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 A Note on the Solutions for a Higher-Order Convective Cahn–Hilliard-Type Equation

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1835
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Type Equation ◽  
Higher Order ◽  
Crystal Surfaces ◽  
Hilliard Equation ◽  
The Cauchy Problem ◽  
The Mean ◽  
Oil Water ◽  
Cahn Hilliard Equation ◽  
Well Posed ◽  
Dynamics Of Phase Transitions

The higher-order convective Cahn-Hilliard equation describes the evolution of crystal surfaces faceting through surface electromigration, the growing surface faceting, and the evolution of dynamics of phase transitions in ternary oil-water-surfactant systems. In this paper, we study the H3 solutions of the Cauchy problem and prove, under different assumptions on the constants appearing in the equation and on the mean of the initial datum, that they are well-posed.

PHASE SEPARATION DYNAMICS AND EXTERNAL FORCE FIELD

1988 ◽  
Vol 02 (06) ◽  
pp. 765-771 ◽  
Author(s):  
K. KITAHARA ◽  
Y. OONO ◽  
DAVID JASNOW
Keyword(s):  
Dynamical System ◽  
Phase Separation ◽  
Force Field ◽  
External Force ◽  
System Modeling ◽  
Bulk Phase ◽  
Hilliard Equation ◽  
External Force Field ◽  
Dynamical System Modeling ◽  
Cahn Hilliard Equation

If spinodal decomposition is modeled by the Cahn-Hilliard (-Cook) equation, the effect of a uniform external force such as gravitation does not appear in the bulk phase kinetics. In contrast, in the Kawasaki exchange modeling of the local dynamics of binary alloys, this effect directly modifies the bulk phase kinetics. We resolve this paradox through the cell-dynamical-system modeling of the Kawasaki exchange dynamics. Its continuum version has turned out to be a modified Cahn-Hilliard equation already proposed by Langer et al. about ten years ago. We demonstrate some examples in which the correction to the Cahn-Hilliard equation is significant.

scholarly journals A coupled surface-Cahn–Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes

2016 ◽  
Vol 26 (06) ◽  
pp. 1149-1189 ◽  
Author(s):  
Harald Garcke ◽  
Johannes Kampmann ◽  
Andreas Rätz ◽  
Matthias Röger
Keyword(s):  
Numerical Simulations ◽  
Lipid Raft ◽  
System Modeling ◽  
Bulk Diffusion ◽  
Lipid Phase ◽  
Qualitative Behavior ◽  
Time Behavior ◽  
Relaxation Dynamics ◽  
Unsaturated Lipids ◽  
Long Time

We propose and investigate a model for lipid raft formation and dynamics in biological membranes. The model describes the lipid composition of the membrane and an interaction with cholesterol. To account for cholesterol exchange between cytosol and cell membrane we couple a bulk-diffusion to an evolution equation on the membrane. The latter describes the relaxation dynamics for an energy which takes lipid–phase separation and lipid–cholesterol interaction energy into account. It takes the form of an (extended) Cahn–Hilliard equation. Different laws for the exchange term represent equilibrium and nonequilibrium models. We present a thermodynamic justification, analyze the respective qualitative behavior and derive asymptotic reductions of the model. In particular we present a formal asymptotic expansion near the sharp interface limit, where the membrane is separated into two pure phases of saturated and unsaturated lipids, respectively. Finally we perform numerical simulations and investigate the long-time behavior of the model and its parameter dependence. Both the mathematical analysis and the numerical simulations show the emergence of raft-like structures in the nonequilibrium case whereas in the equilibrium case only macrodomains survive in the long-time evolution.

scholarly journals Global attractor for the Cahn-Hilliard system

1994 ◽  
Vol 49 (2) ◽  
pp. 277-292 ◽  
Author(s):  
Jan W. Cholewa ◽  
Tomasz Dlotko
Keyword(s):  
Phase Space ◽  
Global Attractor ◽  
Lyapunov Functional ◽  
Natural Extension ◽  
Multicomponent Alloys ◽  
Hilliard Equation ◽  
System A ◽  
Cahn Hilliard Equation ◽  
Dissipative Semigroup

The Cahn-Hilliard system, a natural extension of the single Cahn-Hilliard equation in the case of multicomponent alloys, will be shown to generate a dissipative semigroup on the phase space ℋ = [H2(Ω)]m. Following Hale's ideas and based on the existence and form of the Lyapunov functional, our main result will be the existence of a global attractor on a subset of ℋ. New difficulties specific to the system case make our problem interesting.

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

scholarly journals $$H^1$$ solutions for a Kuramoto–Sinelshchikov–Cahn–Hilliard type equation

2021 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Surface Tension ◽  
Cauchy Problem ◽  
Type Equation ◽  
Well Posedness ◽  
Crystal Surfaces ◽  
Spatiotemporal Evolution ◽  
Hilliard Equation ◽  
Anisotropic Surface ◽  
The Cauchy Problem ◽  
Cahn Hilliard Equation

AbstractThe Kuramoto–Sinelshchikov–Cahn–Hilliard equation models the spinodal decomposition of phase separating systems in an external field, the spatiotemporal evolution of the morphology of steps on crystal surfaces and the growth of thermodynamically unstable crystal surfaces with strongly anisotropic surface tension. In this paper, we prove the well-posedness of the Cauchy problem, associated with this equation.

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.

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