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1. 2 1 snag S2 1 2 88 1 1 84 122 85 1 6 ae ah 8 RNI Chapter 3 Inertial Manifolds Contents Basic Equation and Concept of Inertial Manifold Integral Equation for Determination of Inertial Manifold Existence and Properties of Inertial Manifolds Continuous Dependence of Inertial Manifold on Problem Parameters 0 0 0 000 00 ccc eee eee eee Examples and Discussion 0 0 000000 e eee eee Approximate Inertial Manifolds for Semilinear Parabolic Equations 4 Inertial Manifold for Second Order in Time Equations Approximate Inertial Manifolds for Second Order in Time Equations 0 00 ce eee Idea of Nonlinear Galerkin Method References ennnen enea eee eet et eee ee Ae If an infinite dimensional dynamical system possesses a global attractor of finite dimension see the definitions in Chapter 1 then there is at least theoretically a possibility to reduce the study of its asymptotic regimes to the investigation of pro perties of a finite dimensional system However as the structure of attractor cannot be described in details for the most interesting cases the constructive investigation of this finite dimensional system cannot be carried out In this respect some ideas related to the method of integral manifolds and to the reduction principle are very useful They have led to appearance and intensive use of the concep
2. 9 Idea of Nonlinear Galerkin Method Approximate inertial manifolds have proved to be applicable to the computational study of the asymptotic behaviour of infinite dimensional dissipative dynamical sys tems for example see the discussion and the references in 8 Their usage leads to the appearance of the so called nonlinear Galerkin method 20 based on the re placement of the original problem by its approximate inertial form In this section we discuss the main features of this method using the following example of a second or der in time equation of type 8 1 d2u di du dt 7 Gt Au B u uj Ug 9 1 0 t 0 If all conditions on A and B given in the previous section are fulfilled then Theorem 8 2 is valid It guarantees the existence of a family of mappings hys ly from PH x PH into QH possessing the properties 1 there exist constants M M n p and Lj L n p j 1 2 such that Ar 2 Bo lt M a ln Po poll lt Mge 9 2 A 2 P1 H1 AaB H lt L A ps 42 8 a 9 3 A 1p p1 P1 ln P2 be S L A P3 AY By 2 9 4 for all Pj and Dj from PH such that Mo M s j 0 1 p gt o 2 for any solution u t to problem 9 1 which lies in L p for m 2 the es timate 209 210 Inertial Manifolds C h 1 2 2 2 n 2 40 uP Jul BOPP lt Cet 0 t is valid see Theorem 8 2 for all n lt m 1 and large enough Here u t P t plp
3. Here u t u t uo B uy uo u 9 The linear operator A and the mapping B U t are defined by the equations AU u Augt 2 u D A D A x D Al 7 6 BUY t 0 B ug t U ug 4 Exercise 7 1 Prove that the eigenvalues and eigenvectors of the operator A have the form AX e e u JZ e AZ en m 1 2 TT where u and e are the eigenvalues and eigenvectors of A Exercise 7 2 Display graphically the spectrum of the operator A on the complex plane These exercises show that although problem 7 1 can be represented in the form 7 5 which is formally identical to 0 1 we cannot use Theorem 3 1 here Neverthe less after a small modification the reasoning of Sections 2 4 enables us to prove the existence of IM for problem 7 1 Such a modification based on an idea from 16 is given below First of all we prove the solvability of problem 7 1 Let us first consider the li near problem d2u du 2 Au h ae ee Au t t gt s 7 8 du _ U s Uo a t s These equations can also be rewritten in the form cf 7 5 d qu AU t H t U 7 9 where U t u t w t and H t 0 h t We define a mild solution to problem 7 8 or 7 9 on the segment s s T asa function w t from the class Ls r C s s T Kja NCl s s T H NC s s T Fijo which satisfies equations 7 8 Here Fg D A as before see Chapter 2 One can prove the existence and uniq
4. is large enough C and O are positive constants Exercise 6 4 Consider the abstract form of the two dimensional system of the Navier Stokes equations du a t VAut blu u f t Ul _9 Uo 6 21 see Example 3 5 and Exercises 4 10 and 4 11 of Chapter 2 Assume that A 2 t lt C for t gt 0 Use the dissipativity property for 6 21 and the formula k Co k lt Ne lt Cy k for the eigenvalues of the operator A to show that there exists a col lection of functions p t t 1 from PD A into 1 P D A possessing the properties 187 188 Hoaede FQ Se Inertial Manifolds a A p t lt ca N12 A t PW t lt 2 A P2 for any p P Py PD A b for any solution w t e D A to problem 6 21 there exists t gt 1 such that AQ u t Pu 2 i lt lt c3 exp O N 2 t t c exp o N17 Here P is the orthoprojector onto the first N eigenelements of the operator A Exercise 6 5 Use Theorem 6 1 to construct approximate inertial manifolds for a the nonlocal Burgers equation b the Cahn Hilliard equa tion and c the system of reaction diffusion equations see Sec tions 3 and 4 of Chapter 2 In conclusion of the section we note see 8 9 that in the autonomous case the ap proximate IM can also be built using the equation p Ap PB p p AB p QB p p 6 22 Here pe PH Q I1 P p is the Frech t derivative and D p
5. w is its value at the point p on the element w At least formally equation 6 22 can be ob tained if we substitute the pair p t B p t into equation 6 3 The second of equations 6 3 implicitly contains a small parameter Aw 1 Therefore using 6 22 we can suggest an iteration process of calculation of the sequence giving the approximate IM AQ p QB p Py wW Do P Ap PB p 0 P k21 6 23 where the integers v k are such that 0 lt v k lt k l Ms es i 2 3 One should also choose the zeroth approximation and concretely define the form of the values v k for example we can take v 0 and v k k 1 i 1 2 3 When constructing a sequence of approximate IMs one has to solve only a linear stationary problem on each step From the point of view of concrete calculations this gives certain advantages in comparison with the construction used in Theorem 6 1 However these manifolds have the power order of approximation only for detailed discussion of this construction and for proofs see 9 Exercise 6 6 Prove that the mapping v has the form 6 4 under the condition v 0 Write down the equation for v when V 2 1 V9 2 vg 2 0 Inertial Manifold for Second Order in Time Equations 7 Inertial Manifold for Second Order in Time Equations The approach to the construction of IM given in Sections 2 4 is essentially based on the fact that the system has form 0 1 w
6. Po po t Exercise 1 4 Use the results of Exercises 1 2 and 1 3 to show that if IM M exists then it is strictly invariant i e for any u e M and s lt t there exists ug M such that u w t is a solution to prob lem 1 1 Basic Equation and Concept of Inertial Manifold In the sections to follow the construction of IM is based on a version of the Lyapunov Perron method presented in the paper by Chow Lu 2 This method is based on the following simple fact Lemma 1 1 Let f t be a continuous function on R with the values in H such that IONO lt C teR Then for the mild solution u t Con the whole axis to equation u Au f t 1 11 to be bounded in the subspace Qy Fo it is necessary and sufficient that t t u t e s Ap tsaar a4 0 5 nde 1 12 S 00 for t eR where p is an element from Py and s is an arbitrary real number We note that the solution to problem 1 11 on the whole axis is a function u t e C R H satisfying the equation t u t tS ws e DA F t dt Ss foranys e R Proof It is easy to prove do it yourself that equation 1 12 gives a mild solution to 1 11 with the required property of boundedness Vice versa let u t be a solution to equation 1 11 such that wu lo is bounded Then the func tion q t Qy u t is a bounded solution to equation Sf a t Aa t Quilt Consequently Lemma 2 1 2 implies that t q t e DA Q KH a ar KD Therefore in ord
7. ees Ue 0 5 Year 9 Ul o Mol Here v gt 0 and f x u t isa continuous function of its variables such that for all x e 0 1 t 2 0 and l fire 0 0 ade lt L2 0 where L j are nonnegative numbers As in Example 5 1 we assume that d2 ah _ p 2 n ou A vAZ D A H 0 1 NH 0 1 Blu t sla uw 2 It is evident that Ou OU B t Blua O lt Lifer ual Lely ae Here is the norm in L 0 1 By using the obvious inequality dul2 T 92 1 jj gt ZS hd we Hho 0 we find that 1 l Blu t B uy t lt t Lg 42n u Hence conditions 1 2 and 1 3 are fulfilled with _1 E 0 5 M max dts a zli ths l 178 Hoaede FQ Se Inertial Manifolds Therewith spectral gap condition 2 3 acquires the form 2M JV FQN 1 SF 2N 1 k N 1 where k 5 jemete 2 o Thus the equation m N 1 1Qvv 2 2N 1 21M T m 1l Tay a fF NI IM must be valid for some 0 lt q lt 2 2 We can ensure the fulfilment of this con dition only in the case when rag eo 2 fZ lt m gt 0 2 82 1 or Le if eae Jv J2 1 M mab 4 41 14 lt ant Teak 5 9 Thus in order to apply the above presented theorems to the construction of the inertial manifold for problem 5 6 one should pose some additional conditions see 5 7 and 5 9 on the nonlinear term f x u u x t or require that the diffusion coefficient v
8. holds and m gt N 1 Show that problem 4 7 possesses an invariant manifold of the form M p p s p PH in P H where the function pm p s PH gt P P H is de fined by equation similar to 2 12 174 Roepe SQ Se Inertial Manifolds The following assertion holds Theorem 4 2 Assume that spectral gap condition 3 1 holds Let D p s and O p s be the functions defined by the formulae of the type 2 12 and let these functions give invariant manifolds for problems 1 1 and 4 7 for m gt N 1 respectively Then the estimate C q M 0 Pit Apl 4 p s 00 p s lt 2 7 4 8 Am 1 1 2N 1 Am 1 is valid where the constant D is defined by formula 2 6 Proof It is evident that O p s O p s Q v s p v s p 4 9 where v t p and vl t p are solutions to the integral equations v t Be v w lt t lt s and vn t BaBy v 6 o lt i lt s Here By is defined as in 2 1 Since v t v t I Pao t Pal Bp v Be v 0 we have PEH 0 OI PAF oO AB Te Bp AO The contractiveness property of the operator By leads to the equation 4 v 0 v t Ja Bo alo ol ert In particular this implies that lv v sup els D Aot vm lt a oa Hence with the help of 4 9 we find that A p s P p s lt o s v s lt lv v I lt 4 10 sE ey Continuous Dependence of Ine
9. i 1 i 1 ANA B1 p D Wp s Wai r b lt a B lt Ca pr Med Pi 8 12 i 1 t 1 are valid for all p and from PH such that Ap lt R and A 2p lt R Hereinafter D Bf is the mixed Frech t derivative of the function f of the order with respect to p and of the order B with respect to the values w and wj are from PH Moreover if a 0 or P 0 then the correspon ding products in 8 11 and 8 12 should be omitted Proof We use induction with respect to n It follows from 8 10 and 8 2 that esti mates 8 11 and 8 12 are valid for n 0 1 Assume that 8 11 and 8 12 hold for all n lt k 1 Then the following lemma holds Lemma 8 1 Let F p p B p h p p and let Fo B w D BF p b Wp c Wg p oes wp Then for v lt k 1 and for all integers a B 20 such that a B lt m the estimate a B r gt Pw lt cy pog Mod 8 13 i l i l holds where w j p pePH and Ap lt R A 2p lt R Proof It is evident that F p w is the sum of terms of the type By y BO p h p B s Yp gt YJ s20 Here y is one of the values of the form Y Wet O hyi Wo Ya DETh a Wis s Wo Wi gt We 203 204 Hoaedpe FQ Se Inertial Manifolds Equation 8 2 implies that S BRO lt Ce Jle j 1 Therefore the induction hypothesis gives us 8 13 Let us prove 8 12 The induction hypothesis implies that it is sufficient to estimate the derivatives of
