Spinodal Decomposition¶for the Cahn–Hilliard–Cook Equation

We consider the dynamics of a nonlinear partial differential equation perturbed by additive noise. Assuming that the underlying deterministic equation has an unstable equilibrium, we show that the nonlinear stochastic partial differential equation exhibits essentially linear dynamics far from equilibrium. More precisely, we show that most trajectories starting at the unstable equilibrium are driven away in two stages. After passing through a cylindrical region, most trajectories diverge from the deterministic equilibrium through a cone-shaped region which is centered around a finite-dimensional subspace corresponding to strongly unstable eigenfunctions of the linearized equation, and on which the influence of the nonlinearity is surprisingly small. This abstract result is then applied to explain spinodal decomposition in the stochastic Cahn-Hilliard-Cook equation on a domain G. This equation depends on a small interaction parameter e > 0, and one is generally interested in asymptotic results as e → 0. Specifically, we show that linear behavior dominates the dynamics up to distances from the deterministic equilibrium which can reach e -2+dim G/2 with respect to the H 2 (G)-norm.

[1]  T. Kurtz,et al.  Stochastic equations in infinite dimensions , 2006 .

[2]  Evelyn Sander,et al.  Monte Carlo Simulations for Spinodal Decomposition , 1999 .

[3]  D. Blömker.  Nonhomogeneous Noise and Q-Wiener Processes on Bounded Domains , 2005 .

[4]  Giuseppe Da Prato,et al.  Stochastic Cahn-Hilliard equation , 1996 .

[5]  H. E. Cook,et al.  Brownian motion in spinodal decomposition , 1970 .

[6]  R. Courant,et al.  Methods of Mathematical Physics, Vol. I , 1954 .

[7]  Jan Seidler,et al.  Exponential Integrability of Stochastic Convolutions , 2003 .

[8]  Thomas Wanner,et al.  Maximum norms of random sums and transient pattern formation , 2003 .

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

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

[11]  Evelyn Sander,et al.  Unexpectedly Linear Behavior for the Cahn-Hilliard Equation , 2000, SIAM J. Appl. Math..

[12]  J. M. Ball,et al.  GEOMETRIC THEORY OF SEMILINEAR PARABOLIC EQUATIONS (Lecture Notes in Mathematics, 840) , 1982 .

[13]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[14]  Konstantin Mischaikow,et al.  Evolution of pattern complexity in the Cahn–Hilliard theory of phase separation , 2005 .

[15]  James S. Langer,et al.  Theory of spinodal decomposition in alloys , 1971 .

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

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

[18]  Dirk Blömker,et al.  Spinodal Decomposition¶for the Cahn–Hilliard–Cook Equation , 2001 .

[19]  David E. Edmunds,et al.  Spectral Theory and Differential Operators , 1987, Oxford Scholarship Online.

[20]  Thomas Wanner,et al.  Spinodal Decomposition for the¶Cahn-Hilliard Equation in Higher Dimensions:¶Nonlinear Dynamics , 2000 .

[21]  Jonathan P. Desi,et al.  Complex transient patterns on the disk , 2006 .

[22]  Thomas Wanner,et al.  Spinodal Decomposition for the Cahn–Hilliard Equation in Higher Dimensions.¶Part I: Probability and Wavelength Estimate , 1998 .

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