Skip to main content
SpringerLink
Log in

Global Existence of Weak Solutions to a Higher-Order Parabolic System

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

In this paper, we study the coupling of the non-Newtonian Navier–Stokes equation and the oil–water–surfactant equation which stand for a model of a multi-phase incompressible dipolar viscous non-Newtonian fluid under shear. Based on Galerkin approximation and Simon’s compactness results, we obtain the existence and uniqueness of weak solutions of the system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abels, H., Feireisl, E.: On a diffuse interface model for a two-phase flow of compressible viscous fluids. Indiana Univ. Math. J. 57(2), 659–698 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abels, H., Diening, L., Terasawa, Y.: Existence of weak solutions for a diffuse interface model of non-Newtonian two-phase flows. Nonlinear Anal. Real World Appl. 15, 149–157 (2014)

  3. Boyer, F.: Mathematical study of multi-phase flow under shear through order parameter formulation. Asymptot Anal. 20, 175–212 (1999)

    MathSciNet  MATH  Google Scholar 

  4. Ding, S., Li, Y., Luo, W.: Global solutions for a coupled compressible Navier–Stokes/Allen–Cahn system in 1D. J. Math. Fluid Mech. 15(2), 335–360 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Frigeri, S., Grasselli, M.: Global and trajectory attractors for a nonlocal Cahn–Hilliard–Navier-Stokes system. J. Dyn. Diff. Equat. 24, 827–856 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gal, C.G., Grasselli, M.: Asymptotic behavior of a Cahn–Hilliard–Navier–Stokes system in \(2D\). Ann. Inst. H. Poincaré Anal. Non Linéaire 27, 401–436 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gal, C.G., Grasselli, M.: Instability of two-phase flows: a lower bound on the dimension of global attractor of the Cahn–Hilliard–Navier–Stokes system. Phys. D 240, 629–635 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gompper, G., Goos, J.: Fluctuating interfaces in microemulsion and sponge phases. Phys. Rev. E 50, 1325–1335 (1994)

    Article  Google Scholar 

  9. Gompper, G., Kraus, M.: Ginzburg–Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations. Phys. Rev. E 47, 4301–4312 (1993)

    Article  Google Scholar 

  10. Gpurtin, M.E., Polignone, D., Viñala, J.: Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci. 6, 8–15 (1996)

    MathSciNet  Google Scholar 

  11. Korzec, M.D., Evans, P.L., Münch, A., Wagner, B.: Stationary solutions of driven fourth-and sixth-order Cahn–Hilliard-type equations. SIAM J. Appl. Math. 69(2), 348–374 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Liu, C.: Regularity of solutions for a sixth order nonlinear parabolic equation in two space dimensions. Ann. Polon. Math 107(3), 271–291 (2013)

  13. Liu, C., Wang, Z.: Time periodic solutions for a sixth order nonlinear parabolic equation in two space dimensions. Commun. Pure Appl. Anal. 13(3), 1087–1104 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu, C., Wang, Z.: Optimal control for a sixth order nonlinear parabolic equation. Math. Methods Appl. Sci. 38(2), 247–262 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Liu, C., Tang, H.: Study of weak solutions for a higher order nonlinear degenerate parabolic equation. Appl. Anal. 95(9), 1845–1872 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Málek, J., Nečas, J., Rokyta, M., Ru̇žika, M.: Weak and Measure Valued Solutions to Evolutionary PDES, Applied Mathematics and Mathematical Computation, vol. 13. Chapman & Hall, London (1996)

    Google Scholar 

  17. Pawłow, I., Zajączkowski, W.: A sixth order Cahn–Hilliard type equation arising in oil-water-surfactant mixtures. Commun. Pure Appl. Anal. 10(6), 1823–1847 (2011)

  18. Colli, P., Frigeri, S., Grasselli, M.: Global existence of weak solutions to a nonlocal Cahn–Hilliard–Navier–Stokes system. J. Math. Anal. Appl. 386, 428–444 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Simon, J.: Compact sets in the space \(L^p(0, T;B)\). Ann. Mat. Pura Appl. 146(4), 65–96 (1987)

    MathSciNet  MATH  Google Scholar 

  20. Zhou, Y., Fan, J.: The vanishing viscosity limit for a \(2D\) Cahn–Hilliard–Navier–Stokes system with a slip boundary condition. Nonlinear Anal. RWA 14(2), 1130–1134 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to express their deep thanks to the referees’ valuable suggestions for the revision and improvement of the manuscript.

Author information

Authors and Affiliations

  1. School of Mathematics, Liaoning Normal University, Dalian, 116029, China

    Aibo Liu

  2. Department of Mathematics, Jilin University, Changchun, 130012, China

    Changchun Liu

Authors
  1. Aibo Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Changchun Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Changchun Liu.

Additional information

Communicated by Norhashidah Hj. Mohd. Ali.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, A., Liu, C. Global Existence of Weak Solutions to a Higher-Order Parabolic System. Bull. Malays. Math. Sci. Soc. 42, 1237–1254 (2019). https://doi.org/10.1007/s40840-017-0543-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-017-0543-3

Keywords

Mathematics Subject Classification

Search

Navigation