10. t lt M In the general case equation 6 10 is a variant of the dissipativity property Exercise 6 3 Let He Vy eC s be a function from C 9 s L s Assume that i L ii vp Vp slt s Bp vn 1 0 Waa eax and Di 0 s QU s P w OI Bias Show that the assertions of Theorem 6 1 remain true for the function 4 p s if we add the term Approximate Inertial Manifolds for Semilinear Parabolic Equations q Wol CoP 2 to the right hand sides of equations 6 9 and 6 11 Here q is de fined by equality 6 5 and ols is the norm of the function vo in the space C g s L s Therefore the function D p s generates a collection of approximate inertial manifolds of the exponential with respect to N order for n large enough Example 6 1 Let us consider the nonlinear heat equation in a bounded domain Q c R2 OU Aut f u Vu EQ t gt s ot 6 20 Uso 0 ul _ Ug 2 Assume that the function f w possesses the properties f u4 1 fue Sa lt Cy uy U9 Sy If u E lt C3 We use Theorem 6 1 and the asymptotic formula Ay CyN2 4 N gt for the eigenvalues of the operator A in Q c R to obtain that in the Sobolev space Hy Q for any N there exists a finite dimensional Lipschitzian surface My of the dimension N such that dist ig aU My lt C exp o N 4 t t Cyexp o N14 for t t and for any mild in H4 Q solution u t to problem 6 20 Here t
11. t pp 9 6 T t 0 p t l P t p t An 41 isthe N 1 th eigenvalue of A and R is the constant from 8 4 The family hy l is defined with the help of a quite simple procedure see 8 8 and 8 9 which can be reduced to the process of solving of stationary equations of the type Av g in the subspace QH Moreover holy p 0 hyp A1QB p 1 p 0 0 O7 In particular estimates 9 5 and 9 6 mean see Corollary 8 1 that trajectories U t u t 6 u t of system 9 1 are attracted by a small for N large enough vicinity of the manifold M p h p p l p P p e PH 9 8 The sequence of mappings h p D generates a family of approximate inertial forms of problem 9 1 6 p yo p Ap PB p h p p 9 9 A finite dimensional dynamical system in PH which approximates in some sense the original system corresponds to each form For n 0 equation 9 9 transforms into the standard Galerkin approximation of problem 9 1 due to 9 7 If 7 gt 0 then we obtain a class of numerical methods which can be naturally called the non linear Galerkin methods However we cannot use equation 9 9 in the computa tional study directly The point is that first in the calculation of h D D we have to solve a linear equation in the infinite dimensional space QH and second we can lose the dissipativity property Therefore we need additional regularization It can be done as f
12. the operator Be maps C into itself and is contractive aTherstore there exists a unique feed point v alts p Evi dently it possesses properties 2 4 and 2 5 Lemma 2 1 is proved Lemma 2 1 enables us to define a collection of manifolds Mh by the formula M p 4 p s p e PH where O p s e D4 QB v t t dt Qv s p 2 12 s L Here v t v t p is the solution to integral equation 2 1 Some properties of the manifolds Mz and the function O p s are given in the following assertion Theorem 2 1 Assume that at least one of two conditions 2 2 and 2 3 is satisfied Then the mapping o S from PH into QH possesses the properties a A O p s lt Dy a 1 471D lA pl 2 13 Jor any p PH hereinafter D is defined by formula 2 6 and Dy M 1 k Ay 2 14 b the manifold ML is a Lipschitzian surface and A 4 p s po s lt 5 48 p p 2 15 forall p p PH and s e R c f u t u t s p 4 p is the solution to problem 1 1 with the initial data ug p p s pe PH then Qu t Pu t t for L Incase of L lt the inequality P Qu t Pu t tJ lt lt Da 1 4 e q 1 4 26M SD Apl 2 16 Integral Equation for Determination of Inertial Manifold holds for alls lt t lt s L where y is an arbitrary number from the segment Ay Any 1 Y 2 2 is fulfilled and y y 2M q ak when 2 3 is fulfilled d f B u t B
13. 0 q u 2 a9 p 8 p lt 1 6 5 In this case equations 2 2 hold Hence Lemma 2 1 enables us to construct a collec tion of manifolds M4 for L p RA with the help of the formula ME p p s p e PH 6 6 where s L p s f e S DAQB v t t dt Qu s p 6 7 s L Here v t v t p is a solution to integral equation 2 1 and L paR 183 184 roevpe FQ Se Inertial Manifolds Exercise 6 1 Show that both the function 4 p s and the surface M do not depend on s in the autonomous case B u t B The following assertion is valid Theorem 6 1 There exist positive numbers p p M O A and A A M 9 such that if 0 or L pin i 0 lt PEP 6 8 then the mappings L s PH gt QH defined by equation 6 7 possess the property A Qu t B4 Pu t t lt o lt ow exp ee Ge ae c exp l 5 WA 6 9 for all t gt t L 2 Here gt 0 is an absolute constant and u t is a mild solution to problem 1 1 such that A u t lt R for teft 6 10 If B u t lt M then estimate 6 9 can be rewritten as follows A Qu t amp 4 Pu t 1 lt O z lt Cpexp F Mal bd exp P N 1 6 11 where D is defined by equality 2 14 Proof Let G t U t s Pu s 4 Pu s s t lt s lt t where U t s v is a mild solution to problem 1 1 with the initial condition v e D A at the moment s Therewith w t U t 0 wo It is evident th
14. 1 p gt 0 k gt 7 44 N UN 1 for some sequence N k which tends to infinity and satisfies the estimate 8 2 y p8 Hn 17 HN k 2 a Mi HN k 1 gt 0 lt q lt 2 42 Then there exists amp gt 0 such that the assertions of Theorem 7 1 hold for all E 2 p Proof Equation 7 44 implies that there exists kg such that the intervals 2 Hng 2 Ung 1 tbls K gt ko cover some semiaxis E0 0 Indeed otherwise there would appear a subse quence N k such that HNK S 2 UN k 1 1 i HN 1 But that is impossible due to 7 44 Consequently for any 9 there exists N N such that equations 7 42 7 43 as well as 7 39 hold Exercise 7 18 Consider problem 7 3 with the function f x t u Guy f x t u possessing the property 2 t U f x t Uy lt Llu Us Use Theorem 7 1 to find a domain in the plane of the parameters L for which one can guarantee the existence of an inertial ma nifold 199 200 Hoaedvpe FQ Se Inertial Manifolds 8 Approximate Inertial Manifolds for Second Order in Time Equations As seen from the results of Section 7 in order to guarantee the existence of IM for a problem of the type d u du _ F y tAu B u 8 1 Ul 9 Yo m bleh we have to require that the parameter y 2 gt 0 be large enough and the spectral gap condition see 7 41 be valid for the operator A Therefore as in the case with parabolic eq
15. 6 2 In this case M is a finite dimensional subspace in F whereas inertial form 6 1 turns into the standard Galerkin approximation of problem 0 1 corresponding to this subspace One can find the simplest non trivial approximation M using for mula 6 2 and assuming that O p t p t A 1 1 P B p t 6 4 The consideration of system 0 1 on M leads to the second equation of equa tions 6 3 being replaced by the equality Aq 1 P B p t The results of the computer simulation see the references in survey 8 show that the use of just the first approximation to IM has a number of advantages in comparison with the tradi tional Galerkin method some peculiarities of the qualitative behaviour of the system can be observed for a smaller number of modes There exist several methods of the construction of an approximate IM We present the approach based on Lemma 2 1 which enables us to construct an approximate IM of the exponential order i e the surfaces in the phase space H such that their expo nentially small with respect to the parameter y 1 Vicinities uniformly attract all the trajectories of the system For the first time this approach was used in paper 15 for a class of stochastic equations in the Hilbert space Here we give its deterministic version Let us consider the integral equation see 2 1 v t B o s Lstss and assume that L p IRA where the parameter p possesses the property
16. FQ Se Inertial Manifolds s Cr Pm Pm ul a Dn Pm u ess sup Pm Pm RO t e s s T holds for all t s s T where T gt 0 is an arbitrary number Exercise 7 7 Using the results of Exercises 7 5 and 7 6 show that we can pass to the limit n in equations 7 11 and prove the existence and uniqueness of mild solutions to problem 7 8 on every segment s s T under the condition 7 12 Exercise 7 8 For a mild solution u t to problem 7 8 prove the energy equation 1 la 14 2u e P 2ef Jit Pat Ss pap u faa ia aawa 1 2 n rs In particular the exercises above show that for h t 0 problem 7 8 generates a linear evolutionary semigroup eA in the space J6 D A 2 x H by the formula e A ug u u t t 7 15 where u t is a mild solution to problem 7 8 for h t 0 Equation 7 14 implies that the semigroup e A is contractive for gt 0 Exercise 79 Assume that conditions 7 12 are fulfilled Show that the mild solution to problem 7 8 can be presented in the form t t u t e ug my fet 9A 0 h T dt 7 16 S tA is defined by equation 7 15 where the semigroup e Let us now consider nonlinear problem 7 1 and define its mild solution as a function U t u t u t C s s T J satisfying the integral equation t U t e8 U ABU t dt 7 17 S on s s T Here B U t t 0 B u t t and Uy ug u1 Iner
17. by formula 2 14 Using Lemma 2 1 under condition 2 3 with q instead of q we obtain that Pols a 0 D lAl ri t lt D and ry lt e where D is given by formula 2 6 Therefore finaly we have that ae ts tLy A Ls L A b Va t s lt e AN Dat oa 1 D A pll for all t e s L s Consequently sup ere 0 i s t e s L s lt yL i YL lt e Hops Tera e1 D Pb Existence and Properties of Inertial Manifolds 169 Dite K i K Therefore since B 1 is a contractive operator in C g s Lj s equation 3 25 gives us that Q a aa AG L s s 1 l q JUSTE 8 lt ep ys pie r Dy AP Here we also use the equality q 30 q Hence estimate 3 23 follows from 8 24 Theorem 3 3 is proved Exercise 3 5 Show that in the case when B w t lt M equation 8 23 can be replaced by the inequality 1 7 yl A p s 2 p s lt Da 1 a eN mn Exercise 3 6 Assume that the hypotheses of Theorem 3 1 hold Then the estimate A Qu t 4 Pu t D lt z a Auo owl 40 TOINI holds for t gt t and for any solution w t to problem 1 1 possess ing the dissipativity property A9u t ay lt R fort gt t 2s and for some R and t Here yy Ay Ajy and the constant a gt 0 does not depend on N Therefore if the hypotheses of Theorem 3 1 hold then a hounded solution to prob lem 1 1 gets into the exponentially small with respect to 19 and L vicinity
18. corresponding sub space KH Using 7 18 we define a new inner product and a norm in by the equalities U V U Vi Uo Vag U U gy2 where U U U and V V V are decompositions of the elements U and V into the orthogonal terms V U e 9 i 1 2 194 Hoaedpe FQ Se Inertial Manifolds Lemma 7 1 The estimates U 2 gt a fe uyu U ug u H 7 19 N lUl gt KA Sy s 4uol U uo uy 7 20 N 1 hold for 0 lt O lt 1 2 Here Sy Uy41 min 1 E Hno 7 21 z i HN 1 Proof Let U ug u 6 It is evident that in this case ABuo lt u Juo for any B gt 0 Therefore IU gt eju Aud gt iR E uyu i e equation 7 19 holds Let U 6 Then using the inequality Abu gt u Jeol B gt 0 uoe Linfe k N 1 7 22 for 0 lt 6 lt 1 we find that ulg gt 52 A Zuo e 1 8 Wa 1 o If we take 6 On 5 figs and use 7 22 then we obtain estimate 7 20 The lemma is proved In particular this lemma implies the estimate al Auo lt ue 41 Oy elUl 7 23 for any U ug u1 where 0 lt O lt 1 2 and dy has the form 7 21 Exercise 7 13 Prove the equivalence of the norm and the norm generated by the inner product 7 4 Exercise 7 14 Show that we can take N e je Uy41 for 0 0 in 7 20 and 7 23 Exercise 7 15 Prove that the eigenvectors f of the operator A see 7 7
