Well-posedness and long-time behavior of a bulk-surface Cahn--Hilliard model with non-degenerate mobility

Abstract

We study a bulk-surface Cahn--Hilliard model with non-degenerate mobility and singular potentials in two dimensions. Following the ideas of the recent work by Conti, Galimberti, Gatti, and Giorgini [Calc. Var. Partial Differential Equations, 64(3):Paper No. 87, 32, 2025] for the Cahn--Hilliard equation with homogeneous Neumann boundary conditions, we show the uniqueness of weak solutions together with a continuous dependence estimate for sufficiently regular mobility functions. Next, under weaker assumptions on the mobility functions, we show the existence of a weak solution that exhibits the propagation of uniform-in-time regularity and satisfies the instantaneous separation property. Lastly, we consider the long-time behavior and prove that the unique weak solution converges to a solution of the stationary bulk-surface Cahn--Hilliard equation. Our approach for the uniqueness proof relies on a new well-posedness and regularity theory for a bulk-surface elliptic system with non-constant coefficients, which may be of independent interest.

References in corpus (10)

• An Energetic Variational Approach for the Cahn--Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis
• 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
• On the the wave equation with hyperbolic dynamical boundary conditions, interior and boundary damping and source
• Convergence of a Robin boundary approximation for a Cahn--Hilliard system with dynamic boundary conditions
• On a transmission problem for equation and dynamic boundary condition of Cahn-Hilliard type with nonsmooth potentials
• Two-phase flows with bulk-surface interaction: thermodynamically consistent Navier-Stokes-Cahn-Hilliard models with dynamic boundary conditions
• On a coupled bulk-surface Allen-Cahn system with an affine linear transmission condition and its approximation by a Robin boundary condition
• Two-phase flows through porous media described by a Cahn--Hilliard--Brinkman model with dynamic boundary conditions
• Strong well-posedness and separation properties for a bulk-surface convective Cahn--Hilliard system with singular potentials