An asymptotic analysis for a nonstandard Cahn--Hilliard system with viscosity
- Colli, Pierluigi
- Gilardi, Gianni
- Podio-Guidugli, Paolo
- Sprekels, Jürgen
2010 Mathematics Subject Classification
- 74A15 35K55 35A05 35B40
- viscous Cahn-Hilliard system, phase field model, asymptotic limit, existence of solutions
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, $varepsilon,partial_tmu$ and $delta,partial_trho$ -- 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 , 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 $eps$ tends to 0. We prove convergence of $(varepsilon,delta)-$solutions to the corresponding solutions for the case $eps$ =0, whose long-time behavior we characterize; in the proofs, we employ compactness and monotonicity arguments.
- Discrete Contin. Dyn. Syst. Ser. S, 6 (2013) pp. 353--368.