ISRN Mathematical AnalysisVolume 2013 (2013), Article ID 291823, 29 pageshttp://dx.doi.org/10.1155/2013/291823

Review Article

Attractors and Finite-Dimensional Behaviour in the 2D Navier-Stokes Equations

James C. Robinson

Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

Received 10 March 2013; Accepted 29 April 2013

Academic Editors: I. Fragala, G. Mantica, and A. Peris

Copyright © 2013 James C. Robinson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The purpose of this review is to give a broad outline of the dynamical systems approach to the two-dimensional Navier-Stokes equations. This example has led to much of the theory of infinite-dimensional dynamical systems, which is now well developed. A second aim of this review is to highlight a selection of interesting open problems, both in the analysis of the two-dimensional Navier-Stokes equations and in the wider field of infinite-dimensional dynamical systems.

1. Introduction

The Navier-Stokes equations are the fundamental mathematical model of fluid flow; a physical derivation of the equations can be found in Batchelor [1] or Doering and Gibbon [2], for example. Their rigorous analysis goes back to Leray [3], who proved the global existence of weak (-valued) solutions in 3D and local existence of strong (-valued) solutions; similar results were obtained by Hopf [4] for bonded domains. Global existence and uniqueness of weak solutions in the 2D case was first shown by Ladyzhenskaya [5].

The dynamical systems approach to the Navier-Stokes equations was developed over a number of years, notably by Ladyzhenskaya [6] and Foias, Constantin, Temam, and coauthors; see Constantin et al. [7], for example. Delay differential equations provided a stimulus for the development of the theory from a different but related viewpoint; see Hale et al. [8], for example.

Since rigorous existence and uniqueness results are only available for the 2D equations, we confine ourselves here to this case.

The Navier-Stokes equations are posed on a spatial domain , supplemented with appropriate boundary conditions. Here is the two-component velocity, the parameter is the kinematic viscosity, and is the scalar pressure, which serves to enforce the divergence-free condition . The right-hand side is a (somewhat artificial) “body force,” which serves to maintain some nontrivial motion of the fluid.

For simplicity we will treat the equations on a periodic domain , so that , where are unit vectors parallel to the coordinate axes. In addition we will make the simplifying assumption that and have zero average over , and that is divergence-free ().

Although we will generally confine our analysis to the case of periodic boundary conditions, many results are also true for the case of Dirichlet boundary conditions, and we will occasionally comment on this case in what follows.

Note that while much of the existence and uniqueness theory, particularly in the 3D case, is carried out in the whole space setting (which allows one to use the tools of harmonic analysis; see e.g., Lemarié-Rieusset [9] or Cannone [10]), this is not convenient in the dynamical systems approach. Even when , the decay of solutions to zero is a delicate matter. We discuss this briefly in Section 3.

Periodic boundary conditions are particularly useful for analysis, since in this case we can expand as a Fourier series where denotes the complex conjugate of (this condition enforces the reality of ).

One of the main reasons for the rigorous study of the Navier-Stokes equations is the attempt to gain further understanding of fluid turbulence. However, even at the modelling level there are questions to answer. The standard heuristic model of turbulence (see Frisch [11]) requires the fluid to be subject to the injection of energy at a constant rate at a particular “scale.” One might attempt to model this by using a forcing , where is some combination of Fourier modes associated with the lengthscale . But if one multiplies the equation by and integrates then the resulting energy equation shows that the rate of energy injection (the right-hand side) depends on the velocity field . Of course, one could take a time-dependent forcing (in the hope that for all ) but this seems very unnatural.

Open Question  1. What is a reasonable model equation for homogeneous fully developed turbulence?

For an interesting example of a nonstandard velocity-dependent forcing, see Cheskidov et al. [12].

2. The Navier-Stokes Equations

2.1. The Navier-Stokes Equations in Functional Form

We now rewrite the Navier-Stokes equations in a more convenient way. The ideas go back to Leray [3] and are now standard—see, for example, Constantin and Foias [13], Temam [14] or [15], Ladyzhenskaya [5], and Robinson [16].

The main idea is to remove the pressure by projecting onto the space of all divergence-free vector fields; under appropriate boundary conditions gradients and divergence-free functions are orthogonal, since If we denote by the orthogonal projection in onto the space of all such divergence-free functions (the “Leray projector”), then , and so we obtain an equation for alone as, where is known as the Stokes operator (recall that we assumed that was divergence-free).

Let let be the completion of in the norm of which we denote by , and let be the completion of in the norm. (We use , for spaces of scalar or two-component functions interchangeably—what is meant should be clear from the context.) Since we are assuming that has zero average, we have the Poincaré inequality (for the definition of see later), and so we can (and will) use for the norm on (It is common practice in papers on the Navier-Stokes equations to use for the norm in and for the norm in . We will not adopt this practice here, preferring over for the sake of clarity.) We denote by the dual of , and denote the norm of by

By we denote the domain of , that is, all those for which is finite. Note that in the case of periodic boundary conditions this has a simple characterisation as those Fourier series (4) for which (cf. [15]). Note that . (In the case of periodic boundary conditions we have for all ; see [13].)

By we denote the eigenvalues of , ordered so that ; we denote the corresponding eigenfunctions by , so that . The constant in the Poincaré inequality (10) is given by the first eigenvalue .

We now define a bilinear form by setting (One can also proceed in a more roundabout way, defining by and then defining via the Riesz Representation Theorem.) The following properties of will be useful throughout all that follows. First we have two orthogonality properties; in two and three dimensions we have (this follows from an integration by parts and cancellations due to the boundary conditions), and—only for periodic boundary conditions in two dimensions—we also have (the proof is by expansion, rearrangement, and cancellation of terms using the fact that is divergence-free).

In our analysis we frequently require inequalities for . Rather than listing them now, we will derive them when required, since the ingredients are very simple: the three-exponent Hölder inequality and the Sobolev embeddings for any in 2D. In addition, in 2D we will need Ladyzhenskaya’s inequality (this follows from interpolating between and and then using the 2D embedding along with the Poincaré inequality).

With these definitions we can rewrite the Navier-Stokes equations in the functional form

2.2. Weak Solutions

Suppose that . If we assume that is smooth, then we can multiply the equation by , and then using we obtain the simple estimate which yields the differential inequality from which we derive the energy inequality These formal calculations show that one would expect solutions to satisfy Indeed, this essentially yields the definition of a weak solution.

Definition 1. A weak solution of the Navier-Stokes equations is a function such that (18) holds as an equality in for almost every . Note that in the 2D case using the 2D Ladyzhenskaya inequality . It follows, since , that we have , and, hence, that if a weak solution exists, .For a rigorous proof of the existence of weak solutions one replaces (18) by the Galerkin approximations where is the orthogonal projection onto the first eigenfunctions of , These approximations are essentially finite-dimensional systems of Lipschitz ordinary differential equations and so have solutions that exist while their norm stays finite. By repeating the previous calculations for (25), which are now rigorously justifiable since is smooth, one can find uniform bounds on in . Similarly, is uniformly bounded in . Using all these uniform bounds one can extract a subsequence that converges to some sufficiently strongly that every term in (25) also converges to the required term from the full equation. For details see Constantin and Foias [13] Robinson [16], or Temam [15], for example.

We state formally a result about the existence and uniqueness of weak solutions in the 2D case. In 3D weak solutions are known to exist (following the same argument), but their time derivative is less regular [] and continuity into and uniqueness are not known. It is also not known if every 3D weak solution must satisfy the energy inequality (22).

Theorem 2. If and , the 2D Navier-Stokes equations have a unique weak solution that exists for all ; for any and consequently . All such solutions satisfy the energy inequality (22).

