Selected Title s i n Thi s Serie s 49 Vladimi r V. Chepyzho v an d Mar k I. Vishik, Attractor s fo r equation s o f mathematical physics, 200 2 48 Yoa v Benyamin i an d Jora m Lindenstrauss, Geometri c nonlinea r functiona l analysis, Volume 1, 200 0 47 Yur i I. Manin, Frobeniu s manifolds, quantu m cohomology, an d modul i spaces, 199 9 46 J. Bourgain, Globa l solution s o f nonlinear Schrodinge r equations, 199 9 45 Nichola s M. Kat z an d Pete r Sarnak, Rando m matrices, Frobeniu s eigenvalues, an d monodromy, 199 9 44 Max-Alber t Knus, Alexande r Merkurjev, an d Marku s Rost, Th e boo k o f involutions, 199 8 43 Lui s A. Caffarell i an d Xavie r Cabre, Full y nonlinea r ellipti c equations, 199 5 42 Victo r Guillemi n an d Shlom o Sternberg, Variation s o n a theme b y Kepler, 199 0 41 Alfre d Tarsk i an d Steve n Givant, A formalization o f set theor y withou t variables, 198 7 40 R. H. Bing, Th e geometri c topolog y o f 3-manifolds, 198 3 39 N. Jacobson, Structur e an d representation s o f Jordan algebras, 196 8 38 O. Ore, Theor y o f graphs, 196 2 37 N. Jacobson, Structur e o f rings, 195 6 36 W. H. Gottschal k an d G. A. Hedlund, Topologica l dynamics, 195 5 35 A. C. Schaeffe r an d D. C. Spencer, Coefficien t region s fo r Schlich t functions, 195 0 34 J. L. Walsh, Th e locatio n o f critica l point s o f analyti c an d harmoni c functions, 195 0 33 J. F. Ritt, Differentia l algebra, 195 0 32 R. L. Wilder, Topolog y o f manifolds, 194 9 31 E. Hill e an d R. S. Phillips, Functiona l analysi s an d semigroups, 195 7 30 T. Rado, Lengt h an d area, 194 8 29 A. Weil, Foundation s o f algebraic geometry, 194 6 28 G. T. Whyburn, Analyti c topology, 194 2 27 S. Lefschetz, Algebrai c topology, 194 2 26 N. Levinson, Ga p an d densit y theorems, 194 0 25 Garret t Birkhoff, Lattic e theory, 194 0 24 A. A. Albert, Structur e o f algebras, 193 9 23 G. Szego, Orthogona l polynomials, 193 9 22 C. N. Moore, Summabl e serie s an d convergenc e factors, 193 8 21 J. M. Thomas, Differentia l systems, 193 7 20 J. L. Walsh, Interpolatio n an d approximatio n b y rationa l function s i n th e comple x domain, 193 5 19 R. E. A. C. Pale y an d N. Wiener, Fourie r transform s i n the comple x domain, 193 4 18 M. Morse, Th e calculu s o f variations i n the large, 193 4 17 J. M. Wedderburn, Lecture s o n matrices, 193 4 16 G. A. Bliss, Algebrai c functions, 193 3 15 M. H. Stone, Linea r transformation s i n Hilber t spac e an d thei r application s t o analysis, 1932 14 J. F. Ritt, Differentia l equation s fro m th e algebrai c standpoint, 193 2 13 R. L. Moore, Foundation s o f point se t theory, 193 2 12 S. Lefschetz, Topology, 193 0 For a complet e lis t o f titles i n this series, visi t th e AMS Bookstor e a t www.ams.org/bookstore/.
