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On the Existence of Global Attractor for 3D Viscous Cahn-Hilliard Equation

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Abstract

This paper is concerned with the global dynamic for the solution semiflow of the 3D viscous Cahn-Hilliard equation on a bounded domain Ω. We show that the equation has a global attractor in H 4(Ω) when the initial value belongs to H 1(Ω).

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References

  1. Babin, A.V., Vishik, M.I.: Attractor of partial differential evolution equations in an unbounded domain. Proc. R. Soc. Edinb., Sect. A, Math. 116, 221–243 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bai, F., Elliott, C.M., Gardiner, A., Spence, A., Stuart, A.M.: The viscous Cahn-Hilliard equation. I. Computations. Nonlinearity 8, 131–160 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bonfoh, A.S.: Exponential attractors for the viscous Cahn-Hilliard equation in an unbounded domain. Int. J. Evol. Equ. 4, 113–119 (2009)

    MATH  MathSciNet  Google Scholar 

  4. Carvalho, A.N., Dlotko, T.: Dynamics of the viscous Cahn-Hilliard equation. J. Math. Anal. Appl. 344, 703–725 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Choo, S.M., Chung, S.K., Lee, Y.J.: A conservative difference scheme for the viscous Cahn-Hilliard equation with a nonconstant gradient energy coefficient. Appl. Numer. Math. 51, 207–219 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dlotko, T.: Global attractor for the Cahn-Hilliard equation in H 2 and H 3. J. Differ. Equ. 113, 381–393 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  7. Elliott, C.M., Kostin, I.N.: Lower semicontinuity of a non-hyperbolic attractor for the viscous Cahn-Hilliard equation. Nonlinearity 9, 687–702 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Elliott, C.M., Zheng, S.: On the Cahn-Hilliard equation. Arch. Ration. Mech. Anal. 96, 339–357 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  9. Grinfeld, M., Novick-Cohen, A.: The viscous Cahn-Hilliard equation: Morse decomposition and structure of the global attractor. Trans. Am. Math. Soc. 351, 2375–2406 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Hale, J.K.: Asymptotic Behaviour of Dissipative Systems. Am. Math. Soc., Providence (1988)

    Google Scholar 

  11. Ke, Y., Yin, J.: A note on the viscous Cahn-Hilliard equation. Northeast. Math. J. 20, 101–108 (2004)

    MATH  MathSciNet  Google Scholar 

  12. Li, D., Zhong, C.: Global attractor for the Cahn-Hilliard system with fast growing nonlinearity. J. Differ. Equ. 149, 191–210 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  13. Liu, C., Yin, J.: Some properties of solutions for viscous Cahn-Hilliard equation. Northeast. Math. J. 14, 455–466 (1998)

    MATH  MathSciNet  Google Scholar 

  14. Rossi, R.: On two classes of generalized viscous Cahn-Hilliard equations. Commun. Pure Appl. Anal. 4, 405–430 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Sell, R., You, Y.: Dynamics of Evolutionary Equations. Springer, New York (2002)

    Book  MATH  Google Scholar 

  16. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1988)

    Book  MATH  Google Scholar 

  17. Wang, B., Lin, S.: Existence of global attractors for the three-dimensional Brinkman-Forchheimer equation. Math. Methods Appl. Sci. 31, 1479–1495 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  18. Zhao, X., Liu, C.: Optimal control problem for viscous Cahn-Hilliard equation. Nonlinear Anal. 74, 6348–6357 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  19. Zheng, S.: Asymptotic behavior of solution to the Cahn-Hillard equation. Appl. Anal. 21, 165–184 (1986)

    Google Scholar 

  20. Zheng, S., Milani, A.: Global attractors for singular perturbations of the Cahn-Hilliard equations. J. Differ. Equ. 209, 101–139 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

This paper is supported by the “Natural Science Foundation of Jiangsu Province of China for Young Scholar (No. BK20140130)” and the “Fundamental Research Funds for the Central Universities (No. JUSRP11407)”.

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Authors and Affiliations

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

    Xiaopeng Zhao

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

    Changchun Liu

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  1. Xiaopeng Zhao
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  2. Changchun Liu
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Correspondence to Xiaopeng Zhao.

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Zhao, X., Liu, C. On the Existence of Global Attractor for 3D Viscous Cahn-Hilliard Equation. Acta Appl Math 138, 199–212 (2015). https://doi.org/10.1007/s10440-014-9963-3

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  • DOI: https://doi.org/10.1007/s10440-014-9963-3

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