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

Li, Chun; Wu, Hao

doi:10.1007/s00205-019-01356-x

Cited by 101 publications

(126 citation statements)

References 84 publications

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“…Liu & Wu [19], Hyon et al [14] Du et al [6], Eisenberg et al [9] and Koba et al [17]. Its application to the Wright-Fisher model has been studied in [7].The detailed structures of EnVarA can be found in [7,14,19,20].…”

Section: The Energetic Variational Approachmentioning

confidence: 99%

Numerical methods for porous medium equation by an energetic variational approach

Duan

Wang

et al. 2019

Journal of Computational Physics

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We study numerical methods for porous media equation (PME). There are two important characteristics: the finite speed propagation of the free boundary and the potential waiting time, which make the problem not easy to handle. Based on different dissipative energy laws, we develop two numerical schemes by an energetic variational approach. Firstly, based on f log f as the total energy form of the dissipative law, we obtain the trajectory equation, and then construct a fully discrete scheme. It is proved that the scheme is uniquely solvable on an admissible convex set by taking the advantage of the singularity of the total energy. Next, based on 1 2f as the total energy form of the dissipation law, we construct a linear numerical scheme for the corresponding trajectory equation. Both schemes preserve the corresponding discrete dissipation law. Meanwhile, under some smoothness assumption, it is proved, by a higher order expansion technique, that both schemes are second-order convergent in space and first-order convergent in time. Each scheme yields a good approximation for the solution and the free boundary. No oscillation is observed for the numerical solution around the free boundary. Furthermore, the waiting time problem could be naturally treated, which has been a well-known difficult issue for all the existence methods. Due to its linear nature, the second scheme is more efficient.The porous medium equation (PME) can be found in many physical and biological phenomena, such as the flow of an isentropic gas through a porous medium [18], the viscous gravity currents [12], nonlinear heat transfer and image processing; e.g., see [33]. The aim of this paper is to provide numerical methods for the PMEwhere f := f (x, t) is a non-negative scalar function of space x ∈ R d and the time t ∈ R, the space dimension is given by d ≥ 1, and m is a constant larger than 1.The PME is a nonlinear degenerate parabolic equation since the diffusivity D(f ) = mf m−1 = 0 at points where f = 0. In turn, the PME has a special feature: the finite speed of propagation, called finite propagation [33]. If the initial data has a compact support, the solution of Cauchy problem of the PME will have a compact support at any given time t > 0. In comparison with the heat equation, which can smooth out the initial data, the solution of the PME becomes non-smooth even if the initial data is smooth with compact support. If an initial data is zero in some open domain in Ω, it causes the appearance of the free boundary (in some cases, called interface) that separates the regions where the solution is positive from the regions where the value is zero in the domain. Moreover, for certain initial data, the solution of the PME can exhibit a waiting time phenomenon where the free boundary remains stationary until a finite positive time (called waiting time).After that time instant, the interface begins to move with a finite speed. Many theoretical analyses have been available in the existing literature, including the earlier works by Oleǐnik et al. [25], Ka...

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Section: The Energetic Variational Approachmentioning

confidence: 99%

Numerical methods for porous medium equation by an energetic variational approach

Duan

Wang

et al. 2019

Journal of Computational Physics

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“…Existence and uniqueness of weak and strong solutions of the system (CH) have been established by C. Liu and H. Wu in [24]. The idea of their proof is to construct solutions of a regularized system where the equations for µ and µ Γ are replaced by…”

Section: Introductionmentioning

confidence: 99%

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

Garcke¹,

Knopf²

2020

SIAM J. Math. Anal.

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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.

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“…This motivates the current section to provide a derivation of these systems of equations from balance laws. We are aware that the recent work of Liu and Wu [26] also provides a mathematical derivation of a coupled bulk-surface Cahn-Hilliard system (which can obtained as the limit K → 0 of the system (2.7) with h(s) = s, m(u) = 1, n(φ) = 1 and replacing M ∂ ν λ u + λ u = λ φ with ∂ ν λ u = 0 on Σ as a boundary condition). This is done by means of an energetic variational approach that combines the least action principle and Onsager's principle of maximum energy dissipation.…”

Section: Remark 21 (Limiting Transmission Conditions/fast Reaction Lmentioning

confidence: 99%

On a coupled bulk–surface Allen–Cahn system with an affine linear transmission condition and its approximation by a Robin boundary condition

2019

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We study a coupled bulk-surface Allen-Cahn system with an affine linear transmission condition, that is, the trace values of the bulk variable and the values of the surface variable are connected via an affine relation, and this serves to generalize the usual dynamic boundary conditions. We tackle the problem of well-posedness via a penalization method using Robin boundary conditions. In particular, for the relaxation problem, the strong well-posedness and long-time behavior of solutions can be shown for more general and possibly nonlinear relations. New difficulties arise since the surface variable is no longer the trace of the bulk variable, and uniform estimates in the relaxation parameter are scarce. Nevertheless, weak convergence to the original problem with affine linear relations can be shown. Using the approach of Colli and Fukao (Math. Models Appl. Sci. 2015), we show strong existence to the original problem with affine linear relations, and derive an error estimate between solutions to the relaxed and original problems.

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

Cited by 101 publications

References 84 publications

Numerical methods for porous medium equation by an energetic variational approach

Numerical methods for porous medium equation by an energetic variational approach

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

On a coupled bulk–surface Allen–Cahn system with an affine linear transmission condition and its approximation by a Robin boundary condition

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