Colli, Pierluigi; Gilardi, Gianni; Podio-Guidugli, Paolo; Sprekels, Jürgen An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity. (English) Zbl 1277.35201 Discrete Contin. Dyn. Syst., Ser. S 6, No. 2, 353-368 (2013). The authors study a diffusion model of phase-field type. This model was developed in a series of papers by E. Fried and M. E. Gurtin [Physica D 68, No. 3–4, 326–343 (1993; Zbl 0793.35049)], M. E. Gurtin [Physica D 92, No. 3–4, 178–192 (1996; Zbl 0885.35121)] and P. Podio-Guidugli [Ric. Mat. 55, No. 1, 105–118 (2006; Zbl 1150.74091]. There are two unknowns in the problem: the order parameter \(\rho\) and the chemical potential \(\mu\). The system consists of two parabolic partial differential equations, interpreted as balances of micro-forces and micro-energy. As a two parameter regularization each equation includes a viscosity term, of the type \(\epsilon\partial_t\mu\) and \(\delta\partial_t\rho\) with two positive parameters \(\epsilon\) and \(\delta\). The model is complemented by Neumann homogeneous boundary conditions and suitable initial conditions.In [SIAM J. Appl. Math. 71, No. 6, 1849–1870 (2011; Zbl 1331.74011)] the authors verified that for positive \(\epsilon\) and \(\delta\) the problem is well-posed. Furthermore, they studied the long-time behaviour of solutions. This paper studies the asymptotic limit of the system as \(\epsilon\) tends to 0 and the equations become degenerate. The main result is the convergence of solutions for positive parameters to the corresponding solutions for \(\epsilon=0\). Moreover the long-time behaviour for \(\epsilon=0\) is characterized.The key tools in the proofs are compactness and monotonicity arguments. Reviewer: Dirk Blömker (Augsburg) Cited in 1 ReviewCited in 12 Documents MSC: 35K51 Initial-boundary value problems for second-order parabolic systems 35B40 Asymptotic behavior of solutions to PDEs 35K55 Nonlinear parabolic equations 74A15 Thermodynamics in solid mechanics 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35K65 Degenerate parabolic equations Keywords:viscous Cahn-Hilliard system; phase field model; asymptotic limit; continuous dependence; two parameter regularization Citations:Zbl 0793.35049; Zbl 0885.35121; Zbl 1150.74091; Zbl 1331.74011 PDFBibTeX XMLCite \textit{P. Colli} et al., Discrete Contin. Dyn. Syst., Ser. S 6, No. 2, 353--368 (2013; Zbl 1277.35201) Full Text: DOI arXiv