Thomas Wanner
Department of Mathematical Sciences
George Mason University
4400 University Drive, MS 3F2
Fairfax, Virginia 22030, USA

 

Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: Nonlinear dynamics

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  1. Stanislaus Maier-Paape, Thomas Wanner:
    Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: Nonlinear dynamics
    Archive for Rational Mechanics and Analysis 151(3), pp. 187-219, 2000.

Abstract

This paper addresses the phenomenon of spinodal decomposition for the Cahn-Hilliard equation ut=Δ(ϵ2Δu+f(u))u_t = -\Delta(\epsilon^2 \Delta u + f(u)) in Ω\Omega, subject to uν=Δuν=0\frac{\partial u}{\partial \nu} = \frac{\partial \Delta u}{\partial \nu} = 0 on Ω\partial \Omega, where ΩRn\Omega \subset \mathbb{R}^n, n=1,2,3n = 1,2,3, is a bounded domain with sufficiently smooth boundary, and ff is cubic-like, for example f(u)=uu3f(u) = u - u^3. Based on the results of Maier-Paape, Wanner (1998) the nonlinear Cahn-Hilliard equation will be discussed. This equation generates a nonlinear semiflow in certain affine subspaces of H2(Ω)H^2(\Omega). In a neighborhood UϵU_{\epsilon} with size proportional to ϵn\epsilon^n around the constant solution μ\mu, where μ\mu lies in the spinodal region, we observe the following behavior. Within a local inertial manifold NϵN_\epsilon containing μ\mu there exists a finite-dimensional invariant manifold MϵM_\epsilon which dominates the behavior of all solutions starting with initial conditions from a small ball around μ\mu with probability almost 11. The dimension of MϵM_\epsilon is proportional to ϵn\epsilon^{-n} and the elements of MϵM_\epsilon exhibit a common geometric quantity which is strongly related to a characteristic wavelength \ell proportional to ϵ\epsilon.

The published version of the paper can be found at https://doi.org/10.1007/s002050050196.

Bibtex

@article{maier:wanner:00a,
   author = {Stanislaus Maier-Paape and Thomas Wanner},
   title = {Spinodal decomposition for the {C}ahn-{H}illiard equation in
            higher dimensions: {N}onlinear dynamics},
   journal = {Archive for Rational Mechanics and Analysis},
   year = 2000,
   volume = 151,
   number = 3,
   pages = {187--219},
   doi = {10.1007/s002050050196}
   }