Global existence for a strongly coupled Cahn-Hilliard system with viscosity

TL;DR

This paper proves the existence of solutions for a complex coupled nonlinear diffusion system modeling two-species phase segregation on a lattice, extending previous models with more general phase-field and nonlinear diffusion equations.

Contribution

It introduces a more general phase-field equation coupled with a nonlinear diffusion equation, expanding the mathematical understanding of phase segregation models.

Findings

Existence of solutions for the coupled system is established.

The model includes nonlinear dependence of conductivity on phase and chemical potential.

The results extend previous models to more general nonlinear settings.

Abstract

An existence result is proved 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 is meant to model two-species phase segregation on an atomic lattice under the presence of diffusion. A similar system has been recently introduced and analyzed in the paper arXiv:1103.4585 . Both systems conform to the general theory developed in [P. Podio-Guidugli, Models of phase segregation and diffusion of atomic species on a lattice, Ric. Mat. 55 (2006) 105-118]: two parabolic PDEs, interpreted as balances of microforces and microenergy, are to be solved for the order parameter and the chemical potential. In the system studied in this note, a phase-field equation fairly more general than in arXiv:1103.4585 is coupled with a highly nonlinear diffusion equation for the chemical potential, in which the conductivity coefficient is allowed to depend nonlinearly on both variables.