Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: Nonlinear dynamics
type: Item
instance of: scholarly article
title: Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: Nonlinear dynamics
zbMATH Open document ID: 0954.35089
author: Stanislaus Maier-Paape
author: Thomas Wanner
published in: Archive for Rational Mechanics and Analysis
publication date: 2001-02-01T00:00:00Z
review text: The authors study the spinodal decomposition for the Cahn-Hilliard equation \[ u_t=-\Delta(\varepsilon^2\Delta u +f(u)), \;\text{ in } \Omega,\qquad {\partial u\over \partial \nu}={\partial\Delta u\over\partial \nu}=0 \;\text{ on } \partial \Omega, \] where \(\Omega\subset \mathbb{R}^n\), \(n\in \{1,2,3\}\), is bounded domain, \(\nu \) is the normal vector field on \(\partial \Omega\), and \(f\) is a cubic like nonlinearity. The equation models the phase separation of a binary alloy and the spinodal decomposition refers to the first stage of this process, in which the mixture of two metallic components becomes inhomogeneous forming a fine-grained structure with a characteristic length scale. The authors offer an interesting mathematical explanation of this phenomenon in terms of solutions of the Cahn-Hilliard equation. They show, roughly speaking, that in any neighborhood of a homogeneous equilibrium, the behavior of almost all solutions is governed by solutions contained in a finite-dimensional attractive invariant manifold, and the spatial structure of the solutions on this manifold is in a good agreement with the structure of the mixture experimentally observed in the spinodal decomposition.
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B42
Mathematics Subject Classification ID: 35B25
zbMATH DE Number: 1441232
zbMATH Keywords: cubic like nonlinearity
zbMATH Keywords: phase separation of a binary alloy
zbMATH Keywords: finite-dimensional attractive invariant manifold
reviewed by: Peter Poláčik
MaRDI profile type: Publication
full work available at URL: s002050050196
OpenAlex ID: W2061444804
DOI: 10.1007/S002050050196
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description: scientific article; zbMATH DE number 1441232