Global Well-posedness for the Density-Dependent Incompressible Flow of Liquid Crystals

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Abstract

In the present paper, we consider the global well-posedness of the density-dependent incompressible flow of liquid crystals in \(\mathbb{R}^{2}\). The local existence and uniqueness of the system are obtained without the assumption of small density variation. The global well-posedness is proved when the initial density and liquid crystal orientation are small. However, the initial velocity field is allowed to be arbitrarily large.

Keywords

Global well-posedness Liquid crystal flow Besov space 

Mathematics Subject Classification (2010)

35Q35 35Q30 76D03 

Notes

Acknowledgement

This work is supported by NSFC under grant numbers 11601533 and 11571240.

References

  1. 1.
    Abidi, H.: Équation de Navier-Stokes avec densité et viscosité variables dans l’espace critique. Rev. Mat. Iberoam. 23(2), 537–586 (2007) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bahouri, H., Chemin, J.Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren Math. Wiss., vol. 343. Springer, Berlin (2011) MATHGoogle Scholar
  3. 3.
    Danchin, R., Mucha, P.B.: A Lagrangian approach for the incompressible Navier-Stokes equations with variable density. Commun. Pure Appl. Math. 65(10), 1458–1480 (2012) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Danchin, R., Mucha, P.B.: Incompressible flows with piecewise constant density. Arch. Ration. Mech. Anal. 207(3), 991–1023 (2013) MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    De Anna, F.: Global solvability of the inhomogeneous Ericksen-Leslie system with only bounded density. Anal. Appl. 15(06), 863–913 (2017) MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ericksen, J.L.: Hydrostatic theory of liquid crystal. Arch. Ration. Mech. Anal. 9(1), 371–378 (1962) MathSciNetMATHGoogle Scholar
  7. 7.
    Fan, J., Li, F., Nakamurac, G.: Global strong solution to the 2D density-dependent liquid crystal flows with vacuum. Nonlinear Anal. 97, 185–190 (2014) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hineman, J., Wang, C.: Well-posedness of nematic liquid crystal flow in \(L^{3}_{loc}(\mathbb{R}^{3})\). Arch. Ration. Mech. Anal. 210(1), 177–218 (2013) MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Huang, J., Paicu, M., Zhang, P.: Global solutions to 2-D inhomogeneous Navier-Stokes system with general velocity. J. Math. Pures Appl. 100(1), 806–831 (2013) MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Huang, T., Wang, C.: Blow up criterion for nematic liquid crystal flows. Commun. Partial Differ. Equ. 37(5), 875–884 (2012) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Leslie, F.: Theory of flow phenomenum in liquid crystals. In: The Theory of Liquid Crystals, vol. 4, pp. 1–81. Academic Press, New York (1979) Google Scholar
  12. 12.
    Li, X., Wang, D.: Global solution to the incompressible flow of liquid crystal. J. Differ. Equ. 252(1), 745–767 (2012) MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Li, X., Wang, D.: Global strong solution to the density-dependent incompressible flow of liquid crystals. Trans. Am. Math. Soc. 367(4), 2301–2338 (2015) MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lin, F., Lin, J.: Liquid crystal flow in two dimensions. Arch. Ration. Mech. Anal. 197(5), 297–336 (2010) MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lin, F., Liu, C.: Nonparabolic dissipative systems modeling the flow of liquid crystals. Commun. Pure Appl. Math. 48(5), 501–537 (1995) MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Lin, F., Liu, C.: Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals. Discrete Contin. Dyn. Syst. 2(1), 1–22 (1996) Google Scholar
  17. 17.
    Liu, Q., Liu, S., Tan, W., Zhong, X.: Global well-posedness of the 2D nonhomogeneous incompressible nematic liquid crystal flows. J. Differ. Equ. 261(11), 6521–6569 (2016) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Liu, Q., Zhang, T., Zhao, J.: Global solutions to the 3D incompressible nematic liquid crystal system. J. Differ. Equ. 258(5), 1519–1547 (2015) MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Liu, Q., Zhang, T., Zhao, J.: Well-posedness for the 3D incompressible nematic liquid crystal system in the critical \(L^{p}\) framework. Discrete Contin. Dyn. Syst., Ser. A 36(1), 371–402 (2016) CrossRefMATHGoogle Scholar
  20. 20.
    Liu, S., Zhang, J.: Global well-posedness for the two-dimensional equations of nonhomogeneous incompressible liquid crystal flows with nonnegative density. Discrete Contin. Dyn. Syst., Ser. B 21(8), 2631–2648 (2016) MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Paicu, M., Zhang, P.: Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system. J. Funct. Anal. 262(8), 3556–3584 (2012) MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Wang, C.: Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data. Arch. Ration. Mech. Anal. 200(1), 1–19 (2011) MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Wen, H., Ding, S.: Solutions of incompressible hydrodynamic flow of liquid crystals. Nonlinear Anal., Real World Appl. 12(3), 1510–1531 (2011) MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Xu, F., Hao, S., Yuan, J.: Well-posedness for the density-dependent incompressible flow of liquid crystals. Math. Methods Appl. Sci. 38(12), 2680–2702 (2015) MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Xu, H., Li, Y., Zhai, X.: On the well-posedness of 2-D incompressible Navier-Stokes equations with variable viscosity in critical spaces. J. Differ. Equ. 260(8), 6604–6637 (2016) MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Zhai, X., Li, Y., Yan, W.: Global solution to the 3-D density-dependent incompressible flow of liquid crystals. Nonlinear Anal. 156, 249–274 (2017) MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Zhai, X., Yin, Z.: On the well-posedness of 3-D inhomogeneous incompressible Navier-Stokes equations with variable viscosity. J. Differ. Equ. 264(3), 2407–2447 (2018) MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsShenzhen UniversityShenzhenChina

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