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

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.

References in corpus (13)

• An Energetic Variational Approach for the Cahn--Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis
• On the Cahn-Hilliard equation with dynamic boundary conditions and a dominating boundary potential
• Phase-field dynamics with transfer of materials: The Cahn--Hilliard equation with reaction rate dependent dynamic boundary conditions
• A Review on the Cahn-Hilliard Equation: Classical Results and Recent Advances in Dynamic Boundary Conditions
• Weak solutions of the Cahn-Hilliard system with dynamic boundary conditions: A gradient flow approach
• Convergence of a Robin boundary approximation for a Cahn--Hilliard system with dynamic boundary conditions
• On the nonlocal Cahn--Hilliard equation with nonlocal dynamic boundary condition and boundary penalization
• On a transmission problem for equation and dynamic boundary condition of Cahn-Hilliard type with nonsmooth potentials
• An efficient and convergent finite element scheme for Cahn--Hilliard equations with dynamic boundary conditions
• A convergent SAV scheme for Cahn--Hilliard equations with dynamic boundary conditions
• Two-phase flows with bulk-surface interaction: thermodynamically consistent Navier-Stokes-Cahn-Hilliard models with dynamic boundary conditions
• Long-time dynamics of the Cahn--Hilliard equation with kinetic rate dependent dynamic boundary conditions
• Two-phase flows through porous media described by a Cahn--Hilliard--Brinkman model with dynamic boundary conditions

Cited by in corpus (1)

• Strong well-posedness and separation properties for a bulk-surface convective Cahn--Hilliard system with singular potentials