Neocompact sets and stochastic Navier-Stokes equations.

Nigel J. Cutland and H. Jerome Keisler

Abstract

We give a detailed exposition of the use of neocompact sets in proving existence of solutions to stochastic Navier-Stokes equations. These methods yield new results concerning optimality of solutions.

1 Introduction

In this paper we give a detailed exposition of the way in which the recent work of S. Fajardo and H. J. Keisler [6] can be used to establish existence of solutions to stochastic Navier-Stokes equations. Fajardo & Keisler [6] develop general methods for proving existence theorems in analysis, with the aim of embracing the many particular existence theorems that can be proved rather easily using nonstandard analysis. The machinery developed centres round the notion of a neocompact set - which is a weakening of the notion of a compact set of random variables with values in a metric space M - and the notion of a rich adapted probability space, in which any countable chain of nonempty neocompact sets has a nonempty intersection.

In the papers [2, 3] Capiński & Cutland used nonstandard methods to greatly simplify some known existence proofs for the deterministic Navier-Stokes equations and (using similar methods) solved a longstanding problem concerning existence of solutions to general stochastic Navier-Stokes equations. The aim here is to show how the main results of these papers can be obtained using the neocompactness methods developed in [6]. In addition, these methods yield additional information concerning the nature of the set of solutions and existence of optimal solutions.

For the convenience of the reader we begin with a brief summary of the notions and results we shall need from [6].

2 Neocompact sets.

In this section we review the notion of a neocompact set from the paper [6]. Neocompact sets share many of the useful properties of compact sets. We shall introduce the notion in two contexts – probability spaces and adapted spaces.

If Ω Ω = (Ω, P, G) is a probability space and (M, ρ_{M}) is a complete metric space, then L^{0}(Ω, M) will denote the metric space of all measurable functions from Ω into M with the metric of convergence in probability,

ρ(x, y) = inf{ϵ > 0 : P [ρ_{M}(x(ω), y(ω)) < ϵ] ≥ ϵ}.

For any set A we write A^{ϵ} = {x : ρ(x, A) ≤ ϵ}.

We let Meas(M) be the space of all Borel probability measures on M with the Prohorov metric, and let law : L^{0}(Ω, M) → Meas(M) be the function mapping each random variable x to the measure on M induced by x. This function is continuous.

Definition 2.1 Let Ω Ω be a probability space and let M, M′ denote complete separable metric spaces. A set B ⊂ L^{0}(Ω, M) is called basic if either

(1) B is compact, or

(2) B = {x : law(x) ∈ C} for some compact set C ⊂ Meas(M).

By the family of neocompact sets over Ω we mean the collection of all subsets of L^{0}(Ω, M) obtained by repeated application of the following rules:

(a) Every basic set is neocompact.

(b) Finite unions of neocompact sets are neocompact.

(c) Finite and countable intersections of neocompact sets are neocompact.

(d) Finite cartesian products of neocompact sets are neocompact (where we identify L^{0}(Ω, M)× L^{0}(Ω, M′) with L^{0}(Ω, M × M′) in the natural way).

(e) If C ⊂ L^{0}(Ω, M × M′) is neocompact, then the set

{x : (∃y)(x, y) ∈ C}

is neocompact.

(f) If C ⊂ L^{0}(Ω, M × M′) is neocompact and D ⊂ L^{0}(Ω, M′) is basic neocompact and

nonempty, then the set

{x : (∀y ∈ D)(x, y) ∈ C}

is neocompact.

It is not hard to see that the family of compact sets is closed under all of the rules (a)–(f). In fact, the family of compact sets is closed under arbitrary intersections, and condition (f) holds for arbitrary nonempty sets D. One of the reasons that compact sets are useful in proving existence theorems is that they have the following property:

If C is a set of compact sets such that any finite subset of C has a nonempty intersection, then C has a nonempty intersection.

We define a rich probability space as one in which the neocompact sets have a weaker form of this property, called the countable compactness property.

Definition 2.2 A collection C of sets has the countable compactness property if the intersection of any countable decreasing chain C_{1} ⊃ C_{2} ⊃ ⋯ of nonempty sets in C is nonempty.

Definition 2.3 A probability space Ω Ω is said to be rich if it is atomless and for any complete separable metric space M, the collection of neocompact sets in L^{0}(Ω, M) has the countable compactness property.

We now turn to neocompact sets in adapted spaces. By an adapted space we shall mean a structure Ω Ω = (Ω, P, G, G_{t}) where t runs over the dyadic rationals and the G_{t} are σ-subalgebras of G which increase in t. For each real s, we let F_{s} be the P -completion of ⋂_{t>s} G_{t}.

Definition 2.4 Let Ω Ω be an adapted space. The families of basic sets and neocompact sets for the adapted space are defined exactly as for the case of a probability space except that we add to the family of basic sets all sets B of the form

(3) B = {x ∈ L^{0}(Ω, M) : law(x) ∈ C and x is G_{t}-measurable},

where C is compact in Meas(M) and t is a dyadic rational.

Definition 2.5 An adapted space Ω Ω = (Ω, P, G, G_{t}) is said to be rich if the probability space (Ω, P, G_{0}) is atomless, Ω Ω admits a Brownian motion, and for any complete separable metric space M, the collection of neocompact sets in L^{0}(Ω, M) has the countable compactness property.

The following fact is implicit in the paper [12] and will be proved explicitly in [7].

Theorem 2.6 Rich probability spaces and rich adapted spaces exist.

This is proved by showing that the adapted Loeb spaces, which were the underlying spaces used in [2] and [3], are rich. Every rich adapted space is also rich as a probability space.

The analogues of closed sets and continuous functions are defined in terms of neocompact sets in the following way.

Definition 2.7 A set C ⊂ L^{0}(Ω, M) is neoclosed if C ∩ D is neocompact for every neocompact set D.

A function f mapping a neoclosed set C ⊂ L^{0}(Ω, M) into L^{0}(Ω, N) is neocontinuous if for every neocompact set D ⊂ C, the graph of f restricted to D is neocompact.

It is shown in the paper [6] that for a rich adapted space, every neoclosed set is closed, and every neocontinuous function is continuous. Parallel to the classical case, images of neocompact sets under neocontinuous functions are neocompact, and preimages of neoclosed sets under neocontinuous functions are neoclosed. We shall identify N with the set of constant functions from Ω into N. With this identification, each closed subset of N becomes a neoclosed subset of L^{0}(Ω, N), and we obtain the notions of a neocontinuous function from L^{0}(Ω, M) into N, and a neocontinuous function from a closed subset of M into L^{0}(Ω, N).

In the paper [6], several important sets and functions are shown to be neocompact, neoclosed, or neocontinuous for a rich adapted space.

For example, it is shown that the set of Brownian motions on Ω Ω is neocompact, the set of stopping times between 0 and 1 for Ω Ω is neocompact, and the set of all F_{t}-adapted stochastic processes with values in M is neoclosed.

