Convergent families of inertial manifolds for convergent approximations

Numerical Algorithms 14 (1997) 179–188 179

Convergent families of inertial manifolds forconvergent approximations

James C. Robinson

Department of Applied Mathematics and Theoretical Physics, Silver Street,Cambridge CB3 9EW, UK

E-mail: [email protected]

This paper shows that a sequence of (suitably uniform) inertial manifolds for a family ofapproximations converges to an inertial manifold for the limiting problem, without imposingany additional assumptions.

Keywords: inertial manifolds, Galerkin approximations, numerical approximations.

AMS subject classification: 35B40, 35K22, 65M12.

1. Introduction

The theory of inertial manifolds [4] provides a rigorous method to connect the poten-tially infinite-dimensional behaviour of a partial differential equation to the dynamicsof a finite-dimensional system, and explains the low-dimensional structure seen, forexample, in numerical investigations of the Kuramoto–Sivashinksy equation [6].

In the standard theory, the existence of an inertial manifold can be proved underthe so-called spectral gap condition [1]. Many papers then consider the existence of in-ertial manifolds for approximations of the original equation – either through Galerkinapproximations [2, 4, 10], temporal discretisations [3], finite difference approxima-tions [7], or general approximations in [8] and [9], the final two papers obtaining C1

convergence of the manifolds.Generally, these amount to perturbation results, in that the approximation can be

viewed as a C1 perturbation of the original equation, and the spectral gap conditionenforces some normal hyperbolicity which ensures that the manifolds persist underperturbation ([8, 9], cf. classical finite dimensional persistence results such as thosein [17]). The structure of such results can be summarised in

Vague theorem 1. Under the spectral gap condition the equation has an inertial man-ifold, and so does a family of approximations. The approximate inertial manifoldsconverge to the inertial manifold for the exact problem.

J.C. Baltzer AG, Science Publishers

180 J. C. Robinson / Convergent inertial manifolds

The aim of this paper is to prove that

Vague theorem 2. If a set of convergent approximations have inertial manifolds (withsome uniform properties) then they converge to an inertial manifold for the full prob-lem.

The idea is to remove any assumption on the full problem, and to deduce theexistence of an inertial manifold there from the existence in the approximations. Thisremoves the possibility that every approximation has an inertial manifold even thoughthe exact problem does not. Furthermore, this general convergence result removes thenecessity to prove convergence for each particular approximation scheme to whichthe first “theorem” could be applied. The results are a generalisation of those in [14],which cover examples 1 and 2 below. Note, however, that with the assumption ofthe spectral gap condition, C1 convergence of the manifolds can be obtained (e.g.,[8, 10]).

2. Standard theory

The object of study is the equation on a Hilbert space H (usually L2)

du/dt+Au+ f(u) = 0, (1)

where A is a linear operator which is positive, self-adjoint, unbounded, and has acompact inverse. The nonlinearity is taken to be a globally bounded Lipschitz mapfrom D(Aα) into D(Aβ) (with 0 6 α− β < 1) and with support contained in a ballΩρ in D(Aα),

supp(f) ⊂ Ωρ ≡u: |u|α 6 ρ

, sup

u∈Ωρ

∣∣f(u)∣∣β6 C0, ∀ u ∈ D(Aα),

and ∣∣f(u1)− f(u2)∣∣β6 C1|u1 − u2|α, ∀ u1, u2 ∈ D(Aα),

where |v|θ = |Aθv| and | · | is the norm on H . Standard existence and uniquenessresults [5] ensure that equation (1) generates a strongly continuous semigroup S(t) onD(Aα).

Since A has a compact inverse, there is a set of eigenfunctions wn of A,

Awn = λnwn, λn+1 > λn,

which form an orthonormal basis for H . These eigenfunctions can be used to producea natural decomposition of H into low and high wavenumbers via the projectionoperators

PNu =N∑j=1

(u,wj)wj , QNu =∞∑

j=N+1

(u,wj)wj . (2)

J. C. Robinson / Convergent inertial manifolds 181

An inertial manifold M is the graph of a Lipschitz function φ :PNH → QNHwhich is positively invariant and attracts all trajectories exponentially,

dist(u(t),M

)6 C(u0)e−kt,

where C(u0) can be chosen uniformly for all u0 ∈ X, a bounded set in D(Aα).The standard existence results require a large gap in the spectrum ofA [1, 4, 13]:

Theorem 1. If the eigenvalues of A satisfy the spectral gap condition

λN+1 − λN > 2C1(λα−βN+1 + λα−βN

)(3)

then there exists an inertial manifold given as the graph of a function φ :PNH →QNH , with supp(φ) ∈ Ωρ, the Lipschitz constant of φ bounded by 1, and the rate ofattraction bounded by

k > λn. (4)

3. Approximate schemes

It is useful to develop an abstract approximation scheme which will include manyparticular examples.

