Infinite-dimensional trajectory attractors of elliptic boundary-value problems in cylindrical domains

Mielke, Alexander; Zelik, S.

doi:10.1070/rm2002v057n04abeh000550

Cited by 20 publications

(13 citation statements)

References 35 publications

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“…The next result based on the logarithmic convexity for elliptic/parabolic equations, see [1,45] and references therein indicates that the log-Lipschitz continuity naturally appears under the study of equations (2.1).…”

Section: Proof 1) ⇒ 2) To Verify This We First Note That the Set Omentioning

confidence: 99%

Inertial manifolds and finite-dimensional reduction for dissipative PDEs

Zelik

2014

Proceedings of the Royal Society of Edinburgh: Section A Mathem

243

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Abstract. These notes are devoted to the problem of finite-dimensional reduction for parabolic PDEs. We give a detailed exposition of the classical theory of inertial manifolds as well as various attempts to generalize it based on the so-called Mané projection theorems. The recent counterexamples which show that the underlying dynamics may be in a sense infinite-dimensional if the spectral gap condition is violated as well as the discussion on the most important open problems are also included.

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Section: Proof 1) ⇒ 2) To Verify This We First Note That the Set Omentioning

confidence: 99%

Inertial manifolds and finite-dimensional reduction for dissipative PDEs

Zelik

2014

Proceedings of the Royal Society of Edinburgh: Section A Mathem

243

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“…For example, most of the results stated above remain valid in the general case of a nonscalar diffusion matrix a; see [43,44]. Moreover, partial analogues to them were obtained in [4,21,42] for dissipative wave equations, and in [8], for elliptic equations in cylindrical domains.…”

Section: A Smentioning

confidence: 94%

Multiparameter semigroups and attractors of reaction-diffusion equations in ${\mathbb R}^n$

Zelik

2004

Trans. Moscow Math. Soc.

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Abstract. The space-time dynamics generated by a system of reaction-diffusion equations in R n on its global attractor are studied in this paper. To describe these dynamics the extended (n + 1)-parameter semigroup generated by the solution operator of the system and the n-parameter group of spatial translations is introduced and their dynamic properties are studied. In particular, several new dynamic characteristics of the action of this semigroup on the attractor are constructed, generalizing the notions of fractal dimension and topological entropy, and relations between them are studied. Moreover, under certain natural conditions a description of the dynamics is obtained in terms of homeomorphic embeddings of multidimensional Bernoulli schemes with infinitely many symbols.

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“…We also note that embeddings of Bernoulli shifts with the infinite number of symbols (analogous to (0.17)) seem to be natural (and universal) for the spatio-temporal dynamics generated by PDEs in unbounded domains. In fact, the embeddings of the symbolic dynamics of this type into the temporal dynamics generated by RDEs of the form (0.1) are constructed in [39]; see also [32] for the analogous results in the case of elliptic problems in cylindrical domains.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, estimate (0.8) may be considered as a natural generalization of this principle to the case of unbounded domains; see also [17,32,40]). The rest of the paper is devoted to a more comprehensive study of the spatially homogeneous case of equation (0.1) ( = R n , g ≡ const).…”

Section: Introductionmentioning

confidence: 99%

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Attractors of reaction‐diffusion systems in unbounded domains and their spatial complexity

Zelik¹

2003

Comm Pure Appl Math

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The nonlinear reaction-diffusion system in an unbounded domain is studied. It is proven that, under some natural assumptions on the nonlinear term and on the diffusion matrix, this system possesses a global attractor A in the corresponding phase space. Since the dimension of the attractor happens to be infinite, we study its Kolmogorov's ε-entropy. Upper and lower bounds of this entropy are obtained.Moreover, we give a more detailed study of the attractor for the spatially homogeneous RDE in R n . In this case, a group of spatial shifts acts on the attractor. In order to study the spatial complexity of the attractor, we interpret this group as a dynamical system (with multidimensional "time" if n > 1) acting on a phase space A. It is proven that the dynamical system thus obtained is chaotic and has infinite topological entropy.In order to clarify the nature of this chaos, we suggest a new model dynamical system that generalizes the symbolic dynamics to the case of the infinite entropy and construct the homeomorphic (and even Lipschitz-continuous) embedding of this system into the spatial shifts on the attractor.Finally, we consider also the temporal evolution of the spatially chaotic structures in the attractor and prove that the spatial chaos is preserved under this evolution.

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Infinite-dimensional trajectory attractors of elliptic boundary-value problems in cylindrical domains

Cited by 20 publications

References 35 publications

Inertial manifolds and finite-dimensional reduction for dissipative PDEs

Inertial manifolds and finite-dimensional reduction for dissipative PDEs

Multiparameter semigroups and attractors of reaction-diffusion equations in ${\mathbb R}^n$

Attractors of reaction‐diffusion systems in unbounded domains and their spatial complexity

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