Some examples of neocontinuous functions which will be used in this paper are the distance function ρ for L^{0}(Ω, M), the function x(·) ↦ f(x(·)) where f : M → N is continuous, the expected value function x(·) ↦ E(x(·)) restricted to a uniformly integrable subset of L^{0}(Ω, IR), and the stochastic integral function x ↦ ∫ x db where b is a Brownian motion and x is in a bounded set of adapted processes.

Moreover, compositions of neocontinuous functions are neocontinuous.

In the paper [6] the notion of neo-lower semicontinuity, abbreviated neo-lsc, is defined as a useful generalisation of the classical notion of lowersemi-continuity. Recall that a function f : M → \overline{IR} is lower semicontinuous (lsc) if whenever x_{n} → x in M then \underline{lim}_{n→∞}f(x_{n}) ≥ f(x); equivalently, f is lsc if for every compact C ⊆ M, the upper graph

\{(x, r) ∈ C × \overline{IR} : f(x) ≤ r\}

is compact. This is the definition that is generalised in [6]. Regard \overline{IR} = [−∞, ∞] as a compact metric space with the metric ρ(r, s) = | arctan(r) − arctan(s)| and we have:

Definition 2.8 Let D ⊆ L^{0}(Ω, M); a function f : D → L^{0}(Ω, \overline{IR}) is neo-lsc if for every neocompact set C ⊆ D the upper graph

\{(x, y) ∈ C × L^{0}(Ω, \overline{IR}) : f(x) ≤ y a.s.\}

is neocompact.

If f : D → \overline{IR} it is easy to check that f is neo-lsc if and only if the upper graph \{(x, r) ∈ C × \overline{IR} : f(x) ≤ r\} is neocompact for each neocompact C.

It is shown in [6] that the expectation operator restricted to positive random variables is neo-lsc, and that a composition g ∘ f of two neo-lsc functions is neo-lsc provided that g is monotone. Also, if f : M → \overline{IR} is lsc then the function g : L^{0}(Ω, M) → L^{0}(Ω, \overline{IR}) defined by g(x)(ω) = f(x(ω)) is neo-lsc.

We shall need the following important lemma from [6].

Lemma 2.9 (Closure Under Diagonal Intersections.) Let Ω Ω be either a rich probability space or a rich adapted space. Let A_{n} be neocompact in L^{0}(Ω, M) for each n ∈ IN, and let ϵ_{n} ↘ 0. Then the set A = ⋂_{n} ((A_{n})^{ϵ_{n}}) is neocompact in L^{0}(Ω, M).

The key result from [6] which we shall apply in this paper is the following theorem.

Theorem 2.10 (Approximation Theorem.) Let Ω Ω be either a rich probability space or a rich adapted space. Let A be neoclosed in L^{0}(Ω, M) and let f : L^{0}(Ω, M) → L^{0}(Ω, N) be neocontinuous. Let B ⊂ L^{0}(Ω, M) and D ⊂ L^{0}(Ω, N) be neocompact. Suppose that for each ϵ > 0

(∃x ∈ A ∩ B^{ϵ}) f(x) ∈ D^{ϵ}.

Then

(∃x ∈ A ∩ B) f(x) ∈ D.

It is easy to check that the analogous result holds with compact, closed, and continuous in place of neocompact, neoclosed, and neocontinuous. We shall use this classical analogue as a warmup in Section 4, and then apply the Approximation Theorem for rich probability spaces and rich adapted spaces later in this paper.

3 The Navier-Stokes equations.

The classical Navier-Stokes equations describe the evolution in time of the velocity field u : D → IR^{n} of an incompressible fluid in a domain D ⊆ IR^{n}, so we are thinking of a function u(x, t) : D × [0, T] → IR^{n} given by:

\{ \begin{matrix}\frac{∂u}{∂t} − νΔu + ⟨u, ∇⟩u + ∇p = f \\ div u = 0 \end{matrix}  (1)

(where ⟨, ⟩ denotes the inner product in IR^{n}). For convenience we will assume a fixed finite time horizon T but there is no difficulty in extending to the time set [0, ∞). The domain D is bounded with boundary of class C^{2} and we will work with the homogeneous Dirichlet boundary condition u |_{∂D}= 0. Of course in important applications, n = 3 but from the mathematical point of view we can allow n ≤ 4. In this equation, p denotes the pressure, and f denotes the external forces.

The usual setting for these equations involves the function spaces H, V which are obtained by closing the set {u ∈ C_{0}^{∞}(D, IR^{n}) : div u = 0} in the norms | · | and | · | + ∥ · ∥ respectively, where

\begin{matrix}|u| = (u, u)^{\frac{1}{2}} ; & (u, v) = ∑_{j=1}^{n} ∫_{D} u^{j}(ξ)v^{j}(ξ)dξ, \\ ∥u∥ = ((u, u))^{\frac{1}{2}} ; & ((u, v)) = ∑_{j=1}^{n} (\frac{∂u}{∂ξ_{j}} , \frac{∂v}{∂ξ_{j}}). \end{matrix}

H, V are Hilbert spaces. We fix an orthonormal basis (e_{n})_{n∈IN} for H consisting of eigenvectors of the operator −Δ with eigenvalues (λ_{n})_{n∈IN} . For u ∈ H we write u_{k} = (u, e_{k}). We let V′ be the dual space of V with respect to the | · | norm, and let (·, ·) denote the duality between V′ and V extending the scalar product in H. In the equation (1) it is usual to take the force f ∈ L^{2}(0, T; V′), and then the equation is understood as a Bochner integral equation in V′. i.e. for each v ∈ V:

(u(t), v) − (u_{0}, v) = ∫_{0}^{t}[−ν((u(s), v)) − b(u(s), u(s), v) + (f(s), v)]ds  (2)

where u_{0} is the given initial condition. The pressure vanishes in this weak formulation because ⟨∇p, v⟩ = −⟨p, div v⟩ = 0, but of course p can be recovered from a solution to the equation (2). The trilinear form b is the nonlinear term in (1), so that we have:

b(u, v, w) = ∑_{i,j=1}^{n} ∫_{D} u^{j}(ξ)\frac{∂v^{i}}{∂ξ_{j}} (ξ)w^{i}(ξ)dξ = (⟨u, ∇⟩v, w).

Note that for u, v, z ∈ V we have b(u, v, z) = −b(u, z, v) so that b(u, v, v) = 0. There are a number of well known inequalities giving continuity of b in various topologies (see [13] and [14] p.12 for example) and we list here those (for n ≤ 4) that we shall need.

|b(u, v, z)| ≤ c∥u∥ ∥v∥ ∥z∥  (3)
|b(u, v, z)| ≤ c|u| ∥v∥ |Az|  (4)
|b(u, v, z)| ≤ c|u| |Av| ∥z∥  (5)

It is customary to write A for the self-adjoint extension of the operator −Δ on H, and to write B(u) = b(u, u, ·) ∈ V′ for appropriate u. Then it is an easy consequence of (4) and (5) that:

Proposition 3.1 For all m, B(·) ∈ C(K_{m}, V′_{weak}), where K_{m} is the compact subset of H given by K_{m} = {u ∈ H : ∥u∥ ≤ m} with the H-topology. □

Proof See Proposition 3.4 of [3].