19. d 1 a number of reaction diffusion equations the Swift Hohenberg equa tion and a non local version of the Burgers equation The corresponding references and an extended list of equations can be found in survey 8 In conclusion of this section we give one more interesting application of the theorem on the existence of an inertial manifold 179 180 Hoaedpe FQ Se Inertial Manifolds Example 5 3 Let us consider the system of reaction diffusion equations OU _ ou a vAu f u Vu an 0 5 11 ina bounded domain Q c R Here u Uj gt Um and the function f u satisfies the global Lipschitz condition If u 6 F v nl lt L lw ol a YP 6 12 where u v e R n e R and L gt 0 We also assume that f 0 0 lt L Problem 5 11 can be rewritten in the form 0 1 in the space H L Q if we suppose Au vAut u B u u f u Vu It is clear that the operator A is positive in its natural domain and it has a dis crete spectrum Equation 5 12 implies that the relation 1 2 B u B v lt kiu v v oye Ju vl lt 1 2 lt f bia 4 le v v V u oF is valid for B w Thus B u B v lt MIA u v where 1 M L Lax 4 Therefore problem 5 11 generates an evolutionary semigroup S see Chap ter 2 in the space D A 2 An important property of S is the following the subspace L which consists of constant vectors is invariant with respect to this s
20. e M t s such that A9 u t ae lt Cent gt 00 t gt s In this case the solution t is said to be an induced trajectory for u t onthe manifold M In particular the existence of induced trajectories means that the so lution to original infinite dimensional problem 1 1 can be naturally associated with the solution to the system of ordinary differential equations 1 10 161 162 Hoaede FQ Se Inertial Manifolds Theorem 3 1 Assume that spectral gap condition 3 1 is valid for some q lt 2 ae Then the manifold M s e R given by formula 3 2 is inertial for prob lem 1 1 Moreover for any solution u t U t s ug there exists an in duced trajectory u t U t s ug such that u t M for t s and lu w t lt OCA CAA Oue where y hy EMAR and t 2s 3 4 Proof Obviously it is sufficient just to prove the existence of an induced trajectory u t e M possessing property 3 4 Let u t be a mild solution to problem 1 1 u t U t s ug We construct the induced trajectory u t U t s up for w t in the form u t u t w t where w t lies in the space Cf C s 00 D A of continuous functions on the semiaxis s 00 such that lw sup fet s Aw t lt 3 5 t2s where y y 2M q a9 We introduce the notation F w t B u t w t B u t 3 6 and consider the integral equation cf 1 15 t w t BY w t q w aoe QF w t
21. ip s 0 p s lt l q L D A pl e Nmn 3 23 lt D 1 e YN Tmin is valid with Lmin min L1 La 0 lt Ly Lo lt the constants D and D are defined by equations 2 6 and 2 14 2M 40 _ 2M 1 4 0 Yn Ant FAN wede N Proof Let 0 lt L lt La lt Definition 2 12 implies that Dp s p s Qw s v3 s 3 24 where v t is a solution to integral equation 2 1 with L Lj j 1 2 The ope rator B La acting in Cy o S L s see 2 1 can be represented in the form By vu t By v b v t s te s L s where v l amp 168 Hoaedpe FQ Se Inertial Manifolds and v t is an arbitrary element in C 9 s Lo Therefore if v t is a solution to problem 2 1 with L L then s L s L v t v t Bp v B vo bv t s 3 25 for all s L lt t lt s Let us estimate the value b vs t s As before it is easy to verify that 4 va t s Sg Vo Ea r s L1 1 s Ly Pals pa for allt e s L s where t r t M let A Qldc t r t ul APe t 1A Qh oS Dar 3 and the norm val _ is defined using the constants q pied and y Ay 2M 19 by the formula 2 Wols SUP eM aost te s Lo sj Evidently spectral gap condition 8 1 implies the same equation with the parameter q instead of q Therefore simple calculations based on 1 8 give us that y s t q 2 T where D is defined
22. p D ph p ph yp Ap PB p h p p Hereinafter p J and pf are the Frech t derivatives of the function f p p with respect to p and Oe f w and E Jf w are values of the corresponding deri vatives on an element w Using these formal equations we can suggest the following iteration process to determine the class of functions hy l giving the sequence of approximate IMs with the help of 8 7 Ah p QB p hy _1 p D Vy Uy P z Op lena pe 8 5b 13 Yp Ap PB pth _ p D 8 8 where k 1 2 3 and the integers v k should be choosen such that k 1 lt lt v k lt k Here l p p is defined by the formula lp p D Oy 13 D 6 213 y SAD PB p t hy_1 D 8 9 where k 1 2 3 We also assume that holp lop 0 8 10 Exercise 8 amp 7 Find the form of h p p and l p for v 1 0 and for v 1 1 The following assertion contains information on the smoothness properties of the functions h and l which will be necessary further Theorem 8 1 Assume that the class of functions Migs Let is defined according to 8 8 8 10 Then for each n the functions h and l belong to the class C as mappings from PH x PH into QH and for all integers a P 0 such that a P lt m the estimates Approximate Inertial Manifolds for Second Order in Time Equations ACD Bh Dp D Wy o Wes Wiss p lt a b lt Ca pr Med Wod 811
23. p t Therefore equation 2 12 implies the invariance property Qu to Pu ty to Let us estimate the value 2 19 If we reason in the same way as in the proof of Lemma 2 1 then we obtain that t t L ar y s t t L A bz to s t lt a Oto ws ai toL q s tg L e pa seh where q s t is defined by formula 2 7 and A 0 p q s t M f FA A i e Natar 2 20 s L Therefore simple calculations give us that t t L xr tyotL A b to s o lt eT Ann o s 0 abe 2 21 where D is defined by formula 2 14 Let vy g t be the solution to integral equa tion 2 1 for s t and p Pu t Then using 2 12 2 18 and 2 1 we find that Qu to B Pu to to Q w to va to 2 22 However for all t ty L t we have that by L ty L w t v t Blo BE Lee by lto 83 t ty L Therefore the contractibility property of the operator Bp gives us that i u me te lg oD a aCe e al Hence it follows from 2 21 and 2 22 that A Qu to D4 Pu to to lt A wto v Cto 5 q le Un lt nay eteo ae This and equation 2 5 imply 2 16 Therefore assertion c is proved In order to prove assertion d it should be kept in mind that if B u t wu then the structure of the operator B L enables us to state that sd s h L By v 7 B on for s h L lt t lt s h where v t v t h Therefore if v t C 9 s L s is a so
24. see Section 3 of Chapter 2 Therewith d2 dx where H5 0 l is the Sobolev space of the order s The mapping B t given by the formula u x gt f x u x t satisfies conditions 1 2 and 1 3 with 0 0 In this case spectral gap condition 2 3 has the form A v D A H 0 L N H2 0 1 T2 2 4M vig N 1 N gt a Thus problem 5 1 5 3 possesses an inertial manifold of the dimension N provided that bo 2 42M an N gt 5 4 Dik bo for some q lt 2 2 Exercise 5 1 Find the conditions under which the inertial manifold of prob lem 5 1 5 3 is one dimensional What is the structure of the cor responding inertial form Exercise 5 2 Consider problem 5 1 and 5 3 with the Neumann bounda ry conditions ou Ox _ u a 0 5 5 ka Show that problem 5 1 5 3 and 5 5 has an inertial manifold of the dimension N 1 provided condition 5 4 holds for some x l Examples and Discussion 177 N gt 0 Hint A v d dx with condition 5 5 B u t gu f x u t where gt 0 is small enough Exercise 5 39 Find the conditions on the parameters of problem 5 1 5 8 and 5 5 under which there exists a one dimensional inertial mani fold Show that if f x u t f u t then the corresponding iner tial form is of the type p t f 1 Ply Po Example 5 2 Consider the problem Ou _ 8u Ou Be age oe O lt x lt l t gt 0
25. the second term in the right hand side of 8 9 It has the form phe 13 D p p 8 14 where D p P yp Ap PB pt hy_1 p P The Frech t derivatives of value 8 14 D BS hy 13 Dyp Do Wy vs Was Wy os Wp are sums of the terms of the type G o T Dott hy ap Wii sag We Ww 1 Yo where a O p T aye ra 7 Yo r D B D p Wop Wag gy Ppp gt Won Here 0 lt o lt a 0 lt T lt f and the sets of indices possess the following proper ties OF saha Jes N Q Bera Ooh Fisted oh Oya steg Ogee TE args we ES i N P ad Pp fib eee i U Pis Pass 1 2s irea B The induction hypothesis implies that AG o tT lt Th Ab aaa A J4 2y d Using the induction hypothesis again as well as Lemma 8 1 and the inequality AM Pr lt Ay IPhl we obtain an estimate of the following form if o amp or T then the correspon ding product should be considered to be equal to 1 a o B t ave d lt COHN T T Po TI od 0 1 O 1 Approximate Inertial Manifolds for Second Order in Time Equations Hereinafter A is the k th eigenvalue of the operator A Thus it is possible to state that AG o tI lt 14a eve ia J 8 15 Using the inequality lOn lt Ay IAQnl s gt 0 8 16 and equation 8 15 it is easy to find that estimates 8 12 are valid for n k If we use 8 8 8 12 and follow a similar line of reasoning we can easily obtain 8 11 Theorem 8 1
26. u does not depend on t then DB p s B p ie O p t is independent of t Proof Equations 2 12 and 1 8 imply that s 9 Be A D p s lt M f S A le Anat 91 40v z f at lt s L S 0 EE lt M 55 taka aE Parta 9s s L By virtue of 2 9 we have that q s s lt q Therefore when we change the vari able in integration Ay s T with the help of equation 2 5 we obtain 2 13 Similarly using 2 4 and 1 8 one can prove property 2 15 Let us prove assertion c We fix t s s L and assume that w t is a function on the segment s s L such that w t u t for t s t and w t v t fort e s L s Here v t is the solution to integral equation 2 1 Using equations 1 4 and 2 1 we obtain that t w t e A p 4 OL y s fet 1A B w t t dt S t t o 5 Ap 4 je A PB w t t dt e DAQB w t dt 2 17 S s L for s lt t lt t Evidently equation 2 17 also remains true for t s L s Equa tion 1 4 gives us that S s tp A fe p e t plio fe s 9A PB w t t dt to Therefore the substitution in 2 17 gives us that ty w t Bpi W t bz to s t 2 18 forall t t L to where p t Pu t and t L blige s t f e DA QB v 1 T dt 2 19 s L 159 160 Hoaede FQ Se Inertial Manifolds In particular if L equation 2 18 turns into equation 2 1 with s t and p
27. y is a positive number A is a positive selfadjoint operator with discrete spectrum and B is a nonlinear mapping from the domain D A of the operator A12 into H such that for some integer m gt 2 the function B w lies in C asa mapping from D A into H and for every p gt 0 the following estimates hold 8 2 k KBO u w wp lt Co Mey j 1 Approximate Inertial Manifolds for Second Order in Time Equations 201 BO u B ur wy w lt CLAY wj 8 3 1 where k 0 1 is a norm in the space H ie lt p Al 2 wl lt P and wj D A zy ee B u is the Frech t derivative of the order k of B w and BU w W 1 Wz is its value on the elements wy Wy Let Lim pR be a class of solutions to problem 8 1 possessing the following properties of regularity D fork 0 1 m 1 and forall T gt 0 ulk t e C 0 T D A and ul t e C 0 T D AM2 ul Dt e C 0 T H where C 0 T V is the space of strongly continuous functions on 0 T with the values in V hereinafter w t ak u t II for any u e Lm p the estimate lutt D AM 2 ul a Aut DA lt R2 8 4 holds for k 1 m and for t gt t where t depends on ug and wy only In fact the classes A _R are studied in 18 This paper contains necessary and sufficient conditions which guarantee that a solution belongs to a class Lm R It should be noted that in 18 the nonline
28. 1 lt lt e1 9 Mv v pall P1 4 4 for T lt s where lw ess sup fere oe Law 4 5 i lt s and y y EMN as before Hence after simple calculations as in Section 2 we find that A p s p s lt 1b o P2 wil 3 P a 4 6 N 1 Let us estimate the value v v Since v and v are fixed points of the correspon ding operator B we have that Continuous Dependence of Inertial Manifold on Problem Parameters 173 A u t v lt le 4 Pl lao t B v t that f A e 94 QI IB lT 1 B 0 1 Dhar Therefore by using spectral gap condition 3 1 and estimate 4 4 as above it is easy to find that f a _ P2 1 l k lea lt qw v ls lel P t i N N 1 Consequently q P2 Pi 2 k Therefore equation 4 6 implies that 0 g a P2 2 4 2 kk Hence estimate 2 5 gives us the inequality Pit Po 2 k q P2 A p s p s lt Apl 7 2 41 6 _ a M lag ae 2 1 This implies the assertion of Theorem 4 1 Let us now consider the Galerkin approximations Uy t of problem 1 1 We re mind see Chapter 2 that the Galerkin approximation of the order m is defined as a function w with the values in PH this function being a solution to the problem du ap tA Ym EnB Um um _ 5 Uom 4 7 Here P is the orthoprojector onto the span of elements fej wats Cm in H Exercise 4 7 Assume that spectral gap condition 3 1