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Attractor s for Equation s of Mathematica l Physic s
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http://dx.doi.org/10.1090/coll/049 America n Mathematica l Societ y Colloquiu m Publication s Volum e 49 Attractor s for Equation s of Mathematica l Physic s Vladimi r V. Chepyzho v Mar k I. Vishi k 5jiEM47y America n Mathematica l Societ y Providence, Rhod e Islan d
Editorial Boar d Susan J. Friedlander, Chai r Yuri Mani n Peter Sarna k 2000 Mathematics Subject Classification. Primar y 35K90, 35B40, 37C70, 37L30 ; Secondary 35Q99, 35Q30, 35L70, 35K57. ABSTRACT. The author s stud y ne w problem s relate d t o the theory o f infinite-dimensional dynam - ical systems that wer e intensively develope d durin g th e las t fe w years. The y construc t th e attrac - tors an d stud y thei r propertie s fo r variou s non-autonomou s equation s o f mathematica l physics : the 2 D an d 3 D Navier-Stoke s systems, reaction-diffusio n systems, dissipativ e wav e equations, the comple x Ginzburg-Landa u equation, an d others. Sinc e th e attractor s usuall y hav e infinit e dimension, th e researc h i s focuse d o n th e Kolmogoro v -entrop y o f attractors. Uppe r estimate s for th e ^-entrop y o f unifor m attractor s o f non-autonomou s equation s i n term s o f e-entrop y o f time-dependent coefficient s o f the equatio n ar e proved. The author s als o construc t attractor s fo r thos e equation s o f mathematica l physic s fo r whic h the solutio n o f the correspondin g Cauch y proble m i s no t uniqu e o r th e uniquenes s i s no t know n (for example, th e 3 D Navier-Stoke s system). Th e theor y o f th e trajector y attractor s fo r thes e equations i s developed, whic h i s late r use d t o construc t globa l attractor s fo r equation s withou t uniqueness. Th e metho d o f trajector y attractor s i s applie d t o th e stud y o f finite-dimensiona l approximations o f attractors. Th e perturbatio n theor y fo r trajector y an d globa l attractor s i s developed an d use d i n th e stud y o f th e attractor s o f equation s wit h term s rapidl y oscillatin g with respec t t o spatia l an d tim e variables. I t i s show n tha t th e attractor s o f these equation s ar e contained i n a thin neighbourhoo d o f the attracto r o f the average d equation. Library o f Congres s Cataloging-in-Publicatio n Dat a Chepyzhov, Vladimi r V., 1962 - Attractors fo r equation s o f mathematical physic s / Vladimi r V. Chepyzhov, Mar k I. Vishik. p. cm. (Colloquiu m publication s / America n Mathematica l Society, ISS N 0065-925 8 ; v. 49 ) Includes bibliographica l reference s an d index. ISBN 0-8218-2950-5 (alk. paper ) 1. Attractor s (Mathematics ) 2. Evolutio n equations-numerica l solutions. 3. Mathemati - cal physics. I. Vishik, M. I. II. Title. III. Colloquiu m publication s (America n Mathematica l Society) ; v. 49. QA614.813.C46 200 1 515'.353-dc21 200104640 6 Copying an d reprinting. Individua l reader s o f thi s publication, an d nonprofi t librarie s acting fo r them, ar e permitte d t o mak e fai r us e o f the material, suc h a s to cop y a chapter fo r us e in teachin g o r research. Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f the sourc e i s given. Republication, systemati c copying, o r multiple reproduction o f any material i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society. Request s fo r suc h permission shoul d be addressed to the Assistant t o the Publisher, America n Mathematical Society, P. O. Bo x 6248, Providence, Rhod e Islan d 02940-6248. Request s ca n als o b e mad e b y e-mai l t o reprint-permissionoams.org. 200 2 b y the America n Mathematica l Society. Al l rights reserved. The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government. Printed i n the Unite d State s o f America. @ Th e pape r use d i n this boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability. Visit th e AM S hom e pag e a t URL : http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 0 7 0 6 0 5 0 4 0 3 0 2
Dedicated to our wives, Katya and Asya
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Contents Introduction 1 Part 1. Attractor s o f Autonomous Equation s 1 5 Chapter I. Attractor s o f Autonomous Ordinar y Differentia l Equation s 1 7 1. Semigroup s an d attractor s 1 7 2. Example s o f ordinary differentia l equation s an d thei r attractor s 2 1 Chapter II. Attractor s o f Autonomous Partia l Differentia l Equation s 2 7 1. Functio n space s an d embeddin g theorem s 2 8 2. Operato r semigroups. Basi c notion s 3 5 3. Attractor s o f semigroups 3 7 4. Reaction-diffusio n system s 3 8 5. 2 D Navier-Stokes syste m 4 6 6. Hyperboli c equatio n wit h dissipatio n 4 9 Chapter III. Dimensio n o f Attractors 5 1 1. Fracta l an d Hausdorf f dimensio n 5 1 2. Dimensio n o f invariant set s 5 3 3. Optimizatio n o f the boun d fo r th e fracta l dimensio n 5 9 4. Applicatio n t o semigroup s 6 2 5. Application s t o evolutio n equation s 6 5 6. Lowe r bound s fo r th e dimensio n o f attractors 7 3 Part 2. Attractor s o f Non-autonomous Equation s 7 7 Chapter IV. Processe s an d Attractor s 7 9 1. Symbol s o f non-autonomous equation s 8 0 2. Cauch y proble m an d processe s 8 2 3. Unifor m attractor s 8 3 4. Haraux' s exampl e 8 5 5. Th e reduction t o a semigroup 8 6 6. O n unifor m (w.r.t. rgl) attractor s 9 2 Chapter V. Translatio n Compac t Function s 9 5 1. Almos t periodi c function s 9 5 2. Translatio n compac t function s i n C(IR ; M) 9 7 3. Translatio n compac t function s i n L l c (R; ) 10 1 4. Translatio n compac t function s i n L ^(R; ) 10 4 5. Othe r translatio n compac t function s 10 6
x CONTENT S Chapter VI. Attractor s o f Non-autonomous Partia l Differentia l Equation s 10 7 1. 2 D Navier-Stokes syste m 10 7 2. Non-autonomou s reaction-diffusio n system s 11 4 3. Non-autonomou s Ginzburg-Landa u equatio n an d other s 11 8 4. Non-autonomou s dampe d hyperboli c equation s 11 9 Chapter VII. Semiprocesse s an d Attractor s 12 9 1. Familie s o f semiprocesses an d thei r attractor s 12 9 2. O n the reductio n t o the semigrou p 13 2 3. Non-autonomou s equation s wit h tr.c. o n R + symbol s 13 5 4. Prolongation s o f semiprocesses t o processe s 13 7 5. Asymptoticall y almos t periodi c function s 14 0 6. Non-autonomou s equation s wit h a.a.p. symbol s 14 3 7. Cascad e system s an d thei r attractor s 14 6 Chapter VIII. Kernel s o f Processes 14 9 1. Propertie s o f kernels 14 9 2. O n the dimensio n o f connected set s 15 3 3. Dimensio n estimate s fo r kerne l section s 15 5 4. Application s t o non-autonomous equation s 15 7 Chapter IX. Kolmogoro v ^-Entrop y o f Attractors 16 3 1. Estimate s o f the e-entrop y 16 3 2. Fracta l dimensio n o f attractors 17 3 3. Functiona l dimensio n an d metri c orde r 17 6 4. Application s t o evolutio n equation s 17 7 5. 