7 |
[2] | B. Q. Dong, Z. F. Zhang, The BKM criterion for the 3D Navier-Stokes equations via two velocity components, Nonlinear Anal. Real World Appl., 11 (2010), 2415–2421. https://doi.org/10.1016/j.nonrwa.2009.07.013 doi: 10.1016/j.nonrwa.2009.07.013 |
[3] | S. Gala, M. A. Ragusa, A new regularity criterion for the Navier-Stokes equations in terms of the two components of the velocity, Electron. J. Qual. Theory Differ. Equ., 2016 (2016), 1–9. https://doi.org/10.14232/ejqtde.2016.1.26 doi: 10.14232/ejqtde.2016.1.26 |
[4] | T. Huang, C. Y. Wang, Blow up criterion for nematic liquid crystal flows, Commmun. Part. Differ. Equ., 37 (2012), 875–884. https://doi.org/10.1080/03605302.2012.659366 doi: 10.1080/03605302.2012.659366 |
[5] | 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. https://doi.org/10.1007/s002090100332 doi: 10.1007/s002090100332 |
[6] | T. Kato, G. Ponce, Commutator estimates and the Euler and the Navier-Stokes equations, Commun. Pure Appl. Math., 41 (1988), 891–907. https://doi.org/10.1002/cpa.3160410704 doi: 10.1002/cpa.3160410704 |
[7] | Q. Li, B. Q. Yuan, Blow-up criterion for the 3D nematic liquid crystal flows via one velocity and vorticity components and molecular orientations, AIMS Math., 5 (2020), 619–628. https://doi.org/10.3934/math.2020041 doi: 10.3934/math.2020041 |
[8] | Q. Li, B. Q. Yuan, A regularity criterion for liquid crystal flows in terms of the component of velocity and the horizontal derivative components of orientation field, AIMS Math., 7 (2022), 4168–4175. https://doi.org/10.3934/math.2022231 doi: 10.3934/math.2022231 |
[9] | F. H. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena, Commun. Pure Appl. Math., 42 (1989), 789–814. https://doi.org/10.1002/cpa.3160420605 doi: 10.1002/cpa.3160420605 |
[10] | Q. Liu, P. Wang, The 3D nematic liquid crystal equations with blow-up criteria in terms of pressure, Nonlinear Anal. Real World Appl., 40 (2018), 290–306. https://doi.org/10.1016/j.nonrwa.2017.08.008 doi: 10.1016/j.nonrwa.2017.08.008 |
[11] | A. J. Majda, A. L. Bertozzi, Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, Cambridge: Cambridge University Press, 2002. https: //doi.org/10.1115/1.1483363 |
[12] | B. Q. Yuan, Q. Li, Note on global regular solution to the 3D liquid crystal equations, Appl. Math. Lett., 109 (2020), 106491. https://doi.org/10.1016/j.aml.2020.106491 doi: 10.1016/j.aml.2020.106491 |
[13] | B. Q. Yuan, X. Li, Regularity of weak solutions to the 3D magneto-micropolar equations in Besov spaces, Acta Appl. Math., 163 (2019), 207–223. https://doi.org/10.1007/s10440-018-0220-z doi: 10.1007/s10440-018-0220-z |
[14] | 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. |
[15] | 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. https://doi.org/10.1016/j.jmaa.2018.07.047 doi: 10.1016/j.jmaa.2018.07.047 |
[16] | 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. https://doi.org/10.1002/mma.4014 doi: 10.1002/mma.4014 |
[17] | X. X. Zheng, A regularity criterion for the tridimensional Navier-Stokes equations in term of one velocity component, J. Differ. Equ., 256 (2014), 283–309. https://doi.org/10.1016/j.jde.2013.09.002 doi: 10.1016/j.jde.2013.09.002 |