Diego Chamorro
email: diego.chamorro@univ-evry.fr

Oscar Jarrín
email: oscar.jarrin@udla.edu.ec

A turbulent study for a damped Navier-Stokes equation: turbulence and problems

Keywords: Navier-Stokes equations, Kolmogorov's dissipation law, Turbulence theory. MSC2020: 35Q30, 76D05

In this article we consider a damped version of the incompressible Navier-Stokes equations in the whole three-dimensional space with a divergence-free and time-independent external force. Within the framework of a well-prepared force and with a particular choice of the damping parameter, when the Grashof numbers are large enough, we are able to prove some estimates from below and from above between the fluid characteristic velocity and the energy dissipation rate according to the Kolmogorov dissipation law. Precisely, our main contribution concerns the estimate from below which is not often studied in the existing literature. Moreover, we address some remarks which open the door to a deep discussion on the validity of this theory of turbulence.

Introduction

The study of turbulence is an open and difficult problem in the field of fluid dynamics which has many implications in the mathematical, physical and technical world. In mathematics, different techniques have been used to try to solve this problem: indeed, from the stochastic point of view we can cite the works [START_REF] Chevillard | A stochastic representation of the local structure of turbulence[END_REF], [START_REF] Poursina | Stochastic Approach to Turbulence: A Comprehensive Review[END_REF] and [START_REF] Pozoroski | Filtered particle tracking in isotropic turbulence and stochastic modeling of subgrid-scale dispersion[END_REF] and from the deterministic point of view we can cite [START_REF] Chamorro | On the Kolmogorov dissipation law in a damped Navier-Stokes equation[END_REF], [START_REF] Doering | Energy dissipation in body-forced turbulence[END_REF], [START_REF] Foias | Estimates for the energy cascade in threedimensional turbulent flows[END_REF], [START_REF] Foias | Kolmogorov theory via finite-time averages[END_REF] and [START_REF] Vigneron | Free turbulence on R3 and T3[END_REF].

In his celebrated 1941 theory, Andrey Kolmogorov [START_REF] Kolmogorov | The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers[END_REF] stated some simple laws in order to describe the turbulent regime and in the present article we will study the Kolmogorov dissipation law in the deterministic framework. Let us first recall the global framework of the theory: we consider here the classical 3D Navier-Stokes equations

   ∂ t u = ν∆ u -( u • ∇) u -∇p + f , div( u) = 0, ν > 0, u(0, •) = u 0 , (1.1)
where u : [0, +∞[×R 3 -→ R 3 is the velocity of the fluid, p : [0, +∞[×R 3 -→ R is the pressure, ν > 0 is the fluid viscosity parameter and f ∈ L 2 (R 3 )∩ Ḣ-1 (R 3 ) is a given divergence-free, time-independent external force. Now, from a divergence-free initial data u 0 ∈ L 2 (R 3 ), we can construct Leray solutions u ∈ L ∞ loc ([0, +∞[, L 2 (R 3 )) ∩ L 2 loc ([0, +∞[, Ḣ1 (R 3 )) which satisfy the energy inequality

u(t, •) 2 L 2 + 2ν t 0 u(s, •) 2 Ḣ1 ds ≤ u 0 2 L 2 + 2 t 0 f , u(s, •) Ḣ-1 × Ḣ1 ds. (1.2)
In this context, the time-independent external force f will introduce a constant supply of energy (at some fixed length scale 0 which represents the radius of a 3D ball where the force acts effectively over the fluid) and this permanent energy contribution is meant to lead the system to a turbulent state. Within this framework we define the averaged in time quantity

U = lim sup T →+∞ 1 T T 0 u(t, •) 2 L 2 dt 3 0 1 2 , (1.3)
which represents the fluid characteristic velocity and associated to this quantity we define the dimensionless Reynolds number (see [START_REF] Constantin | Euler equations Navier-Stokes equations and turbulence[END_REF], [START_REF] Doering | Energy dissipation in body-forced turbulence[END_REF]) by

Re = U 0 ν . (1.4)
Finally, we introduce the energy dissipation rate by the following averaged expression