77-entrop y an d metri c orde r o f E 18 8 6. ^-entrop y i n the extende d phas e spac e 19 2 Part 3. Trajector y Attractor s 19 7 Chapter X. Trajector y Attractor s o f Autonomou s Ordinar y Differentia l Equations 19 9 1. Preliminar y proposition s 20 0 2. Constructio n o f the trajectory attracto r 20 3 3. Example s o f equations 20 5 4. Dependenc e o n a parameter 20 7 Chapter XI. Attractor s i n Hausdorf f Space s 21 1 1. Som e topological preliminarie s 21 1 2. Semigroup s i n topological space s an d attractor s 21 4 3. Application s t o {M., X)-attractors 21 8 Chapter XII. Trajector y Attractor s o f Autonomous Equation s 21 9 1. Trajector y space s o f evolution equation s 21 9 2. Existenc e o f trajectory attractor s 22 2 3. Trajector y an d globa l attractor s 22 4 Chapter XIII. Trajector y Attractor s o f Autonomou s Partia l Differentia l Equations 22 9 1. Autonomou s Navier-Stoke s system s 22 9
CONTENTS x i 2. Autonomou s hyperboli c equation s 24 2 3. Hyperboli c equation s dependin g o n a parameter 25 1 Chapter XIV. Trajector y Attractor s o f Non-autonomous Equation s 25 9 1. Non-autonomou s equations, thei r symbols, an d trajector y space s 26 0 2. Existenc e o f uniform trajector y attractor s 26 2 3. Equation s wit h symbol s o n the semiaxi s 26 6 Chapter XV. Trajector y Attractor s o f Non-autonomou s Partia l Differential Equation s 26 9 1. Non-autonomou s Navier-Stoke s system s 26 9 2. Trajector y attracto r fo r 2 D Navier-Stokes syste m 27 8 3. Reac t ion-diffusion system s 28 2 4. Non-autonomou s hyperboli c equation s 29 2 Chapter XVI. Approximatio n o f Trajectory Attractor s 29 9 1. Trajector y attractor s o f non-autonomou s ordinar y differential equation s 29 9 2. Trajector y attractor s o f Galerki n system s 30 2 3. Convergenc e o f trajectory attractor s o f Galerkin system s 30 3 Chapter XVII. Perturbatio n o f Trajectory Attractor s 30 5 1. Trajector y attractor s o f perturbed equation s 30 5 2. Dependenc e o f trajectory attractor s o n a small parameter 30 7 Chapter XVIII. Averagin g o f Attractor s o f Evolutio n Equation s wit h Rapidly Oscillatin g Term s 31 1 1. Averagin g o f rapidly oscillatin g function s 31 1 2. Averagin g o f equations an d system s 32 0 3. Perturbatio n wit h rapidl y oscillatin g term s 34 1 Appendix A. Proof s o f Theorems II. 1.4 an d II. 1.5 34 5 Appendix B. Lattice s an d Covering s 34 9 Bibliography 35 3 Index 361
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Index (E x E, )-continuous family, 8 8 (e^c,s)-closed family, 26 2 e-entropy, 51, 16 5 e-period, 9 5 d-dimensional Hausdorf f measure, 5 2 fc-dimensional torus, 8 2 m-dimensional trace, 6 2 Almost periodi c function, 81, 95 asymptotically, 133, 14 0 in the Stepano v sense, 9 7 Attracting property, 8 4 Attractor, 19, 217, 21 8 (A4,1)-attractor, 21 8 global, 19, 37, 225, 239, 24 9 uniform, 26 5 Lorenz, 6 5 non-uniform, 8 5 trajectory, 203, 22 3 uniform, 26 2 uniform, 84, 9 3 Average in Loo(Q), 31 2 in L P, W (Q), 31 1 time uniform, 31 6 Averaging spatial, 31 1 time, 31 6 Backward uniquenes s property, 13 8 Belousov-Zhabotinsky equations, 4 3 Bochner-Amerio criterion, 9 6 Cascade