A Stable FE Method For the Space-Time Solution of the 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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Large deviations for a stochastic Cahn–Hilliard equation in Hölder norm

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025720500101 ◽

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.

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ANALYTICAL APPROXIMATE SOLUTION FOR NONLINEAR SPACE-TIME FRACTIONAL CAHN-HILLIARD EQUATION

International Electronic Journal of Pure and Applied Mathematics ◽

10.12732/iejpam.v7i4.1 ◽

2014 ◽

Vol 7 (3) ◽

Cited By ~ 2

Author(s):

Mohamed S. Mohamed ◽

Kh.S. Mekheimer

Keyword(s):

Approximate Solution ◽

Space Time ◽

Analytical Approximate Solution ◽

Hilliard Equation ◽

Cahn Hilliard Equation

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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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Error estimates for the scalar auxiliary variable (SAV) schemes to the viscous Cahn-Hilliard equation with hyperbolic relaxation

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2021.125002 ◽

2021 ◽

Vol 499 (1) ◽

pp. 125002

Author(s):

Hongtao Chen

Keyword(s):

Error Estimates ◽

Auxiliary Variable ◽

Hilliard Equation ◽

Cahn Hilliard Equation

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Existence and Continuity of Inertial Manifolds for the Hyperbolic Relaxation of the Viscous Cahn–Hilliard Equation

Applied Mathematics & Optimization ◽

10.1007/s00245-021-09749-9 ◽

2021 ◽

Author(s):

Ahmed Bonfoh

Keyword(s):

Inertial Manifolds ◽

Hilliard Equation ◽

Cahn Hilliard Equation

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On global dynamics of 2D convective Cahn–Hilliard equation

Boundary Value Problems ◽

10.1186/s13661-020-01477-3 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Xiaopeng Zhao

Keyword(s):

Initial Boundary ◽

Initial Boundary Value Problem ◽

Global Dynamics ◽

Time Behavior ◽

Long Time Behavior ◽

Initial Value ◽

Hilliard Equation ◽

Long Time ◽

Cahn Hilliard Equation ◽

Initial Boundary Value

AbstractIn this paper, we study the long time behavior of solution for the initial-boundary value problem of convective Cahn–Hilliard equation in a 2D case. We show that the equation has a global attractor in $H^{4}(\Omega )$ H 4 ( Ω ) when the initial value belongs to $H^{1}(\Omega )$ H 1 ( Ω ) .

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