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ISRN Mathematical Analysis
Volume 2013 (2013), Article ID 291823, 29 pages
Attractors and Finite-Dimensional Behaviour in the 2D Navier-Stokes Equations
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.
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.
The Navier-Stokes equations are the fundamental mathematical model of fluid flow; a physical derivation of the equations can be found in Batchelor  or Doering and Gibbon , for example. Their rigorous analysis goes back to Leray , who proved the global existence of weak (-valued) solutions in 3D and local existence of strong (-valued) solutions; similar results were obtained by Hopf  for bonded domains. Global existence and uniqueness of weak solutions in the 2D case was first shown by Ladyzhenskaya .
The dynamical systems approach to the Navier-Stokes equations was developed over a number of years, notably by Ladyzhenskaya  and Foias, Constantin, Temam, and coauthors; see Constantin et al. , for example. Delay differential equations provided a stimulus for the development of the theory from a different but related viewpoint; see Hale et al. , 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  or Cannone ), 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 ) 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. .
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  and are now standard—see, for example, Constantin and Foias , Temam  or , Ladyzhenskaya , and Robinson .
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. ). Note that . (In the case of periodic boundary conditions we have for all ; see .)
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  Robinson , or Temam , 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  or Robinson ). 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 (, Chapter III Section 6) which shows that is injective, and an observation in Chapter 11 of Constantin and Foias .
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 ), 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.
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 ). 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 , e.g., for the whole space and  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 . 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 , 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 , 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  (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  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 , Chepyzhov and Vishik , Chueshov , Hale , Ladyzhenskaya , Robinson , Temam , and Vishik .
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  for random dynamical systems; see also Hale  and Babin and Visihik . (The first result along these lines seems to be due to Billotti and LaSalle .)
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  remark that one can construct a flow for which Bing’s pseudo-arc  is the global attractor, and this set is not simply connected. Langa and Robinson  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?
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.  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 .
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  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 ; making use of an analysis due to Doering and Foias  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 , the proof of the existence of a global attractor for the 2D Navier-Stokes equations was first published by Ladyzhenskaya in 1972 (see , for an English translation) and later, along with many other important results, by Foias and Temam .
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 . 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 , 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. .
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  or Temam .
5.3. Gevrey Regularity
We now use the theory of Gevrey regularity, developed by Foias and Temam , 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 , 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.
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 ; 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
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 .
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 , 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 .
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 ,
6.2. Dimension Estimates
Techniques to show that invariant sets have finite box-counting dimension date back to Mallet-Paret . His argument, valid for invariant subsets of Hilbert spaces, admits generalisation to Banach spaces (see , recently updated in ). However, the method that gives the best estimates is restricted to Hilbert spaces and was developed in infinite-dimensional spaces by Constantin and Foias  and Constantin et al.  [see also  or ], after the finite-dimensional approach of Douady and Oesterlé .
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 ,