A Cahn–Hilliard model in a domain with non-permeable walls

Goldstein, Gisèle Ruiz; Miranville, Alain; Schimperna, Giulio

doi:10.1016/j.physd.2010.12.007

Cited by 87 publications

(112 citation statements)

References 28 publications

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“…We refer the reader to [120] for the study of this problem (see also [96], [97] and [102] for similar dynamic boundary conditions in the case of semipermeable walls).…”

Section: Cahn-hilliard Equation 565mentioning

confidence: 99%

“…As already mentioned, as the limit is not a classical solution to (3.5) in general, we first need to define a proper weak formulation of the problem. More precisely, this will be done by considering a variational inequality (see also [120] and [127] for a very close approach, but at an abstract level, i.e., based on duality arguments).…”

Section: Variational Solutionsmentioning

confidence: 99%

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The Cahn-Hilliard Equation with Logarithmic Potentials

2011

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Our aim in this article is to discuss recent issues related with the Cahn- Hilliard equation in phase separation with the thermodynamically relevant logarithmic potentials; in particular, we are interested in the well-posedness and the study of the asymptotic behavior of the solutions (and, more precisely, the existence of finite-dimensional attractors). We first consider the usual Neumann boundary conditions and then dynamic boundary conditions which account for the interactions with the walls in confined systems and have recently been proposed by physicists. We also present, in the case of dynamic boundary conditions, some numerical results

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“…We refer the reader to [120] for the study of this problem (see also [96], [97] and [102] for similar dynamic boundary conditions in the case of semipermeable walls).…”

Section: Cahn-hilliard Equation 565mentioning

confidence: 99%

Section: Variational Solutionsmentioning

confidence: 99%

The Cahn-Hilliard Equation with Logarithmic Potentials

2011

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“…Now, in order to define dynamic boundary conditions, we introduce, in addition to the bulk free energy ψ Ω , a surface free energy ψ Γ and write

\begin{matrix} aligned \\ right Ψ = \int_{Ω} ψ_{Ω} dx + \int_{Γ} ψ_{Γ} dσ = \int_{Ω} (\frac{1}{2} | \nabla u |^{2} + F MathClass-open(u MathClass-close)) dx + \int_{Γ} (\frac{1}{2} | \nabla_{Γ} u |^{2} + F_{Γ} MathClass-open(u MathClass-close)) dσ \end{matrix}

where ∇ Γ is the surface gradient and F Γ is a surface potential, with derivative f Γ (typically,

F_{Γ} (s) MathClass-rel= \frac{1}{2} a_{Γ} s^{2} MathClass-bin+ b_{Γ} s MathClass-punc, a_{Γ} MathClass-rel> 0

; hence, f Γ ( s ) = a Γ s + b Γ , see ).…”

Section: Introductionmentioning

confidence: 99%

“…Actually, we will follow here another approach, proposed in , consisting instead in writing that we have the total (in the bulk and on the bounadry) mass conservation

\frac{d}{dt} ((), {MathClass-op\int}_{Ω} u dx MathClass-bin+ {MathClass-op\int}_{Γ} u dσ) MathClass-rel= 0

and that

w MathClass-rel= \frac{δ ψ_{Ω}}{δu} in Ω

and

w MathClass-rel= \frac{δ ψ_{Γ}}{δu} on Γ

As we will see in the succeeding text, one interest of this approach is that it allows to define and study dynamic boundary conditions in a systematic way (see also for phase‐field models). Here, this yields the boundary conditions

\frac{∂u}{∂t} MathClass-rel= Δw MathClass-bin- \frac{∂w}{∂ν} on Γ

(actually, we can take, more generally,

\frac{∂u}{∂t} MathClass-rel= κΔw MathClass-bin- \frac{∂w}{∂ν} on Γ

κ ≥ 0) and

w MathClass-rel= MathClass-bin- Δ_{Γ} u MathClass-bin+ \frac{∂u}{∂ν} MathClass-bin+ f_{Γ} (u) on Γ

…”

Section: Introductionmentioning

confidence: 99%

Sixth-order Cahn-Hilliard systems with dynamic boundary conditions

Miranville

2014

Math. Meth. Appl. Sci.

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“…From a theoretical point of view, this system has already been studied (see for exemple [6] and the references therein). From a numerical point of view, we have several results.…”

Section: Introductionmentioning

confidence: 99%

Finite-Volume Analysis for the Cahn-Hilliard Equation with Dynamic Boundary Conditions

Nabet¹

2014

Springer Proceedings in Mathematics &Amp; Statistics

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This work is devoted to the convergence analysis of a finite-volume approximation of the 2D Cahn-Hilliard equation with dynamic boundary conditions. The method that we propose couples a 2d-finite-volume method in a bounded, smooth domain Ω ⊂ R 2 and a 1d-finite-volume method on ∂ Ω . We prove convergence of the sequence of approximate solutions. One of the main ingredient is a suitable space translation estimate that gives a limit in L ∞ 0, T, H 1 (Ω ) whose trace is in L ∞ 0, T, H 1 (∂ Ω ) .

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A Cahn–Hilliard model in a domain with non-permeable walls

Cited by 87 publications

References 28 publications

The Cahn-Hilliard Equation with Logarithmic Potentials

The Cahn-Hilliard Equation with Logarithmic Potentials

Sixth-order Cahn-Hilliard systems with dynamic boundary conditions

Finite-Volume Analysis for the Cahn-Hilliard Equation with Dynamic Boundary Conditions

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