Smooth extensions for inertial manifolds of semilinear parabolic equations
aa r X i v : . [ m a t h . A P ] F e b SMOOTH EXTENSIONS FOR INERTIAL MANIFOLDS OFSEMILINEAR PARABOLIC EQUATIONS
ANNA KOSTIANKO AND SERGEY ZELIK , Abstract.
The paper is devoted to a comprehensive study of smooth-ness of inertial manifolds for abstract semilinear parabolic problems. Itis well known that in general we cannot expect more than C ,ε -regularityfor such manifolds (for some positive, but small ε ). Nevertheless, asshown in the paper, under the natural assumptions, the obstacles to theexistence of a C n -smooth inertial manifold (where n ∈ N is any givennumber) can be removed by increasing the dimension and by modifyingproperly the nonlinearity outside of the global attractor (or even out-side the C ,ε -smooth IM of a minimal dimension). The proof is stronglybased on the Whitney extension theorem. Contents
1. Introduction 12. Preliminaries I: Taylor expansions and Whitney ExtensionTheorem 63. Preliminaries II: Spectral gaps and the construction of an inertialmanifold 104. Main result 185. Verifying the compatibility conditions 256. Examples and concluding remarks 30References 341.
Introduction
It is believed that in many cases the long-time behaviour of infinite dimen-sional dissipative dynamical systems generated by evolutionary PDEs (atleast in bounded domains) can be effectively described by finitely many pa-rameters (the so-called order parameters in the terminology of I. Prigogine)which obey a system of ODEs. This system of ODEs (if exists) is usually re-ferred as an Inertial Form (IF) of the considered PDE, see [13, 31, 33, 40, 42]and references therein for more details. However, despite the fundamentalsignificance of this reduction from both theoretical and applied points of
Mathematics Subject Classification.
Key words and phrases.
Inertial manifolds, finite-dimensional reduction, smoothness,Whitney extension theorem.This work is partially supported by the RSF grant 19-71-30004 as well as the EPSRCgrant EP/P024920/1. The authors also would like to thank Dmitry Turaev for manyfruitful discussions. view and big interest during the last 50 years, the nature of such a reduc-tion and its rigorous justification remains a mystery.Indeed, it is well understood now that the key question of the theory ishow smooth the desired IF can/should be. For instance, in the case of
H¨older continuous IFs, there is a highly developed machinery for constructing thembased on the so-called attractors theory and the Mane projection theorem.We recall that, by definition, a global attractor is a compact invariant setin the phase space of the dissipative system considered which attracts astime goes to infinity the images of bounded sets under the evolutionarysemigroup related with the considered problem. Thus, on the one hand, aglobal attractor (if it exists) contains all of the non-trivial dynamics and, onthe other hand, it is usually essentially ”smaller” than the initial phase spaceand this second property allows us to speak about the reduction of degreesof freedom in the limit dynamics. In particular, one of the main results ofthe attractors theory tells us that, under relatively weak assumptions on adissipative PDE (in a bounded domain), the global attractor exists and hasfinite Hausdorff and fractal dimensions. In turn, due to the Mane projectiontheorem, this finite-dimensionality guarantees that this attractor can beprojected one-to-one to a generic finite-dimensional plane of the phase spaceand that the inverse map is H¨older continuous. Finally, this scheme givesus an IF with H¨older continuous vector field defined on some compact set of R N which is treated as a rigorous justification of the above mentioned finite-dimensional reduction. This approach works, for instance, for 2D Navier-Stokes equations, reaction-diffusion systems, pattern formation equations,damped wave equations, etc., see [1, 3, 4, 13, 14, 16, 30, 33, 38, 40] andreferences therein.However, the above described scheme has a very essential intrinsic draw-back which prevents us to treat it as a satisfactory solution of the finite-dimensional reduction problem. Namely, the vector field in the IF thusconstructed is H¨older continuous only and there is no way in general to geteven its Lipschitz continuity. As a result, we may lose the uniqueness ofsolutions for the obtained IF and have to use the initial infinite-dimensionalsystem at least in order to select the correct solution of the reduced IF.Other drawback is that the Mane projection theorem is not constructive,so it is not clear how to choose this ”generic” plane for projection in appli-cations, in addition, the IF constructed in such a way is defined only on acomplicated compact set (the image of the attractor under the projection)and it is not clear how to extend it on the whole R N preserving the dynamics(surprisingly, this is also a deep open problem, some partial solution of it isgiven in [31], see also the references therein).It is also worth noting that the restriction for IF to be only H¨older con-tinuous is far from being just a technical problem here. As relatively sim-ple counterexamples show (see [7, 23, 28, 36, 42]) the fractal dimension ofthe global attractor may be finite and not big, but the attractor cannotbe embedded into any finite-dimensional Lipschitz (or even log-Lipschitz)finite-dimensional sub-manifold of the phase space. What is even more im-portant, the dynamics on this attractor does not look as finite-dimensional XTENSIONS OF INERTIAL MANIFOLDS 3 at all (despite the existence of a
H¨older continuous (with the H¨older expo-nent arbitrarily close to one) IF provided by the Mane projection theorem).For instance, it may contain limit cycles with super-exponential rate of at-traction, decaying travelling waves in Fourier space and other phenomenawhich are impossible in the classical dynamics generated by smooth ODEs.These examples suggest that, in contradiction to the widespread paradigm,H¨older continuous IF is probably not an appropriate tool for distinguish-ing between finite and infinite dimensional limit behavior and, as a result,fractal-dimension is not so good for estimating the number of degrees offreedom for the reduced dynamics, see [7, 23, 42] for more details.An alternative, probably more transparent approach to the finite-dimen-sional reduction problem which has been suggested in [10] is related withthe concept of an Inertial Manifold (IM). By definition, an IM is a finite-dimensional smooth (at least Lipschitz) invariant sub-manifold of the phasespace which is globally exponentially stable and possesses the so called ex-ponential tracking property (=existence of asymptotic phase). Usually thismanifold is C ,ε -smooth for some positive ε and is normally hyperbolic, sothe exponential tracking is an immediate corollary of normal hyperbolicity.Then the corresponding IF is just a restriction of the initial PDE to IMand is also C ,ε -smooth. However, being a sort of center manifold, an IMrequires a separation of the dependent variable to the ”slow” and ”fast”components and this, in turn, leads to extra rather restrictive assumptionswhich are usually formulated in terms of spectral gap conditions. Namely,let us consider the following abstract semilinear parabolic equation in a realHilbert space H :(1.1) ∂ t u + Au = F ( u ) , u (cid:12)(cid:12) t =0 = u , where A : D ( A ) → H is a self-adjoint positive operator such that A − iscompact and F : H → H is a given nonlinearity which is globally Lips-chitz in H with Lipschitz constant L . Let also 0 < λ ≤ λ ≤ · · · be theeigenvalues of A enumerated in the non-decreasing order and { e n } ∞ n =1 be thecorresponding eigenvectors. Then, the sufficient condition for the existenceof N -dimensional IM reads(1.2) λ N +1 − λ N > L. If this condition is satisfied, the desired IM M N is actually a graph of aLipschitz function M N : H N → ( H N ) ⊥ , where H N = span { e , · · · , e N } is aspectral subspace spanned by the first N eigenvectors, and the correspondingIF has the form(1.3) ddt u N + Au N = P N F ( u N + M N ( u N )) , u N ∈ H N ∼ R N , where P N is the orthoprojector to H N , see [5, 6, 10, 19, 29, 35, 37, 42] andalso § F and M N are. We mentionthat although the spectral gap condition (1.2) is rather restrictive (e.g. inthe case where A is a Laplacian in a bounded domain, it is satisfied in 1D A. KOSTIANKO AND S. ZELIK case only) and is known to be sharp in the class of abstract semilinear par-abolic equations (see [7, 29, 35, 42] for more details), it can be relaxed forsome concrete classes of PDEs. For instance, for scalar 3D reaction-diffusionequations (using the so-called spatial averaging principle, see [27]), for 1Dreaction-diffusion-advection systems (using the proper integral transforms,see [23, 24]), for 3D Cahn-Hilliard equations and various modifications of 3DNavier-Stokes equations (using various modifications of spatial-averaging,see [11, 20, 22, 26]), for 3D complex Ginzburg-Landau equation (using theso-called spatio-temporal averaging, see [21]), etc. Note also that the globalLipschitz continuity assumption for the non-linearity F is not an essentialextra restriction since usually one proves the well-posedness and dissipativ-ity of the PDE under consideration before constructing the IM. Cutting offthe non-linearity outside the absorbing ball does not affect the limit dy-namics, but reduces the case of locally Lipschitz continuous non-linearity(satisfying the proper dissipativity restrictions) to the model case where thenon-linearity is globally Lipschitz continuous. Of course, this cut-off proce-dure is not unique and as we will see below, the right choice of it is extremelyimportant in the theory of IMs.The main aim of the present paper is to study the smoothness of theIFs for semilinear parabolic equations (1.1) in the ideal situation where thenon-linearity F is smooth and the spectral gap condition (1.2) is satisfied.As we have already mentioned, in this case we have C ,ε -smooth IM M N for some ε > C ,ε -smooth, see[42] and references therein. But, unfortunately, the exponent ε > C -regularity of the IM. Thespectral gap condition for C -regular IM is(1.4) λ N +1 − λ N > L and such exponentially big spectral gaps are not available if A is a finite orderelliptic operator in a bounded domain. The corresponding counterexampleswere given in [5], see also Example 3.11 below. Thus, the existing IM theorydoes not allow us, even in the ideal situation, to construct more regular than C ,ε IFs (where ε > smooth
PDE using the standard methods may bemore effective than simulations based on the reduced non-smooth ODEs);2) C ,ε -regularity is not enough to build up normal forms and/or studythe bifurcations properly (for instance, the simplest saddle-node bifurcationrequires C -smoothness, the Hopf bifurcation needs C , etc., see [17, 18] formore details) and, therefore, we need to return back to the initial PDE tostudy these bifurcations. Thus, the natural question” Is it possible to construct a smooth ( C k -smooth for any finite k ) or toextend the existing C ,ε -smooth IF to a more regular one ?”become crucial for the theory of inertial manifolds. XTENSIONS OF INERTIAL MANIFOLDS 5
In the present paper we give an affirmative answer on this question underslightly stronger spectral gap assumption(1.5) lim sup N →∞ ( λ N +1 − λ N ) = ∞ . In contrast to (1.4), this assumption does not require exponentially big spec-tral gaps (and is satisfied for the most part of examples where the IMs exist),but guarantees the existence of infinitely-many spectral gaps of size largerthan 2 L and, consequently, the existence of an infinite tower of the embeddedIMs:(1.6) M N ⊂ M N ⊂ · · · ⊂ M N n ⊂ · · · and the corresponding IFs(1.7) ddt u N n + Au N n = P N n F ( u N n + M N n ( u N n )) , u N n ∈ H N n . Let n ∈ N be given. We say that a C n,ε -smooth submanifold f M N n of thephase space H (which is a graph of C n,ε -smooth f M N n : H N n → ( H N n ) ⊥ ) isa C n,ε -smooth extension of the initial IM M N for some ε > M N ⊂ f M N n ;2) The manifold f M N n is C b -close to the IM M N n .Then, the first condition guarantees that the C n,ε -smooth system of ODEs(1.8) ddt u N n + Au N n = P N n F ( u N n + f M N n ( u N n )) , u N n ∈ H N n will possess the initial IM P N n M N as an invariant submanifold. The sec-ond condition together with the robustness theorem for normally hyperbolicmanifolds ensures us that this manifold will be globally exponentially sta-ble and normally hyperbolic (in particular, it will possess an exponentialtracking property in H N n ). In this case we refer to the system (1.8) as a C n,ε -smooth extension of the corresponding IF (1.3), see § C n -smooth on the one hand and, on theother hand, its limit dynamics coincides with the dynamics of the IF whichcorresponds to the IM M N and, in turn, coincides with the limit dynamicsof the initial abstract parabolic problem (1.1). Note that the manifold f M N n is not necessarily invariant under the solution semigroup S ( t ) generated bythe initial equation (1.1) and this allows us to overcome the standard ob-stacles to the smoothness of an invariant manifold (e.g. such as resonances,see Examples 3.11 and 6.6 below).The main result of the paper is the following theorem which suggests asolution of the smoothness problem for IMs. Theorem 1.1.
Let the nonlinearity F ∈ C ∞ b ( H, H ) and let the operator A satisfy the spectral gap conditions (1.5) . Let also N ∈ N be the smallestnumber for which the spectral gap condition (1.2) is satisfied and M N be thecorresponding IM. Then, for every n ∈ N , there exists ε = ε n > and the C n,ε -smooth extension of the IM M N as well as the C n,ε -smooth extensionof the corresponding IF in the sense described above. The proof of this theorem is given in § §
5. To construct the de-sired extension f M N n , we first define it on the manifold P N n M N only in a A. KOSTIANKO AND S. ZELIK natural way f M N n ( p ) = (1 − P N n ) M N ( P N p ). Then, we present an explicitconstruction of Taylor jets of order n for this function via some inductiveprocedure, see §
4. Finally, we check (in §
5) the compatibility conditions forthe constructed Taylor jets and get the desired extension by the Whitneyextension theorem.Our main result can be reformulated in the following way.
