Finite Dimensional Dynamics on Attractors Alp Eden Bogazici University, Mathematics Department, B... more Finite Dimensional Dynamics on Attractors Alp Eden Bogazici University, Mathematics Department, Bebek, Istanbul, Turkey 15 May 2001 Abstract We construct a finite dimensional generalized dynamical system on the finite dimensional attractors of damped hyperbolic ...
The paper deals with the Cauchy problem for semilinear wave equations in separable Hilbert spaces... more The paper deals with the Cauchy problem for semilinear wave equations in separable Hilbert spaces. Using a method inspired from O. A. Ladyzhenskaya [Usp. Mat. Nauk 42, No. 6, 25-60 (1987; Zbl 0687.35072)], the authors give sufficient conditions for that the corresponding continuous semigroup (assumed to exist) has the so-called “discrete squeezing property”. No examples and applications are included.
ABSTRACT An improvement in the original constructions of exponential attractors is indicated. Nam... more ABSTRACT An improvement in the original constructions of exponential attractors is indicated. Namely, when the solution semigroup is -contractive and satisfies the discrete squeezing property, then even when the invariant set on which the semigroup acts is not compact, the original constructions carries through. We obtain the same conclusion for the construction with Lyapunov dimension for -constructions.
ABSTRACT This paper is a study of global attractors of abstract semilinear parabolic equations an... more ABSTRACT This paper is a study of global attractors of abstract semilinear parabolic equations and their embeddings in finite-dimensional manifolds. As is well known, a sufficient condition for the existence of smooth (at least -smooth) finite-dimensional inertial manifolds containing a global attractor is the so-called spectral gap condition for the corresponding linear operator. There are also a number of examples showing that if there is no gap in the spectrum, then a -smooth inertial manifold may not exist. On the other hand, since an attractor usually has finite fractal dimension, by Mañé's theorem it projects bijectively and Hölder-homeomorphically into a finite-dimensional generic plane if its dimension is large enough. It is shown here that if there are no gaps in the spectrum, then there exist attractors that cannot be embedded in any Lipschitz or even log-Lipschitz finite-dimensional manifold. Thus, if there are no gaps in the spectrum, then in the general case the inverse Mañé projection of the attractor cannot be expected to be Lipschitz or log-Lipschitz. Furthermore, examples of attractors with finite Hausdorff and infinite fractal dimension are constructed in the class of non-linearities of finite smoothness.
Journal of Physics A: Mathematical and Theoretical, 2009
Page 1. A note on the global existence of small amplitude solutions to a generalized DaveyStewar... more Page 1. A note on the global existence of small amplitude solutions to a generalized DaveyStewartson system This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2009 J. Phys. A: Math. Theor. 42 245208 ...
Finite Dimensional Dynamics on Attractors Alp Eden Bogazici University, Mathematics Department, B... more Finite Dimensional Dynamics on Attractors Alp Eden Bogazici University, Mathematics Department, Bebek, Istanbul, Turkey 15 May 2001 Abstract We construct a finite dimensional generalized dynamical system on the finite dimensional attractors of damped hyperbolic ...
The paper deals with the Cauchy problem for semilinear wave equations in separable Hilbert spaces... more The paper deals with the Cauchy problem for semilinear wave equations in separable Hilbert spaces. Using a method inspired from O. A. Ladyzhenskaya [Usp. Mat. Nauk 42, No. 6, 25-60 (1987; Zbl 0687.35072)], the authors give sufficient conditions for that the corresponding continuous semigroup (assumed to exist) has the so-called “discrete squeezing property”. No examples and applications are included.
ABSTRACT An improvement in the original constructions of exponential attractors is indicated. Nam... more ABSTRACT An improvement in the original constructions of exponential attractors is indicated. Namely, when the solution semigroup is -contractive and satisfies the discrete squeezing property, then even when the invariant set on which the semigroup acts is not compact, the original constructions carries through. We obtain the same conclusion for the construction with Lyapunov dimension for -constructions.
ABSTRACT This paper is a study of global attractors of abstract semilinear parabolic equations an... more ABSTRACT This paper is a study of global attractors of abstract semilinear parabolic equations and their embeddings in finite-dimensional manifolds. As is well known, a sufficient condition for the existence of smooth (at least -smooth) finite-dimensional inertial manifolds containing a global attractor is the so-called spectral gap condition for the corresponding linear operator. There are also a number of examples showing that if there is no gap in the spectrum, then a -smooth inertial manifold may not exist. On the other hand, since an attractor usually has finite fractal dimension, by Mañé's theorem it projects bijectively and Hölder-homeomorphically into a finite-dimensional generic plane if its dimension is large enough. It is shown here that if there are no gaps in the spectrum, then there exist attractors that cannot be embedded in any Lipschitz or even log-Lipschitz finite-dimensional manifold. Thus, if there are no gaps in the spectrum, then in the general case the inverse Mañé projection of the attractor cannot be expected to be Lipschitz or log-Lipschitz. Furthermore, examples of attractors with finite Hausdorff and infinite fractal dimension are constructed in the class of non-linearities of finite smoothness.
Journal of Physics A: Mathematical and Theoretical, 2009
Page 1. A note on the global existence of small amplitude solutions to a generalized DaveyStewar... more Page 1. A note on the global existence of small amplitude solutions to a generalized DaveyStewartson system This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2009 J. Phys. A: Math. Theor. 42 245208 ...
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