We can now make precise what is taken to be a weak solution to (1)

Definition 3.2 Given u_{0} ∈ H and f ∈ L^{2}(0, T; V′) the function u : [0, T) → H is a weak solution of the Navier-Stokes equations if

(i) u ∈ L^{2}(0, T; V) ∩ L^{∞}(0, T; H) ,
(ii) for all v ∈ V, for all t ≥ 0, u satisfies equation (2)

The regularity condition (i) emerges naturally by consideration of the time evolution of the energy \frac{1}{2}|u|^{2}.

The spaces H and V involved in the formulation of the Navier-Stokes equations are two from the spectrum of Hilbert spaces H^{α} given by

H^{α} = {u ∈ H : ∑_{k=1}^{∞} λ_{k}^{α}u_{k}^{2} < ∞}

for α ≥ 0, with norm |u|_{α} = ( ∑_{k=1}^{∞} λ_{k}^{α}u_{k}^{2})^{\frac{1}{2}} . The dual spaces H^{−α} are represented by the sets

H^{−α} = {u ∈ IRIN : ∑_{k=1}^{∞} λ_{k}^{−α}u_{k}^{2} < ∞}

with the corresponding norms |u|_{−α}. It is easily checked that H = H^{0}, V = H^{1} and V′ = H^{−1}.

We write H_{n} = span{e_{1}, …, e_{n}} and denote the projection from H onto H_{n} by Π_{n}. For u ∈ H, u^{(n)} denotes Π_{n}u.

4 Existence of weak solutions

In this section we prove

Theorem 4.1 For every u_{0} ∈ H there is a weak solution u to the Navier-Stokes equations with u(0) = u_{0} and with u in the space M_{0} defined below.

This is, of course, a classical result. We shall present a proof using the classical analogue of the Approximation Theorem (Theorem 2.10) and compare it with a classical proof. The point of doing this is to show how the neocompactness machinery allows a direct generalization of our proof to give the existence result for the stochastic Navier-Stokes equations [3], for which there is no classical proof using compactness.

For the development here and in subsequent sections we define some spaces and auxiliary functions that will be important for the construction of solutions. We put

M = C([0, T], H^{−2}) ∩ {y : y(0) ∈ H}

which is a complete metric space with the metric | · |_{M} given by

|y_{1} − y_{2}|_{M} = sup_{t≤T} |y_{1}(t) − y_{2}(t)|_{−2} + |y_{1}(0) − y_{2}(0)|.

For y ∈ M define θ(y) by

θ(y) = ν ∫_{0}^{T} ∥y(t)∥^{2}dt + sup_{t≤T} |y(t)|^{2}.

Let M_{0} be the subset of M given by

M_{0} = M ∩ {y : θ(y) ≤ K + |y(0)|^{2}}

where K = ν^{−1} ∫_{0}^{T} |f(t)|V′^{2}dt.

Proposition 4.2 M_{0} is a closed subspace of M.

Proof This follows easily from the fact that if u, u^{n} ∈ H^{−2} with u^{n} → u in H^{−2}, then |u|_{α} ≤ \underline{lim}|u^{n}|_{α} for any α. Apply this to |y^{n}(t)| = |y^{n}(t)|_{0} and ∥y^{n}(t)∥ = |y^{n}(t)|_{1} for any sequence y^{n} ∈ M_{0} with y^{n} → y ∈ M, and use Fatou’s lemma (∫_{0}^{T} \underline{lim}∥y^{n}(t)∥^{2}dt ≤ \underline{lim} ∫_{0}^{T} ∥y^{n}(t)∥^{2}dt) and the fact that sup_{t≤T} \underline{lim}|y^{n}(t)|^{2} ≤ \underline{lim} sup_{t≤T} |y^{n}(t)|^{2}. □

Remark It is easy to check that M_{0} ⊆ C([0, T], H^{−α}) for all α > 0, although we will not need this. It is a consequence of the fact that for y ∈ M_{0} each y_{k}(t) is continuous, and |y(t)| is bounded.

Now for each k > 0 define

M_{k} = M ∩ {y : θ(y) ≤ k}

Clearly each M_{k} is closed in M and M_{0} ⊆ ⋃_{k>0} M_{k} = M_{∞}, say.

We now define a function γ : M_{∞} → M by

γ(y)(s) = ∫_{0}^{s}[−νAy(t) − B(y(t)) + f(t)]dt.

To see that γ takes its values in M, use the following key facts about A and B:

\begin{matrix}(i) |Au|_{−1} = ∥u∥ \\ (ii) |B(u)|_{−1} ≤ c∥u∥^{2} \end{matrix}  (6)

with (ii) holding by (3). These properties of A and B ensure that for y ∈ M_{∞} we have

γ(y) ∈ C([0, T], H_{weak}^{−1}) ∩ L^{∞}(0, T; H^{−1})
⊆ C([0, T], H^{−α})

for any α > 1 and in particular for α = 2.

Remark This shows that the choice of H^{−2} in the definition of M was somewhat arbitrary. We could have taken H^{−α} for any α > 1.

We observe that u = y is a weak solution to the Navier-Stokes equations with initial condition u_{0} = x if and only if y = x + γ(y).

The next two facts are the key to the construction of a solution.

Proposition 4.3 The function γ is continuous on each set M_{k} (in the topology of M).

Proof This is routine Bochner integration theory using the facts (6) above. □

Proposition 4.4 For each k > 0 the set γ(M_{k}) is relatively compact in M.

Proof The facts (6) can be used to show that there are constants d and (d_{m})_{m≥1} such that

γ(M_{k}) ⊆ {z ∈ M : z(0) = 0 & sup_{t≤T} |z(t)|_{−1} ≤ d
& |z_{m}(s) − z_{m}(t)|^{2} ≤ d_{m}|s − t| for all s, t ≤ T and m ∈ IN}.

It is then routine to check that the set on the right is compact in M. □

As with classical proofs, we now consider the finite dimensional approximations to (2) - known as the Galerkin approximations. For this purpose we define a sequence of functions γ_{n} : M_{∞} → M by

γ_{n}(y)(s) = ∫_{0}^{s}[−νAy^{(n)}(t) − B^{(n)}(y(t)) + f^{(n)}(t)]dt
= Π_{n}γ(y)(s).

It is clear that γ_{n}(y) ∈ C([0, T], H_{n}).

Classical ODE theory (see [13] for example) shows:

Theorem 4.5 For each n > 0 and x ∈ H there is a unique y ∈ M_{0} such that y = x^{(n)}+γ_{n}(y) (so y ∈ M_{K+|x|^{2}}).

