**Colloquium Publications**

Volume: 49;
2002;
363 pp;
Hardcover

MSC: Primary 35; 37;

Print ISBN: 978-0-8218-2950-9

Product Code: COLL/49

List Price: $84.00

AMS Member Price: $67.20

MAA Member Price: $75.60

**Electronic ISBN: 978-1-4704-3195-2
Product Code: COLL/49.E**

List Price: $84.00

AMS Member Price: $67.20

MAA Member Price: $75.60

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# Attractors for Equations of Mathematical Physics

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*Vladimir V. Chepyzhov; Mark I. Vishik*

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
infinite-dimensional dynamical 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. Upper estimates 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 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 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 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 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.

#### Readership

Graduate students and research mathematicians interested in partial differential equations, dynamical systems and ergodic theory.

#### Reviews & Endorsements

In general, let me say that this book is a must for every mathematician who works on attractors.

-- translated from Jahresbericht der Deutschen Mathematiker-Vereinigung

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

#### Table of Contents

# Table of Contents

## Attractors for Equations of Mathematical Physics

- Cover Cover11
- Title page v6
- Dedication vii8
- Contents ix10
- Introduction 114
- Attractors of autonomous equations 1528
- Attractors of autonomous ordinary differential equations 1730
- Attractors of autonomous partial differential equations 2740
- Dimension of attractors 5164
- Attractors of non-autonomous equations 7790
- Processes and attractors 7992
- Translation compact functions 95108
- Attractors of non-autonomous partial differential equations 107120
- Semiprocesses and attractors 129142
- Kernels of processes 149162
- Kolmogorov 𝜖-entropy of attractors 163176
- Trajectory attractors 197210
- Trajectory attractors of autonomous ordinary differential equations 199212
- Attractors in Hausdorff spaces 211224
- Trajectory attractors of autonomous equations 219232
- Trajectory attractors of autonomous partial differential equations 229242
- Trajectory attractors of non-autonomous equations 259272
- Trajectory attractors of non-autonomous partial differential equations 269282
- Approximation of trajectory attractors 299312
- Perturbation of trajectory attractors 305318
- Averaging of attractors of evolution equations with rapidly oscillating terms 311324
- Appendix A. Proofs of Theorems II.1.4 and II.1.5 345358
- Appendix B. Lattices and coverings 349362
- Bibliography 353366
- Index 361374
- Back Cover Back Cover1377