29. 20 where the constant y gt 0 does not depend on u t and is the Sobolev norm of the first order Exercise 5 7 Consider the problem ou vow f x u VRB Rs uj og Ul 0 5 16 where the function f x u has the form f x u g Uj Ug sine ga uj Ug sin22 Here T 2 cat uj z u a sinjarde j l 2 0 and g u Ug are continuous functions such that gU Uz gWr Va lt 1 2 lt L e o Jus v g 0 0 0 Show that if 2 2 2 we DA 181 182 Hoaedvpe FQ Se Inertial Manifolds then the dynamical system generated by problem 5 16 has the two dimensional flat inertial manifold M p sing p sin2x py Pz R and the corresponding inertial form is Pit VP MPi Po Bgt4VPD Go Py Po Exercise 5 8 Study the question on the existence of an inertial manifold for the Hopf model of turbulence appearance see Section 7 of Chap ter 2 6 Approximate Inertial Manifolds for Semilinear Parabolic Equations Even in the cases when the existence of IM can be proved the question concerning the effective use of the inertial form 0p Ap PB p p t t 6 1 is not simple The fact is that it is not practically possible to find a more or less ex plicit solution to the integral equation for D p t even in the finite dimensional case In this connection we face the problem of approximate or asymptotic construc tion of an invariant inertial manifold Various aspe
30. 8 23 s 0 and j 1 2 d JA O81 PEV lt Cay gt PESO 8 24 s 0 are valid for t large enough and for each v 2 0 where j 0 1 m 1 Proof Let f h or f l It is clear that the value Rea ee PE is the algebraic sum of terms of the form BHF pY Ya pe o s D PHF p ren DOS D T wg DD P 0 PE Therefore Theorem 8 1 and Lemma 8 3 imply 8 23 and 8 24 Lemma 8 4 is proved We use Lemmata 8 3 and 8 4 as well as inequality 8 16 to obtain that j 2 j l Axo Ss oria DMs axel al s 0 s 0 j dy Ani EROE 8 25 s 0 where j 0 1 m 2 and the numbers c F and dy 5 do not depend on N 207 208 Hoaede FQ Se Inertial Manifolds If we now assume that 8 19 holds for n lt k 1 then equation 8 25 implies 8 19 for n k and for k lt m 1 Using 8 22 and 8 23 we obtain equation 8 20 Theorem 8 2 is proved Corollary 8 1 Let the manifold M have the form 8 7 with h p pb h p p and l p 1 p D We also assume that U t u t u t where u t is the solution to problem 0 1 from the class Lm p Then dist ane acne OO M S Cars WSO ame Thus the thickness of the layer that attracts the trajectories in the phase space has the power order with respect to y 4 unlike the semilinear parabolic equations of Section 6 Example amp 1 Let us consider the nonlinear wave equation 8 5 Let d dimQ lt 2 We as sume the fol
31. Then the function p s given by equality 7 38 satisfies the Lip schitz condition q P p s po s lt NE P17 Pol 7 40 and the manifold M is invariant with respect to the evolutionary opera tor S t T generated by the formula S t T Ug u t w t ts in H where u t is a solution to problem 7 1 with the initial condition Uy ug u1 Moreover if 0 lt q lt 2 42 then there exist initial conditions Uj ug u M such that S t 8 Up S t s UQ lt C e 7 9 QU B PU s for t gt s where y 5 Ay n The proof of the theorem is based on Lemma 7 4 and estimates 7 29 and 7 30 It almost entirely repeats the corresponding reasonings in Sections 2 and 3 We give the reader an oppotunity to recover the details of the reasonings as an exercise Inertial Manifold for Second Order in Time Equations Let us analyse condition 7 39 Equation 7 31 implies that 7 39 holds if UN 1 7 Un 4 0 1 2 Payas eee Om una 7 41 However if we assume that Cape Piet Hya Vines 7 42 then for condition 7 41 to be fulfilled it is sufficient to require that 2Uy4 ETD a Pl 7 43 Thus if for some N conditions 7 42 and 7 48 hold then the assertions of Theo rem 7 1 are valid for system 7 1 This enables us to formulate the assertion on the existence of IM as follows Theorem 7 2 Assume that the eigenvalues Uy of the operator A possess the properties inf EN gt 0 and Un k 1 CokP 1 0
32. a t Then J EPO lt Ce 5 YJ GOW PO i 0 Jor j 0 1 m and for t large enough The main result of this section is the following assertion Theorem 8 2 Let u t be a solution to problem 8 1 lying in Lm p with m gt 2 Assume that h p and l p are defined by 8 8 8 10 Then the es timates JA u t un t lt Ch R AWE 8 19 AY af u t oie erie 8 20 are valid for n lt m 1 and for t large a Here 0 lt j lt sm n l U t and t are defined by 8 18 and hy is the N 1 th eigenva lue of the operator A Proof Let us consider the difference between the solution w t and the trajectory in duced by this solution X t u t u t X t z u t U t s20 where U t and w t are defined by formula 8 18 Since y t q t equation 8 4 implies that JAv2 8 Day Ayo lt C j 0 1 2 m l 821 Approximate Inertial Manifolds for Second Order in Time Equations for t large enough Equations 8 8 8 10 also give us that AXi X Y X t QEo t We use Lemma 8 3 and equation 8 21 to find that AxP lt CARB 7 0 1 m2 for t large enough Therefore equation 8 19 holds for n 0 1 and for t large enough From equations 8 6 8 8 and 8 9 it is easy to find that AX 8X1 YOK ier VO py ey v3 PE v 7 plg 1 PE 1 QE 4 and Xn i Xk 1 phy PE 1 8 22 Lemma 8 4 The estimates j Aa Spry PEV lt CAN D PO
33. and a simple transformation it is easy to obtain the estimate or 0 M A p t l_ lt C s N a exn iy TEJE 9 3 21 Obviously for q lt 2 J2 we have that MAN 2M 40 y IN lt y dys Mag Therefore equations 3 21 and 8 20 imply 8 19 Vice versa we assume that equation 3 19 holds for the solution u t Then B u t lt eST MA u tss 8 22 It is evident that q t eVS 9Qu t is a bounded on s solution to the equation Si A 1 a FO Existence and Properties of Inertial Manifolds 167 where F t exp y s QB u t By virtue of 8 22 the function F t is bounded in QH It is also clear that A A y isa positive operator with discrete spectrum in QH Therefore Lemma 1 1 is applicable It gives t Qu t e A QB u t dt Using the equation for Pu t it is now easy to find that u t Bo u t t lt s where p Pu s and B u is the integral operator similar to the one in 2 1 Hence we have that Qu s Pu s s accoring to definition 2 12 of the function p s D p s Thus Theorem 3 2 is proved The following assertion shows that IM M M can be approximated by the mani folds M L lt with the exponential accuracy see 2 12 Theorem 3 3 Assume that spectral gap condition 3 1 is fulfilled with q lt 1 We also assume that the function L p s is defined by equality 2 12 for 0 lt L lt Then the estimate AE
34. and approximate inertial manifolds for von Karman evolution equations Math Meth in Appl Sci 1994 Vol 17 P 667 680 MARION M TEMAM R Nonlinear Galerkin methods SIAM J Num Anal 1989 Vol 26 P 1139 1157
35. ar wave equation of the type afu y u Au g u f x xLeQ t gt 0 8 5 UpQ 9 Ul o5 4 9 M 2 5 serves as the main example Here y gt 0 f x e C Q and the conditions set on the function g s from C R are such that we can take g w sinu or g u u2Ptl where p 0 1 2 for d dimQ lt 2 and p 0 1 for d 3 In this example the classes Ly Rp are nonempty for all m Other examples will be given in Chapter 4 We fix an integer N and assume P Py to be the projector in H onto the sub space generated by the first N eigenvectors of the operator A Let Q P If we apply the projectors P and Q to equation 8 1 then we obtain the following sys tem of two equations for p t PU t and q t Qu t 62p y0 p Ap PB p q 0 a y0 4 Aq QB p q The reasoning below is formal Its goal is to obtain an iteration scheme for the deter mination of an approximate IM We assume that system 8 6 has an invariant mani fold of the form 8 6 M pt h p b D 1 p D p PH 8 7 202 Hoaede FQ Se Inertial Manifolds in the phase space D A x H Here h and l are smooth mappings from PH x PH into QD A If we substitute q t h p t 0 p t and q t l p t 6 p t in the second equality of 8 6 then we obtain the following equation 6 13 P 0 1 yp Apt PB pt h p p yl p p Ah p p QOB pt h p p The compatibility condition I p t 6 t 3 hG epl gives us that I
36. ariable t with the values in 6 Lemma 7 4 The operator B maps the space C into itself and possesses the pro perties 1 1 4Ky IS IVII lt bla Se re J I 7 33 i N N 1 Ane An and 4K ISM S lt Ko Ivi vl 7 34 N 1 N Proof Let us prove 7 34 Evidently equations 7 29 and 7 30 imply that BO BO lt Ky enO va var j E t Ky owl iy T Ve a far oo Since v t Vala lt eS OV v it is evident that BHO O lt ge Val with Ss _ t E q Ky fen Par E f g OnT T ar t oo Simple calculations show that q lt 4Ky Ay4 Ay Jere Consequently equa tion 7 34 holds Equation 7 33 can be proved similarly Lemma 7 4 is proved 197 198 Roepe SQ Se Inertial Manifolds Thus if for some g lt 1 the condition 7 4Ky Ani 7 N gt aa 7 35 holds then equation 7 32 is uniquely solvable in C and its solution V can be esti mated as follows zi 1 1 IVI lt 1 4 pl M 2 7 36 N N 1 Therefore we can define a collection of manifolds M in the space 6 by the for mula M p p s p PH 7 37 where p s e t A QB V t t dt 7 38 Here V t is a solution to integral equation 7 32 The main result of this section is the following assertion Theorem 7 1 Assume that _ Z 4K e gt uya and ANTAN r 7 39 for some 0 lt q lt 1 where je2 U and Ky is defined by formula 7 31
37. at 186 Hoaedpe FQ Se Inertial Manifolds A tns1 74 t By L L 2 2 0 1 2 After iterations we find that d t lt 2d t 2By L 2 n 0 1 2 6 19 Equation 6 17 also gives us that a t lt 4a t By L E 1 2 lt t lt t L Therefore it follows from 6 19 that d t lt 2 opi t t m2 d t 2 By L 2 for all t gt t L 2 This implies 6 9 and 6 11 if we take y Awad in the equa tion for By L L 2 Thus Theorem 6 1 is proved In particular it should be noted that relations 6 9 and 6 11 also mean that a solu tion to problem 0 1 possessing the property 6 10 reaches the layer of the thick ness Ey C exp Co AN n adjacent to the surface ME given by equation 6 6 for large enough Moreover it is clear that if problem 0 1 is autonomous B u t B u and if it possesses a global attractor then the attractor lies in this layer In the autonomous case ME does not depend on t see Exercise 6 1 These observations give us some information about the position of the attractor in the phase space Sometimes they enable us to establish the so called localization theo rems for the global attractor Exercise 6 2 Let B u lt M Use equations 1 4 and 1 8 to show that t s lua lt amp Pug Ro where Ry M 1 k Aq In particular the result of this exercise means that assumption 6 10 holds for any R gt Rp and for t large enough under the condition B w
38. at Qu t O4 Pu t t Q u t a t oac oL Pii t orea t O Pu t 612 Let us estimate each term in this decomposition Equation 1 6 implies that ILUA 0 lt Gy t s A Qu s O4 Pu s s 6 13 Approximate Inertial Manifolds for Semilinear Parabolic Equations 185 where y t e NAA M 1 kja ARL e2 Using 2 16 we find that A9 Qa s O PA s s lt By L t s 6 14 where By L t D 1 a te V2 q 1 R D e moreover the second term in By L T can be omitted if B w t lt M see Exer cise 2 1 At last equations 2 15 and 1 5 imply that A 4 Pa t t BA Pu t t lt lt ag et 49 Qu s O4 Pu s s 6 10 Thus equations 6 12 6 15 give us the inequality d t lt aylt s d s By L t s 6 16 for t 2 s 2 t where d t A9 Qu t 4 Pu t O and a q Got a ow a M 1 k a e It follows from 6 16 that under the condition s L 2 lt t lt s L the equation d t lt Oy a s By L L 2 6 17 holds with L oa N 1 1 0 q aL On 7 E 2 a M 1 k y N 1 t 6 s It is clear that Oy z lt 1 2 if Anal gt 42 ARAP gt 16a M 1 k and aL lt m2 q lt 1 16a 6 18 Let p P M O A be such that equation 6 18 holds for L Phares and for the parameter q of the form 6 5 with O lt p lt p Then equation 6 8 with A 4 1 4a M 1 k p implies that a z lt 1 2 Let t t 1 2 nL Then it follows from 6 17 th