Proof. We have sketched the proof of existence of weak solutions. It remains to prove the continuity into and uniqueness of the solution. Both rely on the regularity of .The bounds on in and on in are sufficient to guarantee that is continuous into ; essentially this is enough regularity to ensure that which on integrating yields continuity of into (see e.g., Evans [17] or Robinson [16]). So any weak solution (in 2D) will also be continuous into , that is, will trace out a meaningful trajectory in the energy space.It remains to prove the uniqueness. To do this, consider two solutions and and their difference . Then the equation for is Since and , there is enough regularity to take the inner product with and obtain (recalling that ) After an application of Young’s inequality this yields Thus in particular if then for all , since .

2.3. Strong Solutions and Regularisation

We now suppose that . We show that if then we obtain a more regular type of solution and deduce that if then for every .

Suppose that we now take the inner product of (18) with . Since we are working with periodic boundary conditions we can use the orthogonality property to make our analysis a little easier, and we obtain which after an application of Young’s inequality yields An integration in time shows that provided that the solution satisfies A strong solution is a weak solution that has this additional regularity.

Proposition 3. If and then there exists a unique strong solution such that for any In addition and .

We can deduce from this result that weak solutions immediately regularise, that is, become strong.

Corollary 4. If and , then for all .

Proof. Let be the weak solution of (18), and fix some . Since , there exists an such that . The solution of the NSE with is strong, and so . But the function is a weak solution of (18); by uniqueness it is the solution with this initial data. Thus as claimed.

For an alternative and perhaps more elegant proof one can take the inner product of (18) with and follow the previous calculations above. This yields more information, including the rate of regularisation of the norm (like ) as follows: using (22).

We note that in 3D one can only prove the local existence of strong solutions; that is, solutions remain bounded in on some time interval , where depends on ; these are unique in the class of weak solutions satisfying the energy inequality (22).

Open Question  2 (Clay Millennium Problem). Do the three-dimensional Navier-Stokes equations have strong solutions for all ?

2.4. The Solution Semigroup

We define a dynamical system using the Navier-Stokes equations on the phase space , although we could also use or indeed for any . Given an initial condition , we have seen that the equation has a unique solution for all positive times. In this case, we can define a semigroup of solution operators , for , by These operators satisfy and we can consider the semidynamical system

We define the semigroup only for since, as for many parabolic PDEs, it is unnatural to consider . For example, backwards solutions of the heat equation () can blow up instantaneously unless the initial data is analytic.

2.5. Backwards Uniqueness

We cannot solve the equations backwards, but we can show that solutions enjoy the backwards uniqueness property, that is, that is injective for every : We do this by combining the argument in Temam ([18], Chapter III Section 6) which shows that is injective, and an observation in Chapter 11 of Constantin and Foias [13].

First we prove backwards uniqueness for an abstract problem when the difference of two solutions satisfies where for for some .

The proof uses the “Dirichlet quotient” By differentiating the expression for , substituting for , using (43), and observing that (see Lemma  III.6.1 in [18]), one can deduce that from which it follows that

Lemma 5. If and for some , then for all .

Proof. For a contradiction assume that for some . Then since is continuous into , there exists an such that on . Let be the largest time such that on , and note that .On we have Integrating between and yields whence is bounded as , a contradiction.

This argument has been extended by Kukavica [19] to allow for additional logarithmic terms in the estimate (44).

Corollary 6. The semigroup is injective.

Proof. Suppose that and . Since has a regularising effect (Corollary 4), for any we know that and are elements of . It follows from Proposition 3 that and are elements of .If then Noting that it follows that satisfies the condition of Lemma 5, and, hence, that , that is, for every .Finally we simply observe that and are continuous from into , and it follows that .

2.6. The Pressure

It is easy to start to think of the “Navier-Stokes equations” as the equations in their functional form (18) and, hence, of the velocity as the only dependent variable of interest. Although this is a mathematically convenient point of view, it is often the pressure that is of interest in physical problems.

Thus one must ask whether can be recovered given . If we take the divergence of (1) then we obtain a Poisson equation for the pressure (we took ). If we impose the additional condition that , then this equation has a unique solution (see Simon [20]). On or one can in fact obtain bounds on the pressure in any Lebesgue space This follows since , where denotes the Riesz transform in the th component, using the Calderon-Zygmund Theorem (see [21], e.g., for the whole space and [22] for its periodic variant).

Open Question  3. Does the pressure estimate (53) hold in bounded domains?

One could also wonder about the converse question: does the pressure determine the velocity? In a two-dimensional incompressible flow this seems at least plausible; one can determine the velocity by a single scalar streamfunction, and the pressure is a single scalar. To formalise this, we write and then the pressure equation (52) becomes that is, Thus the stream function satisfies the Monge-Ampère equation (, where denotes the Hessian of ), as was observed by Larchevêque [23]. This equation is well understood when the right-hand side is positive, since in this case the equation provides a canonical example of a “nonlinear elliptic problem,” see Chapter 11 of Gilbarg and Trudinger [24], for example. However, little is known when the right-hand side changes sign since in this case the problem is of mixed (nonlinear) elliptic/hyperbolic type.

Open Question 4 (Gero Friesecke). For 2D incompressible flow with periodic (or Dirichlet) boundary conditions, does the pressure determine the velocity uniquely?

If so one could reduce the 2D Navier-Stokes dynamical system to a dynamical system for the scalar .

3. “Trivial Dynamics” When

When the forcing is zero all solutions decay to zero and the asymptotic dynamics are trivial. This is simple to show on any domain in which the Poincaré inequality (10) is valid, in particular on a bounded or periodic domain (with the zero average condition on and ). Indeed, from (21) we have which shows that converges to zero (in ) exponentially fast. A more detailed analysis due to Foias and Saut [25], shows that in fact where is an eigenvalue of the Stokes operator, and where is an eigenfunction of the Stokes operator with eigenvalue .

On the whole space the analysis is significantly more involved. Following Schonbek [26] (who considered the harder problem of decay of weak solutions in ) one can consider the energy equation in Fourier space where . This “Fourier splitting” technique allows one to estimate both integrals, using the fact that for , and thus deduce that

In the two-dimensional case Gallay and Wayne [27] were able to go much further and recover a whole-space version of the results of Foias and Saut. By considering the equation for the vorticity and making the change of variables the linear part of the equation becomes the Schrödinger operator whose spectral properties are well understood. As a consequence, they could show that the solutions decay to zero like a particular solution, the Oseen vortex, with a magnitude depending on the total initial vorticity .

Thus when solutions all decay to zero. It seems reasonable to claim that is the “attractor” of the dynamical system in this case. We now make this notion more precise and then examine the more interesting case when .

4. The Global Attractor

We now leave the Navier-Stokes equations for a time and give the definition of the global attractor for an abstract semiflow on a Hilbert space and prove some properties that all such attractors share. General treatments of the theory of attractors are given in Babin and Vishik [28], Chepyzhov and Vishik [29], Chueshov [30], Hale [31], Ladyzhenskaya [6], Robinson [16], Temam [18], and Vishik [32].

4.1. Existence of the Global Attractor

We say that a set attracts if where is the Hausdorff semidistance between two sets, Note that this distance does not define a metric—indeed, if then one only has . To obtain a metric on subsets of , we need to use the symmetric Hausdorff distance

Definition 7. The global attractor is the maximal compact invariant set, that attracts all bounded sets (is “attracting”): for any bounded set .

That is maximal means that if is a bounded invariant set then . Equation (68) says that attracts all orbits, at a rate uniform on any bounded set. Without the compactness condition we could just take . Note that while is the maximal compact invariant set, it is also the minimal set that attracts all bounded sets (the proofs are simple). Confusion is possible, since various authors refer to as the “minimal attractor” and others as the “maximal attractor.”

We give a result on the existence of global attractors in a version inspired by a similar result due to Crauel [33] for random dynamical systems; see also Hale [31] and Babin and Visihik [28]. (The first result along these lines seems to be due to Billotti and LaSalle [34].)

First we define the -limit set of a set , which consists of all the limit points of the orbit of as follows: This can be also be characterised as in some sense captures all the recurrent dynamics of the orbit through .