The exact problem generates a semigroup S(t) on a spaceX, with u(t) = S(t)u0.A family of approximate schemes Sµ will be considered, where each Sµ generatesa forward trajectory on a space Xµ, at a perhaps discrete set of times Tµ.

To compare the dynamics of Sµ and S, the existence of a projection operatorPµ :X → Xµ and an injection operator

iµ :Xµ → X with∥∥iµx1 − iµx2

∥∥X6 C‖x1 − x2‖Xµ and Pµiµ = id (5)

is assumed. Then a convergent approximate scheme can be defined as follows.

Definition 2. An approximate scheme is a sequence of approximate evolutions Sµ(t)defined for t ∈ Tµ and with Sµ(t) :Xµ → Xµ. The scheme is said to converge toS(t) as µ→ 0 if, given any ε > 0 and T > ε, for all t ∈ (ε, T ] there exist sequencestµ with tµ ∈ Tµ such that

(i) tµ → t as µ→ 0, and

(ii) for any bounded set B ∈ X,

supt∈(ε,T ]

supu0∈B

∥∥iµSµ(tµ)Pµu1 − S(t)u0∥∥X→ 0, (6)

as µ→ 0 and ‖u1 − u0‖X → 0.


Part (i) of the definition can be thought of heuristically as “Tµ → R+”. In thesame way it would be natural to expect that “Xµ → X”, i.e., that given x ∈ X, thereexists a sequence xµ ∈ Xµ such that iµxµ → x. Although not explicitly contained inthe definitions, this in fact follows from property (ii). Indeed, since S(t) is a stronglycontinuous semigroup on X, for t small enough one can ensure that∥∥S(t)x− x

∥∥X6 ε/2, t 6 τ.

Now choose µ small enough that (with tµ → t)

supt∈(τ/2,2τ ]

∥∥iµSµ(tµ)Pµx− S(t)x∥∥X6 ε/2,

and so, letting xµ = Sµ(tµ)Pµx ∈ Xµ,

‖iµxµ − x‖X 6 ε,

which is “Xµ → X”.Part (ii) of the definition is essentially a classical error estimate showing that

the trajectories can be approximated uniformly on bounded time intervals – this willclearly involve choosing an appropriate sequence of times tµ for each t ∈ (ε, T ].

A selection of examples will show the wide applicability of the rather abstractdefinition used above.

Example 1. The familydu/dt+Au+ fµ(u) = 0, (7)

where fµ satisfies the same conditions as f above, and fµ → f uniformly.

Take Xµ = X = D(Aα) (so Pµ = iµ = id), Tµ = R+, and Sµ(t) the semigroup onD(Aα) generated by (7). The convergence result (6) follows from [5] or [14] (and infact convergence is uniform on [0, T ]).

Example 2. The Galerkin approximations

duN/dt+AuN + Pnf(uN ) = 0, uN ∈ PNH, uN (0) = PNu(0). (8)

Here consider N → ∞ rather than µ → 0. Take XN = PNH , SN (t) the semigroupgenerated by the ODE (8) on PNH , and TN = R+. The projection PN is exactly theFourier projection PN defined in (2) and iN is the identity. The convergence resultin (6) – where the ε is necessary to deal with the non-uniform convergence of PNuto u for u ∈ B – can be found in [12] and [14], the latter proof taking advantage ofthe particularly simple form assumed for f .

Example 3. Time-discrete approximations of the Galerkin approximations, for exam-ple

un+1 = e−Aδt[un − δt fN (un)

](9)

orun+1 = (I +Aδt)−1[un − δt fN (un)

]. (10)


There can be problems with convergence of such methods in infinite-dimensionalspaces [15], hence the restriction to finite-dimensional approximations. In this casetake X∆t = X (= PNH , for example) but T∆t = n∆t. The convergence result (6)is then straightforward (see [15] for (9) and [3] or [8] for (10)).

