Exponential attractors and inertial manifolds for singular perturbations of the Cahn-Hilliard equations
type: Item
instance of: scholarly article
title: Exponential attractors and inertial manifolds for singular perturbations of the Cahn-Hilliard equations
zbMATH Open document ID: 1055.35028
author: Songmu Zheng
author: Albert Milani
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2004-10-01T00:00:00Z
review text: The authors study the viscous Cahn-Hilliard equation, i.e., \[ \varepsilon u_{tt} +u_t+\Delta (\Delta u-u^3+u-\delta u_t)=0,\quad \Delta=\partial^2_x,\;x\in \mathbb{R}.\tag{1} \] The distinguish two cases: (a) \(\varepsilon>0\), \(\delta>0\), (b) \(\varepsilon >0\), \(\delta=0\). The aim is to study (1) in a suitable function space and to prove the existence of an exponential attractor resp. of an inertial manifold. The functional frame for (1) is based on the Sobolev spaces \(H^m=H^m(0,\pi)\cap H_0^1(0, \pi)\), \(H_0^m=H_0^m(0,\pi)\) and \(H^{-m}=(H_0^m)'\); \(\|\;\|_m\) is the norm in \(H^m\). The basic space for (1) is \(X=H^1\times H^{-1}\), i.e., \(u\in H^1\), \(u_t\in H^{-1}\). In Sect. 2, known results are recalled. Thus Theorem 2.1 asserts among others global existence of solutions \((u,u_t)\in C_b([0, \infty);X)\) for initial data \((u_0,u_1 )\in X\), and moreover the existence of an absorbing set \(B=\{(u,v)\in X/ \varphi_0(u,v)\leq R\}\), some \(R>0\) where \(\varphi_0\) is a certain functional of somewhat involved structure which is bounded from below. Theorem 2.2 then asserts the existence of global attractors \(A_{\varepsilon \delta}\) and \(A_{\varepsilon 0}\) for the two cases (a), (b). In Sect. (3) the notion of exponential attractor is explained and, based on \textit{A. Eden}, \textit{C. Foias}, \textit{B. Nicolaenko} and \textit{R. Temam} [Exponential attractors for dissipative evolution equations. (Chichester: Wiley) (1994; Zbl 0842.58056)] in Theorems 2.1, 2.2, the existence of an exponential attractor for (1) is proved. The difficulty is to show that the attractor is compact. This is achieved by restricting the flow \(S(t)\) for (1) to more regular data, inducing more regular trajectories and thus allowing for compact embeddings. Moreover the so-called discrete squeezing property has to be checked. The second half of the paper is devoted to the proof that (1) admits an inertial manifold. The proof splits into two steps. First it is assumed that…
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35B42
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35B25
zbMATH DE Number: 2104892
zbMATH Keywords: global attractor
zbMATH Keywords: discrete squeezing property
reviewed by: Bruno Scarpellini
MaRDI profile type: Publication
full work available at URL: j.na.2004.03.023
OpenAlex ID: W2092084328
cites work: Global dynamics of the viscous Cahn-Hilliard equation
cites work: Global solutions to a generalized Cahn-Hilliard equation with viscosity
cites work: Free Energy of a Nonuniform System. I. Interfacial Free Energy
cites work: Integral manifolds and inertial manifolds for dissipative partial differential equations
cites work: Q5688748
cites work: The viscous Cahn-Hilliard equation as a limit of the phase field model: Lower semicontinuity of the attractor
cites work: Q4851736
cites work: Lower semicontinuity of a non-hyperbolic attractor for the viscous Cahn - Hilliard equation
cites work: Viscous Cahn-Hilliard equation. II: Analysis
cites work: On the Cahn-Hilliard equation
cites work: Uniform exponential attractors for a singularly perturbed damped wave equation.
cites work: A construction of a robust family of exponential attractors
cites work: Q4039920
cites work: Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation
cites work: Lower semicontinuity of the attractor for a singularly perturbed hyperbolic equation
cites work: Inertial manifolds associated to partly dissipative reaction-diffusion systems
cites work: Q3784535
cites work: Q3784536
cites work: Some global dynamical properties of a class of pattern formation equations
cites work: Q3770238
cites work: Steady states of the one-dimensional Cahn–Hilliard equation
cites work: Infinite-dimensional dynamical systems. An introduction to dissipative parabolic PDEs and the theory of global attractors
cites work: Convergence of solutions to cahn-hilliard equation
cites work: Infinite-dimensional dynamical systems in mechanics and physics
cites work: Asymptotic behavior of solution to the Cahn-Hillard equation
cites work: Global attractors for singular perturbations of the Cahn-Hilliard equations
DOI: 10.1016/J.NA.2004.03.023
description: scientific article; zbMATH DE number 2104892