scholarly journals Geometric evolution of bilayers under the functionalized Cahn–Hilliard equation

2013 ◽  
Vol 469 (2153) ◽  
pp. 20120505 ◽  
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.

scholarly journals Sharp Interface Limit of Stochastic Cahn-Hilliard Equation with Singular Noise

2022 ◽  
Author(s):  
Ľubomír Baňas ◽  
Huanyu Yang ◽  
Rongchan Zhu
Keyword(s):  
White Noise ◽  
Space Time ◽  
Sharp Interface ◽  
Two Dimensional ◽  
Sharp Interface Limit ◽  
Hilliard Equation ◽  
Stochastic Problems ◽  
Divergence Type ◽  
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.

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

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.

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

Analysis ◽  
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.

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

2016 ◽  
Vol 76 (2) ◽  
pp. 433-456 ◽  
Author(s):  
Alpha Albert Lee ◽  
Andreas Münch ◽  
Endre Süli
Keyword(s):  
Sharp Interface ◽  
Hilliard Equation ◽  
Degenerate Mobility ◽  
Cahn Hilliard Equation

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

2015 ◽  
Vol 48 (1) ◽  
pp. 401-402
Author(s):  
Alpha A Lee ◽  
Andreas Munch ◽  
Endre Suli
Keyword(s):  
Sharp Interface ◽  
Hilliard Equation ◽  
Degenerate Mobility ◽  
Cahn Hilliard Equation

The Cahn–Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature

1996 ◽  
Vol 7 (3) ◽  
pp. 287-301 ◽  
Author(s):  
J. W. Cahn ◽  
C. M. Elliott ◽  
A. Novick-Cohen
Keyword(s):  
Mean Curvature ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Normal Velocity ◽  
Hilliard Equation ◽  
Formal Asymptotics ◽  
Gradient Energy ◽  
The Mean ◽  
Cahn Hilliard Equation ◽  
Image Position

We show by using formal asymptotics that the zero level set of the solution to the Cahn–Hilliard equation with a concentration dependent mobility approximates to lowest order in ɛ. an interface evolving according to the geometric motion,(where V is the normal velocity, Δ8 is the surface Laplacian and κ is the mean curvature of the interface), both in the deep quench limit and when the temperature θ is where є2 is the coefficient of gradient energy. Equation (0.1) may be viewed as motion by surface diffusion, and as a higher-order analogue of motion by mean curvature predicted by the bistable reaction-diffusion equation.

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