Journal of Mathematical Analysis and Applications, 1992
We consider the initial value problem for a class of second order evolution equations that includ... more We consider the initial value problem for a class of second order evolution equations that includes, among others, the 3D sine-Gordon equation with damping and the 3D Klein-Gordon type equations with damping. We show the existence of a set with finite fractal dimension that contains the global attractor and attracts all smooth solutions at an exponential rate.
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.
In the present study, we consider a generalized Davey-Stewartson (GDS) system consisting of a non... more In the present study, we consider a generalized Davey-Stewartson (GDS) system consisting of a nonlinear Schrödinger (NLS) type equation and two asymmetrically coupled linear wave equations. We obtain integral representation of solutions to the coupled linear wave equations. As applications, we present some localized solutions to the GDS system for a special choice of parameters and find some estimates of the solutions.
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.
This article appeared in a journal published by Elsevier. The attached copy is furnished to the a... more This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are
Mathematical Methods in the Applied Sciences, 2013
We give a detailed study of the infinite-energy solutions of the Cahn-Hilliard equation in the 3D... more We 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.
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 ...
We study initial boundary value problems for the convective Cahn-Hilliard equation ∂ t u + ∂ 4 x ... more We study initial boundary value problems for the convective Cahn-Hilliard equation ∂ t u + ∂ 4 x u + u∂ x u + ∂ 2 x (|u| p u) = 0. It is well-known that without the convective term, the solutions of this equation may blow up in finite time for any p > 0. In contrast to that, we show that the presence of the convective term u∂ x u in the Cahn-Hilliard equation prevents blow up at least for 0 < p < 4 9. We also show that the blowing up solutions still exist if p is large enough (p ≥ 2). The related equations like Kolmogorov-Sivashinsky-Spiegel equation, sixth order convective Cahn-Hilliard equation, are also considered.
Journal of Mathematical Analysis and Applications, 2005
We find conditions on data guaranteeing global nonexistence of solutions to an inverse source pro... more We find conditions on data guaranteeing global nonexistence of solutions to an inverse source problem for a class of nonlinear parabolic equations. We also establish a stability result on a bounded domain for a problem with the opposite sign on the power type nonlinearity.
Journal of Mathematical Analysis and Applications, 1991
The Lieb-Thirring inequalities give a sharp upper bound for the LP-norm of a function which is th... more The Lieb-Thirring inequalities give a sharp upper bound for the LP-norm of a function which is the pointwise sum of the squares of a finite orthonormal sequence of functions that are elements of a suitable Sobolev space [LT]. Originally proven for the functions defined on the whole n-dimensional Euclidean space, they were later extended to bounded domains and to suborthogonal sequences of functions [GMT]. Here, we present a simple proof of these inequalities for bounded intervals in one space dimension utilizing simple Sobolev inequalities and standard results from Hilbert space theory. 0 1991 Academic PI~SS, hc.
Journal of Dynamics and Differential Equations, 1994
In this paper we present a new construction of exponential attractors based on the control of Lya... more In this paper we present a new construction of exponential attractors based on the control of Lyapunov exponents over a compact, invariant set. The fractal dimension estimate of the exponential attractor thus obtained is of the same order as the one for global attractors estimated through Lyapunov exponents. We discuss various applications to Navier-Stokes systems.
Journal of Dynamics and Differential Equations, 1991
Various properties of Local and Global Lyapunov exponents are related by redefining them as the s... more Various properties of Local and Global Lyapunov exponents are related by redefining them as the spectral radii of some positive operators on a space of continuous functions and utilizing the theory developed by Choquet and Foias. These results are then applied to the problem of estimating the Hausdorff dimension of the global attractor and the existence of a critical trajectory, along which the Lyapunov dimension is majorized, is established. Using this new estimate, the existing dimension estimate for the global attractor of the Lorenz system is improved. Along the way a simple relation between topological entropy and the fractal dimension is obtained.
Journal of Dynamics and Differential Equations, 1991
In this paper, we study a class of doubly nonlinear parabolic PDEs, where, in addition to some we... more In this paper, we study a class of doubly nonlinear parabolic PDEs, where, in addition to some weak nonlinearities, also some mild nonlinearities of porous media type are allowed inside the time derivative. In order to formulate the equations as dynamical systems, some existence and uniqueness results are proved. Then the existence of a compact attractor is shown for a class of nonlinear PDEs that include doubly nonlinear porous medium-type equations. Under stronger smoothness assumptions on the nonlinearities, the finiteness of the fractal dimension of the attractor is also obtained.
There is a sizable and growing literature on scholars who fled from the Nazi regime, a literature... more There is a sizable and growing literature on scholars who fled from the Nazi regime, a literature which often focuses on the periods before leaving Germany and after settling permanently in the USA, but relatively less work on the interim period in which many of them found temporary homes in countries such as Turkey. In this article we would like to discuss the scholarly work, activities and the impact of mathematicians Richard von Mises, William Prager and Hilda Geiringer during their stay in Turkey. We argue that the establishment and the development of applied mathematics and mechanics in Turkey owe much to them.
Following the approach described in [1], we define a semigroup S(t) associated to an autonomous s... more Following the approach described in [1], we define a semigroup S(t) associated to an autonomous system, and then prove, using an energy functional, that S(t) is an a-contraction and satisfies the squeezing property. (~) 2000 Elsevier Science Ltd. All rights reserved.
