Connecting a direct and a Galerkin approach to slow manifolds in infinite dimensions
aa r X i v : . [ m a t h . D S ] F e b CONNECTING A DIRECT AND A GALERKIN APPROACH TOSLOW MANIFOLDS IN INFINITE DIMENSIONS
MAXIMILIAN ENGEL, FELIX HUMMEL, AND CHRISTIAN KUEHN
Abstract.
In this paper, we study slow manifolds for infinite-dimensionalevolution equations. We compare two approaches: an abstract evolution equa-tion framework and a finite-dimensional spectral Galerkin approximation. Weprove that the slow manifolds constructed within each approach are asymp-totically close under suitable conditions. The proof is based upon Lyapunov-Perron methods and a comparison of the local graphs for the slow manifolds inscales of Banach spaces. In summary, our main result allows us to change be-tween different characterizations of slow invariant manifolds, depending uponthe technical challenges posed by particular fast-slow systems. Introduction
The perturbation theory of normally hyperbolic invariant manifolds introducedby Fenichel [4, 9] has proved to be a useful tool in the theory of dynamical systems.One important consequence of Fenichel’s works is that they provide a suitableframework for the treatment of fast-slow systems [5, 7] of the form ε∂ t u ε = Au ε + f ( u ε , v ε ) ,∂ t v ε = Bv ε + g ( u ε , v ε ) , (1.1)where 0 ≤ ε ≪ A, B are matrices, and f, g are differentiablenonlinearities. The unknown functions u ε and v ε are called fast and slow variable,respectively. System (1.1) is already written in a variant of (local) Fenichel normalform [5, 7] separating matrices A, B and the nonlinearities f, g , which is also aquite natural form in the PDE context to be considered below. For the classicalfinite-dimensional case, Fenichel’s techniques are also known as geometric singularperturbation theory. The main result is that – under suitable assumptions – for
Date : February 10, 2021.2020
Mathematics Subject Classification.
Primary 37L15, 37L25, 37L65; Secondary 34E15,35K57.
Key words and phrases.
Fast-slow systems, reaction-diffusion equations, Galerkin discretiza-tion, infinite-dimensional dynamics.ME was supported by Germany’s Excellence Strategy – The Berlin Mathematics ResearchCenter MATH+ (EXC-2046/1, project ID: 390685689).FH acknowledges partial support via the SFB/TR109 “Discretization in Geometry and Dy-namics” as well as partial support of the EU within the TiPES project funded the EuropeanUnions Horizon 2020 research and innovation programme un der grant agreement No. 820970.CK acknowledges support via a Lichtenberg Professorship as well as support via theSFB/TR109 “Discretization in Geometry and Dynamics” as well as partial support of the EUwithin the TiPES project funded the European Unions Horizon 2020 research and innovationprogramme un der grant agreement No. 820970.This project is TiPES contribution all ε > S ε which is locally invariant under theflow generated by (1.1) and which can be written as a graph over the slow variable.More precisely, one may write S ε := { ( h ε ( v ) , v ) : v ∈ Y } , where X and Y are the finite-dimensional vector spaces u ε and v ε , respectively,take values in, and h ε : Y → X is a Lipschitz continuous function. These manifolds,which are called slow manifolds, are ε -close over compact subsets in Y to the criticalmanifold S := { ( h ( v ) , v ) : v ∈ Y } , where h ( v ) denotes the unique solution of0 = Ah ( v ) + f ( h ( v ) , v ) . Moreover, the flow on S ε converges to the slow flow on S which is defined to bethe flow which is generated by the singular limit of (1.1) as ε →
0, that is0 = Au + f ( u , v ) ,∂ t v = Bv + g ( u , v ) . (1.2)The existence of such slow manifolds is usually taken as a formal justification forthe intuitive idea, that after a short initial time the dynamics of (1.1) only evolveon the slow time scale and are described well by the slow subsystem (1.2). Since0 = Au + f ( u, v )is supposed to have the unique solution u = h ( v ), one may rewrite (1.2) as ∂ t v = Bv + g ( h ( v ) , v ) , u = h ( v ) . (1.3)Altogether, we can then reduce (1.1) to (1.3). The advantage of (1.3) is that thefast variable is now uniquely determined by the slow variable, i.e., the dimensionof the dynamical problem (1.1) has been reduced.It has been an open problem for a few decades, how to generalize Fenichel theory tothe infinite-dimensional setting, with fast-slow systems of partial differential equa-tions as an important application. Even though persistence of normally hyperbolicinvariant manifolds in Banach spaces was derived by Bates, Lu and Zeng in [2] forbounded semiflow perturbations, the existence of slow manifolds for PDEs, involv-ing spatial differential operators in the slow variable equations, had only been shownin very special cases such as for the Maxwell-Bloch equations [8]. Recently, therehave been two new attempts to provide techniques for a geometric singular pertur-bation theory in infinite dimensions: In [3], slow manifolds in infinite dimensionswere approximated by finite-dimensional slow manifolds within a Galerkin proce-dure, paving the way for an extension of geometric blow-up from ODEs to PDEs.A more direct approach was taken in [6], where a two-parameter family S ε,ζ of slowmanifolds was contructed via a Lyapunov-Perron argument. The main ingredientof the latter procedure is a splitting of the slow variable space Y = Y ζF ⊕ Y ζS into aquickly decaying part and a part on which the linear dynamics are invertible. Wewill introduce both approaches in Section 2 and provide a precise comparison resultin Section 3, relating the two types of slow manifolds to each other via estimatesfor their distance and its decay in ε, ζ . Finally, in Section 4, we exemplify this mainresult at the hand of a slow-fast PDE with fast reaction-diffusion dynamics, alsodiscussing intricacies of the Galerkin limits. LOW MANIFOLDS IN INFINITE DIMENSIONS 3 The two approaches
Assumptions.