Example 4. More general finite difference schemes are discussed in [11] for the com-plex Ginzburg–Landau equation, and in [7] for the Kuramoto–Sivashinky equation.The analysis is performed in terms of embedding and projection operators similar toPµ and iµ above.

Example 5. Finite element schemes are discussed in [8] and [16], where the conver-gence is shown to fit into the form of (6).

4. Main theorem

Consider the “imperfect semigroup” Sµ defined on X by

Sµ(t)u0 = iµSµ(t)Pµu0.

AlthoughSµ(t)Sµ(s) = Sµ(t+ s)

on X (because Pµiµ = id), Sµ(t)x does not converge to x as t goes to zero (becauseiµPµ 6= id). However, it will be convenient to consider the dynamics on X generatedby Sµ, though of course Sµ in fact defines a semigroup on the subspace of X that isiµXµ.

Using the Sµ notation, the convergence criterion (6) can be rewritten compactlyas ∥∥Sµ(tµ)u1 − S(t)u0

∥∥X→ 0 (11)

as µ+ ‖u0 − u1‖X → 0.Suppose that the approximate scheme Sµ(t) has an inertial manifold Σµ, i.e., for

all t ∈ Tµ,distXµ

(Sµ(t)uµ0 ,Σµ

)6 C

(uµ0

)e−kt.

Then since the norm inequality for the embedding iµ (5) shows that

distX(iµSµ(t)uµ0 , iµΣµ

)6 D

(iµu

µ0

)e−kt,

the imperfect semigroup Sµ has an inertial manifold iµΣµ in X with

dist(Sµ(t)u0, iµΣµ

)6 C(u0)e−kt. (12)

Uniformity assumptions on the properties of the embedded manifolds iµΣµ andthe rate of convergence will enable the proof of the limit theorem.


Assume that the injections of Σµ in X can be given as graphs over a fixedfinite-dimensional subspace of X, ΠX, so that

Σµ = iµΣµ = G[φµ], (13)

whereφµ : ΠX → ΘX

(Θ+Π = id, (Θu,Πu) = 0 for all u ∈ X) with the Lipschitz constant of the functionsφµ uniform, ∥∥φµ(p1)− φµ(p2)

∥∥X6 l‖p1 − p2‖X for all p1, p2 ∈ ΠX. (14)

Assume further that the rate of attraction to the manifolds Σµ is uniform, so that (12)holds for all µ. Finally, assume that

supp(φµ) ⊂ ΠΩρ (15)

for some large ball Ωρ in X.Notice that by combining (14) and (15), it is possible to choose % such that

supp(φµ) ⊂ ΠΩ% (16)

and alsoG[φµ] ⊂ Ω%. (17)

Simply choose % = (1+ l)ρ, so that (16) follows immediately and the Lipschitz boundon φµ (14) yields (17).

Theorem 3. Suppose that a convergent family of approximations has inertial manifoldssatisfying (12), (13), (14), and (17), and that given % > 0 and [T1, T2] it is possible topick R > 0 such that, uniformly in µ,[

Sµ(t)(∂ΠΩR)]∩Ω% = ∅ for all t ∈ [T1, T2] ∩ Tµ. (18)

Then the limiting semigroup S(t) has an inertial manifold given as G[φ], attractingtrajectories of S(t) according to (12). Moreover, φµ → φ as µ→ 0.

Note that (18) is equivalent to: given % > 0 and T > 0 it is possible to pickR > 0 such that [

S(T )(∂ΠΩR)]∩Ω% = ∅,

which follows easily in the standard setup since the equation is truncated to be linearoutside Ωρ.

Proof. The first step in the proof is the hardest, that is, to show that the φµ form aCauchy sequence in the sup norm,

‖φ‖∞ = supp∈ΠX

∥∥φ(p)∥∥X.


Once φµ → φ it is straightforward to show that φ is invariant and G[φ] attractsexponentially.

Note that from the convergence criterion (6), for any T > ε > 0 and boundedset B ∈ X, we can find a µ0 such that for each t ∈ (ε, T ], provided µ and ν are bothless than µ0, ∥∥Sµ(tµ)u0 − Sν(tν)u0

∥∥X6 ε,

where tλ is the appropriate sequence from (6) with tλ → t. It is thus sufficient toshow that if two semiflows are close enough together than their inertial manifolds areclose.