39. be large enough Exercise 5 4 Assume that f x u E t ef a u 6 t in 5 6 where the function f possesses properties 5 7 and 5 8 with arbitrary Lj gt 0 Show that problem 5 6 has an inertial manifold for any 0 lt lt Eg where JV 2 1 if l E 2T max La L L 0 l 2 2 4 Jn voy Jt 1 2 Characterize the dependence of the dimension of inertial manifold on Exercise 5 5 Study the question on the existence of an inertial manifold for problem 5 6 in which the Dirichlet boundary condition is replaced by the Neumann boundary condition 5 5 Examples and Discussion It should be noted that A Cyn4 1 0 1 noo d dimaQ where n are the eigenvalues of the linear part of the equation of the type ou vAu f x u Vu t weQ t gt 0 in a multidimensional bounded domain Q Therefore we can not expect that Theo rem 3 1 is directly applicable in this case In this connection we point out the paper 3 in which the existence of IM for the nonlinear heat equation is proved in a boun ded domain Q c R4 d lt 3 that satisfies the so called principle of spatial ave raging the class of these domains contains two and three dimensional cubes It is evident that the most severe constraint that essentially restricts an applica tion of Theorem 3 1 is spectral gap condition 3 1 In some cases it is possible to weaken or modify it a little In this connection we mention papers 6 and 7 in
40. cise 1 2 the in variance of the collection M and the fact that equality 8 8 is equivalent to the equation u s M Theorem 3 1 is completely proved Exercise 3 3 Show that if the hypotheses of Theorem 3 1 hold then the in duced trajectory u t is uniquely defined in the following sense if there exists a trajectory w t such that w t M for t gt s and A u t u t lt Cees with y Ay 2M ay then u t u t for t2s Existence and Properties of Inertial Manifolds The construction presented in the proof of Theorem 3 1 shows that in order to build the induced trajectory for a solution w t with the exponential order of decrease y given it is necessary to have the information on the behaviour of the solution u t for all values t gt s In this connection the following simple fact on the exponential closeness of the solution w t to its projection Pu t Pwu t t onto the mani fold appears to be useful sometimes Exercise 3 4 Show that if the hypotheses of Theorem 3 1 hold then the es timate A Qu t B Pu t lt 2 y t s 40 _ is valid for any solution u t to problem 1 1 Here y Ay 2M q X and t gt s Hint add the value Pu t t Qu t 0 to the expression under the norm sign in the left hand side Here w t is the induced trajectory for w t It is evident that the inertial manifold M consists of the solutions u t to problem 1 1 which are de
41. cts of this problem related to fi nite dimensional systems are presented in the book by Ya Baris and O Lykova 14 For infinite dimensional systems the problem of construction of an approxi mate IM can be interpreted as a problem of reduction i e as a problem of construc tive description of finite dimensional projectors P and functions t PH gt 1 P H such that an equation of form 6 1 inherits of course this needs to be specified all the peculiarities of the long time behaviour of the original system 0 1 It is clear that the manifolds arising in this case have to be close in some sense to the real IM in fact the dynamics on IM reproduces all the essential features of the qualitative behaviour of the original system Under such a formulation a prob lem of construction of IM acquires secondary importance so one can directly con struct a sequence of approximate IMs Usually see the references in survey 8 the problem of the construction of an approximate IM can be formulated as follows find a surface of the form M p p t pe PH 6 2 which attracts all the trajectories of the system in its small vicinity The character of closeness is determined by the parameter hig 1 related to the decomposition Approximate Inertial Manifolds for Semilinear Parabolic Equations oP Ap PB p q t 6 3 Aq 1 P B p 4 t We obtain the trivial approximate IM mM if we put D p t p t 0 in
42. e BS o t e t 9 4p fet 4 P80 0 t dt t t e t DA QB v t t dt s L Hereinafter the index N of the projectors Py and Qy is omitted i e P is the ortho projector onto Lin e EAE en and Q 1 P It should be noted that the most sig nificant case for the construction of IM is when L 156 Hoaedpe FQ Se Inertial Manifolds Lemma 2 1 Let at least one of two conditions be fulfilled 08 1 0 0 or0 lt L lt and 6 0 Ayri Ay EH k tan O lt a lt 2 3 where k is defined by equation 1 7 Then for any fixed s e R there ere a unique function v t p C satisfying equation 2 1 for all e s L s where y is an arbitrary number Tom the segment An a in the case of 2 2 and y Ay 2M q 28 in the case of 2 3 Moreover PC pD pls lt O o P2 2 4 and Wd lt h ayh D 14 a 2 5 where Dy M 1tk Ayyi May 2 6 Proof Let us apply the fixed point method to equation 2 1 Using 1 8 it is easy to check similar estimates are given in Chapter 2 that EBE OO By v2 lt Ss lt eN 40 v p f peat Mp T va T gdT t t o ie i 4 Arar PoE P Mol valad lt s L lt NEL Ap p a s t a s t gee Plo zoe where t q s t M f ea N e Aw TNU gr 2 7 s L Integral Equation for Determination of Inertial Manifold and S qo s t m fag A Sa 2 8 t Therefore if the estimate q s t
43. emigroup The dimension of this subspace is equal to m The action of the semigroup in this subspace is generated by a system of ordinary differential equations f u 0 u t eL 5 18 Exercise 5 6 Assume that equation 5 12 holds for amp n 0 Show that equation 5 13 is uniquely solvable on the whole time axis for any initial condition and the equation t lt s sup fe s Ju s lt 5 14 holds for any s R Examples and Discussion The subspace L consists of the eigenvectors of the operator A corresponding to the eigenvalue A 1 The next eigenvalue has the form A vu 1 where u is the first nonzero eigenvalue of the Laplace operator with the Neumann boundary condition on 0Q Therefore spectral gap equation 38 1 can be rewritten in the form vu gt a i rmah 4 1 JZ mei 1 5 15 for N m and 0 1 2 where 0 lt q lt 2 J2 It is clear that there exists Vo gt 0 such that equation 5 15 holds for all v gt v Therefore we can apply Theorem 3 1 to find that if v is large enough then there exists IM of the type M p 0 v eL LHL The invariance of the subspace L and estimate 5 14 enable us to use Theorem 3 2 and to state that L c M This easily implies that p 0 ie M L Thus Theorem 3 1 gives us that for any solution u t to problem 5 11 there exists a so lution w t to the system of ordinary differential equations 5 13 such that Ju t u t l lt Ce t
44. en we find that PBF wo lt t79 1 499 wl 00 t Lae DA Ql F w t tdt A97 A Pl F w t t ar s t for t gt s Therefore 8 9 8 11 and 8 12 give us that t s EBE eo Q lt CNA AR k Anas poke Magy mera f 164 Hoaedpe FQ Se Inertial Manifolds Since y Ay 2M q 28 spectral gap condition 3 1 implies that A B w 1 lt ef S Ana A q w gem y Hols 4 3 14 Similarly with the help of 3 10 8 12 we have that 4 Bs w BS fw 0 lt lt CTAN A qw q0 qe w 7 4 3 15 for any w W Cy From equations 3 8 3 9 and 2 15 we obtain that 00 M A a w Quo P Pug s lt e A e amp 1 A p eves ae lwls 7 Ss Therefore 3 11 implies that q2 Aal lt AQuo OCP rg zi bal Similarly we have that 2 40 q lt Ww 0 3 17 A a w a I aca Dls 3 17 It follows from 8 14 8 17 that BS w A Quy 9 Pug s i z Tog lls 3 18 2 Bel Bylo gt lt F T M Tle Therefore if q lt 2 2 then the operator B is continuous and contractive in c Estimate 3 13 of its fixed point follows from 3 18 Lemma 3 1 is proved In order to complete the proof of Theorem 3 1 we must prove that the function u t u t w t is a mild solution to problem 1 1 lying in M t s here w t is a solution to integral equation 3 7 We can do that by using the result of Exer
45. er to prove 1 12 it is sufficient to use the constant variation formula for a solution to the finite dimensional equation oP Ap Py f t p t pult Thus Lemma 1 1 is proved 153 154 Hoaedpe FQ Se Inertial Manifolds Lemma 1 1 enables us to obtain an equation to determine the function D p t Indeed let us assume that B w t is bounded and there exists M with the func tion p t possessing the property A p t lt C forall p e PH andteR Then the solution to problem 1 1 lying in M has the form u t p t p t t It is bounded in the subspace QH and therefore it satisfies the equation of the form u t e t s Ap t t pfen DAP B u t t dt e 04Q B u t t dt te R 1 13 S 00 Moreover S O p s Qyu s e l DA Qu B u t T dt 1 14 Actually it is this fact that forms the core of the Lyapunov Perron method It is proved below that under some conditions i integral equation 1 13 is uniquely solvable for any p PyH and ii the function p s defined by equality 1 14 gives IM In the construction of IM with the help of the Lyapunov Perron method an im portant role is also played by the results given in the following exercises Exercise 7 5 Assume that sup eV 8 9 F 0 t lt s lt oo where y is any number from the interval Ay Ay41 and s R Let w t be a mild solution on the whole axis to equation 1 11 Show that w t pos sesses the property su
46. fined for all real t see Exercises 1 3 and 1 4 These solutions can be characterized as follows Theorem 3 2 Assume that spectral gap condition 3 1 holds with q lt 2 J2 and M is the inertial manifold for problem 1 1 constructed in Theorem 3 1 Then for a solution u t to problem 1 1 defined for all te R to lie in the inertial manifold u t M it is necessary and sufficient that lul sup e77 57D An 2 lt t lt s lt 3 19 for each s e R where y Ay aM A Proof If u t e M then w t Pu t Pu t t Therefore equation 2 13 implies that q D Aul lt Da E rh MPU 3 20 The function p t Pu t satisfies the equation t p t e t 5 4p s PBU t dt S for all real t and s Therefore we have that 165 166 Hoaedpe FQ Se Inertial Manifolds S A p 2 lt e 2 A p s M28 el TOAN 1 Aullar for t lt s With the help of 8 20 we find that mag A p 2 lt C s N gje TN N k e7 PAN Ap r at t for t lt s where qD C s N q A u i D 74 shy N Hence the inequality ue PA lt C s N ae foar t holds for the function t A p t e075 An and t lt s If we introduce the func tion y t i p t dt then the last inequlity can be rewritten in the form M19 VO a 2 C s N q t lt s F mr mre a w t exp Tog gt C s N q exp iz ie t lt s After the integration over the segment t s
47. in Russian CHEPYZHOV V V GORITSKY A YU Global integral manifolds with exponential tra cking for nonautonomous equations Russian J Math Phys 1997 Vol 5 1 MITROPOLSKIJ YU A LYKOVA O B Integral manifolds in non linear mechanics Moskow Nauka 1973 in Russian HENRY D Geometric theory of semilinear parabolic equations Lect Notes in Math 840 Berlin Springer 1981 BOUTET DE MONVEL L CHUESHOV I D REZOUNENKO A V Inertial manifolds for retarded semilinear parabolic equations Nonlinear Analysis 1998 Vol 34 P 907 925 BARIS YA S LYKOVA O B Approximate integral manifolds Kiev Naukova dumka 1993 in Russian CHUESHOV I D Approximate inertial manifolds of exponential order for semi linear parabolic equations subjected to additive white noise J of Dyn and Diff Eqs 1995 Vol 7 4 P 549 566 Mora X Finite dimensional attracting invariant manifolds for damped semi linear wave equations Res Notes in Math 1987 Vol 155 P 172 183 CHUESHOV I D On a construction of approximate inertial manifolds for second order in time evolution equations Nonlinear Analysis TMA 1996 Vol 26 5 P 1007 1021 GHIDAGLIA J M TEMAM R Regularity of the solutions of second order evolution equations and their attractors Ann della Scuola Norm Sup Pisa 1987 Vol 14 P 485 511 CHUESHOV I D Regularity of solutions