In the following, we discuss in detail the assumptions forthe subsequent statements. It is, in fact, one of the main difficulties in infinite-dimensional geometric singular perturbation theory to find conditions, which allowfor the construction of slow manifolds and are, at the same time, satisfied in manyimportant applications. Although the list of assumptions we impose is quite long,it has already been demonstrated in [6] that the conditions are satisfied for a largeclass of PDEs, e.g. reaction-diffusion systems; in comparison to [6], we add a fewassumptions which allow us to trade regularity for better estimates. Moreover, wealso add a splitting in the fast variable space so that we can define an appropriateGalerkin approximation.In the following, let n ∈ N .2.1.1. Assumption ( A n ) . We consider the fast-slow system (1.1) on Banach spaces X and Y , supplemented by the initial conditions u ε (0) = u ∈ X n , v ε (0) = v ∈ Y n , (2.1)where X n , Y n are elements of the interpolation-extrapolation scales introduced here-after (see also Appendix A) and we have 0 = Au + f ( u , v ) for ε = 0. As-sume further that the nonlinearities satisfy f (0 ,
0) = 0 and g (0 ,
0) = 0. Then thefollowing conditions ensure that (1.1) together with (2.1) has a unique solution( u ε , v ε ) ∈ C ([0 , ∞ ); X n − × Y n − ) ∩ C ([0 , ∞ ); X n × Y n ) which is approximatedwell by the slow flow in a sense which we will make precise later.(i) Generation of semigroups : the closed linear operator A : X ⊃ D ( A ) → X generates an exponentially stable C -semigroup (e tA ) t ≥ ⊂ B ( X ) on theBanach space X . The closed linear operator B : Y ⊃ D ( B ) → Y is thegenerator of a C -semigroup (e tB ) t ≥ ⊂ B ( Y ) on the Banach space Y .(ii) Generation of Banach scales : the interpolation-extrapolation scales gen-erated by (
X, A ) and (
Y, B ) (see Appendix A) are — up to uniform equiv-alence of norms for each fixed α ∈ [ − , ∞ ) and all α ∈ [ − , α ] — givenby ( X α ) α ∈ [ − , ∞ ) and ( Y α ) α ∈ [ − , ∞ ) . If 0 / ∈ ρ ( B ), then ( Y α ) α ∈ [ − , ∞ ) shall beequivalent to the interpolation-extrapolation scale generated by B − λ for some λ ∈ ρ ( B ).(iii) Bounded Fr´echet derivatives : let γ X ∈ (0 ,
1] if (e tA ) t ≥ ⊂ B ( X ) is holo-morphic and γ X = 1 otherwise. In addition, we choose δ X ∈ [1 − γ X , δ Y ∈ (0 ,
1] if (e tB ) t ≥ ⊂ B ( Y ) is holomorphic and δ Y = 1 otherwise.The nonlinearities f : X n − δ X × Y n − δ X → X n − and g : X n × Y n → Y n − δ Y are continuously differentiable and there are constants L f , L g > n ) such that k Df ( x, y ) k B ( X n × Y n ,X n − γX ) < L f ( x ∈ X n , y ∈ Y n ) , k Df ( x, y ) k B ( X n − δX × Y n − ,X n − δX ) < L f ( x ∈ X X n − δX , y ∈ Y n − ) , k Dg ( x, y ) k B ( X n × Y n ,Y n − δY ) < L g ( x ∈ X n , y ∈ Y n ) . (iv) Bounds for semigroups : we choose constants M A , M B , C A , C B > n ) as well as ω A < ω B ∈ R (which do not depend on n ) MAXIMILIAN ENGEL, FELIX HUMMEL, AND CHRISTIAN KUEHN such that for all t > k e tA k B ( X n ) ≤ M A e ω A t , k e tA k B ( X n − γX ,X n ) ≤ C A t γ X − e ω A t , k e tA k B ( X n − δX ,X n ) ≤ C A t δ X − e ω A t and k e tB k B ( Y n ) ≤ M B e ω B t , k e tB k B ( Y n − δY ,Y n ) ≤ C B t δ Y − e ω B t . (v) Relation of constants : we define ω f := ω A + (2 C A L f ) γX ( γ X ) − γXγX if γ X ∈ (0 ,
1) and take ω f > ω A + C A L f if γ X = 1. Moreover, we assume ω f < ,L f max {k A − k B ( X γX ,X ) , k A − k B ( X δX − ,X δX ) } < , Remark . The conditions of Assumption ( A n ) are almost identical to the ones in[6, Section 4]. Here they are slightly simplified in the sense that the differentiabilityof the nonlinearities, which is assumed here, is not necessary for all results in [6].2.1.2. Assumption ( B n ) . This assumption is sufficient for obtaining a two-parameterfamily of slow manifolds S ε,ζ [6], in particular specifying the role of the second pa-rameter ζ : we assume that for each small ζ > Y = Y ζF ⊕ Y ζS ,independently from n , into a fast part Y ζF and a slow part Y ζS such that the pro-jections pr Y ζF and pr Y ζS commute with B on Y n .The crucial characterization of the fast part is that Y ζF ∩ Y n − δ Y contains theparts of Y n − δ Y that decay under the semigroup (e tB ) t ≥ almost as fast as thespace X n under (e ζ − tA ) t ≥ ; analogously, the slow space Y ζS ∩ Y n contains the partsof Y n which do not decay or which only decay slowly under the semigroup (e tB ) t ≥ compared to X n under (e ζ − tA ) t ≥ . This idea is expressed in point (v) of thefollowing assumptions:(i) Closed subspaces : the spaces Y ζF ∩ Y β and Y ζS ∩ Y β are closed in Y β for all β ≥ k · k Y β .