Download or read book Finite and Infinite Dimensional Attractors for Evolution Equations of Mathematical Physics written by Messoud Efendiev and published by . This book was released on 2010-11 with total page 239 pages. Available in PDF, EPUB and Kindle. Book excerpt:
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.
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.
Download or read book Handbook of Differential Equations Evolutionary Equations written by C.M. Dafermos and published by Elsevier. This book was released on 2008-10-06 with total page 609 pages. Available in PDF, EPUB and Kindle. Book excerpt: The material collected in this volume discusses the present as well as expected future directions of development of the field with particular emphasis on applications. The seven survey articles present different topics in Evolutionary PDE’s, written by leading experts. - Review of new results in the area- Continuation of previous volumes in the handbook series covering Evolutionary PDEs- Written by leading experts
Download or read book Infinite Dimensional Dynamical Systems written by James C. Robinson and published by Cambridge University Press. This book was released on 2001-04-23 with total page 488 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book treats the theory of global attractors, a recent development in the theory of partial differential equations, in a way that also includes much of the traditional elements of the subject. As such it gives a quick but directed introduction to some fundamental concepts, and by the end proceeds to current research problems. Since the subject is relatively new, this is the first book to attempt to treat these various topics in a unified and didactic way. It is intended to be suitable for first year graduate students.
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
Download or read book From Finite to Infinite Dimensional Dynamical Systems written by James Robinson and published by Springer Science & Business Media. This book was released on 2001-05-31 with total page 236 pages. Available in PDF, EPUB and Kindle. Book excerpt: Proceedings of the NATO Advanced Study Institute, Cambridge, UK, 21 August-1 September 1995
Download or read book Attractors for Semigroups and Evolution Equations written by Olga A. Ladyzhenskaya and published by Cambridge University Press. This book was released on 2022-06-09 with total page 97 pages. Available in PDF, EPUB and Kindle. Book excerpt: First published 1992; Re-issued 2008; Reprinted with Introduction 2022.
Download or read book Infinite Dimensional Dynamical Systems in Mechanics and Physics written by Roger Temam and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 517 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first attempt at a systematic study of infinite dimensional dynamical systems generated by dissipative evolution partial differential equations arising in mechanics and physics. Other areas of science and technology are included where appropriate. The relation between infinite and finite dimensional systems is presented from a synthetic viewpoint and equations considered include reaction-diffusion, Navier-Stokes and other fluid mechanics equations, magnetohydrodynamics, thermohydraulics, pattern formation, Ginzburg-Landau, damped wave and an introduction to inertial manifolds.
Download or read book Exponential Attractors for Dissipative Evolution Equations written by A. Eden and published by . This book was released on 1994 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt: Covering a pioneering area of dynamical systems, this monograph includes references, Navier-Stokes equations and many applications which should be of particular interest to those working in the field of fluid mechanics.
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.
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).
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 577 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
Download or read book Studies in Evolution Equations and Related Topics written by Gaston M. N'Guérékata and published by Springer Nature. This book was released on 2021-10-27 with total page 275 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume features recent development and techniques in evolution equations by renown experts in the field. Each contribution emphasizes the relevance and depth of this important area of mathematics and its expanding reach into the physical, biological, social, and computational sciences as well as into engineering and technology. The reader will find an accessible summary of a wide range of active research topics, along with exciting new results. Topics include: Impulsive implicit Caputo fractional q-difference equations in finite and infinite dimensional Banach spaces; optimal control of averaged state of a population dynamic model; structural stability of nonlinear elliptic p(u)-Laplacian problem with Robin-type boundary condition; exponential dichotomy and partial neutral functional differential equations, stable and center-stable manifolds of admissible class; global attractor in Alpha-norm for some partial functional differential equations of neutral and retarded type; and more. Researchers in mathematical sciences, biosciences, computational sciences and related fields, will benefit from the rich and useful resources provided. Upper undergraduate and graduate students may be inspired to contribute to this active and stimulating field.
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 2012-09-26 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.
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:
Download or read book Dynamics of Evolutionary Equations written by George R. Sell and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 680 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory and applications of infinite dimensional dynamical systems have attracted the attention of scientists for quite some time. This book serves as an entrée for scholars beginning their journey into the world of dynamical systems, especially infinite dimensional spaces. The main approach involves the theory of evolutionary equations.