Proof The equation y = x^{(n)} + γ_{n}(y) for x ∈ H is simply the Galerkin approximation to the Navier-Stokes equations in dimension n, with initial condition x^{(n)}, which has a unique solution y ∈ C([0, T], H_{n}); consideration of the time evolution of the energy |y(t)|^{2} shows that

|y(s)|^{2} + ν ∫_{0}^{s} ∥y(t)∥^{2}dt ≤ |y(0)|^{2} + ν^{−1} ∫_{0}^{s} |f(t)|V′^{2}dt

so that y ∈ M_{0}. Clearly, y ∈ M_{K+|x|^{2}}. □

The final result in preparation for the fundamental existence result Theorem 4.1 is:

Proposition 4.6 For k > 0, γ_{n}(y) → γ(y) uniformly on M_{k}.

Proof By Proposition 4.4, \overline{γ(M_{k})} is a compact subset of M. Dini’s theorem tells us that Π_{n}z → z uniformly on any any compact set. □

We now complete the proof of 4.1 by using the classical analogue of the Approximation Theorem (2.10).

Proof of Theorem 4.1 Using the classical analogue of the Approximation Theorem, it is sufficient to prove that there is a compact D ⊆ M such that

(∀ε > 0)(∃y ∈ D^{ε} ∩ M_{0})(|x + γ(y) − y|_{M} < ε).

For this we take D = x + \overline{γ(M_{k})} where k = K + |x|^{2}, which is compact by Proposition 4.4. Now, given ε > 0 take n such that |γ_{n}(y) − γ(y)| < \frac{1}{2}ε for all y ∈ M_{k} (using Proposition 4.6), and |x − x^{(n)}| < \frac{1}{2}ε. Take the unique y ∈ M_{0} with y = x^{(n)} + γ_{n}(y) ∈ M_{k}, given by Proposition 4.5. Then

|x + γ(y) − y| ≤ |γ(y) − γ_{n}(y)| + |x − x^{(n)}| < ε

and since x + γ(y) ∈ D we have y ∈ D^{ε} and the proof is complete. □

This proof is really not very different from a classical proof using the preliminary results Propositions 4.4, 4.5 and 4.6 above, which runs as follows:

For x ∈ H, take y_{n} = x^{(n)} + γ_{n}(y_{n}) ∈ M_{0} ∩ M_{k} as given by Proposition 4.5, and let z_{n} = x + γ(y_{n}) ∈ D. By Proposition 4.4, there is a convergent subsequence z_{n_{i}} → z, say with z ∈ D. Then

|y_{n_{i}} − z| ≤ |x^{(n_{i})} + γ_{n_{i}}(y_{n_{i}}) − (x + γ(y_{n_{i}}))| + |z_{n_{i}} − z| → 0

using Proposition 4.6. So z belongs to the closed set M_{0} . By Proposition 4.3 we have

x + γ(z) = x + lim γ(y_{n_{i}}) = lim z_{n_{i}} = z,

so that z is the required solution.

5 Existence of statistical solutions

The idea of a statistical solution to the Navier-Stokes equations was developed by Foias [8], and the idea is as follows. Suppose that in the Navier-Stokes equations (1) or (2) the initial condition is given by a probability measure μ_{0} on H, with the informal idea that there is some underlying probability P such that for A ⊆ H

μ_{0}(A) = P (u_{0} ∈ A).

Then, informally, as time evolves we can think of measures μ_{t} given by

μ_{t}(A) = P (u(t) ∈ A).  (7)

However, this idea is only heuristic, because we do not have any meaning for the value u(t) for a random initial condition u_{0} - since in dimension n ≥ 3 the uniqueness problem for equation (2) is still open. If we did have uniqueness of solutions then equation (7) could be made precise by writing S_{t}(u) for the value at time t of the solution to (2) with initial condition u, and then the family of measures μ_{t} would be given by

μ_{t}(A) = μ_{0}(S_{t}^{−1}(A)).

Even though there is no such function S_{t} to make this precise, by arguing informally Foias [8] derived the following equation that would be satisfied by the family μ_{t} if S_{t} did exist:

∫H φ(u)dμ_{t}(u) − ∫H φ(u)dμ(u) =

∫_{0}^{t} ∫H[−ν((u, φ′(u))) − b(u, u, φ′(u)) + (f(s), φ′(u))]dμ_{s}(u)ds  (8)

where φ is any test function of the form φ(u) = exp^{i(u,v)} with v ∈ V. This is called the Foias equation, and it makes sense because there is no reference to the possibly non-existent function S_{t} that was used informally in Foias’ derivation. Solutions to the Foias equation (see below) are called statistical solutions to the Navier-Stokes equations. In dimensions n ≥ 3 Foias’ derivation of equation (8) is only heuristic so it does not guarantee the existence of statistical solutions, and these must be constructed by some other means, as Foias did in [8].

Here is the precise definition of a statistical solution.

Definition 5.1 Suppose that a Borel probability measure μ on H is given, with ∫H |u|^{2}dμ < ∞. Then a family of probability measures (μ_{t})_{t≥0} is a statistical solution of the Navier-Stokes equations with initial condition μ if μ_{0} = μ and

(i) the function t ↦ ∫_{H} |u|^{2}dμ_{t}(u) is L^{∞}(0, T) for all T < ∞,
(ii)  ∫_{0}^{T} ∫_{H} ∥u∥^{2}dμ_{t}(u)dt < ∞ for all T < ∞,
(iii) for all t ≥ 0 and test functions φ as above equation (8) holds.

Foias’ proof [8] of existence of statistical solutions to the Navier-Stokes equations in dimensions n ≤ 4 is long and complicated. A new and very short proof in [2] was based on the uniqueness of solutions to the Galerkin approximation in an infinite hyperfinite dimension N, which made Foias’ heuristic derivation completely rigorous in this setting. Essentially what was proved there was the existence of a random solution to (2) for a given random initial condition. This is easily proved using the neocompactness methods from [6], together with the results of the previous section, as follows.

Theorem 5.2 Let x(ω) ∈ H be a random variable defined on a rich probability space (Ω, P). Then there is a random variable y : Ω → M_{0} such that for almost all ω ∈ Ω, y(ω) is a (weak) solution to the Navier-Stokes equations with initial condition x(ω).

Proof Since H is separable there is an increasing sequence of compact sets C_{n} with x(ω) ∈ ∪_{n≥1}C_{n} a.s. Let D_{n} = (C_{n} + \overline{γ(M_{k_{n}})}) ∩ M_{0}, where k_{n} = K + sup_{z∈C_{n}} |z|^{2}. Then by Proposition 4.4, D_{n} is compact, and by Theorem 4.1 we have

(∀z ∈ C_{n})(∃y ∈ D_{n})[y = z + γ(y)].  (9)

By [6], the function

y(·) ↦ γ(y(·))

is neocontinuous and the set

D = {y(·) : (∀n)(P [y(ω) ∈ D_{n}] ≥ P [x(ω) ∈ C_{n}])}

is neocompact. It follows that there are sequences x_{n}(·) and y_{n}(·) of simple random variables (i.e. random variables with finite ranges) such that x_{n} takes its values in C_{n}, x_{n}(ω) → x(ω) almost everywhere, y_{n}(·) ∈ D, and

y_{n}(ω) = x_{n}(ω) + γ(y_{n}(ω))

for almost all ω ∈ Ω.

