On the stochastic Cahn–Hilliard equation with a singular double-well potential

We derive a new, effective macroscopic Cahn–Hilliard equation whose homogeneous free energy is represented by fourth-order polynomials, which form the frequently applied double-well potential. This upscaling is done for perforated/strongly heterogeneous domains. To the best knowledge of the authors, this seems to be the first attempt of upscaling the Cahn–Hilliard equation in such domains. The new homogenized equation should have a broad range of applicability owing to the well-known versatility of phase-field models. The additionally introduced feature of systematically and reliably accounting for confined geometries by homogenization allows for new modelling and numerical perspectives in both science and engineering. Our results are applied to wetting dynamics in porous media and to a single channel with strongly heterogeneous walls.

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The convective viscous Cahn–Hilliard equation: Exact solutions

European Journal of Applied Mathematics ◽

10.1017/s0956792515000285 ◽

2015 ◽

Vol 27 (1) ◽

pp. 42-65 ◽

Cited By ~ 5

Author(s):

P. O. MCHEDLOV-PETROSYAN

Keyword(s):

Exact Solutions ◽

Applied Mathematics ◽

Travelling Wave ◽

Additional Constraint ◽

Travelling Wave Solutions ◽

Convective Term ◽

Nonlinear Convection ◽

Hilliard Equation ◽

Cahn Hilliard Equation ◽

Double Well Potential

In this paper, we give exact solutions for the convective viscous Cahn--Hilliard equation. This equation with a general symmetric double-well potential and Burgers-type convective term was introduced by T. P. Witelski (1996 Studies in Applied Mathematics96, 277–300) to study the joint effects of nonlinear convection and viscosity. We consider this equation with a polynomial, generally asymmetric potential. We also consider both Burgers-type and cubic convective terms. We obtained exact travelling-wave solutions for both cases. For the former case, with an additional constraint on nonlinearity and viscosity, we also obtained an exact two-wave solution.

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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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