Luigi C Berselli
email: luigi.carlo.berselli@unipi.it

Roger Lewandowski
email: roger.lewandowski@univ-rennes1.fr

On the Reynolds time-averaged equations and the long-time behavior of Leray-Hopf weak solutions, with applications to ensemble averages

Keywords: Navier-Stokes equations, time-averaging, Reynolds equations, Boussinesq hypothesis. 2010 MSC: Primary 35Q30. Secondary: 76D05, 76D06, 76F05. 1

We consider the three dimensional incompressible Navier-Stokes equations with non stationary source terms f chosen in a suitable space. We prove the existence of Leray-Hopf weak solutions and that it is possible to characterize (up to sub-sequences) their long-time averages, which satisfy the Reynolds averaged equations, involving a Reynolds stress. Moreover, we show that the turbulent dissipation is bounded by the sum of the Reynolds stress work and of the external turbulent fluxes, without any additional assumption, than that of dealing with Leray-Hopf weak solutions.

Finally, in the last section we consider ensemble averages of solutions, associated to a set of different forces and we prove that the fluctuations continue to have a dissipative effect on the mean flow.

Introduction

Let us consider the 3D homogeneous incompressible Navier-Stokes equations (NSE in the sequel),

         v t + (v • ∇) v -ν∆v + ∇p = f in ]0, +∞[×Ω, ∇ • v = 0 in ]0, +∞[×Ω, v = 0 on ]0, +∞[×Γ, v(0, x) = v 0 (x) in Ω, (1.1)
where Ω ⊂ R 3 is a bounded Lipschitz domain, Γ = ∂Ω its boundary, v = v(t, x)

denotes the fluid velocity, p = p(t, x) the pressure, f = f (t, x) the external source term, and (t, x) ∈]0, +∞[×Ω. The main aim of this paper is to study the longtime averages of weak solutions to the NSE (1.1), namely

v(x) := lim t→∞ M t (v), where M t (v) := 1 t t 0 v(s, x) ds, (1.2)
when the source term f is time dependent, and to link it to the Reynolds averaged equations (see (4.6) below). We also will consider the problem of ensemble averages, which is closely related to long-time averages. Long-time average for "tumultuous" flows, today turbulent flows, seem to have been considered first by G. Stokes [START_REF] Stokes | On the Effect of the Internal Friction of Fluids on the Motion of Pendulums[END_REF] and then by O. Reynolds [START_REF] Reynolds | On the dynamic theory of the incompressible viscous fluids and the determination of the criterion[END_REF]. The idea is that steady-state turbulent flows are oscillating around a stationary flow, which can be expressed through long-time averages. L. Prandtl used them to introduce the legendary "Prandtl mixing length" (see [START_REF] Prandtl | Über die ausgebildeten Turbulenz[END_REF] and in [START_REF] Prandtl | Guide à travers la mécanique des fluides[END_REF]Ch. 3,Sec. 4]) to model a turbulent boundary layer over a plate. However, although long-time averaging plays a central role in turbulence modeling (since it is natural as well as "ensemble averaging") it is not clear whether the limit (1.2) is well defined, or not.

The mathematical problem of properly defining long-time averages and to investigate the connection with the Reynolds equations was already studied before, but when the source term does not depend on time, namely f = f (x).

So far as we know, the first who considered this issue is C. Foias [START_REF] Foias | Statistical study of Navier-Stokes equations. I, II[END_REF]Sec. 8], when f ∈ H, where

H := {u ∈ L 2 (Ω) 3 s.t. ∇ • u = 0, u • n = 0 on Γ}. (1.3)
Foias analysis is based on the notion of "statistical solution" introduced in [START_REF] Foias | Statistical study of Navier-Stokes equations. I, II[END_REF]Sec. 3] and he was able to prove that for any given Leray-Hopf solution v = v(t, x) to (1.1) then:

-There exists v ∈ H, such that, up to a sub-sequence, M t (v) → v in H as t → ∞; -There is a stationary statistical solution µ to the NSE, which is a probability measure on H, such that it holds v = H w dµ(w);

-There exists a random variable on (H, µ) called v (in the notation of Foias called δv) such that

v = H v dµ = 0 and ∇v 2 = H ∇v 2 dµ < ∞,
such that (v, v ) is a solution to the Reynolds Equations given by (4.6) below. If µ is a statistical solution, v is the random variable expressed by its probability law, Prob(v ∈ F ) = µ(v + F ), (1.4) for any Borel set F ∈ H.

-According to our notations, the Reynolds stress σ (r) given by

∇ • σ (r) = H ∇ • (v -v) ⊗ (v -v) dµ(v),
is dissipative on the mean flow, which is one of the main challenges of such analysis, because of the Boussinesq assumption (see in [START_REF] Rebollo | Mathematical and numerical foundations of turbulence models and applications[END_REF]Ch. 4,Sec. 4.4.3.1]).

In fact this is much more better, since Foias in [11, Sec. 8-2-a, Prop. 1, p. 99] was able to prove that the turbulent dissipation ε is bounded by the work of the Reynolds stress on the mean flow,

ε := ν ∇v 2 ≤ (∇ • σ (r) , v), (1.5)
where (•, •) and . denote the standard L 2 (Ω) scalar product and norm, respectively. (Sometimes norm is normalized by |Ω|, the measure of the domain, but this is inessential). This analysis is partially reported in [START_REF] Foias | Encyclopedia of Mathematics and its Applications[END_REF]Ch. 3,Sec. 3], where the limit in (1.2) is replaced by the abstract Banach limit, but without any link to the Reynolds equations and dissipation inequality such as (1.5).

The original Foias analysis is very deep and essential to the field. However, it is worth noting that in this approach: i) The natural time filter used to determine the Reynolds stress (initially suggested by Prandtl, when the limit makes sense)

σ (r) = σ (r) (x) := lim t→∞ 1 t t 0 [(v -v) ⊗ (v -v)](s, x) ds,
is replaced by an abstract probability measure that it is not possible to calculate in practical simulations, although the ergodic assumption -which remains to be proved-would mean that they coincide (see for instance Frisch [START_REF] Frisch | Turbulence: The legacy of A[END_REF]); ii) The fluctuation given by (1.4), when the force is time independent, is a time independent random variable, which may be questionable from the physical point of view. In fact, when v is given by (1.2), one cannot conclude that this v yields the Reynolds decomposition

v(t, x) = v(x) + v (t, x),
in which the fluctuation is time dependent for a realistic non stationary flow.

iii) Concerning items i) and ii) above, we can suggest that probably there are still work to be done in the interesting field of statistical solutions, if one wants to use this mathematical tool to face the tough closure problem in the Reynolds equations, and apply to realistic problem. Moreover, the case of a time dependent source term also remains to be considered in connection with the scope of statistical solutions (see the warning in [START_REF] Foias | Statistical study of Navier-Stokes equations. I, II[END_REF]Part I,Sec. 5,p. 313]).

Still in the case of a stationary source term, the long-time averaged problem has been also considered more recently in [START_REF] Lewandowski | Long-time turbulence model deduced from the Navier-Stokes equations[END_REF] and at the time the author was not aware yet of the connection between time-averaging and statistical solutions in Foias work. In [START_REF] Lewandowski | Long-time turbulence model deduced from the Navier-Stokes equations[END_REF], he studied the equation satisfies by M t (v) (see equations (5.1) below) for a Leray-Hopf weak solution of the NSE, and he took the limit in this equation when the domain if of class C 9/4,1 and f = f (x) ∈ L 5/4 (Ω) 3 ∩ V , where

V := {u ∈ H 1 0 (Ω) 3 s.t. ∇ • u = 0}, (1.6)
and V denotes the dual space of V , with duality pairing < . , . >. The analysis is based on the energy inequality, which yields a uniform estimate in time of the L 2 norm of v, and on a L p -regularity result by Amrouche and Girault [START_REF] Amrouche | On the existence and Regularity of the solutions of Stokes Problem in arbitrary dimension[END_REF] about the steady Stokes problems, valid in C k,α domains. It is shown in [START_REF] Lewandowski | Long-time turbulence model deduced from the Navier-Stokes equations[END_REF] that there exists σ (r) ∈ L 5/3 (Ω) 9 and p ∈ W 1,5/4 (Ω)/R such that, up to a sub-sequence, when t → ∞, M t (v) converges to a field v ∈ W 2,5/4 (Ω) 3 , which satisfies in the sense of the distributions the closed Reynolds equations:

     (v • ∇) v -ν∆v + ∇p + ∇ • σ (r) = f in Ω, ∇ • v = 0 in Ω, v = 0 on Γ. (1.7)
Moreover, it is also shown that σ (r) is dissipative on the mean flow, namely

0 ≤ (∇ • σ (r) , v), (1.8)
which is weaker than (1.5).

