A regularity criterion for liquid crystal flows in terms of the component of velocity and the horizontal derivative components of orientation field

Qiang Li; Baoquan Yuan; Qiang Li; Baoquan Yuan

doi:10.3934/math.2022231

AIMS Mathematics

2022, Volume 7, Issue 3: 4168-4175. doi: 10.3934/math.2022231

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Research article

A regularity criterion for liquid crystal flows in terms of the component of velocity and the horizontal derivative components of orientation field

Qiang Li ¹,
Baoquan Yuan ^{2
,
,}

1.
School of Mathematics and Statistics, Xinyang College, Henan 464000, China
2.
School of Mathematics and Information Science, Henan Polytechnic University, Henan 454000, China

Received: 14 October 2021 Revised: 06 December 2021 Accepted: 07 December 2021 Published: 17 December 2021
MSC : 35B65, 35Q35, 76A15

In this paper, we establish a regularity criterion for the 3D nematic liquid crystal flows. More precisely, we prove that the local smooth solution $ (u, d) $ is regular provided that velocity component $ u_{3} $, vorticity component $ \omega_{3} $ and the horizontal derivative components of the orientation field $ \nabla_{h}d $ satisfy

$ \begin{eqnarray*} \int_{0}^{T}|| u_{3}||_{L^{p}}^{\frac{2p}{p-3}}+||\omega_{3}||_{L^{q}}^{\frac{2q}{2q-3}}+||\nabla_{h} d||_{L^{a}}^{\frac{2a}{a-3}} \mbox{d} t<\infty,\nonumber \\ with\ \ 3< p\leq\infty,\ \frac{3}{2}< q\leq\infty,\ 3< a\leq\infty. \end{eqnarray*} $
- liquid crystal flow,
- velocity component,
- regularity criterion
Citation: Qiang Li, Baoquan Yuan. A regularity criterion for liquid crystal flows in terms of the component of velocity and the horizontal derivative components of orientation field[J]. AIMS Mathematics, 2022, 7(3): 4168-4175. doi: 10.3934/math.2022231

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Abstract

In this paper, we establish a regularity criterion for the 3D nematic liquid crystal flows. More precisely, we prove that the local smooth solution $ (u, d) $ is regular provided that velocity component $ u_{3} $, vorticity component $ \omega_{3} $ and the horizontal derivative components of the orientation field $ \nabla_{h}d $ satisfy

$ \begin{eqnarray*} \int_{0}^{T}|| u_{3}||_{L^{p}}^{\frac{2p}{p-3}}+||\omega_{3}||_{L^{q}}^{\frac{2q}{2q-3}}+||\nabla_{h} d||_{L^{a}}^{\frac{2a}{a-3}} \mbox{d} t<\infty,\nonumber \\ with\ \ 3< p\leq\infty,\ \frac{3}{2}< q\leq\infty,\ 3< a\leq\infty. \end{eqnarray*} $

References

[1]	C. S. Cao, Sufficient conditions for the regularity to the 3D Navier-Stokes equations, Discrete Contin. Dyn. Syst., 26 (2010), 1141–1151. http://dx.doi.org/10.3934/dcds.2010.26.1141 doi: 10.3934/dcds.2010.26.1141
[2]	C. S. Cao, E. S. Titi, Regularity criteria for the three-dimensional Navier-Stokes equations, Indiana Univ. Math. J., 57 (2008), 2643–2661.
[3]	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
[4]	J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Rational Mech. Anal., 9 (1962), 371–378. https://doi.org/10.1007/BF00253358 doi: 10.1007/BF00253358
[5]	J. S. Fan, B. L. Guo, Regularity criterion to some liquid crystal models and the Landau-Lifshitz equations in $\mathbb{R}^{3}$, Sci. China Ser. A-Math., 51 (2008), 1787–1797. https://doi.org/10.1007/s11425-008-0013-3 doi: 10.1007/s11425-008-0013-3
[6]	W. J. Gu, B. Samet, Y. Zhou, A regularity criterion for a simplified non-isothermal model for nematic liquid crystals, Funkcial. Ekvac., 63 (2020), 247–258.
[7]	X. J. Jia, Y. Zhou, Remarks on regularity criteria for the Navier-Stokes equations via one velocity component, Nonlinear Anal.: Real World Appl., 15 (2014), 239–245. https://doi.org/10.1016/j.nonrwa.2013.08.002 doi: 10.1016/j.nonrwa.2013.08.002
[8]	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
[9]	F. H. Lin, C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Commun. Pure Appl. Math., 48 (1995), 501–537. https://doi.org/10.1002/cpa.3160480503 doi: 10.1002/cpa.3160480503
[10]	F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265–283. https://doi.org/10.1007/BF00251810 doi: 10.1007/BF00251810
[11]	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
[12]	Q. Liu, J. H. Zhao, S. B. Cui, A regularity criterion for the three-dimensional nematic liquid crystal flow in terms of one directional derivative of the velocity, J. Math. Phys., 52 (2011), 033102. https://doi.org/10.1063/1.3567170 doi: 10.1063/1.3567170
[13]	C. Y. Qian, Remarks on the regularity criterion for the nematic liquid crystal flows in $\mathbb{R}^{3}$, Appl. Math. Lett., 274 (2016), 679–689. https://doi.org/10.1016/j.amc.2015.11.007 doi: 10.1016/j.amc.2015.11.007
[14]	C. Y. Qian, A remark on the global regularity for the 3D Navier-Stokes equations, Appl. Math. Comput., 57 (2016), 126–131. https://doi.org/10.1016/j.aml.2016.01.016 doi: 10.1016/j.aml.2016.01.016
[15]	C. Y. Qian, A further note on the regularity criterion for the 3D nematic liquid crystal flows, Appl. Math. Comput., 290 (2016), 258–266. https://doi.org/10.1016/j.amc.2016.06.011 doi: 10.1016/j.amc.2016.06.011
[16]	R. Y. Wei, Y. Li, Z. A. Yao, Two new regularity criteria for nematic liquid crystal flows, J. Math. Anal. Appl., 424 (2015), 636–650. https://doi.org/10.1016/j.jmaa.2014.10.089 doi: 10.1016/j.jmaa.2014.10.089
[17]	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
[18]	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. http://dx.doi.org/10.22436/jnsa.010.06.17 doi: 10.22436/jnsa.010.06.17
[19]	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
[20]	Z. J. Zhang, Serrin-type regularity criterion for the Navier-Stokes equations involving one velocity and one vorticity component, Czech. Math. J., 68 (2018), 219–225. https://doi.org/10.21136/CMJ.2017.0419-16 doi: 10.21136/CMJ.2017.0419-16
[21]	Z. J. Zhang, F. Alzahrani, T. Hayat, Y. Zhou, Two new regularity criteria for the Navier-Stokes equations via two entries of the velocity Hessian tensor, Appl. Math. Lett., 37 (2014), 124–130. https://doi.org/10.1016/j.aml.2014.06.011 doi: 10.1016/j.aml.2014.06.011
[22]	L. L. Zhao, F. Q. Li, On the regularity criteria for 3-D liquid crystal flows in terms of the horizontal derivative components of the pressure, J. Math. Rese. Appl., 40 (2020), 165–168.
[23]	Y. Zhou, M. Pokorný, On the regularity to the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity, 23 (2010), 1097–1107.
[24]	Z. J. Zhang, W. J. Yuan, Y. Zhou, Some remarks on the Navier-Stokes equations with regularity in one direction, Appl. Math., 64 (2019), 301–308. https://doi.org/10.21136/AM.2019.0264-18 doi: 10.21136/AM.2019.0264-18

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