Attractors and neo-attractors for 3D stochastic Navier-Stokes equations

Nigel J. Cutland and H.Jerome Keisler

July 31, 2004

Abstract

In [14] nonstandard analysis was used to construct a (standard) global attractor for the 3D stochastic Navier–Stokes equations with general multiplicative noise, living on a Loeb space, using Sell’s approach [26]. The attractor had somewhat ad hoc attracting and compactness properties. We strengthen this result by showing that the attractor has stronger properties making it a neo-attractor – a notion introduced here that arises naturally from the Keisler-Fajardo theory of neometric spaces [18].

To set this result in context we first survey the use of Loeb space and nonstandard techniques in the study of attractors, with special emphasis on results obtained for the Navier–Stokes equations both deterministic and stochastic, showing that such methods are well-suited to this enterprise.

KEYWORDS attractor, neo-attractor, stochastic Navier–Stokes equations, Loeb space

1 Introduction

The chief topic of this paper is the study of attractors for the time-homogeneous stochastic Navier-Stokes (sNS) equations in a bounded domain 𝔻 ⊂ ℝ^{d}, principally for d = 3. These are equations that describe the velocity of an incompressible fluid at each point in 𝔻; a general version of the sNS equations is

\{ \begin{matrix}du = [νΔu − ⟨u, ∇⟩u + f(u) − ∇p]dt + g(u)dw_{t} \\ div u = 0 \end{matrix}  (1)

Here u = u(t, ·, ·) where u(t, x, ω) is the (random) velocity of the fluid at the location x ∈ 𝔻 at time t, so that we have

u : [0, ∞) × 𝔻 × Ω → ℝ^{d}

where Ω is the domain of an underlying probability space. The initial condition u(0, ·, ·) = u_{0} is prescribed (and may be random); the boundary condition is either u(t, x, ω) = 0 for x ∈ ∂𝔻 or, occasionally, when d = 2 we assume periodic boundary conditions. The term p denotes the pressure and ν is the viscosity. The terms f and g represent external forces influencing the fluid, which allow for feedback involving the velocity field u but for the present discussion are homogeneous in time.

The deterministic Navier–Stokes equations are the result of taking g = 0 and u non-random – so we simply have u = u(t, x).

Although the theory of attractors for 2D deterministic Navier-Stokes equations is well understood (see [28] for an exposition), in higher dimensions, or when noise is introduced into the system, or both, even the existence of attractors is still problematic. Nonstandard techniques have prove to be useful in the study of the problems to be addressed. For example, in the paper [14], a new notion of a process attractor was introduced. A process attractor A ⊆ X was constructed for a family X of solutions to the 3D stochastic Navier-Stokes equations with a general multiplicative noise. This made essential use of a filtered Loeb probability space using methods from nonstandard analysis, and followed earlier papers [6], [7], [9] that show the usefulness of nonstandard analysis in the study of attractors for Navier-Stokes equations – in all cases giving completely standard results. Such techniques are natural because attractors have to do with the behaviour of a system “at infinity” – a notion which can be handled easily in the extended framework of nonstandard analysis. In the case of the Navier–Stokes equations a further advantage stems from the fact that nonstandard analysis has proved to be an effective tool for the study of these equations – both deterministic [3] and stochastic [4]; see the book [8] for a complete exposition.

The purpose of the current paper is two-fold. The first (Part I) is to amplify the remark above by giving a brief survey of the way in which nonstandard analysis provides a useful tool for the study of attractors in general and specifically for the Navier-Stokes equations – both deterministic and stochastic. We include a new and simple proof of Sell’s result establishing the existence of an attractor for 3D Navier–Stokes equations, and its characterization in terms of two-sided solutions. This then provides the context for Part II of the paper.

The second purpose, in Part II, making this a sequel to [14], is to strengthen the results of that paper by introducing the notion of a neo-attractor for a system of stochastic Navier-Stokes equations. This is a stronger notion than that discussed in [14], and we show that the attractor A of that paper is actually a neo-attractor. The upshot of this is the existence of an attractor for the 3D stochastic Navier–Stokes equations that has new, stronger and more natural compactness and attraction properties. The general theory of neoattractors that is developed also gives the existence of a neoattractor for any specialized subset Y of the family X having natural closure properties.

The authors would like to thank the editor for suggesting the expansion of an earlier version of this paper to include the survey to put the new results in context.

1.1 Outline of the paper

In Part I (Section 2) we take a general definition of an attractor for a dynamical system and show how nonstandard analysis can be used to prove one of the basic existence theorems. This is applied in Section 3 to give a simple proof of Sell’s existence theorem for an attractor for the 3D deterministic Navier-Stokes equations.

Part II begins with a discussion of the general problem of defining attractors for stochastic systems (Section 4) followed by an outline of the existence results for these that was established in the papers [7], [9] and [14].

We then proceed to new ideas and results. In [14] the compactness and attraction properties of the attractor 𝔸 for general 3D stochastic Navier-Stokes equations were somewhat ad hoc and unsatisfactory. The class of open neighbourhoods of 𝔸 that absorb bounded sets was rather unnatural, while the compactness property required of an attractor was defined in terms of laws of the processes in 𝔸 rather than in terms of the set 𝔸 itself. In Section 5 we use the Keisler-Fajardo theory of neometric spaces from [18] to define a stronger and more natural notion called a neo-attractor. The open neighbourhoods of a neo-attractor that absorb bounded sets are the neo-open sets, while the attractor itself is required to be neo-compact. We prove a general abstract existence theorem that isolates sufficient conditions for the existence of a neo-attractor. In Section 7 it is shown that the family of solutions X in the paper [14] satisfies these conditions, and that the attractor A of that paper is a neo-attractor in the sense of this paper. The general existence theorem also gives existence of a neo-attractor for any subset Y of solutions having natural closure properties.

Part II depends in an essential way on [14] although we will outline in Section 6 the main ideas to make this paper as self-contained as possible.

1.2 Preliminaries

In Section 2 below we outline the basic ideas of global attractors in a deterministic setting. The book [28] is a good reference for a general background on attractors.

The book [8] for background on stochastic Navier-Stokes equations, and any of the books [1, 2, 13] for more background on nonstandard analysis and Loeb measures. The appendices provide a brief summary of the notions needed from nonstandard analysis, and a short account of the theory of neometric spaces that is needed here. In order to assist the reader we use sans serif symbols (x, y, u, v, T etc) for nonstandard objects and the symbols σ, τ to denote nonstandard time.

PART I: DETERMINISTIC SYSTEMS

2 Attractors

A setting for the study of attractors that includes all particular instances in this paper is that of a dynamical system in a metric space M described by a semigroup (S_{t})_{t≥0} of operators, which are assumed to be continuous. To make this precise we have (drawing largely on the exposition in [28])

Definition 2.1 A semiflow on a metric space M is a semigroup of continuous operators S_{t} : M → M (t ≥ 0) such that

(a) S_{0} = I (the identity mapping),

(b) S_{t+s} = S_{t} ∘ S_{s} for all s, t ≥ 0,

(c) S_{t} is continuous for all t ≥ 0.

(Some authors require the resulting mapping S : [0, ∞) × M → M to be continuous, but we do not need this.)

The notion of an attractor for such a system is concerned with the asymptotic behaviour of its trajectories. There is a variety of notions of an attractor in the literature, each with its own rationale. For our purposes we adopt the following fairly strong definition.

Definition 2.2 A global attractor for a semiflow (S_{t})_{t≥0} is a set 𝔸 ⊆ M such that

(a) (Invariance) S_{t}𝔸 = 𝔸 for all t.

(b) 𝔸 is compact

(c) (Attraction) For every open neighbourhood U of 𝔸 and every bounded

set B ⊆ M,

S_{t}B ⊆ U

eventually (meaning that there is t_{0} = t_{0}(B, U) such that S_{t}B ⊆ U for all t ≥ t_{0}).

Remarks

1. Some authors describe such an attractor as a (global) set attractor (in contrast to a point attractor, where the attraction property (c) applies only to singleton sets B = {x}).

2. If there is a global attractor it is unique. To see this consider x ∈ 𝔸′ \ 𝔸 where 𝔸′ is another global attractor. Take an open neighbourhood U of 𝔸 that excludes x. Since 𝔸′ is bounded there is t with S_{t}𝔸′ ⊆ U, which is impossible since S_{t}𝔸′ = 𝔸′ for all t.

A fairly general existence result for global attractors (as given in [28] for example) involves the notion of an absorbing set. From the various definitions that appear in the literature the following is appropriate for our needs.

Definition 2.3 A set E is an absorbing set for the semiflow(S_{t})_{t≥0} if for every bounded set B ⊆ M

S_{t}B ⊆ E

eventually.

To illustrate the applicability of nonstandard techniques for the study of attractors we give a proof of the following variation of a theorem that appears in [28] (Theorem 1.1) .

Theorem 2.4 Suppose that M is a metric space with a semiflow (S_{t})_{t≥0} having a bounded absorbing set E. Assume further that there is some t > 0 such that S_{t}E is relatively compact (i.e. has compact closure). Then there is a global 𝔸 attractor for the semiflow given by

𝔸 = ⋂_{t∈[0,∞)} S_{t}(E)  (2)

Remarks 1. It is clear that for any two bounded absorbing sets E, E′ we have

⋂_{t∈[0,∞)} S_{t}(E) = ⋂_{t∈[0,∞)} S_{t}(E′)

2. It is easy to see that the assumption on the absorbing set E implies the following apparently stronger condition that often appears in the literature:

for each bounded set B ⊆ M there is t > 0 such that S_{t}B is relatively compact.

3. If S_{t}E ⊆ K with K compact then clearly K is a compact absorbing set – so this apparently stronger hypothesis is also implicit.

Proof. First, as noted above, we may assume that E is compact, so that ∗E ⊆ns(M). Let

𝔸 = ⋂_{t∈[0,∞)} S_{t}(E)

Then each S_{t}(E) is compact (since S_{t} is continuous) so 𝔸 is also compact. Write T = ^{∗}S, so that we have T_{τ} for all τ ≥ 0, τ ∈^{∗} ℝ, and observe that: (♯) if x ∈ ^{∗}E and τ is infinite then T_{τ}x is nearstandard and ^{∘}T_{τ}x ∈ 𝔸.

To see this we have

T_{σ}x ∈ ^{∗}E

and so

∘(T_{σ}x) ∈ E

for any infinite σ by the absorbing property of E and so for any finite t

∘(T_{τ}x) = ^{∘}(T_{t}T_{τ−t}x)
= S_{t}^{∘}(T_{τ−t}x) ∈ S_{t}E

where we have used the continuity of S_{t}, and hence ^{∘}(T_{τ}x) ∈ 𝔸.

To see that 𝔸 is invariant take any a ∈ 𝔸. Then a = T_{τ}x for some infinite τ and x ∈ ^{∗}E, so (♯) and the continuity of S_{t} gives

S_{t}a = ^{∘}T_{t}a =^{∘} (T_{t}T_{τ}x) = ^{∘}(T_{τ+t}x) ∈ 𝔸

and

a = ^{∘}a =^{∘} (T_{t}T_{τ−t}x) = S_{t}^{∘}(T_{τ−t}x) ∈ S_{t}𝔸

which shows that 𝔸 =S_{t}𝔸.

Finally we show that 𝔸 attracts bounded sets. Take an open neighbourhood U of 𝔸; it is sufficient to show that S_{t}E ⊆ U eventually. If this fails then there is a sequence of points x_{n} ∈ E with S_{n}x_{n} ∉ U. Then for any infinite N we have ∘T_{N}x_{N} ∈ 𝔸 but T_{N}x_{N} ∉ ^{∗}U; since U is open this means that ^{∘}T_{N}x_{N} ∉ U, a contradiction.

Remark It can also be shown that the attractor 𝔸 is given by

𝔸 = ^{∘} ⋃_{τ−infinite} T_{τ}^{∗}E =^{∘} ⋂_{n∈ℕ} ⋃_{τ≥n} T_{τ}^{∗}E

which was essentially the way an attractor was defined in the papers [7],[9] for example.

Theorem 2.4 has a counterpart in the theory of neo-attractors to be developed later. The same is true for the next result, which gives necessary and sufficient conditions for the existence of a global attractor. We have not been able to find this stated explicitly in the literature although it is easy to derive using standard techniques. For interest we provide a nonstandard proof, using the fact that a sequence (s_{n}) is relatively compact in M iff and only if ^{∗}s_{N} is nearstandard for each infinite N.

Theorem 2.5 Suppose that M is a metric space with a semiflow (S_{t})_{t≥0}. The the following are equivalent.

(a) There is a global attractor for the semiflow (S_{t})_{t≥0}; (b) There is a bounded absorbing set E such that for any sequence (x_{n}) in E

and t_{n} → ∞ the set {S_{t_{n}}x_{n}} is relatively compact.

If (a) (or (b)) holds the global attractor 𝔸 is given by

𝔸 = ⋂_{t≥0}S_{t}E  (3)

for any bounded absorbing set E.

Proof. (a)⇒(b) For each n let

E_{n} = 𝔸^{\frac{1}{n}} = \{x : d(x, 𝔸) < \frac{1}{n}\}

which is open and therefore an absorbing set that is bounded (because 𝔸 is compact hence bounded).

Let E = E_{1}, and let T = ^{∗}S. Take a sequence x_{n} ∈ E and t_{n} → ∞ and consider an infinite N. We need to show that T_{t_{N}} x_{N} is nearstandard. Since each E_{m} is absorbing, T_{t_{N}} x_{N} ∈ ^{∗}E_{m} for each finite m and so T_{t_{N}} x_{N} ∈ ^{∗}E_{M} for some infinite M. Then d(T_{t_{N}} x_{N}, z) < \frac{1}{M} ≈ 0 for some z ∈ ^{∗}𝔸 ⊆ns(M) and so T_{t_{N}} x_{N} is nearstandard as required.

(b)⇒(a) Take a bounded absorbing set E as given by (b) and r_{0} such that S_{t}Ē ⊆ E for t ≥ r_{0}. Make the following observation.

(♯♯) If x_{n} ∈ E for all n and t_{n} → ∞ then T_{t_{N}} x_{N} is nearstandard and ∘T_{t_{N}} x_{N} ∈ 𝔸 for all infinite N.

To see this, the relative compactness of {S_{t_{n}}x_{n}} means that T_{t_{N}} x_{N} is nearstandard for all infinite N. Then

∘T_{t_{N}} x_{N} = ^{∘}(T_{t}T_{r_{0}}T_{t_{N} −t−r_{0}}x_{N}) = S_{t}S_{r_{0}}^{∘}(T_{t_{N} −t−r_{0}}x_{N}) ∈ S_{t}(S_{r_{0}} Ē) ⊆ S_{t}E

since T_{t_{N} −t−r_{0}}x_{N} ∈ ^{∗}E (by absorbtion) and so ^{∘}(T_{t_{N} −t−r_{0}}x_{N}) ∈ Ē.

For the compactness of 𝔸 take a_{n} = S_{n}x_{n} ∈ 𝔸 and by (♯♯) we have ^{∘}a_{N} ∈ 𝔸 for all infinite N, which shows that 𝔸 is sequentially compact, hence compact.

For invariance let a ∈ 𝔸. There is x_{n} ∈ E with a = S_{n}x_{n} for all n and so a = T_{N}x_{N} and

S_{t}a = ^{∘}T_{t}T_{N}x_{N}=^{∘} (T_{N+t}x_{N}) ∈ 𝔸

by (♯♯) with t_{n} = n + t, giving S_{t}𝔸 ⊆ 𝔸. Conversely

a = ^{∘}a =^{∘} (T_{t}T_{N−t}x_{N}) = S_{t}^{∘}(T_{N−t}x_{N}) ∈ S_{t}𝔸

again by (♯♯), with t_{n} = n − t, giving 𝔸 ⊆S_{t}𝔸.

