Space–Time Decay Estimates for the Incompressible Flow of Liquid Crystals

  • Qunyi Bie
  • Xuemei DengEmail author
  • Deyi Ma


In this paper, we study decay in space and time for derivatives of solutions to the incompressible flow of liquid crystals. Based on the known time decay results, we further obtain space–time decay rates agreed with those of the heat equation. More precisely, we firstly derive weighted bounds of the director field by weighted energy estimate, and then by regarding the nonlinear term \(\mathrm{div}{(}\nabla {d}\odot \nabla {d}{)}\) as a forcing term in the velocity equation, we could get the weighted bounds for the vorticity, which implies the corresponding estimates for the velocity field. Our results show that the weighted \(L^p\) decay for the velocity is not as good as that of both the director field and the vorticity. This is basically due to the presence of the pressure term in the velocity equation, which leads to the restriction on the exponent of the weight.


Liquid crystal flow Space–time decay Weighted estimates 

Mathematics Subject Classification

35Q35 35K15 35B40 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Amrouche, C., Girault, V., Schonbek, M.E., Schonbek, T.P.: Pointwise decay of solutions and of higher derivatives to Navier–Stokes equations. SIAM J. Math. Anal. 31, 740–753 (2000)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Caffarelli, L., Kohn, R., Nirenberg, L.: First order interpolation inequalities with weights. Compos. Math. 53, 259–275 (1984)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Dai, M., Schonbek, M.: Asymptotic behavior of solutions to the liquid crystal system in \({H}^m({R}^3)\). SIAM J. Math. Anal. 46, 3131–3150 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ericksen, J.L.: Conservation laws for liquid crystals. Trans. Soc. Rheol. 5, 23–34 (1961)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ericksen, J.L.: Hydrostatic theory of liquid crystals. Arch. Ration. Mech. Anal. 9, 371–378 (1962)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gao, J., Tao, Q., Yao, Z.-A.: Asymptotic behavior of solution to the incompressible nematic liquid crystal flows in \({R}^3\). arXiv:1412.0498v2
  7. 7.
    Guo, Y., Wang, Y.: Decay of dissipative equations and negative Sobolev spaces. Commun. Partial Differ. Equ. 37, 2165–2208 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hong, M.-C.: Global existence of solutions of the simplified Ericksen–Leslie system in dimension two. Calc. Var. Partial Differ. Equ. 40, 15–36 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kawashima, S., Nishibata, S., Nishikawa, M.: \({L}^p\) energy method for multi-dimensional viscous conservation laws and application to the stability of planar waves. J. Hyperbolic Differ. Equ. 1, 581–603 (2004)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kukavica, I.: Space-time decay for solutions of the Navier–Stokes equations. Indiana Univ. Math. J. 50, 205–222 (2001)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kukavica, I.: On the weighted decay for solutions of the Navier–Stokes system. Nonlinear Anal. Theory Methods Appl. 70, 2466–2470 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kukavica, I., Torres, J.J.: Weighted bounds for the velocity and the vorticity for the Navier–Stokes equations. Nonlinearity 19, 293–303 (2006)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Kukavica, I., Torres, J.J.: Weighted \({L}^p\) decay for solutions of the Navier–Stokes equations. Commun. Partial Differ. Equ. 32, 819–831 (2007)CrossRefGoogle Scholar
  14. 14.
    Leslie, F.M.: Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal. 28, 265–283 (1968)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lin, F.: Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena. Commun. Pure Appl. Math. 42, 789–814 (1989)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lin, F., Lin, J., Wang, C.: Liquid crystal flows in two dimensions. Arch. Ration. Mech. Anal. 197, 297–336 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lin, F., Liu, C.: Nonparabolic dissipative systems modeling the flow of liquid crystals. Commun. Pure Appl. Math. 48, 501–537 (1995)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lin, F., Wang, C.: Global existence of weak solutions of the nematic liquid crystal flow in dimension three. Commun. Pure Appl. Math. 69, 101–139 (2015)MathSciNetGoogle Scholar
  19. 19.
    Miyakawa, T.: On space-time decay properties of nonstationary incompressible Navier–Stokes flows in \(\mathbb{R}^n\). Funkcial Ekvac. 43, 541–558 (2000)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Schonbek, M.E., Schonbek, T.P.: On the boundedness and decay of moments of solutions to the Navier–Stokes equations. Adv. Differ. Equ. 5, 861–898 (2000)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Takahashi, S.: A weighted equation approach to decay rate estimates for the Navier–Stokes equations. Nonlinear Anal. 37, 751–789 (1999)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Weng, S.: Space-time decay estimates for the incompressible viscous resistive MHD and Hall-MHD equations. J. Funct. Anal. 270, 2168–2187 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Wiegner, M.: Decay results for weak solutions of the Navier–Stokes equations on \({R}^n\). J. Lond. Math. Soc. 35, 303–313 (1987)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.College of Science and Three Gorges Mathematical Research CenterChina Three Gorges UniversityYichangPeople’s Republic of China

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