Well-posedness of a bulk-surface convective Cahn--Hilliard system with dynamic boundary conditions

TL;DR

This paper proves the existence, regularity, and uniqueness of solutions for a complex bulk-surface convective Cahn--Hilliard system with dynamic boundary conditions, accounting for dynamic contact angles and boundary interactions.

Contribution

It introduces a comprehensive analysis of a generalized Cahn--Hilliard model with dynamic boundary conditions, including existence, asymptotic limits, and regularity results.

Findings

Existence of weak solutions for K,L in (0,∞)

Weak solutions exist for all boundary condition types via asymptotic limits

Higher regularity and uniqueness under constant mobility functions

Abstract

We consider a general class of bulk-surface convective Cahn--Hilliard systems with dynamic boundary conditions. In contrast to classical Neumann boundary conditions, the dynamic boundary conditions of Cahn--Hilliard type allow for dynamic changes of the contact angle between the diffuse interface and the boundary, a convection-induced motion of the contact line as well as absorption of material by the boundary. The coupling conditions for bulk and surface quantities involve parameters $K,L\in[0,\infty]$, whose choice declares whether these conditions are of Dirichlet, Robin or Neumann type. We first prove the existence of a weak solution to our model in the case $K,L\in (0,\infty)$ by means of a Faedo--Galerkin approach. For all other cases, the existence of a weak solution is then shown by means of the asymptotic limits, where $K$ and $L$ are sent to zero or to infinity, respectively. Eventually, we establish higher regularity for the phase-fields, and we prove the uniqueness of weak solutions given that the mobility functions are constant.