E = ν lim sup T →+∞ 1 T T 0 u(t, •) 2 Ḣ1 dt 3 0 . (1.5)
We recall that the turbulent regime of a fluid is usually characterized by the large values of the Reynolds numbers ( [START_REF] Houël | Estimates on the energy dissipation rate in kinetic turbulence[END_REF], [START_REF] Tennekes | A First Course in Turbulence[END_REF]), then the Kolmogorov dissipation law [START_REF] Kolmogorov | The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers[END_REF] in the turbulent state reads as follows: [START_REF] Beirão | On suitable weak solutions to the Navier-Stokes equations in the whole space[END_REF]. (1.6) This seemingly simple relationship raises some very deep issues. First note that the Reynolds number (1.4) is built from the characteristic velocity U which can be seen as an a posteriori measurement and it will be interesting to replace the condition Re 1 by another one which does not rely on U . Second, and far more annoying, we observe (as pointed out in [START_REF] Jarrín | Descriptions déterministes de la turbulence dans les équations de Navier-Stokes[END_REF]Section 1.1.3]) that since we consider the Navier-Stokes equations in the whole three-dimensional space, we can not ensure that the term U given in (1.3) is a well-defined (bounded) quantity in the setting of Leray solutions and this last fact makes very hard to establish the relationship (1.6) rigorously. Note however that -by the energy inequality (1.2)-the quantity E given in (1.5) is well defined, see the Appendix A for the details.

E ∼ U 3 0 ( Re
Let us mention that in a previous work [START_REF] Chamorro | On the Kolmogorov dissipation law in a damped Navier-Stokes equation[END_REF], we introduced a suitable frequency truncation which allowed us to give a rigorous sense of these quantities. However, this modification appeared to be too strong: although the Reynolds number are large and that we have the relationship (1.6), we could proof that the fluid is not in a turbulent regime (see the details in [START_REF] Chamorro | On the Kolmogorov dissipation law in a damped Navier-Stokes equation[END_REF]).

To overcome some of these issues, we propose in this article the following modified Navier-Stokes equations:

   ∂ t u = ν∆ u -( u • ∇) u -∇p + f -β u, div( u) = 0, (β, ν > 0), u(0, •) = u 0 , (1.7)
where a damping term -β u is added. This perturbation of the Navier-Stokes equation is a more relevant model than a brutal truncation in the Fourier variable: indeed, the parameter 0 < β is known as the Rayleigh (or Ekman) friction coefficient and the term -β u models the bottom friction in ocean models and is the main energy sink in large scale atmospheric models. See the book [START_REF] Pedlosky | Geophysical Fluid Dynamics[END_REF]Chapter 5] for more details about this model. Of course when β = 0 we recover the classical Navier-Stokes equations, but as we shall see the term -β u will play a crucial role in our computations: in particular it will helps us to define correctly the quantity U given in (1.3).

Next, we introduce the dimensionless Grashof number (studied in [START_REF] Doering | Energy dissipation in body-forced turbulence[END_REF] and [START_REF] Tennekes | A First Course in Turbulence[END_REF]) by the expression

Gr = f L 2 3 2 0 ν 2 .
(1.8)

Note that the Grashof number depends only on the external force f , its length scale 0 and on the viscosity parameter ν but it is not related to the velocity field u. We thus shall see in the Corollary 4.1 below (under a particular choice of the external force f and of the damping parameter β) that a large Grashof number (Gr 1) implies that we have a large Reynolds number (Re 1).

In the context of equation (1.7) and with the help of the Grashof numbers defined in (1.8) we can prove (under some particular hypotheses over the external force f ) the following version of the Kolmogorov dissipation law (1.6)

E ∼ U 3 0 (Gr 1), (1.9)
and the proof of the previous relationship constitutes the main result of this article (see Theorem 2 below).

We continue by considering another modification of the classical Navier-Stokes equations (1.1) and for 3 7 < α < 3 (the lower and upper bounds here are only technical, see the Remark 7.2 below) we study the following fractional damped Navier-Stokes equations

   ∂ t u = -ν(-∆) α 2 u -( u • ∇) u -∇p + f -β u, div( u) = 0, (β, ν > 0), u(0, •) = u 0 , (1.10)
where the operator (-∆) α 2 can be defined in the Fourier level by its symbol |ξ| α . The fractional diffusion term -ν(-∆) α 2 u has been successfully employed to model anomalous reaction-diffusion process in porous media models [START_REF] Meerschaert | Multidimensional advection and fractional dispersion[END_REF][START_REF] Meerschaert | Operator Lévy motion and multiscaling anomalous diffusion[END_REF] and in computational turbulence models [START_REF] Pope | Turbulent Flows[END_REF]Chapter 13.2]. In these last models, the term -ν(-∆) α 2 is used to characterize anomalous viscous diffusion effects in turbulent fluids which are driven by the parameters α and ν.

