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

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

Pierluigi Colli 1, , Gianni Gilardi 1, , Paolo Podio-Guidugli 2, and  Jürgen Sprekels 3,

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.
Keywords: asymptotic limit, phase field model, Viscous Cahn-Hilliard system, existence of solutions..
Mathematics Subject Classification: Primary: 35K55; Secondary: 35A05, 35B40, 74A1.
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:
[1]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Noordhoff, Leyden, 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), 167-178.  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), 519-541.  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), 219-234.  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), 1849-1870.  Google Scholar

[6]

E. DiBenedetto, "Degenerate Parabolic Equations," Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2_6.  Google Scholar

[7]

M. Frémond, "Non-smooth Thermomechanics," Springer-Verlag, Berlin, 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), 326-343.  Google Scholar

[9]

M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192.  Google Scholar

[10]

J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires," Dunod Gauthier-Villars, Paris, 1969.  Google Scholar

[11]

P. Podio-Guidugli, Models of phase segregation and diffusion of atomic species on a lattice, Ric. Mat., 55 (2006), 105-118.  Google Scholar

[12]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura. Appl., 146 (1987), 65-96.  Google Scholar

show all references

References:
[1]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Noordhoff, Leyden, 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), 167-178.  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), 519-541.  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), 219-234.  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), 1849-1870.  Google Scholar

[6]

E. DiBenedetto, "Degenerate Parabolic Equations," Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2_6.  Google Scholar

[7]

M. Frémond, "Non-smooth Thermomechanics," Springer-Verlag, Berlin, 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), 326-343.  Google Scholar

[9]

M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192.  Google Scholar

[10]

J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires," Dunod Gauthier-Villars, Paris, 1969.  Google Scholar

[11]

P. Podio-Guidugli, Models of phase segregation and diffusion of atomic species on a lattice, Ric. Mat., 55 (2006), 105-118.  Google Scholar

[12]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura. Appl., 146 (1987), 65-96.  Google Scholar

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