Finally the attraction property is proved exactly as in the previous theorem.

Remarks (1) We could also have given a further equivalent condition (b+) (i) There is a bounded absorbing set E;

(ii) For any bounded set B, any sequence (x_{n}) in B, and t_{n} → ∞, the set {S_{t_{n}}x_{n}} is relatively compact.

It is clear that (b+) implies (b); and for (b)⇒(b+) simply note that if B is bounded then taking r with S_{r}B ⊆ E gives S_{r}x_{n} ∈ E for all n and {S_{t_{n}}x_{n}} = {S_{t_{n}−r}S_{r}x_{n}} is relatively compact (we may assume that t_{n} ≥ r for all n).

(2) It can also be shown that the attractor 𝔸 has the description

𝔸 = ^{∘}(ns(M) ∩ ⋃_{τ−infinite} T_{τ}^{∗}E)

and an alternative proof can be given starting from this definition of 𝔸.

The following notion of a of a subflow and the corollary below will be useful in later discussion.

Definition 2.6 Let (S_{t})_{t≥0} be a semiflow on a metric space M. By a subflow of (S_{t})_{t≥0} we mean the restriction of (S_{t})_{t≥0} to a closed subspace N of M such that S_{t}(N) ⊆ N for each t ≥ 0.

Corollary 2.7 Assume the hypotheses of Theorem 2.4 or (b) of Theorem 2.5. Then every subflow of (S_{t})_{t≥0} has a global attractor.

Proof. F = E ∩ N is a bounded absorbing set for the subflow, and any subset of N that is relatively compact in M is relatively compact in N, so the preceding theorems apply to the restriction of the flow S_{t} to the subspace N.

Before moving on to questions concerning attractors for stochastic systems, we give a simple proof of Sell’s theorem [26] showing the existence of a global attractor for the deterministic 3D Navier–Stokes equations. Here nonstandard techniques are useful in verifying the hypotheses in order to apply Theorem 2.4.

3 Attractors for 3D Navier-Stokes equations

3.1 The 3D Navier–Stokes equations

In this section we consider the time homogeneous deterministic Navier–Stokes equations

\{ \begin{matrix}du(t) = [νΔu(t) − ⟨u(t), ∇⟩u(t) + f(u(t)) − ∇p(t)] dt \\ div u = 0 \end{matrix}  (4)

in a bounded domain 𝔻 ⊆ ℝ^{3} with boundary of class C^{2}. Recall from the introduction that the term f(u(t)) denotes external forces while p(t) is the pressure. The function u = u(t, x) is the velocity of a fluid at the point x ∈ 𝔻 at time t, and in the above equation u(t) = u(t, ·) denotes the whole velocity field at time t. This can be regarded as a functional equation, and we adopt the conventional Hilbert space approach to it as follows. Let H be the closure of the set

{u ∈ C_{0}^{∞}(𝔻, ℝ^{3}) : div u = 0}

in the L^{2} norm |u| = (u, u)^{1/2}, where u = ⟨u_{1}, u_{2}, u_{3}⟩ and

(u, v) = ∑_{j=1}^{3} ∫_{𝔻} u_{j}(x)v_{j}(x)dx.

The letters u, v, w will be used for elements of H. The subspace V is the closure of the set {u ∈ C_{0}^{∞}(𝔻, ℝ^{3}) : div u = 0} in the stronger norm |u| + ∥u∥ where ∥u∥ = ((u, u))^{1/2} and

((u, v)) = ∑_{j=1}^{3} (\frac{∂u}{∂x_{j}} , \frac{∂v}{∂x_{j}} ) .

H and V are Hilbert spaces with scalar products (·, ·) and ((·, ·)) respectively, and | · | ≤ c∥ · ∥ for some constant c.

By A we denote the self adjoint extension of the projection of −Δ in H. Classical theory shows that there is an orthonormal basis {e_{k} : k ∈ ℕ} of eigenfunctions of A with corresponding eigenvalues λ_{k} > 0 such that λ_{k} ↗ ∞. For u ∈ H we write u_{k} = (u, e_{k}), and write Pr_{m} for the projection of H on the subspace H_{m} spanned by {e_{1}, …, e_{m}}. Since each e_{k} ∈ V, then H_{m} ⊆ V. If u = ∑ u_{k}e_{k} ∈ V then ∥u∥^{2} = ∑ λ_{k}u_{k}^{2}, so that the constant c above is λ1^{\frac{1}{2}} .

A trilinear form b is defined by

b(u, v, w) = ∑_{i,j=1}^{3} ∫_{𝔻} u_{j}(x)\frac{∂v_{i}}{∂x_{j}} (x)w_{i}(x)dx = (⟨u, ∇⟩v, w)

whenever the integrals make sense. Note the following well-known properties of the trilinear form b, where c is a real constant.

b(u, v, w) = −b(u, w, v),
b(u, v, v) = 0,
|b(u, v, w)| ≤ c∥u∥ ∥v∥ ∥w∥
|b(u, v, w)| ≤ c|u|^{\frac{1}{4}} ||u∥^{\frac{3}{4}} |v|^{\frac{1}{4}} ∥v∥^{\frac{3}{4}} ∥w∥  (5)

The last inequalities are two of the many continuity properties of b that are used in the study of the Navier–Stokes equations, and are proved in [27] for example.

From (5) we have the following crucial lemma (a slight extension of the Crucial Lemma (Lemma 2.7.7) of the book [8]).

Lemma 3.1 (Crucial Lemma) If u, v ∈ ^{∗}V with ||u|| and ||v|| both finite, and z ∈ V then

∗b(u, v,^{∗} z) ≈ b(u, v, z)

where u = ^{∘}u and v = ^{∘}v (with u, v ∈ V.)

Proof. The finiteness of ||u|| and ||v|| means that |u −^{∗}u| ≈ 0 ≈ |v−^{∗}v|. We have b(u, v, z) = ^{∗}b(^{∗}u,^{∗}v,^{∗}z) and

|^{∗}b(u, v,^{∗} z) −^{∗}b(^{∗}u,^{∗}v,^{∗}z)| ≤|^{∗}b(u −^{∗}u, v,^{∗} z)|+|^{∗}b(^{∗}u, v −^{∗}v,^{∗}z)|

Using (5) gives

|^{∗}b(u −^{∗}u, v,^{∗} z)| ≤c|u −^{∗}u|^{\frac{1}{4}} ||u −^{∗}u∥^{\frac{3}{4}} |v|^{\frac{1}{4}} ∥v∥^{\frac{3}{4}} ∥z∥ ≈ 0

and similarly |^{∗}b(^{∗}u, v −^{∗}v,^{∗}z)| ≈ 0 which gives the result.

3.2 Functional formulation of the Navier–Stokes equations

In the above framework, the deterministic Navier–Stokes equations may be formulated as the following differential equation in V′ (the dual of V):

du = [−νAu − B(u) + f(u)]dt,  (6)

where B(u) = b(u, u, ·). Note that the pressure has disappeared, because ∇p = 0 in V′ (using div v = 0 in V and an integration by parts). Although equation (6) is regarded as an equation in V′, it turns out that solutions can be found that live in H (and in fact in V for almost all times).

The equation (6) is really an integral equation, with the integral being the Bochner integral. Thus, when we write

u(t_{1}) = u(t_{0}) + ∫_{t_{0}}^{t_{1}}[−νAu(t) − B(u(t)) + f(u(t))]dt

we mean that for all v ∈ V we have

(u(t_{1}), v) = (u(t_{0}), v) + ∫_{t_{0}}^{t_{1}}[−ν(Au(t), v) − (B(u(t)), v) + (f(u(t)), v)]dt

as an equation in ℝ.

The existence of solutions to these equations was established by Leray in the 1930’s using the method of finite dimensional (“Galerkin”) approximations, a method that is still one of the principal tools in use. An efficient proof of existence was given in [3] using nonstandard methods, using a hyperfinitedimensional approximation in H_{N} for an infinite N, and thereby avoiding the need for specialized compactness results. In all cases the solutions are so called weak solutions, and the question of uniqueness of these is still open. (By contrast, so called strong solutions that live in V for all time can only be shown to exist for short time scales, but for these uniqueness is known.)

3.3 Attractors for 3D Navier–Stokes equations

The problem with the very definition of an attractor for the 3D Navier–Stokes equations is a consequence of the problem of uniqueness. If uniqueness was known then a semigroup could be defined on H by setting S_{t}v = u(t) where u(·) is the unique solution with initial condition u(0) = v. To overcome this possible non-uniqueness, Sell’s radical approach in [26] was to take a space of weak solutions W defined below as the arena of activity rather than the space H. We will show how nonstandard techniques provide a simple proof of Sell’s existence theorem for an attractor in this setting. The solutions constructed by the Galerkin method have certain other properties that can be derived heuristically from the equations themselves (and the same is true for the solutions constructed in [3]). This leads to Sell’s definition of a weak solution [26].

Definition 3.2 (Sell [26]) Let f : H → H such that |f(u)| ≤ c + d|u| with d < νλ_{1}. A weak solution to the Navier–Stokes equations (6) is a function u : [0, ∞) → H such that

(W1) u ∈ L^{∞}(0, ∞; H) ∩ L^{2}(0, T; V) for all T

(W2) for all t ≥ t_{0} ≥ 0

u(t) = u(t_{0}) + ∫_{t_{0}}^{t}[−νAu(s) − B(u(s)) + f(u(s))]ds  (7)

as a Bochner integral equation in V′.

(W3) for almost all t_{0} > 0 and all t > t_{0}

|u(t)|^{2} ≤ |u(t_{0})|^{2} exp(−k_{1}(t − t_{0})) + k_{2}  (8)

where k_{1} = νλ_{1} − d and k_{2} = c^{2}k_{1}^{−2}.

(W4) for almost all t_{0} > 0 and all t > t_{0}

|u(t)|^{2} + 2ν ∫_{t_{0}}^{t} ||u(s)||^{2} ds ≤ |u(t_{0})|^{2} + 2 ∫_{t_{0}}^{t}(f(u(s)), u(s))ds (9)

Denote by W = W(f) the set of all weak solutions for a given f as above. A norm is defined on W by

|u| = (∫_{0}^{∞} |u(t)|^{2} exp(−t)dt)^{\frac{1}{2}} = (∫_{0}^{∞} |u(t)|^{2}μ(dt))^{\frac{1}{2}}

where μ(dt) = exp(−t)dt

Thus W is a subspace of the complete metric space M of measurable functions ξ : [0, ∞) → H with norm

|ξ| = (∫_{0}^{∞} ξ(t)^{2} exp(−t)dt)^{\frac{1}{2}} = (∫_{0}^{∞} ξ(t)^{2}μ(dt))^{\frac{1}{2}}

where μ(dt) = exp(−t)dt. That is, M = L^{2}(0, ∞; H; μ), which is actually a separable Hilbert space.

Remarks

1. It is well known that a weak solution is weakly continuous in H; this follows from (7).

2. Sell [26] has f ∈ H so that c = |f| and d = 0, giving k_{1} = νλ_{1} and k_{2} = |f|^{2}(νλ_{1})^{−2} so (W3) is consistent with Sell’s corresponding condition.

3. Sell [26] adds the condition that D_{t}u ∈ L_{loc}^{p}[0, V′) for p = \frac{4}{3}. This follows however from the fact that

D_{t}u = −νAu − B(u) + f

together with:

|Au|_{V′} = ||u||
|B(u)|_{V′} ≤ c|u|^{\frac{1}{2}} ||u||^{\frac{3}{2}}

the latter inequality using (5).

4. Sell also defines a wider set of generalised weak solutions in order to have a complete metric space. Here however completeness is not needed; we will see below how the attractor we construct for the class W is also an attractor for the wider class of generalised weak solutions.

5. The above norm is equivalent to the metric used by Sell in [26], and has the same bounded sets.

6. It is well known (see for example [28]) that if f : H → H such that |f(u)| ≤ c_{0} + d_{1}|u| and d_{1} < νλ_{1} then for any initial condition u_{0} ∈ H there is a weak solution u ∈ W(f) with u(0) = u_{0}.

3.4 The semiflow and a global attractor

A semiflow S_{t} is defined on W as follows.

Definition 3.3 (Sell [26]) For u ∈ W and t ≥ 0 define S_{t}u ∈ L^{∞}(0, ∞; H) by

S_{t}u(s) = u(t + s)

for s ≥ 0. That is, S_{t}u is obtained by translating u to the left by t and retaining only the portion defined for positive time.

It is clear that S_{t} maps W into W and has the semigroup property. It is easy to see that each S_{t} is continuous (in fact Lipschitz continuous) because if u, u′ ∈ W then

|S_{t}(u − u′)|^{2} = ∫_{0}^{∞} |(u − u′)(t + s)|^{2} exp(−s)ds
= exp(t) ∫_{t}^{∞} |(u − u′)(s)|^{2} exp(−s)ds
≤ exp(t).|u − u′|^{2}

Theorem 3.4 (Sell [26]) Suppose that f : H → H is continuous with |f(u)| ≤ c_{0} + d_{1}|u|, where d_{1} < νλ_{1}. Then there is a global attractor 𝔸 for the set W = W(f) of weak solutions to the Navier–Stokes equations with forcing term f, and any subflow also has a global attractor.

Proof. We check that the hypotheses of Theorem 2.4 are satisfied for the metric space W. The result will then follow from Theorem 2.4.

The set E = {u ∈ W : |u|^{2} ≤ k_{2} + 1} is a bounded absorbing set. This follows easily from the energy inequality (8) in the definition of W, which shows that for any u ∈ W

|S_{t}u|^{2} ≤ |u|^{2} exp(−k_{1}t) + k_{2}.

So if B = {u ∈ W : |u| ≤ k} = B(k), say, we have

S_{t}B ⊆ E

whenever t ≥ 2(k_{1})^{−1} log k.

To check the compactness property of the maps S_{t}, we show that if B is bounded then S_{1}B is relatively compact. Here the technology of nonstandard analysis proves useful, particularly Loeb measure and integration theory.

Let B = B(k); it is sufficient to show that each u ∈ ^{∗}S_{1}^{∗}B is nearstandard to a point u ∈ W. Let u = ^{∗}S_{1}v with v ∈ ^{∗}B. That is

u(τ) = v (τ + 1)

for τ ≥ 0. Then (using τ and σ to denote nonstandard times)

∫_{0}^{1} |v(τ)|^{2}dτ ≤ e ∫_{0}^{1} |v(τ)|^{2} exp(−τ)dτ ≤ ek^{2}

and so there is some τ_{0} ∈ ^{∗}(0, 1) with |v(τ_{0})| finite and with ^{∗}(8) and ^{∗}(9) holding for all τ ≥ τ_{0}. (In fact we may take τ_{0} with |v(τ_{0})| ≤ ek^{2} + 1.)

Writing k_{3} = |v(τ_{0})|^{2} + k_{2} this means that

|v(τ)|^{2} ≤ k_{3} < ∞

for all τ ≥ τ_{0}. So

|u(τ)| ≤ \sqrt{k_{3}}  (10)

for all τ ≥ 0. Thus u(τ) is weakly nearstandard for each τ ≥ 0 and we can define the weak standard part ^{∘}u(τ) ∈ H. If τ_{1} ≈ τ_{2} then from (7) we have, taking v = e_{n} for finite n

u_{n}(τ_{2}) − u_{n}(τ_{1}) = ∫_{t_{0}}^{t_{1}}[−νλ_{n}u_{n}(τ) − (^{∗}B(u(τ)),^{∗}e_{n}) + (^{∗}f(u(τ)),^{∗}e_{n})]dτ ≈ 0

since the integrand is S-integrable (see below). Thus^{∘}u(τ_{2}) = ^{∘}u(τ_{1}), so for real t ≥ 0 we may define u(t) = ^{∘}u(τ) for any τ ≈ t, and u(t) is weakly continuous. We will see that u ∈ W and u ≈ u in the norm of W.

