Displaying similar documents to “On the viscous Allen-Cahn and Cahn-Hilliard systems with Willmore regularization”

Some models of Cahn-Hilliard equations in nonisotropic media

Alain Miranville (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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We derive in this article some models of Cahn-Hilliard equations in nonisotropic media. These models, based on constitutive equations introduced by Gurtin in [19], take the work of internal microforces and also the deformations of the material into account. We then study the existence and uniqueness of solutions and obtain the existence of finite dimensional attractors.

Global Attractor for the Convective Cahn-Hilliard Equation

Xiaopeng Zhao, Changchun Liu (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

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This paper is concerned with the convective Cahn-Hilliard equation. We use a classical theorem on existence of a global attractor to derive that the convective Cahn-Hilliard equation possesses a global attractor on some subset of H².

Remarks on the Fractal Dimension of Bi-Space Global and Exponential Attractors

Jan W. Cholewa, Radoslaw Czaja, Gianluca Mola (2008)

Bollettino dell'Unione Matematica Italiana

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Bi-space global and exponential attractors for the time continuous dynamical systems are considered and the bounds on their fractal dimension are discussed in the context of the smoothing properties of the system between appropriately chosen function spaces. The case when the system exhibits merely some partial smoothing properties is also considered and applications to the sample problems are given.

Attractors for general operators

Alain Miranville (2003)

Applications of Mathematics

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In this article we introduce the notion of a minimal attractor for families of operators that do not necessarily form semigroups. We then obtain some results on the existence of the minimal attractor. We also consider the nonautonomous case. As an application, we obtain the existence of the minimal attractor for models of Cahn-Hilliard equations in deformable elastic continua.

Stabilization of the Kawahara equation with localized damping

Carlos F. Vasconcellos, Patricia N. da Silva (2011)

ESAIM: Control, Optimisation and Calculus of Variations

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We study the stabilization of global solutions of the Kawahara (K) equation in a bounded interval, under the effect of a localized damping mechanism. The Kawahara equation is a model for small amplitude long waves. Using multiplier techniques and compactness arguments we prove the exponential decay of the solutions of the (K) model. The proof requires of a unique continuation theorem and the smoothing effect of the (K) equation on the real line, which are proved in this work. ...

Stabilization of the Kawahara equation with localized damping

Carlos F. Vasconcellos, Patricia N. da Silva (2011)

ESAIM: Control, Optimisation and Calculus of Variations

Similarity:

We study the stabilization of global solutions of the Kawahara (K) equation in a bounded interval, under the effect of a localized damping mechanism. The Kawahara equation is a model for small amplitude long waves. Using multiplier techniques and compactness arguments we prove the exponential decay of the solutions of the (K) model. The proof requires of a unique continuation theorem and the smoothing effect of the (K) equation on the real line, which are proved in this work. ...

The existence of an exponential attractor in magneto-micropolar fluid flow via the ℓ-trajectories method

Piotr Orliński (2013)

Colloquium Mathematicae

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We consider the magneto-micropolar fluid flow in a bounded domain Ω ⊂ ℝ². The flow is modelled by a system of PDEs, a generalisation of the two-dimensional Navier-Stokes equations. Using the Galerkin method we prove the existence and uniqueness of weak solutions and then using the ℓ-trajectories method we prove the existence of the exponential attractor in the dynamical system associated with the model.

Asymptotic behavior of a sixth-order Cahn-Hilliard system

Alain Miranville (2014)

Open Mathematics

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Our aim in this paper is to study the asymptotic behavior, in terms of finite-dimensional attractors, of a sixth-order Cahn-Hilliard system. This system is based on a modification of the Ginzburg-Landau free energy proposed in [Torabi S., Lowengrub J., Voigt A., Wise S., A new phase-field model for strongly anisotropic systems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2009, 465(2105), 1337–1359], assuming isotropy.

On the dynamics of equations with infinite delay

Dalibor Pražák (2006)

Open Mathematics

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We consider a system of ordinary differential equations with infinite delay. We study large time dynamics in the phase space of functions with an exponentially decaying weight. The existence of an exponential attractor is proved under the abstract assumption that the right-hand side is Lipschitz continuous. The dimension of the attractor is explicitly estimated.

On the definition of strange nonchaotic attractor

Lluís Alsedà, Sara Costa (2009)

Fundamenta Mathematicae

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The aim of this paper is twofold. On the one hand, we want to discuss some methodological issues related to the notion of strange nonchaotic attractor. On the other hand, we want to formulate a precise definition of this kind of attractor, which is "observable" in the physical sense and, in the two-dimensional setting, includes the well known models proposed by Grebogi et al. and by Keller, and a wide range of other examples proposed in the literature. Furthermore, we analytically prove...