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.

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.

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.

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 ON THE HYPERBOLIC RELAXATION OF THE CAHN–HILLIARD EQUATION IN 3D: APPROXIMATION AND LONG TIME BEHAVIOUR

2007 ◽  
Vol 17 (03) ◽  
pp. 411-437 ◽  
Author(s):  
ANTONIO SEGATTI
Keyword(s):  
Diffusion Flux ◽  
Relative Concentration ◽  
Galerkin Approximation ◽  
Alloy System ◽  
Global Attractors ◽  
Uniqueness Result ◽  
Point Of View ◽  
Hilliard Equation ◽  
Long Time ◽  
Cahn Hilliard Equation

In this paper we consider the hyperbolic relaxation of the Cahn–Hilliard equation ruling the evolution of the relative concentration u of one component of a binary alloy system located in a bounded and regular domain Ω of ℝ3. This equation is characterized by the presence of the additional inertial term ∊uttthat accounts for the relaxation of the diffusion flux. For this equation we address the problem of the long time stability from the point of view of global attractors. The main difficulty in dealing with this system is the low regularity of its weak solutions, which prevents us from proving a uniqueness result and a proper energy identity for the solutions. We overcome this difficulty by using a density argument based on a Faedo–Galerkin approximation scheme and the recent J. M. Ball's theory of generalized semiflows. Moreover, we address the problem of the approximation of the attractor of the continuous problem with the one of Faedo–Galerkin scheme. Finally, we show that the same type of results hold also for the damped semilinear wave equation when the nonlinearity ϕ is not Lipschitz continuous and has a super critical growth.

scholarly journals On a mixture of an MGT viscous material and an elastic solid

2022 ◽  
Author(s):  
José R. Fernández ◽  
Ramón Quintanilla
Keyword(s):  
Existence And Uniqueness ◽  
Elastic Solid ◽  
Uniqueness Result ◽  
Linear Operators ◽  
System Of Equations ◽  
Viscous Material ◽  
Decay Of Solutions ◽  
Exponentially Stable ◽  
Extra Assumption

AbstractA lot of attention has been paid recently to the study of mixtures and also to the Moore–Gibson–Thompson (MGT) type equations or systems. In fact, the MGT proposition can be used to describe viscoelastic materials. In this paper, we analyze a problem involving a mixture composed by a MGT viscoelastic type material and an elastic solid. To this end, we first derive the system of equations governing the deformations of such material. We give the suitable assumptions to obtain an existence and uniqueness result. The semigroups theory of linear operators is used. The paper concludes by proving the exponential decay of solutions with the help of a characterization of the exponentially stable semigroups of contractions and introducing an extra assumption. The impossibility of location is also shown.

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 A Uniqueness Result for a Simple Superlinear Eigenvalue Problem

2021 ◽  
Vol 31 (2) ◽  
Author(s):  
Michael Herrmann ◽  
Karsten Matthies
Keyword(s):  
Eigenvalue Problem ◽  
Numerical Simulations ◽  
Convolution Operator ◽  
Existence And Uniqueness ◽  
Uniqueness Result ◽  
Constitutive Laws ◽  
Parameter Family ◽  
Shape Constraint ◽  
Special Case ◽  
Nonlinear Eigenfunctions

AbstractWe study the eigenvalue problem for a superlinear convolution operator in the special case of bilinear constitutive laws and establish the existence and uniqueness of a one-parameter family of nonlinear eigenfunctions under a topological shape constraint. Our proof uses a nonlinear change of scalar parameters and applies Krein–Rutman arguments to a linear substitute problem. We also present numerical simulations and discuss the asymptotics of two limiting cases.

Sign in / Sign up

Export Citation Format

Share Document