Journal of Mathematical Analysis and Applications, 1992
We consider the initial value problem for a class of second order evolution equations that includ... more We consider the initial value problem for a class of second order evolution equations that includes, among others, the 3D sine-Gordon equation with damping and the 3D Klein-Gordon type equations with damping. We show the existence of a set with finite fractal dimension that contains the global attractor and attracts all smooth solutions at an exponential rate.
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.
In the present study, we consider a generalized Davey-Stewartson (GDS) system consisting of a non... more In the present study, we consider a generalized Davey-Stewartson (GDS) system consisting of a nonlinear Schrödinger (NLS) type equation and two asymmetrically coupled linear wave equations. We obtain integral representation of solutions to the coupled linear wave equations. As applications, we present some localized solutions to the GDS system for a special choice of parameters and find some estimates of the solutions.
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&#39;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.
This article appeared in a journal published by Elsevier. The attached copy is furnished to the a... more This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are
Mathematical Methods in the Applied Sciences, 2013
We give a detailed study of the infinite-energy solutions of the Cahn-Hilliard equation in the 3D... more We 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.
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 ...
We study initial boundary value problems for the convective Cahn-Hilliard equation ∂ t u + ∂ 4 x ... more We study initial boundary value problems for the convective Cahn-Hilliard equation ∂ t u + ∂ 4 x u + u∂ x u + ∂ 2 x (|u| p u) = 0. It is well-known that without the convective term, the solutions of this equation may blow up in finite time for any p > 0. In contrast to that, we show that the presence of the convective term u∂ x u in the Cahn-Hilliard equation prevents blow up at least for 0 < p < 4 9. We also show that the blowing up solutions still exist if p is large enough (p ≥ 2). The related equations like Kolmogorov-Sivashinsky-Spiegel equation, sixth order convective Cahn-Hilliard equation, are also considered.
Journal of Mathematical Analysis and Applications, 2005
We find conditions on data guaranteeing global nonexistence of solutions to an inverse source pro... more We find conditions on data guaranteeing global nonexistence of solutions to an inverse source problem for a class of nonlinear parabolic equations. We also establish a stability result on a bounded domain for a problem with the opposite sign on the power type nonlinearity.
Journal of Mathematical Analysis and Applications, 1991
The Lieb-Thirring inequalities give a sharp upper bound for the LP-norm of a function which is th... more The Lieb-Thirring inequalities give a sharp upper bound for the LP-norm of a function which is the pointwise sum of the squares of a finite orthonormal sequence of functions that are elements of a suitable Sobolev space [LT]. Originally proven for the functions defined on the whole n-dimensional Euclidean space, they were later extended to bounded domains and to suborthogonal sequences of functions [GMT]. Here, we present a simple proof of these inequalities for bounded intervals in one space dimension utilizing simple Sobolev inequalities and standard results from Hilbert space theory. 0 1991 Academic PI~SS, hc.
Journal of Dynamics and Differential Equations, 1994
In this paper we present a new construction of exponential attractors based on the control of Lya... more In this paper we present a new construction of exponential attractors based on the control of Lyapunov exponents over a compact, invariant set. The fractal dimension estimate of the exponential attractor thus obtained is of the same order as the one for global attractors estimated through Lyapunov exponents. We discuss various applications to Navier-Stokes systems.
Journal of Dynamics and Differential Equations, 1991
Various properties of Local and Global Lyapunov exponents are related by redefining them as the s... more Various properties of Local and Global Lyapunov exponents are related by redefining them as the spectral radii of some positive operators on a space of continuous functions and utilizing the theory developed by Choquet and Foias. These results are then applied to the problem of estimating the Hausdorff dimension of the global attractor and the existence of a critical trajectory, along which the Lyapunov dimension is majorized, is established. Using this new estimate, the existing dimension estimate for the global attractor of the Lorenz system is improved. Along the way a simple relation between topological entropy and the fractal dimension is obtained.
Journal of Dynamics and Differential Equations, 1991
In this paper, we study a class of doubly nonlinear parabolic PDEs, where, in addition to some we... more In this paper, we study a class of doubly nonlinear parabolic PDEs, where, in addition to some weak nonlinearities, also some mild nonlinearities of porous media type are allowed inside the time derivative. In order to formulate the equations as dynamical systems, some existence and uniqueness results are proved. Then the existence of a compact attractor is shown for a class of nonlinear PDEs that include doubly nonlinear porous medium-type equations. Under stronger smoothness assumptions on the nonlinearities, the finiteness of the fractal dimension of the attractor is also obtained.
There is a sizable and growing literature on scholars who fled from the Nazi regime, a literature... more There is a sizable and growing literature on scholars who fled from the Nazi regime, a literature which often focuses on the periods before leaving Germany and after settling permanently in the USA, but relatively less work on the interim period in which many of them found temporary homes in countries such as Turkey. In this article we would like to discuss the scholarly work, activities and the impact of mathematicians Richard von Mises, William Prager and Hilda Geiringer during their stay in Turkey. We argue that the establishment and the development of applied mathematics and mechanics in Turkey owe much to them.
Following the approach described in [1], we define a semigroup S(t) associated to an autonomous s... more Following the approach described in [1], we define a semigroup S(t) associated to an autonomous system, and then prove, using an energy functional, that S(t) is an a-contraction and satisfies the squeezing property. (~) 2000 Elsevier Science Ltd. All rights reserved.
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