# On a Cahn-Hilliard system with convection and dynamic boundary conditions

## TL;DR

This paper studies a complex Cahn-Hilliard system with convection and dynamic boundary conditions, proving key properties like existence, uniqueness, and regularity of solutions, and introducing a rigorous approximation scheme.

## Contribution

It introduces a comprehensive analysis of a Cahn-Hilliard system with convection and dynamic boundary conditions, including existence, uniqueness, regularity, and an approximation method.

## Key findings

- Proved existence and uniqueness of solutions.
- Established regularity and boundedness properties.
- Developed a rigorous approximation scheme.

## Abstract

This paper deals with an initial and boundary value problem for a system coupling equation and boundary condition both of Cahn-Hilliard type; an additional convective term with a forced velocity field, which could act as a control on the system, is also present in the equation. Either regular or singular potentials are admitted in the bulk and on the boundary. Both the viscous and pure Cahn-Hilliard cases are investigated, and a number of results is proven about existence of solutions, uniqueness, regularity, continuous dependence, uniform boundedness of solutions, strict separation property. A complete approximation of the problem, based on the regularization of maximal monotone graphs and the use of a Faedo-Galerkin scheme, is introduced and rigorously discussed.