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.

HYPERBOLIC RELAXATION OF THE VISCOUS CAHN–HILLIARD EQUATION IN 3-D

2005 ◽  
Vol 15 (02) ◽  
pp. 165-198 ◽  
Author(s):  
STEFANIA GATTI ◽  
MAURIZIO GRASSELLI ◽  
VITTORINO PATA ◽  
ALAIN MIRANVILLE
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Diffusion Flux ◽  
Relative Concentration ◽  
General Theorem ◽  
Viscosity Coefficient ◽  
Exponential Attractor ◽  
Exponential Attractors ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

We consider a modified version of the viscous Cahn–Hilliard equation governing the relative concentration u of one component of a binary system. This equation is characterized by the presence of the additional inertial term ωuttthat accounts for the relaxation of the diffusion flux. Here ω≥0 is an inertial parameter which is supposed to be dominated from above by the viscosity coefficient δ. Endowing the equation with suitable boundary conditions, we show that it generates a dissipative dynamical system acting on a certain phase-space depending on ω. This system is shown to possess a global attractor that is upper semicontinuous at ω = δ = 0. Then, we construct a family of exponential attractors εω,δ, which is a robust perturbation of an exponential attractor of the Cahn–Hilliard equation, namely the symmetric Hausdorff distance between εω,δand ε0, 0goes to 0 as (ω, δ) goes to (0, 0) in an explicitly controlled way. This is done by using a general theorem which requires the construction of another dynamical system, strictly related to the original one, but acting on a different phase-space depending on both ω and δ.

scholarly journals On global dynamics of 2D convective Cahn–Hilliard equation

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 ) .

scholarly journals Relaxation of the Cahn–Hilliard equation with singular single-well potential and degenerate mobility

2020 ◽  
Vol 32 (1) ◽  
pp. 89-112
Author(s):  
BENOÎT PERTHAME ◽  
ALEXANDRE POULAIN
Keyword(s):  
Hilliard Equation ◽  
Degenerate Mobility ◽  
Long Time ◽  
Single Well ◽  
Cahn Hilliard Equation ◽  
Long Time Asymptotics ◽  
Living Tissues ◽  
Entropy Structure ◽  
Second Order Equations ◽  
Relaxed System

The degenerate Cahn–Hilliard equation is a standard model to describe living tissues. It takes into account cell populations undergoing short-range attraction and long-range repulsion effects. In this framework, we consider the usual Cahn–Hilliard equation with a singular single-well potential and degenerate mobility. These degeneracy and singularity induce numerous difficulties, in particular for its numerical simulation. To overcome these issues, we propose a relaxation system formed of two second-order equations which can be solved with standard packages. This system is endowed with an energy and an entropy structure compatible with the limiting equation. Here, we study the theoretical properties of this system: global existence and convergence of the relaxed system to the degenerate Cahn–Hilliard equation. We also study the long-time asymptotics which interest relies on the numerous possible steady states with given mass.

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