Corollary 1.2.
Let the assumptions of Theorem 1.1 hold. Then, for every n ∈ N , there exists ε = ε n > and C n − ,ε -smooth ”correction” e F n ( u ) of theinitial nonlinearity F such that1) e F n ( u ) = F ( u ) for all u ∈ M N and M N is an IM for the modifiedequation (1.9) ∂ t u + Au = e F n ( u ) , u (cid:12)(cid:12) t =0 = u as well. In particular, the dynamics of (1.9) on M N coincides with theinitial dynamics (generated by (1.1) ) and M N possesses an exponentialtracking property for solutions of (1.9) .2) The extended manifold f M N n constructed in Theorem 1.1 is an IM (ofsmoothness C n,ε ) for the modified equation (1.9) , see Corollary 6.4 below. In this interpretation, the modified nonlinearity e F n can be considered asa ”cutted-off” version of the initial function F and the main result claimsthat all obstacles for the existence of C n -smooth IM can be removed byincreasing the dimension of the IM and using the properly chosen cut-offprocedure.To conclude, we note that the main aim of this paper is to verify theprincipal possibility to get smooth extensions of IM rather than to obtainthe optimal bounds for the dimensions N n of the constructed extensions. Bythis reason, the obtained bounds look far from being optimal, but we believethat they can be essentially improved, see Remark 6.7 for the discussion ofthis problem.The paper is organized as follows. In § § §
4. The proof of it is also given there by modulo ofcompatibility conditions for Whitney extension theorem which are verifiedin §
5. Finally, the applications of the proved theorem as well as a discussionof open problems and related topics are given in § Preliminaries I: Taylor expansions and Whitney ExtensionTheorem
In this section we briefly recall the standard results on Taylor expansionsof smooth functions in Banach spaces and related Whitney extension the-orem as well as prepare some technical tools which will be used later. Westart with some basic facts from multi-linear algebra, see e.g. [12] for a moredetailed exposition. Let X and Y be two normed spaces. For any n ∈ N , we XTENSIONS OF INERTIAL MANIFOLDS 7 denote by L s ( X n , Y ) the space of multi-linear continuous symmetric mapsfrom X n to Y endowed by the standard norm k M k L s ( X n ,Y ) := sup ξ i ∈ X, ξ i =0 (cid:26) k M ( ξ , · · · , ξ n ) kk ξ k · · · k ξ n k (cid:27) . Every element M ∈ L s ( X n , Y ) defines a homogeneous continuous polyno-mial P M of order n on X with values in Y via P M ( ξ ) := M ( { ξ } n ) , where { ξ } n := ξ, · · · , ξ | {z } n -times . Vice versa, the multi-linear symmetric map M = M P can be restored in aunique way if the corresponding homogeneous polynomial is known via thepolarization equality: M P ( ξ , · · · , ξ n ) = 12 n n ! X ε i = ± ,i =1 , ··· ,n ε · · · ε n P ( a + n X j =1 ε j ξ j )for all a, ξ , · · · , ξ n ∈ X , see e.g. [12]. Thus, there is a one-to-one corre-spondence between homogeneous polynomials and multi-linear symmetricmaps. Moreover, if we introduce the following norm on the space P n ( X, Y )of n -homogeneous polynomials k P k P n ( X,Y ) := sup ξ =0 (cid:26) k P ( ξ ) kk ξ k n (cid:27) , this correspondence becomes an isometry. By this reason, we will iden-tify below multi-linear forms and the corresponding homogeneous polyno-mials where this does not lead to misunderstandings. We also mentionhere the generalization of the Newton binomial formula, namely, for any P ∈ P n ( X, Y ) and ξ, η ∈ X , we have(2.1) P ( ξ + η ) = n X j =0 C jn P ( { ξ } j , { η } n − j ) , C jn := n ! j !( n − j )! , see e.g. [12]. Finally, we denote by P n ( X, Y ) the space of all continuouspolynomials of order less than or equal to n on X with values in Y , i.e. P ( ξ ) ∈ P n ( X, Y ) if P ( ξ ) = n X j =0 j ! P j ( ξ ) , P j ( ξ ) ∈ P j ( X, Y ) . The following standard result is crucial for our purposes.
Lemma 2.1.
For every n ∈ N there exist real numbers a kj ∈ R , k, j ∈{ , · · · , n } , such that for every P = P nk =0 1 k ! P k , P k ∈ P k ( X, Y ) and every k ∈ { , · · · , n } , we have (2.2) P k ( ξ ) = n X j =0 a kj P (cid:18) jn ξ (cid:19) and, therefore, (2.3) k P k ( ξ ) k ≤ K n,k max j =0 , ··· ,n k P (cid:18) jn ξ (cid:19) k A. KOSTIANKO AND S. ZELIK for some constants K n,k which are independent of P . For the proof of this lemma see [12].
Corollary 2.2.
Let P ( ξ, δ ) ∈ P n ( X, Y ) be a family of polynomials of ξ depending on a parameter δ ∈ B where B is a set in X containing zero.Assume that (2.4) k P ( ξ, δ ) k ≤ C ( k ξ k + k δ k ) n + α , ξ ∈ X, δ ∈ B for some α ≥ . Then, for any k ∈ { , · · · , n } , (2.5) k P k ( · , δ ) k P k ( X,Y ) ≤ C k k δ k n − k + α for some constants C k depending on C , n and k .Proof. Indeed, according to (2.3) and (2.4), we have k P k ( ξ, δ ) k ≤ C ′ ( k ξ k + k δ k ) n + α . Assuming that δ = 0 (there is nothing to prove otherwise), replacing ξ by k δ k ξ and using that P k is homogeneous of order k , we get k P k ( ξ, δ ) k ≤ C ′ (1 + k ξ k ) n + α k δ k n − k + α . Using once more that P k is homogeneous of order k in ξ , we finally arrive at k P k ( ξ, δ ) k ≤ C ′′ k ξ k k k δ k n − k + α which gives (2.5) and finishes the proof. (cid:3) Let now U ⊂ X be an open set and let F : U → Y be a map. As usual, forany u ∈ U , we denote by F ′ ( u ) ∈ L ( X, Y ) the Frechet derivative of F at u (if it exists). Analogously, for any n ∈ N , we denote by F ( n ) ( u ) ∈ L s ( X n , Y )its n th Frechet derivative. The space of all functions F : U → Y such that F ( n ) ( u ) exists and continuous as a function from U to L s ( X n , Y ) is denotedby C n ( U, Y ). For any α ∈ (0 , C n,α ( U, Y ) the space offunctions F ∈ C n ( U, Y ) such that F ( n ) is H¨older continuous with exponent α on U . The action of F ( n ) ( u ) to vectors ξ , · · · , ξ n ∈ X is denoted by F ( n ) ( u )[ ξ , · · · , ξ n ]. The Taylor jet of length n + 1 of the function F at point u and vector ξ ∈ X will be denoted by J nξ F ( u ):(2.6) J nξ F ( u ) := F ( u ) + 11! F ′ ( u ) ξ + 12! F ′′ ( u )[ ξ, ξ ] + · · · + 1 n ! F ( n ) ( u )[ { ξ } n ] . Obviously, the function ξ → J nξ F ( u ) ∈ P n ( X, Y ) for every u ∈ U . We willalso systematically use the truncated Taylor jets(2.7) j nξ F ( u ) := 11! F ′ ( u ) ξ + 12! F ′′ ( u )[ ξ, ξ ] + · · · + 1 n ! F ( n ) ( u )[ { ξ } n ]which do not contain zero order term. Theorem 2.3 (Direct Taylor theorem) . Let F ∈ C n ( U, Y ) and u , u ∈ U be such that u t := tu + (1 − t ) u ∈ U for all t ∈ [0 , . Let also ξ := u − u .Then (2.8) F ( u ) = J nξ F ( u )++ 1 n ! Z (1 − s ) n − (cid:16) F ( n ) ( u + sξ ) − F ( n ) ( u ) (cid:17) ds [ { ξ } n ] . XTENSIONS OF INERTIAL MANIFOLDS 9
In particular, if F ∈ C n,α ( U, Y ) , then (2.9) k F ( u ) − J nξ F ( u ) k ≤ C k ξ k n + α for some positive C . For the proof of this classical result see e.g. [12]. We also mention thatin terms of truncated jets formula (2.9) reads(2.10) F ( u ) − F ( u ) = j nξ F ( u ) + O ( k ξ k n + α ) , ξ := u − u . The above theorem can be inverted as follows.
Theorem 2.4 (Converse Taylor theorem) . Let function F be such that, forany u ∈ U there exists a polynomial ξ → P ( ξ, u ) ∈ P n ( X, Y ) such that, forall u , u ∈ U , (2.11) k F ( u ) − P ( ξ, u ) k ≤ C k ξ k n + α , ξ := u − u for some C > and α ∈ (0 , . Then, F ∈ C n ( U, Y ) , P ( ξ, u ) = J nξ F ( u ) for all u ∈ U and F ( n ) ( u ) is locally H¨older continuous in U with exponent α . If, in addition, U is convex, then F ∈ C n ( U, Y ) and k F ( n ) ( u ) − F ( n ) ( u ) k ≤ C ′ k u − u k α , where C ′ depends only on n , α and the constant C from (2.11) . For the proof of this theorem see [12].Keeping in mind the Whitney extension problem, we recall that arbitrarilychosen set of polynomials P ( ξ, u ), u ∈ U , does not define in general a C n,α -smooth function, but some compatibility conditions must be satisfied forthat. Indeed, let u ∈ U and let δ, ξ ∈ X be such that u := u + δ ∈ U and u := u + δ + ξ = u + ξ ∈ X . Then, from (2.9), we have k F ( u ) − P ( ξ + δ, u ) k ≤ C k ξ + δ k n + α and k F ( u ) − P ( ξ, u + δ ) k ≤ C k ξ k n + α . Therefore,(2.12) k P ( ξ + δ, u ) − P ( ξ, u + δ ) k ≤ C ( k ξ k + k δ k ) n + α . These are the desired compatibility conditions. In other words, if we aregiven a set V ⊂ X and a family of polynomials { P ( ξ, u ) , u ∈ V } ⊂ P n ( X, Y )and want to find a function F ∈ C n,α ( X, Y ) such that J nξ F ( u ) = P ( ξ, u ) forall u ∈ V , then the compatibility conditions (2.12) must be satisfied for all u , u + δ ∈ V and all ξ ∈ X .Inequalities (2.12) can be rewritten in a more standard form which usuallyappears in the statement of Whitney extension theorem. Namely, using(2.1), we see that P ( ξ + δ, u ) = n X l =0 l ! n X k = l k − l )! P k ([ { ξ } l , { δ } k − l ] , u ) where P ( ξ, u ) = P nl =0 1 l ! P l ([ { ξ } l ] , u ), P l ( · , u ) ∈ P l ( X, Y ). Applying nowCorollary 2.2 to (2.12), we get the desired alternative form of the compati-bility conditions:(2.13) k P l ( { ξ } l , u + δ ) − n − l X k =0 k ! P l + k ([ { ξ } l , { δ } k ] , u ) k ≤ C k ξ k l k δ k n − l + α for l = { , · · · , n } . Compatibility conditions (2.13) have natural interpre-tation: if P k ( { ξ } , u ) = F ( k ) ( u )[ { ξ } k ] as we expect, then (2.13) is nothingelse than Taylor expansions of F ( l ) ( u + δ )[ { ξ } l ] at u .The next theorem shows that the introduced compatibility conditions aresufficient for the existence of F in the case when X is finite-dimensional. Theorem 2.5 (Whitney extension theorem) . Let dim
X < ∞ and let V be an arbitrary subset of X . Assume also that we are given a family ofpolynomials { P ( ξ, u ) , u ∈ V } ⊂ P n ( X, Y ) which satisfies the compatibilityconditions (2.12) with some α ∈ (0 , . Then, there exists a function F ∈ C n,α ( X, Y ) such that J nξ F ( u ) = P ( ξ, u ) for all u ∈ V . For the proof of this theorem see [39] or [8]. Note that the theorem fails ifthe dimension of X is infinite, but there are no restrictions on the dimensionof the space Y , see [41].3. Preliminaries II: Spectral gaps and the construction of aninertial manifold
In this section we briefly discuss the classical theory of inertial manifoldsfor semilinear parabolic equations, see e.g. [42] for a more detailed exposi-tion.Let H be an infinite-dimensional real Hilbert space. Let us consider anabstract parabolic equation in H :(3.1) ∂ t u + Au = F ( u ) , u (cid:12)(cid:12) t =0 = u , where A : D ( A ) → H is a linear self-adjoint positive operator in H withcompact inverse and F ∈ C ∞ b ( H, H ) is a smooth bounded function on H such that all its derivatives are also bounded on H .It is well-known that under the above assumptions equation (3.1) is glob-ally well-posed for any u ∈ H in the class of solutions u ∈ C ([0 , T ] , H ) forall T > H :(3.2) S ( t ) : H → H, t ≥ , S ( t ) u := u ( t ) . Moreover, the solution operators S ( t ) ∈ C ∞ ( H, H ) for every fixed t ≥
0, see[14, 42] for the details.Let 0 < λ ≤ λ ≤ · · · be the eigenvalues of the operator A enumerated inthe non-decreasing order and let { e n } ∞ n =1 be the corresponding orthonormalsystem of eigenvectors. Then, by the Parseval equality, for every u ∈ H , wehave k u k H = ∞ X n =1 ( u, e n ) , u = ∞ X n =1 ( u, e n ) e n , XTENSIONS OF INERTIAL MANIFOLDS 11 where ( · , · ) is an inner product in H . For a given N ∈ N , we denote by P N and Q N the orthoprojectors on the first N and the rest of eigenvectors of A respectively: P N u := N X n =1 ( u, e n ) e n , Q N u := ∞ X n = N +1 ( u, e n ) e n . We are now ready to introduce the main object of study in this paper - aninertial manifold (IM).