From ^{∗}(9) for τ_{0} we see that ∫_{τ_{0}}^{τ} ||v(σ)||^{2} dσ is finite for all finite τ and so ||v(τ)|| is finite for a.a. finite τ ≥ τ_{0} (with respect to the Loeb measure d_{L}τ). Thus, for a.a. finite τ ≥ 0, ||u(τ)|| is finite so that u(τ) is strongly nearstandard and

|^{∘}u(τ)| = ^{∘}|u(τ)|  (11)

Take any standard finite time T. We will show that u has properties (W3) and (W4) of Definition 3.2 on [0, T]. Let Y be the set of all τ ∈ ^{∗}[0, T] with the following properties:

(i) ||u(τ)|| is finite;

(ii) ^{∗}(8) and ^{∗}(9) hold in ^{∗}[0, T] with t_{0} = τ. That is, for all τ_{1} ∈ ^{∗}[τ, T]

|u(τ_{1})|^{2} ≤ |u(τ)|^{2} exp(−k_{1}(τ_{1} − τ)) + k_{2}

and

|u(τ_{1})|^{2} + 2ν ∫_{τ}^{τ_{1}} ||u(σ)||^{2} dσ ≤ |u(τ)|^{2} + 2 ∫_{τ}^{τ_{1}}(^{∗}f(u(σ)), u(σ))dσ

Then Y has Loeb measure T and so the set D = st(Y) is a subset of [0, T] with full measure (because st^{−1}(D) ∩^{∗}[0, T] ⊇ Y ). Now we see that (8) and (9) hold in [0, T] for all t_{0} ∈ D. Take any such t_{0} = ^{∘}τ with τ ∈ Y. Then for t ∈ (t_{0}, T] we have

|u(t)|^{2} = |^{∘}u(t)|^{2} ≤ ^{∘}|u(t)|^{2} ≤ ^{∘} (|u(τ)|^{2} exp(−k_{1}(t − τ)) + k_{2})
= ^{∘}|u(τ)|^{2} exp(−k_{1}(t − t_{0})) + k_{2}
= |u(t_{0})|^{2} exp(−k_{1}(t − t_{0})) + k_{2}

using (11) to give ^{∘}|u(τ)| = |^{∘}u(τ)|= |u(t_{0})|.

Similar reasoning shows that for t ∈ (t_{0}, T]

|u(t)|^{2} + 2ν ∫_{t_{0}}^{t} ||u(s)||^{2} ds ≤ |u(t_{0})|^{2} + 2^{∘}∫_{τ}^{t}(^{∗}f(u(σ)), u(σ))dσ

Now note that (^{∗}f(u(σ)), u(σ)) is S-integrable on [t_{0}, t] since it is bounded, and so

∘∫_{τ}^{t}(^{∗}f(u(σ)), u(σ))dσ = ∫_{τ}^{t} ^{∘}(^{∗}f(u(σ)), u(σ))d_{L}σ = ∫_{τ}^{t}(^{∘}(^{∗}f(u(σ))),^{∘}u(σ))d_{L}σ
= ∫_{τ}^{t}(f(^{∘}u(σ)), u(^{∘}σ))d_{L}σ = ∫_{t_{0}}^{t}(f(u(s)), u(s))ds

where we have used (11) and the continuity of f. This establishes (9) on [0, T].

Next we show that u satisfies the equation (7). Let v ∈ V. We know that for all t ≥ t_{0} ≥ 0

(u(t),^{∗}v) = (u(t_{0}),^{∗}v) + ∫_{t_{0}}^{t}[−ν((u(σ),^{∗}v)) − (^{∗}B(u(σ)),^{∗}v) + (^{∗}f(u(σ)),^{∗}v)]dσ

(12)

We have ^{∘}(u(t),^{∗}v) = (u(t), v) and ^{∘}(u(t_{0}),^{∗}v) = (u(t_{0}), v) . For the integral, we need to show that each integrand is S-integrable on [t_{0}, t], using Lindstrøm’s lemma for the first two terms. For the first term we have

∫_{t_{0}}^{t}((u(σ),^{∗}v))^{2}dσ ≤ ∫_{t_{0}}^{t} ||u(σ)||^{2}dσ. ∫_{t_{0}}^{t} ||^{∗}v||^{2}dσ < ∞

so ((u(σ),^{∗}v)) is S-integrable. For the second term we have, using (5), |(^{∗}B(u(σ)),^{∗}v)| ≤ c|u(σ)|^{\frac{1}{2}} ||u(σ)||^{\frac{3}{2}} ||v|| and so

∫_{t_{0}}^{t} |(^{∗}B(u(σ)),^{∗}v)|^{\frac{4}{3}} dσ ≤ (c||v||)^{\frac{4}{3}} k3^{\frac{1}{3}} ∫_{t_{0}}^{t} ||u(σ)||^{2}dσ < ∞

where we have used (10), showing that (^{∗}B(u(σ)),^{∗}v) is S-integrable. The final term is S-integrable since it is bounded. So the theory of Loeb integration gives

(u(t), v) − (u(t_{0}), v) = ^{∘}[(u(t),^{∗}v) − (u(t_{0}),^{∗}v)]

= ^{∘} [∫_{t_{0}}^{t}[−ν((u(σ),^{∗}v)) − (^{∗}B(u(σ)),^{∗}v) + (^{∗}f(u(σ)),^{∗}v)]dσ]

= ∫_{t_{0}}^{t}[−ν^{∘}((u(σ),^{∗}v)) −^{∘}(^{∗}B(u(σ)),^{∗}v) +^{∘}(^{∗}f(u(σ)),^{∗}v)]d_{L}σ

= ∫_{t_{0}}^{t}[−ν((u(^{∘}σ), v)) − (B(u(^{∘}σ)), v) + (f(u(^{∘}σ)), v)]d_{L}σ

= ∫_{t_{0}}^{t}[−ν((u(s), v)) − (B(u(s)), v) + (f(u(s)), v)]ds

where the penultimate equality is because for a.a. σ, ||u(σ)|| is finite and hence (a) u(σ) is weakly nearstandard in V and so ((u(σ),^{∗}v)) ≈ (^{∘}u(σ), v);

(b) (^{∗}B(u(σ)),^{∗}v) ≈ (B(^{∘}u(σ)), v) using Lemma 3.1; and similarly (^{∗}f(u(σ)),^{∗}v) ≈ (f(u(^{∘}σ)), v)

(c) ^{∘}u(σ) = u(^{∘}σ)

(d) (^{∗}f(u(σ)),^{∗}v) ≈ (f(u(^{∘}σ)), v) using the continuity of f and (11). The final equality holds because the standard part mapping ^{∘} : ^{∗}[t, t_{0}] → [t, t_{0}] is measure preserving.

We are now done provided we can show that |u−^{∗}u| ≈ 0. We have |u(τ)|, |^{∗}u(τ)| ≤ c_{1} for all τ and for a.a. τ ∈ ^{∗}[0, T] we have ||u(τ)|| and ||^{∗}u(τ)|| finite so that ^{∘}|u(τ) −^{∗}u(τ)| = |^{∘}u(τ) −^{∘}(^{∗}u(τ))| for such τ. The weak continuity of u means that ^{∘}(^{∗}u(τ)) = u(^{∘}τ) for all τ and since ^{∘}u(τ) = u(^{∘}τ) also we obtain, for finite T,

∫_{0}^{T} |u(τ) −^{∗}u(τ)|^{2} exp(−τ)dτ ≈ ∫_{0}^{T} ^{∘}|u(τ) −^{∗}u(τ)|^{2}^{∘} exp(−τ)d_{L}τ
= ∫_{0}^{T} |^{∘}u(τ) −^{∘}(^{∗}u(τ))|^{2} exp(−^{∘}τ)d_{L}τ = 0

Thus

∫_{0}^{T} |u(τ) −^{∗}u(τ)|^{2} exp(−τ)dτ ≤ T^{−1}

for all finite T and by overflow we have the same for some infinite time T. This is enough to show that |u −^{∗}u| ≈ 0 and so u ≈ u as required.

Remarks (1) This proof is very close in many respects to the original nonstandard proof of existence of weak solutions to the Navier–Stokes equations given in the paper [3]. In that paper the existence of solutions was established by constructing internal solutions to the Galerkin approximation in dimension N where N is infinite. The process of taking standard parts and showing that this gives a standard weak solution is almost identical to the above proof that ∘u(t) is a solution.

(2) In [26] Sell proves the existence of a global attractor for a wider class of generalized weak solutions – these are solutions that may have a singularity at t = 0 but are weak solutions away from 0. If u is a generalized weak solution then S_{t}u ∈ W for any t > 0, so it is clear that 𝔸 is a global attractor for this larger class of solutions.

(3) If the definition of a weak solution (Definition 3.2) is tightened by adding to the list (W1)-(W4) further equalities or inequalities that are preserved under S_{t} then we will have a subflow provided the resulting set is closed in W, and this will then have a global attractor. In Part II when discussing solutions to the stochastic Navier–Stokes equations we will need to impose additional conditions on the set of weak stochastic solutions in just this way.

3.5 Two-sided solutions

A neat characterisation of the global attractor above is given by considering two-sided solutions. In [26] Sell remarked that each solution u in the global attractor 𝔸 is the restriction to non-negative time of a solution defined for all time – which we call a two-sided solution. Here we use the ideas in the previous section to show the converse – so that 𝔸 is precisely the set of restrictions of two-sided solutions. First we must make the necessary definitions.

Definition 3.5 Let f : H → H such that |f(u)| ≤ c + d|u| with d < νλ_{1}. A two-sided weak solution to the Navier–Stokes equations (6) is a function v ∈ L^{∞}(−∞, ∞; H) ∩ L^{2}(−T, T; V) for all T with the properties (W2)-(W4) of Definition 3.2 holding for all t_{0} ∈ (−∞, ∞).

Denote by W̄ = W̄(f) the set of all weak solutions for a given f as above. A norm is defined on W̄ by

|v| = (∫_{−∞}^{∞} |v(t)|^{2} exp(−|t|)dt)^{\frac{1}{2}} = (∫_{−∞}^{∞} |v(t)|^{2}μ(dt))^{\frac{1}{2}}

where μ(dt) = exp(−|t|)dt

Thus W̄ is a subset of the Hilbert space M̄ = L^{2}(−∞, ∞; H; μ̄) where μ̄(dt) = exp(−|t|)dt. It will be useful to define the right-shift operator R_{t} on M̄ for any t ∈ ℝ by

(R_{t}v)(s) = v(s − t)

for any function v ∈ M̄. For future reference notice that for t > 0

S_{t}((R_{t}v) ↾ [0, ∞)) = v ↾ [0, ∞)  (13)

Now we have:

Theorem 3.6 Let f : H → H such that |f(u)| ≤ c + d|u| with d < νλ_{1}. The set W̄ = W̄(f) of all two-sided weak solutions is bounded and compact, and the global attractor 𝔸 for one-sided solutions (Theorem 3.4) is given by

𝔸 = W̄ ↾ [0, ∞)

Proof. From (8) it follows that for any v ∈ W̄

|u(t)| ≤ \sqrt{k_{2}}

for all t. This is because v ∈ L^{∞}(−∞, ∞; H), so we can find arbitrarily large negative t_{0} with |v(t_{0})| ≤ |v|_{∞} and (8) holding, so that for a given t we can make |v(t_{0})|^{2} exp(−k_{1}(t − t_{0})) arbitrarily small. So W̄ is bounded – in fact |v| ≤ 2|v|_{∞} ≤ 2\sqrt{k_{2}}.

Notice also that if v ∈ W̄ then R_{t} ∈ W̄ also, and v ↾ [0, ∞) ∈ E. So if u = v ↾ [0, ∞) then from (13)

u = S_{t}((R_{t}v) ↾ [0, ∞)) ∈ S_{t}E

and thus u ∈ 𝔸, showing that W̄ ↾ [0, ∞) ⊆ 𝔸.

For the converse, take u ∈ 𝔸 so that ^{∗}u ∈ T_{τ}^{∗}E for some infinite τ. Let ∗u = T_{τ}v with v ∈ ^{∗}E and consider the left translate v̄ : ^{∗}[−τ, ∞) → ^{∗}H of v given by

v̄(σ) = v(σ + τ)

Then |v̄(τ_{0})| is finite for almost all τ_{0} ∈ ^{∗}[−τ, −τ + 1] and from this it follows using ^{∗}(8) that

∘|v̄(τ)| ≤ \sqrt{k_{2}}

for all finite times τ. So we may define v(t) = ^{∘}v̄(t) for real times t, and then

|v(t)| ≤ \sqrt{k_{2}}

for all t.

Adapting the proof of Theorem 3.4 shows that v ∈ W and for t ≥ 0 we have

v(t) = ^{∘}v̄(t) = ^{∘}v(t + τ) = ^{∘}T_{τ}v(t) = ^{∘}(^{∗}u(t)) = u(t)

so

u = v ↾ [0, ∞)

showing that 𝔸 ⊆ W̄ ↾ [0, ∞).

The proof that W̄ is compact follows the same lines as the proof that S_{1}B is relatively compact in the proof of Theorem 3.4, and is omitted.

3.5.1 Another approach

In the paper [6] an alternative way to overcome the problem of lack of uniqueness in order to define a notion of attractor for 3D Navier–Stokes equations. The idea was to work with internal solutions to the Galerkin approximation on H_{N} for some fixed infinite N. Here we have uniqueness of solutions, so there is a well defined internal semigroup T_{τ} defined on H_{N}. This was used to define various multi-valued semiflows on H by means of somewhat ad hoc standard part operations, leading to existence of compact attractors in H, but in a rather weak sense – so as is to be expected, the results are less pleasing than in dimension 2. Rather more satisfactory results were obtained for an approach that used small initial pieces of trajectories of solutions as phase space – this is an idea that is intermediate between the above and that of Sell. For full details, and further variations on this theme, see [6].

PART II: STOCHASTIC SYSTEMS

4 Stochastic attractors for Navier–Stokes equations

For stochastic systems the problem of defining attractors stems from the continual injection of noise into the system as it evolves – so it is unreasonable to expect there to be any set that attracts the random paths as t → ∞. However, there are several ways to formulate the idea of an attractor for a system of stochastic differential equations in a way that circumvents this problem. One is to consider measure attractors (see [7],[25] for example); another is to work with the notion of stochastic attractor developed by Crauel & Flandoli [10]

— but only in 2D. In the case of 3D stochastic Navier–Stokes equations these approaches are not available because of the possible nonuniqueness. Here it makes sense to consider extending the approach of Sell [26] discussed in Section 3 that was used for 3D deterministic Navier–Stokes equations.

We will briefly review results that have been achieved using the first two of these approaches for the 2D equations, and then present new results concerning the extension of Sell’s approach to 3D stochastic Navier–Stokes equations. In each case, to avoid unnecessary additional complications, the drift and noise coefficients f, g in (1) are taken to be time-independent, so the equations considered are

du = [−νAu − B(u) + f(u)]dt + g(u)dw_{t}  (14)

Here, as noted in the introduction u is a stochastic process, so in the functional formulation we have

u : [0, ∞) × Ω → H

where Ω is the domain of an underlying probability space. The equation (14) is an infinite dimensional stochastic differential equation (SDE), or SPDE, with w = w(t, ω) a Wiener process that models noise and provides the random forcing term g(u)dw_{t}. The theory of SDEs and SPDEs requires that the underlying probability space (Ω, F,P) has some extra structure – namely a filtration (F_{t})_{t≥0} satisfying the usual conditions, such that w is adapted; that is w(t, ·) is F_{t}-measurable for each t. A solution u(t, ω) is also required to be adapted.

