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Hardcover ISBN:  9780821829509 
Product Code:  COLL/49 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 
eBook ISBN:  9781470431952 
Product Code:  COLL/49.E 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
Hardcover ISBN:  9780821829509 
eBook ISBN:  9781470431952 
Product Code:  COLL/49.B 
List Price:  $188.00 $143.50 
MAA Member Price:  $169.20 $129.15 
AMS Member Price:  $150.40 $114.80 

Book DetailsColloquium PublicationsVolume: 49; 2002; 363 ppMSC: Primary 35; 37
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 a number 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 infinitedimensional dynamical systems that were intensively developed during the last 20 years. They construct the attractors and study their properties for various nonautonomous equations of mathematical physics: the 2D and 3D NavierStokes systems, reactiondiffusion systems, dissipative wave equations, the complex GinzburgLandau 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. Upper estimates for the \(\varepsilon\)entropy of uniform attractors of nonautonomous equations in terms of \(\varepsilon\)entropy of timedependent coefficients are proved.
Also, the authors construct attractors for those equations of mathematical physics for which the solution of the corresponding Cauchy problem 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 finitedimensional 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 to spatial 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 nonautonomous 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.
ReadershipGraduate students and research mathematicians interested in partial differential equations, dynamical systems and ergodic theory.

Table of Contents

Chapters

Introduction

Attractors of autonomous equations

Chapter 1. Attractors of autonomous ordinary differential equations

Chapter 2. Attractors of autonomous partial differential equations

Chapter 3. Dimension of attractors

Attractors of nonautonomous equations

Chapter 4. Processes and attractors

Chapter 5. Translation compact functions

Chapter 6. Attractors of nonautonomous partial differential equations

Chapter 7. Semiprocesses and attractors

Chapter 8. Kernels of processes

Chapter 9. Kolmogorov $\varepsilon $entropy of attractors

Trajectory attractors

Chapter 10. Trajectory attractors of autonomous ordinary differential equations

Chapter 11. Attractors in Hausdorff spaces

Chapter 12. Trajectory attractors of autonomous equations

Chapter 13. Trajectory attractors of autonomous partial differential equations

Chapter 14. Trajectory attractors of nonautonomous equations

Chapter 15. Trajectory attractors of nonautonomous partial differential equations

Chapter 16. Approximation of trajectory attractors

Chapter 17. Perturbation of trajectory attractors

Chapter 18. Averaging of attractors of evolution equations with rapidly oscillating terms

Appendix A. Proofs of Theorems II.1.4 and II.1.5

Appendix B. Lattices and coverings


Reviews

In general, let me say that this book is a must for every mathematician who works on attractors.
translated from Jahresbericht der Deutschen MathematikerVereinigung 
A collection of a number of results obtained recently by the authors, two of the leading researchers on the subject ... new results are also included.
Mathematical Reviews


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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 a number 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 infinitedimensional dynamical systems that were intensively developed during the last 20 years. They construct the attractors and study their properties for various nonautonomous equations of mathematical physics: the 2D and 3D NavierStokes systems, reactiondiffusion systems, dissipative wave equations, the complex GinzburgLandau 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. Upper estimates for the \(\varepsilon\)entropy of uniform attractors of nonautonomous equations in terms of \(\varepsilon\)entropy of timedependent coefficients are proved.
Also, the authors construct attractors for those equations of mathematical physics for which the solution of the corresponding Cauchy problem 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 finitedimensional 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 to spatial 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 nonautonomous 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.
Graduate students and research mathematicians interested in partial differential equations, dynamical systems and ergodic theory.

Chapters

Introduction

Attractors of autonomous equations

Chapter 1. Attractors of autonomous ordinary differential equations

Chapter 2. Attractors of autonomous partial differential equations

Chapter 3. Dimension of attractors

Attractors of nonautonomous equations

Chapter 4. Processes and attractors

Chapter 5. Translation compact functions

Chapter 6. Attractors of nonautonomous partial differential equations

Chapter 7. Semiprocesses and attractors

Chapter 8. Kernels of processes

Chapter 9. Kolmogorov $\varepsilon $entropy of attractors

Trajectory attractors

Chapter 10. Trajectory attractors of autonomous ordinary differential equations

Chapter 11. Attractors in Hausdorff spaces

Chapter 12. Trajectory attractors of autonomous equations

Chapter 13. Trajectory attractors of autonomous partial differential equations

Chapter 14. Trajectory attractors of nonautonomous equations

Chapter 15. Trajectory attractors of nonautonomous partial differential equations

Chapter 16. Approximation of trajectory attractors

Chapter 17. Perturbation of trajectory attractors

Chapter 18. Averaging of attractors of evolution equations with rapidly oscillating terms

Appendix A. Proofs of Theorems II.1.4 and II.1.5

Appendix B. Lattices and coverings

In general, let me say that this book is a must for every mathematician who works on attractors.
translated from Jahresbericht der Deutschen MathematikerVereinigung 
A collection of a number of results obtained recently by the authors, two of the leading researchers on the subject ... new results are also included.
Mathematical Reviews