Definition 3.1.
A set M = M N is an inertial manifold of dimension N forproblem (3.1) (with the base H N := P N H ) if1. M is invariant with respect to the semigroup S ( t ): S ( t ) M = M .2. M is a graph of a Lipschitz continuous function M : H N → Q N H : M = { p + M ( p ) , p ∈ H N } . M possesses an exponential tracking property, namely, for every tra-jectory u ( t ) of (3.1) there exists a trace solution ¯ u ( t ) ∈ M such that(3.3) k u ( t ) − ¯ u ( t ) k ≤ Ce − θt , t ≥ θ > λ N and constant C = C u which depends on u .Note that, although only Lipschitz continuity is traditionally required inthe definition, usually IMs are C ,ε -smooth for some ε > M is governed by thesystem of ODEs:(3.4) ddt u N + Au N = P N F ( u N + M ( u N )) , u N := P N u ∈ R N which is called an inertial form (IF) associated with equation (3.1). Inthe case where the spectral subspace H N is used as a base for IM (like inDefinition 3.1), the regularity of the corresponding vector field in the IF isdetermined by the regularity of the IM only.The following theorem is the key result in the theory of IMs. Theorem 3.2.
Let the function F in equation (3.1) be globally Lipschitzcontinuous with Lipschitz constant L and let, for some N ∈ N , the followingspectral gap condition (3.5) λ N +1 − λ N > L be satisfied. Then equation (3.1) possesses an IM M N of dimension N .Proof. Although this statement is classical, see e.g. [29, 35, 42], the elementsof its proof will be crucially used in what follows, so we sketch them below.To construct the IM, we will use the so-called Perron method, namely, wewill prove that, for every p ∈ H N the problem(3.6) ∂ t u + Au = F ( u ) , t ≤ , P N u (cid:12)(cid:12) t =0 = p possesses a unique backward solution u ( t ) = V ( p, t ), t ≤
0, belonging to theproper weighted space, and then define the desired map M : H N → Q N H via(3.7) M ( p ) := Q N V ( p, . To solve (3.6) we use the Banach contraction theorem treating the nonlin-earity F as a perturbation. To this end we need the following two lemmas. Lemma 3.3.
Let θ ∈ ( λ N , λ N +1 ) and let us consider the equation (3.8) ∂ t v + Av = h ( t ) , t ∈ R , h ∈ L e θt ( R , H ) , where the space L e θt ( R , H ) is defined via the weighted norm (3.9) k h k L eθt ( R ,H ) := Z t ∈ R e θt k h ( t ) k dt < ∞ . Then, problem (3.8) possesses a unique solution u ∈ L e θt ( R , H ) and thesolution operator T : L e θt → L e θt , u := T h satisfies: (3.10) kT k L ( L eθt ,L eθt ) = 1min { θ − λ N , λ N +1 − θ } . The proof of this identity is just a straightforward calculation based ondecomposition of the solution u ( t ) with respect to the base { e n } ∞ n =1 andsolving the corresponding ODEs, see [42].The second lemma gives the analogue of this formula for the linear equa-tion on negative semi-axis. Lemma 3.4.
Let θ ∈ ( λ N , λ N +1 ) . Then, for any p ∈ H N and any h ∈ L e θt ( R − , H ) , the problem (3.11) ∂ t v + Av = h ( t ) , t ≤ , P N v (cid:12)(cid:12) t =0 = p possesses a unique solution v ∈ L e θt ( R − , H ) . This solution can be writtenin the form v = T h + H p, where T is exactly the solution operator constructed in Lemma 3.3 appliedto the extension of the function h ( t ) by zero for t ≥ and H : H N → L e θt ( R − , H ) is a solution operator for the problem with zero right-hand side: H ( p, t ) := N X n =1 ( p, e n ) e − λ n t . Indeed, this lemma is an easy corollary of Lemma 3.3, see [42].We are now ready to prove the theorem. To this end, we fix an optimalvalue θ = λ N +1 + λ N and write the equation (3.6) as a fixed point problem(3.12) u = T ◦ F ( u ) + H ( p )in the space L e θt ( R − , H ). Since the norm of the operator T is equal to λ N +1 − λ N and the Lipschitz constant of F is L , the spectral gap condition(3.5) guarantees that the right-hand side of (3.12) is contraction for every p ∈ H N . Thus, by the Banach contraction theorem, for every p ∈ H N ,there exists a unique solution u ( t ) = V ( p, t ) of problem (3.6) belonging to XTENSIONS OF INERTIAL MANIFOLDS 13 L e θt ( R − , H ) and the map p → V ( p, · ) is Lipschitz continuous. Due to theparabolic smoothing property, we know that k u (0) k ≤ C (1 + k u k L ([ − , ,H ) ) and k u (0) − w (0) k ≤ C k u k L ([ − , ,H ) for any two backward solutions u, w of (3.1), see e.g. [42]. In particular,these formulas show that the solution V ( p, t ) is continuous in time ( V ( p, · ) ∈ C e θt ( R − , H ), where the weighted space of continuous functions is definedanalogously to (3.9)) and the map p → V ( p, · ) is Lipschitz continuous asa map from H N to C e θt ( R − , H ). Thus, formula (3.7), defines indeed aLipschitz manifold of dimension N over the base H N as graph of Lipschitzcontinuous function M : H N → Q N H .The invariance of this manifold follows by the construction, so we onlyneed to verify the exponential tracking property.Let u ( t ) = S ( t ) u be an arbitrary solution of problem (3.1) and let φ ( t ) ∈ C ∞ ( R ) be a cut-off function such that φ ( t ) ≡ t ≤ φ ( t ) ≡ t ≥
1. Then the function φ ( t ) u ( t ) is defined for all t ∈ R . We seek forthe desired solution ¯ u ( t ) ∈ M (by the construction of M such solutions aredefined for all t ∈ R ) in the form(3.13) ¯ u ( t ) = φ ( t ) u ( t ) + v ( t ) . Inserting this anzatz to (3.1), we end up with the equation for v ( t ):(3.14) ∂ t v + Av = F ( φu + v ) − φF ( u ) − φ ′ u. Let v ∈ L e θt ( R , H ) be a solution of this equation. Then, since ¯ u = v for t ≤
0, we necessarily have ¯ u ∈ M by the construction of the IM. On theother hand, for t ≥
1, we have v = ¯ u − u ∈ L e θt ([1 , ∞ ) , H ) and using theparabolic smoothing again, we get the desired estimate (3.3). Thus, we onlyneed to find such a solution v ( t ). To this end, we invert the linear part ofequation (3.14) to get the fixed point equation(3.15) v = T ( F ( φu + v ) − φF ( u ) − φ ′ u ) . It is straightforward to verify using Lemma 3.3 that the right-hand side of(3.15) is a contraction on the space L e θt ( R , H ) if the spectral gap conditionholds, see [42]. Thus, the Banach contraction theorem finishes the proof ofexponential tracking. (cid:3) Remark 3.5.
It is well-known that the spectral gap condition (3.5) is sharpin the sense that if it is violated for some N and L , one can find a nonlinearity F such that equation (3.1) does not possess an IM of dimension N with base H N , see [35].More recent examples show that if this condition is violated for all N :sup N ∈ N { λ N +1 − λ N } < L, one can construct a smooth nonlinearity F such that equation (3.1) doesnot possess any Lipschitz or even Log-Lipschitz finite-dimensional manifold(not necessarily invariant) which contain the global attractor, see [7, 42]. Remark 3.6.
Theorem 3.2 guarantees the existence of an IM M N for every N such that the spectral gap condition (3.5) is satisfied. Typically, this N isnot unique, instead, we have a whole sequence { N k } ∞ k =1 of N s satisfying the spectral gap condition. Therefore, according to the theorem, we will have asequence of IMs {M N k } ∞ k =1 of increasing dimensions: N < N < N < · · · .Moreover, from the explicit description of an IM using backward solutionsof (3.6), we see that(3.16) M N ⊂ M N ⊂ M N ⊂ · · · In this case it can be also proved that M N k − is a normally hyperbolicsubmanifold of M N k .Let us now discuss the further regularity of the IM M . To this end, weneed one more auxiliary statement. Proposition 3.7.
Let the spectral gap condition (3.5) hold and let u ( t ) ∈ C ( R − , H ) be an arbitrary function. Let also the exponent θ ∈ ( λ N , λ N +1 ) satisfy (3.17) θ − := L + λ N < θ < λ N +1 − L := θ + . Then, for any h ∈ L e θt ( R − , H ) and every p ∈ H N the corresponding equationof variations (3.18) ∂ t v + Av − F ′ ( u ( t )) v = h ( t ) , t ≤ , P N v (cid:12)(cid:12) t =0 = p possesses a unique solution v ∈ L e θt ( R − , H ) ∩ C e θt ( R − , H ) and the followingestimate holds: (3.19) k v k C eθt ( R − ,H ) ≤ C k v k L eθt ( R − ,H ) ≤ C L,θ (cid:16) k h k L eθt ( R − ,H ) + k p k (cid:17) , where the constant C L,θ is independent of u , h and p . Indeed, equation (3.18) can be solved via the Banach contraction theoremtreating the term F ′ ( u ) v as a perturbation analogously to the non-linearcase. Inequalities (3.17) guarantee that the map T F ′ ( u ) v is a contractionon L e θt ( R − , H ), due to (3.10). Corollary 3.8.
Let the assumptions of Theorem 3.2 hold and let, in addi-tion, the exponent ε ∈ (0 , be such that (3.20) λ N +1 − (1 + ε ) λ N > (2 + ε ) L. Assume also that F ∈ C ,ε ( H, H ) . Then the associated IM M N is C ,ε -smooth, for any p, ξ ∈ H N , the derivative M ′ ( p ) ξ can be found as the valueof the Q N projection of V ′ ( t ) = V ′ ( p, t ) ξ at t = 0 , where the function V ′ solves the equation of variations: (3.21) ∂ t V ′ + AV ′ − F ′ ( u ( t )) V ′ = 0 , t ≤ , P N V ′ (cid:12)(cid:12) t =0 = ξ, u ( t ) := V ( p, t ) and k M ′ ( p ) − M ′ ( p ) k L ( H N ,H ) ≤ C k p − p k ε for some constant C independent of p , p ∈ H N .Proof. Let p , p ∈ H N and u i ( t ) := V ( p i , t ) be the corresponding trajecto-ries belonging to the IM. Let also v ( t ) := u ( t ) − u ( t ) and ξ := p − p .Then v solves(3.22) ∂ t v + Av − L u ,u ( t ) v = 0 , t ≤ , P N v (cid:12)(cid:12) t =0 = ξ, XTENSIONS OF INERTIAL MANIFOLDS 15 where L u ,u ( t ) := R F ′ ( su ( t )+(1 − s ) u ( t )) ds . Since the norm of L u ,u ( t )does not exceed L , Proposition 3.7 is applicable to equation (3.22) and,therefore, for every θ satisfying (3.17), we have the estimate(3.23) k v k C eθt ( R − ,H ) ≤ C k v k L eθt ( R − ,H ) ≤ C θ k p − p k . Note also that the function V ′ ( p, t ) ξ is well-defined for all p, ξ ∈ H N dueto Proposition 3.7 and satisfy the analogue of (3.23). Let w ( t ) := v ( t ) − V ′ ( p , t ) ξ with ξ := p − p . Then, this function solves(3.24) ∂ t w + Aw − F ′ ( u ) w == F ( u ) − F ( u ) − F ′ ( u ) v := h u ,u ( t ) , P N w (cid:12)(cid:12) t =0 = 0 . Since F ∈ C ,ε ( H, H ), by the Taylor theorem, we have k h u ,u ( t ) k ≤ C k v ( t ) k ε which, due to (3.23), gives k h u ,u k L e (1+ ε ) θt ( R − ,H ) ≤ C k v k L eθt ( R − ,H ) k v k εC eθt ( R − ,H ) ≤ C ′ k ξ k ε . Fixing now θ in such a way that θ > θ − and (1 + ε ) θ < θ + (this is possibleto do due to assumption (3.20)) and applying Proposition 3.7 to equation(3.24), we finally arrive at k M ( p ) − M ( p ) − M ′ ( p ) ξ k = k w (0) k ≤ C k w k L e (1+ ε ) θt ( R − ,H ) ≤ C k ξ k ε and the converse Taylor theorem finishes the proof of the corollary. (cid:3) The next corollary claims that the constructed manifold M is actuallylives in higher regular space H := D ( A ). Corollary 3.9.