48. is proved Theorem 8 1 and equation 8 4 imply the following lemma Lemma 8 2 Assume that u t is a solution to problem 8 1 lying in L ml Let p t Pu t and let a t h0 P t Is t LPE OP 8 17 Then the estimates 42 QM OP Jago with 0 lt j lt m l and aOR ao are valid for t large enough m R lA Cr mMm lA Cr sm Proof It should be noted that q t is the sum of terms of the form j 1 1 DY Bh p ap PEPE p POD PEDO PPO where a B tseko lo tirs Tg are nonnegative integers such that 1SA PSj iyt tig tT Tg J Similar equation also holds for gq t Further one should use Theorem 8 1 and the estimates PDE JA pa lap DA lt R2 t2t l lt k lt m which follow from 8 4 Let us define the induced trajectories of the system by the formula U t u t T where s 0 1 2 and u t p t aslt H t P t l 8 18 205 206 Roepe FQ Se Inertial Manifolds Here p t Pu t u t isa solution to problem 8 1 q t and g t are defined with the help of 8 17 Assume that w t lies in L p Then Lemma 8 2 implies that the induced trajectories can be estimated as follows ja a x 4 Jau Pe lt CRs US Sips POOPHA uO lt Cr for t large enough Using 8 3 8 4 and the last estimates it is easy to prove the following assertion do it yourself Lemma 8 3 Let E t B p t a t B w t
49. ith a selfadjoint positive operator A How ever there exists a wide class of problems which cannot be reduced to this form From the point of view of applications the important representatives of this class are second order in time systems arising in the theory of nonlinear oscillations d u du 2 eet T B u t t gt s gt 0 7 1 u u du u lis 0 dt Js t s In this section we study the existence of IM for problem 7 1 We assume that A is a selfadjoint positive operator with discrete spectrum u and e are the cor responding eigenvalues and eigenelements and the mapping B w t possesses the properties of the type 1 2 and 1 3 for 0 lt O lt 1 2 ie B u t is a continuous mapping from D A x R into H such that B 0 t lt Mo B uy 1 B ug t lt M Au uo 7 2 where 0 lt O lt 1 2 and u ug D A GH The simplest example of a system of the form 7 1 is the following nonlinear wave equation with dissipation au u u u Ae oe ae Bae Dt le Be 0 0 lt x lt L t gt s Ule o Ul age 0 7 3 ul Uo 2 OL Let J D A x H It is clear that is a separable Hilbert space with the inner product U V Aug Vvo u1 04 7 4 where U ug uj and V vo v are elements of In the space prob lem 7 1 can be rewritten as a system of the first order U AU BU t t gt s Ui Uo 7 5 189 190 Hoaedpe FQ Se Inertial Manifolds
50. lowing cf 18 about the function g s S lim s a o d0 gt 0 s gt there exists C gt 0 such that Ss lim s sg s 0 fotod gt 0 s gt 0 for any m there exists B m gt 0 such that g s lt Ca 1 s80 8 26 Under these assumptions the solution u t lies in L p for R gt 0 large enough if and only if the initial data satisfy some compatibility conditions 18 Moreover the global attractor of system 8 5 exists and any trajectory lying in possesses properties 8 4 for allt e R and k 1 2 18 It is easy to see that Theorem 8 2 is applicable here the form of A B and H is evi dent in this case In particular Theorem 8 2 gives us that for a trajectory U t u t u t of problem 8 5 which lies in the global attractor 4 the estimate 1 2 jaoje 1 u O JA20 O ult E So holds for all n 1 2 all j 1 2 and all t e R Here t and u t are defined with the help of 8 18 Therewith Idea of Nonlinear Galerkin Method sup dist U M Ue A lt me ee n 1 2 8 27 where M is a manifold of the type 8 7 with k h p p and l l p Here dist U M is the distance between U and M in the space D A x D A Equation 8 27 gives us some information on the location of the global attractor in the phase space Other examples of usage of the construction given here can be found in papers 17 and 19 see also Section 9 of Chapter 4
51. lution to integral equation 2 1 then the function v t v t h eC g sth L sth Existence and Properties of Inertial Manifolds is its solution when s h is written instead of s Consequently equation 2 12 gives us that O p s h Qu s h Qvu s O4 p s Thus Theorem 2 1 is proved Exercise 21 Show that if B u t lt M then inequalities 2 13 and 2 16 can be replaced by the relations A p s lt Dz 2 23 A9 Qu t B4 Pu t O lt Dgd ayle 2 24 where D is defined by formula 2 14 3 Existence and Properties of Inertial Manifolds In particular assertion c of Theorem 2 1 shows that if the spectral gap condition Ayai aw Atk Aguitay 0 lt q lt 1 8 1 is fulfilled then the collection of surfaces M p 0 p s pe PH seR 3 2 is invariant i e U t s M cM o lt s lt t lt o 3 3 Here O p s p s is defined by formula 2 12 for L and U t s is the evolutionary operator corresponding to problem 1 1 It is defined by the for mula U t s ug w t where w t is a mild solution to problem 1 1 In this section we show that collection 3 2 possesses the property of exponen tial uniform attraction Hence M is an inertial manifold for problem 1 1 More over Theorem 3 1 below states that M is an exponentially asymptotically complete IM i e for any solution w t U t s ug there exists a solution w t U t 8 Uo lying in the manifold w t
52. m g w Other applications of Theorem 9 1 can also be pointed out 213 214 Hoaede FQ Se 10 M 12 13 14 15 16 17 18 19 20 Inertial Manifolds References FOIAS C SELL G R TEMAM R Inertial manifolds for nonlinear evolutionary equations J Diff Eq 1988 Vol 73 P 309 353 CHOW S N LU K Invariant manifolds for flows in Banach spaces J Diff Eq 1988 Vol 74 P 285 317 MALLET PARET J SELL G R Inertial manifolds for reaction diffusion equations in higher space dimensions J Amer Math Soc 1988 Vol 1 P 805 866 CONSTANTIN P Foras C NICOLAENKO B TEMAM R Integral manifolds and inertial manifolds for dissipative partial differential equations New York Springer 1989 TEMAM R Infinite dimensional dynamical systems in Mechanics and Physics New York Springer 1988 MIKLAV I M A sharp condition for existence of an inertial manifold J of Dyn and Diff Eqs 1991 Vol 3 P 437 456 ROMANOV A V Exact estimates of dimention of inertial manifold for non linear parabolic equations Izvestiya RAN 1993 Vol 57 4 P 36 54 in Russian CHUESHOV I D Global attractors for non linear problems of mathematical physics Russian Math Surveys 1993 Vol 48 3 P 133 161 CHUESHOV I D Introduction to the theory of inertial manifolds Kharkov Kharkov Univ Press 1992
53. nd 1 3 are not assumed to be uniform with respect touwe D A8 The dissipativity property enables us to restrict ourselves to the con sideration of the trajectories lying in a vicinity of the absorbing set when we study the asymptotic behaviour of solutions to problem 0 1 In this case it is convenient to modify the original problem Assume that the mapping B u t is continuous with respect to its arguments and possesses the properties B u t lt Co B u t B Ug 9 lt C A uy ug 3 26 for any p gt 0 and for all u uj and ug lying in the ball By v A u lt p Let s be an infinitely differentiable function on R 0 such that xy s 1 Vessels x s 0 s22 O lt x s lt 1 Ix s lt 2 seR We define the mapping Bp u t by assuming that Bp u t x R7 A ul B w t we D A 3 27 Exercise 3 10 Show that the mapping Bp w t possesses the properties A Bp u H lt M Balu t Bp uy t lt MA uo 8 28 where M C3 pR 1 2 R and C is a constant from 3 26 Let us now assume that B u t satisfies condition 3 26 and the problem du ai Au B u t Ul _9 Uo 3 29 has a unique mild solution on any segment s S T and possesses the following dissipativity property there exists Ry gt 0 such that for any R gt 0 the relation Continuous Dependence of Inertial Manifold on Problem Parameters A u t s uo lt Ry forall t s gt to R 3 30 holds
54. nd slow in the subspace WH motions is possible Moreover the subspace of slow motions turns out to be finite dimensional It should be noted in advance that such separation is not unique However if the global attractor exists then every IM contains it When constructing IM we usually use the methods developed in the theory of integral manifolds for central and central unstable cases see 11 12 If the inertial manifold exists then it continuously depends on 1 i e jim A p 8 t 0 for any p PyH and s e R Indeed let w t be the solution to problem 1 1 with ug p P p s p RyH Then u t M for t gt s and hence u t Pyu t Pyu t t Therefore D p t O p s P p t P Pyult t u t uo p Pyult Consequently Lipschitz condition 1 9 leads to the estimate A p 5 p D lt C A u t uo Since u t e C s D A this estimate gives us the required continuity pro perty of p t 151 152 Hoaedvpe FQ Se Inertial Manifolds Exercise 1 17 Prove that the estimate AP p t 5 O p t lt Cp p N o P holds for O p t when0 lt so lt 1 0 lt P lt 0 teR The notion of the inertial manifold is closely related to the notion of the inertial form If we rewrite the solution u t in the form u t p t q t where p t Pyu t g t Qyu t and Qy Py then equation 1 1 can be re written as a system of two equations gl tADl
55. of the manifold M lt s lt oo at an exponential velocity According to 2 12 in order to build an approximation MZ of the inertial manifold M we should solve integral equation 2 1 for L large enough This equation has the same structure both for L lt oo and for L o Therefore it is im possible to use the surfaces M directly for the effective approximation of M However by virtue of contractiveness of the operator B in the space C Cy 9 2 s its fixed point v t which determines M can be found with the help of iterations This fact enables us to construct the collection M _ s of appro ximations for M as follows Let vg Vo 5 t p be an element of C We take Dn Vp s6 P BS ry gM n l Bn and define the surfaces M by the formula M s p p s pe PH where p s QU s P s wa 254 170 Hoaedpe FQ Se Inertial Manifolds Exercise 37 Let vy p andlet B u t B u Show that p s 0 and p s A QB p Exercise 3 8 Assume that spectral gap condition 8 1 is fulfilled Show that Ep 8 s lt a lo a D 4 l where D is defined by formula 2 6 and p s is the function that determines the inertial manifold Exercise 39 Prove the assertion for p s similar to the one in Exer cise 3 5 Theorems represented above can also be used in the case when the original system is dissipative and estimates 1 2 a
56. ollows Assume that f p stands for one of the functions h p or 1 p p We define the value i 1 2 F B fy m n D x R lApl 1A aI Buso D 10 where y s is an infinitely differentiable function on R such that a 0 lt y s lt 1 b y s 1 fr0 lt s lt 1 c x s 0 fors 2 R isthe radius of dissipativity see 8 4 for k 0 of system 9 1 Py is the orthoprojector in H onto the sub space generated by the first M eigenvectors of the operator A M gt N We consider the following N dimensional evolutionary equation in the subspace Ay i Idea of Nonlinear Galerkin Method 211 a p yd p Ap PyB p h p a p 9 11 Plo Enuo 9 P 9 v1 Exercise 9 7 Prove that problem 9 11 has a unique solution for t gt 0 and the corresponding dynamical system is dissipative in PYH x PyH We call problem 9 11 a nonlinear Galerkin n N M approximation of problem 9 1 The following assertion is valid Theorem 9 1 Assume that the mappings h p p and l p D satisfy equations 9 2 9 5 for n lt m 1 and for some m 2 Moreover we assume that 9 5 is valid for all t gt 0 Let h and I be defined by 9 10 with the help of h and l and let uh t P O h PO OP T t O p t 1 w t 0 0 where p t is a solution to problem 9 11 Then the estimate 1 2 fje u t u 2 00 m lt a ARTD ag An exp Bt 9 12 holds where u t is a solution to