Now let C = {x(·)}, which is neocompact. Then we have proved

(∀ε > 0)(∃y ∈ D^{ε})(∃z ∈ C^{ε})(z + γ(y) = y)

(simply take z = x_{n} within ε of x and y = y_{n}).

The Approximation Theorem gives a random variable y(·) ∈ D with

y(ω) = x(ω) + γ(y(ω))

for almost all ω, which is as required. □

To see that this is sufficient to give statistical solutions, we have

Theorem 5.3 Suppose that y is an M_{0}-valued random variable on a probability space (not necessarily a rich space) with

y(ω) = x(ω) + γ(y(ω))

for a.a. ω, and such that

E(|x(ω)|^{2}) < ∞.

Then the family of measures μ_{t} on H given by μ_{t}(A) = P (y(ω, t) ∈ A) is a statistical solution to the Navier-Stokes equations.

Proof Simply follow through Foias’ heuristic, as in [2]. □

6 Stochastic Navier-Stokes equations

We show here how the existence proof for stochastic Navier-Stokes equations in dimensions n ≤ 4 can be presented using the machinery of [6]. The pattern of the proof is similar to the corresponding existence result for the deterministic case (Sec. 4), that used the classical analogue of this machinery. In that case we showed how the essential facts that made this work also led to a simple proof that avoids that machinery. In the stochastic case however, it seems unlikely that the proof below can be recast in a way that avoids the use of some form of the neocompactness machinery, because of the need for some kind of enriched probability space to carry the solution in general.

We begin by reviewing from [3] the formulation of the stochastic Navier-Stokes equations. Suppose that Q : H → H is a linear nonnegative trace class operator and that w(t), t ≥ 0 is an H-valued Wiener process with covariance Q, defined on an adapted probability space Ω Ω = (Ω, F, (F_{t})_{t≥0}, P). If f : [0, T] × V → V′ and g : [0, T] × V → L(H, V′) then the stochastic Navier-Stokes equations with full feed-back take the form

du(t) = [−νAu(t) − B(u(t)) + f(t, u(t))]dt + g(t, u(t))dw(t)  (10)

which is to be understood as an integral equation, using the Bochner integral for the drift term and the stochastic integral of Ichikawa [10] for the noise term. The initial condition u_{0} can be random in H, although to begin with we consider only fixed initial conditions u_{0} ∈ H. Thus we have the following definition:

Definition 6.1 A stochastic process u(t, ω) on Ω Ω is a solution to the stochastic Navier-Stokes equations (10) if it is adapted and has almost all paths in the space

Z = C(0, T; H_{weak}) ∩ L^{2}(0, T; V) ∩ L^{∞}(0, T; H)

and the integral equation

u(t) = u_{0} + ∫_{0}^{t}[−νAu(s) − B(u(s)) + f(s, u(s))]ds + ∫_{0}^{t} g(s, u(s))dw(s)  (11)

holds as an identity in V′.

Now take a rich adapted probability space Ω Ω = (Ω, P, G, (G_{t})_{t≥0}) carrying an H-valued Wiener process w with covariance Q. A process is said to be adapted for Ω Ω if it is adapted with respect to the filtration (F_{t})_{t≥0} where F_{t} is the P -completion of ⋂_{s>t} G_{s}.

Recall that K_{m} is the compact subset of H given by K_{m} = {u ∈ H : ∥u∥ ≤ m} with the H-topology.

The existence theorem proved in [3], in the special case that Ω Ω is an adapted Loeb space, is

Theorem 6.2 Let Ω Ω be a rich adapted space. Suppose that u_{0} ∈ H and f : [0, T] × V → V′ and g : [0, T] × V → L(H, H) are jointly measurable functions with the following properties:

(i) f(t, ·) ∈ C(K_{m}, V′_{weak}) for all m,

(ii) g(t, ·) ∈ C(K_{m}, L(H, H )_{weak}) for all m,

(iii) |f(t, u)|V′ + |g(t, u)|_{H,H} ≤ a(t)(1 + |u|), for some function a ∈ L^{2}(0, T).

Then for each fixed u_{0} ∈ H the stochastic Navier-Stokes equation (11) has a solution u on Ω Ω, with u independent of G_{0}, and such that

E (sup_{t≤T} |u(t)|^{2} + ∫_{0}^{T} ∥u(t)∥^{2}dt) < ∞.

(In fact u is in the space N_{0} defined below).

We present a proof of this using the neocompactness machinery; first we set up some notation.

Let N be the set

N = {y ∈ L^{0}(Ω, M) : y is adapted},

and let G^{⊥} be the set

G^{⊥} = {y ∈ L^{0}(Ω, M) : y is independent of G_{0}}.

Let

E(y) = ∫_{0}^{T} ∥y(t)∥^{2}dt + sup_{t≤T} |y(t)|^{2},

and define

N_{0} = N ∩ {y : y(0) ∈ H & E(E(y)) ≤ K_{1} + K_{2}|y(0)|^{2} }

where K_{1} and K_{2} are constants to be specified below. (In this definition by “y(0) ∈ H” we mean that y(0) is non-random and in H.)

Proposition 6.3 N, G^{⊥}, and N_{0} are neoclosed subsets of L^{0}(Ω, M).

Proof It is shown in [6] that the set of adapted processes is neoclosed; i.e. N is neoclosed. To show that G^{⊥} is neoclosed, first show that the set

G_{0}^{⊥} = {x ∈ L^{0}(Ω, IR) : |x| ≤ π/2 and x is independent of G_{0}}

is neoclosed. We have

G_{0}^{⊥} = {x : (∀z ∈ C)(E(xz) = E(x)E(z) and |x| ≤ π/2)}

where C is the set of G_{0}-measurable indicator functions, which is a basic neocompact set. Then use the universal quantifier rule (f) and the fact that the expectation E is neocontinous on uniformly bounded sets of random variables. For G^{⊥} itself, note that

y ∈ G^{⊥} ⇔ (∀k)(∀q ∈ [0, T])(arctan(y_{k}(q, ·)) ∈ G_{0}^{⊥})

(q ranging over rationals). The function y(·) ↦ arctan(y_{k}(q, ·)) is neocontinuous. Now use the facts that neoclosed sets are closed under countable intersections and preimages by neocontinuous functions.

The set of y with y(0) ∈ H, constant, is neoclosed because the function y ↦ y(0) is neocontinuous and the space H is closed and separable (so the set of constant random variables in L^{0}(Ω, H) is neoclosed).