Let the assumptions of Corollary 3.8 hold. Then the man-ifold M is simultaneously a C ,ε -smooth IM for equation (3.1) in the phasespace H = D ( A ) .Proof. This is an almost immediate corollary of the parabolic smoothingproperty. Indeed, let us first check that
M ∈ H . To this end, it is enoughto check that the backward solution (3.6) actually belongs to C e θt ( R − , H ).First, using the L ( H )-maximal regularity for the solutions of a linear par-abolic equation(3.25) ∂ t v + Av = h ( t ) , t ≤ , namely, that(3.26) k v k C α ( − , H ) + k ∂ t v k L ( − , H ) + k Av k L ( − , H ) ≤≤ C α (cid:0) k h k L ( − , ,H ) + k v k L ( − , ,H ) (cid:1) , where α ∈ (0 , ), we end up with the estimate(3.27) k u k C α ( − , H ) ≤ C α (cid:0) k F ( u ) k L ( − , H ) + k u k L ( − , H ) (cid:1) ≤≤ C α,θ (1 + k u k L eθt ( R − ,H ) ) ≤ C (1 + k p k ) , where α ∈ (0 , ). Second, using the C α ( H )-maximal regularity for solutionsof (3.25) and the obvious estimate k F ( u ) k C α ( − , H ) ≤ k F k C α ( H,H ) (1 + k u k αC α ( − , H ) ) , we arrive at(3.28) k ∂ t u k C α ( − , H ) + k Au k C α ( − , H ) ≤≤ C (cid:0) k F ( u ) k C α ( − , H ) + k u k C α ( − , H ) (cid:1) ≤≤ C (cid:0) k u k C α ( − , H ) (cid:1) ≤ C (1 + k p k )and the fact that M ( p ) belongs to H is proved. The fact that M is C ,ε -smooth as a map from H N to H can be verified analogously and the corol-lary is proved. (cid:3) Remark 3.10.
The analogue of Corollary 3.8 holds for higher derivativesas well. For instance, if we want to have C n,ε -smooth IM, we need to requirethat(3.29) λ N +1 − ( n + ε ) λ N > ( n + 1 + ε ) L. To verify this, we just need to define the higher order Taylor jets for the IM M using second, third, etc., equations of variations for (3.6) and use againProposition 3.7. For instance, the second derivative V ′′ = V ′′ ( p, t )[ ξ, ξ ]solves(3.30) ∂ t V ′′ + AV ′′ − F ′ ( u ( t )) V ′′ = F ′′ ( u ( t ))[ V ′ ( p, t ) ξ, V ′ ( p, t ) ξ ] , P N V ′′ (cid:12)(cid:12) t =0 = 0 , u ( t ) := V ( p, t ) . According to Proposition 3.7, in order to be able to solve this equation,we need θ + > θ − (since V ′ ∈ L e θt with θ > θ − and the right-hand side F ′′ ( u )[ V ′ , V ′ ] ∈ L e θt ) which gives (3.29) for n = 2.We believe that sufficient condition (3.29) for the existence of C n,ε -smoothIM is sharp for any n and ε , but we restrict ourselves by recalling below theclassical counterexample of G. Sell to the existence of C -smooth IM whichdemonstrates the sharpness of (3.29) for n = 2, see [5]. Example 3.11.
Let H := l (space of square summable sequences with thestandard inner product) and let us consider the following particular case ofequation (3.1):(3.31) ddt u + u = 0 , ddt u n + 2 n − u n = u n − , n = 2 , , · · · Here λ n = 2 n − and we have a set of resonances 2 λ n = λ n +1 which preventthe existence of any finite-dimensional invariant local manifold of dimensiongreater than zero which is C -smooth and contains zero. Note that thenon-linearity here is locally smooth near zero and since we are interestedin local invariant manifolds near zero, the behaviour of it outside the smallneighbourhood of zero is not important (we may always cut-off it outsideof the neighbourhood to get global Lipschitz continuity). Moreover, since F ′ (0) = 0, decreasing the size of the neighbourhood we may make the Lip-schitz constant L as small as we want. Thus, according to Corollary 3.8, XTENSIONS OF INERTIAL MANIFOLDS 17 for any N ∈ N , there exists a local invariant manifold M N of dimension N with the base H N which is C ,ε -smooth for any ε < C -smooth invariant local manifold does not exist. In-deed, let M N be such a manifold of dimension N . Then, since the tangentplane T M N (0) to this manifold at zero is invariant with respect to A (dueto the fact that F ′ (0) = 0), we must have H ′ N := T M N (0) = span { e n , · · · , e n N } for some n < n < · · · < n N . Thus, the manifold M N can be presentedlocally near zero as a graph of C -function M : H ′ N → ( H ′ N ) ⊥ such that M (0) = M ′ (0) = 0. In particular, expanding M in Taylor series near zero,we have u n N +1 = ( M ( u n , · · · , u n N ) , e n N +1 ) = cu n N + · · · Let us try to compute the constant c . Inserting this in the ( n N + 1)-thequation and using the invariance, we get(3.32) ∂ t u n N +1 + 2 n N u n N +1 = 2 c∂ t u n N u n N + 2 n N cu n N + · · · == − c n N − u n N + 2 n N cu n N + · · · = 0 + · · · = u n N which gives 0 = 1. Thus, the manifold M N cannot be C -smooth. Remark 3.12.
Note that in the case where A is an elliptic operator of order2 k in a bounded domain Ω of R d , we have λ n ∼ Cn k/d due to the Weylasymptotic. Thus, one may expect in general only the gaps of the size(3.33) λ N +1 − λ N ∼ CN kd − ∼ C ′ λ − d k N which is much weaker than (3.29) with n >
1. Sometimes the exponentin the right-hand side of (3.33) may be improved due to big multiplicityof eigenvalues (e.g. for the Laplace-Beltrami operator on a sphere S d , wehave λ / N there for all d ), but this exponent is always less than one in allmore or less realistic examples. Thus, the existence of C n -smooth IMs with n > λ , · · · , λ N are close to zero, λ N +1 is of order one and L is small.In contrast to this, if the spectral gap conditions (3.5) are satisfied forsome N , we always can find a small positive ε = ε N such that (3.20) will bealso satisfied. Thus, if the nonlinearity F is smooth enough, we automati-cally get a C ,ε -smooth IM for some small ε depending on N and L . Remark 3.13.
Let ¯ u ( t ) be a trajectory of (3.1) belonging to the IM, i.e. Q N ¯ u ( t ) ≡ M N ( P N ¯ u ( t ))and let ¯ u N := P N ¯ u ( t ). Then, we may write a linearization near the trajec-tory ¯ u ( t ) in two natural ways. First, we may just linearize equation (3.1)without using the fact that ¯ u ∈ M N . This gives the equation(3.34) ∂ t v + Av − F ′ (¯ u ) v = h ( t )which we have used above to get the existence of the IM, its smoothnessand exponential tracking. Alternatively, we may linearize the reduced ODEs (3.4):(3.35) ∂ t v N + Av N − F ′ (¯ u )( v N + M ′ N (¯ u ) v N ) = h N ( t ) . Of course, these two equations are closely related. Namely, if v N ( t ) solves(3.35), then the function(3.36) v ( t ) := v N ( t ) + M ′ N (¯ u ( t )) v N ( t )solves (3.34) with(3.37) h ( t ) := h N ( t ) + M ′ N (¯ u ( t )) h N ( t ) . Vice versa, if h ( t ) satisfies (3.37) and the solution v ( t ) of (3.34) satisfies(3.36) for some t , then it satisfies (3.36) for all t and v N ( t ) := P N v ( t ) solves(3.35).This equivalence is a straightforward corollary of the invariance of themanifold M N and we leave its rigorous proof to the reader.4. Main result
In this section we develop an alternative approach for constructing C n -smooth IFs which does not require huge spectral gaps. The key idea is torequire instead the existence of many spectral gaps and to use the secondspectral gap in order to solve equation (3.30) for the second derivative, thethird gap to solve the appropriate equation for the third derivative, etc. Ofcourse, this will not allow us to construct C n -smooth IM (we know that itmay not exist for n >
1, see Example 3.11). Instead, for every p ∈ M N andthe corresponding trajectory u = V ( p, t ), we construct the correspondingTaylor jet J nξ V ( p, t ) of length n +1 belonging to the space P n ( H N n , H ) for all t ≤
0, where N k is the dimension of the IM M N k built up on the k th spectralgap. These jets must be constructed in such a way that the compatibilityconditions are satisfied. Then, the Whitney embedding theorem will give usthe desired smooth extension of the initial IM. To be more precise, we givethe following definition of such a smooth extension. Definition 4.1.
Let equation (3.1) possess at least two spectral gaps whichcorresponds to the dimensions K and K and let ε > M K and M K respectively, the corre-sponding C ,ε -functions generating these manifolds are denoted by M K and M K respectively. A C n,ε -smooth submanifold f M K (not necessarily invari-ant) of dimension K is called a C n -extension of the IM M K if the followingconditions hold:1) f M K is a graph of a C n,ε -smooth function f M K : P K H → Q K H .2) f M K (cid:12)(cid:12) P K M K = Q K M K and therefore M K ⊂ f M K .3) f M K is µ -close in the C b -norm to M K for a sufficiently small µ . Remark 4.2.
The C n,ε dynamics on the extended IM f M K is naturallydefined via(4.1) ∂ t u K + Au K = P K F ( u K + f M K ( u K )) , u K ∈ H K and u ( t ) := u K ( t )+ f M K ( u K ( t )). Obviously, the manifold f M K is invariantwith respect to the dynamical system thus defined. Moreover, due to the XTENSIONS OF INERTIAL MANIFOLDS 19 second condition of Definition 4.1, the C ,ε -submanifold P K M K ⊂ H K isinvariant with respect to equation (4.1) and the restriction of (4.1) coincideswith the initial IF (3.4) generated by the IM M K . Thus, system of ODEs(4.1) is indeed a smooth extension of the IF (3.4).Finally, the 3rd condition of Definition 4.1 guarantees that P K M K is anormally hyperbolic stable invariant manifold for (4.1) (since it is so for theIF generated by the function M K ). This means that P K M K also pos-sesses an exponential tracking property. Thus, the limit dynamics generatedby the extended IF coincides with the one generated by the initial abstractparabolic equation (3.1).We are now ready to state the main result of the paper. Theorem 4.3.