Theorem 8. A semigroup has a global attractor if and only if it has a compact attracting set , and then .

The proof requires the following simple lemma.

Lemma 9. If is a compact set and is a sequence such that then has a convergent subsequence whose limit lies in .

Proof. For each find a point such that . Since is compact there is a subsequence , and it follows that .

As a first step to proving Theorem 8 we prove the following properties of -limit sets.

Proposition 10. If there exists a compact attracting set then the -limit set of any bounded set is a nonempty, invariant, closed subset of . Furthermore attracts .

Proof. To see that is nonempty choose some point . Then since is attracting It follows that for some sequence As the intersection of a decreasing sequence of closed sets, is clearly closed. To show that suppose that , and Then since is attracting implying that a subsequence of converges to a point in . Since the sequence itself converges it follows that . So is compact.Now suppose that does not attract . Then there exists a and a sequence of such that and, hence, such that However, the previous argument shows that a subsequence of converges to some point . By (77) we should have while by definition . So attracts .

Now observe that and that since is invariant

Proof of Theorem 8. It follows from the previous proposition that is nonempty, compact, invariant, and attracts . So all we have to prove is that attracts . Since attracts it suffices to show that . But this follows immediately from (79) and (80). The “only if” part is clear, taking .

In order to apply Theorem 8 we often prove something stronger than the existence of a compact attracting set, namely the existence of a compact absorbing set. We say that a set absorbs if there is a time such that We say that is absorbing in if it absorbs every bounded subset of .

4.2. Structure of the Attractor

We now want to examine the attractor itself in more detail. We show that it is connected, consists of all complete bounded orbits, and contains the unstable manifolds of all fixed points and periodic orbits. This gives us a better idea of the kind of dynamics we can expect to understand if we restrict our attention to the attractor.

4.2.1. The Global Attractor is Connected

Proposition 11. If is the global attractor of a semigroup on a Hilbert space then is connected.

Proof. If is not connected then is the disjoint union of two nonempty compact sets and , which are therefore separated by a distance . Let be a ball that contains ; since attracts there exists a such that for all . Since is connected so is , from which it follows that either or for all . This contradicts the fact that both and are nonempty.

In general it is not possible to show that the attractor is path connected (two points can be joined by a curve); there are (perhaps artificial) examples of sets that are not path connected that can be global attractors. Günther and Segal [35] remark that one can construct a flow for which Bing’s pseudo-arc [36] is the global attractor, and this set is not simply connected. Langa and Robinson [37] showed that invariant sets that enjoy a certain property related to normal hyperbolicity will be simply connected (and connected in other senses too), but such a condition is very strong and would be very hard to check in examples.

Open Question  5. Are there natural conditions under which the global attractor is path connected?

Other topological properties of global attractors are known, for example, a global attractor of a semiflow on a linear space has “trivial shape,” see Garay [38] or Robinson and Sánchez-Gabites [39].

4.2.2. An Analytic Characterisation of the Global Attractor

A “complete” orbit is a solution of the PDE (or ODE) which is defined for all . In general we do not expect the solutions of a PDE to lie on a complete orbit, since we cannot define for . We say that a complete orbit is bounded if there is some such that for all .

The global attractor consists of all bounded complete orbits. This is a noteworthy result, since it gives an analytic characterisation of the global attractor as the set of a particular class of solutions, even though our original definition was a dynamical one.

Theorem 12. The global attractor is the union of all the complete bounded orbits.

Proof. Let be a complete bounded orbit, and assume that is not contained in ; then for some there is a point with . However, since attracts bounded sets, for large enough Since is a complete orbit, for some ; (82) now gives a contradiction and shows that all complete bounded orbits lie in .Now take ; for it is immediate that is bounded since it lies in by invariance of . We now have to construct for . Since is invariant, there exists an such that (this need not be unique unless is injective, see Theorem 13, below). Let for . Now find such that , and set for . Continue inductively in this way to define for all . That is a trajectory follows from the continuity of and the semigroup property.In this way we have shown that every lies on a complete bounded orbit, and, hence, that is precisely the union of all such orbits.

It is interesting to note that in certain situations there are complete orbits that are not bounded. For example, Constantin et al. [40] explore the sets of solutions of the 2D Navier-Stokes equations (with periodic boundary conditions) that exist for all time and grow at the same rate as solutions of the linear Stokes problem as .

4.2.3. A Dynamical System on the Attractor

If the semigroup is injective on (in the sense of Section 2.5) then the dynamics, restricted to , actually define a dynamical system; that is, makes sense for all , not just for . This is one good reason for investigating the dynamics on the attractor. The importance of this result is emphasised in Hale [31].

Theorem 13. If the semigroup is injective on then every trajectory on is defined for all , and (67) holds for all . In particular, is a dynamical system.

Proof. For each we know that , and so there exists a unique with . We define to give for all and, hence, obtain (67) for also. Since is compact, it follows that as defined here is continuous on . Thus is a continuous map from into itself for all , and it is easy to check that for all .

4.2.4. Unstable Manifolds in the Attractor

To investigate the structure of the attractor further, we need to recall the definition of unstable manifolds.

Definition 14. The unstable manifold of an invariant set is the set

Now, the unstable manifold of any invariant set (in particular of any fixed point or periodic orbit) is contained in the attractor.

Theorem 15. If is a compact invariant set, then

Proof. Let ; then by definition (Definition 14) lies on the complete orbit . As we know that , and as we know that , so the orbit is bounded. Thus lies on a complete bounded orbit, and by Theorem 12, .

5. Asymptotic Bounds on Solutions

Central to proving results on existence of an attractor for the Navier-Stokes equations (and for other PDEs) are various bounds on the norms of solutions. In order to prove the existence of solutions for all time, we have to prove that some norm of the solution is bounded for all time. Because of the strong dissipation in many parabolic problems, it is often a short step from these bounds to time-asymptotic bounds that are independent of the initial conditions, and these are essentially what we require for the existence of an attractor.

We will give our estimates in terms of the dimensionless Grashof number which measures the relative strength of the forcing and viscosity and is defined as (recall that we use to denote the norm of ). For an alternative definition that uses the norm of in rather than in see Robinson [41] and Section 6.4.

Although the Grashof number is mathematically convenient (it only makes use of terms that occur explicitly in the equation), it is more conventional to discuss qualitative properties of fluid flows in terms of the Reynolds Number, , where is a temporal and spatial average of the velocity () and is the forcing scale. This issue is discussed in detail by Gibbon and Pavliotis [42]; making use of an analysis due to Doering and Foias [43] they show that when is large for an appropriate definition of the Reynolds number Re.

5.1. A Compact Absorbing Set When

Our aim here is to show that when and there is a bounded set in that is absorbing. Since is compactly embedded in , this yields a compact absorbing set in . We obtain such a set in two stages. First we show that there is a bounded absorbing set in and then use this (and an auxiliary estimate) to prove the existence of an absorbing set in .

Although the existence of an absorbing set in for the 2D equations was first shown (in different terminology) by Foias and Prodi [44], the proof of the existence of a global attractor for the 2D Navier-Stokes equations was first published by Ladyzhenskaya in 1972 (see [45], for an English translation) and later, along with many other important results, by Foias and Temam [46].

5.1.1. A bounded Absorbing Set in

To prove the existence of an absorbing set we will need the following simple lemma. To prove this one simply multiplies by the integrating factor and integrates.

Lemma 16 (Gronwall).   If is positive almost everywhere and then

We now prove the existence of an absorbing set in and an asymptotic bound on the integral (in time) of the norm in .

Proposition 17. Given let Then for any and there exists a time such that that is, the ball in of radius is absorbing. One also has

Proof. We take the inner product of with to obtain Since this gives We now use the Poincaré inequality on the term and Young’s inequality on the right-hand side to write Tidying this up gives and then from Gronwall’s inequality (Lemma 16) From this it is clear that given any there exists a time , which depends only on and such that If we return to (96) and use the Poincaré inequality on the term on the right-hand side we can then apply Young’s inequality to obtain and integrating from to yields which implies that for yielding (107). A similar integration yields (93).