The set of y with E(E(y)) ≤ K_{1} + K_{2}|y(0)|^{2} is neoclosed since the function E : M → IR is lsc and so E(E(y(·)) is neo-lsc, and the function y ↦ y(0) is neocontinuous.

N_{0} is the intersection of these two sets with N and thus is also neoclosed. □

Now for each k > 0 define

N_{k} = N ∩ {y : E(E(y)) ≤ k}.

Clearly each N_{k} is neoclosed in N, and N_{0} ⊆ ∪_{k>0}N_{k} = N_{∞} say.

We now define a function γ̃ : N_{∞} → N by

γ̃(y)(s) = ∫_{0}^{s}[−νAy(t) − B(y(t)) + f(t, y(t))]dt + ∫_{0}^{s} g(t, y(t))dw(t)  (12)

Note that u = y is a solution to the stochastic Navier-Stokes equations with initial condition u_{0} = x if and only if y = x + γ̃(y). Moreover, we have

γ̃ : G^{⊥} ∩ N_{∞} → G^{⊥} ∩ N,

because the Wiener process w has increments that are independent of F_{0} (by definition).

We wish to prove neocompact analogues of Propositions 4.4 and 4.3 for the sets N_{k}. For this we must deal with the two integrals in γ̃ separately. We may write

γ̃(y) = h(y) + I(y)

where I(y) denotes the infinite dimensional stochastic integral

I(y) = ∫_{0}^{·} g(s, y(s))dw(s).

and h(y) is the Bochner integral term

h(y)(s, ω) = ∫_{0}^{s}[−νAy(t, ω) − B(y(t, ω)) + f(t, y(t, ω))]dt = γ(y(ω))(s).

Here γ : M_{∞} → M is slightly more general than the function γ in Section 4 because of the feedback in f, but Propositions 4.3 and 4.4 are still valid; i.e. γ here is also continuous on M_{k} and γ(M_{k}) is relatively compact for each k > 0 (with the obvious modifications to the definition of M_{k}).

First we deal with h. Define the set

N̂_{k} = {y ∈ L^{0}(Ω, M) : E(E(y)) ≤ k}

which is neoclosed, and let

M̂_{m} = {y ∈ L^{0}(Ω, M) : E(y(·)) ≤ m a.s. } = L^{0}(Ω, M_{m}).

First we note that:

Lemma 6.4 Assume the hypotheses of Theorem 6.2. Then for each m > 0 the function h is neocontinuous from M̂_{m} into L^{0}(Ω, M) and h(M̂_{m}) is contained in a neocompact subset of L^{0}(Ω, M).

Proof Since h(y)(ω) = γ(y(ω)) and γ is continuous, it follows that h is neocontinuous on each M̂_{m}. Since γ(M_{m}) is relatively compact, it follows that h(M̂_{m}) is contained in a neocompact set. □

Lemma 6.5 Assume the hypotheses of Theorem 6.2. Then for each k > 0 the function h is neocontinuous from N̂_{k} into L^{0}(Ω, M) and h(N̂_{k}) is contained in a neocompact subset of L^{0}(Ω, M).

Proof For each m let D_{m} be a neocompact set with h(M̂_{m}) ⊆ D_{m}. Fix k, and take y ∈ N̂_{k}. By Chebyshev’s inequality, for each m we have P (E(y) ≤ m) ≥ 1 − \frac{k}{m}. Define y^{m} by

y^{m} = \{ \begin{matrix}y & on the set \{E(y) ≤ m\} \\ 0 & otherwise \end{matrix}

Then we have y^{m} ∈ M̂_{m} and h(y) = h(y^{m}) on \{E(y) ≤ m\}. Hence ρ(h(y), h(y^{m})) ≤ \frac{k}{m} and so h(y) ∈ (D_{m})^{k/m}. Thus h(N̂_{k}) ⊆ D where

D = ⋂_{m}(D_{m})^{\frac{k}{m}}

which is neocompact by closure under diagonal intersections.

To see that h is neocontinuous on N̂_{k}, take a neocompact set C ⊆ N̂_{k} and for each m let C^{m} = {y^{m} : y ∈ C} ⊆ M̂_{m}. The function y ↦ y^{m} is neocontinuous, and so C^{m} is neocompact. The graph of h restricted to C is the neocompact set

⋂_{m}\{(y, v) ∈ C × D : (∃z ∈ C^{m})(ρ(y, z) + ρ(v, h(z)) ≤ \frac{2k}{m} )\}.

□

Corollary 6.6 Assume the hypotheses of Theorem 6.2. Then for each k the function h is neocontinuous from N_{k} into N, and h(N_{k}) is contained in a neocompact subset of N with respect to L^{0}(Ω, M).

Proof This follows immediately from Lemmas 6.4 and 6.5, since N_{k} ⊆ N̂_{k}. □

To obtain the same result for I we must first prepare the way by dealing with a class of bounded integrands. For each m > 0 define

J_{m} = N ∩ {y : sup_{t≤T} |y(t)|^{2} ≤ m a.s.};

clearly J_{m} is neoclosed.

Lemma 6.7 Assume the hypotheses of Theorem 6.2. Then for each m the function I is neocontinuous from J_{m} into N, and I(J_{m}) is contained in a neocompact subset of N with respect to L^{0}(Ω, M).

Proof Let I_{n}(y) denote the finite dimensional stochastic integral

I_{n}(y) = ( ∫_{0}^{·} g(s, y(s))dw(s))^{(n)}.

The results of [6] show that for each n and m, the function I_{n} is neocontinuous from J_{m} into L^{0}(Ω, M). The bound (iii) on g insures that for fixed m the set ∪_{n}law(I_{n}(J_{m})) is tight, so the function I maps J_{m} into a neocompact set D in L^{0}(Ω, M).

For each neocompact set C ⊆ J_{m} in L^{0}(Ω, M), the graph of I|C is the neocompact set

⋂_{n} {(y, z) ∈ C × D : Π_{n}(z) = I_{n}(y)}.

Therefore I is neocontinuous on J_{m}. □

Now we must extend Lemma 6.7 to N_{k}.

Let ST be the set of all stopping times τ ∈ L^{0}(Ω, [0, T]) and for any process y let y^{τ} denote the stopped process y(ω)(t ∧ τ). It is shown in [FK] that ST is neocompact, and that the function (y, τ) ↦ y^{τ} is neocontinuous from N × ST into N.

Proposition 6.8 For each k > 0 the set I(N_{k}) is contained in a neocompact subset of N with respect to L^{0}(Ω, M).

Proof By Lemma 6.7, for each m ∈ IN there is a neocompact set D_{m} ⊆ N such that

I(J_{m}) ⊆ D_{m}.

If y ∈ N_{k} then E(sup_{t≤T} |y(t)|^{2}) ≤ k, and by Chebyshev’s inequality we have

P (sup_{t≤T} |y(t)|^{2} ≤ m) ≥ 1 − \frac{k}{m}.

