A doubly nonlinear Cahn-Hilliard system with nonlinear viscosity

TL;DR

This paper investigates a complex viscous Cahn-Hilliard system with nonlinear viscosity, proving existence, and under certain conditions, uniqueness and continuous dependence of solutions for this highly nonlinear model.

Contribution

It introduces a new family of viscous Cahn-Hilliard equations with non-smooth viscosity and establishes fundamental well-posedness results.

Findings

Existence of solutions for the nonlinear system

Conditions for uniqueness and continuous dependence

The system models a 'forward-backward' parabolic approximation

Abstract

In this paper we discuss a family of viscous Cahn-Hilliard equations with a non-smooth viscosity term. This system may be viewed as an approximation of a "forward-backward" parabolic equation. The resulting problem is highly nonlinear, coupling in the same equation two nonlinearities with the diffusion term. In particular, we prove existence of solutions for the related initial and boundary value problem. Under suitable assumptions, we also state uniqueness and continuous dependence on data.