Attractors For Equations Of Mathematical Physics

Author : Vladimir V. Chepyzhov
Publisher : American Mathematical Soc.
ISBN 13 : 0821829505
Total Pages : 377 pages
Book Rating : 4.09/5 ( download)

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Download or read book Attractors for Equations of Mathematical Physics written by Vladimir V. Chepyzhov and published by American Mathematical Soc.. This book was released on 2002 with total page 377 pages. Available in PDF, EPUB and Kindle. Book excerpt: One of the major problems in the study of evolution equations of mathematical physics is the investigation of the behavior of the solutions to these equations when time is large or tends to infinity. The related important questions concern the stability of solutions or the character of the instability if a solution is unstable. In the last few decades, considerable progress in this area has been achieved in the study of autonomous evolution partial differential equations. For anumber of basic evolution equations of mathematical physics, it was shown that the long time behavior of their solutions can be characterized by a very important notion of a global attractor of the equation. In this book, the authors study new problems related to the theory of infinite-dimensionaldynamical systems that were intensively developed during the last 20 years. They construct the attractors and study their properties for various non-autonomous equations of mathematical physics: the 2D and 3D Navier-Stokes systems, reaction-diffusion systems, dissipative wave equations, the complex Ginzburg-Landau equation, and others. Since, as it is shown, the attractors usually have infinite dimension, the research is focused on the Kolmogorov $\varepsilon$-entropy of attractors. Upperestimates for the $\varepsilon$-entropy of uniform attractors of non-autonomous equations in terms of $\varepsilon$-entropy of time-dependent coefficients are proved. Also, the authors construct attractors for those equations of mathematical physics for which the solution of the corresponding Cauchyproblem is not unique or the uniqueness is not proved. The theory of the trajectory attractors for these equations is developed, which is later used to construct global attractors for equations without uniqueness. The method of trajectory attractors is applied to the study of finite-dimensional approximations of attractors. The perturbation theory for trajectory and global attractors is developed and used in the study of the attractors of equations with terms rapidly oscillating with respect tospatial and time variables. It is shown that the attractors of these equations are contained in a thin neighborhood of the attractor of the averaged equation. The book gives systematic treatment to the theory of attractors of autonomous and non-autonomous evolution equations of mathematical physics.It can be used both by specialists and by those who want to get acquainted with this rapidly growing and important area of mathematics.

Author : A.V. Babin
Publisher : Elsevier
ISBN 13 : 0080875467
Total Pages : 543 pages
Book Rating : 4.60/5 ( download)

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Download or read book Attractors of Evolution Equations written by A.V. Babin and published by Elsevier. This book was released on 1992-03-09 with total page 543 pages. Available in PDF, EPUB and Kindle. Book excerpt: Problems, ideas and notions from the theory of finite-dimensional dynamical systems have penetrated deeply into the theory of infinite-dimensional systems and partial differential equations. From the standpoint of the theory of the dynamical systems, many scientists have investigated the evolutionary equations of mathematical physics. Such equations include the Navier-Stokes system, magneto-hydrodynamics equations, reaction-diffusion equations, and damped semilinear wave equations. Due to the recent efforts of many mathematicians, it has been established that the attractor of the Navier-Stokes system, which attracts (in an appropriate functional space) as t - ∞ all trajectories of this system, is a compact finite-dimensional (in the sense of Hausdorff) set. Upper and lower bounds (in terms of the Reynolds number) for the dimension of the attractor were found. These results for the Navier-Stokes system have stimulated investigations of attractors of other equations of mathematical physics. For certain problems, in particular for reaction-diffusion systems and nonlinear damped wave equations, mathematicians have established the existence of the attractors and their basic properties; furthermore, they proved that, as t - +∞, an infinite-dimensional dynamics described by these equations and systems uniformly approaches a finite-dimensional dynamics on the attractor U, which, in the case being considered, is the union of smooth manifolds. This book is devoted to these and several other topics related to the behaviour as t - ∞ of solutions for evolutionary equations.

