Global attractor of the Cahn–Hilliard equation in spaces (original) (raw)
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Global Attractor for the Weak Solutions of a Class of Viscous Cahn-Hilliard Equations
Series on Advances in Mathematics for Applied Sciences, 2006
We address the long-time behaviour of a class of viscous Cahn-Hilliard equations, modelling phase separation in mixtures and alloys. Specifically, we prove the existence of (a suitable notion of) the global attractor for the weak solutions of the so-called generalized viscous Cahn-Hilliard equation. November 4, 2005 17:43 Dissipative Phase Transitions P. Colli, N. Kenmochi, J. Sprekels Attractor for generalized viscous Cahn-Hilliard equations 5
Exponential attractors for the Cahn–Hilliard equation with dynamic boundary conditions
We consider in this article the Cahn-Hilliard equation endowed with dynamic boundary conditions. By interpreting these boundary conditions as a parabolic equation on the boundary and by considering a regularized problem, we obtain, by the Leray-Schauder principle, the existence and uniqueness of solutions. We then construct a robust family of exponential attractors.
Nonlinear Analysis: Real World Applications, 2009
We consider a model of non-isothermal phase separation taking place in a confined container. The order parameter φ is governed by a viscous or non-viscous Cahn-Hilliard type equation which is coupled with a heat equation for the temperature θ. The former is subject to a non-linear dynamic boundary condition recently proposed by physicists to account for interactions with the walls, while the latter is endowed with a standard (Dirichlet, Neumann or Robin) boundary condition. We indicate by α the viscosity coefficient, by ε a (small) relaxation parameter multiplying ∂ t θ in the heat equation and by δ a small latent heat coefficient (satisfying δ ≤ λα, λ > 0) multiplying ∆θ in the Cahn-Hilliard equation and ∂ t φ in the heat equation. We analyze the asymptotic behavior of the solutions within the theory of infinite-dimensional dynamical systems. We first prove that the model generates a strongly continuous semigroup on a suitable phase space Y α K (depending on the choice of the boundary conditions) which possesses the global attractor A ε,δ,α . Our main results allow us to show that a proper lifting A 0,0,α , α > 0, of the global attractor of the wellknown viscous Cahn-Hilliard equation (that is, the system corresponding to (ε, δ) = (0, 0)) is upper semicontinuous at (0, 0) with respect to the family A ε,δ,α ε,δ,α>0 . We also establish that the global attractor A 0,0,0 of the non-viscous Cahn-Hilliard equation (corresponding to (ε, α) = (0, 0)) is upper semicontinuous at (0, 0) with respect to the same family of global attractors. Finally, the existence of exponential attractors M ε,δ,α is also obtained in the cases ε = 0, δ = 0, α = 0, (0, δ, α) , δ = 0, α = 0 and (ε, δ, α) = (0, 0, α) , α ≥ 0, respectively. This allows us to infer that, for each (ε, δ, α) ∈ 0, ε 0 × 0, δ 0 × 0, α 0 , A ε,δ,α has finite fractal dimension and this dimension is bounded, uniformly with respect to ε, δ and α.
Trajectory and smooth attractors for Cahn–Hilliard equations with inertial term
Nonlinearity, 2010
This paper is devoted to a modification of the classical Cahn-Hilliard equation proposed by some physicists. This modification is obtained by adding the second time derivative of the order parameter multiplied by an inertial coefficient ε > 0, which is usually small in comparison with the other physical constants. The main feature of this equation is the fact that even a globally bounded nonlinearity is 'supercritical' in the case of two and three space dimensions. Thus, the standard methods used for studying semilinear hyperbolic equations are not very effective in the present case. Nevertheless, we have recently proven the global existence and dissipativity of strong solutions in the 2D case (with a cubic controlled growth nonlinearity) and for the 3D case with small ε and arbitrary growth rate of the nonlinearity (see Grasselli et al 2009 J. Evol. Eqns 9 371-404, Grasselli et al 2009 Commun. Partial Diff. Eqns 34 137-70)). The present contribution studies the long-time behaviour of rather weak (energy) solutions of that equation and it is a natural complement of the results of our previous papers (Grasselli et al 2009 J. Evol. Eqns 9 371-404, Grasselli et al 2009 Commun. Partial Diff. Eqns 34 137-70).
