Transactions of the American Mathematical Society

The viscous Cahn-Hilliard equation: Morse decomposition and structure of the global attractor

Author(s): M. Grinfeld; A. Novick-Cohen
Journal: Trans. Amer. Math. Soc. 351 (1999), 2375-2406.
MSC (1991): Primary 35K22, 35K30, 58F12, 58F39
Posted: February 15, 1999
Retrieve article in: PDF DVI PostScript
This article is available free of charge

Abstract: In this paper we establish a Morse decomposition of the stationary solutions of the one-dimensional viscous Cahn-Hilliard equation by explicit energy calculations. Strong non-degeneracy of the stationary solutions is proven away from turning points and points of bifurcation from the homogeneous state and the dimension of the unstable manifold is calculated for all stationary states. In the unstable case, the flow on the global attractor is shown to be semi-conjugate to the flow on the global attractor of the Chaffee-Infante equation, and in the metastable case close to the nonlocal reaction-diffusion limit, a partial description of the structure of the global attractor is obtained by connection matrix arguments, employing a partial energy ordering and the existence of a weak lap number principle.

1.: N. Alikakos, P.W. Bates and G. Fusco, Slow motion for Cahn-Hilliard equation in one space dimension, J. Diff. Equations, 90 (1991), 81-134. MR 92a:35152
2.: S. Allen and J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall. 27 (1979), 1084-1095.
3.: N. Alikakos and G. Fusco, Equilibrium and dynamics of bubbles for the Cahn-Hilliard equation, Collection: Inter. Conf. on Differential Equations, vol 1,2 (Barcelona, 1991), 59-67, World Sci. Publish., River Edge, NJ (1993). MR 95b:35111
4.: N. Alikakos and G. Fusco, Slow dynamics for the Cahn-Hilliard equation in higher space dimensions. Part I: Spectral estimates, Comm. in PDEs 19 (1994), 1397-1447. MR 95j:35163
5.: S. B. Angenent, The Morse-Smale property for a semilinear parabolic equation, JDE 62 (1986), 427-442. MR 87e:58115
6.: F. Bai, C. M. Elliott, A. Gardiner, A. Spence and A. M. Stuart, The viscous Cahn-Hilliard equation. Part I: Computations, Nonlinearity 8 (1995), 131-160. MR 95m:35082
7.: P. W. Bates and P. C. Fife, Spectral comparison principles for the Cahn-Hilliard and phase field equations, and time scales for coarsening, Physica 43D (1990), 335-348. MR 91h:80001
8.: P. W. Bates and P. C. Fife, The dynamics of nucleation for the Cahn-Hilliard equation, SIAM J. Appl. Math, 53 (1993), 990-1008. MR 94g:82034
9.: P. W. Bates and P. J. Xun, Metastable patterns for the Cahn-Hilliard equation, Part II, J. Diff. Equations, 117, (1995), 165-216. MR 95m:35188
10.: D. Brochet, D. Hilhorst, and A. Novick-Cohen, Maximal attractor and inertial sets for a conserved phase field model, Advances in PDE 1(1996), 547-578. MR 97e:35189
11.: L. Bronsard and D. Hilhorst, On the slow dynamics for the Cahn-Hilliard equation in one space dimension, Proc. Roy. Soc. London, Series A, 439 (1992), 669-682. MR 93k:35210
12.: J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28 (1958), 258-267.
13.: J. Carr, M. E. Gurtin and M. Slemrod, Structured phase transitions on a finite interval, Arch. Rat. Mech. Anal. 86 (1984), 317-354. MR 86i:80001
14.: C. C. Conley, Isolated Invariant Sets and Morse Index, AMS Regional Conference Series in Mathematics 38 AMS, Providence, RI (1978). MR 80c:58009
15.: A. Eden, C. Foias, B. Nicolaenko, and R. Temam, Inertial sets for dissipative evolution equations. Part I: Construction and application, IMA Preprint 812 (1991).
16.: J. C. Eilbeck, J. E. Furter and M. Grinfeld, On a stationary state characterization of transition from spinodal decomposition to nucleation behavior in the Cahn-Hilliard model of phase separation, Phys. Lett. A 139 (1989), 42-46. MR 89k:80015
17.: C. M. Elliott A. M. Stuart, The viscous Cahn-Hilliard equation. Part II: Analysis, J. Diff. Equations 128 (1996), 387-414. MR 97c:35080
18.: C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal. 96 (1986), 339-357. MR 87k:80007
19.: B. Fiedler, Global attractors of one-dimensional parabolic equations: 16 examples, Tatra Mount. Math. Publ. 4 (1994), 67-92. MR 95g:35089
20.: B. Fiedler and C. Rocha, Heteroclinic orbits of semilinear parabolic equations, J. Diff. Equations 125, (1996) 239-281. MR 96k:58200
21.: P.C. Fife, Models for phase separation and their mathematics, in Nonlinear Partial Differential Equations and Applications, M. Mimura and T. Nishida, eds., Kinokuniya Pubs., Tokyo, in press.
22.: R. D. Franzosa, The continuation theory for Morse decompositions and connection matrices, Trans. AMS 310 (1988), 781-803. MR 90g:58111
23.: R. D. Franzosa, The connection matrix theory for Morse decompositions, Trans. AMS 311 (1989), 561-592. MR 90a:58149