(Note that one can obtain a similar result with the weaker assumption that , by replacing the right-hand side of (96) by and continuing similarly.)

5.1.2. A Bounded Absorbing Set in

The existence of an absorbing set in for the 2D equations was first shown (in a different terminology) by Foias and Prodi [44]. This is the crucial ingredient for proving the existence of a global attractor. Although in the proof we use the orthogonality property , which is only valid for periodic boundary conditions in 2D, the same result (with a slightly more involved argument and weaker estimates) holds for Dirichlet boundary conditions.

Proposition 18. Given and let Then for any where is the same as in Proposition 17; that is, the ball in of radius is absorbing. One also has

Note that by an appropriate choice of we can take any where

Proof. To prove the existence of this absorbing set we use a “trick”—a double integration in time—which can be formalised as the “uniform Gronwall lemma” (see Lemma 1.1 in Chapter III of Temam [18], e.g., although the statement of this as a formal lemma somewhat obscures the underlying idea). We take the inner product of (18) with to give We now use an orthogonality property and the Cauchy-Schwarz inequality to rewrite this as Dropping the terms we have We now use the double integration trick. First integrate this equation between and , with , which gives (since ). We now integrate both sides with respect to between and and obtain Now provided that we can use (107) to give We note also that if we return to (111) then we have and integrating from to we obtain which yields (107) and (108).

5.2. Smoothness When

In fact with higher regularity of we can obtain much better bounds on the functions in the attractor. We will use the following estimates on the nonlinear term, which follow easily from the fact that is an algebra for . We use to denote the norm in .

Lemma 19. For any

We use this to prove better asymptotic regularity of solutions when is more regular.

Corollary 20. If then is bounded in .

Proof. Suppose as an inductive hypothesis that for any , Note that this holds for since the attractor in bounded in ; the integral bound follows from (117). We show that while it follows that (119) holds with replaced by .Now, for any , we have for some , since is invariant. It follows from (119) that there exists a such that We now consider the solution starting at , noting that .Take the inner product of (18) with to obtain using Lemma 19. After using Young’s inequality on the right-hand side this becomes If we drop the second term on the LHS and use Gronwall’s inequality we can deduce that Now return to (122) and integrate from to , starting at , to obtain

Open Question 7. The previous bounds are very crude. Can one find the optimal dependence of the bounds on the attractor in terms of norms of ?

For one approach to this, based on the time analyticity of solutions, see Foias et al. [47].

We say that a function is smooth if for every .

Corollary 21. If is smooth then the Navier-Stokes attractor consists of smooth functions.

To close this section we note that it is in fact possible to obtain asymptotic bounds on when (the previous proof requires ), by making estimates on the time derivative of the equation [16, 48]. We will use the existence of an absorbing set in in Sections 8 and 9. For another approach to higher regularity see Guillopé [49] or Temam [18].

5.3. Gevrey Regularity

We now use the theory of Gevrey regularity, developed by Foias and Temam [50], to show that if is real analytic then the functions on the attractor are all real analytic, in a uniform way.

A function is real analytic, that is, it can be represented locally by its Taylor series expansion, if and only if its derivatives satisfy for some and see John [51], for example. This motivates the definition of the analytic Gevrey class ; this consists of functions such that where is defined using the power series for exponentials Writing as a Fourier expansion we have In particular, therefore, if the Fourier coefficients of must decay exponentially fast.

Foias and Temam [50] (see also Doering and Gibbon [2] for an alternative proof) showed that if for some then is bounded in , and and depend only on .

We give the proof here, following Foias and Temam’s paper closely. We assume the following inequality, which is Lemma 2.1 in Foias and Temam [50]; if for some then for some . In order to make the notation more compact, we can write and . The previous inequality is now

Theorem 22. If then for one has where .

Proof . Taking the scalar product with in leads to an equation for like . Not only do the solutions of this equation blow up in a finite time, but also we need to control in order to control ; we would need to start with analyticity in order to prove it.The trick to get round this is to define , and take the scalar product of with to obtain The left-hand side of the equation can be bound as We therefore have Now we can set and we have with The solution of (140) is and so for . Since , we have and so for , we have and the theorem follows.

The following is an immediate corollary of Theorem 22.

Corollary 23. Suppose that for some . Then there exists a and a constant that depends only on such that

Proof. Let from Theorem 22. Take . Then for some . Since , , and so we obtain uniformly over , using (144).

The in the exponent reflects the size of the radius of analyticity of . Some interesting work on obtaining the maximum radius of analyticity can be found in Kukavica [52].

We will use the real analyticity of solutions on the attractor in Section 8.2.2 to show that elements of the attractor can be distinguished by their values at a finite number of points in .

Open Question 8. If then one can show (cf. (125)) that see Friz and Robinson [53], for example. Can one obtain sufficiently good bounds in answer to Open Question to deduce Gevrey regularity for the attractor given these bounds on ?

6. Finite-Dimensional Attractors

We now show that the attractor of the 2D Navier-Stokes equations is a finite-dimensional subset of the infinite-dimensional phase space . This was first shown by Ladzyhenskaya [54].

6.1. The (Upper) Box-Counting Dimension

The box-counting dimension, which we will write as , is based on counting the number of closed balls of a fixed radius needed to cover .

We denote the minimum number of balls in such a cover by . If were a line, we would expect that , if were a surface, we would have , and for a (3−) volume, we would have . So one possible method for obtaining a general measure of dimension would be to say that has dimension if . Accordingly, we make the following definition.

Definition 24. The box-counting dimension of , , is given by where we allow the limit in (147) to take the value .

Note that it follows from the definition that if , then for sufficiently small ,

For further discussion of the box-counting dimension see Eden et al. [55], Falconer [56, 57], or Robinson [16, 58].

6.2. Dimension Estimates

Techniques to show that invariant sets have finite box-counting dimension date back to Mallet-Paret [59]. His argument, valid for invariant subsets of Hilbert spaces, admits generalisation to Banach spaces (see [60], recently updated in [61]). However, the method that gives the best estimates is restricted to Hilbert spaces and was developed in infinite-dimensional spaces by Constantin and Foias [62] and Constantin et al. [7] [see also [63] or [18]], after the finite-dimensional approach of Douady and Oesterlé [64].

The idea is to study the evolution of infinitesimal -dimensional volumes as they evolve under the flow and try to find the smallest dimension at which we can guarantee that all such -volumes contract asymptotically. We will not give the analysis in detail but merely in outline.

We will consider an abstract problem, written as with contained in a Hilbert space , whose norm we denote by . We assume that the equation has unique solutions given by and a compact global attractor .

We want to start off with an orthogonal set of infinitesimal displacements near an initial point and then watch how the volume they form evolves under the flow.

To study the evolution of this volume we have to study the evolution of a set of infinitesimal displacements about the trajectory . We suppose that the evolution of these displacements is given by the linearised equation which we write as

The validity of such a linearisation is one of the main points to check when applying this theory rigorously. To this end we make the following definition.

Definition 25. We say that is uniformly differentiable on if for every there exists a linear operator , such that, for all ,

Although this is straightforward to check for ordinary differential equations, its proof in the PDE context will often involve technical difficulties.

Heuristically speaking the growth rate of each infinitesimal displacement will be related to the eigenvalues of . In particular, the length of an infinitesimal displacement in the -eigendirection is at time ; the growth rate is , associated with the eigendirection of with eigenvalue . The size of a small -volume with sides in two different eigendirections would be and so the growth “rate” is . The growth rate of an -volume made of infinitesimal directions in eigendirections would be If we can make sure that this growth rate must be negative then we know that -volumes contract. It is true, though by no means immediate, that if all -volumes contract then the dimension of the attractor must be smaller than . In the finite-dimensional setting this result is due to Douady and Oesterlé [64], while in the infinite-dimensional case it was proved by Constantin and Foias [62] and Constantin et al. [7].

