Exponential attractors for a singularly perturbed Cahn‐Hilliard system

Efendiev, Messoud; Miranville, Alain; Zelik, Sergey

doi:10.1002/mana.200310186

Cited by 91 publications

(116 citation statements)

References 37 publications

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“…The proof of this theorem is completely analogous to those of theorems 1.1 and 1.2 and we leave it to the reader (see also [11] and [13]). In this section, we extend the results of Section 1 to nonautonomous dynamical systems.…”

Section: Theorem 12mentioning

confidence: 88%

“…To this end, we need to control the distance between E k and S(n)B. We emphasize that, in contrast to [11] and [13], we do not "project" the sets E k onto S(k)B and, consequently, we do not have the embedding E k ⊂ S(k)B. Nevertheless, instead of this embedding, we now have the following estimate.…”

Section: ) For Every Points Hmentioning

confidence: 99%

“…Moreover, it was shown in [11] that, under natural assumptions, it is possible to construct a one-parametrical family of exponential attractors M ε , ε ∈ [0, 1], associated with the one-parametrical family of semigroups S ε satisfying (0.5) uniformly with respect to ε, such that…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems

Efendiev

Zelik

Miranville

2005

Proceedings of the Royal Society of Edinburgh: Section A Mathem

123

134

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Abstract. We suggest in this article a new explicit algorithm allowing to construct exponential attractors which are uniformly Hölder continuous with respect to the variation of the dynamical system in some natural large class. Moreover, we extend this construction to nonautonomous dynamical systems (dynamical processes) treating in that case the exponential attractor as a uniformly exponentially attracting finite-dimensional time-dependent set in the phase space. In particular, this result shows that, for a wide class of nonautonomous equations of mathematical physics, the limit dynamics remains finite-dimensional no matter how complicated the dependence of the external forces on time is. We illustrate the main results of this article on the model example of a nonautonomous reaction-diffusion system in a bounded domain.Introduction.

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Section: Theorem 12mentioning

confidence: 88%

Section: ) For Every Points Hmentioning

confidence: 99%

See 1 more Smart Citation

Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems

Efendiev

Zelik

Miranville

2005

Proceedings of the Royal Society of Edinburgh: Section A Mathem

123

134

View full text Add to dashboard Cite

show abstract

“…also [9]). In particular, the new strategy devised in [11] allows to construct a robust (under perturbations) family of exponential attractors (see also [12][13][14]24,34] and references therein). This family is characterized by an explicit estimate on the symmetric distance between exponential attractors of the unperturbed and perturbed problems, and the continuity with respect to the perturbations does not involve time shifts as in the previous results.…”

Section: Introductionmentioning

confidence: 99%

Singularly perturbed 1D Cahn–Hilliard equation revisited

Bonfoh

Grasselli

Miranville

2010

Nonlinear Differ. Equ. Appl.

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Abstract. We consider a singular perturbation of the one-dimensional Cahn-Hilliard equation subject to periodic boundary conditions. We construct a family of exponential attractors {M }, ≥ 0 being the perturbation parameter, such that the map → M is Hölder continuous. Besides, the continuity at = 0 is obtained with respect to a metric independent of . Continuity properties of global attractors and inertial manifolds are also examined. Mathematics Subject Classification (2000). 35B25, 35B40, 37L25, 82C26.

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“…Such equations, where f is the derivative of some double-well potential F , are generalizations of the Cahn-Hilliard equation which is very important in material science and models the qualitative behaviours of two phase systems (see [7], [8], [9], [11], [12], [13] and [14]). The layout of this paper is as follows.…”

mentioning

confidence: 99%

Finite-dimensional attractor for the viscous Cahn-Hilliard equation in an unbounded domain

Bonfoh

2006

Quart. Appl. Math.

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Abstract.We consider the viscous Cahn-Hilliard equation in an infinite domain. Due to the noncompactness of operators, we use weighted Sobolev spaces to prove that the semigroup generated by this equation has the global attractor which has finite Hausdorff dimension.1. Introduction. Many equations arising from mechanics and physics possess a global attractor, which is a compact invariant set which uniformly attracts the trajectories as time goes to infinity, and thus appears as a suitable object for the study of the asymptotic behaviour of the system. An important issue is then to study the dimension, in the sense of the Hausdorff or fractal dimension, of the global attractor. A finite bound of the dimension of the attractor means that the system has an asymptotic behaviour determined by a finite number of degrees of freedom, indeed a remarkable improvement compared to the a priori infinite-dimensional dynamics (see [5] and [15]).For the equations on bounded domains, the known constructions of global attractors make use of some compactness properties in an essential manner, and more specifically of the compact embedding of H m 1 into H m 2 , when m 1 > m 2 . Such properties are no longer valid for equations on unbounded domains, and it is thus more difficult to develop a general theory of existence of the global attractor in this case. A possibility then consists of working in weighted Sobolev spaces (see In this article, we study, on an unbounded domain, the existence of global attractors and their Hausdorff dimensions for the viscous Cahn-Hilliard equation of the form

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Exponential attractors for a singularly perturbed Cahn‐Hilliard system

Cited by 91 publications

References 37 publications

Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems

Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems

Singularly perturbed 1D Cahn–Hilliard equation revisited

Finite-dimensional attractor for the viscous Cahn-Hilliard equation in an unbounded domain

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