Consider an ε/2 neighbourhood Nε/2 of Σµ. Then, uniformly in µ, Nε/2 absorbsΩ% in some time T . Let Tλ be the sequence used in (6) to approximate T . Thensince Σµ is flat outside Ω%, Σµ ∩ ΩR = (Σµ ∩ Ω%) ∪ ΠΩR for all µ, and it can beguaranteed by (18) that

Nε/2 ⊃ Sµ(Tµ)(Ω% ∪ΠΩR) ⊃ Σµ ∩Ω%.

Now, if Sµ and Sν are ε/2 close on [T/2, 2T ] (say), then

Nε ⊃ Sν(Tν)(Ω% ∪ΠΩR) ⊃ Σν ∩Ω%,

and sodist(Σν , Σµ

)6 ε. (19)

It remains to show that this implies that

‖φµ − φν‖∞ 6 Kε,

where K does not depend on µ. It is this stage that requires the uniform Lipschitzcontinuity of φµ (14). This is unfortunate, as ideally one would like to show that evenif the Lipschitz constants of φµ blew up as µ→ 0, a C0 invariant manifold could beobtained in the limit. However, with a more refined analysis one can obtain interestingresults even in this case [14].

For any point v ∈ X, and Mφ = G[φ], where φ has Lipschitz constant lφ,

dist(v,Mφ)2 = infp∈ΠX

(‖Πv − p‖2

X +∥∥Θv − φ(p)

∥∥2X

)and ∥∥Θv − φ(Πv)

∥∥2X

=∥∥Θv − φ(p) + φ(p)− φ(Πv)

∥∥2X

6 2∥∥Θv − φ(p)

∥∥2X

+ 2∥∥φ(p)− φ(Πv)

∥∥2X

6 2∥∥Θv − φ(p)

∥∥2X

+ 2lφ‖Πv − p‖2X

6 c2(∥∥Θv − φ(p)∥∥2X

+ ‖Πv − p‖2X

)


for all p ∈ ΠX, where c2 = 2 max(lφ, 1). Therefore∥∥Θv − φ(Πv)∥∥X6 c dist(v,Mφ).

Setting v = p+ ψ(p) ∈Mψ,∥∥ψ(p)− φ(p)∥∥X6 c dist(v,Mφ)

and so‖φ− ψ‖∞ 6 c dist(Mφ,Mψ). (20)

Since lφµ is uniform over µ, combining (19) with (20) shows that for µ, ν smallenough

‖φµ − φν‖∞ 6 Kε,with K2 = 2 max(l, 1), and so the sequence φµ is Cauchy, with φµ → φ. Since theconvergence is uniform, φ : ΠX → ΘX also has Lipschitz constant l.

To show invariance, consider a trajectory with initial condition u0 = p0 +φ(p0).Then at time t > 0 invariance implies that

S(t)u0 = p(t) + φ(p(t)

), (21)

where p(t) = Πu(t). Now consider the trajectories of the imperfect semigroups Sµthrough uµ0 = p+ φµ(p), and times tµ → t, so that

Sµ(tµ)uµ0 = pµ(tµ) + φµ(pµ(tµ)

).

By the convergence of the approximate schemes,

Sµ(tµ)uµ0 → S(t)u0,

but using the uniform convergence of the functions φµ to φ along with the convergenceof the Π components of uµ(tµ), pµ(tµ)→ p(t),

pµ(tµ) + φµ(pµ(tµ)

)→ p(t) + φ

(p(t)

),

so thatS(t)u0 = p(t) + φ

(p(t)

),

ensuring that (21) holds.Exponential attraction follows from an easy application of the triangle inequality.

Consider a fixed time interval, (ε, T ]. Then for t ∈ (ε, T ) and a sequence tµ (tµ ∈ Tµwith tµ → t),

dist(S(t)u0,M

)6∥∥S(t)u0, Sµ(tµ)u0

∥∥X

+ dist(Sµ(tµ)u0,Σµ

)+ dist(Σµ,M)

6∥∥S(t)u0, Sµ(tµ)u0

∥∥X

+ C(u0)e−ktµ + ‖φµ − φ‖.


Now, given η > 0, choose µ small enough that∥∥S(t)u0, Sµ(t)u0∥∥X6 ηC(u0)e−kT , ‖φµ − φ‖ 6 ηC(u0)e−kT ,

andC(u0)e−ktµ 6 [1 + η]C(u0)e−kt.