Let the nonlinearity F : H → H in equation (3.1) be smoothand all its derivatives be globally bounded. Let also the following form ofspectral gap conditions be satisfied: (4.2) lim sup N →∞ ( λ N +1 − λ N ) = ∞ . Then, for any n ∈ N and any µ > , equation (3.1) possesses a C n,ε -smoothextension f M N n of the initial IM M N (where N is the first N which satisfiesthe spectral gap condition (3.5) and ε > is small enough) such that f M N n is µ -close to the IM M N n in the C b -norm.Proof for n = 2 . Let N be the first N for which the spectral gap condition(3.5) is satisfied with L := k F ′ k C b ( H, L ( H,H )) and let the corresponding M be C ,ε -smooth IM which exists due to Theorem 3.2 and Corollary 3.8. Recallthat for any p ∈ H , we have a solution V ( p, t ) of problem (3.6) (where p isreplaced by P N p ) and its Frechet derivative V ′ ξ ( t ) := V ′ ( p, t ) ξ in p satisfiesequation of variations (3.21) and belongs to the space L e θ t ( R − , H ) for any θ satisfying (3.17). Moreover, for any other p ∈ H , we have the estimate(4.3) k V ( p , t ) − V ( p, t ) − V ′ ξ ( t ) k L eθ ε ) t ( R − ,H ) ≤ C k P N ( p − p ) k ε , where ε > ξ := p − p and C is independent of p and p .Let now N > N be the first N which satisfies(4.4) λ N +1 − λ N − λ N > L (such N exists due to condition (4.2)). Then, we have the corresponding C ,ε -smooth IM M N . Let us denote by W ( p, t ), p ∈ H , the correspondingsolution of (3.6) (where N is replaced by N and p is replaced by P N p ). Thissolution belongs to L e θ t ( R − , H ) with θ satisfying (3.17) (with N replacedby N ). Moreover, analogously to (4.3), we have(4.5) k W ( p , t ) − W ( p, t ) − W ′ ξ ( t ) k L eθ ε ) t ( R − ,H ) ≤ C k P N ( p − p ) k ε , where W ′ ξ ( t ) = W ′ ( p, t ) ξ solves (3.21) with N replaced by N . We also knowthat V ( p, t ) = W ( p, t ) if p ∈ M N and, therefore, due to (4.3) and (4.5),(4.6) k V ′ ( p, · ) ξ − W ′ ( p, · ) ξ k L eθ ε ) t ( R − ,H ) ≤≤ C k P N ξ k ε , ξ = p − p, p, p ∈ M N . Let us define for every p ∈ M N and every ξ ∈ H the ”second derivative” W ′′ ξ = W ′′ ( p, t )[ ξ, ξ ] of the trajectory u ( t ) = W ( p, t ) = V ( p, t ) as a solutionof the following problem(4.7) ∂ t W ′′ ξ + AW ′′ ξ − F ′ ( V ( p, t )) W ′′ ξ == 2 F ′′ ( V ( p, t ))[ V ′ ξ , W ′ ξ ] − F ′′ ( V ( p, t ))[ V ′ ξ , V ′ ξ ] , P N W ′′ ξ (cid:12)(cid:12) t =0 = 0 . Note that the right-hand side of this equation belongs to the weighted space L e ( θ θ t ( R − , H ), where the exponents θ and θ satisfy assumption (3.17)with N = N and N = N respectively. Moreover, due to assumption (4.4),it is possible to fix θ and θ in such a way that the exponent θ + θ stillsatisfies (3.5) with N = N . Thus, by Proposition 3.7, there exists a uniquesolution of (4.7) belonging to the space L e ( θ θ t ( R − , H ) and the function W ′′ ξ is well-defined and satisfies k W ′′ ξ k C e ( θ θ t ( R − ,H ) ≤ C k W ′′ ξ k L e ( θ θ t ( R − ,H ) ≤ C k ξ k , where C is independent of p .Let us define the desired quadratic polynomial ξ → J ξ W ( p, t ), p ∈ M N as follows:(4.8) J ξ W ( p, t ) := V ( p, t ) + W ′ ( p, t ) ξ + 12 W ′′ ( p, t )[ ξ, ξ ] , ξ ∈ H. We need to verify the compatibility conditions for these ”Taylor jets” on p ∈ M N . It is straightforward to check using F ∈ C ,ε , V, W ∈ C ,ε andProposition 3.7 that k W ′′ ( p , · )[ ξ, ξ ] − W ′′ ( p, · )[ ξ, ξ ] k L e ( θ θ ε ) t ( R − ,H ) ≤ C k ξ k k p − p k ε for p, p ∈ M N . This gives us the desired compatibility condition for thesecond derivative, see (2.13) for n = l = 2.Let us now verify the compatibility conditions for the first derivative( l = 1, n = 2 in (2.13)). To this end, we need to expand the difference w ( t ) := W ′ ( p , t ) ξ − W ′ ( p, t ) ξ , p, p ∈ M N in terms of δ = p − p . By thedefinition of W ′ , this function satisfies the equation(4.9) ∂ t w + Aw − F ′ ( V ( p, t )) w = ( F ′ ( V ( p , t )) − F ′ ( V ( p, t ))) W ′ ( p , t ) ξ == F ′′ ( V ( p, t ))[ V ′ ( p, t ) δ, W ′ ( p, t ) ξ ] + h ( t ) , P N w (cid:12)(cid:12) t =0 = 0 , where the reminder h satisfies k h k L e ( θ θ ε ) t ( R − ,H ) ≤ C k δ k ε k ξ k for sufficiently small positive ε (this also follows from the fact that F issmooth and V, W ∈ C ,ε ). Thus, the reminder h in the right-hand side of(4.9) is of higher order in δ and, by this reason, is not essential, so we needto study the bilinear form (w.r.t. δ, ξ ) in the right-hand side. Note that, incontrast to the case where the IM is C , this form is even not symmetric, XTENSIONS OF INERTIAL MANIFOLDS 21 so it should be corrected. Namely, we write the identity(4.10) F ′′ ( V ( p, t ))[ V ′ ( p, t ) δ, W ′ ( p, t ) ξ ] == (cid:8) F ′′ ( V ( p, t ))[ V ′ ( p, t ) δ, W ′ ( p, t ) ξ ] + F ′′ ( V ( p, t ))[ V ′ ( p, t ) ξ, W ′ ( p, t ) δ ] −− F ′′ ( V ( p, t ))[ V ′ ( p, t ) δ, V ′ ( p, t ) ξ ] (cid:9) −− F ′′ ( V ( p, t ))[ V ′ ( p, t ) ξ, W ′ ( p, t ) δ − V ′ ( p, t ) δ ]and note that the first term in the right-hand side is nothing more than thesymmetric bilinear form which corresponds to the quadratic form2 F ′′ ( V ( p, t ))[ V ′ ( p, t ) ξ, W ′ ( p, t ) ξ ] − F ′′ ( V ( p, t ))[ V ′ ( p, t ) ξ, V ′ ( p, t ) ξ ]used in (4.7) to define W ′′ and the second term is of order k δ k ε k ξ k due toestimate (4.6) (where ξ is replaced by δ ) and the growth rate of this termdoes not exceed e − ( θ + θ + ε ) t as t → −∞ . Thus, by Proposition 3.7, we have k w − W ′′ ( p, · )[ δ, ξ ] k L e ( θ θ ε ) t ( R − ,H ) ≤ C k δ k ε k ξ k and the compatibility condition for l = 1 is verified.Finally, let us check the zero order compatibility condition ( l = 0, n = 2in (2.13)). Let R ( t ) := V ( p , t ) − V ( p, t ) − W ′ ( p, t ) δ − W ′′ ( p, t )[ δ, δ ] . Then, as elementary computations show, this function satisfies the equation(4.11) ∂ t R + AR − F ′ ( V ( p, t )) R == (cid:8) F ( V ( p , t )) − F ( V ( p, t )) − F ′ ( V ( p, t ))( V ( p , t ) − V ( p, t )) (cid:9) −− (cid:0) F ′′ ( V ( p, t ))[ V ′ ( p, t ) δ, W ′ ( p, t ) δ ] − F ′′ ( V ( p, t ))[ V ′ ( p, t ) δ, V ′ ( p, t ) δ ] (cid:1) , P N (cid:12)(cid:12) t =0 R = 0 . Since F ∈ C ,ε and V ∈ C ,ε , the first term in the right-hand side equals to(4.12) 12! F ′′ ( V ( p, t ))[ V ′ ( p, t ) δ, V ′ ( p, t ) δ ]up to the controllable in L e ( θ θ ε ) t ( R − , H )-norm remainder of order k δ k ε .The second term can be simplified using (4.6) and also equals to (4.12) upto higher order terms. Thus, the right-hand side of (4.11) vanishes up toterms of order k δ k ε and Proposition 3.7 gives us that(4.13) k R k L e ( θ θ ε ) t ( R − ,H ) ≤ C k δ k ε for some positive ε . This finishes the verification of the compatibility con-ditions.We are now ready to use Whitney extension theorem. To this end, wefirst recall that the IM M N is a graph of the C ,ε -function M N : P N H → Q N H which is defined via M N ( p ) := Q N W ( p, p ∈ P N H = H N (allfunctions V, W, W ′ , W ′′ defined above depend only on P N -component of p ∈ H , so without loss of generality we may assume that p, ξ, δ ∈ H N (wetook them from H in order to simplify the notations only). Thus, projectingthe constructed Taylor jets to t = 0 and Q N H , we get the C ,ε -function M N ( p ) restricted to the invariant set p ∈ P N M N and a family of quadraticpolynomials J ξ M N ( p ) := Q N J ξ W ( p, p ∈ P N M N . Therefore, since H N is finite-dimensional, Whitney extension theorem gives the existence ofa C ,ε -function c M N : P N H → Q N H such that J ξ c M N ( p ) = J ξ M N ( p ) , p ∈ P N M N . Thus, the desired C ε -extension of the IM M N is ”almost” constructed.It only remains to take care about the closeness in the C -norm. To thisend, for any small ν >
0, we introduce a cut-off function ρ ν ∈ C ∞ ( H N , R )such that ρ ( p ) ≡ p belongs to the ν -neighbourhood O ν of P N M N and ρ ( p ) ≡ p / ∈ O ν . Moreover, since P N M N is C ,ε -smooth, we mayrequire also that(4.14) |∇ p ρ ( p ) | ≤ Cν − , where the constant C is independent of ν . Finally, we define(4.15) f M N ( p ) := (1 − ρ ν ( p )) c M N ( p ) + ρ ν ( p )( S ν M N )( p ) , where S µ is a standard mollifying operator:( S µ f )( p ) := Z R N β µ ( p − q ) f ( q ) dq and the kernel β µ ( p ) = µ N β ( p/µ ) and β ( p ) is a smooth, non-negativefunction with compact support satisfying R R N β ( p ) dp = 1.We claim that f M N is a desired extension. Indeed, f M N ( p ) ≡ c M N ( p ) in O ν and therefore f M N and M N coincide on P N M N . Obviously, f M N is C ,ε -smooth. To verify closeness, we note that(4.16) f M N ( p ) − M N ( p ) = (1 − ρ ν ( p ))( c M N ( p ) − M N ( p ))++ ρ ν ( p )(( S ν M N )( p ) − M N ( p ))Using the fact that M N ∈ C ,ε together with the standard estimates for themollifying operator, we get k ( S ν M N )( p ) − M N ( p ) k ≤ Cν , k∇ p ( S ν M N )( p ) − ∇ p M N ( p ) k ≤ Cν ε which together with (4.14) shows that the C -norm of the second term inthe right-hand side of (4.16) is of order ν ε . To estimate the first term, weuse that both functions c M N ( p ) and M N ( p ) are at least C ,ε -smooth and c M N ( p ) = M N ( p ) , ∇ p c M N ( p ) = ∇ p M N ( p ) , p ∈ P N M N . By this reason, k c M N ( p ) − M N ( p ) k ≤ Cν ε , k∇ p c M N ( p ) − ∇ p M N ( p ) k ≤ Cν ε for all p ∈ O ν . Thus, using (4.14) again, we see that k f M N ( · ) − M N ( · ) k C b ( H N ,H ) ≤ Cν ε . This finishes the proof of the theorem for the case n = 2. (cid:3) XTENSIONS OF INERTIAL MANIFOLDS 23
Proof for general n ∈ N . We will proceed by induction with respect to n .Assume that for some n ∈ N , we have already constructed the C ,ε -smoothinertial manifold M N n which is a graph of a map M N n : P N n H → Q N n H and this map is constructed via the solution V ( p, t ), t ≤ p ∈ H of thebackward problem (3.6) where N is replaced by N n . Recall that this mani-fold is constructed using the n th spectral gap. Assume also that, for every p ∈ P N n M N , we have already constructed the n th Taylor jet J nξ V ( p, t ) suchthat the compatibility conditions up to order n are satisfied. In contrast tothe proof for the case n = 2, it is convenient for us to write these conditionsin the form of (2.12):(4.17) k J nξ V ( p , · ) − J nξ + δ V ( p, · ) k L e ( θn +( n − θn − ε ) t ( R − ,H ) ≤≤ C ( k δ k + k ξ k ) n + ε . Here ξ ∈ H is arbitrary, δ := p − p , ε > θ < θ · · · < θ n are theexponents which satisfy conditions (3.17) for N = N , · · · , N n . In order tosimplify notations, we will write below(4.18) J nξ V ( p ) − J nξ + δ V ( p ) = O θ n +( n − θ n − + ε (cid:0) ( k δ k + k ξ k ) n + ε (cid:1) instead of (4.17) and also in similar situations. Rewriting (4.18) in terms oftruncated jets, we have(4.19) j nξ V ( p ) + j nδ V ( p ) − j nξ + δ V ( p ) = O nθ n + ε (cid:0) ( k δ k + k ξ k ) n + ε (cid:1) , where we have used that θ n − < θ n . We also need the induction assumptionthat (4.19) holds for every m ≤ n , namely,(4.20) J mξ V ( p ) − J mξ + δ V ( p ) = O mθ n + ε (cid:0) ( k δ k + k ξ k ) m + ε (cid:1) . Let us now consider the ( n + 1)th spectral gap at N = N n +1 which is thefirst N satisfying(4.21) λ N n +1 + L + n ( λ N n +1 − L ) < λ N n +1 +1 − L. Let M N n +1 be the corresponding IM which is generated by the backwardsolution W ( p, t ) of problem (3.6) with N replaced by N n +1 . We need todefine the ( n + 1)th Taylor jet J n +1 ξ W ( p, t ) for the function W ( p, t ):(4.22) J n +1 ξ W ( p, t ) = W ( p, t ) + n +1 X k =1 k ! W ( k ) ( p, t )[ { ξ } k ] ,ξ ∈ H and p ∈ P N n +1 M N and to verify the compatibility conditions oforder n + 1. Keeping in mind already considered case n = 1 and n = 2,we introduce the required jet (4.22) as a backward solution of the followingequation:(4.23) ∂ t J n +1 ξ W ( p ) + AJ n +1 ξ W ( p ) = F [ n +1] ( p, ξ ) , P N n +1 J n +1 ξ ( p ) (cid:12)(cid:12) t =0 = P N n +1 ( p + ξ ) , where(4.24) F [ n +1] ( p, ξ, t ) := F ( W ( p, t )) + F ′ ( W ( p, t )) j n +1 ξ W ( p, t )++ n +1 X k =2 k ! (cid:16) kF ( k ) ( W ( p, t ))[ { j nξ V ( p, t ) } k − , j nξ W ( p, t )] −− ( k − F ( k ) ( W ( p, t ))[ { j nξ V ( p, t ) } k ] (cid:17) . Symbol ”[ n + 1]” means that we have dropped out all terms of order greaterthan n + 1 from the right-hand side, so F [ n +1] is a polynomial of order n + 1in ξ ∈ H . Alternatively, the dropping out procedure means that we replace(4.25) { j nξ V ( p ) } k → X n + ··· + n k ≤ n +1 n i ∈ N B n , ··· ,n k { j n ξ V ( p ) , · · · , j n k ξ V ( p ) } , where the numbers B n , ··· ,n k ∈ R are chosen in such a way that polynomialsin the left and right-hand side of (4.25) coincide up to order { ξ } n +1 inclu-sively and the term [ { j nξ V ( p, t ) } k − , j nξ W ( p, t )] is treated analogously. Theexplicit expressions for these coefficients can be found using the formulas forhigher order chain rule (Faa di Bruno type formulas, see e.g. [34, 12]), butthese expressions are lengthy and not essential for what follows, so we omitthem.Note also that the truncated jets j nξ V ( p, t ) are taken from the induc-tion assumption. We seek for the solution of equation (4.23) belonging to L e nθn + θn +1 ( R − , H ) for some θ n +1 satisfying (3.17) with N replaced by N n +1 .Expanding (4.24) in series with respect to ξ , we get the recurrent equationsfor finding the ”derivatives” W ( k ) ξ ( p, t ) := W ( k ) ( p, t )[ { ξ } k ]:(4.26) ∂ t W ( k ) ξ + AW ( k ) ξ − F ′ ( W ( p )) W ( k ) ξ == Φ( j k − ξ W, j k − ξ V ) , P N n +1 W ( k ) ξ (cid:12)(cid:12) t =0 = 0for k ≥
2, where Φ is polynomial of order k in ξ which does not contain W ( l ) ξ with l ≥ k . Thus, the functions W ( k ) ξ can be, indeed, found recur-sively. Moreover, the spectral gap assumption (4.21) guarantees that wecan find θ n +1 satisfying (3.17) with N = N n +1 such that θ n +1 + nθ n alsosatisfies this condition. Therefore, Proposition 3.7 guarantees the existenceand uniqueness of the homogeneous polynomials W ( k ) ξ ( p ) satisfying(4.27) k W ( k ) ξ ( p ) k L e ( θn +1+ kθn ) t ( R − ,H ) ≤ C k ξ k k , for k = 1 , · · · , n + 1.To complete the proof of the theorem, we only need to verify that thejet J ξ W ( p, t ) satisfies the compatibility conditions of order n + 1. If thisis verified, the rest of the proof coincides with the one given above for thecase n = 2. We postpone this verification till the next section. Thus, thetheorem is proved by modulo of compatibility conditions. (cid:3) Corollary 4.4.