Author : Messoud Efendiev
Publisher : American Mathematical Soc.
ISBN 13 : 1470409852
Total Pages : 233 pages
Book Rating : 4.52/5 ( download)

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Download or read book Attractors for Degenerate Parabolic Type Equations written by Messoud Efendiev and published by American Mathematical Soc.. This book was released on 2013-09-26 with total page 233 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book deals with the long-time behavior of solutions of degenerate parabolic dissipative equations arising in the study of biological, ecological, and physical problems. Examples include porous media equations, -Laplacian and doubly nonlinear equations, as well as degenerate diffusion equations with chemotaxis and ODE-PDE coupling systems. For the first time, the long-time dynamics of various classes of degenerate parabolic equations, both semilinear and quasilinear, are systematically studied in terms of their global and exponential attractors. The long-time behavior of many dissipative systems generated by evolution equations of mathematical physics can be described in terms of global attractors. In the case of dissipative PDEs in bounded domains, this attractor usually has finite Hausdorff and fractal dimension. Hence, if the global attractor exists, its defining property guarantees that the dynamical system reduced to the attractor contains all of the nontrivial dynamics of the original system. Moreover, the reduced phase space is really "thinner" than the initial phase space. However, in contrast to nondegenerate parabolic type equations, for a quite large class of degenerate parabolic type equations, their global attractors can have infinite fractal dimension. The main goal of the present book is to give a detailed and systematic study of the well-posedness and the dynamics of the semigroup associated to important degenerate parabolic equations in terms of their global and exponential attractors. Fundamental topics include existence of attractors, convergence of the dynamics and the rate of convergence, as well as the determination of the fractal dimension and the Kolmogorov entropy of corresponding attractors. The analysis and results in this book show that there are new effects related to the attractor of such degenerate equations that cannot be observed in the case of nondegenerate equations in bounded domains. This book is published in cooperation with Real Sociedad Matemática Española (RSME).

Author : Boling Guo
Publisher : Walter de Gruyter GmbH & Co KG
ISBN 13 : 3110549654
Total Pages : 438 pages
Book Rating : 4.52/5 ( download)

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Download or read book Attractors and Inertial Manifolds written by Boling Guo and published by Walter de Gruyter GmbH & Co KG. This book was released on 2018-07-09 with total page 438 pages. Available in PDF, EPUB and Kindle. Book excerpt: This two-volume work presents state-of-the-art mathematical theories and results on infinite-dimensional dynamical systems. Inertial manifolds, approximate inertial manifolds, discrete attractors and the dynamics of small dissipation are discussed in detail. The unique combination of mathematical rigor and physical background makes this work an essential reference for researchers and graduate students in applied mathematics and physics. The main emphasis in the first volume is on the mathematical analysis of attractors and inertial manifolds. This volume deals with the existence of global attractors, inertial manifolds and with the estimation of Hausdorff fractal dimension for some dissipative nonlinear evolution equations in modern physics. Known as well as many new results about the existence, regularity and properties of inertial manifolds and approximate inertial manifolds are also presented in the first volume. The second volume will be devoted to modern analytical tools and methods in infinite-dimensional dynamical systems. Contents Attractor and its dimension estimation Inertial manifold The approximate inertial manifold

Author : Boling Guo
Publisher : Walter de Gruyter GmbH & Co KG
ISBN 13 : 3110587262
Total Pages : 413 pages
Book Rating : 4.65/5 ( download)