Then, for any η > 0,

dist(S(t)u0,M

)6 (1 + 3η)C(u0)e−kt,

so if t ∈ (ε, T ]

dist(S(t)u0,M

)6 C(u0)e−kt. (22)

But ε and T are arbitrary, so (22) holds for all t, which is the exponential convergenceproperty as required.

5. Two quick applications

Two immediate applications are now given, one more or less frivolous but the otherwith important implications for numerical experiments. The first is that the strictinequality in the spectral gap condition (3) can be turned into

λN+1 − λN > 2C1(λα−βN+1 + λα−βN

),

since the Lipschitz constant and rate of contraction provided by the strict inequalityare uniform.

The second observation is much more important, and that is that if inertial mani-folds are observed to exist in a series of numerical experiments with increasing refinedresolution, then this is good evidence of the existence of an inertial manifold for thetrue problem. Without a result like the theorem presented here, independent of as-sumptions on the limiting equation, such an assertion would be impossible.

6. Conclusion

It has been shown that the inertial manifolds that can be found for a set of approxi-mating schemes automatically converge to an inertial manifold for the exact equation.In standard results, where the existence of an inertial manifold for the exact equa-tion is given by a spectral gap condition, this shows that if inertial manifolds can bepredicted for sequences of approximations then no case-by-case analysis is needed toobtain convergence.


Acknowledgements

Many thanks to Trinity College for all their support, and to Gabriel Lord whosework [11] set me thinking about generalising previous results. Thanks also to ananonymous referee for reading the manuscript closely and making some helpful sug-gestions.

References

[1] S.-N. Chow, K. Lu and G. R. Sell, Smoothness of inertial manifolds, J. Math. Anal. Appl. 169(1992) 283–312.

[2] P. Constantin, C. Foias, B. Nicolaenko and R. Temam, Integral Manifolds and Inertial Manifoldsfor Dissipative Partial Differential Equations, Appl. Math. Sci. 70 (Springer, New York, 1988).

[3] F. Demengel and J. M. Ghidaglia, Inertial manifolds for partial differential evolution equationsunder time discretization: existence, convergence, and applications, J. Math. Anal. Appl. 155(1991) 177–225.

[4] C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolution equations, J. Diff.Eq. 73 (1988) 309–353.

[5] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Vol. 840(Springer, New York, 1981).

[6] J. M. Hyman, B. Nicolaenko and S. Zaleski, Order and complexity in the Kuramoto–Sivashinksymodel of weakly turbulent interfaces, Physica 18D (1986) 265–292.

[7] D. A. Jones, On the behavior of attractors under finite-difference approximation, Num. Func. Anal.Opt. 16 (1995) 1155–1180.

[8] D. A. Jones and A. M. Stuart, Attractive invariant manifolds under approximation – inertial man-ifolds, J. Diff. Eq. 123 (1995) 588–637.

[9] D. A. Jones, A. M. Stuart and E. S. Titi, Persistence of invariant sets for partial differentialequations, J. Diff. Eq., to appear.

[10] D. A. Jones and E. S. Titi, C1 approximation of inertial manifolds for dissipative nonlinear equa-tions, J. Diff. Eq. 127 (1996) 54–86.

[11] G. Lord, Attractors and inertial manifolds for finite difference approximations of the complexGinzburg–Landau equation, Preprint 95/15, School of Mathematical Sciences, Bath (1995).

[12] P. D. Miletta, Approximation to solutions of evolution equations, Math. Meth. Appl. Sci. 17 (1994)753–763.

[13] J. C. Robinson, Inertial manifolds and the strong squeezing property, in: Nonlinear EvolutionEquations & Dynamical Systems, NEEDS ’94, eds. V. G. Makhanov, A. R. Bishop and D. D. Holm(World Scientific, Singapore, 1995) pp. 178–187.

[14] J. C. Robinson, Some closure results for inertial manifolds, J. Dyn. Diff. Eq., to appear.[15] J. C. Robinson, Computing inertial manifolds, SIAM J. Math. Anal., in preparation.[16] T. Shardlow, Numer. Algorithms 14 (1997), this issue.[17] S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, Appl. Math. Sci. 105

(Springer, New York, 1994).

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Convergent families of inertial manifolds for convergent approximations