57. or the class of solutions under consideration Therefore equation 9 16 implies that d fy os 1 L 49 HOP HAYOP lt Trl Ar Cn r EY CA Hence Gronwall s lemma gives us that a C r P Al2r P lt By Ue oy mri de This and equations 9 13 and He a estimate 9 12 Theorem 9 1 is proved If we take n 0 and N M in Theorem 9 1 then estimate 9 12 changes into the accuracy estimate of the standard Galerkin method of the order N Therefore if the Idea of Nonlinear Galerkin Method parameters N M and n are compatible such that A M 1 ARTA then the error of the corresponding nonlinear Galerkin method has the same order of smallness as in the standard Galerkin method which uses M basis functions However if we use the nonlinear method we have to solve a number of linear algebraic systems of the order M N and the Cauchy problem for system 9 11 which consists of N equations In particular in order to determine the value A 1p we must solve the equation Ah p p Fy Pn QB p for n 1 and choose the numbers N and M such that Ay lt Me 41 Moreover if A k 1 0 1 gt 0 as k then the values N and M must be com patible such that M lt c N We note that Theorem 9 1 as well as the corresponding variant of the nonlinear Galerkin method can be used in the study of the asymptotic properties of solutions to the nonlinear wave equation 8 5 under some conditions on the nonlinear ter
58. ositive numbers depending on 0 A 1 2 1 2 1 and M only Hereinafter Qy I Py where Py is the orthoprojector onto the first N eigenvectors of the operator A Moreover we use the notation k 0 getag fr 0 gt 0 and k 0 for 0 0 1 7 0 Further we will also use the following so called dichotomy estimates proved in Lemma 1 1 of Chapter 2 Joep lt Ag e N teR ko eg a tz 0 1 8 Basic Equation and Concept of Inertial Manifold eton lt 0 t AR Je ntt t gt 0 O gt 0 The inertial manifold IM of problem 1 1 is a collection of surfaces M t e R in H of the form M p p t p e PyH p t e 1 Py F where p t is a mapping from A H x R into 1 A satisfying the Lipschitz condition A P p t po t C 4v P 1 9 with the constant C independent of Pj and t We also require the fulfillment of the invariance condition if ug M then the solution u t to problem 1 1 posses ses the property u t M t s and the condition of the uniform exponential attraction of bounded sets there exists y gt 0 such that for any bounded set B c H there exist numbers Cp and tp gt s such that sup ait ult Uo M ug a lt C go V t ty for all t gt tp Here u t ug is a mild solution to problem 1 1 From the point of view of applications the existence of an inertial manifold IM means that a regular separation of fast in the subspace I Py H a
59. p e7V s D APUN lt t lt s if and only if equation 1 12 holds for t lt s Hint consider the new unknown function w t u t instead of u t Exercise 1 6 Assume that f t is a continuous function on the semiaxis s 00 with the values in H such that for some y from the interval Ay Any 41 the equation sup eO t e s 00 lt holds Prove that for a mild solution u t to equation 1 11 on the semiaxis s 00 to possess the property Integral Equation for Determination of Inertial Manifold sup e716 D Au t t e s 00 lt it is necessary and sufficient that t u t qe ala 0 94 Qy f t dt S 00 z f e AP f t dt 1 15 t where gt s and q is an element of Qy D A9 Hint see the hint to Exercise 1 5 2 Integral Equation for Determination of Inertial Manifold In this section we study the solvability and the properties of solutions to a class of in tegral equations which contains equation 1 13 as a limit case Broader treatment of the equation of the type 1 13 is useful in connection with some problems of the ap proximation theory for IM For s e R and 0 lt L lt we define the space C C g s L s as the set of continuous functions v t on the segment s L s with the values in D A and such that y s 40 ad lol ao Luol lt Here y is a positive number In this space we consider the integral equation v t BS v t s L lt t lt s 2 1 wher
60. possess the following orthogonal properties Chieti his S Sed Oe RAM ft fp 0 1 lt k lt N 7 24 Inertial Manifold for Second Order in Time Equations Note that the last of these equations is one of the reasons of introducing a new inner product Let Py be the orthoprojector onto the subspace 6 in J6 i 1 2 l Lemma 7 2 The equality e Pa NHI gt 0 7 25 is valid Here is the operator norm which is induced by the corres ponding vector norm Proof Let U H We consider the function y t le AtU Since Hy is inva riant with respect to e7At the equation 2 W t Au t w t 6 2uy 41 u t w t let eu holds where u t is a solution to problem 7 8 for h t 0 After simple cal culations we obtain that W zey 4 uy Eu u It is evident that 2 fe uyy eu u lt 2 uy plul teul lt we Therefore dy a t2ey lt 2 67 Ua V Consequently w t lt eN w0 t gt 0 7 26 If we now notice that Ae cma exp At fya 5 Nt Fras then equation 7 26 implies 7 25 Thus Lemma 7 2 is proved Let us consider the subspaces HE Lin fg k lt N Equation 7 24 gives us that the subspaces are orthogonal to each other and there fore F6 He H Using 7 24 it is easy to prove do it yourself that 195 196 Hoaedpe FQ Se Inertial Manifolds Jen Ber lt oN U teR 7 27 eet Ay lel Joe lt e NU t g
61. problem 9 1 which lies in L p for m gt 2 and possesses property 8 4 for k 1 and for all t gt 0 Here n lt lt m 1 Qj A and B are positive constants independent of M and N Ar is the k th eigenvalue of the operator A Proof Let p t Py w t We consider the values u t un t pE P t Qyult Ralo t apt hn PC 6 P t hi PO 6 P and 0 u t U t 0 p t p t Qn 0 u t a P t Op t 1 e t pE aa l 2 r The equalities Pyan p t hy w t amp p t 212 Hoaede FQ Se Inertial Manifolds and Pln p t amp p t p are valid for the class of solutions under consideration Therefore we use 9 5 to find that A w t e lt C C lt c lea POl pe ar o a 0 13 M MW and ru 8 un S C4 C R lt C 3 I42 v t p Ol ap t 0 p 1 SIG Pees 9 14 AM Anal Therefore we must compare the solution p t to problem 9 11 with the value p t Pyu t which satisfies the equation O2pt yd ptAp QyB p t Qyu 9 15 with the same initial conditions as the function p t Let 7 t p t p t Then it follows from 9 11 and 9 15 that 2r t yd r t Ar t F t p u r 0 0 r 0 0 9 16 where F t p U Qy B u t B u 4 Due to the dissipativity of problems 9 11 and 9 15 we use 9 18 to obtain 1 2 1 2 IF os wh lt Cp IAl2r P P 0 p MPD 00 f
62. provided that Aull lt R Here u t S Uug is the solution to problem 3 29 0 0 Exercise 3 11 Show that the asymptotic behaviour of solutions to problem 3 29 completely coincides with the asymptotic behaviour of solu tions to the problem du dt where By Ro is defined by formula 3 27 and R is the constant from equation 8 30 Au Bor u t Ul Uo 3 31 Exercise 3 12 Assume that for a solution to problem 8 29 the invariance property of the absorbing ball is fulfilled if Awl lt Ry then A u ce Ss uo lt R forall t lt s Let M be the invariant manifold of problem 3 31 Then the set Meo M N u A ul lt Ro is in variant for problem 3 29 if ug M o then w t s ug M o ts Thus if the appropriate spectral gap condition for problem 8 29 is fulfilled then there exists a finite dimensional surface which is a locally invariant exponentially at tracting set In conclusion of this section we note that the version of the Lyapunov Perron me thod represented here can also be used for the construction see 13 of inertial manifolds for retarded semilinear parabolic equations similar to the ones considered in Section 8 of Chapter 2 In this case both the smallness of retardation and the fulfil ment of the spectral gap condition of the form 3 1 are required 4 Continuous Dependence of Inertial Manifold on Problem Parameters Let us consider the Cauchy problem du _ p atau SB U
63. q s t lt q s L lt t lt s 2 9 holds then Ji L Bo v Bs ve lt A p P av v 2 10 Let us estimate the values q s t and q3 s t Assume that 2 2 is fulfilled Then it is evident that a s t lt M0 t 1 9dt MAG t s L s L M t s L 9 MA8 t s L 08 1 0 and do s t lt MAR s t lt MAS s t for Ay lt Y lt Ay4 Therefore 98 E 1 0 0 q1 s t q9 s t lt M 30V s L 9 N L Consequently equation 2 2 implies 2 9 Now let the spectral condition 2 3 be fulfilled Then i 0 q s t S MOP Ora hE jrg Mona i Cao Anim for all y lt Ay We change the variable in integration Ay y t T and find that 0 Mk 4 M Mn ygi O AnaY where the constant k is defined by 1 7 It is also evident that MA y An provided that y gt Ay Equation 2 3 implies that y Ay 2M q A9 lies in the interval Ay Ay 1 If we choose the parameter y in such way then we get a s t lt q2 s t lt 157 158 Roepe FQ Se Inertial Manifolds M 1 k b q lt 4 q s t 9 s t lt E iN Hence equation 2 3 implies 2 9 Therefore estimate 2 10 is valid provi ded that the hypotheses of the lemma hold Moreover similar reasoning enables us to show that B t ol lt Di l pl atls 2 11 where D is defined by formula 2 6 In particular cputates 2 10 and 2 11 mean that when s L and p are fixed
64. rtial Manifold on Problem Parameters 175 Let us estimate the value 1 F v It is clear that t 1 P v t e t 1 A 1 P B v 1 dt 00 Therefore Lemma 2 1 1 see also 1 8 gives us that t A R O lt M f E ga e ar t 8 a 5 T Aasa e mei 0 7 el DY ar lo e1578 oo where 2M 40 Y Aut TANS Ana lt Amoi as above Simple calculations analogous to the ones in Lemma 2 1 imply that 0 ae eet nt el 8 4 Wl 31 mt 1 4 P v Q lt where the constant k has the form 1 7 Consequently using 2 5 we obtain 1 2 0 lt KELO 1 a lt PARNE m 1 M 1 k An 1 Y i 0 lt 1 14 1 1 q D A pl Se Ae This and 4 10 imply estimate 4 8 Theorem 4 2 is proved Exercise 4 2 In addition assume that the hypotheses of Theorem 4 2 hold and B w t lt M Show that in this case estimate 4 8 has the form A p s p s lt Clq M BAZ LTS m 1 176 Hoaede FQ Se Inertial Manifolds 5 Examples and Discussion Example 5 1 Let us consider the nonlinear heat equation du _ d2u a age tI t Oe ly t gt 0 5 1 ulo Ul a 0 i 5 2 EE 5 3 Assume that v is a positive parameter and f x U t is a continuous function of its variables which possesses the properties f a Uj b Sf hy Ug t lt Mu Us f x 0 lt M Jt Problem 5 1 5 3 generates a dynamical system in L7 0 1
65. t Ul s uo SE R 4 1 in the space H together with problem 1 1 Assume that B u t is a nonlinear mapping from D A x R into H possessing properties 1 2 and 1 3 with the same constant M as in problem 1 1 If spectral gap condition 8 1 is fulfilled then problem 4 1 as well as 1 1 possesses an invariant manifold 171 172 Roepe SQ Se Inertial Manifolds M p p s pePH seR 4 2 The aim of this section is to obtain an estimate for the distance between the manifolds M and M The main result is the following assertion Theorem 4 1 Assume that conditions 1 2 1 3 and 8 1 are fulfilled both for problems 1 1 and 4 1 We also assume that B v t B w tl lt py palo 4 3 for all v e D A and t R where p and p gt are positive numbers Then the equation sup A p s p s lt C a 9 Pa PaO 0 M p A pl My is valid for the functions p s and p s which give the invariant manifolds for problems 1 1 and 4 1 respectively Here the numbers C q 0 and Ca q 9 M do not depend on N and p Proof Equation 2 12 with L implies that A0 p s amp p s lt f A86 D4 QI IB T 1 B0 dlar where v t and v t are solutions to the integral equations of the type 2 1 cor responding to problems 1 1 and 4 1 respectively Equations 1 3 and 4 3 give us that IBC 1 B 2 DL lt MAOC CTl hp py l4 o O