To extract the “growth rate,” we consider over all possible orthonormal collections of elements of . The idea is, essentially, that the maximum over all choices of gives the largest possible growth rate, that is, the sum of the largest eigenvalues of . A more compact notation for (155) is where denotes the trace in and is the orthogonal projection onto the space spanned by the .

The following theorem is given in a form suitable for calculations.

Theorem 26. Suppose that is uniformly differentiable on , and there exists a such that is compact for all . Let and assume that . If then .

The result of Constantin and Foias gives (where denotes the Hausdorff dimension) and bounds on the box-counting dimension under stronger conditions on the . A proof of this theorem, essentially due to Brian Hunt, is given in Appendix B in Robinson [16]; see also Chepyzhov & Ilyin [65]. A simpler proof of the same result can be obtained under the assumption that is a concave function of [66].

Before we apply this result to give dimension bounds for the attractor of the 2D Navier-Stokes equations we will need an auxiliary lemma. For a proof see Lemma 4.21 in Carvalho et al. [63].

Lemma 27. Let be a positive unbounded linear self-adjoint operator on a Hilbert space , and assume that has a compact inverse. Denote its eigenvalues by , ordered such that . Then for any choice of orthonormal elements in ,

Note that when is the Stokes operator in (on a periodic domain or a bounded domain with Dirichlet boundary conditions), one can obtain an explicit bound, since its eigenvalues satisfy and then

6.3. Dimension Estimate for the Navier-Stokes Equations

In order to apply the previous theory to the Navier-Stokes equations we must first guarantee the differentiability of solutions. We only state the result here; a proof can be found in Robinson [16], Theorem 13.20, and Exercise 13.10.

Theorem 28. The solutions of the Navier-Stokes equations in 2D satisfy (152) with the solution of the equation Furthermore, is compact for all .

We can now use the trace formula to find a bound on the dimension of the attractor.

Theorem 29. The attractor for the 2D periodic Navier-Stokes equations is finite dimensional, with where denotes the time-average .

The result, due to Constantin et al. [7], is also valid as stated for Dirichlet boundary conditions.

Proof . The correct form of the linearised equation is given in (161), and so Thus the time-averaged trace is bounded by In order to control the contribution from the nonlinear term we could use the simple bound (using the Ladyhenskaya inequality and the fact that ), but this would lead to an estimate worse than (162) by a factor of (see [16]). A better estimate can be obtained as follows.Note that we have and we have for each It follows use the Cauchy-Schwarz inequality that where An inequality due to Lieb & Thirring [67], adapted appropriately to this case (details are given in [18]), allows us to bound by It follows that and so, using the Cauchy-Schwarz inequality we obtain Now taking the time average and using Lemma 27 we obtain We therefore have provided that . The bound in (162) now follows from (93).

The above argument provides the best bound known for the case of Dirichlet boundary conditions. By working with the equation for and using the identity , Constantin et al. [68] were able to improve this for periodic boundary conditions to Modulo the logarithm this bound is known to be sharp (see Babin and Vishik [69], Liu [70], and Ziane [71]). (A simpler proof of (174) can be found in Doering & Gibbon [2].)

Open Question 9. Is the dimension estimate sharp in bounded domains?

There are still other interesting open questions regarding the dimension of Navier-Stokes attractors, in particular obtaining bounds for physically relevant problems (or at least two-dimensional versions of them).

6.4. Reflecting the Lengthscales in the Forcing

Note that reflects only the amount of energy being put into the flow and says nothing of the scales at which the energy is supplied. The following simple calculation, inspired by the paper of Olson & Titi [72], shows that it is in fact possible to improve on the dimension estimate in the Dirichlet boundary condition case, as well as on the estimate (174) in the periodic case when the forcing is at very small scales, by making a small modification to the previous argument (see [41, 73]). If we return to our previous analysis of (95) but estimate the right-hand side differently we can obtain then if we use Young’s inequality on the right-hand side and take the time average we will obtain where now we have defined an alternative Grashof number based on , as It follows that

Noting that the constant is the same as that in (162), that this provides an improvement is guaranteed by the following lemma.

Lemma 30. If then .

Proof. If has expansion , where are the eigenfunctions of , then it follows that Since it follows that , and so

In the case of periodic boundary conditions, the eigenvalue corresponding to the eigenfunction with spatial dependence is it follows that , which has the dimension of a length, is essentially the lengthscale on which the eigenfunction varies. Carrying this idea over to the case of Dirichlet boundary conditions, it is a convenient and suggestive shorthand to describe a forcing of the form as being confined to lengthscales between and ; it is clear from (180) that for a forcing of this form

More suggestively one can say that if the forcing is confined to lengthscales between and then Using as a reference length, this gives

Returning to (180), the ratio appears as an average of squared lengthscales, weighted according to the amount of energy injected at each scale. Accordingly, it is natural to define an effective lengthscale of the forcing by in which case the relationship is essentially a tautology. (Ratios of successive Sobolev norms are a natural way to define possible lengthscales; for example, a whole hierarchy of lengths is defined in Doering & Gibbon [2] using (essentially) the ratios of successive norms of the solution .)

Note that (178) therefore implies that the dimension of the attractor is less than one if the forcing is applied at sufficiently small scales. Since the attractor is a compact connected set, this in fact implies that the attractor is a point [56], so that the dynamics are trivial (no matter how ‘complicated’ the forcing is at these scales).

6.5. Physical Interpretation of the Attractor Dimension

One way of interpreting the physical significance of an attractor is as a means of giving a rigorous notion of the number of independent “degrees of freedom” of the system, a notion introduced by Landau & Lifshitz [74].

They supposed that there is a smallest physically relevant length-scale in the problem, so that interactions on scales of less than are irrelevant for the dynamics (e.g., in a fluid the viscosity has a large effect on the very small scales, and one might expect that fluctuations on these scales have a negligible effect). A heuristic indication of the number of degrees of freedom would then be given by how many “boxes” of side fit into the volume within which the system is confined, yielding If we assume that this is a good estimate of the number of degrees of freedom, and that this in turn is well estimated by the attractor dimension, we can isolate a length scale, given in terms of by [cf. [68] or [2]].

Notice that upper bounds on raise the estimate of the length scale . Recall that the best current estimate for the dimension of the 2D Navier-Stokes attractor is in the case of periodic boundary conditions. It is striking that the bound in (191) corresponds via (190) to a length that satisfies to within logarithmic corrections ( denotes the size of one side of our 2D periodic domain). This length scale in (192) is precisely the “Kraichnan length”, derived by other (also heuristic) methods as the natural minimum scale in two-dimensional turbulent flows [75]. This links the rigorous analytical bound on the attractor dimension with an “intuitive” estimate from fluid dynamics. (The material in this section is discussed in more detail in Robinson [76].)

We will see in Section 8.2.2 that our heuristic derivation of the relationship in (190) can be justified rigorously via an appropriate parametrisation of the attractor, at least when the forcing is analytic.

7. Embedding and Parametrisation

In this section we will state formally a theorem that guarantees that any finite-dimensional set (and in particular a finite-dimensional global attractor) can be embedded into if is large enough (roughly twice the dimension of ). This result can be considered in two ways: (i)we can take the set “out” of the infinite-dimensional space and map it, using some linear map , homeomorphically onto a subset of ; (ii) provides a way of parametrising the attractor using a finite set of coordinates.

We will make use of both interpretations in what follows.

The embedding theorem we give below is due essentially to Hunt & Kaloshin [77]. The first such embedding result for finite-dimensional sets was proved by Mañé [60]. Ben-Artzi et al. [78] showed that the inverse of the projection is Hölder continuous in the finite-dimensional case, along with strict bounds on the Hölder exponent. Foias & Olson [79] subsequently proved that the inverse of the projections is Hölder continuous in the infinite-dimensional case, but without bounds on the Hölder exponent, and Hunt & Kaloshin introduced the notion of the “thickness” of a set and gave the strict bounds on the exponent given in the following theorem.