(ii) Lipschitz bound : using the notation f ( x, y F , y S ) := f ( x, y F + y S ) and g ( x, y F , y S ) := g ( x, y F + y S ), the nonlinearity g satisfies k pr Y ζS g ( x − ˜ x, y F − ˜ y F , y S − ˜ y S ) k Y n ≤ L g ζ δ Y − (cid:0) k x − ˜ x k X n + k y F − ˜ y F k Y n + k y S − ˜ y S k Y n (cid:1) . (iii) Semigroup in slow subspace : the realization of B in Y ζS ∩ Y n − , i.e. B Y ζS ∩ Y n − : Y ζS ∩ Y n − ⊃ D ( B Y εS ∩ Y n − ) → Y ζS ∩ Y n − , v Bv with D ( B Y ζS ∩ Y n − ) := { v ∈ Y ζS ∩ Y n : Bv ∈ Y ζS ∩ Y n − } generates a C -group (e tB Y ζS ∩ Yn − ) t ∈ R ⊂ B (( Y ζS ∩ Y n − , k · k Y n − )) which sat-isfies e tB Y ζS ∩ Yn − = e tB on Y ζS ∩ Y n − for t ≥
0. For the sake of readability, wewill still write B instead of B Y ζS ∩ Y n − . LOW MANIFOLDS IN INFINITE DIMENSIONS 5 (iv)
Semigroup in fast subspace : the realization of B in Y ζF ∩ Y n − , i.e. B Y ζF ∩ Y n − : Y ζF ∩ Y n − ⊃ D ( B Y ζF ∩ Y n ) → Y ζF ∩ Y n − , v Bv with D ( B Y ζF ) := { v ∈ Y ζF ∩ Y n : Bv ∈ Y ζF ∩ Y n − } has 0 in its resolvent set. For the sake of readability, we will still write B instead of B Y ζF ∩ Y n − .(v) Speed of decay in Y ζF and Y ζS : there are constants C B , M B > ζ > ≤ N ζF < N ζS < − ζ − ω A suchthat for all t > y F ∈ Y ζF ∩ Y n − δ Y and y S ∈ Y ζS ∩ Y n we have the estimates k e tB y F k Y n ≤ C B t δ Y − e ( N ζF + ζ − ω A ) t k y F k Y n − δY , k e − tB y S k Y n ≤ M B e − ( N ζS + ζ − ω A ) t k y S k Y n . (vi) Estimate for contraction property in Lyapunov-Perron argument :the parameters and constants introduced above satisfy2 γ X L f C A Γ( γ X ) (cid:0) εζ − − ω A + ε ( N ζS + N ζF ) (cid:1) γ X + 2 δ Y L g C B Γ( δ Y )( N ζS − N ζF ) δ Y + 2 ζ δ Y − L g M B N ζS − N ζF < , (2.2) where Γ denotes the gamma function. Remark . Assumption ( B n ) is identical to the conditions in [6, Section 5] exceptfor the fact that in [6, Section 5] it is only assumed for n = 1. Here, we make use ofadditional regularity in certain estimates and therefore formulate the assumptionfor n ∈ N .2.1.3. Assumption ( C n ) . If we want to use a Galerkin approximation in both theslow and the fast variable, then it is useful to also impose similar conditions on X , i.e. that there is a splitting X = X ζF ⊕ X ζS such that the conditions (i)-(v) inAssumption ( B n ) hold with Y and B being replaced by X and A , respectively.2.1.4. Assumption ( D ) . This assumption will enable us to trade regularity for ad-ditional decay behavior. We assume that for 0 ≤ α ≤ β there is a constant C α,β such that, for all x ∈ X ζF ∩ X β , y ∈ Y ζF ∩ Y β , we have the estimates k y k Y α ≤ C α,β ζ β − α k y k Y β , k x k X α ≤ C α,β ζ β − α k x k X β . (2.3) Remark . Let us give an example of a situation in which Assumption ( D ) issatisfied. We define the Bessel potential space on the torus T by H s ( T ) := ( u ∈ D ′ ( T ) : X l ∈ Z (1 + | l | ) s/ h u, e l i L ( T ) e l ∈ L ( T ) ) , where e l = [ x e πilx ], and endow the space with the norm k u k H s ( T ) := k ((1 + | l | ) s/ h u, e l i L ( T ) ) l ∈ Z k ℓ ( Z ) . Consider for example X = Y = L ( T ) and A = B = ∆ − D ( A ) = D ( B ) = Y = X = H ( T ). The interpolation-extrapolation scales are then givenby Y α = X α = H α ( T ). Y ζF and X ζF will be the subspaces of L ( T ) such that the MAXIMILIAN ENGEL, FELIX HUMMEL, AND CHRISTIAN KUEHN l -th Fourier coefficients with ( | l | − ≤ ζ − are equal to 0. With this choice weobtain for y ∈ Y ζF ∩ Y β k y k Y α = k y k H α ( T ) = X l ∈ Z , | l | ≥ ζ − (1 + | l | ) α h y, e l i L ( T ) / ≤ (1 + ζ − ) α − β X l ∈ Z , | l | ≥ ζ − (1 + | l | ) β h y, e l i L ( T ) / . ζ β − α k y k Y β , and the same for x ∈ X ζF ∩ X β . This is estimate (2.3) with C α,β = 1.2.2. The direct approach.
Let us now briefly collect the main results of [6, Sec-tion 5]. Under the assumptions ( A n ) and ( B n ), one can rewrite (1.1) together with(2.1) as ε∂ t u ε ( t ) = Au ε ( t ) + f ( u ε ( t ) , v εF ( t ) , v εS ( t )) ,∂ t v εF ( t ) = Bv εF ( t ) + pr Y ζF g ( u ε ( t ) , v εF ( t ) , v εS ( t )) ,∂ t v εS ( t ) = Bv εS ( t ) + pr Y ζS g ( u ε ( t ) , v εF ( t ) , v εS ( t )) ,u ε (0) = u , v εF (0) = pr Y ζF v , v εS (0) = pr Y ζS v . (2.4)For this equation, we can formulate the following theorem, which is a collection ofthe results in [6, Section 5]. Theorem 2.4.