66. t 0 7 28 ZA We use the following pair of orthogonal with respect to the inner product projectors in the space 96 Palgi A Fit Foe to construct the inertial manifold of problem 7 1 or 7 5 Lemma 7 2 and equa tions 7 27 and 7 28 imply the dichotomy equations leAtp lt N teR eAtql lt enai t0 7 29 We remind that Ay e p and gt Uy 4 The assertion below plays an important role in the estimates to follow Lemma 7 3 Let B U t 0 B ug t where U ug uy H and Bug pos sesses properties 7 2 Then IB U i lt M KylUl Ue B U t B U lt Ky U U Up U 7 30 where H Ky Musi max 1 sive 7 31 l UN 1 The proof of this lemma follows from the structure of the mapping B U t and from estimates 7 2 and 7 23 Exercise 7 16 Show that one can take Ky M 2 uy for 0 0 in 7 30 Hint see Exercise 7 14 Let us now consider the integral equation cf 2 1 for L V t B V 0 s t e t s An oe APB M 7 T dt e t NAQB V t t dt 7 32 t oo Inertial Manifold for Second Order in Time Equations in the space C of continuous vector functions U t on s with the values in 96 such that the norm ae hea 7 IUI supe VS D U L lt Y 5Awiit An is finite Here p e PH and t e o s Exercise 7 17 Show that the right hand side of equation 7 32 is a continu ous function of the v
67. t dt S 00 e DA PF w t t dt t E s 00 3 7 t in the space C Here the value q w QD A is chosen from the condition u s u s w s M i e such that Quo t Qw s Pug Pw s s Therefore by virtue of 3 7 we have 00 q w Quo Puy f e DA Pp F w t t dt s 8 8 sS Thus in order to prove inequality 3 4 it is sufficient to prove the solvability of inte gral equation 8 7 in the space Cy and to obtain the estimate of the solution The preparatory steps for doing this are hidden in the following exercises Existence and Properties of Inertial Manifolds 163 Exercise 3 1 Assume that F w t has the form 3 6 Show that for any w t W t Cf Cs S D A and for s the following inequalities hold F w t Hl lt eM Mwy 5 3 9 F w t t F t Ol lt eI Mw Tl 4 8 10 Exercise 3 2 Using 1 8 prove that the equations 00 0 f Aet D4pjer s ar lt AX y t SN eo V s 311 t l4 e A Ql o V S az lt 0 30 2 k Any 1 Y FANA g s lt 8 12 Ans 77 hold for Ay lt y lt Ay and t gt s Here k is defined by formula 1 7 Lemma 3 1 Assume that spectral gap condition 8 1 holds with q lt 2 2 Then B is a continuous contractive mapping of the space Cy into itself The unique fixed point w of this mapping satisfies the estimate lwl lt o p aleu HP ro s 3 13 Proof If we use 8 7 th
68. t of inertial ma nifold of an infinite dimensional dynamical system see 1 8 and the references therein This manifold is a finite dimensional invariant surface it contains a global attractor and attracts trajectories exponentially fast Moreover there is a possibility to reduce the study of limit regimes of the original infinite dimensional system to solving of a similar problem for a class of ordinary differential equations In this chapter we present one of the approaches to the construction of inertial manifolds IM for an evolutionary equation of the type du Pac AEB t ulo Uo 0 1 Here w t is a function of the real variable t with the values in a separable Hilbert space H We pay the main attention to the case when A is a positive linear operator with discrete spectrum and B u t isa nonlinear mapping of H subordinated to A in some sense The approach used here for the construction of inertial manifolds is based on a variant of the Lyapunov Perron method presented in the paper 2 Other approaches can be found in 1 8 7 9 and 10 However it should be noted that all the methods for the construction of IM known at present time require a quite strong condition on the spectrum of the operator A the difference hy s hy of two neighbouring eigenvalues of the operator A should grow sufficiently fast as N gt oo 1 Basic Equation and Concept of Inertial Manifold In a separable Hilbert space H we consider a Ca
69. tial Manifold for Second Order in Time Equations Exercise 7 10 Show that the estimates IBCU thle lt M IUl B t B U3 t lt MIU Vella hold in the space J6 D A x H Here M is a positive constant Exercise 7 11 Follow the reasoning used in the proof of Theorems 2 1 and 2 3 of Chapter 2 to prove the existence and uniqueness of a mild so lution to problem 7 1 on any segment s s T Thus in the space 6 there exists a continuous evolutionary family of operators S t s possessing the properties S t t I S8tt o 8 tys 8 s and S t s Ug u t t where u t is a mild solution to problem 7 1 with the initial condition Up ug u1 Let condition 2 gt ya 1 hold for some integer N We consider the decomposi tion of the space into the orthogonal sum H HD Hy where H Lin e 0 0 ep k 1 2 N and Jb is defined as the closure of the set Lin e 0 0 e k gt N 1 Exercise 7 12 Show that the subspaces 36 and J6 are invariant with re spect to the operator A Find the spectrum of the restrictions of the operator A to each of these spaces Let us introduce the following inner products in the spaces J6 and b the pur pose of this introduction will become apparent further U V E ug Vo Aug V9 EUQt Uy EVQ ie U Vyg Aug Vo 8 2p 1 Uo Vg EUgt Uy EVot V Here U ug u and V vo v are elements from the
70. uations there arises a problem of construction of an approximate inertial manifold without any assumptions on the behaviour of the spectrum of the operator A and the parameter y gt 0 which characterizes the resistance force Unfortunately the approach presented in Section 6 is not applicable to the equation of the type 8 1 without any additional assumptions on yY First of all it is connected with the fact that the regularizing effect which takes place in the case of parabolic equations does not hold for second order equations of the type 8 1 in the parabolic case the solution at the moment t gt 0 is smoother than its initial con dition In this section see also 17 we suggest an iteration scheme that enables us to construct an approximate IM as a solution to a class of linear problems For the sake of simplicity we restrict ourselves to the case of autonomous equations Bu t B u The suggested scheme is based on the equation in functional derivatives such that the function giving the original true IM should satisfy it This approach was developed for the parabolic equation in 9 see also 8 Unfortunately this ap proach has two defects First approximate IMs have the power order not the expo nential one as in Section 6 and second we cannot prove the convergence of approximate IMs to the exact one when the latter exists Thus in a separable Hilbert space H we consider a differential equation of the type 8 1 where
71. uchy problem of the type du q tAu Blu t t gt s U SER 1 1 where A is a positive operator with discrete spectrum for the definition see Section 1 of Chapter 2 and B is a nonlinear continuous mapping from D A xR 150 Hoaedpe FQ Se Inertial Manifolds into H 0 lt O lt 1 possessing the properties B u t lt M 1 A uf 1 2 and B u t B ug t lt M u u 1 3 for all u w and ug from the domain D A of the operator A Here M is a positive constant independent of t and is the norm in the space H Further it is assumed that e x is the orthonormal basis in H consisting of the eigenfunctions of the operator A Aep Apep O lt A Sios jim A5 Theorem 2 3 of Chapter 2 implies that for any initial condition ug Fg prob lem 1 1 has a unique mild in p solution u t on every half interval s s T i e there exists a unique function w t e C s s 7 Fg which satisfies the inte gral equation t iG O e VAB u t t dt 1 4 for all t e s s T This solution possesses the property see 2 6 in Chapter 2 AP u t o u t l lt Co 8 B O0 lt P lt for 0 lt o lt 1 and t gt s Moreover for any pair of mild solutions w and wo t to problem 1 1 the following inequalities hold see 2 2 15 Lu lt aye Aul ts 1 5 and cf 2 2 18 v APu t lt aes 4M k a dn ara A u s l 1 6 where u t u t wo t a and as are p
72. ueness of mild solutions to 7 8 using the Galerkin method for example The approximate Galerkin solution of the order m is defined as a function Inertial Manifold for Second Order in Time Equations mMm S N k 1 satisfying the equations m t j 2E m t 6j Aun t 6 A t t gt s 7 10 Um S s e Uo mlS e uy e for j 1 2 m Moreover we assume that g t Cl s s T and g t is absolutely continuous Hereinafter we use the notation v t dv dt Evidently equations 7 10 can be rewritten in the form Um t 26 Up t AU t Dy h E 7 11 Um _ Pm o gt m _ Pme1 y where P is the orthoprojector onto Lin e see Cm in H In the exercises given below it is assumed that h t e L R H ug e D A wu eH 7 12 Exercise 7 3 Show that problem 7 10 is uniquely solvable on any segment s s T and u t Ls r Exercise 7 4 Show that the energy equality fem Aule 28 flmo ar 3 t emul pnu fao Um T dt 7 13 holds for any solution to problem 7 10 Exercise 7 5 Using 7 11 and 7 13 prove the a priori estimate A mO fne Aunt lt CCT ug w for the approximate Galerkin solution w to problem 7 8 Exercise 7 6 Using the linearity of problem 7 11 show that for every two approximate solutions w t and u t the estimate AV mlt m tin t tim O AY Un tw D lt 191 192 Hoaedpe
73. which spectral gap condition 3 1 is given with the parameters q 2 and k 0 for 0 lt O lt 1 Besides it is not necessary to assume that the spectrum of the opera tor A is discrete It is sufficient just to require that the selfadjoint operator A pos sess a gap in the positive part of the spectrum such that for its edges the spectral condition holds We can also assume the operator A to be sectorial rather than self adjoint for example see 6 Unfortunately we cannot get rid of the spectral conditions in the construction of the inertial manifold One of the approaches to overcome this difficulty runs as follows let us consider the regularization of problem 0 1 of the form t Au eA u Blu t ul Here gt 0 and the number m gt 0 is chosen such that the operator A A A possesses spectral gap condition 3 1 Therewith IM for problem 5 10 should be naturally called an approximate IM for system 0 1 Other approaches to the con struction of the approximate IM are presented below eae 5 10 It should also be noted that in spite of the arising difficulties the number of equations of mathematical physics for which it is possible to prove the existence of IM is large enough Among these equations we can name the Cahn Hillard equations in the do main Q 0 L d dimQ lt 2 the Ginzburg Landau equations Q 0 L d lt 2 the Kuramoto Sivashinsky equation some equations of the theory of oscilla tions
74. yB a a t Aa t QyB w t a t Pli Po PnMo W Yn By virtue of the invariance property of IM the condition Po Qo eM s implies that p t a t M i e the equality qo po s implies that q t p t t Therefore if we know the function p t that gives IM then the solution u t lying in M can be found in two stages at first we solve the problem p t Ap t WB v t O P 0 t P _g Por AO and then we take u t p t p t t Thus the qualitative behaviour of solu tions w t lying in IM is completely determined by the properties of differential equation 1 10 in the finite dimensional space Ay H Equation 1 10 is said to be the inertial form IF of problem 1 1 In the autonomous case B u t B w one can use the attraction property for IM and the reduction principle see Theorem 7 4 of Chapter 1 in order to state that the finite dimensional IF completely deter mines the asymptotic behaviour of the dynamical system generated by problem 1 1 Exercise 1 2 Let p t give the inertial manifold for problem 1 1 Show that IF 1 10 is uniquely solvable on the whole real axis i e there exists a unique function p t e C PzH such that equation 1 10 holds Exercise 1 3 Let p t bea solution to IF 1 10 defined for all IR Prove that w t p t p t t isa mild solution to problem 1 1 de fined on the whole time axis and such that u _ _

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