ALP EDEN
Bogazici University, Mathematic, Faculty Member
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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... 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 ...
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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... 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.
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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... 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.
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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)... 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.
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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.... 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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Hydrodynamic turbulence - a 19th-century problem with a challenge for the 21st century exponents of bulk heat transfer in convective turbulence hierarchical structures and scalings in turbulence intermittency of passive scalars in... more
Hydrodynamic turbulence - a 19th-century problem with a challenge for the 21st century exponents of bulk heat transfer in convective turbulence hierarchical structures and scalings in turbulence intermittency of passive scalars in delta-correlated flow - introduction to recent work a minimal model for intermittency of passive scalars generalized scaling in turbulent flows about the interaction between vorticity and stretching in coherent structures pressure and intermittency in the inertial range of turbulence time-periodic statistical solutions of the Navier-Stokes equations local exact controllability for the 2-D Navier-Stokes equations nonexistence of global solutions to nonlinear wave equations applications of direct numerical simulation to complex turbulent flows computational aspects of three-dimensional vortex element methods - applications with vortex rings boundary layer turbulence modelling and vorticity dynamics - a kangaroo-process mixing model of boundary layer turbulence - towards a theory of turbulent shear flow?
Research Interests: Physics, Turbulence, Vorticity, Vortex, Intermittency, and 3 moreBoundary Layer, Springer, and Springer Ebooks
We will start with a typical example. Let X be a compact Hausdorff space, and let a : X → and ϕ : X → X both be continuous functions. Define T : C(X; ) → C(X; ) by (Tf)(x) = a(x)f(ϕ(x)), if a > 0 on X, then the operator T is linear and... more
We will start with a typical example. Let X be a compact Hausdorff space, and let a : X → and ϕ : X → X both be continuous functions. Define T : C(X; ) → C(X; ) by (Tf)(x) = a(x)f(ϕ(x)), if a > 0 on X, then the operator T is linear and positive. For any continuous, positive function f : ...
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... a cubic nonlinearity: they have similar scaling properties and the interaction between these two nonlinear terms determine the global behaviour of ... (3) and (4) are hyperbolic. In [1] a virial type identity was derived for the... more
... a cubic nonlinearity: they have similar scaling properties and the interaction between these two nonlinear terms determine the global behaviour of ... (3) and (4) are hyperbolic. In [1] a virial type identity was derived for the solutions of the initial value problem that live in weighted ...
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... Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering. doi:10.3934/cpaa.2009.8.1803 Full text: (333.0K) Alp Eden - Department of Mathematics, Bogaziçi University, Bebek 34342, Istanbul, Turkey (email). ...
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In the present study, we consider a generalized (2 + 1) DaveyStewartson (GDS) system consisting of a nonlinear Schrödinger (NLS) type equation for the complex amplitude of a short wave and two asymmetrically coupled linear wave equations... more
In the present study, we consider a generalized (2 + 1) DaveyStewartson (GDS) system consisting of a nonlinear Schrödinger (NLS) type equation for the complex amplitude of a short wave and two asymmetrically coupled linear wave equations for long waves propagating in an ...
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ABSTRACT We define the focusing and the defocusing cases for the purely elliptic generalized Davey-Stewartson system. These cases are mutually exclusive and exhaustive and therefore close the gap that was left in the previous studies. In... more
ABSTRACT We define the focusing and the defocusing cases for the purely elliptic generalized Davey-Stewartson system. These cases are mutually exclusive and exhaustive and therefore close the gap that was left in the previous studies. In the defocusing case, all solutions exist globally. In the focusing case, any initial data can be scaled to one with negative energy. The solution with the scaled initial data then blows up in finite time. We also show the existence of standing waves and the global existence and scattering of solutions with subminimal mass. Our results equally apply to the elliptic almost-cubic non-linear Schrödinger equation as described in [Commun. Pure Appl. Anal. 8, No. 6, 1803–1823 (2009; Zbl 1180.35480)].