The Lipschitz deviation, used in the statement of the theorem given here, was introduced by Olson & Robinson [80], with the definition revised and investigated further by Pinto de Moura & Robinson [81].

Definition 31. Let be a compact subset of a real Hilbert space . Let be the smallest dimension of a linear subspace such that for some -Lipschitz function ; that is, where is orthogonal complement of in and is the graph of over , Let the Lipschitz deviation of is given by

Note that one could make a simpler definition by not allowing the extra flexibility that comes from including the function , that is, measuring only how well can be approximated by linear subspaces . This quantity was introduced by Hunt & Kaloshin and termed the “thickness exponent” of , .

Theorem 32 (after Hunt & Kaloshin). If is a compact subset of a Hilbert space then, provided that is an integer with and a dense set of bounded linear maps from into have the following properties: (i) they are injective on , (ii) their inverse is Hölder continuous from into with exponent ,

(In fact a “prevalent” set of linear maps are injective with Hölder inverse; see Hunt et al. [82]. One can prove a similar result for subsets of Banach spaces; see Robinson [83].)

It is relatively easy to show that if is a compact subset of that is bounded in then when ; see Friz & Robinson [84]. Since , this gives a class of sets (“smooth attractors”, i.e., those bounded in for every ) that have zero thickness and, hence, zero Lipschitz deviation.

We have shown that when is smooth the attractor of the 2D Navier-Stokes equations with forcing consists of smooth functions (Corollary 21), so we immediately have the following result.

Lemma 33. If is smooth then , where is the attractor of the 2D Navier-Stokes equations with forcing .

A natural conjecture, due to Ott et al. [85], is that ‘many of the attractors associated with the evolution equations of mathematical physics have thickness exponent zero’. Specialising this to the Navier-Stokes equations, one has the following interesting question.

Open Question 10. For less regular , does the attractor of the two-dimensional Navier-Stokes equations have zero thickness?

While this question is open, the Lipschitz deviation can be shown to be zero for the attractors of a very large class of PDEs, as we will see in Section 9.4.

Note that even when a set has zero Lipschitz deviation, Theorem 32 only guarantees that one can find a linear map whose inverse is Hölder continuous with any exponent strictly less than one. This is not ideal, for many reasons. Later, in Section 10, we will consider the problem of constructing a finite-dimensional dynamical system that reproduces the dynamics on the attractor, and this turns out to be critical. But also one might naturally try to compute various quantities that depend on lengths of vectors (e.g., Lyapunov exponents) from observations of , and these will be distorted unless the inverse of is Lipschitz.

The bi-Lipschitz embedding problem, to find conditions that guarantee the existence of an embedding with Lipschitz inverse, has attracted much attention (particularly in the theory of metric spaces; see Heinonen [86]), and is still open.

Open Question 11. What assumptions are required on a set ,or more generally a metric space , to guarantee the existence of a bi-Lipschitz embedding of into some ?

(Olson and Robinson showed in [80] that if is a subset of a Hilbert space with finite Assouad dimension then one can find an embedding that is bi-Lipschitz to within logarithmic corrections; that a similar result holds for subsets of Banach spaces was shown by Robinson [83]. See Robinson [58] for further discussion.)

7.1. Time-Delayed Observations

Theorem 32 provides “abstract” embeddings of a finite-dimensional attractor into a finite-dimensional space. But in an experimental situation one would rather be able to take a more specific measurement than some general linear map.

In 1981 Takens showed that for a smooth iterated map (satisfying some generic assumptions) on smooth finite-dimensional manifold , the repeated observations provide a one-to-one mapping of into for a generic set of , provided that .

The following theorem generalises this result to allow fractal subsets of infinite-dimensional spaces. The key element of the proof is a result by Sauer et al. [87] that treats the same situation for fractal subsets of finite-dimensional spaces.

Theorem 34 (see [88]). Let be a compact subset of a Hilbert space with upper box-counting dimension and thickness exponent zero. Suppose that is an invariant set for a Lipschitz map . Fix an integer and suppose further that the set of -periodic points of satisfies for all . Then a prevalent set of Lipschitz maps make the -fold observation map one-to-one between and its image.

Proof (Sketch). Sauer et al. [87] proved the same result for a compact set and a Lipschitz map . Their proof can be generalised to the case of a map such that is a -Hölder function for any , provided and the set of -periodic points of satisfies for all .We now combine this with the embedding result of Theorem 32. If provides an embedding of into , note that the induced map on is given by . In particular, and is -Hölder for every .

Note that while we have obtained an embedding of into , the result as stated, unlike that of Theorem 32, says nothing about the continuity of the resulting parametrisation of . Such results are important if one wishes to obtain information about properties of the original dynamical system from measurements via .

Open Question 12. Can one improve Theorem 34 to obtain information about the continuity of ?

8. Determining Nodes and Parametrisation by Nodal Values

In this section we return to the particular example of the 2D Navier-Stokes equations and introduce a different approach to finite-dimensional behaviour, in a form due first to Foias & Temam [89]. We show that if two solutions converge as on a sufficiently large collection of points (or “nodes”), then they must converge throughout the domain. This collection of points is referred to as a set of “determining nodes.”

It is as important to stress what such results do not say as what they do say. If you take two solutions of the full equations and know that they converge on some set of nodes, you can deduce that the full solutions are converging. However, it does not say a priori that the knowledge of the nodal values at any instant will determine the full solution.

However, it is natural to conjecture (as did Foias & Temam in [89]) that if one restricts to the attractor—that is, if one has already taken a “dynamical limit” as —then in fact the instantaneous value of the solution at a sufficient number of points will be enough to distinguish different solutions. We prove a generalised version of a result due to Foias & Titi [90], which in the context of the Navier-Stokes equations is conditional, that guarantees this, and we state a result due to Friz and Robinson [53] that this is indeed the case when the forcing is analytic.

8.1. Determining Nodes

We choose a finite set of points, or nodes, in , , and set so that for every there is an such that We say that is a set of determining nodes, if, whenever we have Since, by Agmon’s inequality in 2D, and we remarked at the end of Section 5.2 that there is an absorbing set in , it suffices to show that In fact, we will show that which clearly gives (207) (and, hence, (205)) since .

Fundamental to the proof is the following lemma, relating a bound on to a bound on .

Lemma 35. If and one sets then

Proof. Recall the Sobolev embedding theorem , where is the set of continuous functions on with Hölder exponent one half; since and (this follows straightforwardly in the case of periodic boundary conditions) we have The expression (210) follows immediately from this and the definition of and .

We will also need the following simple lemma.

Lemma 36. Suppose that is nonnegative and satisfies where and that for all . Then as .

Proof. Choose . Then there exists a such that for all . So for , By Gronwall’s inequality, and so choosing large enough that we have so that .

We now use these two results to study the time evolution of , where .

Theorem 37 (determining nodes). There exists such that if , then are a set of determining nodes.

Proof. The equation for is and taking the inner product of this with and using (15) and the three-term identity which follows by differentiating, we obtain Thus so therefore Now, we know that we have absorbing sets in and , so that for large enough and therefore Now, choose such that Then we have By assumption, we know that , and we also know that for large enough .It follows from Lemma 36 that as , and so and the nodes are determining.

Here we have made no attempt to obtain the best estimate for the separation , and indeed, the estimate derived from (225) is very coarse. The bound was improved in a series of papers, Foias & Titi [90] and Jones & Titi [91], until the best current results which are due to Jones & Titi [92] (their paper gives the best bounds for determining nodes, modes, and volume elements (see next section)), as compare this with the length scale from the Kraichnan theory discussed in Section 6.5, where

This motivates the following.

Open Question 13. Can one obtain a bound on the number of determining nodes that agrees with the attractor dimension estimate, that is, ?

