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

TL;DR

This paper analyzes the asymptotic behavior of a nonstandard Cahn-Hilliard system with viscosity, showing solution convergence as viscosity tends to zero and characterizing the limit problem's long-term dynamics.

Contribution

It provides the first rigorous analysis of the asymptotic limit of a viscous Cahn-Hilliard system, including convergence results and long-time behavior characterization.

Findings

Solutions converge to the limit problem as viscosity approaches zero

The long-time behavior of the limit problem is characterized

Compactness and monotonicity methods are employed in proofs

Abstract

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 and the chemical potential; each equation includes a viscosity term and the field equations are complemented by Neumann homogeneous boundary conditions and suitable initial conditions. In the recent paper arXiv:1103.4585v1 [math.AP] we proved that this problem is well posed and investigated the long-time behavior of its solutions. Here we discuss the asymptotic limit of the system as the viscosity coefficient of the order parameter equation tends to 0. We prove convergence of solutions to the corresponding solutions for the limit problem, whose long-time behavior we characterize; in the proofs, we employ compactness and monotonicity arguments.