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

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]  Matthias Winter,et al.  Stationary solutions for the Cahn-Hilliard equation , 1998 .

[2]  J. Cahn,et al.  A microscopic theory for antiphase boundary motion and its application to antiphase domain coasening , 1979 .

[3]  C. Conley.  Isolated Invariant Sets and the Morse Index , 1978 .

[4]  B. Fiedler.  Global attractors of one-dimensional parabolic equations: sixteen examples , 1994 .

[5]  Kevin Zumbrun,et al.  Connectivity of Phase Boundaries in Strictly Convex Domains , 1998 .

[6]  R. Schaaf.  Global Solution Branches of Two Point Boundary Value Problems , 1991 .

[7]  M. Gurtin,et al.  Structured phase transitions on a finite interval , 1984 .

[8]  Morton E. Gurtin,et al.  On the structure of equilibrium phase transitions within the gradient theory of fluids , 1988 .

[9]  Konstantin Mischaikow,et al.  A dynamical system approach to a phase transition problem , 1991 .

[10]  Pedro Frettas,et al.  STABILITY OF STATIONARY SOLUTIONS FOR A SCALAR NON-LOCAL REACTION-DIFFUSION EQUATION , 1995 .

[11]  M. Grinfeld,et al.  Counting stationary solutions of the Cahn–Hilliard equation by transversality arguments , 1995, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[12]  Danielle Hilhorst,et al.  On the slow dynamics for the Cahn–Hilliard equation in one space dimension , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[13]  Amy Novick-Cohen,et al.  On Cahn-Hilliard type equations , 1990 .

[14]  Peter W. Bates,et al.  Slow motion for the Cahn-Hilliard equation in one space dimension , 1991 .

[15]  Paul C. Fife,et al.  Models for phase separation and their mathematics. , 2000 .

[16]  James F. Reineck.  Connecting orbits in one-parameter families of flows , 1988 .

[17]  J. Ball,et al.  Material Instabilities in Continuum Mechanics, and Related Mathematical Problems , 1990 .

[18]  N. Alikakos,et al.  Slow Dynamics for the Cahn‐Hilliard Equation in Higher Space Dimensions: The Motion of Bubbles , 1998 .

[19]  Hiroshi Matano,et al.  Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation , 1982 .

[20]  J. Furter,et al.  A monotonicity theorem and its application to stationary solutions of the phase field model , 1992 .

[21]  Andrew M. Stuart,et al.  The viscous Cahn-Hilliard equation. I. Computations , 1995 .

[22]  J. Rubinstein,et al.  Nonlocal reaction−diffusion equations and nucleation , 1992 .

[23]  J. Furter,et al.  On a stationary state characterization of transition from spinodal decomposition to nucleation behaviour in the Cahn-Hilliard model of phase separation , 1989 .

[24]  Peter W. Bates,et al.  Spectral comparison principles for the Cahn-Hilliard and phase-field equations, and time scales for coarsening , 1990 .

[25]  B. Fiedler,et al.  Heteroclinic Orbits of Semilinear Parabolic Equations , 1996 .

[26]  B. Niethammer.  Existence and uniqueness of radially symmetric stationary points within the gradient theory of phase transitions , 1995, European Journal of Applied Mathematics.

[27]  B. Nicolaenko,et al.  Inertial sets for dissipative evolution equations , 1990 .

[28]  Roger Temam,et al.  Some Global Dynamical Properties of a Class of Pattern Formation Equations , 1989 .

[29]  J. Hale.  Asymptotic Behavior of Dissipative Systems , 1988 .

[30]  Jack K. Hale,et al.  Slow-motion manifolds, dormant instability, and singular perturbations , 1989 .

[31]  A. Novick-Cohen.  The nonlinear Cahn-Hilliard equation: Transition from spinodal decomposition to nucleation behavior , 1985 .

[32]  The steady states of the one-dimensional Cahn-hilliard equation , 1992 .

[33]  Andrew M. Stuart,et al.  Viscous Cahn–Hilliard Equation II. Analysis , 1996 .

[34]  J. Mallet-Paret.  Morse Decompositions for delay-differential equations , 1988 .

[35]  Peter W. Bates,et al.  The Dynamics of Nucleation for the Cahn-Hilliard Equation , 1993, SIAM J. Appl. Math..

[36]  Jack K. Hale,et al.  Lower semicontinuity of attractors of gradient systems and applications , 1989 .

[37]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[38]  Robert D. Franzosa.  The continuation theory for Morse decompositions and connection matrices , 1988 .

[39]  L. Segel,et al.  Nonlinear aspects of the Cahn-Hilliard equation , 1984 .

[40]  Yasumasa Nishiura,et al.  Spectral comparison between the second and the fourth order equations of conservative type with non-local terms , 1998 .

[41]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

[42]  Konstantin Mischaikow,et al.  Global asymptotic dynamics of gradient-like bistable equations , 1995 .

[43]  C M Elliott,et al.  The viscous Cahn-Hilliard equation. I. Computations , 1995 .

[44]  Peter W. Bates,et al.  Metastable Patterns for the Cahn-Hilliard Equation: Part II. Layer Dynamics and Slow Invariant Manifold , 1995 .

[45]  Daniel B. Henry,et al.  Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations , 1985 .

[46]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[47]  N. Alikakos,et al.  Slow dynamics for the cahn-hilliard equation in higher space dimension part i: spectral estimates ∗ , 1994 .

[48]  Sigurd B. Angenent,et al.  The Morse-Smale property for a semi-linear parabolic equation , 1986 .

[49]  J. M. Ball,et al.  ASYMPTOTIC BEHAVIOUR OF DISSIPATIVE SYSTEMS (Mathematical Surveys and Monographs 25) , 1990 .

[50]  Robert D. Franzosa.  The connection matrix theory for Morse decompositions , 1989 .

[51]  K. Sato,et al.  [ERG]. , 2020, Rinsho ganka. Japanese journal of clinical ophthalmology.

[52]  Zheng Songmu,et al.  Asymptotic behavior of solution to the Cahn-Hillard equation , 1986 .