Attractors of evolution equations
by Babin, A. V., (Anatoliĭ Vladimirovich)
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 r…
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- Full title :Attractors of evolution equations / A.V. Babin and M.I. Vishik.
Summary
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 - & infin; 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 - + & infin;, 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 - & infin; of solutions for evolutionary equations.
Read More ↓Edition Statements
- Publication Date:1992.
Publication Statements
- Publisher(s):Distributors for the U.S. and Canada, Elsevier Science Pub. Co, North-Holland
- Place of Publication:Amsterdam
- Language:eng
- Publication Date:1992.
Notes
Translation of: Attraktory ėvoli︠u︡t︠s︡ionnykh uravneniĭ.
The Physical Object
- Description:1 online resource (x, 532 pages)
Edition Identifiers
- ISBN:9780444890047, 0444890041, 9780080875460, 0080875467, 1281789577, 9781281789570, 9786611789572, 661178957X
- oclc:ocn316572125
- LC Classification:QA374
- Dewey Decimal:515/.353
Table of Contents
Front Cover; Attractors of Evolution Equations; Copyright Page; PREFACE; Contents; INTRODUCTION; Chapter 1. QUASILINEAR EVOLUTIONARY EQUATIONS AND SEMIGROUPS GENERATED BY THEM; Chapter 2. MAXIMAL ATTRACTORS OF SEMIGROUPS; Chapter 3. ATTRACTORS AND UNSTABLE SETS; Chapter 4. SOME INFORMATION ON SEMIGROUPS OF LINEAR OPERATORS; Chapter 5. INVARIANT MANIFOLDS OF SEMIGROUPS AND MAPPINGS AT EQUILIBRIUM POINTS; Chapter 6. STEADY-STATE SOLUTIONS; Chapter 7. DIFFERENTIABILITY OF OPERATORS OF SEMIGROUPS GENERATED BY PARTIAL DIFFERENTIAL EQUATIONS; Chapter 8. SEMIGROUPS DEPENDING ON A PARAMETER.
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