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A vanishing diffusion limit in a nonstandard system of phase field equations (Articolo in rivista)

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A vanishing diffusion limit in a nonstandard system of phase field equations (Articolo in rivista) (literal)

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Colli, Pierluigi; Gilardi, Gianni; Krejci, Pavel; Sprekels, Juergen (2014)
A vanishing diffusion limit in a nonstandard system of phase field equations
in Evolution Equations and Control Theory (EECT); American Institute of Mathematical Sciences, Springfield [MO] (Stati Uniti d'America)
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Dipartimento di Matematica \"F. Casorati\", Università di Pavia, via Ferrata 1, 27100 Pavia, Italy; Institute of Mathematics, Czech Academy of Sciences,?Zitná 25, CZ-11567 Praha 1, Czech Republic; Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin, Germany (literal)

Titolo

A vanishing diffusion limit in a nonstandard system of phase field equations (literal)

Abstract

We are concerned with a nonstandard phase field model of Cahn-Hilliard type. The model, which was introduced by Podio-Guidugli (Ric. Mat. 2006), describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been recently investigated by Colli, Gilardi, Podio-Guidugli, and Sprekels in a series of papers: see, in particular, SIAM J. Appl. Math. 2011 and Boll. Unione Mat. Ital. 2012. In the latter contribution, the authors can treat the very general case in which the diffusivity coefficient of the parabolic PDE is allowed to depend nonlinearly on both variables. In the same framework, this paper investigates the asymptotic limit of the solutions to the initial-boundary value problems as the diffusion coefficient a in the equation governing the evolution of the order parameter tends to zero. We prove that such a limit actually exists and solves the limit problem, which couples a nonlinear PDE of parabolic type with an ODE accounting for the phase dynamics. In the case of a constant diffusivity, we are able to show uniqueness and to improve the regularity of the solution. (literal)

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