Let n ∈ N and suppose that Assumption ( A n ) and Assumption ( B n ) hold true. Fix c ∈ (0 , and let < ε < c ω f ω A ζ . Then there is a family of sets S ε,ζ given as graphs S ε,ζ := { ( h ε,ζX ( v ,S ) , h ε,ζY ζF ( v ,S ) , v ,S ) | v ,S ∈ Y ζS ∩ Y n } where ( h ε,ζX , h ε,ζY ζF ) : Y ζS ∩ Y n → X n × ( Y ζF ∩ Y n ) is differentiable such that the followingassertions hold:(a) Invariance : the set S ε,ζ is invariant under the semiflow generated by (2.4) .(b) Distance between S ε,ζ and critical manifold : there is a constant C > ,which is independent of ε and ζ , such that for all v ,S ∈ Y ζS ∩ Y n we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h ε,ζX n ( v ,S ) − h ( v ,S ) h ε,ζY ζF ( v ,S ) !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X n × Y n ≤ C ε + 1( N ζS − N ζF ) δ y ! k v ,S k Y n . (c) Exponentially fast convergence to S ε,ζ : there are ε , ζ > and constants C, c > independent of T such that for all ε ∈ (0 , ε ] , ζ ∈ (0 , ζ ] with < ε Approximation by slow subsystem : the reduced slow subsystem given by Au ζ ( t ) + f ( u ζ ( t ) , v ζ ( t )) , Y ζF v ζ ( t ) ,∂ t v ζ ( t ) = Bv ζ ( t ) + pr Y ζS g ( u ζ ( t ) , v ζ ( t )) ,v ζ (0) = pr Y ζS v . (2.5) has a unique solution ( u ζ , v ζ ) ∈ C ([0 , T ]; X n − × Y n − ) ∩ C ([0 , T ]; X n × Y n ) which approximates the solution of the full fast-slow system.More precisely, there are a constant C > which may depend on T , somesuitably chosen ω g ∈ R and ε , ζ > such that for all ε ∈ (0 , ε ] , ζ ∈ (0 , ζ ] with < ε < c ω f ω A ζ , all t ∈ [0 , T ] and all v ∈ Y n we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) u ε ( t ) − h ( v ζ ( t )) v ε ( t ) − v ζ ( t ) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) Y n ≤ C (cid:18) k pr Y ζF v k Y n + (cid:0) ε + ω g − ζ − ω A − N ζF ) δY (cid:1) k v k Y n + ( ε δ Y + e ε − ω f t ) k u − h ( v ) k X n (cid:19) . If even ( u , pr Y ζF v , pr Y ζS v ) ∈ S ε,ζ , then it holds that (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) u ε ( t ) − h ( v ζ ( t )) v ε ( t ) − v ζ ( t ) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) X n × Y n ≤ C (cid:16) ε + ω g − ζ − ω A − N ζF ) δY + N ζS − N ζF ) δY (cid:17) k v k Y n . The Galerkin approach. Assuming conditions ( A n ), ( B n ) and ( C n ), wemay additionally consider the projection of (1.1) and (2.1) to the slow part in bothvariables, i.e. ε∂ t u εG = Au εG + pr X ζS f ( u εG , v εG ) ,∂ t v εG = Bv εG + pr Y ζS g ( u εG , v εG ) ,u εG (0) = pr X ζS u , v εG (0) = pr Y ζS u . (2.6)Note that, in general, this is not necessarily a finite-dimensional evolution equation.If A and B generate C -groups, one may even have X ζS = X and Y ζS = Y . However,if A and B have eigenvalue expansions with only a finite number of eigenvalues ineach vertical stripe of bounded width within the complex plane, the spaces X ζS and Y ζS are finite-dimensional. This is the situation of many applications such as theLaplacian ∆ on the torus T . In such a case (2.6) indeed coincides with a Galerkinapproximation of (1.1) and (2.1). The spaces X ζS and Y ζS are given as the linearspan of the eigenfunctions associated with the N A and N B eigenvalues, includingmultiplicities, of A and B , respectively, in { z ∈ C : ζ − ω A + N ζS ≤ ℜ ( z ) } .For example, for A = B = ∆ on X = Y = L ( T ) with eigenvalues λ k = − π k , k ∈ Z , and eigenfunctions ( e k ) k ∈ Z , the expansions u ε ( x, t ) = X k ∈ Z u εk ( t ) e k ( x ) , v ε ( x, t ) = X k ∈ Z v εk ( t ) e k ( x )give, upon taking the inner product of (1.1) with each e k , the system of GalerkinODEs ε∂ t u εk = λ k u εk + h f ( u ε , v ε ) , e k i ∂ t v εk = λ k v εk + h g ( u ε , v ε ) , e k i . (2.7) MAXIMILIAN ENGEL, FELIX HUMMEL, AND CHRISTIAN KUEHN A truncation at | k | ≤ k = N ∆ , for ε and ζ given as before, yields the describedcorrespondence to system (2.6). Such a Galerkin approach is very insightful insituations of dynamical interest such as dynamic bifurcations in reaction-diffusionsystems where geometric techniques can be applied to the finite-dimensional ap-proximation and then be extended to the infinite-dimensional limit (see e.g. [3]).Generally, the existence of such an infinite-dimensional limit raises questions: ifwe fix ζ and let ε > G ε,ζ given by G ε,ζ = { ( h ε,ζG ( v ) , v ) : v ∈ Y ζS ∩ Y n } for certain mappings h ε,ζG : Y ζS ∩ Y n → X ζS ∩ X n . If Y ζS ∩ Y n and X ζS ∩ X n are finite-dimensional, then this can be derived by classical finite-dimensional Fenichel theory.If they are infinite-dimensional, then one can still use the results from [6]. It is nowof particular interest to study the behavior of G ε,ζ – or a similar object relatedto it – as ζ → 