Asymptotic limits of viscous Cahn–Hilliard equation with homogeneous Dirichlet boundary condition

Kagawa, Keiichiro; Ôtani, Mitsuharu

doi:10.1016/j.jmaa.2022.126106

Cited by 5 publications

(1 citation statement)

References 15 publications

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“…Letting δ → 0, the target problem is based on the nonlinear diffusion equation ∂ t u − ∆(β(u) − f ) = 0 in Q. Similar asymptotic limits have been applied also in other contexts (see, e.g., [20,21,23,29,32,33,34,45,48]).…”

Section: Introductionmentioning

confidence: 99%

The Cahn--Hilliard Equation with Forward-Backward Dynamic Boundary Condition via Vanishing Viscosity

Colli¹,

Fukao²,

Scarpa

2022

SIAM J. Math. Anal.

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A system with equation and dynamic boundary condition of Cahn-Hilliard type is considered. This system comes from a derivation performed in Liu-Wu (Arch. Ration. Mech. Anal. 233 (2019), 167-247) via an energetic variational approach. Actually, the related problem can be seen as a transmission problem for the phase variable in the bulk and the corresponding variable on the boundary. The asymptotic behavior as the coefficient of the surface diffusion acting on the boundary phase variable goes to 0 is investigated. By this analysis we obtain a forward-backward dynamic boundary condition at the limit. We can deal with a general class of potentials having a double-well structure, including the non-smooth double-obstacle potential. We illustrate that the limit problem is well-posed by also proving a continuous dependence estimate. Moreover, in the case when the two graphs, in the bulk and on the boundary, exhibit the same growth, we show that the solution of the limit problem is more regular and we prove an error estimate for a suitable order of the diffusion parameter.

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Section: Introductionmentioning

confidence: 99%

The Cahn--Hilliard Equation with Forward-Backward Dynamic Boundary Condition via Vanishing Viscosity

Colli¹,

Fukao²,

Scarpa

2022

SIAM J. Math. Anal.

View full text Add to dashboard Cite

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A Cahn–Hilliard system with forward-backward dynamic boundary condition and non-smooth potentials

Colli

Fukao²,

Scarpa³

2022

J. Evol. Equ.

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A system with equation and dynamic boundary condition of Cahn-Hilliard type is considered. This system comes from a derivation performed in Liu-Wu (Arch. Ration. Mech. Anal., 233:167-247, 2019) via an energetic variational approach. Actually, the related problem can be seen as a transmission problem for the phase variable in the bulk and the corresponding variable on the boundary. The asymptotic behavior as the coefficient of the surface diffusion acting on the boundary phase variable goes to 0 is investigated. By this analysis we obtain a forward-backward dynamic boundary condition at the limit. We can deal with a general class of potentials having a double-well structure, including the non-smooth double-obstacle potential. We illustrate that the limit problem is well-posed by also proving a continuous dependence estimate. Moreover, in the case when the two graphs, in the bulk and on the boundary, exhibit the same growth, we show that the solution of the limit problem is more regular and we prove an error estimate for a suitable order of the diffusion parameter.

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The time-periodic problem of the viscous Cahn–Hilliard equation with the homogeneous Dirichlet boundary condition

Kagawa

Ôtani

2023

J. Fixed Point Theory Appl.

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Asymptotic limits of viscous Cahn–Hilliard equation with homogeneous Dirichlet boundary condition

Cited by 5 publications

References 15 publications

The Cahn--Hilliard Equation with Forward-Backward Dynamic Boundary Condition via Vanishing Viscosity

The Cahn--Hilliard Equation with Forward-Backward Dynamic Boundary Condition via Vanishing Viscosity

A Cahn–Hilliard system with forward-backward dynamic boundary condition and non-smooth potentials

The time-periodic problem of the viscous Cahn–Hilliard equation with the homogeneous Dirichlet boundary condition

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