4.1 Measure attractors

This approach is currently applicable only to d = 2 since it is necessary that the equation (14) has a unique solution. The functional formulation of the 2D stochastic Navier–Stokes equations outlined in Sections 3.1 and 3.2 applies equally in the 2D setting. To ensure that for each initial condition u ∈ H there is a unique solution u(t, ω) = v(t, ω; u) with u(0) = u (so v(0, ω; u) = u) it is assumed that f, g satisfy an appropriate Lipschitz condition. A semigroup S_{t} is now defined on M_{1}(H), the set of Borel probability measures on H, where S_{t}μ = μ_{t} is defined for μ ∈ M_{1}(H) by

∫_{H} ϑ(u)dμ_{t}(u) = ∫_{H} [𝔼_{P} ϑ(v(t, ·; u))] dμ(u)

for all bounded weakly continuous functions θ : H → H.

An attractor for the dynamical system (M_{1}(H), S_{t}) is called a measure attractor. The existence of measure attractors for the sNS equations was first investigated by Schmallfuß, in [25] for example. The paper [7] with Capiński establishes existence of a measure attractor for (14) under quite general conditions:

Theorem 4.1 Suppose that f, g are Lipshitz and satisfy an appropriate growth condition^{1}. Then there is a measure attractor 𝔸 ⊂ M_{1}(H) for the stochastic Navier–Stokes equations (14). That is

(a) 𝔸 is weakly compact;

(b) S_{t}𝔸 = 𝔸 for all t;

(c) for each open set O ⊇ 𝔸, and for each r > 0

S_{t}B^{r} ⊆ O

for all sufficiently large t, where B^{r} = {μ ∈ X : ∫ |u|^{2}dμ(u) ≤ r}

The methods in [7] do not make essential use of Loeb spaces although at some points they can be employed to assist the construction.

4.2 Stochastic attractors

For a stochastic system such as (14) the idea of a stochastic attractor developed by Crauel & Flandoli [10] takes into account the fact that at all times new noise is introduced into the evolution of each path of any solution to (14). A stochastic attractor is defined to be a random set 𝔸(ω) that, at time 0, attracts trajectories “starting at −∞” (compared to the usual idea of an attractor being a set “at time ∞” that attracts trajectories starting at time 0).

This idea is spelled out below, and involves the introduction of a one parameter group θ_{t} : Ω → Ω of measure preserving maps, which should be thought of as a shift of the noise to the left by t. In proving the existence of a stochastic attractor for the system (14) the nonstandard framework makes it particularly easy to consider −∞.

1For example, a sufficient condition is that |f(u)|_{−1} ≤ c+δ_{1}∥u∥ and |g(u)|_{H,H} ≤ c+δ_{2}∥u∥ for some δ_{1}, δ_{2} > 0 with 2δ_{1}+δ_{2}^{2}.trQ < 2ν, where Q is the covariance of the H-valued Wiener process w.

Making this precise, suppose that φ is a stochastic flow of solutions to (14). That is, φ is a measurable function

φ : [0, ∞) × H × Ω → H

such that φ(·, ·, ω) is continuous for a.a. ω, and for each fixed initial condition u_{0} the process u(t, ω) = φ(t, u_{0}, ω) is a solution to (14) with u(0, ω) = u_{0}.

The notion of a semigroup in the usual definition of a deterministic attractor, along with the notion of an attractor itself, is now replaced by the following.

Definition 4.2 (i) The flow φ is a crude cocycle if for each s ∈ ℝ^{+} there is a full set Ω_{s} such that for all ω ∈ Ω_{s}

φ(s + t, x, ω) = φ(t, φ(s, x, ω), θ_{s}ω)

holds for each x ∈ H and t ∈ ℝ^{+}.

(ii) A cocycle is perfect if Ω_{s} does not depend on s.

(iii) Given a perfect cocycle φ, a global stochastic attractor is a random compact subset A(ω) of H such that for almost all ω

φ(t, 𝔸(ω), ω) = 𝔸(θ_{t}ω), t ≥ 0,

lim_{t→∞} dist(φ(t, B, θ_{−t}ω), 𝔸(ω)) = 0

for each bounded set B ⊂ H.

Note that the existence of a perfect cocycle is necessary for the possibility of having a stochastic attractor. Constructing a perfect cocycle is difficult for infinite dimensional systems, particularly for those that are truly stochastic (as compared to random dynamical systems in which paths may be treated individually).

4.2.1 Existence of a stochastic attractor for the Navier–Stokes equations

A stochastic attractor was constructed for the stochastic Navier–Stokes equation with d = 2 by Crauel & Flandoli [10], but their version of (14) reduced to a random equation that could be solved pathwise, giving essentially a pathwise construction of the random attractor 𝔸(ω). The first example of a stochastic attractor for a truly stochastic version of the Navier–Stokes equations was constructed in [9] using Loeb space methods, seemingly in an essential way. In the following, for simplicity the Wiener process was taken to be one dimensional.

Theorem 4.3 (Capiński & Cutland[9]) (a) Suppose that (g(u)−g(v), u−v) = 0 and (g(u), u) = 0.^{2} With appropriate Lipschitz and growth conditions on f, g, there is an adapted Loeb space carrying a stochastic flow of solutions to the system (14) that is a perfect cocycle, and there is a stochastic attractor 𝔸(ω) (compact in the strong topology of H) for this system.

2For example g(u) = ⟨h, ∇⟩u for some h ∈ H.

(b) If g has the additional property that ((g(v), v)) = 0 for v ∈ V the stochastic attractor is bounded and weakly compact in V.

The proof of this result is quite long and complicated, and uses heavily the fact that solutions to (14) may be obtained as standard parts of Galerkin approximations of dimension N, infinite. A delicate extension of the Kolmogorov continuity theorem as adapted to a nonstandard setting by Lindstrøm [1] is at the heart of the construction of the perfect cocycle. An outline of the main steps and ideas of the proof is given in Chapter 2 of [13].

4.3 Sell’s approach for stochastic systems

As explained in detail in Section 3, Sell’s radical approach [26] to the problem of attractors for the deterministic Navier–Stokes equations for d = 3, bearing in mind the possible nonuniqueness of solutions, was to replace the phase space H by a space W of entire solutions to the Navier–Stokes equations equipped with the semigroup S_{t} on W defined by

(S_{t}u)(s) = u(t + s).

For the 3-dimensional stochastic case, Sell’s idea was used by Flandoli & Schmalfuß in the paper [15] for the Navier–Stokes equations with a special form of multiplicative noise, using a mild solution concept. The equation considered allowed essentially a pathwise solution, and then a random attractor was obtained by combining Sell’s approach with the idea of pulling back in time to −∞, as developed by Crauel & Flandoli [10]. In a later paper [16] Flandoli & Schmalfuß consider in the same framework the Navier–Stokes equations with an irregular forcing term, but no feedback.

An alternative way to extend Sell’s approach to the stochastic Navier– Stokes equations was developed in [14], and for systems of the form (14) with a very general multiplicative noise. The only assumption on the coefficients f, g is that they are continuous and fulfil a mild growth condition. For simplicity the Wiener process w is taken to be a 1-dimensional Brownian motion but this is not an essential restriction.

The idea is to consider a set X of solutions to (14) on a filtered probability space Ω = (Ω, F, (F_{t})_{t≥0} ,P ) and consider a semigroup (S_{t})_{t≥0} acting on X in the same way as Sell’s semigroup on deterministic solutions. For this to make sense however some additional structure must be assumed for the underlying probability space Ω, because a simple time shift of an adapted process will not be adapted. Thus we now assume that the space Ω is equipped with a family of measure preserving maps θ_{t} : Ω → Ω for t ≥ 0 with the following properties:

(θ1) θ_{0} = identity and θ_{t} ∘ θ_{s} = θ_{t+s};

(θ2) θ_{t}F_{s} = F_{t+s} for all s, t ≥ 0;

(θ3) w(t + s, θ_{t}ω) − w(t, θ_{t}ω) = w(s, ω) for all s ≥ 0.

Note that the property (θ3) tells us that for a fixed t the increments of the process w(t + s, θ_{t}ω) are the same as those of the process w(s, ω). Thus θ_{t} can be thought of as a shift of the noise to the right by t.

The family (θ_{t}) allows the following definition of a semiflow S_{r} of stochastic processes.

Definition 4.4 (Semiflow of Processes) (a) Suppose that u = u(t, ω) is an adapted stochastic process defined for t > 0. Then for any r ≥ 0 the adapted process v = S_{r}u is defined by

v(t, ω) = u(r + t, θ_{r}ω).

(b) By the semiflow (S_{t})_{t≥0} on a filtered space Ω we mean that there is a measure preserving family (θ_{t})_{t≥0} obeying (θ1) − (θ3) from which S_{t} is defined as above.

Note that for any process u the process S_{t}u lives on the same space as u. For any theory of attractors involving the semiflow S_{t} and a set of solutions X of solutions to (14) on a filtered probability space Ω there are a number of difficulties to be addressed in the 3D situation, stemming largely from the possible non-uniqueness of solutions. First, in most standard existence proofs for (14) the space Ω that carries the solution has to be constructed carefully, and both it and the Wiener process depend on the solution itself. Here however we need a single space that carries solutions for all initial conditions (including those that are random); moreover, we need not just one solution for each initial condition but a sufficiently rich supply of solutions. This in itself requires a rather large probability space. Moreover, the space Ω must be rich enough to carry all processes that are in some sense infinite limits under the action of S_{t}. In the paper [14] the underlying space was an adapted Loeb space that was shown to meet all these requirements. (At a deeper level this is a consequence of the fact that Loeb spaces are universal – which can be described informally as saying that anything that happens anywhere happens on a Loeb space).

The need for a very large underlying probability space Ω raises a second set of problems. The resulting set of solutions X on Ω is also large and consequently it is unrealistic to expect an attractor 𝔸 that is compact; in fact various noncompactness results were proved in [14]. However, the attractor 𝔸 in [14] has a compact set of laws, but this is somewhat unsatisfactory. Similarly for the attracting property: in [14] this was established only for an ad hoc family of open neighbourhoods of 𝔸.

In the remaining sections of this paper we to show that the attractor 𝔸 in [14] has stronger compactness and attraction properties that are described naturally using the framework of neometric spaces, in which we develop the notion of a neoattractor – which is neocompact and has the attraction property for all neo-open neighbourhoods of 𝔸. This is the subject of the next section.

5 Neometric spaces and Neoattractors

Before developing the theory of neoattractors, later to be applied to the stochastic Navier–Stokes equations, we recall the basic definitions involved in the theory of neometric spaces, as developed in the papers [17] and [18] in the setting of a nonstandard universe. For the sake of completeness we provide in Appendix 2 a brief summary of the theory of neometric spaces.

5.1 Neometric spaces

Roughly speaking, a neometric space M is a metric space with extra structure given by its parent, which is a ^{∗}metric space M. We start by defining the standard part of an element of the parent space.

Given a ^{∗}metric space (M, δ) (which is internal), the standard part ^{∘}x of an element x ∈ M is defined by

∘x = {y ∈ M : δ(x, y) ≈ 0}.

Note that this is a generalisation of the notion of the standard part of a nearstandard element of a standard metric space. For a subset C ⊆ M, the standard part of C is the set ^{∘}C = {^{∘}x : x ∈ C}. For each point z ∈ M, the nonstandard hull around z is the set

H(M, z) = {^{∘}x : δ(x, z) is finite}

with the metric ρ(^{∘}x,^{∘}y) = st(δ(x, y)). Each nonstandard hull is a complete metric space.

Definition 5.1 By a neometric space we will mean a closed subspace M of the nonstandard hull of a ^{∗}metric space M around some point z. M is called the parent of M.

Note that this notion is relative to a particular nonstandard universe. Each neometric space is a complete metric space. Hereafter, M, N , … will stand for neometric spaces, and M, N, … will be their parents.

The monad of a subset A ⊆ M is the set

monad(A) = {x ∈ M : ^{∘}x ∈ A}.

Note that this is an extension of the idea of the monad of a point in a standard metric space. For any set A ⊆ M, A = ^{∘}(monad(A)). The monad operation commutes with arbitrary unions, intersections, and complements.

A point x ∈ M is near-standard in M, in symbols x ∈ ns(M), if x belongs to the monad of M, that is, ^{∘}x ∈ M. (Thus ns(M) is an alternative notation for monad(M).) Note that the monad of M is contained in the nonstandard hull around a point.

We adopt the usual convention of identifying a point x of a standard metric space M with the standard part of its internal counterpart, ^{∘}(^{∗}x). With this convention, each standard complete metric space M in the original superstructure is a closed subset of a nonstandard hull of ^{∗}M, and thus is a neometric space. This is important for applications; for example, the product of a neometric space with the real line is a neometric space.

Definition 5.2 By a Π_{1}^{0} set we mean the intersection of a countable collection of internal subsets of M.

A set A ⊆ M is said to be countably determined if there is a countable family of internal sets B_{1}, B_{2}, … such that A is a finite or infinite Boolean combination of B_{1}, B_{2}, . . ..

Thus any Π_{1}^{0} set is countably determined. It is clear that the family of countably determined sets in M is closed under complements, and under finite and countable unions and intersections.

The neocompact subsets of a neometric space are analogues of compact sets.

Definition 5.3 Let M be a neometric space.

A set C ⊆ M is neocompact (in M) if C is the standard part of some Π_{1}^{0} set A ⊆ ns(M).

Note that if N ⊆ M, then a set C ⊆ N is neocompact in N if and only if it is neocompact in M. It is easily seen that finite unions of neocompact sets, and finite Cartesian products of neocompact sets, are neocompact; ℵ_{1}-saturation shows that countable intersections of neocompact sets are also neocompact. In a standard separable metric space, the neocompact sets are the same as the compact sets.

We now introduce the neoclosed sets.

Definition 5.4 Let X ⊆ M. A set C ⊆ X is neoclosed in X if C ∩ D is neocompact for every neocompact set D ⊆ X in M. The complement of a neoclosed set in X is called neoopen in X.

Note that if X ⊆ M and C is neoclosed in M, then C ∩ X is neoclosed in X. However, C ∩ X is not necessarily neoclosed in M. If X is itself neoclosed in M then a set D ⊆ X is neoclosed in X if and only if it is neoclosed in M.

Whereas neocompact is weaker than compact (Proposition 9.2(a)), neoclosed is stronger than closed.

Examples 5.5 (a) ([17], Proposition 4.14.) For each neocompact set C and positive real number r, the set C^{≤r} = {x : ρ(x, C) ≤ r} is neoclosed in M.

(b) ([17], Lemma 4.7) For each x ∈ M and r ∈ (0, ∞), the open ball {y : ρ(x, y) < r} is neoopen in M.

The analogue of the continuous functions is as follows.

Definition 5.6 A function f : M → N is said to be neocontinuous if for every neocompact set C in M, the restriction f ↾ C = {(x, f(x)) : x ∈ C} of f to C is neocompact in M × N.

Neocontinuous is a stronger notion than continuous, but many naturally occurring continuous functions are actually neocontinuous.

Examples 5.7 (a) ([18], p. 145.) The distance function ρ is neocontinuous from M × M to ℝ.

(b) The projection function is neocontinuous from M × N to M.