0, which corresponds with N A , N B → ∞ in the situation describedabove. Under a suitable notion of convergence, the potential limiting object G ε, may be considered as a type of slow manifold. The main difficulty for such anapproach is that the existence of G ε,ζ for fixed ε becomes unclear when ζ gets toosmall. In [3], such a procedure was carried out for a particular example, by usingan explicit approximation of the parametrization h ε,ζG whose limit for ζ → ε and ζ and the dynamical interpretation of the different objects, aswe will demonstrate in Example 4.1. Hence, it becomes particularly important tounderstand the relation between G ε,ζ and S ε,ζ in order to measure the quality ofthe Galerkin approximation for infinite-dimensional fast-slow systems.3. The main result In this section, we fix m, n ∈ N , m ≤ n . For our main result we supposethat — as it was derived in [6] — the slow manifolds have been constructed via aLyapunov-Perron approach. Therefore, we have that h ε,ζX ( v ,S ) h ε,ζY ζF ( v ,S ) ! = R −∞ e − ε − sA f (¯ u ( s ) , ¯ v F ( s ) , ¯ v S ( s )) d s R −∞ e − sB pr Y ζF g (¯ u ( s ) , ¯ v F ( s ) , ¯ v S ( s )) d s ! , where v ,S ∈ Y ζS ∩ Y n and where (¯ u, ¯ v F , ¯ v S ) denotes the unique fixed point of theoperator L v ,S ,ε,ζ : C η,n → C η,n , uv F v S t ε − R t −∞ e ε − ( t − s ) A f ( u ( s ) , v F ( s ) , v S ( s )) d s R t −∞ e ( t − s ) B pr Y ζF g ( u ( s ) , v F ( s ) , v S ( s )) d s e tB v ,S + R t e ( t − s ) B pr Y ζS g ( u ( s ) , v F ( s ) , v S ( s )) d s . (3.1)Here, the space C η,n with η := ζ − ω A + N ζS + N ζF u, v F , v S ) ∈ C (( −∞ , X × ( Y ζF ∩ Y n ) × ( Y ζS ∩ Y n )) such that k ( u, v F , v S ) k C η,n := sup t ≤ e − ηt (cid:0) k u ( t ) k X n + k v F ( t ) k Y n + k v S ( t ) k Y n (cid:1) < ∞ . LOW MANIFOLDS IN INFINITE DIMENSIONS 9 The fixed point (¯ u, ¯ v F , ¯ v S ) has been shown to exist in [6, Proposition 5.1]. Likewise,we may write h η,ζG ( v ,S ) = ε − Z −∞ e − ε − sA pr X ζS f (¯ u G ( s ) , ¯ v G ( s )) d s, (3.2)where (¯ u G , ¯ v G ) denotes the unique fixed point of the operator L Gv ,S ,ε,ζ : C Gη,n → C Gη,n , (cid:18) u G v G (cid:19) " t ε − R t −∞ e ε − ( t − s ) A pr X ζS f ( u G ( s ) , v G ( s )) d s e tB v ,S + R t e ( t − s ) B pr Y ζS g ( u G ( s ) , v G ( s )) d s ! . (3.3)Analogously to before, C Gη,n denotes the space of all ( u G , v G ) ∈ C (( −∞ , X ζS ∩ X n ) × ( Y ζS ∩ Y n )) such that k ( u G , v G ) k C Gη,n := sup t ≤ e − ηt (cid:0) k u G ( t ) k X n + k v G ( t ) k Y n (cid:1) < ∞ . With this terminology at hand, we can now formulate our main theorem: Theorem 3.1. We fix m, n ∈ N , m ≤ n and c ∈ (0 , . Suppose that the assump-tions ( A m ) , ( B m ) and ( C m ) as well as ( A n ) , ( B n ) , ( C n ) and ( D ) are satisfied. Thenthere is a constant C > such that for all ε, ζ > small enough with c ω f ω A ζ > ε and all v ,S ∈ Y ζS ∩ Y n , we have k h ε,ζX ( v ,S ) − h ε,ζG ( v ,S ) k X m + k h ε,ζY ζF ( v ,S ) k Y m ≤ C ζ n − m ( N ζS − N ζF ) δ Y + ζ n − m + γ X ! k v ,S k Y n . (3.4) Proof. As above, let (¯ u, ¯ v F , ¯ v S ) be the unique fixed point of L v ,S ,ε,ζ from (3.1)and (¯ u G , ¯ v G ) the one of L Gv ,S ,ε,ζ from (3.3). Then, using assumptions ( A n ), (iii)and (iv), and ( D ), we have e − ηt ( k h ε,ζX (¯ v S ( t )) − h ε,ζG (¯ v G ( t )) k X m ) ≤ e − ηt (cid:13)(cid:13)(cid:13)(cid:13) ε − Z t −∞ e − ε − ( t − s ) A pr X ζF f (¯ u ( s ) , ¯ v F ( s ) , ¯ v S ( s )) ds (cid:13)(cid:13)(cid:13)(cid:13) X m + e − ηt (cid:13)(cid:13)(cid:13)(cid:13) ε − Z t −∞ e − ε − ( t − s ) A pr X ζS (cid:2) f (¯ u ( s ) , ¯ v F ( s ) , ¯ v S ( s )) − f (¯ u G ( s ) , ¯ v G ( s )) (cid:3) ds (cid:13)(cid:13)(cid:13)(cid:13) X m ≤ C m,n ζ n − m e − ηt (cid:13)(cid:13)(cid:13)(cid:13) ε − Z t −∞ e − ε − ( t − s ) A pr X ζF f (¯ u ( s ) , ¯ v F ( s ) , ¯ v S ( s )) ds (cid:13)(cid:13)(cid:13)(cid:13) X n + e − ηt (cid:13)(cid:13)(cid:13)(cid:13) ε − Z t −∞ e − ε − ( t − s ) A pr X ζS (cid:2) f (¯ u ( s ) , ¯ v F ( s ) , ¯ v S ( s )) − f (¯ u G ( s ) , ¯ v G ( s )) (cid:3) ds (cid:13)(cid:13)(cid:13)(cid:13) X m ≤ L f C A C m,n ζ n − m Z t −∞ e ( ε − ζ − ω A − η )( t − s ) ε γ X ( t − s ) − γ X ds k (¯ u, ¯ v F , ¯ v S ) k C η,n + L f C A Z t −∞ e ( ε − ω A − η )( t − s ) ε γ X ( t − s ) − γ X ds k (¯ u − ¯ u G , ¯ v F , ¯ v S − ¯ v G ) k C η,m . It was shown in the proof of [6, Proposition 5.2] that the mapping Y ζS ∩ Y k → C η,k , v ,S (¯ u, ¯ v F , ¯ v S ) is Lipschitz continuous. Let L > Z t −∞ e ( ε − ω A − η )( t − s ) ε γ X ( t − s ) − γ X ds ≤ Γ( γ X )( εη − ω A ) γ X . Hence, we obtain that e − ηt ( k h ε,ζX (¯ v S ( t )) − h ε,ζG (¯ v G ( t )) k X m ) ≤ LL f C A C m,n Γ( γ X )( εη − ζ − ω