Global existence for a nonstandard viscous Cahn-Hilliard system with dynamic boundary condition

TL;DR

This paper proves global existence results for a complex, coupled nonlinear PDE system modeling phase segregation with dynamic boundary conditions involving surface phase transitions.

Contribution

It introduces a novel analysis of a nonstandard viscous Cahn-Hilliard system with dynamic boundary conditions, extending previous work by incorporating surface phase transitions.

Findings

Established well-posedness depending on free energy smoothness

Proved global existence of solutions for the system

Analyzed the impact of boundary conditions on solution behavior

Abstract

In this paper, we study a model for phase segregation taking place in a spatial domain that was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are diffcult to handle analytically. In contrast to the existing literature about this PDE system, we consider here a dynamic boundary condition involving the Laplace-Beltrami operator for the order parameter. This boundary condition models an additional nonconserving phase transition occurring on the surface of the domain. Different well-posedness results are shown, depending on the smoothness properties of the involved bulk and surface free energies.