Let the assumptions of Theorem 4.3 hold with µ > beingsmall enough. Then the invariant manifold P N n M N of the extended IF XTENSIONS OF INERTIAL MANIFOLDS 25 (4.1) possesses an exponential tracking property in H N n , i.e. for every so-lution u N n ( t ) of (4.1) there exists the corresponding solution ¯ u N n belongingto this manifold such that (4.28) k u N n ( t ) − ¯ u N n ( t ) k ≤ Ce − θ t for some positive C and θ .Proof. As we have already mentioned, this is the standard corollary of thefact that M N is normally hyperbolic and, therefore, persists under small C -perturbations, see [2, 9, 15, 17] and references therein. Nevertheless, forthe convenience of the reader, we sketch below a direct proof without formalrefereing to normal hyperbolicity.We first construct an invariant manifold ¯ M N with the base H N in H N n for the extended IF. We do this exactly as in the proof of Theorem 3.2 bysolving the backward problem(4.29) ∂ t u N n + Au N n − P N n F ( u N n + f M N n ( u N n )) = 0 , P N u N n = p in the space L e θt ( R − , H N n ) with θ = ( λ N + λ N +1 ) /
2. This equation is( Cµ )-closed to(4.30) ∂ t ¯ u N n + A ¯ u N n − P N n F (¯ u N n + M N n (¯ u N n )) = 0 , P N ¯ u N n = p in the C -norm (since f M N n is µ -closed to M N n due to Theorem 4.3). Thus,using Remark 3.13 and the Banach contraction theorem, we can constructa unique solution u N n ( t ) of (4.29) in the ( Cµ )-neighbourhood of the corre-sponding solution ¯ u N n of problem (4.30) and vice versa. This gives us theexistence of the manifold ¯ M N which is generated by all backward solutionsof (4.30) belonging to the space L e θt ( R − , H N n ). Since the solutions belong-ing to the invariant manifold P N n M N satisfy exactly the same property,we conclude that ¯ M N = P N n M N .It remains to verify that the manifold ¯ M N possesses an exponentialtracking property. This can be done also as in the proof of Theorem 3.2 byconsidering the analogue of equation (3.14) for system (4.1) and using againthat f M N n is close to M N n in the C -norm. This finishes the proof of thecorollary. (cid:3) Corollary 4.5.
Arguing as in Corollary 3.9, we check that the extended IM f M N n is also C n,ε -submanifold of H := D ( A ) . Verifying the compatibility conditions
The aim of this section is to show that the jets J n +1 ξ W ( p, t ), p ∈ P N n +1 H ,constructed via (4.23), satisfy the compatibility conditions up to order n + 1and, thus, to complete the proof of Theorem 4.3. We will proceed by induc-tion with respect to the order m ≤ n + 1.Indeed, the first order compatibility conditions are trivially satisfied sincethe functions W ( p, t ) are C ,ε -smooth. Assume that the m th order condi-tions are satisfied for some m ≤ n + 1 and for all m ≤ m (5.1) J m ξ W ( p ) − J m δ + ξ W ( p ) = O θ n +1 +( m − θ n (cid:0) ( k δ k + k ξ k ) m + ε (cid:1) , for all ξ ∈ H , p , p ∈ P N n +1 M N , ε > δ := p − p and some constant C which is independent of p, p . Using the fact that V ( p, t ) = W ( p, t ) for all p ∈ P N n +1 M N together with the analogue of (5.1) for the alreadyconstructed jets J mξ V ( p, t ), we end up with(5.2) V ( p ) = W ( p ) = V ( p ) + j m δ V ( p ) + O m θ n + ε ( k δ k m + ε ) == W ( p ) + j m δ W ( p ) + O θ n +1 +( m − θ n + ε ( k δ k m + ε )for all p , p ∈ P N n +1 M N , δ := p − p and, therefore v ( t ) := V ( p , t ) − V ( p, t )satisfies(5.3) v = j m δ V ( p ) + O m θ n + ε ( k δ k m + ε ) == j m δ W ( p ) + O θ n +1 +( m − θ n + ε ( k δ k m + ε ) ,j m δ V ( p ) − j m δ W ( p ) = O θ n +1 +( m − θ n + ε ( k δ k m + ε ) . We now turn to the ( m + 1)th-jets and start with the following lemma whichgives the compatibility conditions in the particular case ξ = 0. Lemma 5.1.
Let the above assumptions hold. Then (5.4) v = W ( p ) − W ( p ) = j m +1 δ W ( p ) + O θ n +1 + mθ n + ε ( k δ k m +1+ ε ) , for all p , p ∈ P N n +1 M N and δ := p − p . Moreover, (5.5) F ( V ( p )) = F [ m +1] ( p, δ ) + O θ n +1 + mθ n + ε ( k δ k m +1+ ε ) for some ε > .Proof. Let R := v − j m +1 δ W ( p ). Then, by the definition (4.23), this functionsolves(5.6) ∂ t R + AR = F ( V ( p )) − F [ m +1] ( p, δ ) , P N n +1 R (cid:12)(cid:12) t =0 = 0 . Let us study the term F [ m +1] ( p, δ ) at the right-hand side (which is definedby (4.24)). Using (5.3) and the trick (4.25), we may replace j mδ V ( p ) and j mδ W ( p ) by v in all terms in (4.24) which contain the second and higherderivatives of F (the error will be of order k δ k m +1+ ε ). Actually, we cannotdo this in the term with the first derivative at the moment since this requires(5.3) for W of order m + 1 which we are now verifying. This, gives(5.7) F [ m +1] ( p, δ ) = F ( V ( p ))++ F ′ ( V ( p )) j m +1 δ W ( p )+ m +1 X k =2 k ! F ( k ) ( V ( p ))[ { v } k ]+ O θ n +1 + mθ n + ε ( k δ k m +1+ ε ) . Indeed, let us consider the terms in (4.24) containing j mδ W only (the termswithout it are analogous, but simpler). Using the analogue of (4.25):(5.8) [ { j mδ V ( p ) } k − , j mδ W ( p )] →→ X n + ··· + n k ≤ m +1 n i ∈ N B ′ n , ··· ,n k { j n δ V ( p ) , · · · , j n k − δ V ( p ) , j n k δ W ( p ) } , the growth exponent of the remainder does not exceed( n + · · · + n k − ) θ n + θ n +1 + ( n k − θ n + ε ≤ θ n +1 + mθ n + ε, where we have implicitly used our induction assumptions (5.3) and decreasedthe exponent ε if necessary. XTENSIONS OF INERTIAL MANIFOLDS 27
Using now the Taylor theorem for F ∈ C m +1 ,ε together with estimate(3.23) for v , we infer that F ( V ( p )) − F [ m +1] ( p, δ ) = F ′ ( V ( p )) R + O θ n +1 + mθ n + ε (cid:0) k δ k m +1+ ε (cid:1) and, therefore, the function R solves(5.9) ∂ t R + AR − F ′ ( V ( p )) R = O θ n +1 + mθ n + ε ( k δ k m +1+ ε ) , P N n +1 R (cid:12)(cid:12) t =0 = 0 . Since by the induction assumption θ n < λ N n +1 − L , assumption (4.21) guar-antees the existence of θ n +1 and ε > θ n +1 + mθ n + ε satisfies(3.17) with N replaced by N n +1 . Thus, Proposition 3.7 gives the estimate k R k L e ( θn +1+ mθn + ε ) t ( R − ,H ) ≤ C k δ k m +1+ ε and (5.4) is proved. Estimate (5.5) is now a straightforward corollary of(5.7) and the Taylor theorem (since we are now allowed to replace j m +1 δ W by v ). Thus, the Lemma is proved. (cid:3) We now turn to the general case ξ = 0. To this end we need the followingkey lemma. Lemma 5.2.