A ) γ X ζ n − m k v k Y n + L f C A Γ( γ X )( εη − ω A ) γ X k (¯ u − ¯ u G , ¯ v F , ¯ v S − ¯ v G ) k C η,m . (3.5)Furthermore, combining [6, (5-3)] with Assumption ( D ) yields e − ηt k h ε,ζY ζF (¯ v S ( t )) k Y m ≤ C m,n ζ n − m e − ηt k h ε,ζY ζF (¯ v S ( t )) k Y n ≤ δ Y C m,n LL g C B Γ( δ Y )( N ζS − N ζF ) δ Y ζ n − m k v k Y n . (3.6)Concerning ¯ v S − ¯ v G , we observe with ( B n ), (ii) and (v), that e − ηt k ¯ v S ( t ) − ¯ v G ( t ) k Y m = e − ηt (cid:13)(cid:13)(cid:13)(cid:13)Z t e ( t − s ) B pr Y ζS (cid:2) g (¯ u ( s ) , ¯ v F ( s ) , ¯ v S ( s )) − g (¯ u G ( s ) , ¯ v G ( s )) (cid:3) ds (cid:13)(cid:13)(cid:13)(cid:13) Y m ≤ L g C B Z t ζ δ Y − e ( t − s )( ζ − ω A − η ) ds k (¯ u − ¯ u G , ¯ v F , ¯ v S − ¯ v G ) k C η,m ≤ L g C B ζ δ Y − N ζS − N ζF k (¯ u − ¯ u G , ¯ v F , ¯ v S − ¯ v G ) k C η,m . (3.7)Summing up (3.5), (3.6) and (3.7) yields e − ηt (cid:2) k h ε,ζX (¯ v S ( t )) − h ε,ζG (¯ v G ( t )) k X m + k h ε,ζY ζF (¯ v S ( t )) k Y m + k ¯ v S ( t ) − ¯ v G ( t ) k Y m (cid:3) ≤ C m,n δ Y LL g C B Γ( δ Y )( N ζS − N ζF ) δ Y + LL f C A Γ( γ X )( εη − ω A ) γ X ! ζ n − m k v k Y n + L f C A Γ( γ X )( εη − ω A ) γ X + L g C B ζ δ Y − N ζS − N ζF ! k ( h ε,ζX (¯ v S ) − h ε,ζG (¯ v G ) , h ε,ζY ζF (¯ v S ( t )) , ¯ v S − ¯ v G ) k C η,m . Therefore, if we write ˜ L := L f C A Γ( γ X )( ω A − εη ) γ X + L g C B ζ δ Y − N ζS − N ζF ! , which is strictly smaller than 1 by Assumption ( B m ), we obtain k ( h ε,ζX (¯ v S ) − h ε,ζG (¯ v G ) , h ε,ζY ζF (¯ v S ) , ¯ v S − ¯ v G ) k C η,m ≤ C m,n − ˜ L δ Y LL g C B Γ( δ Y )( N ζS − N ζF ) δ Y + LL f C A Γ( γ X )( ζ − ω A − εη ) γ X ! ζ n − m k v k Y n . LOW MANIFOLDS IN INFINITE DIMENSIONS 11 The fact that k h ε,ζX ( v ,S ) − h ε,ζG ( v ,S ) k X m + k h ε,ζY ζF ( v ,S ) k Y m ≤ k ( h ε,ζX (¯ v S ) − h ε,ζG (¯ v G ) , h ε,ζY ζF (¯ v S ) , ¯ v S − ¯ v G ) k C η,m finally yields the assertion. (cid:3) Case study of an explicit reaction-diffusion problem As discussed in Section 2.3, in certain situations of interest the spaces X ζS and Y ζS are N A -dimensional and N B -dimensional with N A and N B being the number ofeigenvalues including multiplicities of A and B , respectively, in { z ∈ C : ζ − ω A + N ζS ≤ ℜ ( z ) } . In a Galerkin approach, one usually studies the limit N A , N B →∞ which corresponds to ζ → 0. However, when we fix ε > 0, the condition( c ω A /ω f ) ζ > ε for some c ∈ (0 , ζ → 0. In such a situation, the existence of slow manifolds in the sense of[6] is unclear. Moreover, in the limit ζ → ζ ∈ (0 , ε ω f / ( c ω A )).In a general setting, it is not clear whether this holds true or not. However,in certain situations it is possible to explicitly derive invariant manifolds for (2.6)which resemble slow manifolds from the classical finite-dimensional theory. Usingsuch a computation, we discuss the intricacies of the limit ζ → Example 4.1. (i) Explicit computation of slow manifolds : consider the fol-lowing fast-slow system ε∂ t u ε = ∆ u ε − u ε + ( v ε ) ,∂ t v ε = ∆ v ε − v ε , (4.1)on the torus T . A natural approach for a Galerkin approximation is to truncate toa certain number of Fourier modes. Writing u εk ( t ) := Z T u ε ( t, x ) e − πikx dx, v εk ( t ) := Z T v ε ( t, x ) e − πikx d x ( k ∈ Z ) , we can expand u ε ( t, x ) = X k ∈ Z u εk ( t ) e πikx , v ε ( t, x ) = X k ∈ Z v εk ( t ) e πikx . Applying h· , e πikx i L ( T ) to both sides of (4.1) yields ε∂ t u εk = − (1 + 4 π k ) u εk + X j,l ∈ Z ,j + l = k v εj v εl ,∂ t v εk = − (1 + 4 π k ) v εk (4.2) for all k ∈ Z . Truncating at a certain k ∈ N , we obtain ε∂ t u εk = − (1 + 4 π k ) u εk + X j,l ∈ Z , | j | , | l |≤ k ,j + l = k v εj v εl ,∂ t v εk = − (1 + 4 π k ) v εk , (4.3)which is a finite-dimensional fast-slow ODE for sufficiently small ε > 0. We candirectly solve the slow equation by v εk ( t ) = e − (1+4 π k ) t v εk (0) , and, if [ ε − (1 + 4 π k ) − − π ( j + l )] = 0, the fast equation is solved by u εk ( t ) − e − ε − (1+4 π k ) t u εk (0)= X j,l ∈ Z , | j | , | l |≤ k ,j + l = k ε − Z t e − ε − (1+4 π k )( t − s ) v εj ( s ) v εl ( s ) d s = X j,l ∈ Z , | j | , | l |≤ k ,j + l = k ε − e − ε − (1+4 π k ) t Z t e [ ε − (1+4 π k ) − − π ( j + l )] s v εj (0) v εl (0) d s = X j,l ∈ Z , | j | , | l |≤ k ,j + l = k ε − e − − π ( j + l )] t − e − ε − (1+4 π k ) t ε − (1 + 4 π k ) − − π ( j + l ) v εj (0) v εl (0) . The essential property of a slow manifold is to eliminate the fast dynamics. In fact,cancelling out the terms with the ε − in the exponent, we obtain u εk (0) = X j,l ∈ Z , | j | , | l |≤ k ,j + l = k v εj (0) v εl (0)1 + 4 π k − ε [2 + 4 π ( j + l )] , (4.4)which could be seen as a formula for the slow manifold. The