24.: P. Freitas, Stability of stationary solutions for a nonlocal reaction-diffusion equation, Quart. J. Mech. Appl. Math., 48 (1995), 556-582. MR 97a:35111
25.: G. Fusco and J.K. Hale, Slow motion manifolds, dormant instability and singular perturbations, Dynamics Differential Equations 1 (1989), 75-94. MR 90i:35131
26.: M. Grinfeld, J. E. Furter and J. C. Eilbeck, A monotonicity theorem and its applications to stationary solutions of the phase field model, IMA J. Appl. Math. 49 (1992), 61-72; 50 (1993), 203-204. MR 93j:80003
27.: M. Grinfeld and A. Novick-Cohen, Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments, Proc. Royal Soc. Edin. A 125 (1995), 351-370. MR 96c:58157
28.: M. E. Gurtin and H. Matano, On the structure of equilibrium phase transitions within the gradient theory of fluids, Quart. Appl. Math. 46 (1988), 301-317. MR 89j:49015
29.: J. K. Hale, Asymptotic Behaviour of Dissipative Systems, Math. Surveys and Monographs 25 A.M.S. 1988. MR 89g:58059
30.: H. Hattori and K. Mischaikow, A dynamical system approach to a phase transition problem, J. Diff. Equations 94 (1991), 340-378. MR 93m:58104
31.: J. K. Hale and G. Raugel, Lower semicontinuity of attractors of gradient systems and applications, Ann. di Matem. Pura ed Applic. 154 (1989), 281-326. MR 91f:58087
32.: D. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic differential equations, J. Diff. Equations 59 (1985), 165-205. MR 86m:58080
33.: J. Mallet-Paret, Morse decompositions for delay-differential equations, J. Diff. Equations 72 (1988), 270-315. MR 80m:58112
34.: H. Matano, Nonincrease of the lap number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 29 (1982), 401-441. MR 84m:35060
35.: K. Mischaikow, Global asymptotic dynamics of gradient-like bistable equations, SIAM J. Math. Analysis 26 (1995), 1199-1224. MR 96j:35025
36.: B. Neithammer, Existence and uniqueness of radially symmetric stationary points within the gradient theory of phase transitions, Eur. J. Appl. Math. 6 (1995), 45-68.
37.: B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of a class of pattern formation equations, Comm. PDE 14 (1989), 245-297. MR 90e:35138
38.: A. Novick-Cohen, The nonlinear Cahn-Hilliard equation: transition from spinodal decomposition to nucleation behavior, J. Stat. Phys. 38 (1985), 707-723.
39.: A. Novick-Cohen, On the viscous Cahn-Hilliard equation, in: Material Instabilities in Continuum Mechanics and Related Mathematical Problems, J. M. Ball, ed., Oxford University Press, Oxford 1988. MR 89f:73005
40.: A. Novick-Cohen, On Cahn-Hilliard type equations, Nonlinear Analysis TMA 15 (1990), 797-814. MR 92g:35105
41.: A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modelling perspectives, Advances in Math. Sci. and Appl. 8 (1998), 965-985.
42.: A. Novick-Cohen and L. A. Peletier, Steady states of the one-dimensional Cahn-Hilliard equation, Proc. Royal Soc. Edin. A 123 (1993), 1071-1098. MR 94m:35272
43.: A. Novick-Cohen and L. A. Segel, Nonlinear aspects of the Cahn-Hilliard equation, Physica 10D (1984), 277-298. MR 85k:35120
44.: I. Ohnishi and Y. Nishiura, Spectral comparison between the second and fourth order equations of conservative type with nonlocal terms, Japan J. Indust. Appl. Math. 15 (1998), 253-262.
45.: J. Reineck, Connecting orbits in one-parameter families of flows, Erg. Theory Dynam. Sys. $\textbf{8}^{*}$ (1988), 359-374. MR 89i:58128
46.: J. Rubinstein and P. Sternberg, Non-local reaction-diffusion equations and nucleation, IMA J. Appl. Math. 48 (1992), 249-264. MR 93f:35116
47.: R. Schaaf, Global Solution Branches of Two-Point Boundary Value Problems, Lecture Notes in Mathematics 1458, Springer-Verlag, Berlin 1990. MR 92a:34003
48.: P. Sternberg and K. Zumbrun, , Arch. Rat. Mech. Anal. 141 (1998), 375-400.
49.: R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Math. Sciences 68, Springer-Verlag, New York, 1988. MR 89m:35204
50.: J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poicare, Anal. Non Lin., 15 (1998), 459-492. CMP 98:15
51.: S. Zheng, Asymptotic behaviour of solutions to the Cahn-Hilliard equation, Applic. Analysis 23 (1986), 165-184. MR 88b:35036

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 35K22, 35K30, 58F12, 58F39

M. Grinfeld
Affiliation: Department of Mathematics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, Scotland, United Kingdom
Email: michael@conley.maths.strath.ac.uk

A. Novick-Cohen
Affiliation: Faculty of Mathematics, Technion-IIT, Haifa 32000, Israel
Email: amync@techunix.technion.ac.il

DOI: 10.1090/S0002-9947-99-02445-9
PII: S 0002-9947(99)02445-9
Received by editor(s): September 24, 1996
Posted: February 15, 1999
Copyright of article: Copyright 1999, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS