A vanishing diffusion limit in a nonstandard system of phase field equations

TL;DR

This paper studies the behavior of a complex phase field model as the diffusion coefficient approaches zero, proving the existence of a limit system that combines a nonlinear PDE with an ODE, and establishing uniqueness in certain cases.

Contribution

It provides the first rigorous analysis of the vanishing diffusion limit in a nonstandard, highly nonlinear phase field model, extending previous work to more general diffusivity functions.

Findings

The limit exists and solves a coupled PDE-ODE system.

Uniqueness and regularity are established for constant diffusivity.

The analysis covers general nonlinear diffusivity dependencies.

Abstract

We are concerned with a nonstandard phase field model of Cahn-Hilliard type. The model, which was introduced by Podio-Guidugli (Ric. Mat. 2006), describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been recently investigated by Colli, Gilardi, Podio-Guidugli, and Sprekels in a series of papers: see, in particular, SIAM J. Appl. Math. 2011 and Boll. Unione Mat. Ital. 2012. In the latter contribution, the authors can treat the very general case in which the diffusivity coefficient of the parabolic PDE is allowed to depend nonlinearly on both variables. In the same framework, this paper investigates the asymptotic limit of the solutions to the initial-boundary value problems as the diffusion coefficient in the equation governing the evolution of the order parameter tends to zero. We prove that such a limit actually exists and solves the limit problem, which couples a nonlinear PDE of parabolic type with an ODE accounting for the phase dynamics. In the case of a constant diffusivity, we are able to show uniqueness and to improve the regularity of the solution.