STOCHASTIC CAHN–HILLIARD EQUATION WITH FRACTIONAL NOISE

2008 ◽  
Vol 08 (04) ◽  
pp. 643-665 ◽  
Author(s):  
LIJUN BO ◽  
YIMING JIANG ◽  
YONGJIN WANG
Keyword(s):  
Weak Convergence ◽  
Existence And Uniqueness ◽  
Mild Solutions ◽  
Fractional Noise ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation ◽  
Fractional Noises

We study the existence and uniqueness of global mild solutions to a class of stochastic Cahn–Hilliard equations driven by fractional noises (fractional in time and white in space), through a weak convergence argument.

scholarly journals Global Mild Solutions and Attractors for Stochastic Viscous Cahn-Hilliard Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-22 ◽  
Author(s):  
Xuewei Ju ◽  
Hongli Wang ◽  
Desheng Li ◽  
Jinqiao Duan
Keyword(s):  
Boundary Value Problem ◽  
White Noise ◽  
Global Attractor ◽  
Existence And Uniqueness ◽  
Mild Solutions ◽  
Boundary Value ◽  
Uniqueness Result ◽  
Hilliard Equation ◽  
Bounded Set ◽  
Cahn Hilliard Equation

This paper is devoted to the study of mild solutions for the initial and boundary value problem of stochastic viscous Cahn-Hilliard equation driven by white noise. Under reasonable assumptions we first prove the existence and uniqueness result. Then, we show that the existence of a stochastic global attractor which pullback attracts each bounded set in appropriate phase spaces.

On Cahn—Hilliard systems with elasticity

2003 ◽  
Vol 133 (2) ◽  
pp. 307-331 ◽  
Author(s):  
Harald Garcke
Keyword(s):  
Existence And Uniqueness ◽  
Uniqueness Result ◽  
Separation Process ◽  
Diffusion Equations ◽  
System Of Equations ◽  
Ginzburg Landau ◽  
Elastic Interactions ◽  
Hilliard Equation ◽  
Phase Separation Process ◽  
Cahn Hilliard Equation

Elastic effects can have a pronounced effect on the phase-separation process in solids. The classical Ginzburg—Landau energy can be modified to account for such elastic interactions. The evolution of the system is then governed by diffusion equations for the concentrations of the alloy components and by a quasi-static equilibrium for the mechanical part. The resulting system of equations is elliptic-parabolic and can be understood as a generalization of the Cahn—Hilliard equation. In this paper we give a derivation of the system and prove an existence and uniqueness result for it.

Global attractor for a hyperbolic Cahn–Hilliard equation with a proliferation term

2021 ◽  
pp. 1-23
Author(s):  
Padouette Boubati Badieti Matala ◽  
Daniel Moukoko ◽  
Mayeul Evrard Isseret Goyaud
Keyword(s):  
Boundary Conditions ◽  
Hyperbolic Equation ◽  
Global Attractor ◽  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Hilliard Equation ◽  
Uniqueness Of The Solution ◽  
Cahn Hilliard Equation

In this article, we study a hyperbolic equation of Cahn–Hilliard with a proliferation term and Dirichlet boundary conditions. In particular, we prove the existence and uniqueness of the solution, and also the existence of the global attractor.

scholarly journals A fourth-order parabolic equation with a logarithmic nonlinearlity

2004 ◽  
Vol 69 (1) ◽  
pp. 35-48 ◽  
Author(s):  
Ahmed Bonfoh
Keyword(s):  
Free Energy ◽  
Parabolic Equation ◽  
Fourth Order ◽  
Constitutive Equations ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solutions ◽  
Hilliard Equation ◽  
Finite Dimensional ◽  
Fourth Order Parabolic Equation ◽  
Cahn Hilliard Equation

We consider some generalisations of the Cahn—Hilliard equation based on constitutive equations derived by M. Gurtin in (1996) with a logarithmic free energy. Compared to the classical Cahn—Hilliard equation (see [4, 5]), these models take into account the work of internal microforces and the anisotropy of the material. We obtain the existence and uniqueness of solutions results and then prove the existence of finite dimensional attractors.

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.

Central limit theorem and moderate deviations for a perturbed stochastic Cahn–Hilliard equation

2019 ◽  
Vol 20 (03) ◽  
pp. 2050017
Author(s):  
Ruinan Li ◽  
Xinyu Wang
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Weak Convergence ◽  
White Noise ◽  
Central Limit ◽  
Moderate Deviation ◽  
Hilliard Equation ◽  
Deviation Principle ◽  
Cahn Hilliard Equation ◽  
Weak Convergence Approach

In this paper, we prove a central limit theorem and a moderate deviation principle for a perturbed stochastic Cahn–Hilliard equation defined on [Formula: see text] with [Formula: see text]. This equation is driven by a space-time white noise. The weak convergence approach plays an important role.

Existence results of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Shengli Xie
Keyword(s):  
Differential Equations ◽  
Differential Evolution ◽  
Existence Theorem ◽  
Banach Spaces ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Infinite Delay ◽  
Mild Solutions ◽  
Existence Results ◽  
Order Case

AbstractIn this paper we prove the existence and uniqueness of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay in Banach spaces. We generalize the existence theorem for integer order differential equations to the fractional order case. The results obtained here improve and generalize many known results.

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.

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