8.2. Nodal Parametrisation

As remarked earlier, Foias & Temam [89] conjectured that on the attractor one should be able to take a finite number of nodes and distinguish functions by their value at these nodes at one fixed time; that is, for a set , if then

8.2.1. A Conditional Result

The first result in this direction was due to Foias & Titi [90], who showed, using the existence of an inertial manifold (see Section 9.2), that the result holds for the one-dimensional Kuramoto-Sivashinsky equation.

We give a version of their argument here which is valid for the 2D Navier-Stokes equations, but which requires a strong assumption which is not known to be valid in this case. The main assumption of the lemma is that for some , Combining this with (210) it follows that if is contained in a cube centred at with sides of length then We will return to assumption (230) in Section 9.3. (For the 1D Kuramoto-Sivashinsky equation, the existence of an inertial manifold guarantees that is Lipschitz on the attractor and on a one-dimensional domain functions in are -Hölder, so the conclusion holds without the need for additional assumptions in this case. A more general result is given by Cockburn et al. [93].)

Lemma 38. Consider the 2D Navier-Stokes equations, and suppose that there exists a constant such that Suppose that is sufficiently small that where is the constant in (210), and let be a collection of nodes placed at the corners of squares with sides of length . Then the map , where is a bi-Lipschitz embedding of into , and in particular the nodes are instantaneously determining.

Proof. Suppose that and set . Let be a collection of nodes equally spaced by ; then certainly . Noting that it follows that is Lipschitz.Now we show the reverse inequality for sufficiently small. Split into a collection of squares with sides of length . As in (231) for any we have and so for any . Therefore thus, if is sufficiently small that (233) is satisfied then for some , and, hence, implies that .

8.2.2. Nodal Parametrisation for Analytic

One can also parametrise the attractor by nodal values provided that the attractor consists of analytic functions, with , for almost every choice of points in . This result, originally due to Friz & Robinson [53], has been considerably refined and the most powerful version is given in Kukavica & Robinson [94]. We only state the result here, but note that its proof makes use of the attractor, its finite-dimensionality, and the regularity of its elements.

Theorem 39. Let be a compact subset of with finite dimension that consists of real analytic functions. Then for almost every set of points in makes the map , defined by one-to-one between and its image.

We note here that for the 2D Navier-Stokes equations, if we space our nodes evenly over the domain, then the separation required by our theorem is of the order of , with logarithmic corrections, confirming the Kraichnan length scale by analytically rigorous means. Note also that this is an entirely natural way to produce a length-scale from the equations and ties in with the heuristic definition of the “number of degrees of freedom” due to Landau & Lifshitz, which indicates that one would expect that cf. Section 6.5.

Note that since one can show (under minimal assumptions) that these “instantaneous” determining nodes are “determining” in the sense of Theorem 37 (see [53]), this shows that when is analytic the answer to Open Question 13 is positive.

It is natural to ask whether the result of Theorem 39 can be significantly improved.

Open Question 14. Can one parametrise finite-dimensional sets of functions by nodal values under weaker conditions?

As a first step away from analyticity, note that Kukavica & Robinson [94] showed using a more general version of Theorem 39 based on functions with finite order of vanishing and arguments essentially due to Poon [95, 96] that such a parametrisation occurs for attractors of reaction-diffusion equations when the nonlinearity is only . It is natural to ask whether a similar result holds for the Navier-Stokes equations.

Open Question 15. Can one parametrise the attractor of the 2D NSE by nodal values when is only (and not necessarily analytic)?

9. Determining Modes and the Foias-Temam Conjecture

In a paper from 1967, Foias & Prodi [44] showed that the dynamics are “determined” by a finite number of Fourier modes, in that if and are two solutions of the Navier-Stokes equations, and for sufficiently large then in fact .

Just as with the “determining nodes” discussed in the previous section, it is important that this result does not say that the solutions of the -mode Galerkin truncation determine the solution, nor that knowledge of the modes at any instant will determine the full solution.

While the “natural conjecture” in this case is that on the attractor a sufficiently high-dimensional finite-dimensional Fourier projection will distinguish different elements of the attractor () this is still open.

9.1. Determining Modes

We give a simple proof that enough modes are determining, following Foias et al. [97].

Theorem 40 (Determining modes). There exists a such that the first Fourier modes are determining provided that .

Proof . If and are two solutions then the equation satisfied by is We write and ; take the inner product of this equation with to obtain since and . Therefore and so, since , this becomes If is sufficiently large that then, since we have assumed that as , this is an equation of the form as in the proof of Theorem 37; using Lemma 36 it follows that as , and the proof is complete.

Note that one can obtain a similar result by taking the average the solution over a subgrid of a set of smaller squares, termed “determining volume elements” (introduced in [90]; see also [92, 98]). So, we split into equal squares, of sides , and label them , . We define the average of over a square as and show that if is “large enough”, then implies that

A unified treatment of determining nodes, modes, volume elements, and more general “determining functionals” is given in Cockburn et al. [93].

9.2. Inertial Manifolds

One situation in which a Fourier projection will distinguish elements of the attractor is when the equation admits an inertial manifold. An inertial manifold for a general evolution equation is a finite-dimensional, positively invariant Lipschitz smooth manifold that exponentially attracts all trajectories [99, 100], so that

In order to describe the results further, we assume that is a positive, linear, self-adjoint operator with compact inverse, and is a Lipschitz function from (the domain of ) into , for some . Since has a compact inverse, there is a set of orthonormal eigenfunctions of with corresponding eigenvectors , which one can order such that see Renardy & Rogers [101], for example. One can define the finite-dimensional projection operators and their orthogonal complements by where is the scalar product in .

All current existence proofs give the inertial manifold as a Lipschitz (or smoother) graph over one of the finite-dimensional subspaces ; that is,

For a general evolution equation restricting the flow from (255) to the manifold given by (254) immediately yields the set of ordinary differential equations for , as Since and is Lipschitz, it follows that (256) has unique solutions (see Hartman [102], Section , Theorem 1.1). Clearly the solutions of (256) on are precisely those projected down from ; that is, and since is an invariant manifold in , is the global attractor for the finite-dimensional system (256).

However, there are two problems with the inertial manifold approach. Outstandingly, the conditions known to be sufficient to prove the existence of such an object are restrictive; essentially a large gap is required in the spectrum of the linear operator , Although this is satisfied for some interesting examples (e.g., the Kuramoto-Sivashinsky equation [18, 103, 104], the Ginzburg-Landau equation [18], and reaction-diffusion equations in space dimension 1 [18] (and some special domains in dimensions 2 and 3, Mallet-Paret & Sell [105]), there are many situations in which one can prove the existence of a finite-dimensional global attractor but not (at present) of an inertial manifold—of greatest interest, perhaps, are the 2D Navier-Stokes equations.

The problem here is that the eigenvalues of the Stokes operator satisfy (see (159)), and so the gap between consecutive eigenvalues does not grow as .

Open Question 16. Do the 2D Navier-Stokes equations possess an inertial manifold?

However, on the sphere the eigenvalues of the negative Laplacian are , and so in this case . In the case of the 2D Navier-Stokes equations one can take in (258), and so the right-hand side also behaves like . In this marginal case one may hope to obtain a positive result.

Open Question 17. Do the 2D Navier-Stokes equations on the sphere possess an inertial manifold?

A second problem is that, the dimension of the inertial manifold and, hence, of the differential system (257) can be much greater than that of the attractor. For example, for the Kuramoto-Sivashinsky equation the best estimate of the dimension of the attractor is [18, 106], whereas the best estimate of the dimension of the inertial manifold is [104].

More generally, one can ask if the whole approach is too restrictive.

Open Question 18. Can one obtain inertial manifolds as bona fide manifolds, that is, not given as graphs?