The main part of the present study is in continuation of [START_REF] Lewandowski | Long-time turbulence model deduced from the Navier-Stokes equations[END_REF], bringing substantial improvements. The novelty is that we are considering time dependent source terms f = f (t, x), which was never considered before for this problem, up to our knowledge. Moreover, we do not need extra regularity assumption on the domain Ω ⊂ R 3 and on f . The first main result of this paper, Theorem 2.3 below, is close to that proved in [START_REF] Lewandowski | Long-time turbulence model deduced from the Navier-Stokes equations[END_REF]. Roughly speaking, we will show that M t (v) given by (1.2), converges to some v (up to sub-sequences) and that there are p and σ (r) such that (1.7) holds, at least in D (Ω), in which f is replaced by f .

One key point is the determination of a suitable class for the source term.

Throughout the paper, we will take f : R + → V , made of function for which there is a constant C > 0 such that

∀ t ∈ R + t+1 t f (s) 2 V ds ≤ C.
In this respect we observe our results improve the previous ones also in terms of regularity of the force, not only because we consider a time-dependent one.

The main building block of our work is the derivation of a uniform estimate of the L 2 norm of v, for f as above (see (2.1) and its Corollary (2.2)). This allows to prove the existence of a weak solution on [0, ∞) to the NSE and to pass to the limit in the equation satisfied by M t (v), when t → ∞. Moreover, we are able to generalise Foias result (1.5) by proving that the turbulent dissipation ε is bounded by the sum of the work of the Reynolds stress on the mean flow and of the external turbulent fluxes, namely

ε = ν ∇v 2 ≤ (∇ • σ (r) , v) + < f , v >, (1.9)
which is one of the main features of our result. Note that physically, it is expected that (1.9) becomes an equality for strong solutions. Furthermore, we also prove that when f is "attracted" (in some sense, see (2.5) below) by a stationary force f = f (x) as t → ∞, then the turbulent fluxes < f , v > in (1.9) vanish, so that (1.5) is restored. In particular, the question whether the stress tensor σ (r) is dissipative remains an open problem for general unsteady f , as those for which we still have global existence of weak solutions.

In the second part of the paper, we will also consider ensemble averages, often used in practical experiments. This consists of considering n ∈ N realizations of the flow, {v 1 , . . . , v n } and to evaluate the arithmetic mean

v := 1 n n k=1 v k .
Layton considers such ensemble averages in [START_REF] Layton | The 1877 Boussinesq conjecture: Turbulent fluctuations are dissipative on the mean flow[END_REF], by introducing the corresponding Reynolds stress, written as

R(v, v) = v ⊗ v -v ⊗ v .
He shows that for a fixed stationary source term f and n strong solutions of the NSE, then R(v, v) is dissipative on the ensemble average, in time average. More specifically it holds lim inf

t→∞ M t [(∇ • R(v, v), v )] ≥ 0,
and to prove this inequality he first performs the ensemble average, then it takes the long-time average. In order to remove the additional assumption used in [START_REF] Layton | The 1877 Boussinesq conjecture: Turbulent fluctuations are dissipative on the mean flow[END_REF][START_REF] Jiang | Algorithms and models for turbulence not at statistical equilibrium[END_REF] of having strong solutions, we will carry out an approach very close, but in a reversed order: We first take the long-time average of the realizations for a sequence of time independent source terms {f k } k∈N . Then, we form the ensemble average

S n := 1 n n k=1 v k ,
where v k are weak solutions of the Reynolds equations. Under suitable (but very light) regularity assumption about the f k 's, we show the convergence of {S n } n∈N to some v that satisfies the closed Reynolds equations, and such that dissipativity still holds, that is

0 ≤ (∇ • σ (r) , v ),
which holds for weak solutions (see the specific statement in Theorem 2.4).

Plan of the paper. The paper is organized as follows. We start by giving in Sec. 2 the specific technical statements of the results we prove. Then, in Sec. 3 we give some results about the functional spaces we are working with and we prove in detail the main energy estimates (3.4) and (3.5) for source terms f ∈ L 2 uloc (R + , V ), which yields an existence result of global weak solutions to the NSE for such f . The Sec. 4 is devoted to the Reynolds problem and the develop additional properties of the time average operator M t . Finally, we give the proofs of the two main results on time averages in Sec. 5 and Sec. 6, respectively.

Main results

On the source term and an existence result

Since we aim to consider long-time averages for the NSE, we must consider solutions which are global-in-time (defined for all positive times). Due to the well-known open problems related to the NSE, this enforces us to restrict to weak solutions. By using a most natural setting, we take the initial datum v 0 ∈ H, where H is defined by (1.3).

The classical Leray-Hopf results of existence (but without uniqueness) of a global weak solution v to the NSE holds when f ∈ L 2 (R + ; V ), and the velocity

v satisfies v ∈ L 2 (R + , V ) ∩ L ∞ (R + , H),
where V is defined by (1.6) and V denotes its topological dual. We will also denote by < , > the duality pairing1 between V and V .

Source

terms f ∈ L 2 (R + ; V ) verify ∞ t f (s) 2
V ds → 0 when t → +∞. Therefore, a turbulent motion cannot be maintained for large t, which is not relevant for our purpose. The choice adopted in the previous studies on Reynolds equations was that of a constant force, and we also observe that many estimates could have been easily extended to a uniformly bounded f ∈ L ∞ (R + ; V . On the other hand, we consider a broader class for the source terms. According to the usual folklore in mathematical analysis, we decided to consider the space L 2 uloc (R + ; V ) made of all strongly measurable vector fields f : R + → V such that

f L 2 uloc (R+;V ) := sup t≥0 t+1 t f (s) 2 V ds 1/2
< +∞.

We will see in the following, that the above space, which strictly contains both L 2 (R + ; V ) and L ∞ (R + ; V ) is well suited for our framework. We will prove the following existence result, in order to make the paper self-contained.

Theorem 2.1. Let v 0 ∈ H, and let f ∈ L 2 uloc (R + ; V ). Then, there exists a weak solution v to the NSE (1.1) global-in-time, obtained by Galerkin approximations, such that v ∈ L 2 loc (R + ; V ) ∩ L ∞ (R + ; H), and which satisfies for all t ≥ 0,

v(t) 2 ≤ v 0 2 + 3 + C Ω ν F 2 ν , (2.1)
ν t 0 ∇v(s) 2 ds ≤ v 0 2 + ([t] + 1) F 2 ν , (2.2)
where F := f L 2 uloc (R+;V ) . Remark 2.2. The weak solution v shares most of the properties of the Leray-Hopf weak solutions, with estimates valid for all positive times. Notice that we do not know whether or not this solution is unique. Anyway, it will not get "regular" as t → +∞, which is the feature of interest for our study. As usual by regular we mean that it does not necessarily have the L 2 -norm of the gradient (locally) bounded, hence that it is not a strong solution.

Long-time averaging

This section is devoted to state the main results of the paper about long-time and ensemble averages.

Theorem 2.3. Let be given v 0 ∈ H, f ∈ L 2 uloc (R + ; V )
, and let v a global-intime weak solution to the NSE (1.1). Then, there exist a) a sequence {t n } n∈N such that lim

n→∞ t n = +∞; b) a vector field v ∈ V ; c) vector field f ∈ V ; d) a vector field B ∈ L 3/2 (Ω) 3 ;
e) a second order tensor field σ (r) ∈ L 3 (Ω) 9 ; such that it holds:

i) when n → ∞, M tn (v) v in V, M tn (f ) f in V , M tn (v • ∇) v B in L 3/2 (Ω) 3 , M tn (v ⊗ v ) σ (r) in L 3 (Ω) 9 ; M tn (< f , v >) → < f , v > +< f , v >, where v = v -v, and f = f -f ; ii) the closed Reynolds equations (2.
3) holds true in the weak sense:

     (v • ∇) v -ν∆v + ∇p + ∇ • σ (r) = f in Ω, ∇ • v = 0 in Ω, v = 0 on Γ; (2.3) iii) the following equalities F = B -(v • ∇) v = ∇ • σ (r) are valid in D (Ω);
iv) the following energy balance holds true

ν ∇v 2 + Ω F • v dx = < f , v >;
v) the turbulent dissipation ε is bounded by the sum of the work of the Reynolds stress on the mean flow and the external turbulent fluxes,

ε = ν ∇v 2 ≤ Ω (∇ • σ (r) ) • v dx + < f , v >; (2.4)
vi) if in addition the source term f verifies:

∃ f ∈ V , such that lim t→+∞ t+1 t f (s) -f 2 V ds = 0, (2.5
)

then f = f and < f , v > = 0; in particular, the Reynolds stress σ (r) is dissipative in average, that is (1.8) holds true.
Our second result has to be compared with results in Layton et al. [START_REF] Labovsky | Magnetohydrodynamic flows: Boussinesq conjecture[END_REF][START_REF] Layton | The 1877 Boussinesq conjecture: Turbulent fluctuations are dissipative on the mean flow[END_REF], where the long-time averages are taken for an ensemble of solutions.

Theorem 2.4. Let be given a sequence {f k } k∈N ⊂ L q (Ω) converging weakly to some f in L q (Ω), with q > 6 5 and let {v k } k∈N be the associated long-time average of velocities, whose existence has been proved in Theorem 2.3. Then, the sequence of arithmetic averages of the long-time limits { v n } n∈N , defined as

v n := 1 n n k=1 v k ,
converges weakly, as n → +∞, in V to some v , which satisfies the following system of Reynolds type

     v • ∇ v -ν∆ v + ∇ p + ∇ • σ (r) = f in Ω, ∇ • v = 0 in Ω, v = 0 on Γ,
where σ (r) is dissipative in average, that is more precisely

0 ≤ 1 |Ω| Ω ∇ • σ (r) • v dx.
In this case we do not have a sharp lower bound on the dissipation as in Theorem 2.3, since here the averaging is completely different and the fluctuations are not those emerging in long-time averaging. Nevertheless, the main statement is in the same spirit of the first proved result.

Navier-Stokes equations with uniformly-local source terms

This section is devoted to sketch a proof of Theorem 2.1. Most of the arguments are quite standard and we will give appropriate references at each step, to focus on what seems (at least to us) non-standard when f ∈ L 2 uloc (R + ; V ); especially the proof of the uniform L 2 -estimate (3.4), which is the building block for the results of the present paper. Before doing this, we introduce the function spaces we will use, and precisely define the notion of weak solutions we will deal with.

Functional setting

Let Ω ⊂ R 3 be a bounded open set with Lipschitz boundary ∂Ω. This is a sort of minimal assumption of regularity on the domain, in order to have the usual properties for Sobolev spaces and to characterize in a proper way divergencefree vector fields in the context of Sobolev spaces, see for instance Constantin and Foias [START_REF] Constantin | Navier-Stokes equations[END_REF], Galdi [START_REF] Galdi | An introduction to the Navier-Stokes initial-boundary value problem[END_REF][START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems[END_REF], Girault and Raviart [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF], Tartar [START_REF] Tartar | An introduction to Navier-Stokes equation and oceanography[END_REF].

We use the customary Lebesgue spaces (L p (Ω), . p ) and Sobolev spaces (W 1,p (Ω), . 1,p ). For simplicity, we denote the L 2 -norm simply by . and we write H 1 (Ω) := W 1,2 (Ω). For a given sequence {x n } n∈N ⊂ X, where (X, . X ) is Banach space, we denote by x n → x the strong convergence, while by x n x the weak one.

As usual in mathematical fluid dynamics, we use the following spaces

V = {ϕ ∈ D(Ω) 3 , ∇ • ϕ = 0}, H = v ∈ L 2 (Ω) 3 , ∇ • v = 0, v • n = 0 on Γ , V = v ∈ H 1 0 (Ω) 3 , ∇ • v = 0 ,
and we recall that V is dense in H and V for their respective topologies [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF][START_REF] Tartar | An introduction to Navier-Stokes equation and oceanography[END_REF]. Let (X, . X ) be a Banach space, we use the Bochner spaces L p (I; X), for I = [0, T ] (for some T > 0) or I = R + equipped with the norm

u L p (I;X) :=      I u(s) p X ds 1 p for 1 ≤ p < ∞, ess sup s∈I u(s) X for p = +∞.
The existence of weak solutions for the NSE (1.1) is generally proved in the literature when v 0 ∈ H and the source term f ∈ L 2 (I; V ), or alternatively when the source term is a given constant element of V . In order to study the long-time behavior of weak solutions of the NSE (1.1), we aim to enlarge the class of function spaces allowed for the source term f , to catch a more complex behavior than that coming from constant external forces, as initially developed in [START_REF] Lewandowski | Long-time turbulence model deduced from the Navier-Stokes equations[END_REF]. To do so, we deal with "uniformly-local" spaces, as defined below in the most general setting. Definition 3.1. Let be given p ∈ [1, +∞[. We define L p uloc (R + ; X) as the space of strongly measurable functions f : R + → X such that

f L p uloc (X) := sup t≥0 t+1 t f (s) p X ds 1/p < +∞.
It is easily checked that the spaces L p uloc (R + ; X) are Banach spaces strictly containing both the constant X-valued functions, and also L p (R + ; X), as illustrated by the following elementary lemma. Lemma 3.2. Let be given f ∈ C(R + ; X) converging to a limit ∈ X, when t → +∞. Then, for any p ∈ [1, +∞[, we have that f ∈ L p uloc (R + ; X), and there exists T > 0 such that

f L p uloc (X) ≤ sup t∈[0,T +1] f (t) p X + 2 p-1 (1 + X ) p 1 p
.

Proof. As = lim t→+∞ f (t), there exists T > 0 such that:

∀ t > T , f (t)-X ≤ 1.
In particular, it holds

t+1 t f (s) p X ds ≤ 2 p-1 (1 + X ) p for t > T,
while for all t ∈ [0, T ],

t+1 t f (s) p X ds ≤ sup t∈[0,T +1] f (t) p X ,
hence the result.

However, it easy to find examples of discontinuous functions in L p uloc (R + ; X) which are not converging when t → +∞, and which are not belonging to L p (R + ; X).

Weak solutions

There are many ways of defining weak solutions to the NSE (see also P.-L. Lions [START_REF] Lions | Mathematical topics in fluid mechanics[END_REF]). Since we are considering the incompressible case, the pressure is treated as a Lagrange multiplier. Following the pioneering idea developed by J. Leray [START_REF] Leray | Sur le mouvement d'un liquide visqueux emplissant l'espace[END_REF], the NSE are projected over spaces of divergence-free functions. This is why when we talk about weak solutions the NSE, only the velocity v is mentioned, not the pressure.

As in J.-L. Lions [START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF], we give the following definition of weak solution, see also Temam [30, Ch. III]. Definition 3.3 (Weak solution). Given v 0 ∈ H and f ∈ L 2 (I; V ) we say that v is a weak solution over the interval I = [0, T ] if the following items are fulfilled: i) the vector field v has the following regularity properties

v ∈ L 2 (I; V ) ∩ L ∞ (I; H),
and is weakly continuous from I to H, while lim

t→0 + v(t) -v 0 H = 0; ii) for all ϕ ∈ V, d dt Ω v(t, x) • ϕ(x) dx - Ω v(t, x) ⊗ v(t, x) : ∇ϕ(x) dx + ν Ω ∇v(t, x) : ∇ϕ(x) dx = < f (t), ϕ >,
holds true in D (I);

iii) the energy inequality

d dt 1 2 v(t) 2 + ν ∇v(t) 2 ≤ < f (t), v(t) >, (3.1)
holds in D (I), where we write v(t) instead of v(t, •) for simplicity.

When f ∈ L 2 (0, T ; V ) and v is a weak solution in I = [0, T ], and this holds true for all T > 0, we speak of a "global-in-time solution", or simply a "global solution". In particular, ii) is satisfied in the sense of D (0, +∞).

There are several ways to prove the existence of (at least) a weak solution to the NSE. Among them, in what follows, we will use the Faedo-Galerkin method.

Roughly speaking, let {ϕ n } n∈N denote a Hilbert basis of V , and let, for n ∈ N,

V n := span{ϕ 1 , • • • , ϕ n }. By assuming f ∈ L 2 (I; V ), it
can be proved by the Cauchy-Lipschitz theorem (see [START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF]) the existence of a unique v n ∈ C 1 (I; V n ) such that for all ϕ k , with k = 1, . . . , n it holds

d dt Ω v n (t, x) • ϕ k (x) dx - Ω v n (t, x) ⊗ v n (t, x) : ∇ϕ k (x) dx + ν Ω ∇v n (t, x) : ∇ϕ k (x) dx = < f (t), ϕ k >, (3.2)
and which naturally satisfies the energy balance (equality)

d dt 1 2 v n (t) 2 + ν ∇v n (t) 2 = < f , v n > . (3.3)
It can be also proved (always see again [START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF]) that from the sequence {v n } n∈N one can extract a sub-sequence converging, in an appropriate sense, to a weak solution to the NSE. When I = R + we get a global solution.

However, if assume f ∈ L 2 uloc (R + ; V ), the global result of existence does not work so straightforward. Of course, for any given T > 0, we have

L 2 uloc (R + ; V ) [0, T ] → L 2 ([0, T ]; V ),
where

L 2 uloc (R + ; V ) |[0,T ] denotes the restriction of a function in L 2 uloc (R + ; V ) to [0, T ].
Therefore, no doubt that the construction above holds over any timeinterval [0, T ]. In such case letting T go to +∞ to get a global solution (with some uniform control of the kinetic energy) is not obvious, and we do not know any reference explicitly dealing with this issue, which deserves to be investigated more carefully. This is the aim of the next subsection, where we prove the most relevant a-priori estimates.

A priori estimates

Let be given f ∈ L 2 uloc (R + ; V ), and let v n = v be the solution of the Galerkin projection of the NSE over the finite dimensional space V n . The function v satisfies (3.2) and (3.3) (we do not write the subscript n ∈ N for simplicity), is smooth, unique, and can be constructed by the Cauchy-Lipschitz principle over any finite time interval [0, T ]. Hence, we observe that by uniqueness it can be extended to R + . We then denote F := f L 2 uloc (R+;V ) and then after a delicate manipulation of the energy balance combined with the Poincaré inequality, we get the following lemma. Lemma 3.4. For all t ≥ 0 we have

v(t) 2 ≤ v 0 2 + 3 + C Ω ν F 2 ν , (3.4)
as well as

ν t 0 ∇v(s) 2 ds ≤ v 0 2 + ([t] + 1) F 2 ν , (3.5)
where C Ω denotes the constant in the Poincaré inequality u 2 ≤ C Ω ∇u 2 , valid for all u ∈ V .

Proof. We focus on the proof of the a priori estimate (3.4), the estimate (3.5) being a direct consequence of the energy balance. By the Young inequality we deduce from the the energy inequality,

∀ ξ, τ ∈ R + s.t. 0 ≤ ξ ≤ τ, v(τ ) 2 + ν τ ξ ∇v(s) 2 ds ≤ v(ξ) 2 + 1 ν τ ξ f (s) 2 V ds.
In particular, when 0 ≤ τ -ξ ≤ 1,

v(τ ) 2 + ν τ ξ ∇v(s) 2 ds ≤ v(ξ) 2 + F 2 ν . (3.6)
From this point, we argue step by step. The case 0 ≤ t ≤ 1 is the first step, which is straightforward. The second step is the heart of the proof. The issue is that energy may increase, without control, when the time increases.

We will show that even if this happens, we can still keep the control on it, thanks to (3.6). The last step is the concluding step, carried out by induction on n writing t = τ + n, for τ ∈ [0, 1].

Step 1. t ∈ [0, 1]: take ξ = 0, t = τ ∈ [0, 1]. Then v(t) 2 ≤ v 0 2 + F 2 ν ∀ t ∈ [0, 1].
Step 2. We show that the following implication holds true

v(t) ≤ v(t + 1) ⇒        v(t) ≤ 1 + C Ω ν F 2 ν , v(t + 1) ≤ 2 + C Ω ν F 2 ν .
In the following we will set

C 2 := v 0 2 + F 2 ν + 2 + C Ω ν F 2 ν = v 0 2 + 3 + C Ω ν F 2 ν . (3.7)
Sub-Step 2.1. By the energy inequality with ξ = t, and τ = t + 1, we have, by using the hypothesis on the L 2 -norm at times t and t + 1:

ν t+1 t ∇v(s) 2 ds ≤ v(t + 1) 2 -v(t) 2 + ν t+1 t ∇v(s) 2 ds ≤ F 2 ν .
Hence, by the Poincaré's inequality:

t+1 t v(s) 2 ds ≤ C Ω t+1 t ∇v(s) 2 ds ≤ C Ω F 2 ν 2 , (3.8)
Sub-Step 2.2. Let be given > 0 and let ξ ∈ [t, t + 1] be such that

v(ξ) 2 < inf s∈[t,t+1] v(s) 2 + ≤ v(s) 2 + ∀ s ∈ [t, t + 1].
Let us write:

v(t) 2 ≤ v(t + 1) 2 = v(t + 1) 2 -v(ξ) 2 + v(ξ) 2 = v(t + 1) 2 -v(ξ) 2 + t+1 t v(ξ) 2 ds,
being the integration with respect to the s variable.

To estimate the right-hand side we use the energy inequality with τ = t + 1 to get

v(t + 1) 2 -v(ξ) 2 ≤ F 2 ν .
Moreover, using the estimate (3.8) we get,

t+1 t v(ξ) 2 ds ≤ t+1 t ( v(s) 2 + ) ds ≤ C Ω F 2 ν 2 + ε, therefore, letting ε → 0 yields, v(t) 2 ≤ 1 + C Ω ν F 2 ν .
In addition, we get from the energy inequality

v(t + 1) 2 ≤ v(t) 2 + F 2 ν ≤ 2 + C Ω ν F 2 ν .
Step 3. Conclusion of the proof of (3.4). Any t ≥ 0 can be decomposed as Once we have proved that the uniform (independent of n ∈ N) L 2 -estimate is satisfied by the Galerkin approximate functions, it is rather classical to prove that we can extract a sub-sequence that converges weakly * in L ∞ (0, T ; L 2 (Ω)) (for all positive T ) to a weak solution to the NSE, which inherits the same bound. We refer to the references already mentioned for this point.

t = n + τ,

Reynolds decomposition and time-averaging

We sketch the standard routine, concerning time-averaging, when used in turbulence modeling practice. In particular, we recall the Reynolds decomposition and the Reynolds rules. Then, we give a few technical properties of the timeaveraging operator M t , defined by

M t (ψ) := 1 t t 0 ψ(s) ds,
for a given fixed time t > 0. We need to apply it not only to real functions of a real variable, but also to Banach valued functions, hence we need to deal with the Bochner integral. Before all, we start with the following corollary of Bochner theorem (see Yosida [31, p. 132]). Lemma 4.1. Assume that, for some t > 0 we have ψ ∈ L p ([0, t]; X) (namely ψ is a Bochner p-summable function over [0, t], with values in the Banach space X). Then, it holds

M t (ψ) X ≤ 1 t 1 p ψ L p ([0,t];X) . (4.1)
Estimate (4.1) is the building block to give a sense to the long-time average as

ψ := lim t→∞ M t (ψ), (4.2)
whenever the limit exists.

It is worth noting at this stage that the mapping µ behind the long-time average, defined on the Borel sets of R + by

A → lim t→+∞ 1 t λ(A ∩ [0, t]) := lim t→+∞ M t (1 A ) = µ(A),
where λ the Lebesgue measure, is not -strictly speaking-a probability measure since it is not σ-additive2 . Therefore, the quantity ψ is not rigorously a statistic, even if practitioners could be tempted to write it (in a suggestive and evocative meaningful way) as follows:

ψ(x) = R+ ψ(s, x) dµ(s).

General setup of turbulence modeling

We recall that M t is a linear filtering operator which commutes with differentiation with respect to the space variables (the so called Reynolds rules). In particular, one has the following result (its proof is straightforward), which is essential for our modeling process.

Lemma 4.2. Let be given ψ ∈ L 1 ([0, T ], W 1,p (Ω)), then

DM t (ψ) = M t (Dψ) ∀ t > 0,
for any first order differential operator D acting on the space variables x ∈ Ω.

By denoting the long-time average of any field ψ by ψ as in (4.2), we consider the fluctuations ψ around the mean value, given by the Reynolds decomposition

ψ := ψ + ψ .
Observe that long-time averaging has many convenient formal mathematical properties, recalled in the following.

Lemma 4.3. The following formal properties holds true, provided the long-time averages do exist.

1. The "bar operator" preserves the no-slip boundary condition. In other words, if ψ |Γ = 0, then ψ |Γ = 0;

2. Fluctuation are in the kernel of the bar operator, that is ψ = 0;

3. The bar operator is idempotent, that is ψ = ψ, which also yields ψ ϕ = ψϕ.

Accordingly, the velocity components can be decomposed in the Reynolds decomposition as follows:

v(t, x) = v(x) + v (t, x).
Let us determine (at least formally) the equation satisfied by v. To do so, we use the above Reynolds rules to expand the nonlinear quadratic term into

v ⊗ v = v ⊗ v + v ⊗ v , (4.3)
which follows by observing that v

⊗ v = v ⊗ v = 0.
The above rules allow us to prove the following result showing a certain "orthogonality" between averages and fluctuations. Lemma 4.4. Let be given a linear space X ⊆ L 2 (Ω) with a scalar product ( . , . ). Let in addition be given a function ψ : R + → X such that ψ is well defined. Then it follows that (ψ , ψ ) = (ψ, ψ) -(ψ, ψ), provided all averages are well defined.

Proof. The proof follows by observing that

ψ = ψ -ψ, hence (ψ , ψ ) = (ψ -ψ, ψ -ψ) = (ψ, ψ) -2(ψ, ψ) + (ψ, ψ),
and by the Reynolds rules (ψ, ψ) = (ψ, ψ) and (ψ, ψ) = (ψ, ψ), from which is follows the thesis.

In particular, we will use it for the V scalar product showing that

∇u 2 = ∇u 2 -∇u 2 , (4.4)
for u : R + → V , such that the long-time average exists.

Remark 4.5. Observe that, for weak solutions of the NSE v, the average M t ( ∇v 2 ) is bounded uniformly, by the result of Theorem 2.1 and -up to subsequences-some limit can be identified. Moreover, by using an argument similar to Lemma 4.1 it follows that

M t (∇v) 2 ≤ M t ( ∇v 2 ),
which show that (up to sub-sequences) also the second term from the right-hand side can be properly defined. Consequently, also the average of the squared Vnorm of the fluctuations from the left-hand side is well defined by difference.

Long-time averaging applied to the Navier-Stokes equations (in a strong formulation) gives the following "equilibrium problem" for the long-time average

v(x),      -ν∆v + ∇ • (v ⊗ v) + ∇p = f in Ω, ∇ • v = 0 in Ω, v = 0 on Γ, (4.5)
which we will treat in the next section to make appear the closure problem.

Reynolds stress and Reynolds tensor

The first equation of system (4.5) can be rewritten also as follows (by using the decomposition into averages and fluctuations)

-ν∆v + ∇ • (v ⊗ v) + ∇p = -∇ • (v ⊗ v ) + f , (4.6)
called the Reynolds equations. Beside convergence issues, a relevant point is to characterize the average of product of fluctuations from the right-hand side, which is the divergence of the so called Reynolds stress tensor, defined as follows

σ (r) = v ⊗ v . (4.7)
The Boussinesq hypothesis, formalized in [START_REF] Boussinesq | Theorie de l'ècoulement tourbillant[END_REF] (see also [START_REF] Rebollo | Mathematical and numerical foundations of turbulence models and applications[END_REF]Ch. 3 & 4], for a comprehensive and modern presentation) corresponds then to a closure hypothesis with the following linear constitutive equation:

σ (r) = -ν t ∇v + ∇v T 2 + 2 3 k Id, (4.8)
where ν t ≥ 0 is a scalar coefficient, called turbulent viscosity or eddy-viscosity (sometimes called "effective viscosity"), and

k = 1 2 |v | 2 ,
is the turbulent kinetic energy, see [START_REF] Berselli | Mathematics of Large Eddy Simulation of turbulent flows[END_REF][START_REF] Rebollo | Mathematical and numerical foundations of turbulence models and applications[END_REF]. Formula (4.8) is a linear relation between stress and strain tensors, and shares common formal points with the linear constitutive equation valid for Newtonian fluids. In particular, this assumptions motivates the fact that σ (r) must be dissipative3 on the mean flow. Some recent results in the numerical verification of the hypothesis can be found in the special issue [START_REF] Bois | Joseph Boussinesq, a scientist of bygone days and present times[END_REF] dedicated to Boussinesq. Here, we show that, beside the validity of the modeling assumption (4.8), the Reynolds stress tensor σ (r) is dissipative, under minimal assumptions on the regularity of the data of the problem.

Time-averaging of uniformly-local fields

We list in this section some technical properties of the operator M t acting on uniform-local fields, and the corresponding global weak solutions to the NSE. The first result is the following Lemma 4.6. Let 1 < p < ∞ and let be given f ∈ L p uloc (R + ; X). Then

∀ t ≥ 1, M t (f ) X ≤ 2 f L p uloc (R+;X) .
Proof. Applying (4.1) and some straightforward inequalities yields

M t (f ) X ≤ 1 t t 0 f X ds ≤ 1 t [t]+1 0 f X ds ≤ 1 t [t] k=0 k+1 k f X ds.
Therefore by the Hölder inequality we get:

M t (f ) X ≤ 1 t [t] k=0 k+1 k f p X ds 1/p k+1 k 1 ds 1/p ≤ [t] + 1 t f p L p uloc (X) ≤ 2 f p L p uloc (R+;X) ,
the last inequality being satisfied since [x]+1

x ≤ 2 is valid for all x ≥ 1.

In the next section, we will focus on the case p = 2 and X = V . We will need the following result, which is a consequence of Lemma 3.4. Lemma 4.7. Let be given v 0 ∈ H, f ∈ L 2 uloc (R + ; V ), and let v be a global weak solution to the NSE corresponding to the above data. Then, we have, ∀ t ≥ 1,

M t ( ∇v 2 ) ≤ v 0 2 νt + 2 F 2 ν 2 , (4.9)
M t ( f 2 V ) ≤ 2F 2 , (4.10)
where

F = f L 2 uloc (R+;V ) .
Proof. It suffices to divide estimate (3.5) by νt. Therefore, it follows that

1 t t 0 ∇v(s) 2 ds ≤ v 0 2 νt + 2 F 2 ν 2 .
Estimate (4.10) is straightforward.

In particular the family {M t (v)} t∈R+ is bounded in V . Therefore, we can as of now state the following, which will be recalled in a more precise form in the next sections.

Corollary 4.1.

There exists v such that -up to a sub-sequence-M t (v) → v as t → ∞, in appropriate topologies.

The following result is a direct consequence of (4.9) and (4.10) combined with Cauchy-Schwarz inequality.

Corollary 4.2. The family {M t (< f , v >)} t>0 is bounded uniformly in t, and one has

M t (< f , v >) ≤ √ 2F v 0 2 νt + 2 F 2 ν 2 1 2 . (4.11)
We finish this section with a last technical result, that we will need to prove Item vi) of Theorem 2.3. Lemma 4.8. Let 1 < p < ∞ and let be given f ∈ L p uloc (R + ; X), which satisfies in addition

∃ f ∈ X, such that lim t→+∞ t+1 t f (s) -f p X ds = 0. (4.12)
Then, we have

lim t→+∞ 1 t t 0 f (s) -f p X ds = 0. (4.13)
Moreover, M t (f ) weakly converges to f in X when t → +∞. In particular, we have f = f . Proof. By the hypothesis (4.12), we have that

∀ ε > 0 ∃ M ∈ N : t+1 t f (s) -f p X ds < ε 2 ∀ t > M.
Hence, for t ≥ M , then

1 t t 0 f (s) -f p X ds = 1 t M 0 f (s) -f p X ds + 1 t t M f (s) -f p X ds ≤ M t f p L p uloc (X) + f p X + [t] + 1 -M t ε 2 .
It follows that one can choose M large enough such that

1 t t 0 f (s) -f p X ds < ε ∀ t > M,
hence, being this valid for arbitrary ε > 0, it follows (4.13).

It remains to prove the weak convergence of M t (f ) to f ∈ X when t → +∞. To this end, let be given ϕ ∈ X . Then, we have

< ϕ, M t (f ) > -< ϕ, f > = 1 t t 0 < ϕ, f (s) -f > ds, which leads to < ϕ, M t (f ) > -< ϕ, f > ≤ 1 t t 0 ϕ X f (s) -f X ds,
and by Hölder inequality,

< ϕ, M t (f ) > -< ϕ, f > ≤ ϕ X 1 t t 0 f (s) -f p X ds 1 p , yielding, by (4.13)
, to lim

t→+∞ < ϕ, M t (f ) > = < ϕ, f >, hence concluding the proof.
The following corollary definitively concludes Item vi) of Theorem 2.3.

Corollary 4.3. Let be given v 0 ∈ H, f ∈ L 2 uloc (R + ; V ) that satisfies (4.12), and let v be a global weak solution to the NSE corresponding to the above data. Moreover let v be such that lim

t→∞ M t v = v in V (eventually up to a sub-sequence), then lim t→∞ M t (< f , v >) = < f , v > .
Proof. Let us write the following decomposition:

1 t t 0 < f , v > ds = 1 t t 0 < f -f , v > ds + 1 t t 0 < f , v > ds.
On one hand since f ∈ V is independent of t, we obviously have

1 t t 0 < f , v > ds → < f , v > .
On the other hand, we have also

1 t t 0 < f -f , v > ds ≤ 1 t t 0 f -f V ∇v ds ≤ 1 t t 0 f -f 2 V ds 1/2 1 t t 0 ∇v 2 ds 1/2 . (4.14)
Combining (4.13) with (4.9) shows that the right-hand side in (4.14) vanishes as t → ∞.

Proposition 5.1. Let be given a global solution v to the NSE, corresponding to the data v 0 ∈ H and f ∈ L 2 uloc (R + ; V ). Then, there exist a) a sequence {t n } n∈N that goes to +∞ when n goes to +∞;

b) a vector field f ∈ V ; c) a vector field v ∈ V ; d) a vector field B ∈ L 3/2 (Ω) 3 ;
such that such that it holds when n → ∞:

M tn (f ) f in V , M tn (v) v in V, M tn (v • ∇) v B in L 3/2 (Ω) 3 ⊂ V ,
and for all φ φ φ ∈ V ν

Ω ∇v : ∇φ φ φ dx + Ω B • φ φ φ dx = < f , φ φ φ > . (5.2)
Moreover, by defining

F := B -(v • ∇) v ∈ L 3/2 (Ω) 3 , (5.3)
we can also rewrite (5.2) as follows

ν Ω ∇v : ∇φ φ φ dx + Ω (v • ∇) v • φ φ φ dx + Ω F • φ φ φ dx = < f , φ φ φ >; (5.4) e) writing v = v -v, f = f -f , we also have M tn (< f , v >) → < f , v > +< f , v >.
Proof of Proposition 5.1. As f ∈ L 2 (R + ; V ), we deduce from Lemma 4.6 that {M t (f )} t>0 is bounded in V . Hence, we can use weak pre-compactness of bounded sets in the Hilbert space V to infer the existence of t n and f ∈ V such that M tn (f ) f in V . Next, estimate (4.9) from Lemma 4.7, combined with estimate (4.1) from Lemma 4.1, leads to the bound

∃ c > 0 : ∇M t (v) = M t (∇v) ≤ c ∀ t > 0,
proving (up to a the extraction of a further sub-sequence from {t n }, which we call with the same name) that M tn (v) v in V . Then, we observe that, if v ∈ L ∞ (0, T ; H) ∩ L 2 (0, T ; V ) by classical interpolation

(v • ∇) v ∈ L r (0, T ; L s (Ω)) with 2 r + 3 s = 4, r ∈ [1, 2].
In particular, we get

(v • ∇) v L 3/2 (Ω) ≤ v L 6 ∇v L 2 ≤ C S ∇v 2 ,
where C S is the constant of the Sobolev embedding H 1 0 (Ω) → L 6 (Ω). Hence, by using the bounds (3.4)-(3.5) on the weak solution v we obtain that

∃ c : M t (v • ∇) v) L 3/2 (Ω) ≤ c, ∀ t > 0,
proving that, up to a further sub-sequence relabelled again as {t n },

M tn (v • ∇) v B in L 3/2 (Ω) 3 ,
for some vector field B ∈ L 3/2 (Ω) 3 . Next, we use (3.4) which shows that

Ω v 0 -v(t) t • φ φ φ dx → 0 as t → +∞.
Then, writing the weak formulation and by using the results of weak convergence previously proved, we get (5.2). Then, the identity (5.4) comes simply from the definition (5.3) of F. It remains to prove the last item. We know from (4.11) that the sequence

{M tn (< f , v >)} n∈N is bounded in R.
By extracting again a sub-sequence (still denoted by {t n } n∈N ), we can get a convergent sequence still denoted (after relabelling) by {M tn (< f , v >)} n∈N , and let < f , v > be its limit. Let us write the decomposition

M tn (< f , v >) =< f , v > +M tn (< f , v >) + M tn (< f , v >) + M tn (< f , v >).
(5.5) As M tn (< f , v >) = < M tn (f ), v >, we deduce from the results above that M tn (< f , v >) → 0 as n → ∞. Similarly, we also have M tn (< f , v >) → 0. Hence, we deduce from (5.5) that {M tn (< f , v >)} n∈N is convergent, and if we denote by < f , v > its limit, the following natural decomposition holds true:

< f , v > =< f , v > +< f , v >, (5.6)
concluding the proof.

Reynolds stress, energy balance and dissipation

In the first step we have identified a limit (v, f ) for the time-averages of both velocity and external force (v, f ). We need now to recast this in the setting of the Reynolds equations, in order to address the proof of the Boussinesq assumption.

Proof of Theorem 2.3. Beside the results in Proposition 5.1, in order to complete the proof of Theorem 2.3, we have to prove the following facts:

1) the proper identification of the limits with the Reynolds stress σ (r) ;

2) the energy balance for v;

3) to prove that (2.4) holds, namely

ε = ν ∇v 2 ≤ Ω (∇ • σ (r) ) • v dx + < f , v >.
We proceed in the same order. Item 1. Since v ∈ L 2 (0, T ; V ) ⊂ L 2 (0, T ; L 6 (Ω) 3 ), it follows that v ⊗ v ∈ L 1 (0, T ; L 3 (Ω)). Hence, the same argument as in the previous subsection shows that (possibly up to the extraction of a further sub-sequence) there exists a second order tensor θ ∈ L 3 (Ω) 9 such that

M tn (v ⊗ v) θ in L 3 (Ω) 9 .
Let us set

σ (r) := θ -v ⊗ v.
Since the operator M t commutes with the divergence operator, the equation (5.1) becomes

ν Ω ∇V t : ∇φ φ φ dx - Ω M t (v ⊗ v) : ∇φ φ φ dx = < M t (f ), φ φ φ > + Ω v 0 -v(t) t • φ φ φ dx.
(5.7) Then, by taking the limit along the sequence t n → +∞ in (5.7), we get4 the equality

F = ∇ • σ (r) .
Item 2. We use v ∈ V in (2.3) as test function and we obtain the equality

ν ∇v 2 + Ω (∇ • σ (r) ) • v dx = < f , v > . (5.8)
We observe that due to the absence of the time-variable the following identity concerning the integral over the space variables is valid

Ω (v • ∇) v • v dx = Ω (v • ∇) |v| 2 2 dx = 0 ∀ v ∈ V.
This is one of the main technical facts which are typical of the mathematical analysis of the steady Navier-Stokes equations and which allow to give precise results for the averaged Reynolds equations. On the other hand, we recall that if v(t, x) is a non-steady (Leray-Hopf) weak solution, then the space-time integral

T 0 Ω (v • ∇) v • v dx dt,
is not well defined and consequently the above integral vanishes only formally.

Item 3. From now, we assume that the assumption (2.5) in the statement of Theorem 2.3 holds true. We integrate the energy inequality (3.1) between 0 and t n and we divide the result by t n > 0, which leads to

v(t) 2 2t n + 1 t n tn 0 ∇v(s) 2 ds ≤ v 0 2 2t n + 1 t n tn 0 < f , v > ds.
(5.9)

Recall that by Lemma 3.4

v(t) 2 2t → 0 and v 0 2 2t → 0 as t → +∞.
Therefore, we take the limit in (5.9) and we use (5.6), which yields

ν ∇v 2 ≤ < f , v > = < f , v > + < f , v >.
By (5.8) we then have

ν ∇v 2 ≤ ν ∇v 2 + Ω (∇ • σ (r) ) • v dx + < f , v >,
which yields (2.4) by (4.4), concluding the proof.

On ensemble averages

In this section we show how to use the results of Theorem 2.3 to give new insight to the analysis of ensemble averages of solutions. In this case we study suitable averages of the long-time behavior and not the long-time behavior of statistics, as in Layton [START_REF] Layton | The 1877 Boussinesq conjecture: Turbulent fluctuations are dissipative on the mean flow[END_REF].

Since we first take long-time limits and then we average the Reynolds equations, the initial datum is not so relevant. In fact due to the fact that it holds

v 0 2 t → 0 as t → +∞,
then the mean v is not affected by the initial datum.

As claimed in the introduction, we consider now the problem of having several external forces, say a whole family {f k } k∈N ⊂ V , all independent of time. We can think as different experiments with slightly different external forces, whose difference can be due to errors in measurement or in the uncertainty intrinsic in any measurement method. In particular, one can consider for a given force f and that {f k } will represent small oscillations around it, hence we can freely assume that we have an uniform bound

∃ C > 0 : f k V ≤ C ∀ k ∈ N. (6.1)
Having in mind this physical setting, we denote by v k ∈ V the long-time average of the solution corresponding to the external force f k ∈ V and, as explained before (without loss of generality) to the initial datum v 0 = 0. The vector

v k ∈ V satisfies for all φ φ φ ∈ V the following equivalent equalities for all k ∈ N ν Ω ∇v k : ∇φ φ φ dx + Ω B k • φ φ φ dx = < f k , φ φ φ >, ν Ω ∇v k : ∇φ φ φ dx + Ω (v k • ∇) v k • φ φ φ dx + Ω F k • φ φ φ dx = < f k , φ φ φ >, for appropriate B k , F k ∈ L 3 (Ω) 3/2
. Since both V and V are Hilbert spaces, by using (6.1) it follows that there exists f ∈ V and a sub-sequence (still denoted by

{f k }) such that f k f in V .
Our intention is to characterize, if possible, the limit of {v k } k∈N . If the forces are fluctuations around a mean value, then the field v k will remain bounded in V , but possibly without converging to some limit. From an heuristic point of view one can expect that averaging the sequence of velocities (which corresponds to averaging the result over different realizations) one can identify a proper limit, which retains the "average" effect of the flow. Again, it comes into the system the main idea at the basis of Large Scale methods: The average behavior of solutions seems the only quantity which can be measured or simulated.

It is well-known that one of the most used summability technique is that of Cesàro and consists in taking the mean values, hence we focus on the arithmetic mean of time-averaged velocities

S n := 1 n n k=1 v k .
It is a basic calculus result that if a real sequence {x j } j∈N converges to x ∈ R, then also its Cesàro mean S n = 1 n n j=1 x j will converge to the same value x. On the other hand, the converse is false; sufficient conditions on the sequence {x j } j∈N implying that if the Cesàro mean converges, then the original sequence converges, are known in literature as Tauberian theorems. This is a classical topic in the study of divergent sequences/series. In the case of X-valued sequences {u k } k∈N (the space X being an infinite dimensional Banach space) one has again that if a sequence converges strongly or weakly, then its Cesàro mean will converge to the same value, strongly or weakly in X, respectively.

The fact that averaging generally improves the properties of a sequence, is reflected also in the setting of Banach spaces even if with additional features coming into the theory. Two main results we will consider are two theorems known as Banach-Saks and Banach-Mazur.

Banach and Saks originally in 1930 formulated the result in L p (0, 1), but it is valid in more general Banach spaces. Theorem 6.1 (Banach-Saks). Let be given a bounded sequence {x j } j∈N in a reflexive Banach space X. Then, there exists a sub-sequence {x j k } k∈N such that the sequence {S m } m∈N defined by

S m := 1 m m k=1 x j k ,
converges strongly in X.

The reader can observe that in some cases it is not needed to extract a subsequence (think of any orthonormal set in an Hilbert space, which is weakly converging to zero, and the Cesàro averages converge to zero strongly), but in general one cannot infer that the averages of the full sequence converge strongly. One sufficient condition is that of uniform weak convergence. We recall that {x j } ⊂ X uniformly weakly converges to zero if for any > 0 there exists j ∈ N, such that for all φ ∈ X , with φ X ≤ 1, it holds true that # j ∈ N : |φ(x j )| ≥ ε ≤ j.

See also Brezis [6,p. 168].

Another way of improving the weak convergence to the strong one is by the by the convex-combination theorem (cf. Yosida [31, p.120]). Theorem 6.2 (Banach-Mazur). Let (X, . X ) be a Banach space and let {x j } ⊂ X be a sequence such that x j x as j → +∞.

Then, one can find for each n ∈ N, real coefficients {α n j }, for j = 1, . . . , n such that α n j ≥ 0 and n j=1 α n j = 1, such that n j=1 α n j x j → x in X, as n → +∞, that is we can find a "convex combination" of {x j }, which strongly converges to x ∈ X.

One basic point will be that of considering averages of the external forces, which we will denote by f n and considering the same averages of the solution of the Reynolds equations v n . They are both bounded and hence, weakly converging (up to a sub-sequence) to f ∈ V and v ∈ V , respectively. Then, in Hence, passing to the limit in (6.2), we obtain Then, if we take φ φ φ = v in (6.4) we obtain

ν ∇ v 2 + Ω F • v dx = < f , v > . (6.5)
On the other hand, if we take φ φ φ = v n in (6.3) and by the result of the previous section, we have ν ∇ v n 2 ≤ < f n , v n >, hence passing to the limit, by using the strong convergence of f n in V and the weak convergence of v n in V we have

ν ∇ v 2 ≤ lim inf n→+∞ ν ∇ v n 2 ≤ < f , v > .
If we compare with (6.5) we have finally the dissipativity

1 |Ω| Ω (∇ • σ (r) ) • v dx = 1 |Ω| Ω F • v dx ≥ 0,
that is a sort of ensemble/long-time Boussinesq hypothesis, cf. with the results from Ref. [START_REF] Layton | The 1877 Boussinesq conjecture: Turbulent fluctuations are dissipative on the mean flow[END_REF][START_REF] Labovsky | Magnetohydrodynamic flows: Boussinesq conjecture[END_REF].

In the previous theorem, we have a result which does not concern directly with the ensemble averages, but a selection of special coefficients is required. This is not completely satisfactory from the point of view of the numerical computations, where the full arithmetic mean should be considered. The main result can be obtained at the price of a slight refinement on the hypotheses on the external forces To this end we recall a lemma, which is a sort of Rellich theorem in negative spaces (see also Galdi [15,Thm. II.5.3] and Feireisl [9, Thm. 2.8]). Lemma 6.3. Let Ω ⊂ R n be bounded and let be given 1 < p < n. Let {f k } k∈N be a sequence uniformly bounded in L q (Ω) with q > (p * ) , where p * = np n-p is the exponent in the Sobolev embedding W 1,p 0 (Ω) → L p * (Ω). Then, there exists a sub-sequence {f km } m∈N and f ∈ L q (Ω) such that f km f in L q (Ω),

f km → f in W -1,p (Ω),
or, in other words, the embedding L q (Ω) → → W -1,p (Ω) is compact.

  with n ∈ N and τ ∈ [0, 1[. We argue by induction on n. If n = 0, estimate (3.4) has been proved in Step 1. Assume that (3.4) is satisfied for t := n + τ , for all n ≤ N that is, v(n + τ ) ≤ C n = 0, . . . , N, where the constant C is defined by (3.7). If v(N + 1 + τ ) < v(N + τ ) , then (3.4) holds at the time t = N + 1 + τ , by the inductive hypothesis. If v(N + τ ) ≤ v(N + 1 + τ ) , then the inequality (3.4) is satisfied by Step 2 for t = N + 1 + τ , ending the proof.

νΩ∇F

  v : ∇φ φ φ dx + Ω B • φ φ φ dx = < f , φ φ φ > ∀ φ φ φ ∈ V.By the same reasoning used before we have, forF := B -( v • ∇) v , ν Ω ∇ v : ∇φ φ φ dx + Ω ( v • ∇) v • φ φ φ dx + Ω • φ φ φ dx = < f , φ φ φ > . (6.4)

Generally speaking and when no risk of confusion occurs, we always denote by < , > the duality pairing between any Banach space X and its dual X , without mentioning explicitly which spaces are involved.

The mapping µ satisfies µ(A ∪ B) = µ(A) + µ(B) for A ∩ B = ∅ but, on the other hand, we have ∞ n=0 µ([n, n + 1[) = 0 = 1 = µ( ∞ n=0 [n, n + 1[).

The sign adopted in (4.7) is a convention consistent with our mathematical approach. However, according to the analogy of the Reynolds stress with viscous forces, it is also common to define it as σ (r) := -v ⊗ v , which does not change anything.

According to the formal decomposition (4.3), this suggests that Mt n (v ⊗ v ) → 0, provided that one is able to give a rigorous sense and sufficiently strong bounds on v ⊗ v , for the weak solution v.

Acknowledgments

The research that led to the present paper was partially supported by a grant of the group GNAMPA of INdAM the project the University of Pisa within the grant PRA 2018 52 UNIPI Energy and regularity: New techniques for classical PDE problems.

Remark 4.9. It is important to observe that

is -in some sense-an assumption on the (long-time) behavior of the "covariance" between the external force and the solution itself. Cf. Layton [START_REF] Layton | The 1877 Boussinesq conjecture: Turbulent fluctuations are dissipative on the mean flow[END_REF] for a related result in the case of ensemble averages.

The control of the (average/expectation of ) kinetic energy in terms of the energy input is one of the remarkable features of classes of statistical solutions, making the stochastic Navier-Stokes equations very appealing in this context. See the review, with applications to the determination of the Lilly constant, in Ref. [START_REF] Berselli | On a stochastic approach to eddy viscosity models for turbulent flows[END_REF]. See also [START_REF] Flandoli | An introduction to stochastic 3D fluids[END_REF].

Proof of Theorem 2.3

In all this section we have as before v 0 ∈ H, f ∈ L 2 (R + ; V ), and v is a global weak solution to the NSE (1.1) corresponding to the above data. We split the proof of Theorem 2.3 into two steps. We first apply the operator M t to the NSE, then we extract sub-sequences and take the limit in the equations. In the second step we make the identification with the Reynolds stress σ (r) and show that it is dissipative in average, at least when f satisfies in addition (2.5).

Extracting sub-sequences

We set:

Applying the operator M t on the NSE we see that for almost all t ≥ 0 and for all φ φ φ ∈ V , the field V t is a weak solution of the following steady Stokes problem (where t > 0 is simply a parameter)

(5.1)

The full justification of the equality (5.1) starting from the definition of global weak solutions can be obtained by following a very well-known path used for instance to show with a lemma by Hopf that Leray-Hopf weak solutions can be re-defined on a set of zero Lebesgue measure in [0, t] in such a way that v(s) ∈ H for all s ∈ [0, t], see for instance Galdi [14, Lemma 2.1]. In fact, by following ideas developed among the others by Prodi [START_REF] Prodi | Un teorema di unicità per le equazioni di Navier-Stokes[END_REF], one can take χ The process of extracting sub-sequences, which is the core of the main result, is reported in the following proposition. order to prove that the dissipativity is preserved one has to handle the following limit of the products lim

which cannot be characterized, unless (at least) one of the two terms converges strongly. This is why we have to use special means instead of the simple Cesàro averages The first result of this section is then the following: Proposition 6.1. Let be given {f k } k∈N uniformly bounded in V . Then one can find either a Banach-Saks sub-sequence or a convex combination of {v k } k∈N , which are converging weakly to some v ∈ V , which satisfies a Reynolds system (6.4), with an additional dissipative term.

Proof of Theorem 6.1. We define f n and v n to be either

or alternatively

where the sub-sequence {j k } k∈N or the coefficients {α n j } j, n∈N are chosen accordingly to the Banach-Saks of Banach-Mazur theorems in such a way that in both cases

We define, accordingly to the same rules B n , and we observe that, by linearity, we have

Then, we can define

2) also as follows

(6.3) By the uniform bound on f k V and by results of Section 5.2 on the Reynolds equations it follows that there exists

and we can suppose that (up to sub-sequences) we have weak convergence of the convex combinations

We present the proof for the reader's convenience.

Proof of Lemma 6.3. Since by hypothesis L q (Ω) is reflexive, by the Banach-Alaouglu-Bourbaki theorem we can find a sub-sequence f km such that

and by considering the sequence {f km -f } m∈N we can suppose that f = 0. We then observe that by the Sobolev embedding we have the continuous embeddings

where

is the duality identification, while the second one (W 1,p 0 (Ω))

Next, let be given a sequence {f km } ⊂ W -1,p (Ω), then by reflexivity (since 1 < p < ∞) there exists {φ km } ⊂ W 1,p 0 (Ω) such that

with φ km W 1,p 0

(Ω) = ∇φ km L p (Ω) = 1. Hence, by using the classical Rellich theorem, we can find a sub-sequence {φ kj } j∈N such that

In particular, we fix r = q (observe that q > (p * ) implies q < p * ) and we have

The last term converges to zero, by the definition of weak convergence f km 0, while the first one satisfies

and since f km W -1,p is uniformly bounded and φ km -φ W 1,p 0 goes to zero, then also this one vanishes as j → +∞.

Proof of Theorem 2.4. The proof of this theorem can be obtained by following the same ideas of the Proposition 6.1. In fact, the main improvement is that the weak convergence f j f in L q (Ω) implies (without extracting sub-sequences) that

This follows since from any sub-sequence we can find a further sub-sequence which is converging strongly, by Lemma 6.3. Then, by the weak convergence of the original sequence, the limit is always the same and this implies that the whole sequence {f k } strongly converges to its weak limit.

Hence, we have

and then, since v n v in V , we can infer that

and the rest follows as in Proposition 6.1.


Nicolas Guillaudeux
email: nicolas.guillaudeux@inria.fr

Catherine Belleannée

Samuel Blanquart

Jean-Stéphane Varré

To fill in the knowledge about transcript isoforms expressed from a gene, we have proposed a comparative genomics method allowing to identify orthologous exons shared by a pair of genes [1]. We predict transcript isoforms in human, mouse and in a non-model organism, dog, and we identify 135 conserved genes having common gene structures and common potential transcriptomes.

Predicting isoform transcripts: What does the comparison of known transcripts in human, mouse and dog tell us?

[1] S. Blanquart et al., "Assisted transcriptome reconstruction and splicing orthology.", BMC Genomics 17 (2016).

Phylogenetic interpretation of gene structures

Most analyzed genes encountered structure divergence

Modelling gene structures using comparative genomics to predict isoforms [1]

Our structure of a target gene (fig. 1a

Graph components:

"Ambiguous" components: alternative transcription Similar amount of sites specific in human and mouse: likely independent evolution

Figure 6 .Figure 5 .

 65 Figure 6. Distribution of duplicates (left) and singletons (right) components in functional site graphs.

Figure 4 .Figure 2 .

 42 Figure 4. Functional site graph of CREM gene (left). Analysis of 1,663 considered functional site graphs. 255 graphs reveal genes having common gene structures (right).

Figure 1 .Figure 3 .

 13 Figure 1. CREM gene: pairwise comparison between human (a) and mouse (b) reveals new functional sites or exons (red) on human gene (c). This lead to predict new expressible transcripts (d) in human.

  conservation graphs for each gene: • conserved functional site graph (fig.4) & conserved transcript graph (fig.5) • a graph component shows orthology relationships between species • classification of graph components (3-species case): • shared in the 3 species (fig.3a), in 2 species (fig.3b), specific to a species (fig.3c) • only graphs without "ambiguous" components (fig.3d) are considered for analysis Conservation of functional sites: Conservation of transcripts: each node is a functional site of a gene each node is a transcript of a gene each edge represents an orthology relationship each edge represents an orthology relationship between two functional sites between two transcripts