An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity

Pierluigi Colli; Gianni Gilardi; Paolo Podio-Guidugli; Jürgen Sprekels

doi:10.3934/dcdss.2013.6.353

Article Contents

2013, Volume 6, Issue 2: 353-368. Doi: 10.3934/dcdss.2013.6.353

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An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity

1.
Dipartimento di Matematica "F. Casorati", Università di Pavia, Via Ferrata 1, 27100 Pavia
2.
Dipartimento di Ingegneria Civile, Università di Roma "Tor Vergata", Via del Politecnico 1, 00133 Roma
3.
Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin

Received: September 2011

Revised: January 2012

Published: April 2013

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Abstract

This paper is concerned with a diffusion model of phase-field type, consisting of a parabolic system of two partial differential equations, interpreted as balances of microforces and microenergy, for two unknowns: the problem's order parameter $\rho$ and the chemical potential $\mu$; each equation includes a viscosity term -- respectively, $\varepsilon \,\partial_t\mu$ and $\delta\,\partial_t\rho$ -- with $\varepsilon$ and $\delta$ two positive parameters; the field equations are complemented by Neumann homogeneous boundary conditions and suitable initial conditions. In a recent paper [5], we proved that this problem is well-posed and investigated the long-time behavior of its $(\varepsilon,\delta)-$solutions. Here we discuss the asymptotic limit of the system as $\varepsilon$ tends to $0$. We prove convergence of $(\varepsilon,\delta)-$solutions to the corresponding solutions for the case $\varepsilon =0$, whose long-time behavior we characterize; in the proofs, we employ compactness and monotonicity arguments.

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Mathematics Subject Classification: Primary: 35K55; Secondary: 35A05, 35B40, 74A15.

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