Sixth-order Cahn-Hilliard systems with dynamic boundary conditions

Miranville, Alain

doi:10.1002/mma.3134

Cited by 15 publications

(5 citation statements)

References 48 publications

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“…Furthermore, in one space dimension the existence of time-periodic solutions and stability of traveling waves was studied in [14] and [13], respectively. More recent studies on higher-order Cahn-Hilliard equations include [4,17,19,20].…”

Section: Introductionmentioning

confidence: 99%

Convergence of solutions to a convective Cahn-Hilliard type equation of the sixth order in case of small deposition rates

Rybka¹,

Wheeler²

2022

Preprint

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We show stabilisation of solutions to the sixth-order convective Cahn-Hilliard equation. The problem has the structure of a gradient flow perturbed by a quadratic destabilising term with coefficient δ > 0. Through application of an abstract result by Carvalho-Langa-Robinson we show that for small δ the equation has the structure of gradient flow in a weak sense. On the way we prove a kind of Liouville theorem for eternal solutions to parabolic problems. Finally, the desired stabilisation follows from a powerful theorem due to Hale-Raugel.

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Section: Introductionmentioning

confidence: 99%

Convergence of solutions to a convective Cahn-Hilliard type equation of the sixth order in case of small deposition rates

Rybka¹,

Wheeler²

2022

Preprint

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“…The numerical approximation of polyharmonic problems has been first addressed in the eighties by [18] and, more recently, in [13,40,45,42,36]. It is worth mentioning an increasing interest in the numerical approximation of models involving high-order differential operators, e.g., [46,47,41] in the context of sixth order Cahn-Hilliard equations. To the best of our knowledge, the conforming VEM proposed in this article is the first work addressing the approximation of arbitrary-order polyharmonic problems on polygonal meshes.…”

Section: Introductionmentioning

confidence: 99%

The conforming virtual element method for polyharmonic problems

Antonietti

Manzini

Verani

2020

Computers & Mathematics with Applications

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In this work, we exploit the capability of virtual element methods in accommodating approximation spaces featuring high-order continuity to numerically approximate differential problems of the form ∆ p u = f , p ≥ 1. More specifically, we develop and analyze the conforming virtual element method for the numerical approximation of polyharmonic boundary value problems, and prove an abstract result that states the convergence of the method in the energy norm.

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“…The sixth order Cahn-Hilliard equation model is only a phenomenological model. Hence, various modifications of it, for example, [16,17,19,20] and the reference therein, has been proposed in order to capture the dynamical picture of the phase transition phenomena better.…”

Section: Introductionmentioning

confidence: 99%

On the Cauchy problem of a sixth-order Cahn-Hilliard equation arising in oil-water-surfactant mixtures

Zhao

2019

Preprint

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We study the global well-posedness and asymptotic behavior of solutions for the Cauchy problem of three-dimensional sixth order Cahn-Hilliard equation arising in oil-water-surfactant mixtures. First, by using the pure energy method and a standard continuity argument, we prove that there exists a unique global strong solution provided that the H 2 -norm of initial data is sufficiently small. Moreover, we also establish the suitable negative Sobolev norm estimates and obtain the time decay rate of strong solutions. It is worth pointing out that although the problem we considered is a sixth-order parabolic equation, the time decay rate is equivalent to the decay rate of fourth-order generalized heat equation, which is better than our expect.

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Sixth-order Cahn-Hilliard systems with dynamic boundary conditions

Abstract: Our aim in this paper is to define dynamic boundary conditions for several sixth‐order Cahn–Hilliard systems. We then study the well‐posedness and the dissipativity of the systems derived. Copyright © 2014 John Wiley & Sons, Ltd.

Cited by 15 publications

References 48 publications

Convergence of solutions to a convective Cahn-Hilliard type equation of the sixth order in case of small deposition rates

Convergence of solutions to a convective Cahn-Hilliard type equation of the sixth order in case of small deposition rates

The conforming virtual element method for polyharmonic problems

On the Cauchy problem of a sixth-order Cahn-Hilliard equation arising in oil-water-surfactant mixtures

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