Exponential attractors for a singularly perturbed Cahn-Hilliard system
Mathematische Nachrichten, 2004
Our aim in this article is to give a construction of exponential attractors that are continuous under perturbations of the underlying semigroup. We note that the continuity is obtained without time shifts as it was the case in previous studies. Moreover, we obtain an explicit estimate for the symmetric distance between the perturbed and unperturbed exponential attractors in terms of the perturbation parameter. As an application, we prove the continuity of exponential attractors for a viscous Cahn-Hilliard system to an exponential attractor for the limit Cahn-Hilliard system.
Attractors for non-compact semigroups via energy equations
Nonlinearity, 1998
The energy equation approach used to prove the existence of the global attractor by establishing the so-called asymptotic compactness property of the semigroup is considered, and a general formulation that can handle a number of weakly damped hyperbolic equations and parabolic equations on either bounded or unbounded spatial domains is presented. As examples, three specific and physically relevant problems are considered, namely the flows of a second-grade fluid, the flows of a newtonian fluid in an infinite channel past an obstacle, and a weakly damped, forced Korteweg-de Vries equation on the whole line. Contents 1. Introduction 2. Asymptotic Compactness 3. Abstract Energy Equations 4. Applications 4.1. Fluids of Second Grade 4.2. Flows Past an Obstacle 4.3. Weakly Damped, Forced Korteweg-de Vries Equation References
Discrete and Continuous Dynamical Systems - Series S, 2009
We consider a model of non-isothermal phase separation taking place in a confined container. The order parameter φ is governed by a viscous or non-viscous Cahn-Hilliard type equation which is coupled with a heat equation for the temperature θ. The former is subject to a nonlinear dynamic boundary condition recently proposed by physicists to account for interactions with the walls, while the latter is endowed with a standard (Dirichlet, Neumann or Robin) boundary condition. We indicate by α the viscosity coefficient, by ε a (small) relaxation parameter multiplying ∂tθ in the heat equation and by δ a small latent heat coefficient (satisfying δ ≤ λα, δ ≤ λε, λ, λ > 0) multiplying ∆θ in the Cahn-Hilliard equation and ∂tφ in the heat equation. Then, we construct a family of exponential attractors M ε,δ,α which is a robust perturbation of an exponential attractor M 0,0,α of the (isothermal) viscous (α > 0) Cahn-Hilliard equation, namely, the symmetric Hausdorff distance between M ε,δ,α and M 0,0,α goes to 0, for each fixed value of α > 0, as (ε, δ) goes to (0, 0), in an explicitly controlled way. Moreover, the robustness of this family of exponential attractors M ε,δ,α with respect to (δ, α) → (0, 0) , for each fixed value of ε > 0, is also obtained. Finally, assuming that the nonlinearities are real analytic, with no growth restrictions, the convergence of solutions to single equilibria, as time goes to infinity, is also proved.
On the hyperbolic relaxation of the one-dimensional Cahn–Hilliard equation
Journal of Mathematical Analysis and Applications, 2005
We consider the one-dimensional Cahn-Hilliard equation with an inertial term εu tt , for ε 0. This equation, endowed with proper boundary conditions, generates a strongly continuous semigroup S ε (t) which acts on a suitable phase-space and possesses a global attractor. Our main result is the construction of a robust family of exponential attractors {M ε }, whose common basins of attraction are the whole phase-space. 2005 Elsevier Inc. All rights reserved.
2010
We consider a quasi-linear parabolic equation with nonlinear dynamic boundary conditions occurring as generalizations of semilinear reaction-diffusion equations with dynamic boundary conditions and various other phase-field models, such as the isothermal Allen-Cahn equation with dynamic boundary conditions. We thus formulate a class of initial and boundary value problems whose global existence and uniqueness is proven by means of an appropriate Faedo-Galerkin approximation scheme developed for problems with dynamic boundary conditions. We analyze the asymptotic behavior of the solutions within the theory of infinite-dimensional dynamical systems. In particular, we demonstrate the existence of the global attractor.