Maximal attractor and inertial sets for a conserved phase field model
type: Item
instance of: scholarly article
title: Maximal attractor and inertial sets for a conserved phase field model
zbMATH Open document ID: 0853.35052
author: D. Brochet
author: Danielle Hilhorst
author: Amy Novick-Cohen
published in: Advances in Differential Equations
publication date: 1997-01-05T00:00:00Z
review text: It is the concern of the authors to investigate existence and properties of solutions of an equation due to G. Caginalp, which will not be written out. This equation contains two unknowns, the temperature \(u\) and an order parameter \(\varphi\) coming from thermodynamics, \(\varphi\) describes the phase of the underlying substance. The case \(\varphi\) close to 1 corresponds to the liquid phase, \(\varphi\) close to \(- 1\) to the solid phase. The interphase between solid and liquid is described by the set of points where \(\varphi= 0\). This equation is favorably replaced by another one via the substitution \(\varphi= \varphi\), \(v= u+ {\ell\over 2} \varphi\), \(\ell\) a parameter \(> 0\). The resulting system is: \[ \tau\varphi_t= - \xi^2 \Delta(\xi^2 \Delta \varphi- g(\varphi)- \ell\varphi+ 2v),\;v_t= K\Delta v- K\textstyle{{\ell\over2}} \Delta\varphi.\tag{1} \] The constants \(\tau\), \(\xi\), \(K\), \(\ell\) are all \(> 0\) and have physical meaning, and \(g(s)\) is a polynomial subject to some conditions; in the simplest of all cases \(g(s)= {1\over 2}(s^3- s)\). System (1) is considered on \(\Omega\times \mathbb{R}_+\) with \(\Omega\subseteq \mathbb{R}^n\), \(n= 1,2,3\) smooth and bounded. There are Neumann boundary conditions \(\partial_n \varphi= \partial_n \Delta \varphi= \partial_n v= 0\) with \(\partial_n\) the outer unit normal derivative at \(x\in \partial \Omega\). There are initial conditions \(\varphi(x, 0)= \varphi_0(x)\), \(v(x, 0)= v_0(x)\). A functional analytic frame is imposed on (1). There are reasons to consider (1) on the spaces \(H= H^1(\Omega)'\times L^2(\Omega)\) (\(H^1(\Omega)'= \text{dual}\) of \(H^1(\Omega)\)) and \(V\) which consists of the pairs \((\varphi, v)\) in \(H^2(\Omega)\times H^1(\Omega)\) such that \(\partial_n \varphi= 0\). The fact that \(\int_\Omega v(x, t) dx\), \(\int_\Omega \varphi(x, t) dx\) are conserved gives rise to the set \(H_{\beta\gamma}\), \(\widetilde H_{\beta\gamma}\) where \(H_{\beta\gamma}\) is roughly…
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K35
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 883511
zbMATH Keywords: global attractors
zbMATH Keywords: inertial sets
zbMATH Keywords: Galerkin method
reviewed by: Bruno Scarpellini
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 883511