critical manifold inturn would be given by u k (0) = X j,l ∈ Z , | j | , | l |≤ k ,j + l = k v j (0) v l (0)1 + 4 π k . (4.5)Let M k be the set of all ε ∈ (0 , 1) such that there are ( j, k ) ∈ Z with max {| j | , | l |} ≤ k and k ∈ Z , | k | ≤ k with ε − (1 + 4 π k ) − − π ( j + l ) = 0 , (4.6)i.e., M k contains all ε ∈ (0 , 1) for which there may be singularities in (4.4). Thisset is special since the above procedure of cancelling out the terms with an ε − inthe exponent is not possible for such ε ; note that a similar situation occurs in manydynamical systems in the context of resonances and the small divisor problem [10].Although the existence of invariant manifolds is not clear for ε ∈ M k , we observethat M k is finite if k ∈ N and countable with an accumulation point at 0 if k = ∞ . This shows two things: firstly, for all but countably many ε ∈ (0 , 1) thereexists an invariant manifold as a graph over the whole slow variable space for (4.1).Secondly, it seems like there is no ε such that such an invariant manifold exists for LOW MANIFOLDS IN INFINITE DIMENSIONS 13 all ε ∈ (0 , ε ]. Instead, one has to restrict to a subset of the slow variable space, asalso suggested by the direct approach. In this example, one has to impose X ( j,k ) ∈L k ,k v εj (0) v εl (0) = 0 , where L k ,k denotes the set of all pairs ( j, l ) ∈ Z with j + l = k and | j | , | l | ≤ k such that the denominator in (4.4) is equal 0.Even though an invariant manifold for (4.1) exists for all but countably many ε ∈ (0 , ∞ as dist( ε, M k ) → 0, as can be seen directly from (4.4) and(4.5). However, it is easy to see from equation (4.6) that there are no singularitiesin (4.1) if ε < / (2 + 8 π k ). In this case, the slow manifold from the Galerkin ap-proximation is close to the slow manifold obtained by the direct approach. This canbe checked by computing the slow manifold for (4.1) from the Fourier coefficientsas above. In our abstract framework, we obtain the following precise estimate.(ii) Exemplification of abstract framework : one can choose X = L ( T ) and Y = H ( T ) as underlying spaces and A = ∆ − B = ∆ − D ( A ) = H ( T ) and D ( B ) = H ( T ), respectively. Then, we have X α = H α ( T )and Y α = H α ( T ). As nonlinearities, we choose f ( x, y ) := y and g ( x, y ) = 0with γ X = δ X = δ Y = 1. If n ≥ 1, then f : X n × Y n − → X n , ( x, y ) y is a well-defined and smooth nonlinearity, since H α ( T ) is a Banach algebra for α > − . However, the bounds on the derivatives of f from Assumption ( A n ) areonly satisfied locally and the Lipschitz constant only gets small in a neighborhoodaround 0 in Y n . Formally, one would have to use cutoff techniques as for examplein [6, Section 6] in order to apply our methods. But since globabl stability issuesare not our primary concern, we omit the details here. Instead, we just keep inmind that we have to restrict to a certain neighborhood around 0 in Y n so that L f is small enough for (2.2) to hold.For Assumption ( B n ) we need to introduce a splitting of the slow variable space.Let ω A ∈ ( − , 0) be close to − 1. For k ∈ N with − π ( | k | + 2) < ζ − ω A + 1 ≤ − π ( | k | + 1) , we take X ζS = Y ζS := span { [ x e i πkx ] : k ∈ Z , | k | ≤ | k |} ,Y ζF := cl H ( T ) (cid:0) span { [ x e i πkx ] : k ∈ Z , | k | ≥ | k | + 1 } (cid:1) ,X ζF := cl L ( T ) (cid:0) span { [ x e i πkx ] : k ∈ Z , | k | ≥ | k | + 1 } (cid:1) , where cl T M denotes the closure of a set A ⊂ T in a topological space T . Note thatthe projection to X ζS and Y ζS coinides with the projection to the first k Fouriermodes. Thus, our abstract Galerkin equation (2.6) is consistent with the explicitexample (4.3). Now it is straightforward to check that the assumptions ( B n ), ( C n )and ( D ) are satisfied. Nevertheless, let us specify the choice of N ζF and N ζS . Notethat we have e tB f = e t (∆ − f = " x X k ∈ Z e − (4 π k +1) t ˆ f ( k ) e i πkx . Thus, for y S ∈ Y ζS and t ≥ 0, Plancherel’s Theorem gives k e − tB y S k H n ( T ) ≤ e (4 π | k | +1) t k y S k H n ( T ) , so that we may take N ζS := − ζ − ω A − π | k | − − π ( | k | + 2) < ζ − ω A + 1 ≤ − π ( | k | + 1) , it follows that N ζS > N ζF := − ζ − ω A − π ( | k | + 1) − . With these choices, we observe that formula (4.4) defines the slow manifold whichone also obtains from a Lyapunov-Perron approach. Indeed, the solution of equation(4.3), with initial conditions given by (4.4), reads u εk ( t ) = X j,l ∈ Z , | j | , | l |≤ k ,j + l = k e − [2 − π ( j + l )] t v εj (0) v εl (0)1 + 4 π k − ε [2 + 4 π ( j + l )] ,v εk ( t ) = e − (1+4 π k ) t v εk (0) , also for t ∈ ( −∞ , C Gη,n and hence, (4.4)defines the slow manifold given