Bulletin of Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

SPECTRAL APPROXIMATIONS OF ATTRACTORS FOR CONVECTIVE CAHN-HILLIARD EQUATION IN TWO DIMENSIONS

  • ZHAO, XIAOPENG
  • Received : 2014.02.24
  • Published : 2015.09.30

Abstract

In this paper, the long time behavior of the convective Cahn-Hilliard equation in two dimensions is considered, semidiscrete and completely discrete spectral approximations are constructed, error estimates of optimal order that hold uniformly on the unbounded time interval $0{\leq}t<{\infty}$ are obtained.

Keywords

spectral methods;convective Cahn-Hilliard equation;global attractor;error estimates

File

Download PDF

References

  1. C. Canuto and A. Qarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp. 38 (1982), no. 157, 67-86. https://doi.org/10.1090/S0025-5718-1982-0637287-3
  2. D. S. Cohen and J. D. Murry, A generalized diffusion model for growth and dispersal in a population, J Math. Biol. 12 (1981), no. 2, 237-249. https://doi.org/10.1007/BF00276132
  3. A. Eden and V. K. Kalantarov, The convective Cahn-Hilliard equation, Appl. Math. Lett. 20 (2007), no. 4, 455-461. https://doi.org/10.1016/j.aml.2006.05.014
  4. A. Eden and V. K. Kalantarov, 3D convective Cahn-Hilliard equation, Commun. Pure Appl. Anal. 6 (2007), no. 4, 1075-1086. https://doi.org/10.3934/cpaa.2007.6.1075
  5. C. M. Elliott and S. Larsson, Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation, Math. Comp. 58 (1992), no. 198, 603-630. https://doi.org/10.1090/S0025-5718-1992-1122067-1
  6. C. M. Elliott and S. M. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal. 96 (1986), no. 4, 339-357.
  7. J. K. Hale, X. B. Lin, and G. Raugel, Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Math. Comp. 50 (1988), no. 181, 89-123. https://doi.org/10.1090/S0025-5718-1988-0917820-X
  8. C. C. Liu, On the convective Cahn-Hilliard equation with degenerate mobility, J. Math. Anal. Appl. 344 (2008), no. 1, 124-144. https://doi.org/10.1016/j.jmaa.2008.02.027
  9. S. J. Lu, The spectral method for the long-time behavior of a generalized KdV-Burgers equation, Math. Numer. Sin. 21 (1999), no. 2, 129-138.
  10. S. J. Lu and Q. S. Lu, Fourier spectral approximation to long-time behavior of dissipative generalized Kdv-Burgers equations, SIAM J. Numer. Anal. 44 (2006), no. 2, 561-585. https://doi.org/10.1137/S0036142903426671
  11. V. G. Maeja, Sovolev Space, Springer-Verlag, New York, 1985.
  12. A. Novick-Cohen and L. A. Segel, Nonlinear aspects of the Cahn-Hilliard equation, Phy. D. 10 (1984), no. 3, 277-298. https://doi.org/10.1016/0167-2789(84)90180-5
  13. J. Shen, Long-time stability and convergence for fully discrete nonlinear Galerkin methods, Appl. Anal. 38 (1990), no. 4, 201-209. https://doi.org/10.1080/00036819008839963
  14. R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, Vol 68, Springer Verlag, New York, 1988.
  15. M. A. Zaks, A. Podolny, A. A. Nepomnyashchy, and A. A. Golovin, Periodic stationary patterns governed by a convective Cahn-Hilliard equation, SIAM J. Appl. Math. 66 (2005), no. 2, 700-720. https://doi.org/10.1137/040615766
  16. F. Y. Zhang, Spectral approximations of attractors of generalized KdV-Burgers equations, Numer. Math. J. Chinese Univ. 21 (1999), no. 1, 32-47.
  17. X. P. Zhao and B. Liu, The existence of global attractor for convective Cahn-Hilliard equation, J. Korean Math. Soc. 49 (2012), no. 2, 357-378. https://doi.org/10.4134/JKMS.2012.49.2.357
  18. X. P. Zhao and C. C. Liu, Optimal control of the convective Cahn-Hilliard equation, Appl. Anal. 92 (2013), no. 5, 1028-1045. https://doi.org/10.1080/00036811.2011.643786
  19. X. P. Zhao and C. C. Liu, Optimal control for the convective Cahn-Hilliard equation in 2D case, Appl. Math. Optim. 70 (2014), no. 1, 61-82. https://doi.org/10.1007/s00245-013-9234-0

Acknowledgement

Supported by : Natural Science Foundation of China