Spinodal Decomposition for the Cahn–Hilliard Equation in Higher Dimensions.¶Part I: Probability and Wavelength Estimate

Abstract:This paper is the first in a series of two papers addressing the phenomenon of spinodal decomposition for the Cahn–Hilliard equation where , is a bounded domain with sufficiently smooth boundary, and f is cubic-like, for example f(u) =u−u3. We will present the main ideas of our approach and explain in what way our method differs from known results in one space dimension due to Grant [26]. Furthermore, we derive certain probability and wavelength estimates. The probability estimate is needed to understand why in a neighborhood of a homogeneous equilibrium u0≡μ of the Cahn–Hilliard equation, with mass μ in the spinodal region, a strongly unstable manifold has dominating effects. This is demonstrated for the linearized equation, but will be essential for the nonlinear setting in the second paper [37] as well. Moreover, we introduce the notion of a characteristic wavelength for the strongly unstable directions.

[1]  Desai,et al.  Early stages of spinodal decomposition for the Cahn-Hilliard-Cook model of phase separation. , 1988, Physical review. B, Condensed matter.

[2]  SOLUTIONS OF NONLINEAR PLANAR ELLIPTIC PROBLEMS WITH TRIANGLE SYMMETRY , 1997 .

[3]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[4]  Barbara Stoth,et al.  Convergence of the Cahn-Hilliard Equation to the Mullins-Sekerka Problem in Spherical Symmetry , 1996 .

[5]  C. M. Elliott,et al.  Spinodal decomposition in Fe-Cr alloys: Experimental study at the atomic level and comparison with computer models—I. Introduction and methodology , 1995 .

[6]  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.

[7]  Peter W. Bates,et al.  Convergence of the Cahn-Hilliard equation to the Hele-Shaw model , 1994 .

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

[9]  John W. Cahn,et al.  Free Energy of a Nonuniform System. II. Thermodynamic Basis , 1959 .

[10]  Symmetry-breaking at non-positive solutions of semilinear elliptic equations , 1994 .

[11]  P. Fife,et al.  Perturbation of doubly periodic solution branches with applications to the Cahn-Hilliard equation , 1997 .

[12]  L. Bronsard,et al.  Slow motion in the gradient theory of phase transitions via energy and spectrum , 1997 .

[13]  P. Bates,et al.  Metastable Patterns for the Cahn-Hilliard Equation, Part I , 1994 .

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

[15]  Christopher P. Grant.  SPINODAL DECOMPOSITION FOR THE CAHN-HILLIARD EQUATION , 1993 .

[16]  Charles M. Elliott,et al.  The Cahn-Hilliard Model for the Kinetics of Phase Separation , 1989 .

[17]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[18]  C. M. Elliott,et al.  Numerical Studies of the Cahn-Hilliard Equation for Phase Separation , 1987 .

[19]  John W. Cahn,et al.  Phase Separation by Spinodal Decomposition in Isotropic Systems , 1965 .

[20]  Robert L. Pego,et al.  Front migration in the nonlinear Cahn-Hilliard equation , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[21]  Christopher P. Grant.  SLOW MOTION IN ONE-DIMENSIONAL CAHN-MORRAL SYSTEMS , 1995 .

[22]  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.

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

[24]  B. Gidas,et al.  Symmetry and related properties via the maximum principle , 1979 .

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