Let the above assumptions hold. Then, the following formulais satisfied: (5.10) F [ m +1] ( p , ξ ) − F [ m +1] ( p, ξ + δ ) = F ′ ( V ( p )) (cid:16) j m +1 δ W ( p ) + j m +1 ξ W ( p ) − j m +1 ξ + δ W ( p ) (cid:17) ++ O θ n +1 + mθ n + ε (cid:0) ( k δ k + k ξ k ) m +1+ ε (cid:1) , where ξ ∈ H , p , p ∈ P N n +1 M N and δ = p − p .Proof. Indeed, according to the definition (4.24) and formula (5.5), we have(5.11) F [ m +1] ( p , ξ ) = F [ m +1] ( p, δ ) + F ′ ( V ( p )) j m +1 ξ W ( p )++ m +1 X l =2 l ! (cid:16) lF ( l ) ( V ( p ))[ j mξ W ( p ) , { j mξ V ( p ) } l − ] − ( l − F ( l ) ( V ( p ))[ { j mξ V ( p ) } l ] (cid:17) + O θ n +1 + mθ n + ε (cid:0) ( k ξ k + k δ k ) m +1+ ε (cid:1) . We recall that, according to our agreement and formulas (4.25), the right-hand side does not contain the terms of order larger than m + 1. Expandingnow the derivatives F ( l ) ( V ( p )) into Taylor series around V ( p ) and using(5.3), we get(5.12) F [ m +1] ( p , ξ ) = F [ m +1] ( p, δ ) + F ′ ( V ( p ))( j m +1 ξ W ( p ) − j mξ W ( p ))++ m +1 X l =1 m +1 X k = l l !( k − l )! (cid:16) lF ( k ) ( V ( p ))[ { j mδ V ( p ) } k − l , j mξ W ( p ) , { j mξ V ( p ) } l − ] − ( l − F ( k ) ( V ( p ))[ { j mδ V ( p ) } k − l , { j mξ V ( p ) } l ] (cid:17) ++ O θ n +1 + mθ n + ε (cid:0) ( k ξ k + k δ k ) m +1+ ε (cid:1) . Finally, changing the order of summation, we arrive at(5.13) F [ m +1] ( p , ξ ) = F [ m +1] ( p, δ ) + F ′ ( V ( p )) j m +1 ξ W ( p )++ m +1 X k =2 k ! k X l =1 C lk (cid:16) lF ( k ) ( V ( p ))[ { j mδ V ( p ) } k − l , j mξ W ( p ) , { j mξ V ( p ) } l − ] − ( l − F ( k ) ( V ( p ))[ { j mδ V ( p ) } k − l , { j mξ V ( p ) } l ] (cid:17) ++ O θ n +1 + mθ n + ε (cid:0) ( k ξ k + k δ k ) m +1+ ε (cid:1) Let us now look to the term F [ m +1] ( p, ξ + δ ). According to (4.24), we have(5.14) F [ m +1] ( p, ξ + δ ) = F ( V ( p )) + F ′ ( V ( p )) j m +1 ξ + δ W ( p )++ m +1 X k =2 k ! (cid:16) kF ( k ) ( V ( p ))[ j mξ + δ W ( p ) , { j mξ + δ V ( p ) } k − ] − ( k − F ( k ) ( V ( p ))[ { j mξ + δ V ( p ) } k ] (cid:17) . From the induction assumption, the compatibility assumptions (5.1) holdfor j m ξ + δ W and give j m ξ + δ W ( p ) = j m δ W ( p ) + j m ξ W ( p ) + O θ n +1 +( m − θ n + ε (cid:0) ( k δ k + k ξ k ) m + ε (cid:1) for all m ≤ m and the analogous identities hold also for j m ξ + δ V : j m ξ + δ V ( p ) = j m δ V ( p ) + j m ξ V ( p ) + O m θ n + ε (cid:0) ( k δ k + k ξ k ) m + ε (cid:1) . Moreover, using (5.3), we may also get j m ξ + δ W ( p ) = j m δ V ( p ) + j m ξ W ( p ) + O θ n +1 +( m − θ n + ε (cid:0) ( k δ k + k ξ k ) m + ε (cid:1) for all m ≤ m . Inserting these formulas to (5.14), we arrive at(5.15) F [ m +1] ( p, ξ + δ ) = F ( V ( p )) + F ′ ( V ( p )) j m +1 ξ + δ W ( p )++ m +1 X k =2 k ! (cid:16) kF ( k ) ( V ( p ))[ j mδ V ( p ) + j mξ W ( p ) , { j mδ V ( p ) + j mξ V ( p ) } k − ] − ( k − F ( k ) ( V ( p ))[ { j mδ V ( p ) + j mξ V ( p ) } k ] (cid:17) ++ O θ n +1 + mθ n + ε (cid:0) ( k δ k + k ξ k ) m +1+ ε (cid:1) . XTENSIONS OF INERTIAL MANIFOLDS 29
Using the binomial formula (2.1), we arrive at(5.16) F [ m +1] ( p, ξ + δ ) = F ( V ( p )) + F ′ ( V ( p )) j m +1 ξ + δ W ( p )++ m +1 X k =2 k ! k X l =1 kC l − k − F ( k ) ( V ( p ))[ j mξ W ( p ) , { j mδ V ( p ) } k − l , { j mξ V ( p ) } l − ]++ k − X l =0 kC lk − F ( k ) ( V ( p ))[ j mδ V ( p ) , { j mδ V ( p ) } k − l − , { j mξ V ( p ) } l ] −− k X l =0 ( k − C lk F ( k ) ( V ( p ))[ { j mδ V ( p ) } k − l , { j mξ V ( p ) } l ] ! ++ O θ n +1 + mθ n + ε . (cid:0) ( k δ k + k ξ k ) m +1+ ε (cid:1) . We need to compare (5.13) and (5.16). To this end, we first note that lC lk = kC l − k − and, therefore, the terms containing the jets of W in these two formulascoincide. Thus, we only need to look at the terms without jets of W . Inthe case l = k , we have only one term in the right-hand side of (5.16) whichobviously coincides with the analogous term in (5.13). Let us now look atthe terms with l = 1 , · · · , k −
1. Due to the obvious identity − ( l − C lk = kC lk − − ( k − C lk , these terms again coincide. Thus, it remains to look at the extra termswhich correspond to l = 0 in (5.16) and which are absent in the sums of(5.13). Finally, using (5.3) and (5.5), we get the following identity involvingthese extra terms:(5.17) F ( V ( p )) + m +1 X k =2 k ! F ( k ) ( V ( p ))[ { j mδ V ( p ) } k ] = F [ m +1] ( p, δ ) −− F ′ ( V ( p )) j m +1 δ W ( p ) + O θ n +1 + mθ n + ε (cid:0) ( k δ k + k ξ k ) m +1+ ε (cid:1) . This gives the identity(5.18) F [ m +1] ( p , ξ ) − F ′ ( V ( p )) j m +1 ξ W ( p ) = F [ m +1] ( p, ξ + δ ) −− F ′ ( V ( p )) (cid:16) j m +1 ξ + δ W ( p ) − j m +1 δ W ( p ) (cid:17) + O θ n +1 + mθ n + ε (cid:0) ( k δ k + k ξ k ) m +1+ ε (cid:1) and finishes the proof of the lemma. (cid:3) We are now ready to finish the check of the compatibility conditions. Notethat, due to (5.4), we have(5.19) J m +1 ξ W ( p ) − J m +1 ξ + δ W ( p ) = j m +1 δ W ( p ) + j m +1 ξ W ( p ) − j m +1 ξ + δ W ( p ) + O θ n +1 + mθ n + ε (cid:0) ( k δ k + k ξ k ) m +1+ ε (cid:1) . Let finally U ( t ) := J m +1 ξ W ( p ) − J m +1 ξ + δ W ( p ). Then, according to definition(4.22), Lemma 5.2 and the fact that δ = p − p , this function solves the equation(5.20) ∂ t U + AU − F ′ ( V ( p )) U == O θ n +1 + mθ n + ε (cid:0) k δ k + k ξ k ) m +1+ ε (cid:1) , P N n +1 U (cid:12)(cid:12) t =0 = 0and by Proposition 3.7, we arrive at(5.21) J m +1 ξ W ( p ) − J m +1 ξ + δ W ( p ) = O θ n +1 + mθ n + ε (cid:0) ( k δ k + k ξ k ) m +1+ ε (cid:1) . Thus, the ( m + 1)th order compatibility conditions for J m +1 ξ W ( p ) are veri-fied. The induction with respect to m gives us that J n +1 ξ W ( p ) also satisfiesthe compatibility conditions (of course, we cannot take m > n since weneed the compatibility conditions of order m for J mξ V ( p ) to proceed). Thiscompletes the proof of our main Theorem 4.3.6. Examples and concluding remarks
In this section we give several examples to the proved main theorem aswell as its reinterpretations and state some interesting problems for furtherstudy. We start with the application to 1D reaction-diffusion equation.
Example 6.1.
Let us consider the following reaction-diffusion system in 1Ddomain Ω = ( − π, π ):(6.1) ∂ t u = a∂ x u − f ( u ) , u (cid:12)(cid:12) Ω = 0 , u (cid:12)(cid:12) t =0 = u , where u is an unknown function, a > f ( u ) is a given smooth function satisfying f (0) = 0 and some dissipativityconditions, for instance, f ( u ) u ≥ − C + α | u | , u ∈ R . for some C and α > f ( u ) = u − u as in the case of real Ginzburg-Landau equation). Then, due to the maximum principle, we have the fol-lowing dissipative estimate for the solutions of (6.1):(6.2) k u ( t ) k L ∞ ≤ k u k L ∞ e − αt + C ∗ , where the constant C ∗ is independent of u , see, e.g. [1, 4, 40]. Thus, theassociated solution semigroup S ( t ) acting in the phase space H := H (Ω)possesses an absorbing set in C ( ¯Ω) and cutting-off the nonlinearity outsideof this ball, we may assume without loss of generality that f ∈ C ∞ ( R ).After this transform, equation (6.1) can be considered as an abstractparabolic equation (3.1) in the Sobolev space H = H (Ω). Since this spaceis an algebra with respect to point-wise multiplication (since we have onlyone spatial variable), the corresponding non-linearity F ( u )( x ) := f ( u ( x )) is C ∞ -smooth and all its derivatives are globally bounded.Finally, the linear operator A in this example is A = − a∂ x endowed withthe Dirichlet boundary conditions. Obviously, this operator is self-adjoint,positive definite and its inverse is compact. Moreover, its eigenvalues λ k = ak , k ∈ N satisfy (4.2). Thus, our main theorem 4.3 is applicable here and, therefore,problem (6.1) possesses an IM M N of smoothness C ,ε for some ε > XTENSIONS OF INERTIAL MANIFOLDS 31 for every n ∈ N , this IM can be extended to a manifold f M N n of regularity C n,ε n , ε n >
0, in the sense of Definition 4.1.
Remark 6.2.
Our general theorem is applicable not only for a scalar re-action-diffusion equation (6.1), but also for systems where the analogue of(6.2) is known, for instance, for the case of 1D complex Ginzburg-Landauequation (however, one should be careful in the case where the diffusionmatrix is not self-adjoint and especially when it contains non-trivial Jordancells. In this case, even Lipschitz IM may not exist, see [25] for more details).A bit unusual choice of the phase space H = H (Ω) (instead of the naturalone H = L (Ω) is related with the fact that we need H to be an algebra inorder to define Taylor jets for the nonlinearity F and to verify that it is C ∞ .This however may be relaxed in applications since backward solutions of(3.4) and (3.18) are usually smooth in space and time if the non-linearity f issmooth, so the Taylor jets for V ( p, t ) will be well-defined even if we consider L (Ω) as a phase space and the theory works with minimal changes. Thisobservation may be useful if we want to remove the assumption f (0) = 0 in(6.1), but in order to avoid technicalities, we prefer not to go further in thisdirection here.The restriction to 1D case is motivated by the fact that the spectral gapcondition (4.2) is naturally satisfied by the Laplacian in 1D case only (it isan open problem already in 2D case).If we consider higher-order operators, say bi-Laplacian then the analogousresult holds also in 3D. The typical example here is given by Swift-Hohenbergequation in a bounded domain Ω ⊂ R : ∂ t u = − (∆ + 1) u + u − u , u (cid:12)(cid:12) ∂ Ω = ∆ u (cid:12)(cid:12) ∂ Ω = 0 , where the spectral gap condition (4.2) is also satisfied, see [42] and refer-ences therein. We also note that although our main theorem is stated andproved for the case where F maps H to H , it can be generalized in a verystraightforward way to the case where the operator F decreases smoothnessand maps H to H − s := D ( A − s/ ) for some s ∈ (0 , n →∞ ( λ n +1 − λ n λ s/ n +1 + λ s/ n ) = ∞ . After this extension, our theorem becomes applicable to equations whichcontain spatial derivatives in the non-linearity. Typical example of suchapplications is 1D Kuramoto-Sivashinski equation ∂ t u + a∂ x u + ∂ x u + u∂ x u = 0 , Ω = ( − π, π ) , a > Remark 6.3.
As we have mentioned in the introduction, there is an es-sential recent progress in constructing IMs for concrete classes of parabolicequations which do not satisfy the spectral gap conditions (such as scalar reaction-diffusion equations in higher dimensions, 3D Cahn-Hilliard or com-plex Ginzburg-Landau equations, various modifications of Navier-Stokes sys-tems, 1D reaction-diffusion-advection systems, etc.). The techniques devel-oped in our paper is not directly applicable to such problems (in particular,our technique is strongly based on the Perron method of constructing theIMs and it is not clear how to use the Perron method here since we do nothave the so-called absolute normal hyperbolicity in the most part of equa-tions mentioned above, see [20, 22] for more details). However, we believethat the proper modification of our method would allow to cover these casesas well. We return to this problem somewhere else.We now give an alternative (probably more transparent and more elegant)formulation of Theorem 4.3. We recall that in Theorem 4.3, we have directlyconstructed a smooth extended IF (4.1) for the initial equation (3.1). Thisextended IF captures all non-trivial dynamics of (3.1), but the associatedsmooth extended IM M n is not associated with the ”true” IM of any systemof the form (3.1). This drawback can be easily corrected in more or lessstandard way which leads to the following reformulation of our main result. Corollary 6.4.
Let the assumptions of Theorem 4.3 be satisfied and let M N be the C ,ε -smooth IM of equation (3.1) which corresponds to thefirst spectral gap. Then, for every n ∈ N , n > , there exists a modifiednonlinearity e F : H → H which belongs to C n − ,ε n b ( H, H ) for some ε n > such that1) The initial IM M N is simultaneously an IM for the modified equation (6.3) ∂ t u + Au = e F n ( u ) .
2) Equation (6.3) possesses a C n,ε n -smooth IM f M N n of dimension N n such that the initial IM M is a normally hyperbolic globally stable subman-ifold of f M N n .3) The nonlinearity e F n ( u ) depends on the variable u N n := P N n u only andthe IF associated with the IM f M N n is given by (4.1) where K is replacedby N n .Proof. Indeed, we take the manifold f M N n constructed in Theorem 4.3 anddefine the desired function e F n as follows(6.4) P N n e F n ( u ) := P N n F ( u N n + f M N n ( u N n ))and(6.5) Q N n e F n ( u ) :== f M ′ N n ( u N n )[ − A f M N n ( u N n ) + P N n F ( u N n + f M N n ( u N n ))] + A f M N n ( u N n ) . Then, due to the choice of P N n -component of e F n ( u ), the equation for u N n is decoupled from the equation for the Q N n -component and coincides withthe extended IF for (6.3) constructed in Theorem 4.3. On the other hand,the Q N n -component of e F n is chosen in a form which guarantees that f M N n is an invariant manifold for equation (6.3). Moreover, if u ( t ) solves equation(6.3) with such a nonlinearity and v ( t ) := u ( t ) − P N n u ( t ) − f M N n ( u N n ( t )), XTENSIONS OF INERTIAL MANIFOLDS 33 then this function satisfies ∂ t v + Av = 0 , P N n v ( t ) ≡ k v ( t ) k H ≤ k v (0) k H e − λ Nn +1 t . Thus, f M N n is indeed an IM for problem (6.3) and we only need to check theregularity of the modified function e F n .The P N n component (6.4) is clearly C n,ε n -smooth, but the situation withthe Q N n is a bit more delicate due to the presence of terms A f M N n ( u N n )and f M ′ N n ( u N n ). The first term is not dangerous since we know that f M N n is C n,ε n -smooth as the map from H N n to H . The second term is worseand decreases the smoothness of the e F n till C n − ,ε n . Thus, the corollary isproved. (cid:3) Remark 6.5.