system, 13 3 Chafee-Infante equation, 41, 330 Closure, 21 2 Compactness criterion in C{R;M), 9 8 in L l p oc (R;S), 10 1 inl^(r; ), 10 5 theorems, 3 1 Compactum, 21 4 Complete trajectory, 19, 38, 88, 218, 223, 263 Continuous mapping, 21 3 Convergent sequence, 21 2 *-weakly, 3 2 weakly, 3 2 Covering, 21 2 density, 34 9 radius, 34 9 Derivative i n the distributio n sense, 3 1 Differential inequality, 3 5 Dimension fractal, 52, 17 3 functional, 17 6 Hausdorff, 5 2 local fractal, 17 5 functional, 17 6 Lyapunov, 6 2 Dissipative wav e equation, 33 4 Dissipativity condition, 1 7 Douady-Oesterle theorem, 5 5 Energy norm, 5 0 Equilibrium point, 2 0 First Uryso n theorem, 21 4 Fitz-Hugh-Nagumo equations, 4 1 Frechet-Uryson space, 21 3 Fundamental parallelepiped, 34 9 region, 34 9 Gagliardo-Nirenberg inequality, 3 0 Galerkin approximation, 23, 30 2 method, 231, 284 Ginzburg-Landau equation, 42, 118, 32 8 Grashof number, 47, 23 5 Gronwall's inequality, 3 4 Group, 3 6 Holder's inequality, 3 4 Hahn-Banach theorem, 3 2 Haraux's example, 8 5 Hausdorff dimension, 5 2 space, 21 3 Hull, 81, 96, 132, 13 5 Hyperbolic equatio n 361
362 INDEX damped, 11 9 dissipative, 4 9 with dissipation, 49, 71, 159, 185, 292, 306 Inductive limit, 22 1 Instability index, 7 3 Interpolation inequality, 3 0 Kernel of equation, 20, 223, 26 3 of process, 88, 14 9 of semigroup, 38, 21 8 section, 20, 38, 88, 21 8 Kolmogorov e-entropy, 16 4 Ladyzhenskaya's inequality, 46, 230, 23 5 Lattice, 34 9 cube, 35 1 determinant, 34 9 enerating matrix, 34 9 main Voronoi, 35 1 Lieb-Thirring inequality, 6 9 Lipschitz condition, 16 5 Lorenz attractor, 6 5 system, 2 3 Lotka-Volterra system, 4 4 Lyapunov dimension, 6 2 uniform exponents, 6 1 Metric order, 17 6 local, 17 6 Minimality property, 8 4 Multiplicative properties, 8 3 Navier-Stokes system, 26 9 2D, 46, 68, 74, 107, 157, 177, 239, 278, 323 3D, 229, 305, 32 0 NikoPskii space, 27 9 Periodic orbit, 2 0 Point adherent, 21 1 limit, 21 2 Process, 82, 8 3 bounded, 8 3 family o f processes, 8 4 periodic, 8 7 Quasidifferential, 5 3 Quasiperiodic function, 82, 9 6 solution, 2 0 symbol, 8 8 Reaction-diffusion equation, 3 8 system, 66, 75, 114, 158, 181, 282, 32 5 Second axio m o f count ability, 21 2 Second Uryso n theorem, 21 4 Semigroup, 18, 36, 21 4 (, )-bounded, 3 7 (E, E) -continuous, 3 7 asymptotically compact, 3 7 compact, 3 7 identity, 3 6 Semiprocess, 12 9 Set (.M,1)-attracting, 21 8 u;-limit, 19, 38, 130, 21 5 absorbing, 18, 37, 8 3 attracting, 37, 83, 22 3 count ably precompact, 21 4 local unstable, 7 3 precompact, 21 4 relatively dense, 9 5 uniformly absorbing, 8 4 attracting, 84, 92, 26 2 Sets closed, 21 1 open, 21 1 Sine-Gordon equation, 4 9 Sobolev embeddin g theorem, 2 9 Space compact, 21 4 count ably compact, 21 4 Prechet-Uryson, 21 3 Hausdorff, 21 3 metrizable, 21 4 normal, 21 3 separable, 21 2 topological, 21 1 Symbol of equation, 79, 8 0 of process, 8 4 space, 80, 81, 84 Topology base, 21 2 Trajectory, 220, 26 1 attractor, 203, 22 3 uniform, 26 2 space, 200, 219, 26 0 united, 26 1 Translation bounded function, 10 5 compact function, 81, 105, 13 5 group, 26 0 identity, 83, 86 semigroup, 20 0 Uniformly quasidifferentiabl e map, 5 3 sequence, 15 3 Unstable trajectory, 2 0 Volume contractin g condition, 16 5
INDEX 36 3 Voronoi region, 34 9 Weak solution, 230, 242, 28 3 Young's inequality, 3 4