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Author: Colli, Pierluigi × Author: Podio-Guidugli, Paola × DDC: 510 ×

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    Analysis of a time discretization scheme for a nonstandard viscous Cahn-Hilliard system
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Colli, Pierluigi; Gilardi, Gianni; Krejcí, Pavel; Podio-Guidugli, Paola; Sprekels, Jürgen
    In this paper we propose a time discretization of a system of two parabolic equations describing diffusion-driven atom rearrangement in crystalline matter. The equations express the balances of microforces and microenergy; the two phase fields are the order parameter and the chemical potential. The initial and boundary-value problem for the evolutionary system is known to be well posed. Convergence of the discrete scheme to the solution of the continuous problem is proved by a careful development of uniform estimates, by weak compactness and a suitable treatment of nonlinearities. Moreover, for the difference of discrete and continuous solutions we prove an error estimate of order one with respect to the time step.
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    An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Colli, Pierluigi; Gilardi, Gianni; Podio-Guidugli, Paola; Sprekels, Jürgen
    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, $varepsilon,partial_tmu$ and $delta,partial_trho$ -- 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 $eps$ tends to 0. We prove convergence of $(varepsilon,delta)-$solutions to the corresponding solutions for the case $eps$ =0, whose long-time behavior we characterize; in the proofs, we employ compactness and monotonicity arguments.