Skip to main content
SpringerLink
Log in

A Sixth-Order Phase-Field Equation with Degenerate Mobility

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

Abstract

In this paper, we study the weak solutions of a sixth-order phase-field equation with degenerate phase-dependent diffusion mobility in 2D case. The main features and difficulties of this equation are given by a highly nonlinear sixth-order elliptic term, a strong constraint imposed by the presence of the nonlinear principal part and the lack of maximum principle. Based on the Schauder-type estimates and entropy estimates, we are able to prove the global existence of classical solutions for regularized problems. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions by using approximation and compactness tools. In the end, we study the nonnegativity of solutions for the sixth-order degenerate phase-field equation.

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. Bernis, F., Friedman, A.: Higher order nonlinear degenerate parabolic equations. J. Diff. Equ. 83, 179–206 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bonetti, E., Dreyer, W., Schimperna, G.: Global solutions to a generalized Cahn-Hilliard equation with viscosity. Adv. Dif. Equ. 8, 231–256 (2003)

    MathSciNet  MATH  Google Scholar 

  3. Chen, F., Shen, J.: Efficient energy stable schemes with spectral discretization in space for anisotropic Cahn-Hilliard systems. Commun. Comput. Phys. 13, 1189C1208 (2013)

    MathSciNet  Google Scholar 

  4. Dai, S., Du, Q.: Weak solutions for the Cahn-Hilliard equation with degenerate mobility. Arch. Ration. Mech. Anal. 219, 1161–1184 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Eidelman, S.D.: Parabolic Systems. North-Holland, Amsterdam (1969)

    Google Scholar 

  6. Elliott, C.M., Garcke, H.: On the Cahn-Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27, 404–423 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Evans, J.D., Galaktionov, V.A., King, J.R.: Unstable sixth-order thin film equation: I. Blow-up similarity solutions. Nonlinearity 20, 1799–1841 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Friedman, A.: Interior estimates for parabolic systems of partial differential equations. J. Math. Mech. 7, 393–418 (1958)

    MathSciNet  MATH  Google Scholar 

  9. Huang, R., Yin, J., Wang, L.: Non-blow-up phenomenon for the Cahn-Hilliard equation with non-constant mobility. J. Math. Anal. Appl. 379, 58–64 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jüngel, A., Milisić, J.: A sixth-order nonlinear parabolic equation for quantum systems. SIAM J. Math. Anal. 41, 1472–1490 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Korzec, M.D., Evans, P.L., Munch, A., Wagner, B.: Stationary solutions of driven fourth- and sixth-order Cahn-Hilliard-type equations. SIAM J. Appl. Math. 69, 348–367 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Liu, C.: a sixth-order thin film equation in two space dimensions. Adv. Diff. Equ. 20, 557–580 (2015)

    MathSciNet  MATH  Google Scholar 

  14. Lussardi, L.: A note on a phase-field model for anisotropic systems. Asymptot. Anal. 94, 241C254 (2015)

    MathSciNet  MATH  Google Scholar 

  15. Makki, A., Miranville, A.: Well-posedness for one-dimensional anisotropic Cahn-Hilliard and Allen-Cahn systems. Electron. J. Diff. Equ. 2015(04), 1–15 (2015)

  16. Miranville, A.: Asymptotic behavior of a sixth-order Cahn-Hilliard system. Cent. Eur. J. Math. 12, 141–154 (2014)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Torabi, S., Lowengrub, J., Voigt, A., Wise, S.: A new phase-field model for strongly anisotropic systems. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465, 1337–1359 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Yin, J.: On the existence of nonnegative continuous solutions of the Cahn-Hilliard equation. J. Diff. Equ. 97, 310–327 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  20. Yin, J.: On the Cahn-Hilliard equation with nonlinear principal part. J. Partial Diff. Equ. 7, 77–96 (1994)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are indebted to the referees for careful reading of the paper and helpful suggestions. This work is partially supported by “Fundamental Research Funds for the Central Universities (Grant No. JUSRP116030),” “Natural Science Foundation of China for Young Scholar (Grant No. 11401258)” and “Natural Science Foundation of Jiangsu Province for Young Scholar (Grant No. BK20140130).”

Author information

Authors and Affiliations

  1. School of Science, Jiangnan University, 214122, Wuxi, China

    Ning Duan & Xiaopeng Zhao

  2. P. O. 239, No. 717 Jiefang Road Qiaokou District, 430000, Wuhan, China

    Yujuan Cui

Authors
  1. Ning Duan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yujuan Cui
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaopeng Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaopeng Zhao.

Additional information

Communicated by Yong Zhou.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Duan, N., Cui, Y. & Zhao, X. A Sixth-Order Phase-Field Equation with Degenerate Mobility. Bull. Malays. Math. Sci. Soc. 42, 79–103 (2019). https://doi.org/10.1007/s40840-017-0468-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-017-0468-x

Keywords

Mathematics Subject Classification

Search

Navigation