# American Institute of Mathematical Sciences

April  2013, 6(2): 353-368. doi: 10.3934/dcdss.2013.6.353

## 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, Italy 3 Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin, Germany

Received  September 2011 Revised  January 2012 Published  November 2012

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.
Citation: Pierluigi Colli, Gianni Gilardi, Paolo Podio-Guidugli, Jürgen Sprekels. An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 353-368. doi: 10.3934/dcdss.2013.6.353
##### References:
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##### References:
 [1] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,", Noordhoff, (1976).  doi: 10.2165/00003495-197612040-00004.  Google Scholar [2] B. D. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity,, Arch. Rational Mech. Anal., 13 (1963), 167.   Google Scholar [3] P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Existence and uniqueness of a global-in-time solution to a phase segregation problem of the Allen-Cahn type,, Math. Models Methods Appl. Sci., 20 (2010), 519.   Google Scholar [4] P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, A temperature-dependent phase segregation problem of the Allen-Cahn type,, Adv. Math. Sci. Appl., 20 (2010), 219.   Google Scholar [5] P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Well-posedness and long-time behavior for a nonstandard viscous Cahn-Hilliard system,, SIAM J. Appl. Math., 71 (2011), 1849.   Google Scholar [6] E. DiBenedetto, "Degenerate Parabolic Equations,", Springer-Verlag, (1993).  doi: 10.1007/978-1-4612-0895-2_6.  Google Scholar [7] M. Frémond, "Non-smooth Thermomechanics,", Springer-Verlag, (2002).   Google Scholar [8] E. Fried and M. E. Gurtin, Continuum theory of thermally induced phase transitions based on an order parameter,, Phys. D, 68 (1993), 326.   Google Scholar [9] M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance,, Phys. D, 92 (1996), 178.   Google Scholar [10] J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,", Dunod Gauthier-Villars, (1969).   Google Scholar [11] P. Podio-Guidugli, Models of phase segregation and diffusion of atomic species on a lattice,, Ric. Mat., 55 (2006), 105.   Google Scholar [12] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura. Appl., 146 (1987), 65.   Google Scholar
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