Our aim here is to establish in this setting a fractional version of the Kolmogorov dissipation law (1.9) and to discuss the effects of the parameter 3 7 < α < 3 in the turbulent regime. First, by a study based on dimensionality, we will obtain the following fractional dissipation law:

E ∼ U 3 α-1 0 (Gr 1), (1.11)
and we will see that in the case 3 7 < α < 1 the lower regularizing effect of the operator (-∆) α 2 allows to consider a nicer external force while in the case 1 ≤ α < 3 we need to consider a rougher force to establish the relationship (1.11).

Motivated by the previous study and noting that in the proof of this dissipation law the nonlinear term ( u • ∇) u plays a minor role, we consider now the following damped Stokes equation

   ∂ t u = ν∆ u -∇p + f -β u, div( u) = 0, (β, ν > 0), u(0, •) = u 0 .
(1.12)

If β = 0 we recover the usual Stokes equation which is a linearisation of the Navier-Stokes equation (1.1). This equation is usually meaningful in the case when the Reynolds number is small, however, if we add a time-independent external force f we face to the similar problems when trying to define the quantity U given in (1.3). Thus, by adding the term -β u we easily ensure that this term U is correctly defined. Now with the same external force used to study the damped Navier-Stokes system (1.7) (see Section 3 below) and with β 1 we can establish the dissipation law (1.9) for the damped Stokes equation (1.12). This fact raises some issues: on the one hand, we have a damped Stokes equation which (in principle) should not generate a turbulent behavior and, on the other hand, we have the relationship (1.9) when the Grashof number (associated to the external force f ) is large. This apparent contradiction may probably suggest that the Kolmogorov dissipation law (1.9) is not an accurate model to describe turbulence in the whole space R 3 and perhaps numerical simulations could helpful to a better understanding of the effect of the damping term coupled with the presence of the external force.

The outline of the article is the following. In Section 2 we establish existence results as well as the energy inequalities for the damped Navier-Sokes equation (1.7). In Section 3 we give the precise setting that allows us to deduce the Kolmogorov dissipation law (1.9) and we state our main theorem (Theorem 2). Section 4 is devoted to the proof of this result. In Section 5 we point out some important issues that are observed in our framework. In Section 6 we study the damped Stokes equation (1.12) and finally, in Section 7, we study the turbulence in the setting of the fractional equation (1.11).

Existence of solutions, energy estimates and technical results

We start with an existence result for the system (1.7):

Theorem 1 Consider u 0 ∈ L 2 (R 3 ) be a divergence-free initial data and let f ∈ L 2 (R 3 ) ∩ Ḣ-1 (R 3 ) be a divergence free and time independent external force. Then, for 0 < β, there exists a function

u ∈ L ∞ loc ([0, +∞[, L 2 (R 3 )) ∩ L 2 loc ([0, +∞[, Ḣ1 (R 3
)) which is a weak solution of the equation (1.7) that satisfies the energy inequality:

u(t, •) 2 L 2 ≤ u 0 2 L 2 -2ν t 0 u(s, •) 2 Ḣ1 ds + 2 t 0 f , u(s, •) L 2 ds -2β t 0 u(s, •) 2 L 2 ds. (2.1)
Proof. The proof of this theorem is rather classical and it essentially follows the Leray method for the classical Navier-Stokes equation. Let φ ∈ C ∞ 0 (R 3 ) be a positive function such that

R 3 φ(x)dx = 1. For δ > 0 we define φ δ (x) = 1 δ 3 φ x δ .
Moreover, we denote by h νt (x) the usual heat kernel and with the help of the classical Leray projector P (defined for a suitable function ϕ by P( ϕ) = ϕ + ∇ 1 (-∆) ( ∇ • ϕ) and which is used here to get rid of the pressure p, see the book [START_REF] Lemarié-Rieusset | The Navier-Stokes Problem in the 21st Century[END_REF] for the details), we will solve, due to the Duhamel formula, the following integral equation

u(t, x) = h νt * u 0 (x) + t 0 h ν(t-s) * f (x)ds - t 0 h ν(t-s) * (P(([φ δ * u] • ∇) u)(s, x)ds -β t 0 h ν(t-s) * u(s, x)ds, (2.2) in the functional space L ∞ ([0, T ], L 2 (R 3 )) ∩ L 2 ([0, T ], Ḣ1 (R 3
)) which is endowed, for some fixed T > 0, with the norm

• T = • L ∞ t L 2 x + √ ν • L 2 t Ḣ1
x . We have then:

u T ≤ h νt * u 0 + t 0 h ν(t-s) * f (•)ds T (A) + t 0 h ν(t-s) * (P(([φ δ * u] • ∇) u)(s, •)ds T (B) + β t 0 h ν(t-s) * u(s, •)ds T (C)
.

The terms (A) and (B) are classical to estimate and by [START_REF] Lemarié-Rieusset | The Navier-Stokes Problem in the 21st Century[END_REF], Theorem 12.2, page 352, we have

h νt * u 0 T ≤ c u 0 L 2 and t 0 h ν(t-s) * f (•)ds T ≤ C( 1 √ ν + √ T ) f L 2 t H -1 x , (2.3) but since f ∈ L 2 (R 3 ) ∩ Ḣ-1 (R 3 ) we have: f L 2 t H -1 x ≤ f L 2 t Ḣ-1 x ≤ √ T f L ∞ t Ḣ-1 x ≤ √ T f Ḣ-1
, and thus, for the term (A) above we can write

h νt * u 0 + t 0 h ν(t-s) * f (•)ds T ≤ c u 0 L 2 + C( 1 √ ν + √ T ) √ T f Ḣ-1 . (2.4)
For the term (B) we have (see [START_REF] Lemarié-Rieusset | The Navier-Stokes Problem in the 21st Century[END_REF], Theorem 12.2 for the details):

t 0 h ν(t-s) * (P(([φ δ * u] • ∇) u)(s, •)ds T ≤ C( 1 √ ν + 1) √ T δ -3 2 u T u T . (2.5)
Finally, in order to study the term (C) we use again the estimate (2.3) (with u instead of f ) to write

β t 0 h ν(t-s) * u(s, •)ds T ≤ β C( 1 √ ν + √ T ) u L 2 t H -1 x ≤ C( 1 √ ν + √ T ) u L 2 t L 2 x ≤ β C( 1 √ ν + √ T ) √ T u L ∞ t L 2 x ≤ β C( 1 √ ν + √ T ) √ T u T .
(2.6)

Once we have inequalities (2.4), (2.5) and (2.6) at our disposal, for a time T > 0 small enough and for δ > 0, by the Banach contraction principle we obtain

u δ ∈ L ∞ ([0, T ], L 2 (R 3 )) ∩ L 2 ([0, T ], Ḣ1 (R 3 ))
a local solution of the equations (2.2). Now we will prove that this solution u δ is global. We remark that the function u δ satisfies the regularized equation

∂ t u δ = ν∆ u δ -P(([φ δ * u δ ] • ∇) u δ ) + f -β u δ , (2.7)
where all the terms belong to the space L 2 ([0, T ], H -1 (R 3 )) and then we can write

d dt u δ (t, •) 2 L 2 = 2 ∂ t u δ (t, •), u δ (t, •) H -1 ×H 1 = -2ν u δ (t, •) 2 Ḣ1 + 2 f , u δ (t, •) Ḣ-1 × Ḣ1 -2β u δ (t, •) 2 L 2 .
(2.8)

But -2β u δ (t, •) 2 L 2 is a negative quantity and we get d dt u δ (t, •) 2 L 2 ≤ -2ν u δ (t, •) 2 Ḣ1 + 2 f, u δ (t, •) Ḣ-1 × Ḣ1 ≤ -2ν u δ (t, •) 2 Ḣ1 + ν u δ (t, •) 2 Ḣ1 + 1 ν f 2 Ḣ-1 ≤ -ν u δ (t, •) 2 Ḣ1 + 1 ν f 2 Ḣ-1 .
Thereafter, we integrate on the interval of time [0, t] and we obtain the following control

u δ (t, •) 2 L 2 + ν t 0 u δ (s, •) 2 Ḣ1 ds ≤ u 0 2 L 2 + t ν f 2 Ḣ-1 , (2.9)
which allows us to extend the local solution u δ to the whole interval [0, +∞[.

We study now the convergence to a weak solution of the equations (1.7). Indeed, by the Rellich-Lions lemma (see [START_REF] Lemarié-Rieusset | The Navier-Stokes Problem in the 21st Century[END_REF], Theorem 12.1) there exists a sequence of positive numbers (δ n ) n∈N and a function u ∈ L 2 loc ([0, +∞[×R 3 ) such that the sequence ( u δn ) n∈N converges strongly to u in L 2 loc ([0, +∞[×R 3 ). Moreover, this sequence converges to u in the weak- * topology of the spaces L ∞ ([0, T ], L 2 (R 3 )) and L 2 ([0, T ], Ḣ1 (R 3 )) for all T > 0. From these convergences we can deduce that the sequence

P(([φ δn * u δn ] • ∇) u δn ) n∈N converges to P(( u• ∇) u) in the weak- * topology of the space (L 2 t ) loc (H -3 2
x ) and then, the limit u is a weak solution of equation (1.7).

Finally, in order to obtain the energy inequality (2.1), we get back to the identity (2.8) and we integrate (in the time variable) each term of this identity:

u δn (t, •) 2 L 2 + 2ν t 0 u δn (s, •) 2 Ḣ1 ds = u 0 2 L 2 + 2 t 0 f , u δn (s, •) L 2 ds -2β t 0 u δn (s, •) 2 L 2 ds,
from which we obtain the wished inequality (2.1) by applying classical tools (see the book [START_REF] Lemarié-Rieusset | The Navier-Stokes Problem in the 21st Century[END_REF]). Theorem 1 is proven.

With this theorem at our disposal, we will now show that the characteristic velocity U given in the expression (1.3) is well defined. Indeed, we have: Proposition 2.1 Within the framework of Theorem 1, the solutions of the equation (1.7) verify:

lim sup T →+∞ 1 T T 0 u(t, •) 2 L 2 dt ≤ 1 νβ f 2 Ḣ-1 .
(2.10)

Proof. From the energy inequality (2.1) and since f ∈ Ḣ-1 (R 3 ), we can write

2β T 0 u(t, •) 2 L 2 dt + 2ν T 0 u(t, •) 2 Ḣ1 dt ≤ u 0 2 L 2 + 2 T 0 f , u(t, •) Ḣ-1 × Ḣ1 dt ≤ u 0 2 L 2 + 2 T 0 f Ḣ-1 u(t, •) Ḣ1 dt ≤ u 0 2 L 2 + ν T 0 u(t, •) 2 Ḣ1 dt + T ν f 2 Ḣ-1 .
We thus obtain

β T 0 u(t, •) 2 L 2 dt ≤ u 0 2 L 2 + T ν f 2 Ḣ-1 ,
from which, we divide each term by T and by letting T → +∞, we get the wished estimate (2.10).

Note that due to the definition of U given in (1.3), the inequality (2.10) is equivalent to the estimate

U ≤ 1 √ νβ f Ḣ-1 3 2 0
, thus, as long as we have β > 0 and f ∈ Ḣ-1 (R 3 ) we can give a sense to the quantity U . From now on we will assume that we have

1 < U ≤ 1 √ νβ f Ḣ-1 3 2 0 .
(2.11)

The main result

In order to present the main theorem of this article, we need to consider a particular external force f that satisfies some conditions and we also need to be more precise with the damping parameter β.

Let us start with the external force and for a fixed length scale 0 > 1, we assume from now on that f ∈ L 2 (R 3 ) ∩ Ḣ-1 (R 3 ) verifies the following frequency localization

supp f ⊂ |ξ| < c 0 , (3.1)
where 0 < c is a fixed and dimensionless constant. Only for technical reasons (see the Remark 4.1 below) we shall consider in this section 0 < ν < 2, and then, we will also assume that we have

f L 2 5 2 0 and 9 8 0 f Ḣ-1 √ ν f 7 4 L 2 . (3.2)
Note that all the previous conditions (3.1) and (3.2) can be satisfied simultaneously by considering for example a function which is constant in the Fourier level over the ball B(0, c 0 ) and such that

1 4 0 ≤ f L ∞ ≤ 13 3 0 . (3.3)
We define now the averaged external force by the quantity

F = f L 2 3 2 0 , (3.4)
and we fix once and for all the parameter β > 0 by the relationship

β = F 3 2 . (3.5)
Now, with the fluid viscosity parameter 0 < ν < 2 we define the Grashof number by the expression

Gr = F 3 0 ν 2 = f L 2 3 2 0 ν 2 . (3.6)
We recall that Gr is a dimensionless quantity and we note that this definition is the same as the one given in the expression (1.8) above.

With these preliminaries, we can now state our main result.

Theorem 2 Let u 0 ∈ L 2 (R 3 ) be a divergence-free initial data, let f ∈ L 2 (R 3 ) ∩ Ḣ-1 (R 3
) be the divergence-free external force which verifies (3.1) and (3.2) for some fixed length scale 1 0 and for 0 < ν < 2. Moreover, let 0 < β be the damping parameter fixed in (3.5)

. Let u ∈ L ∞ t L 2 x ∩ (L 2 t ) loc Ḣ1
x be a solution of the equation (1.7) associated to u 0 , f and β.

If the Grashof number is large enough ( i.e. Gr 1), then we have the estimates:

1 2 U 3 0 ≤ E ≤ 101 10 U 3 0 , (3.7)
where the quantities U and E are defined in (1.3) and (1.5) respectively.

Some remarks are in order here. First note that our approach is based on the presence of a particular external force f whose properties rely heavily on the length scale 0 which is assumed to be large.

Since we have f L 2 5 2

0 by the condition (3.2), the Grashof number Gr given in (3.6) will be also large, thus, as the region where the force is effective is consequent (which is precisely the notion of 0 ) and as the energy input is big, it is quite reasonable to obtain the dissipation law (3.7) as predicted by the general theory. We also observe that if we have f L 2 5 2 0 , then the quantity F given in (3.4) is also large and so is the parameter β = F On the other hand, it is interesting to remark that with a well-prepared external force and with some very particular values of the parameters in our model we can also obtain similar estimates as in (3.7) even in the non turbulent case when Gr ∼ 1 and 0 = 1. In Section 5 below we address all the details. This remark yields us to question, on the one hand, the relevance of the damping parameter β in our study and, on the other hand, the validity of this turbulence theory.

For our calculation to work we need a large damping parameter β and perhaps some interference between the damping term and the expected dissipation law can be made explicit, but this point is not completely clear for us and perhaps numerical simulations can bring an interesting insight to the behavior of the fluid in this case. Let us make clear here that we do not claim any optimality in our approach and perhaps a different perturbation of the Navier-Stokes equation can show more explicitly how to obtain the Kolmogorov dissipation law (1.9).

Finally, in the Appendix C we study the Kolmogorov dissipation law (1.9) with another different choice for the damping parameter β (see the expression (C.1 for a precise definition) which does not depend on the quantity f L 2 , and consequently, in this case the damping parameter is not necessary large as in (3.5). However, it is worth emphasizing that this different choice for β does not allow us derive (to the best of our knowledge) an estimate from bellow as in the estimates (3.7). See Theorem 6 in the Appendix C for all the details.

Proof of Theorem 2

Note that in this framework all the quantities ν, 0 , F and U are finite and we need to establish some relationships between them in order to perform our computations. In this sense we have the following result.

Lemma 4.1 Under the hypothesis of the Theorem 2 we have the controls

1 F ≤ U ≤ F 3 2 0 . (4.1)
Proof. By the energy inequality (2.1) we can write

2β T 0 u(t, •) 2 L 2 dt + u(T, •) 2 L 2 ≤ u 0 2 L 2 + 2 T 0 R 3 f • u dx dt -2ν T 0 u(t, •) 2 Ḣ1 dt,
from which we deduce the estimate

2β T 0 u(t, •) 2 L 2 dt ≤ u 0 2 L 2 + 2 T 0 R 3 f • u dx dt ≤ u 0 2 L 2 + 2 T 0 f L 2 u(t, •) L 2 dt.
Applying the Young inequalities with some parameter γ > 0 in the integral above (recall that β is given by (3.5)), we obtain

2β T 0 u(t, •) 2 L 2 dt ≤ u 0 2 L 2 + 2 T 0 1 γβ f 2 L 2 + γβ u(t, •) 2 L 2 dt.
which can be rewritten in the following manner:

(2 -γ)β T 0 u(t, •) 2 L 2 dt ≤ u 0 2 L 2 + T 1 γβ f 2 L 2
Dividing each term above by 3 0 and by T and taking the limit when T → +∞, we thus have

(2 -γ)β lim sup T →+∞ 1 T T 0 u(t, •) 2 L 2 dt 3 0 ≤ 1 γβ f 2 L 2 3 0
, now, by the definition of the quantities U and F given in (1.3) and (3.4) we have

γ(2 -γ)β 2 U 2 ≤ F 2 . (4.2)
We define now γ such that γ(2 -γ) = 1 U 3 < 1 (which is the case since we assumed in (2.11) that U > 1), thus with the fact that β 2 = F 3 (see (3.5)), the estimates above becomes

F ≤ U. If we consider now γ such that γ(2 -γ) = 1 2 0 F 4 < 1 (Note that 1 2 0 F 4 < 1 is equivalent to 0 < f L 2
which is a consequence from the first condition in (3.2) and the fact that 0 > 1), from (4.2) we easily obtain U ≤ 0 F 3 2 and the Lemma 4.1 is proven. Remark 4.1 Note that the condition F ≤ U that we have just proved is compatible with the constraint

U ≤ 1 √ νβ f Ḣ-1 3 2 0
given in (2.11) as long as we have the second condition over the force stated in (3.2) with 0 < ν < 2. In this case we do have the estimates

f L 2 3 2 0 = F ≤ U ≤ 1 √ 2β f Ḣ-1 3 2 0 ≤ 1 √ νβ f Ḣ-1 3 2 0
.

From the point of view of the Grashof and Reynold numbers we have the following consequence of this result: With all these estimates at hand, we can now study the proof of the Theorem 2.

• Lower estimate. By the energy inequality (2.1) we have this time

2β T 0 u(t, •) 2 L 2 dt + u(T, •) 2 L 2 ≤ u 0 2 L 2 + 2 T 0 R 3 f • u dx dt + 2ν T 0 u(t, •) 2 Ḣ1 dt, from which we deduce 2β T 0 u(t, •) 2 L 2 dt ≤ u 0 2 L 2 + 2 T 0 f L 2 u(t, •) L 2 dt + 2ν T 0 u(t, •) 2 Ḣ1 dt,
and by the Young inequalities we have

2β T 0 u(t, •) 2 L 2 dt ≤ u 0 2 L 2 + T 0 f 2 L 2 + u(t, •) 2 L 2 dt + 2ν T 0 u(t, •) 2 Ḣ1 dt,
dividing the previous estimate by 3 0 and T , taking the limit when T → +∞ we obtain 2β lim sup

T →+∞ 1 T T 0 u(t, •) 2 L 2 dt 3 0 ≤ f 2 L 2 3 0 + lim sup T →+∞ 1 T T 0 u(t, •) 2 L 2 dt 3 0 +2ν lim sup T →+∞ 1 T T 0 u(t, •) 2 Ḣ1 dt 3 0
.

Recalling at this point the definition of the quantities U given in (1.3), F given in (3.4) and E given in (1.5), we obtain

2βU 2 ≤ F 2 + U 2 + 2E,
from which we deduce the estimate

β - 1 2 U 2 - 1 2 F 2 ≤ E.
Now, since by the Lemma 4.1 we have the estimate

F 2 ≤ U 2 , we obtain (β -1) U 2 ≤ β - 1 2 U 2 - 1 2 F 2 ≤ E. (4.3)
We thus have,

β 0 U - 0 U U 3 0 ≤ E.
0 U U 3 0 ≤ β 0 U - 0 U U 3 0 ≤ E.
Noting now that by the lower bound of the Lemma 4.1 we have

-1 F ≤ -1 U , we have 1 - 0 F U 3 0 ≤ E.
We use now the definition of the Grashof numbers given in (3.6) to obtain the estimate 1 -

4 0 ν 2 Gr U 3 0 ≤ E. (4.4)
At this point we remark that we have

4 0 ν 2 Gr ≤ 1 2 . Indeed, this control is equivalent by (3.6) to 4 0 ν 2 ≤ Gr 2 = 1 2 f L 2 3 2 0 ν 2
which can be rewritten as

5 2 0 ≤ 1 2 f L 2 .
We only need to remark that by (3.2) we have • Upper estimate. By the energy inequality (2.1) and by the Cauchy-Schwarz inequality (first in the spatial variable and then in the temporal variable) we can write 2ν

T 0 ∇ ⊗ u(t, •) 2 L 2 dt ≤ u 0 2 L 2 + 2 T 0 R 3 f (x) • u(t, x) dx dt ≤ u 0 2 L 2 + 2 T 0 f L 2 u(t, •) L 2 dt.
Now, by the Young inequalities (with ρ > 0), we obtain 2ν

T 0 ∇ ⊗ u(t, •) 2 L 2 dt ≤ u 0 2 L 2 + T 0 1 ρ f 2 L 2 + ρ u(t, •) 2 L 2 dt.
Dividing all this estimate by 3 0 and by T and taking the limit T → +∞ we thus have 2ν lim sup

T →+∞ 1 T T 0 ∇ ⊗ u(t, •) 2 L 2 dt 3 0 ≤ 1 ρ f 2 L 2 3 0 + ρ lim sup T →+∞ 1 T T 0 u(t, •) 2 L 2 dt 3 0
. Now, by definition of the quantities E, F and U given in (1.5), (3.4) and (1.3) respectively, we obtain