9.3. The Foias-Temam Conjecture

As we have already discussed, the existence of an inertial manifold for the 2D Navier-Stokes evolution equation is an important open problem and related to the following conjecture due to Foias & Temam, which provides a weaker version of Open Question 16.

Open Question 19. Does there exist an   such that the solutions on the attractor of the 2D Navier-Stokes equations are determined by their first Fourier modes, that is, if then

We note here the following conditional result that would supply an affirmative answer to this question. We showed in Lemma 38 that the same condition (261) would also guarantee the existence of a set of instantaneous determining nodes.

Proposition 41. Suppose that is Lipschitz continuous on the attractor for some . Then the attractor is a subset of a Lipschitz manifold given as a graph over for some .

Proof. Write for . If is Lipschitz continuous from into then for some . Now split , and observe that we have both Since as , we can choose large enough that , and then write that is, It follows that we can define uniquely for each , and then so that the attractor is a subset of a Lipschitz graph over .

One can obtain some continuity of on (but sadly not Lipschitz continuity) by assuming that consists of regular functions, as the following simple result shows; if is bounded in then is Hölder continuous on .

Lemma 42. If is bounded in , then

Proof. Setting write , where are the eigenfunctions of , and then Now use the Hölder inequality with and , so that which gives (268).

If the attractor is bounded in some Gevrey class, as in Corollary 23, then one can use the resulting bounds in (268) and minimise with respect to to deduce that for some . In the light of Proposition 41 this is somewhat frustrating.

9.4. Zero Lipschitz Deviation

In fact, a stronger result than (271) holds much under weaker conditions. Kukavica [19] showed, using refined methods related to the backwards uniqueness proof of Lemma 5, that on the attractor of the 2D Navier-Stokes equations with periodic boundary conditions and a forcing , that

where .

One can deduce from this [81, 107] that the Lipschitz deviation of the Navier-Stokes attractor is zero, even when we only have .

Proposition 43. If then , where is the attractor of the 2D Navier-Stokes equations.

Proof. Let be the orthogonal projection onto the first eigenfunctions of . Consider a subset of that is maximal for the relation For every with , , define . From (273) this is well defined and We can extend from the closed set to a function , preserving the Lipschitz constant [108].We now show that Indeed, if but then there is a such that and then, noting that , we have from which and (275) follows.The bound in (275) implies that and hence since the eigenvalues satisfy Thus .

10. Finite-Dimensional Dynamics?

Since the attractor can be parametrised by a finite number of parameters and (equivalently) can be “faithfully represented” in a finite-dimensional space , it is a natural question whether one can construct a finite-dimensional dynamical system which has an attractor on which the dynamics are “the same” as those on . In this sense, the question is whether the dynamics are in some sense “asymptotically finite-dimensional.” We can make this precise in the following rather wordy definition.

Definition 44. The dynamics of are asymptotically finite-dimensional if for some , comparable to , there exists a map that is injective on , and a dynamical system defined on the whole of such that the dynamics on are conjugate to those of the finite-dimensional system under ; that is,

Such issues were first discussed explicitly in Eden et al. [55]. Some partial results can be found in Robinson [16, 109]. Romanov [110] makes a similar definition, dropping the requirement that is the attractor for , but requiring to be bi-Lipschitz. He obtains some nice general results about such systems.

Note that this definition does not require (i)that is generated by an ordinary differential equation on , (ii)that is the attractor of the finite-dimensional system, although both would be ideal.

This question is still entirely open.

Open Question 20. If has a finite-dimensional attractor, are its dynamics asymptotically finite-dimensional?

There seem to be some major obstructions to proving such a result in continuous time [111], certainly to the “obvious” approach (and refinements), which attempt to obtain from an appropriate ordinary differential equation. In this approach, take one of the linear maps from Theorem 32 that embeds into , and try to use this to write down a finite-dimensional set of ODEs that reproduces the dynamics. If we write the original governing equation as then the resulting ODE for would be For solutions of this ODE to be unique, standard conditions require to be a function such that where satisfies (see Corollary 6.2 in Chapter III of [102]). Taking yields the standard result that a Lipschitz ODE has unique solutions, but this can be weakened by taking provided that .

The continuity of depends on two factors (since is bounded and linear, this is not an issue): (i)the continuity of on ; (ii)the continuity of . Usually (and in the 2D Navier-Stokes equations) the continuity of will be determined by the continuity of on the attractor, and we have already seen in Lemma 38 and Proposition 41 that Lipschitz continuity of on the attractor has many consequences. Here, then, is another.

Proposition 45. If is Lipschitz on the attractor, then the dynamics are asymptotically finite-dimensional.

Proof. If is Lipschitz continuous on the attractor then so is , since, writing , Proposition 41 shows that the attractor is contained in the graph of a Lipschitz function over for some . Thus the equation satisfied by is the right-hand side of which is Lipschitz continuous.

However, such continuity of is not known. We remarked after the proof of Lemma 42 that one can obtain log-Lipschitz continuity of on the attractor under the assumption that is analytic, but with logarithmic exponent 2. Continuity of on the attractor was shown by Kukavica [19] with logarithmic exponent 1/2 (see (272)), and it was shown by Pinto de Moura & Robinson [107] that a similar argument can be used to show that is log-Lipschitz with logarithmic exponent 1. If we could obtain an embedding with Lipschitz inverse then this would be sufficient; but the embedding Theorem 32 only guarantees that is Hölder continuous. Thus even the log-Lipschitz embedding theorem of Olson & Robinson [80] (which requires strong conditions on that are probably not satisfied by the Navier-Stokes attractor) is not sufficient, since it would yield a function that is log-Lipschitz with logarithmic exponent larger than 1 (see [112] for further discussion of this approach).

Something can be done for iterated homeomorphisms, however. The following result, which uses topological properties of the global attractor, is proved in Robinson & Sánchez-Gabites [39].

Theorem 46. Let be the attractor of a homeomorphism with . For any there exist homeomorphisms such that the dynamics on and are conjugate under , that is, and has an attractor with where

However, even this is not ideal, since it is natural to hope that one could obtain a similar result with the attractor of .

Open Question 21. Can one prove Theorem 46 but ensure that is the attractor of (i.e., )?

11. The 3D Navier-Stokes Equations

We have concentrated throughout this review on the 2D Navier-Stokes equations, since at present we cannot show that the 3D Navier-Stokes equations generate unique weak solutions, nor that the strong solutions, which are unique, exist for all time. Trying to investigate the existence of attractors without the guarantee of a sensible semigroup seems futile.

However, we end with a result that shows that if we are prepared to assume that the equations generate a semigroup on , that is, if we assume the existence of strong solutions, then we can show that the equations must have a global attractor. In fact the result here just shows the existence of an absorbing set bounded in , and to show that there is a global attractor we would need an absorbing set that is compact in . A relatively straightforward argument can be used to prove the existence of an absorbing set that is bounded in once we have the absorbing set in , and, hence, of a global attractor.

What we are doing here is making a physically reasonable assumption in a mathematically precise way and then deducing an entirely mathematical consequence. It allows us to consider the asymptotic regimes of the “true” Navier-Stokes equations and so fully developed turbulence, within a mathematical framework.

Another way to view this theorem, which does not require us to make any “unjustified” assumptions, is as a description of the way in which the 3D Navier-Stokes equations must break down if they are not well posed. The theorem shows that existence and uniqueness fail only if there is some solution such that becomes infinite in some finite time.

Theorem 47. Suppose that the 3D Navier-Stokes equations are well posed on , so that for any and , has a strong solution , that is, a solution with for all . Then there exists an absorbing set in .

The theorem is due to Constantin et al. [7]; the proof also appears in [13] and in Temam [18].

Acknowledgment

JCR is currently an EPSRC Leadership Fellow, grant EP/G007470/1. This review paper is a substantially updated version of lectures given at the Instructional Conference on ‘The Mathematical Analysis of Hydrodynamics’ held at the ICMS in Edinburgh in June 2003; my thanks to the organisers of that meeting for their kind invitation.

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