A Hilbert expansion method for the rigorous sharp interface limit of the generalized Cahn–Hilliard equation

AbstractWe study the sharp interface limit of the two dimensional stochastic Cahn-Hilliard equation driven by two types of singular noise: a space-time white noise and a space-time singular divergence-type noise. We show that with appropriate scaling of the noise the solutions of the stochastic problems converge to the solutions of the determinisitic Mullins-Sekerka/Hele-Shaw problem.

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Numerical approximation of the stochastic Cahn–Hilliard equation near the sharp interface limit

Numerische Mathematik ◽

10.1007/s00211-021-01179-7 ◽

2021 ◽

Author(s):

Dimitra Antonopoulou ◽

Ĺubomír Baňas ◽

Robert Nürnberg ◽

Andreas Prohl

Keyword(s):

Finite Element ◽

Gradient Flow ◽

Sharp Interface ◽

Convergence In Probability ◽

Sharp Interface Limit ◽

Stochastic Version ◽

Fully Discrete ◽

Hilliard Equation ◽

Cahn Hilliard Equation ◽

Discrete Finite Element

AbstractWe consider the stochastic Cahn–Hilliard equation with additive noise term $$\varepsilon ^\gamma g\, {\dot{W}}$$ ε γ g W ˙ ($$\gamma >0$$ γ > 0 ) that scales with the interfacial width parameter $$\varepsilon $$ ε . We verify strong error estimates for a gradient flow structure-inheriting time-implicit discretization, where $$\varepsilon ^{-1}$$ ε - 1 only enters polynomially; the proof is based on higher-moment estimates for iterates, and a (discrete) spectral estimate for its deterministic counterpart. For $$\gamma $$ γ sufficiently large, convergence in probability of iterates towards the deterministic Hele–Shaw/Mullins–Sekerka problem in the sharp-interface limit $$\varepsilon \rightarrow 0$$ ε → 0 is shown. These convergence results are partly generalized to a fully discrete finite element based discretization. We complement the theoretical results by computational studies to provide practical evidence concerning the effect of noise (depending on its ’strength’ $$\gamma $$ γ ) on the geometric evolution in the sharp-interface limit. For this purpose we compare the simulations with those from a fully discrete finite element numerical scheme for the (stochastic) Mullins–Sekerka problem. The computational results indicate that the limit for $$\gamma \ge 1$$ γ ≥ 1 is the deterministic problem, and for $$\gamma =0$$ γ = 0 we obtain agreement with a (new) stochastic version of the Mullins–Sekerka problem.

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Geometric evolution of bilayers under the functionalized Cahn–Hilliard equation

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2012.0505 ◽

2013 ◽

Vol 469 (2153) ◽

pp. 20120505 ◽

Cited By ~ 23

Author(s):

Shibin Dai ◽

Keith Promislow

Keyword(s):

Single Layer ◽

Total Surface Area ◽

Sharp Interface ◽

Normal Velocity ◽

Sharp Interface Limit ◽

Hilliard Equation ◽

Geometric Evolution ◽

Willmore Flow ◽

Cahn Hilliard Equation ◽

Curvature Driven

We use a multi-scale analysis to derive a sharp interface limit for the dynamics of bilayer structures of the functionalized Cahn–Hilliard equation. In contrast to analysis based on single-layer interfaces, we show that the Stefan and Mullins–Sekerka problems derived for the evolution of single-layer interfaces for the Cahn–Hilliard equation are trivial in this context, and the sharp interface limit yields a quenched mean-curvature-driven normal velocity at O ( ε −1 ), whereas on the longer O ( ε −2 ) time scale, it leads to a total surface area preserving Willmore flow. In particular, for space dimension n =2, the constrained Willmore flow drives collections of spherically symmetric vesicles to a common radius, whereas for n =3, the radii are constant, and for n ≥4 the largest vesicle dominates.

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On the sharp interface limit of a model for phase separation on biological membranes

Analysis ◽

10.1515/anly-2019-0019 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Helmut Abels ◽

Johannes Kampmann

Keyword(s):

Lipid Raft ◽

Cell Membranes ◽

System Modeling ◽

Bulk Diffusion ◽

Type Equation ◽

Sharp Interface ◽

Interface Problem ◽

Hilliard Equation ◽

System A ◽

Cahn Hilliard Equation

AbstractWe rigorously prove the convergence of weak solutions to a model for lipid raft formation in cell membranes which was recently proposed in [H. Garcke, J. Kampmann, A. Rätz and M. Röger, A coupled surface-Cahn–Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes, Math. Models Methods Appl. Sci. 26 2016, 6, 1149–1189] to weak (varifold) solutions of the corresponding sharp-interface problem for a suitable subsequence. In the system a Cahn–Hilliard type equation on the boundary of a domain is coupled to a diffusion equation inside the domain. The proof builds on techniques developed in [X. Chen, Global asymptotic limit of solutions of the Cahn–Hilliard equation, J. Differential Geom. 44 1996, 2, 262–311] for the corresponding result for the Cahn–Hilliard equation.

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Sharp-Interface Limits of the Cahn--Hilliard Equation with Degenerate Mobility

SIAM Journal on Applied Mathematics ◽

10.1137/140960189 ◽

2016 ◽

Vol 76 (2) ◽

pp. 433-456 ◽

Cited By ~ 14

Author(s):

Alpha Albert Lee ◽

Andreas Münch ◽

Endre Süli

Keyword(s):

Sharp Interface ◽

Hilliard Equation ◽

Degenerate Mobility ◽

Cahn Hilliard Equation

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Exact Solutions of Fractional Burgers and Cahn-Hilliard Equations Using Extended Fractional Riccati Expansion Method

Mathematical Problems in Engineering ◽

10.1155/2014/104069 ◽

2014 ◽

Vol 2014 ◽

pp. 1-9 ◽

Cited By ~ 4

Author(s):

Wei Li ◽

Huizhang Yang ◽

Bin He

Keyword(s):

Exact Solutions ◽

Fractional Differential Equations ◽

Burgers Equation ◽

Expansion Method ◽

Trigonometric Functions ◽

Hyperbolic Functions ◽

Hilliard Equation ◽

Liouville Derivative ◽

Fractional Differential ◽

Cahn Hilliard Equation

Based on a general fractional Riccati equation and with Jumarie’s modified Riemann-Liouville derivative to an extended fractional Riccati expansion method for solving the time fractional Burgers equation and the space-time fractional Cahn-Hilliard equation, the exact solutions expressed by the hyperbolic functions and trigonometric functions are obtained. The obtained results show that the presented method is effective and appropriate for solving nonlinear fractional differential equations.

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Improved (G'/G)-Expansion Method for the Time-Fractional Biological Population Model and Cahn–Hilliard Equation

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4029254 ◽

2015 ◽

Vol 10 (5) ◽

Cited By ~ 14

Author(s):

Dumitru Baleanu ◽

Yavuz Uğurlu ◽

Mustafa Inc ◽

Bulent Kilic

Keyword(s):

Differential Equation ◽

Ordinary Differential Equation ◽

Partial Differential Equations ◽

Population Model ◽

Expansion Method ◽

Fractional Partial Differential Equations ◽

Basic Variable ◽

Hilliard Equation ◽

Biological Population ◽

Cahn Hilliard Equation

In this paper, we used improved (G'/G)-expansion method to reach the solutions for some nonlinear time-fractional partial differential equations (fPDE). The fPDE is reduced to an ordinary differential equation (ODE) by means of Riemann–Liouille derivative and a basic variable transformation. Various types of functions are obtained for the time-fractional biological population model (fBPM) and Cahn–Hilliard (fCH) equation.

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