E-Book Overview
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.
E-Book Content
STUDIES IN MATHEMATICS AND ITS APPLICATIONS VOLUME 25
Editors: J. L. LIONS, Paris G. PAPANICOLAOU, New York H. FUJITA, Tokyo H. B. KELLER, Pasadena
NORTH-HOLLAND AMSTERDAM LONDON NEW YORK TOKYO
ATTRACTORS OF EVOLUTION EQUATIONS
A. V. BABIN Department of Computational Mathematies Moscow Institute for Railroad Transportation Engineers (MIIT) Moscow, U.S.S.R. and M. I. VISHIK Department of Differential Equations Faculty of Mechanics and Mathematics Moscow State University Moscow, U .S.S.R,
1992 NORTH-HOLLAND AMSTERDAM LONDON NEW YORK TOKYO
V
PREFACE
This book is the translation from Russian of our book A.V.Babin, M.I.Vishik “Attraktory evolutsionnykh uravnenii’l, Nauka, Moscow, 1989. Only minor changes were made during the translation. Some misprints were corrected and former Appendix was transformed into Section 6.6. After the original russian manuscript was prepared for print, many new works have appeared, to some earlier papers our attention was drawn during recentely intensified contacts with foreign mathematicians. Therefore I have added many new items to the bibliography. I shall very briefly mention here this added literature. First, new monographs J.K.Hale [3] and R.Temam [5] have appeared. These books are devoted, generally speaking, to the same subject as our book is, but the reader may see that there are only small intersections in the material of the books and these common parts are treated from different points of view. This shows that the field of dinamical systems generated by PDE*s is very large. Some of the papers which appeared since 1987 and are closely related with topics discussed in our book are mentioned below. Attractors of various equations of mathematical physics are studied in: J.Arrieta, A.Carvalho, J.Hale [l]; A.V.Babin, M.I.Vishik [24]; 1.D.Chueshov [3], [4]; P-Constantin,C.Foias, J.D.Gibbon [l]; A.Eden [l], [2]; A.Eden, C.Foias, B.Nicolaenko, R.Temam [3]; A.Eden, A.Libin [l]; A.Eden, A.J.Milani, B.Nicolaenko [l]; A.Eden, B.Michaux, J.M.Rakotoson [l] ; C.Foias, O.Manley, R-Temam [l]; T.L.Gil1, W.W.Zachary [l], [2]; J.K.Hale, G.Rauge1 [2], [3], [ 4 ] ; A.A.Il*in [l]; N.A.Karazeeva, A-A-Kotsiolis,A.P.Oskolkov [l]; 0.A.Ladyzhenskaya [GI, [71; A.Mahalov, E.S.Titi, S.Leibovich [l]; M.Marion [l]; G.Rauge1, G.R.Sel1 [l]; F.Rothe [2]; M.Taboada,
V i
Preface
Y.You [I]; R.Temam [ 6 ] . Dependence on parameters of attractors of PDE's is considered in: J.K.Hale. G.Rauge1 El]; J.K.Hale. X.- B.Lin, G.Rauge1 [l]; X.Mora, J.Sola-Morales [2]; M.Yu.Skvortsov [l], M.I.Vishik, M.Yu.Skvortsov [I]; M.I.Vishik, E21: V.Yu.Skvortsov [l]. Unbounded,local compact attractors of parabolic and hyperbolic equations are considered in: V.V.Chepyzhov [l], 111; A.Yu.Goritsky [21, [31; V.V.Chepyzhov, A.Yu.Goritsky 1[13, C2l.
Inertial