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

Zelik, S.

doi:10.1090/s0077-1554-04-00145-1

Cited by 12 publications

(14 citation statements)

References 49 publications

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“…Since the spaces W l,q ( y ) are uniformly convex, these family are uniquely defined and, moreover, continuous with respect to y ∈ ‫.ޒ‬ Let us also define the function v(x) as follows Integrating this relation over s ∈ ‫ޒ‬ and using (2.12), we finally obtain 27) We are now ready to finish the proof of the proposition. Indeed, due to (2.23)-(2.25), we have…”

Section: Functional Spacesmentioning

confidence: 94%

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Spatially Nondecaying Solutions of the 2d Navier-Stokes Equation in a Strip

Zelik¹

2007

Glasgow Math. J.

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Section: Functional Spacesmentioning

confidence: 94%

“…This inequality is crucial for obtaining the regularity estimates in weighted spaces (see [9][10][27][28][29][30] and Section 3 below).…”

Section: Functional Spacesmentioning

confidence: 99%

Spatially Nondecaying Solutions of the 2d Navier-Stokes Equation in a Strip

Zelik¹

2007

Glasgow Math. J.

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“…The proof of this theorem is a straightforward application of the essentially unstable manifold theorem for maps stated in [43] and, by this reason, is omitted, see also [21,29,31,40] for applications of that theorem for various equations in unbounded domains.…”

Section: Corollary 54mentioning

confidence: 99%

Infinite‐energy solutions for the Cahn–Hilliard equation in cylindrical domains

Eden

Kalantarov

Zelik

2013

Math Methods in App Sciences

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Communicated by P. ColliWe give a detailed study of the infinite-energy solutions of the Cahn-Hilliard equation in the 3D cylindrical domains in uniformly local phase space. In particular, we establish the well-posedness and dissipativity for the case of regular potentials of arbitrary polynomial growth as well as for the case of sufficiently strong singular potentials. For these cases, we prove the further regularity of solutions and the existence of a global attractor. For the cases where we have failed to prove the uniqueness (e.g., for the logarithmic potentials), we establish the existence of the trajectory attractor and study its properties.

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“…We know from the general theory of dissipative systems in unbounded domains (see e.g. [13,29,42,43,44]) that under some reasonable dissipativity assumptions this entropy is finite for systems of evolutionary PDE's, therefore the Sinai-Bunimovich model carries "enough complexity" to be able to capture certain basic features of spatio-temporal chaos in systems of various nature. In particular, it is well established by now (see e.g.…”

mentioning

confidence: 99%

“…in [29,43]). As we mentioned (see [13,44]), the space-time entropy of the attractor of the Ginzburg-Landau equation is finite:…”

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confidence: 99%

Analytical proof of space-time chaos in Ginzburg-Landau equations

Turaev¹,

Zelik²

2010

Discrete &Amp; Continuous Dynamical Systems - A

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We prove that the attractor of the 1D quintic complex Ginzburg-Landau equation with a broken phase symmetry has strictly positive space-time entropy for an open set of parameter values. The result is obtained by studying chaotic oscillations in grids of weakly interacting solitons in a class of Ginzburg-Landau type equations. We provide an analytic proof for the existence of twosoliton configurations with chaotic temporal behavior, and construct solutions which are closed to a grid of such chaotic soliton pairs, with every pair in the grid well spatially separated from the neighboring ones for all time. The temporal evolution of the well-separated multi-soliton structures is described by a weakly coupled lattice dynamical system (LDS) for the coordinates and phases of the solitons. We develop a version of normal hyperbolicity theory for the weakly coupled LDS's with continuous time and establish for them the existence of space-time chaotic patterns similar to the Sinai-Bunimovich chaos in discrete-time LDS's. While the LDS part of the theory may be of independent interest, the main difficulty addressed in the paper concerns with lifting the space-time chaotic solutions of the LDS back to the initial PDE. The equations we consider here are space-time autonomous, i.e. we impose no spatial or temporal modulation which could prevent the individual solitons in the grid from drifting towards each other and destroying the well-separated grid structure in a finite time. We however manage to show that the set of space-time chaotic solutions for which the random soliton drift is arrested is large enough, so the corresponding space-time entropy is strictly positive.2000 Mathematics Subject Classification. 35Q30, 37L30. Key words and phrases. extended systems, attractors of PDE's in unbounded domains, multipulse solutions, normal hyperbolicity, center-manifold reduction, soliton interaction, lattice dynamical systems.We are grateful to A.Mielke and A.Vladimirov for useful discussions, and to WIAS (Berlin) and BenGurion University for the hospitality. 1713 1714 DIMITRY TURAEV AND SERGEY ZELIKAs a tool for finding the spatially-localized, temporally-chaotic solutions one may try, as we do it here, to look for special types of both spatially and temporally localized solutions. Thus, like Shilnikov homoclinic loop and Lorenz butterfly serve as a criterion for chaos formation in systems of ODE's [35,36,37,39], the existence of the Shilnikov homoclinic loop in the dynamical system generated by the PDE on the space of spatially localized solutions implies the space-time chaos in the extended system that corresponds to uniformly bounded solutions of the same PDE.We do not prove this principle in full generality here. Instead, we decided to show how it works for a class of Ginzburg-Landau equations with a broken phase symmetry. The main motivation for such approach is that despite a huge amount of numerical and experimental data on different types of space-time irregular behavior in various systems, there are very few rigorous mathem...

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

Cited by 12 publications

References 49 publications

Spatially Nondecaying Solutions of the 2d Navier-Stokes Equation in a Strip

Spatially Nondecaying Solutions of the 2d Navier-Stokes Equation in a Strip

Infinite‐energy solutions for the Cahn–Hilliard equation in cylindrical domains

Analytical proof of space-time chaos in Ginzburg-Landau equations

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