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 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 Exponential Attractors for the 3D Fractional-order Bardina Turbulence Model With Memory and Horizontal Filtering

2021 ◽  
Author(s):  
Michele Annese ◽  
Luca Bisconti ◽  
Davide Catania
Keyword(s):  
Phase Space ◽  
Fractional Order ◽  
Turbulence Model ◽  
Memory Term ◽  
Exponential Attractor ◽  
Exponential Attractors ◽  
Regularity Properties ◽  
Hereditary Effects ◽  
With Memory ◽  
Simplified Bardina

AbstractWe consider the 3D simplified Bardina turbulence model with horizontal filtering, fractional dissipation, and the presence of a memory term incorporating hereditary effects. We analyze the regularity properties and the dissipative nature of the considered system and, in our main result, we show the existence of a global exponential attractor in a suitable phase space.

Pullback and Uniform Exponential Attractors for Nonautonomous Boussinesq Lattice System

2015 ◽  
Vol 25 (08) ◽  
pp. 1550100 ◽  
Author(s):  
Min Zhao ◽  
Shengfan Zhou
Keyword(s):  
Dynamical System ◽  
Boussinesq Equations ◽  
Lattice System ◽  
Time Dependent ◽  
Pullback Attractor ◽  
Exponential Attractor ◽  
Uniform Attractor ◽  
Exponential Attractors ◽  
Lattice Dynamical System ◽  
External Forces

We first prove the existence of a pullback attractor and a pullback exponential attractor for a nonautonomous lattice dynamical system of nonlinear Boussinesq equations affected by time-dependent coupled coefficients and forces. Then, we prove the existence of a uniform attractor and a uniform exponential attractor for the system driven by quasi-periodic external forces.

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 The Existence of WeakD-Pullback Exponential Attractor for Nonautonomous Dynamical System

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Yongjun Li ◽  
Xiaona Wei ◽  
Yanhong Zhang
Keyword(s):  
Dynamical System ◽  
Diffusion Equation ◽  
External Force ◽  
Exponential Growth ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Exponential Attractor ◽  
Exponential Attractors ◽  
Nonautonomous Dynamical System

First, for a processU(t,τ)∣t≥τ, we introduce a new concept, called the weakD-pullback exponential attractor, which is a family of setsM(t)∣t≤T, for anyT∈R, satisfying the following: (i)M(t)is compact, (ii)M(t)is positively invariant, that is,U(t,τ)M(τ)⊂M(t), and (iii) there existk,l>0such thatdist(U(t,τ)B(τ),M(t))≤ke-(t-τ); that is,M(t)pullback exponential attractsB(τ). Then we give a method to obtain the existence of weakD-pullback exponential attractors for a process. As an application, we obtain the existence of weakD-pullback exponential attractor for reaction diffusion equation inH01with exponential growth of the external force.

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.

Random uniform exponential attractors for non-autonomous stochastic lattice systems and FitzHugh–Nagumo lattice systems with quasi-periodic forces and multiplicative noise

2020 ◽  
Vol 20 (05) ◽  
pp. 2050036
Author(s):  
Zongfei Han ◽  
Shengfan Zhou
Keyword(s):  
Dynamical System ◽  
Product Space ◽  
Multiplicative Noise ◽  
Weighted Spaces ◽  
Random Dynamical System ◽  
Exponential Attractor ◽  
Lattice Systems ◽  
Exponential Attractors ◽  
Existence Criterion ◽  
Infinite Sequences

We first give an existence criterion for a random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system defined on the product space of [Formula: see text]-weighted spaces of infinite sequences. Then, based on this criterion, we prove the existence of random uniform exponential attractors for stochastic lattice systems and stochastic FitzHugh–Nagumo lattice systems that are both with quasi-periodic forces.

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