Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains

Abergel, Frédéric

doi:10.1016/0022-0396(90)90070-6

Cited by 93 publications

(61 citation statements)

References 5 publications

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“…Furthermore, the norm ∆ξ is equivalent on D(A) to the canonical norm ξ H 2 (Ω) (see [1] and [2]). If we denote q −1 = ∇N (q) , then there exists c 1 , c 2 > 0 such that …”

Section: Setting Of the Problemmentioning

confidence: 99%

“…Lq is a norm which is equivalent on D(L) to the norm ∆q , and therefore, L −1 ∆q is a norm which is equivalent on L 2 (Ω) to the usual L 2 (Ω) norm (see [1] and [2]). Therefore, constants in inequalities deduced from the respective equivalence of these norms restricted to Ω n can be chosen to depend on M only, but not on n. From (3.3), we have…”

Section: Setting Of the Problemmentioning

confidence: 99%

“…Such properties are no longer valid for equations on unbounded domains, and it is thus more difficult to develop a general theory of existence of the global attractor in this case.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 [1], [3], [4], [6], and [10]). …”

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confidence: 99%

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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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“…Furthermore, the norm ∆ξ is equivalent on D(A) to the canonical norm ξ H 2 (Ω) (see [1] and [2]). If we denote q −1 = ∇N (q) , then there exists c 1 , c 2 > 0 such that …”

Section: Setting Of the Problemmentioning

confidence: 99%

Section: Setting Of the Problemmentioning

confidence: 99%

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

Bonfoh

2006

Quart. Appl. Math.

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“…In [6], where they some interesting results, as 0   . In [7,8] Babin, Vishik and Abergel consider maximal attractors of semigroups corresponding to evolution differential equations, existence and finite dimensionality of global attractor for evolution equations on unbounded domains. In [9,10] A. Pazy consider Semigroups of linear operator and application to partial differential equation.…”

Section: Introductionmentioning

confidence: 99%

“…For the mathematical setting of this problem we consider a Hilbert space H (see [8]) which is a close sub- …”

Section: Introductionmentioning

confidence: 99%

On the Incompressible Navier-Stokes Equations with Damping

Zhao¹,

Zheng²

2013

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We consider dynamics system with damping, which are obtained by some transformations from the system of incompressible Navier-Stokes equations. These have similar properties to original Navier-Stokes equations the scaling invariance. Due to the presence of the damping term, conclusions are different with proving the origin of the incompressible Navier-Stokes equations and get some new conclusions. For one form of dynamics system with damping we prove the existence of solution, and get the existence of the attractors. Moreover, we discuss with limit-behavior the deformations of the Navier-Stokes equation.

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The Attractor for a Nonlinear Reaction-Diffusion System in the Unbounded Domain and Kolmogorove's ɛ-Entropy

Zelik

2001

Math. Nachr.

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Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains

Cited by 93 publications

References 5 publications

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

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

On the Incompressible Navier-Stokes Equations with Damping

The Attractor for a Nonlinear Reaction-Diffusion System in the Unbounded Domain and Kolmogorove's ɛ-Entropy

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