For y ∈ N let τ_{m,y}(ω) be the first time t such that either t = T or sup_{s≤t} |y(s)|^{2} ≥ m. Then τ = τ_{m,y} is a stopping time; if y ∈ N_{k}, then y^{τ} belongs to J_{m} and P (y = y^{τ}) ≥ 1 − \frac{k}{m}; i.e. ρ(y, y^{τ}) ≤ \frac{k}{m}. Moreover I(y) = I(y^{τ}) on the set \{y = y^{τ}\} and so ρ(I(y), I(y^{τ})) ≤ \frac{k}{m}. But I(y^{τ}) ∈ D_{m} and hence I(y) ∈ (D_{m})^{k/m} in the space L^{0}(Ω, M). It follows that I(N_{k}) is contained in the set

D = N ∩ ⋂_{m}(D_{m})^{\frac{k}{m}} .

By closure under diagonal intersections, D is neocompact in L^{0}(Ω, M). □

Proposition 6.9 Assume the hypotheses of Theorem 6.2. Then for each k the function I is neocontinuous from N_{k} into N.

Proof Let C be a neocompact subset of N_{k}. By the preceding proposition, there is neocompact D ⊇ I(N_{k}). Since C and ST are neocompact, the set C̃ = {y^{τ} : y ∈ C ∧ τ ∈ ST } is neocompact. Moreover, the argument in the proof of Proposition 6.8 shows that for each y ∈ C, for each m there is a stopping time τ such that y^{τ} ∈ C̃ ∩ J_{m} and ρ(y, y^{τ}) ≤ \frac{k}{m} and ρ(I(y), I(y^{τ})) ≤ \frac{k}{m}. It follows that the graph of I|C is the neocompact set

⋂_{m}\{(y, v) ∈ C × D : (∃z ∈ C̃ ∩ J_{m})(ρ(y, z) + ρ(v, I(z)) ≤ \frac{2k}{m} )\},

so I is neocontinuous on N_{k}. □

The following proposition is the key to the construction of a solution. It is the neocompact analogue of propositions 4.4 and 4.3, and follow immediately from Corollary 6.6 and Propositions 6.8, 6.9.

Proposition 6.10 Assume the hypotheses of Theorem 6.2. Then for each k > 0 the function γ̃ is neocontinuous from N_{k} into N and the set γ̃(N_{k}) is contained in a neocompact subset of N with respect to L^{0}(Ω, M).

Next we define the Galerkin approximation to equation (11) and show that the stochastic version of Theorem 4.5 holds. We set

γ̃_{n}(y) = Π_{n}γ̃(y).

Then we have

Theorem 6.11 There are constants K_{1} and K_{2} such that for each n > 0 and x ∈ H there is y ∈ N_{0} ∩ G^{⊥} such that y = x^{(n)} + γ̃_{n}(y) (so y ∈ G^{⊥} ∩ N_{K_{1}+K_{2}|x|^{2}}) .

Proof The equation y = x^{(n)} + γ̃_{n}(y) is a finite dimensional SDE which can be solved by routine standard methods and has a solution. It was shown in [3] that there are K_{1} and K_{2}, independent of n (and which can be given explicitly in terms of the parameters ν, T, trQ, ∫_{0}^{T} a^{2}), such that E(E(y)) ≤ K_{1} + K_{2}|y(0)|^{2} whenever y = x^{(n)} + γ̃_{n}(y). The proof of this estimate uses the same ideas as are used for estimating the energy for the Galerkin approximation to the deterministic Navier-Stokes equations, coupled with the Burkholder-Davis-Gundy inequality and an application of Gronwall’s lemma. (In [3] it was established for an infinite integer n, by transfer of finite dimensional methods which are used in the present situation.) □

Theorem 6.12 (Neo-Dini’s Theorem) For a rich adapted space Ω Ω, let D ⊆ L^{0}(Ω, M) be a neocompact set and suppose that f_{n} : D → IR is a sequence of neocontinuous functions with f_{n}(x) ↘ 0 monotonically as n → ∞ for each fixed x ∈ D. Then f_{n} → 0 uniformly on D.

Proof Suppose f_{n} does not converge uniformly to 0 on D. Then there is an ϵ > 0 such that for arbitrarily large n ∈ IN, the set

D_{n} = {x ∈ D : |f_{n}(x)| ≥ ϵ}

is nonempty. Since f_{n}(x) converges monotonically for each x, the sets D_{n} form a decreasing chain. Each D_{n} is neocompact because D is neocompact and f_{n} is neocontinuous. By the countable compactness property, ⋂_{n} D_{n} is nonempty, contrary to hypothesis. □

We now use the neo-Dini’s theorem to prove a stochastic analogue of Proposition 4.6.

Proposition 6.13 For k > 0, γ̃_{n}(y) → γ̃(y) uniformly on N_{k}.

Proof By Proposition 6.10, γ̃(N_{k}) ⊆ D for some neocompact set D ⊆ N. The projection function Π_{n} is continuous on M and therefore neocontinuous on L^{0}(Ω, M). By the above neo-Dini’s Theorem, Π_{n}(z) → z uniformly on D, and the result follows. □

Proof of Theorem 6.2 We argue as in the proof of Theorem 4.1. Let x = u_{0}. It suffices to show that there exists y ∈ G^{⊥} ∩ N_{k} where k = K_{1} + K_{2}|x|^{2} such that y = x + γ̃(y). By Proposition 12, there is a neocompact set D ⊆ N such that γ̃(N_{k}) ⊆ D. By 6.11 and 6.13, for each ϵ >0 there exists n ∈ IN and y_{n} ∈ G^{⊥} ∩ N_{k} such that

y_{n} = x^{(n)} + γ̃_{n}(y_{n}),

and in the space N, y_{n} is within ϵ of x + γ̃(y_{n}) which belongs to D. Thus

(∀ϵ > 0)(∃z ∈ D^{ϵ} ∩ G^{⊥} ∩ N_{k})(ρ(z, x + γ̃(z)) ≤ ϵ).

By Proposition 6.10, γ̃ is neocontinuous on N_{k}. The Approximation Theorem now gives the required solution y ∈ G^{⊥} ∩ N_{k} such that y = x + γ̃(y). □

We conclude with an existence theorem for stochastic Navier-Stokes equations with a random initial condition.

Theorem 6.14 Let Ω Ω be a rich adapted space. Suppose that u_{0}(ω) is a G_{0}-measurable random variable on Ω Ω with values in H, and f, g satisfy the hypotheses of Theorem 6.2. Then the stochastic Navier-Stokes equation (11) has a solution u(t, ω) on Ω Ω with initial value u(0, ω) = u_{0}(ω). Moreover, if E [|u_{0}(ω)|^{2}] < ∞, then

E (sup_{t≤T} |u(t)|^{2} + ∫_{0}^{T} ∥u(t)∥^{2}dt) < ∞.  (13)

Proof We shall use the fact that Theorem 6.2 gives us a solution with a deterministic initial value which is independent of G_{0}.