5.2 Neo-attractors

Let M be a neometric space and let X ⊆ M. By a neo-semiflow on X we mean a function S : [0, ∞) × M → M such that S_{t}(·) is neocontinuous for each particular t ∈ [0, ∞), and for all x ∈ X and s, t ∈ [0, ∞), we have S_{t}(x) ∈ X, S_{0}(x) = x, and S_{s+t}(x) = S_{s}(S_{t}(x)).

We assume throughout this section that S is a neo-semiflow on X. Since S_{t} is a neocontinuous function for each t ∈ [0, ∞), it follows from Proposition 9.7 that S_{t}C is neocompact whenever C is neocompact.

In the example in Section 7 and and thereafter, S will actually be a neocontinuous semiflow on X, that is, a neo-semiflow which is a neocontinuous function from [0, ∞) × M into M. However, in this section it is enough for S to be a neo-semiflow.

Definition 5.8 A neo-attractor for S on X in M is a set 𝔸 ⊆ X such that: (a) 𝔸 is neocompact in M.

(b) S_{t}𝔸 = 𝔸 for all t ∈ [0, ∞).

(c) For each bounded set B ⊆ X and neoopen set O ⊇ 𝔸 in X, S_{t}B ⊆ O eventually (that is, there exists r ∈ [0, ∞) such that S_{t}B ⊆ O for all t ∈ [r, ∞).)

It is clear that if X ≠ Ø then every neo-attractor is nonempty, because condition (c) fails when A = O = Ø.

Proposition 5.9 There is at most one neo-attractor for S on X.

Proof. Suppose 𝔸, 𝔸′ are neo-attractors for S on X in M, and let x ∈ 𝔸 \ 𝔸′. Since 𝔸 is neocompact, 𝔸 is bounded. The set O = X \ {x} is a superset of 𝔸′ which is neoopen in X, so S_{t}𝔸 ⊆ O for some t ∈ [0, ∞). But x ∈ 𝔸 = S_{t}𝔸, which contradicts x ∉ O.

Note that the above uniqueness result holds even if condition (a) of Definition 5.8 is weakened to the condition that 𝔸 is bounded.

Definition 5.10 An absorbing set for S on X is a set E ⊆ X such that S_{t}B ⊆ E eventually for each bounded set B ⊆ X.

Lemma 5.11 Let B and E be bounded absorbing sets for S on X. If lim_{k→∞} t_{k} = ∞, then

⋂_{k∈ℕ}S_{t_{k}}E = ⋂_{t≥0} S_{t}E = ⋂_{t≥0} S_{t}B.

Proof. It suffices to prove that the first set is contained in the third. Since B is absorbing and E is bounded, there exists b ≥ 0 such that S_{u}E ⊆ B for all u ≥ b. Let t ∈ [0, ∞). Take k ∈ ℕ such that t_{k} − t ≥ b. Then S_{t_{k}}E = S_{t}S_{t_{k}−t}E ⊆ S_{t}B, so ⋂_{k∈ℕ} S_{t_{k}}E ⊆ ⋂_{t≥0} S_{t}B.

Lemma 5.12 Let E be a bounded absorbing set for S on X, and let 𝔸 = ⋂_{t≥0} S_{t}E. Then S_{t}𝔸 ⊆ 𝔸 for each t ∈ [0, ∞).

Proof. Take b ≥ 0 so that S_{u}E ⊆ E for all u ≥ b. For any t, u ∈ [0, ∞),

𝔸 ⊆ S_{u+b}E, S_{t+b}E ⊆ E,

so

S_{t}𝔸 ⊆ S_{t}S_{u+b}E = S_{u}S_{t+b}E ⊆ S_{u}E.

Since this holds for all u ∈ [0, ∞), it follows that S_{t}𝔸 ⊆ 𝔸.

The next theorem is a neometric generalization of Theorem 2.4. It shows that 𝔸 is a neo-attractor in the case that the absorbing set E is neocompact.

Theorem 5.13 Let S be a neo-semiflow on X with a neocompact absorbing set E. Then the set 𝔸 = ⋂_{u∈[0,∞)} S_{u}E is a neo-attractor for S on X.

Proof. We have 𝔸 ⊆ E ⊆ X. Since E is neocompact, S_{t}E is neocompact for each t ≥ 0. Then by Lemma 5.11, 𝔸 is a countable intersection of neocompact sets, and thus 𝔸 is neocompact.

Take b ≥ 0 such that S_{u}E ⊆ E for all u ≥ b. Now let O be a neoopen set in X such that 𝔸 ⊆ O. To prove that S_{t}B ⊆ O eventually for each bounded set B ⊆ X it suffices to show that S_{t}E ⊆ O eventually. Suppose not. Then there is a sequence t_{k}, k ∈ ℕ such that for each k, S_{t_{k}}E \ O ≠ ∅ and t_{k+1} ≥ t_{k} +b. Each set S_{t_{k}}E is neocompact, and since X \O is neoclosed in X, S_{t_{k}}E \ O is neocompact. Moreover, t = t_{k+1} − t_{k} ≥ b, so S_{t}E ⊆ E and hence S_{t_{k+1}}E = S_{t_{k}}S_{t}E ⊆ S_{t_{k}}E. Therefore {S_{t_{k}}E \ O} is a decreasing chain. Then by Lemma 5.11 and countable compactness, 𝔸 \ O = ⋂_{k∈ℕ}(S_{t_{k}}E \ O) ≠ ∅, contradicting 𝔸 ⊆ O.

Let t ∈ [0, ∞). We have S_{t}𝔸 ⊆ 𝔸 by Lemma 5.12. It remains to prove that 𝔸 ⊆ S_{t}𝔸. Choose r ∈ (b, ∞). For each n ∈ ℕ let C_{n} = S_{n·r}E. Then C_{n} is neocompact. Since S_{r}E ⊆ E, C_{n+1} = S_{n·r}S_{r}E ⊆ S_{n·r}E = C_{n}, so {C_{n}} is a decreasing chain. Since r > 0, n · r → ∞ as n → ∞. By Lemma 5.11, 𝔸 = ⋂_{n∈ℕ} C_{n}. By Proposition 9.9,

S_{t}𝔸 = S_{t} ( ⋂_{n∈ℕ}C_{n}) = ⋂_{n∈ℕ}(S_{t}C_{n}).

Moreover, for each n ∈ ℕ, 𝔸 ⊆ S_{n·r+t}E = S_{t}S_{n·r}E = S_{t}C_{n}. Thus 𝔸 ⊆ S_{t}𝔸.

Definition 5.14 Let S be a neo-semiflow on X. By a neo-subflow of S we mean the restriction S ↾ Z of S to a neoclosed set Z in X such that S_{t}Z ⊆ Z for each t ≥ 0.

Corollary 5.15 Assume the hypotheses of Theorem 5.13. Then every neosubflow of S has a neo-attractor.

The next theorem gives a weaker sufficient condition for S to have a neoattractor on X, which is analogous to the necessary and sufficient condition for the deterministic case in Theorem 2.5. We first prove a lemma.

Lemma 5.16 Let E be a bounded absorbing set for S on X, and let 𝔸 = ⋂_{t≥0} S_{t}E. Suppose that E is neoclosed in X and, whenever lim_{n→∞} t_{n} = ∞, B = {x_{1}, x_{2}, …} ⊆ E, and x_{n} ∈ S_{t_{n}}E for each n ∈ ℕ, there is a neocompact set C such that B ⊆ C ⊆ X.

Then whenever lim_{n→∞} t_{n} = ∞, B = {x_{1}, x_{2}, …}, and x_{n} ∈ S_{t_{n}}E for each n ∈ ℕ, there is a neocompact set D such that B ⊆ D ⊆ 𝔸 ∪ B.

Proof. Suppose lim_{n→∞} t_{n} = ∞, B = {x_{1}, x_{2}, …}, and x_{n} ∈ S_{t_{n}}E for each n ∈ ℕ. Take b such that S_{u}E ⊆ E for all u ≥ b. For each k ∈ ℕ, let B_{k} = {x_{n} : t_{n} ≥ k + b}. Then B \ B_{k} is finite. Now fix k for the moment. For each x_{n} ∈ B_{k}, x_{n} ∈ S_{t_{n}}E = S_{k}S_{t_{n}−k}E so we may choose y_{n} ∈ S_{t_{n}−k}E ⊆ E such that x_{n} = S_{k}(y_{n}). Write C_{k} = {y_{n} : x_{n} ∈ B_{k}} ⊆ E, so that B_{k} = S_{k}C_{k} ⊆ S_{k}E. By hypothesis there is a neocompact set D_{k} such that C_{k} ⊆ D_{k} ⊆ X. Since E is neoclosed in X, we may take D_{k} ⊆ E (or else replace D_{k} by D_{k} ∩ E which is neocompact). Then B_{k} ⊆ S_{k}D_{k} ⊆ S_{k}E. S_{k}D_{k} is neocompact by Proposition 9.7, and since B \ B_{k} is finite, (S_{k}D_{k}) ∪ B is neocompact.

Carrying out this for all k gives the set D = ⋂_{k∈ℕ}((S_{k}D_{k}) ∪ B) which is neocompact. By Lemma 5.11, 𝔸 = ⋂_{k∈ℕ} S_{k}E. Therefore B ⊆ D ⊆ 𝔸 ∪ B.

Theorem 5.17 Let S be a neo-semiflow on X. Assume that:

(a) monad(X) is countably determined,

(b) For each u ≥ 0 there is an internal function T_{u} : M → M such that ∘(T_{u}(x)) = S_{u}(^{∘}x) for all x ∈ monad(X), and

(c) There is a bounded absorbing set E for S such that whenever lim_{n→∞} t_{n} = ∞, B = {x_{1}, x_{2}, …} ⊆ E, and x_{n} ∈ S_{t_{n}}E for each n ∈ ℕ, there is a neocompact set C ⊆ X such that B ⊆ C.

Then the set 𝔸 = ⋂_{u∈[0,∞)} S_{u}E is a neo-attractor for S on X.

Moreover, any neo-subflow S ↾ Z of S such that monad(Z) is countably determined has a neo-attractor.

Proof. It is easily seen that if condition (c) holds for E, then it also holds for D ∩ X where D is any closed ball containing E. We may therefore assume that E = D∩X for some closed ball D in M. Then monad(E) = monad(X)∩D for some Π_{1}^{0} set D, and therefore monad(E) is countably determined. Moreover, D is neoclosed in M, so E is neoclosed in X.

We will use Theorem 9.3 to show that 𝔸 is neocompact. Let B = {x_{n} : n ∈ ℕ} ⊆ 𝔸. Then x_{n} ∈ S_{n}E for each n ∈ ℕ. By hypothesis (c) and Lemma 5.16, there is a neocompact set C such that B ⊆ C ⊆ 𝔸 ∪ B = 𝔸.

We now show that monad(𝔸) is countably determined. By Lemma 5.11, we have 𝔸 = ⋂_{n∈ℕ} S_{n}E. It follows from (b) that

monad(S_{n}E) = ⋂_{m∈ℕ}((T_{n} monad(E))^{1/m}).

Then using Lemma 9.1, we see that monad(S_{n}E) is countably determined for each n ∈ ℕ, and therefore

monad(𝔸) = monad( ⋂_{n∈ℕ}S_{n}E) = ⋂_{n∈ℕ}monad(S_{n}E)

is countably determined. Then by Theorem 9.3, 𝔸 is neocompact.

We next prove that 𝔸 = S_{t}𝔸 for each t ∈ [0, ∞). We have S_{t}𝔸 ⊆ 𝔸 by Lemma 5.12. Let x ∈ 𝔸. Then for each n ∈ ℕ, x ∈ S_{t+n}E = S_{t}S_{n}E, so there exists y_{n} ∈ S_{n}E such that x = S_{t}(y_{n}). For each k ∈ ℕ, let B_{k} = {y_{n} : k ≤ n ∈ ℕ}. By (c) and Lemma 5.16, there is a neocompact set C_{k} such that B_{k} ⊆ C_{k} ⊆ 𝔸 ∪ B_{k}. Then x ∈ S_{t}C_{k}. We may choose the sets C_{k} to be a decreasing chain. Then the set C = ⋂_{k∈ℕ} C_{k} is neocompact, and C ⊆ 𝔸. By Proposition 9.9, S_{t}C = ⋂_{k∈ℕ} S_{t}C_{k}. Then x ∈ S_{t}C ⊆ S_{t}𝔸. It follows that 𝔸 = S_{t}𝔸.

Now let O be a neoopen set in X such that 𝔸 ⊆ O. To prove that S_{t}D ⊆ O eventually for each bounded set D ⊆ X it suffices to show that S_{t}E ⊆ O eventually. Suppose not. Then there is a sequence {t_{k}} such that lim_{k→∞} t_{k} = ∞ and S_{t_{k}}E \ O ≠ ∅ for each k ∈ ℕ. Choose x_{k} ∈ S_{t_{k}}E \ O. For each k ∈ ℕ let B_{k} = {x_{n} : n ≥ k}. By (c) and Lemma 5.16, there is a neocompact set C_{k} such that B_{k} ⊆ C_{k} ⊆ 𝔸 ∪ B_{k}. Since O is neoopen in X and B_{k} ∩ O = ∅, we may take C_{k} disjoint from O. We may also take the sets C_{k} to be a decreasing chain. By countable compactness, the intersection C = ⋂_{k∈ℕ} C_{k} is nonempty. But C ⊆ 𝔸 and C ∩ O = ∅, contradicting 𝔸 ⊆ O. This shows that 𝔸 is a neo-attractor for S on X.

Finally, the hypotheses (a)–(c) also hold for any subflow S ↾ Z such that monad(Z) is countably determined.

Here is a sufficient condition for 𝔸 being a neo-attractor which will be used in the next section.

Corollary 5.18 Let S be a neo-semiflow on X. Assume conditions (a) and (b) of Theorem 5.17, and

(c’) There is a bounded absorbing set E for S and a decreasing chain of internal sets C_{n}, n ∈ ℕ such that ⋂_{n} C_{n} ⊆ monad(X), and for each n ∈ ℕ, S_{t}(E) ⊆ ^{∘}C_{n} eventually.

Then Theorem 5.17 (c) holds for E, so 𝔸 = ⋂_{u∈[0,∞)} S_{u}E is a neo-attractor for S on X.

Proof. Suppose lim_{n→∞} t_{n} = ∞, B = {x_{1}, x_{2}, …} ⊆ E, and x_{n} ∈ S_{t_{n}}E for each n ∈ ℕ. We may assume without loss of generality that C_{1} = M. For each k let n_{k} be the greatest n ≤ k such that S_{t_{k}}(E) ⊆ ^{∘}C_{n}. Then lim_{k→∞} n_{k} = ∞. Take x_{k} ∈ C_{n_{k}} such that ^{∘}x_{k} = x_{k}. Let B = {x_{k} : k ∈ ℕ}. Then B \ C_{n} is finite, so D_{n} = C_{n} ∪ B is a decreasing chain of internal sets with

B ⊆ ⋂_{n∈ℕ}D_{n} ⊆ monad(X).

Then D = ^{∘ ⋂}_{n∈ℕ}D_{n} is neocompact and B ⊆ D ⊆ X.

6 Process attractors for 3D stochastic Navier-Stokes equations

In the paper [14] a set X of solutions to the stochastic Navier–Stokes equations (14) was defined and it was shown that there is a set 𝔸 ⊆X that is an attractor in some sense — but falling short of the requirements of the definition in Part I for deterministic settings (Definition 2.2). In particular 𝔸 is not compact. The main result of the present paper is that 𝔸 is however a neo-attractor – which seems to be the strongest kind of notion likely to be applicable in this framework. Here we outline the results of [14] in preparation for the next section, where we will see how they fit into the framework of neometric spaces.

