Global existence and uniqueness for a singular/degenerate Cahn-Hilliard system with viscosity

TL;DR

This paper proves the global existence and uniqueness of solutions for a complex, nonlinear phase-field system modeling two-species phase segregation, extending previous models with more general equations and novel proof techniques.

Contribution

It introduces a more general coupled phase-field and nonlinear diffusion system and provides new methods for establishing global existence and uniqueness of solutions.

Findings

Global-in-time solutions are proved to exist.

A new uniqueness proof is developed for constant atom mobility.

The system models two-species phase segregation with improved generality.

Abstract

Existence and uniqueness are investigated for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system aims to model two-species phase segregation on an atomic lattice; in the balance equations of microforces and microenergy, the two unknowns are the order parameter and the chemical potential. A simpler version of the same system has recently been discussed in arXiv:1103.4585v1. In this paper, a fairly more general phase-field equation is coupled with a genuinely nonlinear diffusion equation. The existence of a global-in-time solution is proved with the help of suitable a priori estimates. In the case of a constant atom mobility, a new and rather unusual uniqueness proof is given, based on a suitable combination of variables.