On a class of Cahn-Hilliard models with nonlinear diffusion

TL;DR

This paper investigates a class of nonlinear Cahn-Hilliard models, including sixth-order variants, establishing existence, convergence, and regularity results relevant for phase separation in mixtures with surfactants.

Contribution

It provides new existence and convergence results for sixth-order Cahn-Hilliard models with nonlinear diffusion, extending understanding of phase separation dynamics.

Findings

Existence of weak solutions for sixth-order models with singular potentials.

Convergence of sixth-order solutions to fourth-order models as the sixth-order term vanishes.

Existence and regularity results for the fourth-order Cahn-Hilliard system.

Abstract

In the present work, we address a class of Cahn-Hilliard equations characterized by a nonlinear diffusive dynamics and possibly containing an additional sixth order term. This model describes the separation properties of oil-water mixtures, when a substance enforcing the mixing of the phases (a surfactant) is added. However, the model is also closely connected with other Cahn-Hilliard-type equations relevant in different types of applications. We first discuss the existence of a weak solution to the sixth-order model in the case when the configuration potential of the system is of singular (e.g., logarithmic) type. Then, we study the behavior of the solutions in the case when the sixth order term is let tend to 0, proving convergence to solutions of the fourth order system in a special case. The fourth order system is then investigated by a direct approach and existence of a weak solution is shown under very general conditions by means of a fixed point argument. Finally, additional properties of the solutions, like uniqueness and parabolic regularization are discussed, both for the sixth order and for the fourth order model, under more restrictive assumptions on the nonlinear diffusion term.