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Download or read book Attractors and Methods written by Boling Guo and published by Walter de Gruyter GmbH & Co KG. This book was released on 2018-07-09 with total page 413 pages. Available in PDF, EPUB and Kindle. Book excerpt: This two-volume work presents state-of-the-art mathematical theories and results on infinite-dimensional dynamical systems. Inertial manifolds, approximate inertial manifolds, discrete attractors and the dynamics of small dissipation are discussed in detail. The unique combination of mathematical rigor and physical background makes this work an essential reference for researchers and graduate students in applied mathematics and physics. The main emphasis in the fi rst volume is on the existence and properties for attractors and inertial manifolds. This volume highlights the use of modern analytical tools and methods such as the geometric measure method, center manifold theory in infinite dimensions, the Melnihov method, spectral analysis and so on for infinite-dimensional dynamical systems. The second volume includes the properties of global attractors, the calculation of discrete attractors, structures of small dissipative dynamical systems, and the existence and stability of solitary waves. Contents Discrete attractor and approximate calculation Some properties of global attractor Structures of small dissipative dynamical systems Existence and stability of solitary waves

Author : Anatoliĭ Vladimirovich Babin
Publisher : American Mathematical Soc.
ISBN 13 : 9780821841099
Total Pages : 184 pages
Book Rating : 4.92/5 ( download)

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Download or read book Properties of Global Attractors of Partial Differential Equations written by Anatoliĭ Vladimirovich Babin and published by American Mathematical Soc.. This book was released on 1992 with total page 184 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Author : Alexander Komech
Publisher : Cambridge University Press
ISBN 13 : 100903605X
Total Pages : pages
Book Rating : 4.54/5 ( download)

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Download or read book Attractors of Hamiltonian Nonlinear Partial Differential Equations written by Alexander Komech and published by Cambridge University Press. This book was released on 2021-09-30 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is the first to present the theory of global attractors of Hamiltonian partial differential equations. A particular focus is placed on the results obtained in the last three decades, with chapters on the global attraction to stationary states, to solitons, and to stationary orbits. The text includes many physically relevant examples and will be of interest to graduate students and researchers in both mathematics and physics. The proofs involve novel applications of methods of harmonic analysis, including Tauberian theorems, Titchmarsh's convolution theorem, and the theory of quasimeasures. As well as the underlying theory, the authors discuss the results of numerical simulations and formulate open problems to prompt further research.

Author : Eleonora Pinto de Moura
Publisher :
ISBN 13 :
Total Pages : 188 pages
Book Rating : 4.88/5 ( download)

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Download or read book On Embeddings and Dimensions of Global Attractors Associated with Dissipative Partial Differential Equations written by Eleonora Pinto de Moura and published by . This book was released on 2010 with total page 188 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Author : Ian W. Knowles
Publisher : Springer
ISBN 13 : 354047983X
Total Pages : 517 pages
Book Rating : 4.33/5 ( download)

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Download or read book Differential Equations and Mathematical Physics written by Ian W. Knowles and published by Springer. This book was released on 2006-11-14 with total page 517 pages. Available in PDF, EPUB and Kindle. Book excerpt: The meeting in Birmingham, Alabama, provided a forum for the discussion of recent developments in the theory of ordinary and partial differential equations, both linear and non-linear, with particular reference to work relating to the equations of mathematical physics. The meeting was attended by about 250 mathematicians from 22 countries. The papers in this volume all involve new research material, with at least outline proofs; some papers also contain survey material. Topics covered include: Schrödinger theory, scattering and inverse scattering, fluid mechanics (including conservative systems and inertial manifold theory attractors), elasticity, non-linear waves, and feedback control theory.

Author : Alexandre Carvalho
Publisher : Springer Science & Business Media
ISBN 13 : 1461445817
Total Pages : 434 pages
Book Rating : 4.14/5 ( download)

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Download or read book Attractors for infinite-dimensional non-autonomous dynamical systems written by Alexandre Carvalho and published by Springer Science & Business Media. This book was released on 2012-09-25 with total page 434 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book treats the theory of attractors for non-autonomous dynamical systems. The aim of the book is to give a coherent account of the current state of the theory, using the framework of processes to impose the minimum of restrictions on the nature of the non-autonomous dependence. The book is intended as an up-to-date summary of the field, but much of it will be accessible to beginning graduate students. Clear indications will be given as to which material is fundamental and which is more advanced, so that those new to the area can quickly obtain an overview, while those already involved can pursue the topics we cover more deeply.