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

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


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  • [1]

    V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Noordhoff, Leyden, 1976.doi: 10.2165/00003495-197612040-00004.


    B. D. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Rational Mech. Anal., 13 (1963), 167-178.


    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.


    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.


    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.


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


    M. Frémond, "Non-smooth Thermomechanics," Springer-Verlag, Berlin, 2002.


    E. Fried and M. E. Gurtin, Continuum theory of thermally induced phase transitions based on an order parameter, Phys. D, 68 (1993), 326-343.


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


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


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


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

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