as the graph of h ε,ζG from the abstract setting. Inparticular, Theorem 3.1 shows that the distance of the Galerkin slow manifold wecomputed for (4.3) to the actual slow manifold for (4.1) is small if ζ, ε > c ω A ω f ζ > ε are small enough. More precisely, if we fix m, n ∈ N , m ≤ n and c ∈ (0 , C > ε > k ∈ N such that k < p cω f / (4 π ε ) − k h ε,ζX ( v ,S ) − h ε,ζG ( v ,S ) k H m ( T ) < Ck − n − m ) − k v ,S k H n +2 ( T ) . Here, h ε,ζX denotes the mapping describing the slow manifold for (4.1) from thedirect approach and h ε,ζG denotes the slow manifold from the Galerkin approachdefined by (4.4). If k is chosen close enough to p cω f / (4 π ε ) − 2, then we alsoobtain the estimate k h ε,ζX ( v ,S ) − h ε,ζG ( v ,S ) k H m ( T ) . ε n − m + k v ,S k H n +2 ( T ) . In particular, the last estimate provides an illustration of the relevance of our mainresult: in situations where a Galerkin approximation may be the procedure ofchoice due to the need of using ODE techniques or for numerical reasons, we knowthat for sufficiently small ε and suitably chosen k the finite-dimensional Galerkinmanifolds are good approximations of the invariant slow manifolds for the PDE, ifthe appropriate norms are taken. Appendix A. Interpolation-Extrapolation Scales We briefly recall the notion of interpolation-extrapolation scales and relatedresults. As a general reference, see [1, Chapter V]. Let T : X ⊃ D ( T ) → X bea densely defined closed linear operator on a Banach space X with 0 ∈ ρ ( T ),where ρ ( T ) denotes its resolvent set. Moreover, for θ ∈ (0 , 1) let ( · , · ) θ be an exactadmissible interpolation functor, i.e. an exact interpolation functor such that X isdense in ( X , X ) θ whenever X d ֒ → X (i.e. the injection is continuous with dense LOW MANIFOLDS IN INFINITE DIMENSIONS 15 range). We define a family of Banach spaces ( X α ) α ∈ [ − , ∞ ) and a family of operators( T α ) α ∈ [ − , ∞ ) ∈ B ( X α +1 , X α ) as follows: • For n ∈ N we choose X n := D ( T n ) endowed with k x k X n := k T n x k X ( x ∈ D ( T n )). In particular, X = D ( T ) = D (id X ) = X . Moreover, T n := T | X n +1 . • X − is defined as the completion of X = X with respect to the norm k x k X − = k T − x k X . The operator T = T is then closable on X − and T − is defined to be the closure. One can also define ( X − n , T − n ) for n ∈ N by iteration, but we do not go beyond n = − • For n ∈ N ∪ {− } , θ ∈ (0 , 1) and α = n + θ we define X α := ( X n , X n +1 ) θ and T α = T n | D ( T α ) where D ( T α ) = { x ∈ X n +1 : T n x ∈ X α } . The family ( X α , T α ) α ∈ [ − , ∞ ) is a densely injected Banach scale in the sense that X α d ֒ → X β whenever α ≥ β , and T α : X α +1 → X α is an isomorphism for all α ∈ R .Moreover T α : X α ⊃ X α +1 → X α is a densely defined closed linear operator with0 ∈ ρ ( T α ) for all α ∈ R . The family ( X α , T α ) α ∈ R is an interpolation-extrapolationscale. References [1] H. Amann. Linear and quasilinear parabolic problems. Vol. I , volume 89 of Monographs inMathematics . Birkh¨auser Boston, Inc., Boston, MA, 1995. Abstract linear theory.[2] P. W. Bates, K. Lu, and C. Zeng. Existence and persistence of invariant manifolds for semi-flows in Banach space. Mem. Amer. Math. Soc. , 135(645):viii+129, 1998.[3] M. Engel and C. Kuehn. Blow-up analysis of fast-slow PDEs with loss of hyperbolicity. arXiv:2007.09973 , pages 1–35, 2020.[4] N. Fenichel. Persistence and smoothness of invariant manifolds for flows. Indiana U. Math.J. , 21:193–225, 1971.[5] N. Fenichel. Geometric singular perturbation theory for ordinary differential equations. J.Differential Equat. , 31:53–98, 1979.[6] F. Hummel and C. Kuehn. Slow manifolds for infinite-dimensional evolution equations. arXiv:2008.10700 , pages 1–53, 2020.[7] C. Jones. Geometric singular perturbation theory. In Dynamical Systems (MontecatiniTerme, 1994) , volume 1609 of Lect. Notes Math. , pages 44–118. Springer, 1995.[8] G. Menon and G. Haller. Infinite dimensional geometric singular perturbation theory for theMaxwell-Bloch equations. SIAM J. Math. Anal. , 33(2):315–346, 2001.[9] S. Wiggins. Normally Hyperbolic Invariant Manifolds in Dynamical Systems . Springer, 1994.[10] J. Yoccoz. An introduction to small divisors problems. In From Number Theory to Physics ,pages 659–679. Springer, 1992. Department of Mathematics and Computer Science, Freie Universit¨at Berlin, 14195Berlin, Germany Email address : [email protected] Faculty of Mathematics, Technical University of Munich, 85748 Garching b. M¨unchen,Germany Email address : [email protected] Faculty of Mathematics, Technical University of Munich, 85748 Garching b. M¨unchen,Germany Email address ::