Weak Solutions of the Cahn--Hilliard System with Dynamic Boundary Conditions: A Gradient Flow Approach

Abstract

The Cahn-Hilliard equation is the most common model to describe phase separation processes of a mixture of two components. For a better description of short-range interactions of the material with the solid wall, various dynamic boundary conditions have been considered in recent times. New models with dynamic boundary conditions have been proposed recently by C. Liu and H. Wu [24]. We prove the existence of weak solutions to these new models by interpreting the problem as a suitable gradient flow of a total free energy which contains volume as well as surface contributions. The formulation involves an inner product which couples bulk and surface quantities in an appropriate way. We use an implicit time discretization and show that the obtained approximate solutions converge to a weak solution of the Cahn-Hilliard system. This allows us to substantially improve earlier results which needed strong assumptions on the geometry of the domain. Furthermore, we prove that this weak solution is unique.

An Energetic Variational Approach for the Cahn–Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis

“…Remark 3.4. In a very recent preprint [40], the authors proved the existence and uniqueness of global weak solutions to problem (3.2) without using the assumption (3.18) when κ = 0. We note that the notion of weak solution introduced in [40, Definition 2] is weaker than that in our Definition 3.1.…”

“…We note that the notion of weak solution introduced in [40, Definition 2] is weaker than that in our Definition 3.1. Instead of the pointwise expressions (3.12)-(3.13) for the chemical potentials (µ, µ Γ ), due to the lower spatial regularity on (φ, ψ) obtained in [40], there the pair (µ, µ Γ ) only satisfies a suitable weak formulation, which warrants the uniqueness of (µ, µ Γ ) up to some constants.…”

On a transmission problem for equation and dynamic boundary condition of Cahn–Hilliard type with nonsmooth potentials

“…is decreasing in time (see [29]) and furthermore, the system (1.1)-(1.6) can be interpreted as a gradient flow of ( , ) in a suitable dual space (see [18]). In light of (1.1), (1.3) and (1.5), we easily deduce that the following properties on mass conservation:…”

“…In addition, we mention the contributions [6,8,15,21] related to the well-posedness, [9,12,13,16,17,20] for the study of long time behavior and the optimal control problems, [7,14] for numerical analysis and [24] for the maximal regularity theory. Comparing the large number of known results on the previous model [15,21], we are only aware of the recent papers [18,29] that analyze the well-posedness of the system (1.1)-(1.6) with (1.12).…”

Boundary control problem and optimality conditions for the Cahn–Hilliard equation with dynamic boundary conditions

“…In this regard, let us address to [19], where both the viscous and the non-viscous Cahn-Hilliard equations, combined with these kinds of boundary conditions, have been investigated by assuming the boundary potential to be dominant on the bulk one. Furthermore, we have to mention [4,9,13,16,23,25,33,[36][37][38]42], where other problems related to the Cahn-Hilliard equation combined with dynamic boundary conditions have been analyzed, and [3,7,8,11,20,29,35] for the coupling of dynamic boundary conditions with different phase field models such as the Allen-Cahn or the Penrose-Fife model. So, according to [19] we supply the above system (1.1)-(1.2) with ∂ n w = 0 on Σ := Γ × (0, T ), (1.5) ∂ n y + ∂ t y Γ − ∆ Γ y Γ + f ′ Γ (y Γ ) = u Γ on Σ, (1.6) where Γ is the boundary of Ω, y Γ denotes the trace of y, ∆ Γ stands for the Laplace-Beltrami operator on the boundary, and ∂ n represents the outward normal derivative.…”