The equations considered are

du = [−νAu − B(u) + f(u)]dt + g(u)dw_{t}  (15)

in the functional formulation so that

u : [0, ∞) × Ω → H

and for simplicity (though it is not essential) the process w is taken to be a 1-dimensional Wiener process. The only conditions imposed on the timeindependent coefficients f, g are as follows, where c_{0}, d_{1}, d_{2} are positive real constants.

(H1) f : H → H and |f(u)| ≤ c_{0} + d_{1}|u|.

(H2) g : H → H and |g(u)| ≤ c_{0} + d_{2}|u|.

(H3) f and g are continuous.

(H4) 2d_{1} + 3d_{2}^{2} < 2νλ_{1}.

The general theory of stochastic Navier–Stokes equations expounded in [8] shows that the equation (15) can be solved with only the assumptions (H1)– (H3). The additional growth restriction (H4) on f, g is needed here to obtain the attractor.

6.1 The space Ω and the semiflow

The particular space Ω that we use is a filtered Loeb space similar to that used in [8] for the construction of solutions to the stochastic Navier–Stokes equations. Loeb spaces constitute a special class of probability spaces that are very rich – in a sense that can be made precise (see for example [22]). The richness is needed to be able to solve the general stochastic Navier–Stokes equations in dimension 3, and it was needed in [14] to show that the single space Ω Ω has solutions to (15) with the same (prescribed) Wiener process w_{t} for any random initial condition.

For the rest of the paper we fix the following adapted Loeb space. Set Ω = ^{∗}(C_{0}(ℝ)), the internal space of ^{∗}continuous functions ω : ^{∗}ℝ → ^{∗}ℝ with ω(0) = 0, and let Q be the internal ^{∗}Wiener measure on Ω.

Thus the canonical process

W (τ, ω) = ω(τ)

is a two-sided ^{∗}Wiener process under Q. This gives the internal filtered probability space

Ω̄ = (Ω, G, (G_{τ})_{τ∈^{∗}ℝ}, Q),

where G_{τ} = ^{∗}σ({W(τ′) : τ′ ≤ τ}) and G = ⋁_{τ∈^{∗}ℝ} G_{τ}.

A family of internal measure preserving maps Θ_{τ} : Ω → Ω is defined for τ ∈ ^{∗}ℝ by

(Θ_{τ}(ω))(σ) = ω(σ − τ) − ω(−τ).

That is, Θ_{τ} is a shift of the path ω to the right by τ and then adjusted to be 0 at 0.

Now let P = Q_{L} be the Loeb measure obtained from Q with the corresponding Loeb σ-algebra F = L(G), giving the Loeb probability space (Ω, L(G), Q_{L}) = (Ω, F, P), and denote the P -null sets by N.

Definition 6.1

(a) The filtered probability space Ω is

Ω = (Ω, F, (F_{t})_{t≥0}, P),

where the right continuous filtration (F_{t})_{t≥0} is defined by

F_{t} = ⋂_{t<^{∘}τ} σ(G_{τ}) ∨ N .

(b) The Wiener process w(t, ω) on Ω is defined by

w(t, ω) = ^{∘}W (t, ω).  (16)

(c) The family of measure preserving transformations (θ_{t})_{t≥0} is given by

θ_{t} = Θ_{t}.

That is, the restriction of the family (Θ_{τ}) to non-negative standard times.

It is well known that (16) defines an almost surely continuous Wiener process on Ω. It is clear that the family θ_{t} satisfies conditions (θ1), (θ2), (θ3).

With the space Ω and the family (θ_{t})_{t≥0} now fixed, the semiflow of processes S_{t} defined by Definition 4.4 is also fixed for the rest of the paper.

6.2 Solutions to the stochastic Navier–Stokes equations

We define below a particular class X of weak solutions to the stochastic Navier– Stokes equations (15). Each element u of X is an adapted stochastic process (with u(t, ω) ∈ H for all t, ω). The properties required for membership of X are among those that can be deduced heuristically from (15) using elementary stochastic calculus. The definition incorporates some truncation functions ψ_{n}, φ_{n}, defined from the following real C^{2} function ψ : [0, ∞) → [0, 1] which is designed to be concave on [0, 1] and constant (with value 1) on [1, ∞).

Definition 6.2 (Truncation functions) (a)

ψ(x) = \{ \begin{matrix}(x − 1)^{3} + 1 & if 0 ≤ x ≤ 1 \\ 1 & if x ≥ 1 \end{matrix}

(b) For each n ∈ ℕ,

ψ_{n}(x) = ψ(x/n^{2}).

(c) For each n ∈ ℕ a function φ_{n}(u) is defined for u in any Hilbert space (finite or infinite dimensional) by

φ_{n}(u) = u^{2}ψ_{n}(u^{2}),

where we write u^{2} to mean |u|^{2} to ease the notation.

The way the functions φ_{n} figure in the definition of X is to guarantee a kind of uniform integrability of solutions in the time variable, which crops up in the paper [14] in the guise of S-integrability from Loeb integration theory: an internal random vector u(ω), |u|^{2} is S-integrable if and only if ^{∘}𝔼(φ_{n}(u)) → 0 as n → ∞.

Now for the definition of the class of solutions X.

Definition 6.3 (a) Given positive real constants k_{1}, k_{2}, k_{3}, α, β, denote by X the class of adapted stochastic processes u : (0, ∞) × Ω → H with the following properties.

(X1) For a.a. ω the path u(·, ω) belongs to the following spaces:

L_{loc}^{∞}(0, ∞; H) ∩ L_{loc}^{2}[0, ∞; H) ∩ L_{loc}^{2}(0, ∞; V) ∩ C(0, ∞; H_{weak}).

(X2) For all t_{1} ≥ t_{0} > 0

u(t_{1}) = u(t_{0}) + ∫_{t_{0}}^{t_{1}}[−νAu(t) − B(u(t)) + f(u(t))]dt + ∫_{t_{0}}^{t_{1}} g(u(t))dw_{t}.

(X3) For a.a. t_{0} > 0 and all t_{1} ≥ t_{0},

𝔼(|u(t_{1})|^{2}) ≤ 𝔼(|u(t_{0})|^{2}) exp(−k_{1}(t_{1} − t_{0})) + k_{2}.  (17)

(X4) For a.a. t_{0} > 0 and all t_{1} ≥ t_{0},

𝔼 (sup_{t_{0}≤s≤t_{1}}|u(s)|^{2} + ∫_{t_{0}}^{t_{1}} ∥u(s)∥^{2}ds) ≤ α𝔼(|u(t_{0})|^{2}) + β(t_{1} − t_{0}). (18)

(X5) For a.a. t_{0} > 0 and all t_{1} ≥ t_{0} and n ∈ ℕ,

𝔼(φ_{n}(u(t_{1}))) ≤ 𝔼(φ_{n}(u(t_{0})) exp(−k_{3}(t_{1} − t_{0}))) + n^{− \frac{1}{2}} (α𝔼(|u(t_{0})|^{2}) + β). (19)

(X6) 𝔼 ∫_{0}^{1} |u(t)|^{2}dt < ∞.

(b) Denote by X_{k} the set of u ∈ X with

(X6_{k}) 𝔼 ∫_{0}^{1} |u(t)|^{2}dt ≤ k.

Remarks

1. The class X depends on the constants k_{1}, k_{2}, k_{3}, α, β; there is a natural choice of these – see Theorem 6.10 below.

2. The sets X_{k} increase with k.

3. The above conditions tell us nothing about u(t, ω) at t = 0 and there may be a singularity there. In this sense the class X is a class of generalized weak solutions to the stochastic Navier–Stokes equations (cf. [26], p.12).

4. The meaning of “loc” in the path properties (X1) is as follows: L_{loc}^{p}(0, ∞) means L^{p}[1/n, n] for all n, whereas L_{loc}^{p}[0, ∞) means L^{p}[0, n] for all n.

5. The conditions (X5) follow naturally from the Foias equation for the stochastic Navier-Stokes equations (see [5]), which may be deduced heuristically from the equation (15). The choice of the functions φ_{n} makes (X5) a uniform integrability condition for |u(t)|^{2} on any [t_{0}, ∞).

6. The semiflow S_{t} maps X into X.

The paths of any solution u ∈ X lie in the space M = L^{2}(0, ∞; H; μ) where μ(dt) = exp(−t)dt, defined in Section 3.1. Thus

X ⊆ L^{2}(Ω, M).

The following lemma relates the sets X_{k} to bounded sets.

Lemma 6.4 ([14] Lemma 4.2) If B ⊆ X, then B is bounded in L^{2}(Ω, M) if and only if B ⊆ X_{k} for some k ∈ ℕ.

The notion of an attractor that was discussed in [14] is as follows. Here we call it a process attractor to avoid confusion with the notion of Part I (Definition 2.2). It involves laws of processes, and needs the following definitions concerning the laws of solutions viewed as probability distributions on the space of paths M.

Definition 6.5 Let u(t, ω) be a process with paths in M.

(a) law(u) is the probability law on M induced by u; i.e.

law(u)(Z) = ℙ(u(·, ω) ∈ Z)

for Borel Z ⊆ M.

(b) law_{w}(u) = law(u, w), the probability law induced on M × C_{0} by the pair of processes (u(t, ω), w(t, ω)), where C_{0} = C_{0}[0, ∞).

For the space of probability laws M_{1}(S) on a separable metric space S a fundamental metric is the Prohorov metric, which we denote by d_{0}; this makes M_{1}(S) separable. Here we are thinking of S = M and S = M × C_{0}.

There is a natural projection mapping π : M_{1}(M × C_{0}) → M_{1}(M) defined by

π(λ)(Z) = λ(Z × C_{0}).

In the current situation the laws on the space M that we are interested are laws of L^{2} random variables, so it is appropriate to define a stronger metric to reflect this.

Definition 6.6 (a) M_{1,2}(M) = {μ ∈ M_{1}(M) : 𝔼_{μ}(|u|^{2}) < ∞}.

(b) the metric d on M_{1,2}(M) is defined by

d(μ_{1}, μ_{2}) = d_{0}(μ_{1}, μ_{2}) + |𝔼_{μ_{1}}(|u|^{2}) − 𝔼_{μ_{2}}(|u|^{2})| .

(c) M_{1,2}(M × C_{0}) = {λ ∈ M_{1}(M × C_{0}) : π(λ) ∈ M_{1,2}(M)}.

(d) the metric d on M_{1,2}(M × C_{0}) is defined by

d(λ_{1}, λ_{2}) = d_{0}(λ_{1}, λ_{2}) + |𝔼_{μ_{1}}(|u|^{2}) − 𝔼_{μ_{2}}(|u|^{2})| ,

where μ_{i} = π(λ_{i}) (i = 1, 2).

The following lemma is easy to check (where ρ is the L^{2} metric on L^{2}(Ω, M)).

Lemma 6.7 (a) The function law maps L^{2}(Ω, M) into M_{1,2}(M) and is continuous with respect to the metrics ρ and d.

(b) The function law_{w} maps L^{2}(Ω, M) into M_{1,2}(M ×C_{0}) and is continuous with respect to the metrics ρ and d.

(c) The mapping π : M_{1}(M × C_{0}) → M_{1}(M) defined by

π(λ)(Z) = λ(Z × C_{0})

is continuous with respect to the metric d.

Using these notions we have:

Definition 6.8 (a) A set of laws A ⊆ law_{w}(X) is a law-attractor for the semiflow S_{t} on X if:

(i) (Invariance) Ŝ_{t}A = A for all t ≥ 0.

(ii) (Attraction) For any open set O ⊃ A and d-bounded set B ⊆ law_{w}(X),

Ŝ_{t}B ⊆ O

eventually (i.e. for some t_{0} = t_{0}(O, B), this holds for all t ≥ t_{0}).

(iii) (Compactness) A is compact in the metric d.

(b) A process attractor for the semiflow S_{t} on X is a set of processes 𝔸 ⊆ X such that:

(i) law_{w}(𝔸) is a law-attractor (in particular law_{w}(𝔸) is compact in the metric d, and so 𝔸 is bounded).

(ii) (Invariance) S_{t}𝔸 = 𝔸 for all t ≥ 0.

(iii) (Attraction) For any bounded set B ⊆ X and compact set K ⊆ L^{2}(Ω, M),

\underline{lim}_{t→∞}ρ(S_{t}B, K) ≥ ρ(A, K).

(iv) 𝔸 is closed in the space L^{2}(Ω, M).

A smaller class X̂ ⊆ X of solutions defined at 0 is as follows.

Definition 6.9 Denote by X̂ the class of stochastic processes u : [0, ∞)×Ω → H with u ∈ X (that is, the restriction of u to (0, ∞) lies in X) with the following additional properties:

(X̂1) For a.a. ω, the path u(·, ω) is in

L_{loc}^{∞}[0, ∞; H) ∩ L_{loc}^{2}[0, ∞; H) ∩ L_{loc}^{2}[0, ∞; V) ∩ C[0, ∞; H_{weak}).

(X̂2) For all t_{1} ≥ t_{0} ≥ 0,

u(t_{1}) = u(t_{0}) + ∫_{t_{0}}^{t_{1}}[−νAu(t) − B(u(t)) + f(u(t))]dt + g(u(t))dw_{t}.

(X̂3) 𝔼(|u(t)|^{2}) is bounded on [0, ∞).

Note that (X̂1) implies (X1), (X̂2) implies (X2), and (X̂3) implies (X6). (In [14] the symbol Y was used for the subspace X̂.)

The main theorem of the paper [14] is:

Theorem 6.10 ([14] Theorems 4.3, 9.12) There are constants k_{1}, k_{2}, k_{3}, α, β (which are given explicitly in terms of the parameters c_{0,}d_{1}, d_{2}, λ_{1} and ν of the system) such that

(a) for every L^{2} F_{0}-measurable initial condition there is a solution u ∈ X̂;

(b) there is a process attractor 𝔸 for the semiflow S_{t} on X and 𝔸 ⊂ X̂.

7 Neo-attractors for 3D stochastic Navier-Stokes equations

In order to show that the set 𝔸 given by Theorem 6.10 is in fact a neo-attractor we will prove that the conditions of Corollary 5.18 hold for the set X, the semiflow (S_{t})_{t≥0} and an absorbing set E ⊆ X in the context of L^{2}(Ω, M) as a neometric space. For this it is necessary to give some properties of the neometric spaces L^{p}(Ω, M), and then to describe a little more of the detail of the paper [14].

7.1 The neometric spaces L^{p}(Ω, M)

The definitions we review here are applicable to any complete separable metric space (M, d), although we will only need them for the specific space M = L^{2}(0, ∞; H; μ) that figures in previous section (and defined in Section 3.1). Among the most important examples of neometric spaces are the spaces L^{p}(Ω, M), where p ∈ {0} ∪ [1, ∞). The parent of the space M = L^{p}(Ω, M) is

M =(SL^{0}(Ω, M), ρ̄_{p})

where SL^{0}(Ω, M) is the set of all ^{∗}measurable functions x : Ω → ^{∗}M,

ρ̄_{0}(x, y) = ^{∗} inf{ε : Q[^{∗}d(x(ω), y(ω)) ≥ ε] ≤ ε},

and when p ≥ 1,

ρ̄_{p}(x, y) = [∫ (^{∗}d(x(ω), y(ω)))^{p} dQ]^{1/p} .

It is convenient that for all p, the parent spaces are ^{∗}metric spaces on the same set SL^{0}(Ω, M), even though the ^{∗}metrics ρ̄_{p} are different. We have

ns(M) = ns^{p}(Ω, M)
= {x ∈ ns^{0}(Ω, M) : (^{∗}d(x(ω), z))^{p} is S-integrable for all z ∈ M}

where

ns^{0}(Ω, M) = {x ∈ SL^{0}(Ω, M) : x(ω) ∈ ns(M) for a.a. ω}.

