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An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity. (English) Zbl 1277.35201

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.

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
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