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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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