In the following proposition we list some facts we need about these neometric spaces.

Proposition 7.1 Let M, N be complete separable metric spaces and let M_{1}(M) be the space of Borel probability measures on M with the Prohorov metric.

(a) ([18], Propositions 5.7 and 6.10.) For each p ∈ {0} ∪ [, ∞), the law function is neocontinuous from L^{p}(Ω, M) to M_{1}(M).

(b) ([18], Theorem 5.14.) For each compact set C in M_{1}(M), the set

{x ∈ L^{0}(Ω, M) : law(x) ∈ C}

is neocompact in L^{0}(Ω, M).

(c) ([18], Theorem 6.7.) If p ∈ [1, ∞), a set C ⊆ L^{p}(Ω, M) is neocompact in L^{p}(Ω, M) if and only if it is neocompact in L^{0}(Ω, M) and uniformly p-integrable.

(d) ([17], Proposition 9.4) For each p ∈ {0} ∪ [1, ∞), the set of all adapted stochastic processes is neoclosed in L^{p}(Ω, M).

(e) ([17], Lemma 5.19.) For every continuous function f : M → N, the function x(·) → f(x(·)) is neocontinuous from L^{0}(Ω, M) to L^{0}(Ω, N).

From now on all discussion is in the context of the neometric space M = L^{2}(Ω, M) where M = L^{2}(0, ∞; H; μ) as in Section 6.

We now show that S = (S_{t})_{t≥0} is a neo-semiflow on X ⊂ M = L^{2}(Ω, M). In fact, it is a neocontinuous semiflow on X as well as on the larger neometric space M =L^{2}(Ω, M). To do this we define an internal semiflow that represents S. The semiflow S_{t} on X extends naturally to give

S : [0, ∞) × L^{2}(Ω, M) → L^{2}(Ω, M)

which has a natural internal internal counterpart

T : ^{∗}[0, ∞) × SL^{0}(Ω, M) → SL^{0}(Ω, M)

given by

(T_{τ}u)(σ, ω) = u(σ + τ, Θ_{τ}ω)

The next proposition is routine, where we define

NS = ns^{2}(Ω, M) = ns(M)

with M = L^{2}(Ω, M).

Proposition 7.2 For finite τ > 0 and for u ∈ NS

∘(T_{τ}u) = S_{^{∘}τ}^{∘}u.  (20)

Thus S is a neocontinuous semiflow on M and on X.

Proof. Neocontinuity follows from Proposition 9.8

7.2 Internal approximate solutions

In [14] the solutions u ∈ X are represented by internal approximate solutions living in the hyperfinite dimensional space H_{N} ⊂ ^{∗}H. These are carried on the internal filtered probability space Ω̄ = (Ω, G, (G_{τ})_{τ≥0}, Q). The following extracts from [14] the key properties of approximate solutions.

Definition 7.3 (a) For each k ∈ ℕ and n ∈ ^{∗}ℕ the internal set X_{k,n} is a set of ^{∗}-adapted (with respect to (G_{τ})_{τ≥0}) processes

u : ^{∗}[0, ∞) × Ω → H_{N}

such that

(X1) u_{τ}(ω) has paths ^{∗}a.s. in ^{∗}M and u ∈ ^{∗}L^{2}(Ω,^{∗}M) i.e.

𝔼 (∫_{0}^{^{∗}∞} |u_{τ}(ω)|^{2} exp(−τ)dτ) < ^{∗}∞.

(X2_{n}) u is a \frac{1}{n}-approximate solution (that is, u has properties that approximate the conditions (X2)-(X5) of Definition 6.3 to order \frac{1}{n})

(X3_{k,n}) u is bounded by k + \frac{1}{n}; that is

𝔼 ∫_{0}^{1} |u_{τ}|^{2}dτ ≤ k + \frac{1}{n}

(b) For each k ∈ ℕ, define

X_{k} = ⋂_{n∈ℕ}X_{k,n}.

(c) The set of internal approximate solutions is

X = ⋃_{k∈ℕ}X_{k}.

The importance of X the lies in the following result from [14].

Theorem 7.4 ([14] Theorem 9.2) (a) For each k ∈ ℕ,

∘(X_{k} ∩ NS) = X_{k},

and hence

(b)

∘(X ∩ NS) = X.

Corollary 7.5 (a) For each k ∈ ℕ, the monad of X_{k} is a Π_{1}^{0} set. The monad of X is countably determined, and in fact is a countable union of Π_{1}^{0} sets.

(b) Each X_{k} is neoclosed;

(c) X is neoclosed

Proof. (a) For each k, n ∈ ℕ, (X_{k,n})^{1/n} is internal, and monad(X_{k}) = ⋂_{n∈ℕ}((X_{k,n})^{1/n}). We also have monad(X) = ⋃_{k∈ℕ} monad(X_{k}).

(b) By Proposition 9.6 using (a).

(c) Let D be neocompact (in the neometric space M = L^{2}(Ω, M) ) ; then D is bounded and so D ∩ X ⊆ X_{k} for some k so that D ∩ X = D ∩ X_{k} is neocompact.

The internal semiflow T_{τ} has the following properties when restricted to X.

Proposition 7.6 ([14] Lemma 8.3) For finite τ > 0:

(a) T_{τ}X ⊆ X.

(b) T_{τ}(X ∩ NS) ⊆ X ∩ NS.

Next note that there is an S-absorbing set for the internal semiflow T_{τ} on X.

Lemma 7.7 ([14] Lemma 8.4) There is k_{0} ∈ ℕ such that X_{k_{0}} is S-absorbing. That is, for each k ∈ ℕ there is an r(k) ∈ ℕ such that

T_{τ}X_{k} ⊆ X_{k_{0}}

for all finite τ ≥ r(k). In fact we may take any k_{0} > k_{2}.

Now make the following definitions.

Definition 7.8 (a) E = X_{k_{0}};

(b) E_{n} = X_{k_{0},n} (so that E = ⋂_{n∈ℕ} E_{n});

(c) E = X_{k_{0}}

Theorem 7.9 E is a bounded neoclosed absorbing set for S on X.

Proof. E is an absorbing set by Proposition 7.2, Theorem 7.4, Proposition 7.6 and Lemma 7.7. E is bounded by Lemma 6.4. E is neoclosed by Proposition 9.6, since E = ^{∘}(E ∩ NS) and E is a Π_{1}^{0} set.

Now let

𝔸 = ⋂_{t≥0} S_{t}E

The main theorem of [14] (see Theorem 6.10 above) shows that 𝔸 is the process attractor for the semiflow S_{t} on X. To see that it is a neoattractor we show that the conditions of Corollary 5.18 are fulfilled.

.

7.3 𝔸 is a neoattractor

Most of the conditions of Corollary 5.18 follow from the previous section. For the remaining conditions of we continue with some details from [14] From the S-absorbing set E define the following set C, called the S-attractor for the internal semiflow T_{τ} on X.

Definition 7.10 Define sets C and C_{n} (for n ∈ ℕ) as follows.

(a) C_{n} = ⋂_{0≤τ≤n} T_{τ}E_{n}.

(b) C = ⋂_{n∈ℕ} C_{n}.

(Note: The sets C_{n} are those denoted Ĉ_{n} in [14].)

Proposition 7.11 (a) The sets C_{n} are internal and decreasing.

(b) C is a Π_{1}^{0} set.

(c) C ⊆ E.

Now comes a key result of [14], which uses the uniform integrability condition on solutions in X that is coded up by the truncation functions.

Theorem 7.12 ([14] Theorem 9.4(d)) C is nearstandard; that is, C ⊆ NS.

Finally we have

Proposition 7.13 For each n ∈ ℕ

S_{t}(E) ⊆ ^{∘}(C_{n})

eventually.

Proof. Since E is S-absorbing there is finite r_{0} with T_{τ}E ⊆ E for finite τ ≥ r_{0}; then for any finite τ ≥ 0 we have

T_{n+r_{0}+τ}E = T_{n}T_{r_{0}+τ}E ⊆ T_{n}E ⊆ C_{n}.

So by Proposition 7.2 and Theorem 7.4, if t ≥ n + r_{0} then

S_{t}E ⊆ ^{∘}(C_{n})

Gathering this together we have

Theorem 7.14 𝔸 is a neo-attractor for the the neocontinuous semiflow (S_{t})_{t≥0} on the set X of solutions to the stochastic Navier–Stokes equations (15).

Proof. We verify the hypotheses of Corollary 5.18. Hypotheses (a) and (b) hold by Corollary 7.5 and Proposition 7.2. E is a bounded absorbing set by Theorem 7.9. The sets C_{n} form a decreasing chain of internal sets by Proposition 7.11, and C ⊆ monad(X) by Proposition 7.11 (c) and Theorem 7.12. Finally, S_{t}(E) ⊆ ^{∘}C_{n} eventually by Proposition 7.13. The result now follows from Corollary 5.18.

We remark that in [14], the set 𝔸 is defined as 𝔸 = ^{∘}C, and it is then proved that 𝔸 is a process attractor and is equal to ⋂_{t≥0} S_{t}E.

Below (Proposition 7.16) we show that this result is a genuine strengthening of the main result of [14]. Close examination of the results of [14] would show that the full strength of the results of section 5.2 is not essential to prove Theorem 7.14. However, the following result does not follow from the results of [14] and gives a much stronger result than proved there.

The following result shows that one still gets a neo-attractor after adding additional inequalities that are preserved under the semiflow. It gives a scheme for proving the existence of neoattractors for systems of stochastic Navier-Stokes equations with additional specialised properties.

Theorem 7.15 Any neo-subflow S ↾ Y of S such that the monad of Y is countably determined has a neo-attractor.

Proof. By Theorem 5.17 and Corollary 5.18.

Finally, we show that the notion of neo-attractor for the stochastic Navier– Stokes equations is stronger than the notion of a process attractor given in [14] and described above, because the compactness and attraction properties required of a process attractor are special cases of the more general properties possessed by a neo-attractor. Thus Theorem 7.14 improves the main result of [14].

Proposition 7.16 Suppose 𝔸 is a neo-attractor for S on X. Then 𝔸 is a process attractor for S on X.

Proof. The function law_{w}(·) : M → M_{1,2} is neocontinuous, so A = law_{w}(𝔸) is neocompact in M_{1,2}. Since M_{1,2} is separable, A is compact in the metric d. The invariance of A follows at once from the invariance of 𝔸. For any open set O in M_{1,2}, O = law_{w}^{−1}(O) ∩ X is neoopen in X. If A ⊆ O, then 𝔸 ⊆ O. Then for each bounded set B ⊆ law_{w}(X), B = X ∩ law_{w}^{−1}(B) is bounded in X, and S_{t}B ⊆ O eventually, so Ŝ_{t}(B) ⊆ O eventually. This shows that A is a law-attractor for S on X.

For each compact set K ⊆ M and each r ∈ (0, ∞), K is neocompact, so K^{≤r} is neoclosed in M and its complement O_{r} = {x ∈ X : ρ(x, K) > r} is neoopen in X. If ρ(𝔸, K) > r, then 𝔸 ⊆ O_{r}, and thus S_{t}B ⊆ O_{r} eventually for each bounded B ⊆ X. It follows that lim inf_{t→∞} ρ(S_{t}B, K) ≥ ρ(𝔸, K). This shows that 𝔸 has the attraction property of a process attractor (Definition 6.8). Finally, 𝔸 is closed by Proposition 9.5.

7.4 Two-sided solutions

In [14] it is proved that the attractor (now shown to be a neo-attractor) is characterised as the restriction to nonnegative times of the set of all bounded two-sided solutions (that is, solutions defined for all time negative and positive). If X̄ is the set of all such solutions to the stochastic Navier-Stokes equations we have:

Theorem 7.17 ([14] Theorem 10.3)

𝔸 = X̄ ↾ [0, ∞)

We conclude by noting that

Theorem 7.18 X̄ is neo-compact.

Proof. It is shown in [14] that X̄ = ^{∘}X̄ where X̄ is the set of internal approximate two-sided solutions; moreover X̄ is a Π_{1}^{0} set and all members of X̄ are nearstandard. We refer the reader to [14] for details.

8 Appendix 1: Nonstandard preliminaries

We work in an ℵ_{1}-saturated nonstandard universe that contains a nonstandard extension ^{∗}J for every mathematical object J involved in our theory. In particular we have ^{∗}ℝ,^{∗}ℕ,^{∗} H,^{∗}M,^{∗}C_{0}(ℝ),^{∗}Wiener measure, etc. The Appendix of [14] provides a brief introduction to those parts of nonstandard analysis that are needed there and the reader is referred to that paper. Below we mention the most important ideas needed for this paper; for further details see [13] for example or any of the standard references [1],[2],[8],[21] or [23].

Here is a brief description;

8.1 The nonstandard universe

We start with a base set 𝔹 which contains all the standard objects involved in our discussion. In particular, 𝔹 should contain the set of reals ℝ and the linear space H. The following superstructure over 𝔹, denoted by 𝕍 = V (𝔹), is an adequate (standard) mathematical universe for our purposes (where P(A) denotes the set of all subsets of a set A):

V_{0}(𝔹) = 𝔹
V_{n+1}(𝔹) = V_{n}(𝔹) ∪ P(V_{n}(𝔹)), n ∈ ℕ

and

𝕍 = V (𝔹) = ⋃_{n∈ℕ}V_{n}(𝔹).

Next, we use the ultrapower construction to build the nonstandard extension ^{∗}𝔹 ⊃ 𝔹, and at the same time construct a mapping ^{∗} : V (𝔹) → V (^{∗}𝔹) which associates to each set A ∈ 𝕍 a nonstandard counterpart ^{∗}A ∈ V (^{∗}𝔹). At level 0, we simply have ^{∗}b = b for each b ∈ 𝔹. At level 1, for each A ⊂ 𝔹 we have A ⊂ ^{∗}A ⊂ ^{∗}𝔹, with ^{∗}A \ A consisting of “ideal” or “nonstandard” elements. For example ^{∗}ℕ \ ℕ consists of infinite (hyper)natural numbers.

In general, for each set A ∈ 𝕍, the mapping ^{∗} maps A injectively into ^{∗}A. So even for mathematical objects^{3} J at higher levels, ^{∗}J can be regarded as an extension of J.

The resulting nonstandard universe is the collection

∗𝕍 = {x : x ∈ ^{∗}A for some A ∈ 𝕍}

consisting of all members of nonstandard counterparts of sets in 𝕍. Although ∗𝕍 ⊂ V (^{∗}𝔹), it is crucial to realize that ^{∗}𝕍 is not the same as V (^{∗}𝔹). Sets in ∗𝕍 are known as internal sets; a set is external if it is not internal.

The key property of the nonstandard universe that makes it tractable is the Transfer Principle which indicates precisely which properties of the superstructure 𝕍 are inherited by ^{∗}𝕍.

3We are taking the approach that every mathematical object is actually a set.

Theorem 8.1 (The Transfer Principle) Suppose that φ is a bounded quantifier statement. Then φ holds in 𝕍 if and only if ^{∗}φ holds in ^{∗}𝕍.

A bounded quantifier statement (bqs) is simply a statement of mathematics that can be written in such a way that all quantifiers range over a prescribed set. That is, we have subclauses such as ∀x ∈ A and ∃y ∈ B but not unbounded quantifiers such as ∀x and ∃y. Most quantifiers in mathematical practice are bounded (often only implicitly in exposition). A bqs φ may also contain fixed sets M from 𝕍, which will be replaced in ^{∗}φ by ^{∗}M.

Members of internal sets are internal (this follows easily from the construction) and since the sets ^{∗}M are also internal, it follows that the information we obtain from the Transfer Principle is entirely about internal sets.