The modified non-linearity e F n ( u ) can be interpreted as a”clever” cut-off of the initial non-linearity F ( u ) outside of the global attrac-tor (even outside of the IM of minimal dimension). In this sense we may saythat all obstacles for the existence of C n,ε -smooth IM can be removed by theappropriate cutting off the nonlinearity outside of the global attractor whichdoes not affect the dynamics of the initial problem. This demonstrates theimportance of finding the proper cut off procedure in the theory of IMs. Example 6.6.
We now return to the model example of G. Sell introducedin Example 3.11 and show how the problem of smoothness of an invariantmanifold can be resolved. Since the non-linearity for this system is not globally
Lipschitz continuous, the above developed theory is formally notapplicable and we need to cut-off the nonlinearity first. We overcome thisproblem by considering only local manifolds in a small neighbourhood of theorigin.Indeed, it is not difficult to see that system (3.31) has an explicit particularsolution u ( t ) = ± e − t , u n +1 ( t ) = C n e − n t t n − , n > , where the coefficients C n satisfy the recurrent relation C n +1 = 12 n − C n , C = 1 . This solution determines 1D local invariant manifold M = { p + M ( p ) : p ∈ H = R , | p | < β } , where M : R → H is defined by M = (0 , M ( p ) , M ( p ) , · · · , ) and M n +1 ( p ) = C n p n (cid:18) ln 1 | p | (cid:19) n − , n ∈ N which is 1D IM for system (3.31) and β is a sufficiently small positive num-ber. Indeed, since C n ≤ − α n for some positive α , this manifold is well-defined as a local submanifold of H = l (if β > C ,ε -smooth for any ε ∈ (0 , M ( p ) is only C ,ε -smooth and higher components are more regular, in particular, M n ( p ) is C n − − ,ε -smooth. This guesses us how to define the extended manifolds of an arbitrary finite smoothness. Namely, let us fix some n ∈ N and considerthe following manifold:(6.6) f M n := { p + f M n ( p ) , p ∈ H n , | p | < β } , f M n ( p ) := ( { } n , M n +1 ( p ) , M n +2 ( p ) , M n +3 ( p ) , · · · ) . Clearly f M n is C n − ,ε -smooth and M is a submanifold of f M n . Moreover,if we define the modified non-linearity e F n ( u ) as follows:(6.7) e F n ( u ) = (0 , u , u , · · · , u n − , M n +1 ( u ) , M n +2 ( u ) , · · · ) , then it will be C n − ,ε -smooth and the extended manifold f M n will be anIM for the corresponding modified equation (6.3). Finally, the normal hy-perbolicity of M in f M n follows from the fact that any solution on M decays to zero not faster than e − t due to the non-zero first component, if welook to the transversal directions, the smallest decay rate is determined bythe second component and this decay is at least as t e − t . Since our modelsystem is explicitly solvable, we leave verifying of this normal hyperbolicityto the reader. We also note that the extended IF in this case reads ddt u + u = 0 , ddt u k + 2 k − u k = u k − , k = 2 , · · · , n which is nothing more than the Galerkin approximation system to (3.31). Remark 6.7.
We see that, in the toy example of equation (3.31), we can findthe desired extension of the initial IM explicitly without using the Whitneyextension theorem (and even without assuming the global boundedness of F and its derivatives). Moreover, the dependence of smoothness of theextended IM on its dimension is very nice, namely, if we want to have C n -smooth IM, it is enough to take dim f M ∼ log n . Of course, this is partiallyrelated with good exponentially growing spectral gaps, but the main reasonis that we have an extra regularity property for the initial IM, namely,that the smoothness of projections Q k M ( p ) grows with k . Unfortunately,this is not true in a more or less general case which makes the extensionconstruction much more involved. In particular, we do not know how togain more than one unit of smoothness from one spectral gap and have touse n different spectral gaps to get n units of smoothness. This, in turn,leads to extremely fast growth of the dimension of the manifold with respectto the regularity (as not difficult to see, in Example 6.1, the dimension of f M N n grows as a double exponent with respect to n ).We believe that this problem is technical and the estimates for the dimen-sion can be essentially improved. Indeed, if we would be able to get n unitsof extra regularity using one extra (sufficiently large) gap the above men-tioned growth of the dimension would become linear in n in Example 6.1.We expect that this linear growth is optimal, and even able to construct thecorresponding Taylor jets. But these jets do not satisfy the compatibilityconditions and we do not know how to correct them properly. References [1] A. Babin and M. Vishik,
Attractors of evolution equations,
Studies in Mathematicsand its Applications, 25. North-Holland Publishing Co., Amsterdam, 1992.
XTENSIONS OF INERTIAL MANIFOLDS 35 [2] P. Bates, K. Lu and C. Zeng,
Persistence of Overflowing Manifolds for Semiflow.
Comm. Pure Appl. Math., vol. 52, (1999) 983–1046.[3] A. Ben-Artzi, A. Eden, C. Foias, and B. Nicolaenko,
H¨older continuity for the inverseof Mane’s projection.
J. Math. Anal. Appl., vol.178, (1993) 22–29.[4] V. Chepyzhov and M. Vishik,
Attractors for equations of mathematical physics,
Amer-ican Mathematical Society Colloquium Publications, 49. American Mathematical So-ciety, Providence, RI, 2002.[5] S.-N. Chow, K. Lu, and G. Sell,
Smoothness of inertial manifolds , Jour. Math. Anal.and Appl., vol. 169, no. 1 (1992) 283–312.[6] P. Constantin, C. Foias, B. Nicolaenko, and R. Temam,
Inertial Manifolds for Dissipa-tive Partial Differential Equations (Applied Mathematical Sciences, no. 70) , Springer-Verlag, New York, 1989.[7] A. Eden, V. Kalanarov and S. Zelik,
Counterexamples to the regularity of Mane pro-jections and global attractors,
Russian Math Surveys, vol. 68, no. 2, (2013) 199–226.[8] C. Fefferman,
A sharp form of Whitney’s extension theorem , Annals of Mathematics,vol. 161, no.1, (2005) 509–577.[9] N. Fenichel,
Persistence and smoothness of invariant manifolds for flows,
Indiana Univ.Math. J., vol. 21, (1971/1972), 193–226.[10] C. Foias, G. Sell, and R. Temam,
Inertial manifolds for nonlinear evolutionary equa-tions , J. Differential Equations, vol. 73, no. 2, (1988) 309–353.[11] C. Gal and Y. Guo,
Inertial manifolds for the hyperviscous Navier-Stokes equations ,J. Differential Equations, vol. 265, no. 9, (2018) 4335–4374.[12] P. H´ajek and Michal Johanis,
Smooth Analysis in Banach Spaces , In: De GruyterSeries in Nonlinear Analysis and Applications, 19, De Gruyter , 2014.[13] J. Hale,
Asymptotic Behaviour of Dissipative Systems , Math. Surveys and Mon., AMSProvidence, RI, 1987.[14] D. Henry,
Geometric theory of semilinear parabolic equations.
Lecture Notes in Math-ematics, 840. Springer-Verlag, Berlin–New York, 1981.[15] M. Hirsch, C. Pugh, and M. Shub,
Invariant manifolds.
Lecture Notes in Mathemat-ics, Vol. 583. Springer-Verlag, Berlin–New York, 1977.[16] B. Hunt and V. Kaloshin,
Regularity of embeddings of infinite-dimensional fractal setsinto finite-dimensional spaces.
Nonlinearity, vol. 12, (1999) 1263–1275.[17] A. Katok and B. Hasselblatt,
Introduction to the modern theory of dynamical systems.
Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press,Cambridge, 1995.[18] H. Kielh¨ofer,
Bifurcation Theory: An Introduction with Applications to Partial Dif-ferential Equations , Applied Mathematical Sciences 156, Springer-Verlag New York,2012.[19] N. Koksch,
Almost Sharp Conditions for the Existence of Smooth Inertial manifolds. in: Equadiff 9: Conference on Differential Equations and their Applications : Proceed-ings, edited by Z. Dosla, J. Kuben, J. Vosmansky, Masaryk University, Brno, 1998,139–166.[20] A. Kostianko,
Inertial manifolds for the 3D modified-Leray- α model with periodicboundary conditions , J. Dyn. Differ. Equations, vol. 30, no. 1, (2018) 1–24.[21] A. Kostianko, Bi-Lipschitz Man´e projectors and finite-dimensional reduction for com-plex Ginzburg-Landau equation,
Proc. A R. Soc. London, vol. 476, no. 2239, (2020)1–14.[22] A. Kostianko and S. Zelik,
Inertial manifolds for the 3D Cahn-Hilliard equationswith periodic boundary conditions , Commun. Pure Appl. Anal., vol. 14, no. 5, (2015)2069–2094.[23] A. Kostianko and S. Zelik,
Inertial manifolds for 1D reaction-diffusion-advection sys-tems. Part II: Periodic boundary conditions,
Commun. Pure Appl. Anal., vol. 17, no.1, (2018) 265–317.[24] A. Kostianko and S. Zelik,
Inertial manifolds for 1D reaction-diffusion-advection sys-tems. Part I: Dirichlet and Neumann boundary conditions,
Commun. Pure Appl. Anal.,vol. 16, no. 6, (2017) 2357– 2376. [25] A. Kostianko and S. Zelik,
Kwak Transform and Inertial Manifolds revisited , Jour.Dyn. Diff. Eqns, to appear.[26] X. Li and C. Sun,
Inertial manifolds for the 3D modified-Leray- α model , J. DifferentialEquations, vol. 268, no. 4, (2020) 1532–1569.[27] J. Mallet-Paret and G. Sell, Inertial manifolds for reaction diffusion equations inhigher space dimensions , J. Am. Math. Soc., vol. 1, no. 4, (1988) 805–866.[28] J. Mallet-Paret, G. Sell, and Z. Shao,
Obstructions to the existence of normally hy-perbolic inertial manifolds,
Indiana Univ. Math. J., 42, no. 3, (1993) 1027–1055.[29] M. Miklavcic,
A sharp condition for existence of an inertial manifold , J. Dyn. Differ.Equations, vol. 3, no. 3, (1991) 437–456.[30] A. Miranville and S. Zelik,
Attractors for dissipative partial differential equations inbounded and unbounded domains , in: Handbook of Differential Equations: Evolution-ary Equations, vol. IV, Elsevier/North-Holland, Amsterdam, 2008.[31] J. Robinson,
Global Attractors: Topology and Finite-Dimensional Dynamics , Jour.Dyn. Dif. Eqns., vol. 11. (1999) 557–581.[32] J. Robinson,
Infinite-dimensional Dynamical Systems , Cambridge University Press,2001.[33] J. Robinson,
Dimensions, embeddings, and attractors,
Cambridge University Press,Cambridge, 2011.[34] S. Roman,
The Formula of Faa di Bruno , Amer. Math. Monthly, vol. 87, (1980)805–809.[35] A. Romanov,
Sharp estimates for the dimension of inertial manifolds for nonlinearparabolic equations , Izv. Math. vol. 43, no. 1, (1994) 31–47.[36] A. Romanov,
Three counterexamples in the theory of inertial manifolds,
Math. Notes,vol. 68, no. 3–4, (2000) 378–385.[37] R. Rosa and R. Temam,
Inertial manifolds and normal hyperbolicity , Acta Applican-dae Mathematica, vol. 45, (1996) 1–50.[38] G. Sell and Y. You,
Dynamics of evolutionary equations , Springer, New York, 2002.[39] E. Stein,
Singular Integrals and Differentiability Properties of Functions , PrincetonUniv. Press, Princeton, 1970.[40] R. Temam,
Infinite-Dimensional Dynamical systems in Mechanics and Physics , sec-ond edition, Applied Mathematical Sciences, vol 68, Springer-Verlag, New York, 1997.[41] J. Wells,
Differentiable functions on Banach spaces with Lipschitz derivatives , J. Diff.Geom., vol. 8, (1973) 135–152.[42] S. Zelik,
Inertial manifolds and finite-dimensional reduction for dissipative PDEs ,Proc. Royal Soc. Edinburgh 144, vol. 6, (2014) 1245–1327. Department of Mathematics,University of Surrey, GU27XH, Guildford, UK School of Mathematics and Statistics, Lanzhou University, Lanzhou730000, P.R. China
Email address : [email protected] Email address ::