Let x(ω) = u_{0}(ω) be a G_{0}-measurable initial value. Since H is separable there is an increasing sequence of compact sets C_{n} ⊆ H with P [x(ω) ∈ C_{n}] ≥ 1 − 1/n for each n ≥ 1. Let k_{n} = K_{1} + K_{2} sup_{z∈C_{n}} |z|^{2}. Let D_{n} be a neocompact subset of N such that

D_{n} ⊇ γ̃(N_{k_{n}}).

We can take the sets D_{n} to be increasing. Let G_{n} be the neocompact set of all G_{0}-measurable random variables in L^{0}(Ω, C_{n}). Then the set B_{n} = G_{n} + D_{n} is a neocompact subset of N such that

B_{n} ⊇ G_{n} + γ̃(N_{k_{n}}).

By closure under diagonal intersections (Lemma 2.9), the set

B = ⋂_{n≥1} ((B_{n})^{1/n})

is neocompact.

For each n ≥ 1 there is a simple function x_{n} ∈ G_{n} such that for each m ≤ n, whenever x(ω) ∈ C_{m} then x_{n}(ω) ∈ C_{m} and x_{n}(ω) is within 1/n of x(ω). Hence x_{n} ∈ (G_{m})^{1/m} for all m ≤ n.

By piecing together finitely many solutions from Theorem 6.2 which are independent of G_{0}, we see that for each n there is a y_{n} ∈ N_{k_{n}} ∩ B_{n} such that

y_{n} = x_{n} + γ̃(y_{n})  (14)

So y_{n} ∈ (B_{m})^{1/m} for each 1 ≤ m ≤ n, and it follows (since the sets B_{n} are increasing) that y_{n} ∈ B. Thus for each n,

(∃x_{n} ∈ {x}^{1/n})(∃y_{n} ∈ B)[y_{n} = x_{n} + γ̃(y_{n})].

Since the addition function is neocontinuous on random variables, it follows that the function

(z(·), y(·)) ↦ z(·) + γ̃(y(·))

is neocontinuous on L^{0}(Ω, H)×N_{k_{n}}. So by the Approximation Theorem there is a stochastic process y(·) ∈ B with

y = x + γ̃(y)

as required.

Suppose finally that E [|x(ω)|^{2}] = m < ∞. By Theorem 6.2, the y_{n} may be chosen so that y_{n} ∈ N_{k}, where k = K_{1} + K_{2}m. Since N_{k} is neoclosed, we may take the solution y to be in N_{k}, and therefore (13) holds. □

In Theorem 6.14, it would be more natural to allow the initial value u_{0}(ω) to be F_{0}- measurable rather than G_{0}-measurable, where F_{0} is the completion of ⋂_{t>0} G_{t}. The following corollary shows that this can be done if the rich adapted space is good enough. The construction of a rich adapted space in [7] actually produces a rich adapted space

Ω Ω = (Ω, P, G, G_{t})

with the following additional property:

For every complete separable M and every F_{0}-measurable random variable x ∈ L^{0}(Ω, M) there is a σ-algebra G′_{0} ⊆ F_{0} such that x is G′_{0}-measurable and the adapted space Ω Ω with G′_{0} in place of G_{0} is still rich.

Corollary 6.15 If Ω Ω is a rich adapted space with the above additional property, then Theorem 6.14 holds with F_{0} in place of G_{0}.

7 Optimal solutions

We show in this section how the neocompact machinery provides existence of optimal solutions to the stochastic Navier-Stokes equations - where the term ‘optimal’ is capable of a wide variety of interpretations. We shall restrict attention to the case of a fixed initial condition u_{0} = x ∈ H, as in Theorem 6.2. Recall that we proved above that there is a solution y to the equation (11) in the set N_{k(x)} where k(x) = K_{1} + K_{2}|x|^{2} (in fact the solution constructed is in G^{⊥} also). It is natural to define the following solution set S_{x} for the initial condition x ∈ H:

S_{x} = {y ∈ N_{k(x)} : y is a solution to the stochastic Navier-Stokes equation and y(0) = x}

= {y ∈ N_{k(x)} : y = x + γ̃(y)}

The key point now is the observation that

Theorem 7.1 For each x ∈ H the set S_{x} is neocompact.

Proof (i) Let f(y) = y − x − γ̃(y), which is neocontinuous on the neoclosed set N_{k(x)}, by Proposition 6.10. Clearly we have

S_{x} = N_{k(x)} ∩ f^{−1}({0})

which is neoclosed. Moreover, S_{x} ⊆ x + γ̃(N_{k(x)}) which is contained in a neocompact set (by Proposition 6.10 again). Hence S_{x} is neocompact. □

The following is a general optimality result for solutions to the stochastic Navier-Stokes equations; it is a special case of a general optimization result found in [6].

Theorem 7.2 Suppose that g : N_{0} → \overline{IR} is a function that is neo-lsc on N_{k(x)}; then there is a solution ŷ ∈ S_{x} with

g(ŷ) = inf{g(y) : y ∈ S_{x}}

i.e. ŷ is an optimal solution for the quantity g(y).

Proof Since g is neo-lsc and S_{x} is neocompact, the upper graph

A = \{(y, r) ∈ S_{x} × \overline{IR} : g(y) ≤ r\}

is neocompact; so the set

B = \{r ∈ \overline{IR} : (∃y ∈ S_{x})(y, r) ∈ A\}.

is a neocompact subset of \overline{IR}, which means that it is in fact compact. Thus B has a minimum element s, say, and there is ŷ ∈ S_{x} with g(ŷ) ≤ s. On the other hand, for any y ∈ S_{x}, putting r = g(y) we have (y, r) ∈ A and so r ∈ B. Thus s ≤ r = g(y) and the optimality of ŷ for g is established. □

There are several natural functions g for which optimal solutions might be sought - for example the function E(E(y)) that occurs naturally in the proofs of the previous section, and the expected energy integral

E(y) = \frac{1}{2}E(∫_{0}^{T} |y(t)|^{2}dt

and the expected enstrophy integral

S(y) = \frac{1}{2}E(∫_{0}^{T} ∥y(t)∥^{2}dt.

Each of these functions is neo-lsc.

On the face of it, the optimality theorem appears to be specific to the particular rich adapted space on which we are working. However, it has been shown that rich adapted spaces are universal, so that if y′ is a solution on some other adapted space Ω Ω′ then there is y ∈ L^{0}(Ω, M) with the same adapted distribution as y′ and so for any neo-lsc function g of the form g(y) = Eg_{0}(y(·)) where g_{0} is lsc, we have the existence of globally optimal solutions. This applies to the natural examples E, E, and S above.

The optimality result above can be generalised to the case where the initial condition u_{0} is random, or specialised to the deterministic setting (where g = 0).

References

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[13] R.Temam, Navier-Stokes equations, North-Holland, Amsterdam 1979.

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