It is possible (and quite convenient) to take an axiomatic approach to ∗𝕍, which simply postulates the existence of a set ^{∗}𝕍 and a mapping ^{∗} : 𝕍 → ^{∗}𝕍 that obeys the Transfer Principle. For most purposes (and certainly the construction of Loeb measures) the further assumption of ℵ_{1}-saturation is needed — a property that comes with the ultrapower construction.

8.2 ℵ_{1}-saturation

Definition 8.2 A nonstandard universe ^{∗}𝕍 is said to be ℵ_{1}-saturated if the following holds:

if (A_{m})_{m∈ℕ} is a countable decreasing sequence of internal sets with each A_{m} ≠ Ø, then ⋂_{m∈ℕ} A_{m} ≠ Ø.

Theorem 8.3 A nonstandard universe ^{∗}𝕍 constructed as a countable ultrapower is ℵ_{1}-saturated.

ℵ_{1}-saturation is a kind of compactness property that is essential for the Loeb measure construction which plays a central role in this paper.

The basic fact is that for each (internal) ^{∗}probability space Ω = (Ω, G, Q), the finitely additive probability measure^{∘}Q : G →ℝ has a σ-additive extension. (This is an important consequence of ℵ_{1}-saturation.) The unique completion of this σ-additive extension is called the corresponding Loeb space, and is denoted by (Ω, G_{L}, Q_{L}). For convenience, we assume here that Ω is ^{∗}countably additive, although most of the general theory carries over to the ^{∗}finitely additive case.

The theory of the Loeb measure and Loeb integration is assumed in this paper (see [8, 12, 13] for example).

8.3 Standard parts

Given a standard Hausdorff space S ⊂ 𝔹, we have S ⊆ ^{∗}S. If x ∈ S and x ∈ ^{∗}S, we say that x is the standard part of x, in symbols x = ^{∘} x or x ≈ x, if x ∈ ^{∗}O for every open neighborhood O of x. Since S is Hausdorff, each x ∈ ^{∗}S has at most one standard part. An element x ∈ ^{∗}S is said to be near-standard, in symbols x ∈ ns(S), if X has a standard part (in S.). Thus the standard part function maps ns(S) onto S and is the identity on S. The standard part of a set B ⊆ ns(S) is the set ^{∘}B = {^{∘}x : x ∈ B}. Here is a useful immediate consequence of the definition of standard part. Remark 8.4 Suppose x = ^{∘} x in a Hausdorff space S.

(a) If O is open in S, x ∈ O implies x∈ ^{∗}O.

(b) If C is closed in S, x∈ ^{∗}C implies x ∈ C.

In the particular case of a standard metric space (S, ρ), x = ^{∘} x if and only if ^{∗}ρ(x, x) ≈ 0, and two points x, y ∈ ^{∗}S are said to be infinitely close, in symbols x ≈ y, if ^{∗}ρ(x, y) ≈ 0.

The following fundamental result is one of the keys to the power of nonstandard analysis.

Theorem 8.5 Suppose S is a Hausdorff space and C ⊆ S. Then (a) C is compact if and only if ^{∗}C ⊆ns(C)

(b) C is relatively compact if and only if ^{∗}C ⊆ns(S)

This criterion will often be used in conjunction with the fact that for a metric space compactness is equivalent to sequential compactness, as follows.

Theorem 8.6 Suppose S is a metric space and C ⊆ S. Then C is compact if and only if for every sequence (x_{n}) in C, x_{N} ∈ ns(C) for every infinite N.

The nonstandard criterion for continuity is very intuitive.

Theorem 8.7 Let f : S_{1} → S_{2} where S_{i} are topological spaces. Then f is continuous if and only if

∗f(x) ≈ ^{∗}f(y) whenever x ≈ y

The book [8] gives information about the standard part mapping for various topologies on the standard set H. The most important are as follows. Here, ∗H has an internal ^{∗}basis {^{∗}e_{n}}_{n∈^{∗}ℕ}, and we write E_{n} = ^{∗}e_{n}. Thus for each N ∈ ^{∗}ℕ, H_{N} = ^{∗}span{E_{1}, …, E_{N}} ⊆ ^{∗}H. We also write u_{(n)} = (u, e_{n}) for u ∈ H and u_{(n)} = (u, E_{n}) for u ∈ ^{∗}H.

Lemma 8.8 Let u ∈ ^{∗}H. Then:

(a) If |u| < ∞ (i.e. |u| is finite) then u is weakly nearstandard in H, and the weak standard part u = st_{weak}(u) is defined by

u_{(n)} = ^{∘}(u_{(n)}), n ∈ ℕ.

(b) If u is nearstandard in the strong topology of H then |u| < ∞ and

st_{weak}(u) = st(u).

(c) If ∥u∥ < ∞ then u is (strongly) nearstandard in H.

In view of the consistency (b) above we use ^{∘}u to denote the standard part of u whenever |u| is finite.

Near-standard points and standard parts also appear in the more general setting of an internal ^{∗}metric space, and in particular the space SL^{0}(Ω) of ∗random variables on a ^{∗}measure space Ω. This is the starting point of the theory of neometric spaces, which is outlined in Section 5 and applied in Section 7.

9 Appendix 2: Neometric spaces

We give here a brief summary of those parts of the theory of neometric spaces that we need, as developed in the papers [17] and [18]. The fundamental definitions have been given in Section 5 so we do not repeat them here.

We will require the following consequence of ℵ_{1}-saturation for countably determined sets.

Lemma 9.1 ([20]) Projections of countably determined sets in M^{2} are countably determined in M.

The next two results give basic facts about neocompact sets and a theorem that is useful for proving results about neocompact sets, which follows from ([18] Corollary 3.8) and ([19] Proposition 3.1, Theorem 3.3).

Proposition 9.2 (a) Every compact set is neocompact.

(b) If C is neocompact in M × N, then the projection

D = {x ∈ M : (∃y ∈ N)(x, y) ∈ C}

is neocompact in M.

(c) If C is neocompact in M × N and K is a nonempty compact subset of N, then

B = {x ∈ M : (∀y ∈ K)(x, y) ∈ C}

is neocompact in M.

Theorem 9.3 The following are equivalent for a set C ⊆ M:

(a) C is neocompact.

(b) The monad of C is a Π_{1}^{0} set.

(c) The monad of C is countably determined, and each countable subset of C is contained in a neocompact subset of C.

Proof. It is obvious that (b) implies (a). To prove (a) implies (b), we observe that if C = ^{∘ ⋂}_{n∈ℕ}A_{n} where each A_{n} is internal, then monad(C) = ⋂_{n∈ℕ}((A_{n})^{1/n}).

Clearly (a) and (b) implies (c), so it remains to prove that (c) implies (b). Assume (c). Then monad(C) is an infinite Boolean combination of sets from a countable sequence B_{1}, B_{2}, … of internal subsets of M. Let us write x ≡_{n} y if x, y belong to the same sets among B_{1}, …, B_{n}, and let

D_{n} = {x : (∃y ∈ monad(C)) x ≡_{n} y}.

Then each D_{n} is internal because it is the finite union of ≡_{n}equivalence classes. It suffices to prove that

monad(C) = ⋂_{n∈N} D_{n}.

It is clear that the left side is contained in the right side. Suppose x ∈ ⋂_{n∈N} D_{n}. Then for each n ∈ ℕ there exists y_{n} ∈ monad(C) such that y_{n} ≡_{n} x. By hypothesis there is a neocompact set A such that

{^{∘}y_{n} : n ∈ N} ⊆ A ⊆ C.

Since (a) implies (b), there is a decreasing chain A_{n} of internal sets such that monad(A) = ⋂_{n∈ℕ} A_{n}. Then y_{n} ∈ A_{n} for each n ∈ ℕ. By ℵ_{1}-saturation there exists y such that y ∈ A_{n} and y ≡_{n} x for all n ∈ ℕ. Then y ∈ monad(A) ⊆ monad(C). Since monad(C) is an infinite Boolean combination of the sets B_{n}, it follows that x ∈ monad(C).

An important consequence is the following fact, which says that neocompact sets behave like compact sets. It can often be used as a shortcut in place of overspill.

Theorem 9.4 (Countable Compactness) If B_{m} is a decreasing chain of nonempty neocompact sets, then ⋂_{m} B_{m} is a nonempty neocompact set.

Proof. Let B = ⋂_{m} B_{m}. By Lemma 9.3, for each m, monad(B_{m}) is equal to a nonempty Π_{1}^{0} set ⋂_{n} C_{mn}. It is easily seen that

⋂_{m} (⋂_{n} C_{mn}) = ⋂_{m}(monad(B_{m})) = monad(B),

so B is neocompact. For each k there is a point x_{k} which belongs to C_{mn} for all m, n < k. By ℵ_{1}-saturation, there is a point x that belongs to C_{mn} for all m, n ∈ ℕ. Therefore ^{∘}x ∈ B, so B is nonempty.

Relationships between closed, neoclosed and neocompact are given next.

Proposition 9.5 (a) ([17], Lemma 4.6.) Every neocompact set is neoclosed and bounded in M.

(b) ([17], Proposition 4.5.) Every neoclosed set is closed.

Thus, whereas neocompact is weaker than compact (Proposition 9.2(a)), neoclosed is stronger than closed.

For a set D ⊆ M, let ^{∘}D = {^{∘}x : x ∈ D}. Neoclosed sets can often be found using the following proposition.

Proposition 9.6 ([18], Proposition 4.1 and Lemma 3.7) Let D_{n}, n ∈ ℕ be a decreasing chain of internal subsets of M and let D be the Π_{1}^{0} set D = ⋂_{n} D_{n}. Then

(a) ^{∘}D ∩ M is neoclosed in M.

(b)

∘D ∩ M = ⋂_{n}^{∘}(D_{n}) ∩ M = ⋂_{n}^{∘}((D_{n})^{≤1/n}) ∩ M.

Here are the basic properties of neocontinuous functions.

Proposition 9.7 (See [17]) (a) Every neocontinuous function is continuous.

(b) If M is separable then every continuous function f : M → N is neocontinuous.

(c) Compositions of neocontinuous functions are neocontinuous.

(d) Let f : M → N be neocontinuous.

(i) If C is neocompact in M, then f(C) is neocompact in N. (ii) If D is neoclosed in N, then f^{−1}(D) is neoclosed in M.

Neocontinuous functions can often be built using the next proposition.

Proposition 9.8 ([18], Theorems 4.16 and 4.18) . A function f : M → N is neocontinuous if and only if for each neocompact set C in M, there is an internal function F : M → N such that for all x ∈ monad(C), F(x) ∈ monad(N) and ^{∘}(F(x)) = f(^{∘}x).

As examples using this proposition we have the distance and projection functions noted earlier.

We will need the following consequence of countable compactness.

Proposition 9.9 Suppose f : M → N is neocontinuous and {C_{n}} is a countable decreasing chain of neocompact sets in M. Then

f ( ⋂_{n∈ℕ}C_{n}) = ⋂_{n∈ℕ}f(C_{n}).

Proof. We prove the nontrivial direction. Let y ∈ ⋂_{n∈ℕ} f(C_{n}) and let D = f^{−1}{y}. The set {y} is neocompact since it is compact, so D is neoclosed by Proposition 9.7. Then D∩C_{n} is a decreasing chain of nonempty neocompact sets. By countable compactness, there exists x ∈ ⋂_{n∈ℕ}(D ∩ C_{n}). Thus x ∈ ⋂_{n∈ℕ} C_{n} and f(x) = y, as required.

References

[1] S. Albeverio, J.-E. Fenstad, R. Høegh-Krohn, and T.L. Lindstrøm, Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, New York 1986.

[2] L.Arkeryd, N.J.Cutland, and C.W.Henson, Eds. Nonstandard Analysis, Theory and Applications, NATO ASI Series, Series C: Mathematics and Physical Sciences Vol.493, Kluwer Academic Publishers, Dordrecht — Boston — London 1997.

[3] M. Capiński and N.J. Cutland, A simple proof of existence of weak and statistical solutions of Navier–Stokes equations, Proc. Royal. Soc. London A 436 (1992), 1-11.

[4] M. Capiński & N.J. Cutland, Stochastic Navier-Stokes equations, Applicandae Mathematicae 25(1991), 59-85.

[5] M. Capiński & N.J. Cutland, Statistical solutions of stochastic Navier-Stokes equations, Indiana University Math Journal 43(1994), 927-940.

[6] M. Capiński & N.J. Cutland, Attractors for three-dimensional Navier– Stokes equations, Proc. Roy. Soc. Lond. Ser.A, 453(1997), 2413-2426.

[7] M. Capiński & N.J. Cutland, Measure attractors for stochastic Navier– Stokes equations, Electronic J. Prob. 3(1998), Paper 8, 1-15.

[8] M.Capiński & N.J.Cutland, Nonstandard Methods for Stochastic Fluid Mechanics, Series on Advances in Mathematics for Applied Sciences, Vol 27, World Scientific, Singapore 1995.

[9] M. Capiński & N.J. Cutland, Existence of global stochastic flow and attractors for Navier–Stokes equations, Probab. Theory Relat. Fields 115(1999), 121-151.

[10] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probability Theory and Related Fields 100 (1994), 365–393.

[11] N.J. Cutland (ed.), Nonstandard Analysis and its Applications, Cambridge University Press, Cambridge 1988.

[12] N.J. Cutland, Loeb measure theory, in Developments in Nonstandard Mathematics, Eds. N.J.Cutland, F.Oliveira, V.Neves, J.Sousa-Pinto, Pitman Research Notes in Mathematics Series Vol. 336, Longman 1995, 151-177.

[13] N.J. Cutland, Loeb Measures in Practice: Recent Advances, Lecture Notes in Mathematics 1751(2000), Springer.

[14] N.J. Cutland and H.J. Keisler, Global attractors for 3D stochastic Navier-Stokes equations, to appear in J. Dynamics and Diff. Equations.

[15] F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier–Stokes equation with multiplicative white noise, Stochastic & Stochastics Reports, 59(1996), 21-45.

[16] F. Flandoli and B. Schmalfuss, Weak solutions and attractors for threedimensional Navier–Stokes equations with nonregular force, J. Dynamics and Differential Equations, 11(1999), 355-398.

[17] S. Fajardo and H.J. Keisler, Existence theorems in probability theory, Advances in Mathematics, 120(1996), 191-257.

[18] S. Fajardo and H.J.Keisler, Neometric spaces, Advances in Mathematics, 118(1996), 134-175.

[19] S. Fajardo and H.J.Keisler, Long Sequences and Neocompact Sets, in Developments in Nonstandard Analysis, (N. Cutland et.al., Eds., pp. 251-260, Longman, Harlow/New York 1995.

[20] C.W. Henson, Analytic sets, Baire sets, and the standard part map. Canadian Journal of Mathematica 31 (1979), pp. 663-672.

[21] A.E.Hurd and P.A.Loeb, An Introduction to Nonstandard Real Analysis, Academic Press, New York 1985.

[22] H. J. Keisler, Rich and saturated adapted spaces, Advances in Mathematics, 128(1997), 242-288.

[23] T.L.Lindstrøm, An invitation to nonstandard analysis, in [11], 1-105.

[24] D.A.Ross, Loeb measure and probability, in Nonstandard Analysis: Theory and Applications, Eds. L.O. Arkeryd, N.J. Cutland, C.W. Henson, NATO ASI Series C: Vol 493, Kluwer, Dordrecht, 1997, 91-120.

[25] B. Schmallfuss, Measure attractors of the stochastic Navier–Stokes equation, Bremen Report No.258, 1991.

[26] G.R. Sell, Global attractors for the three-dimensional Navier–Stokes equations, J.Dynamics and Diff. Equations 8(1996), 1-33.

[27] R.Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, SIAM, 1983.

[28] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, 1988.