Complex transient patterns on the disk

Discrete and Continuous Dynamical Systems

Volumn 15, Issue 4, 2006, Pages 1049-1078

  Desi, Jonathan P ^a   Sander, Evelyn ^b   Wanner, Thomas ^b  

^a UNIVERSITY OF MARYLAND BALTIMORE COUNTY   (United States)

^b GEORGE MASON UNIVERSITY   (United States)

Author keywords

Cahn hilliard equation; Phase separation; Spinodal decomposition

Indexed keywords

EID: 33745299513     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Conference Paper

References (38)

1. R. A. Adams. Sobolev Spaces. Academic Press, San Diego - London, 1978.
2. R. Aurich, A. Bäcker, R. Schubert, and M. Taglieber. Maximum norms of chaotic quantum eigenstates and random waves. Physica D, 129:1-14, 1999.
3. F. Bai, C. M. Elliott, A. Gardiner, A. Spence, and A. M. Stuart. The viscous Cahn-Hilliard equation. Part I: Computations. Nonlinearity, 8:131-160, 1995.
4. F. Bai, A. Spence, and A. M. Stuart. Numerical computations of coarsening in the one-dimensional Cahn-Hilliard model of phase separation. Physica D, 78:155-165, 1994.
5. H. Bauer. Probability Theory, de Gruyter, Berlin, 1996.
6. J. W. Cahn. Free energy of a nonuniform system. II. Thermodynamic basis. Journal of Chemical Physics, 30:1121-1124, 1959.
7. J. W. Cahn. Phase separation by spinodal decomposition in isotropic systems. Journal of Chemical Physica, 42:93-99, 1965.
8. J. W. Cahn. Spinodal decomposition. Transactions of the Metallurgical Society of AIME, 242:166-180, 1968.
9. J. W. Cahn and J. E. Hilliard. Free energy of a nonuniform system I. Interfacial free energy. Journal of Chemical Physics, 28:258-267, 1958.
10. M. I. M. Copetti. Numerical analysis of nonlinear equations arising in phase transition and thermoelasticity. PhD thesis, University of Sussex, 1991.
11. M. I. M. Copetti and C. M. Elliott. Kinetics of phase decomposition processes: Numerical solutions to Cahn-Hilliard equation. Materials Science and Technology, 6:273-283, 1990.
12. R. Courant and D. Hilbert. Methods of Mathematical Physics. Intersciences, New York, 1953.
13. J. P. Desi. A study of the Cahn-Hilliard equation on the unit disk. Master's thesis, University of Maryland, Baltimore County, 2004.
14. K. R. Elder and R. C. Desai. Role of nonlinearities in off-critical quenches as described by the Cahn-Hilliard model of phase separation. Physical Review B, 40:243-254, 1989.
15. K. R. Elder, T. M. Rogers, and R. C. Desai. Early stages of spinodal decomposition for the Cahn-Hilliard-Cook model of phase separation. Physical Review B, 38:4725-4739, 1988.
16. C. M. Elliott. The Cahn-Hilliard model for the kinetics of phase separation. In J. F. Rodrigues, editor, Mathematical Models for Phase Change Problems, pages 35-73. Birkhäuser, Basel, 1989.
17. C. M. Elliott and D. A. French. Numerical studies of the Cahn-Hilliard equation for phase separation. IMA Journal of Applied Mathematics, 38:97-128, 1987.
18. P. C. Fife. Models for phase separation and their mathematics. Electronic Journal of Differential Equations, 2000(48):1-26, 2000.
19. C. P. Grant. Spinodal decomposition for the Cahn-Hilliard equation. Communications in Partial Differential Equations, 18:453-490, 1993.
20. p bounds for eigenfunctions and spectral projections of the Laplacian near concave boundaries. PhD thesis, University of California, Los Angeles, 1992.
21. D. Grieser. Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary. Communications in Partial Differential Equations, 27(7-8):1283-1299, 2002.
22. D. Henry. Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics. Springer-Verlag, Berlin - Heidelberg - New York, 1981.
23. J. E. Hilliard. Spinodal decomposition. In H. I. Aaronson, editor, Phase Transformations, pages 497-560. American Society for Metals, Metals Park, Ohio, 1970.
24. J. M. Hyde, M. K. Miller, M. G. Hetherington, A. Cerezo, G. D. W. Smith, and C. M. Elliott. Spinodal decomposition in Fe-Cr alloys: Experimental study at the atomic level and comparison with computer models. Acta Metallurgica et Materialia, 43:3385-3426, 1995.
25. J.-P. Kahane, Some Random Series of Functions. Cambridge University Press, Cambridge - London - New York, second edition, 1985.
26. L. J. Landau. Bessel functions: Monotonicity and bounds. Journal of the London Mathematical Society. Second Series, 61(1):197-215, 2000.
27. J. S. Langer. Theory of spinodal decomposition in alloys. Annals of Physics, 65:53-86, 1971.
28. S. Maier-Paape and T. Wanner. Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions. Part I: Probability and wavelength estimate. Communications in Mathematical Physics, 195(2):435-464, 1998.
29. S. Maier-Paape and T. Wanner. Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: Nonlinear dynamics. Archive for Rational Mechanics and Analysis, 151(3):187-219, 2000.
30. R. J. Muirhead. Aspects of Multivariate Statistical Theory. Wiley, New York, 1982.
31. E.-M. Nash. Finite-Elemente und Spektral-Galerkin Verfahren zur numerischen Lösung der Cahn-Hilliard Gleichung und verwandter nichtlinearer Evolutionsgleichungen. PhD thesis, Universität Augsburg, 2000.
32. F. Riesz and B. Sz.-Nagy. Functional Analysis. Dover Publications, New York, 1990.
33. E. Sander and T. Wanner. Monte Carlo simulations for spinodal decomposition. Journal of Statistical Physics, 95(5-6):925-948, 1999.
34. E. Sander and T. Wanner. Unexpectedly linear behavior for the Cahn-Hilliard equation. SIAM Journal on Applied Mathematics, 60(6):2182-2202, 2000.
35. R. Temam. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer-Verlag, New York, second edition, 1997.
36. L. N. Trefethen. Spectral methods in MATLAB. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000.
37. T. Wanner. Maximum norms of random sums and transient pattern formation. Transactions of the American Mathematical Society, 356(6):2251-2279, 2004.
38. G. N. Watson. A treatise on the theory of Bessel functions. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition.