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 Large deviations for a stochastic Cahn–Hilliard equation in Hölder norm

2020 ◽  
Vol 23 (02) ◽  
pp. 2050010
Author(s):  
Lahcen Boulanba ◽  
Mohamed Mellouk
Keyword(s):  
Weak Convergence ◽  
Large Deviations ◽  
White Noise ◽  
Space Time ◽  
Large Deviations Principle ◽  
Hilliard Equation ◽  
Hölder Norm ◽  
The Law ◽  
Cahn Hilliard Equation ◽  
Weak Convergence Approach

We consider a stochastic Cahn–Hilliard equation driven by a space–time white noise. We prove that the law of the solution satisfies a large deviations principle in the Hölder norm. Our proof is based on the weak convergence approach for large deviations.

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 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.

Conservative stochastic two-dimensional Cahn–Hilliard equation

2021 ◽  
Vol 31 (3) ◽  
Author(s):  
Michael Röckner ◽  
Huanyu Yang ◽  
Rongchan Zhu
Keyword(s):  
Two Dimensional ◽  
Hilliard Equation ◽  
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
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