Fractional Cahn–Hilliard, Allen–Cahn and porous medium equations

Akagi, Goro; Schimperna, Giulio; Segatti, Antonio

doi:10.1016/j.jde.2016.05.016

Cited by 111 publications

(114 citation statements)

References 44 publications

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“…Several numerical techniques have been recently developed for space and time non-local versions of equation (1.3), most of them based on finite differences or spectral methods [28,21,35,22,27,7]. Also, numerical methods have been studied for nonlocal versions of related phase separation models, like the Cahn-Hilliard equation [5,6].…”

Section: Introductionmentioning

confidence: 99%

Numerical approximations for a fully fractional Allen–Cahn equation

Acosta

Bersetche

2021

ESAIM: M2AN

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A finite element scheme for an entirely fractional Allen-Cahn equation is introduced and analyzed. In the proposed nonlocal model, the Caputo fractional in-time derivative and the fractional Laplacian replace the standard local operators. Piecewise linear finite elements and convolution quadratures are the basic tools involved in the presented numerical method. Error analysis and implementation issues are addressed together with the needed results of regularity for the continuous model. Also, the asymptotic behavior of solutions, for a vanishing diffusion parameter, is analyzed within the framework of the Γ-convergence theory.On the other hand, even though (1.1) makes perfect sense for n = 1, lateral versions are necessary for dealing with the time variable. In such a case, the so-called 2010 Mathematics Subject Classification. 65R20, 65M60, 35R11.

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Section: Introductionmentioning

confidence: 99%

Numerical approximations for a fully fractional Allen–Cahn equation

Acosta

Bersetche

2021

ESAIM: M2AN

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“…Such an energy dissipation low plays an important role in developing stable numerical methods for dissipation systems due to its importance to the long time numerical simulation, see e.g., [12,13,14,44,36,49,16,46,42] and references therein. In recent years, fractional-type phase-field models have attracted more and more attentions [1,2,3,7,30,38,23]. For instance, consider a fractional type free energy [38] E α (φ) := Ω ε 2 2 |∇ α φ| 2 + F (φ) dx, (1.5) where ∇ α is the fractional gradient ∇ α = ( ∂ α ∂x1 , ..., ∂ α ∂x d ) and { ∂ α ∂x k } k are fractional derivatives.…”

mentioning

confidence: 99%

On Energy Dissipation Theory and Numerical Stability for Time-Fractional Phase-Field Equations

Tang¹,

Yu²,

Zhou³

2019

SIAM J. Sci. Comput.

141

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For the time-fractional phase field models, the corresponding energy dissipation law has not been settled on both the continuous level and the discrete level. In this work, we shall address this open issue. More precisely, we prove for the first time that the time-fractional phase field models indeed admit an energy dissipation law of an integral type. In the discrete level, we propose a class of finite difference schemes that can inherit the theoretical energy stability. Our discussion covers the time-fractional gradient systems, including the time-fractional Allen-Cahn equation, the time-fractional Cahn-Hilliard equation, and the time-fractional molecular beam epitaxy models. Numerical examples are presented to confirm the theoretical results. Moreover, a numerical study of the coarsening rate of random initial states depending on the fractional parameter α reveals that there are several coarsening stages for both time-fractional Cahn-Hilliard equation and timefractional molecular beam epitaxy model, while there exists a −α/3 power law coarsening stage.

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“…In Section 4, we verify Cauchy's criterion of solutions to (P) and prove Theorem 1. In the case that q > 1, the above function appears in the porous media equation (see, eg, previous studies [17][18][19][20] ). In the case that 0 < q < 1, is the function in the fast diffusion equation (see, eg, other studies 19,21,22 ).…”

Section: Introduction and Resultsmentioning

confidence: 98%

“…The estimates (17) to (20) yield that there exist a subsequence { k } k∈N , with k ↘ 0 as k → ∞, and some functions v ∈ H 1 (0, T; V * ) ∩ L ∞ (0, T; H), ∈ L 2 (0, T; V) and ∈ L 2 (0, T; H) satisfying…”

Section: Proof Of Theoremmentioning

confidence: 99%

Nonlinear diffusion equations as asymptotic limits of Cahn‐Hilliard systems on unbounded domains via Cauchy's criterion

Fukao

Kurima

Yokota

2018

Math Methods in App Sciences

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This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial‐boundary problem for the nonlinear diffusion equation in an unbounded domain normalΩ⊂RN( N∈double-struckN), written as ∂u∂t+(−Δ+1)β(u)=ginΩ×(0,T), which represents the porous media, the fast diffusion equations, etc, where β is a single‐valued maximal monotone function on double-struckR, and T>0. In Kurima and Yokota (J Differential Equations 2017; 263:2024‐2050 and Adv Math Sci Appl 2017; 26:221‐242) existence and uniqueness of solutions for were directly proved under a growth condition for β even though the Stefan problem was excluded from examples of . This paper completely removes the growth condition for β by confirming Cauchy's criterion for solutions of the following approximate problem ε with approximate parameter ε>0: ∂uε∂t+(−Δ+1)(ε(−Δ+1)uε+β(uε)+πε(uε))=ginΩ×(0,T), which is called the Cahn‐Hilliard system, even if normalΩ⊂RN( N∈double-struckN) is an unbounded domain. Moreover, it can be seen that the Stefan problem excluded from Kurima and Yokota (J Differential Equations 2017; 263:2024‐2050 and Adv Math Sci Appl 2017; 26:221‐242) is covered in the framework of this paper.

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Fractional Cahn–Hilliard, Allen–Cahn and porous medium equations

Cited by 111 publications

References 44 publications

Numerical approximations for a fully fractional Allen–Cahn equation

Numerical approximations for a fully fractional Allen–Cahn equation

On Energy Dissipation Theory and Numerical Stability for Time-Fractional Phase-Field Equations

Nonlinear diffusion equations as asymptotic limits of Cahn‐Hilliard systems on unbounded domains via Cauchy's criterion

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