2E ≤ 1 ρ F 2 + ρU 2 . (4.5)
If we set ρ = 0 10U we have

E ≤ 10U 0 F 2 + 0 10 U.
By the lower bound of the Lemma 4.1 we have F 2 ≤ U 2 and we can write

E ≤ 10U 3 0 + 0 10 U = U 3 0 10 + 2 0 10U 2 ≤ U 3 0 10 + 2 0 10F 2 .
Using the definition of the Grashof numbers given in (3.6) we finally obtain

E ≤ U 3 0 10 + 8 0 10ν 4 Gr 2 . (4.6) E ≤ U 3 0 10 + 8 0 10ν 4 Gr 2 .
We observe now that we have

8 0 ν 4 Gr 2 ≤ 1, indeed, this is equivalent to 4 0 ν 2 ≤ Gr = f L 2 3 2 0 ν 2
and we thus obtain the condition With these two inequalities, the proof of Theorem 2 is finished.

Kolmogorov type estimates for small Grashof numbers

In this section, we show that we can obtain similar estimates as in (3.7) when Gr ∼ 1. For this we will set the parameters in our model as follows. First, we set the length scale and the viscosity parameter as 0 = 1 and ν = √ 2.

(5.1)

Then, we shall consider a particular external force f defined in the Fourier level as

f (ξ) = 2 5 2 1 |ξ|< 1 2 (ξ), (5.2)
where by the Plancherel identity we have

f L 2 = 2 4π 3 1 2
.

(5.3)

We thus have F =

f L 2 3 2 0 = f L 2 4
and as in the last section we set β = F we have

Gr = f L 2 3 2 0 ν 2 = 4π 3
1 2 2.
This particular external force also verifies the following estimates, which are similar to the ones given in (3.2)

f L 2 ≥ 1 and f Ḣ-1 √ ν f 7 4 L 2 . (5.4)
Indeed, the first estimate is a direct consequence of the first identity in (5.1) and the identity (5.3). For the second condition, on the one hand, by a simple computation we have f Ḣ-1 = 4(4π) 14.16. Now, by following the same ideas in the proof of Theorem 2 we will quickly show that we are able to obtain the estimates E ∼ U 3 0 even if Gr 2 and 0 = 1. First we remark that the estimate (4.1) proven in Lemma 4.1 above is still valid in this particular setting and since 0 = 1 it writes down as

1 < F ≤ U ≤ F 3 2 .
(5.5)

Moreover, as pointed out in Remark 4.1, we shall remark that the condition

F ≤ U is still compatible with the constraint U ≤ 1 √ νβ f Ḣ-1 (with 0 = 1)
given in (2.11). Precisely, in this case we have F U 1 √ νβ f Ḣ-1 which is equivalent to the second condition in (5.4). With the estimates (5.5) at our disposal, we get the estimate from below given in (4.4) and since 0 = 1 we shall write this estimate as

1 - 1 ν 2 Gr U 3 0 ≤ E.
But, since Gr 2 and ν = √ 2 we have 1

ν 2 Gr 1 4
and we thus obtain the estimate from below 3 4

U 3 0 ≤ E.
Always by the estimates (5.5), we also get the estimate from above given in (4.6), which in this particular case ( 0 = 1, ν = √ 2 and Gr 2) writes down as:

E ≤ U 3 0 10 + 1 10ν 4 Gr 2 1601 160 U 3 .
Summarizing, by setting the damping parameter β as in (3.5), on the one hand with the particular external force f defined in (5.2) and with the particular values of the parameters in our model given in (5.1), we obtain the Kolmogorov type estimates E ∼ U 3 0 in the non turbulent regime Gr ∼ 1 and 0 = 1. On the other hand, when we consider the particular force given in the example (3.3), and with 0 1, we also obtain the estimates E ∼ U 3 0 in the turbulent regime Gr 1.

We thus observe that by carefully setting the external force and the parameters of our model we are able to obtain Kolmogorov type estimates in both the turbulent and the non turbulent cases. As mentioned in Section 3, it is thus quite natural to deeply question the significance of the parameter β in our study as well as the validity of this theory of the turbulence.

Turbulence in the Stokes equations

As pointed out in the introduction, we may observe in the proof of the Theorem 2 that the study of the Kolmogorov dissipation law in the deterministic framework of the damped Navier-Stokes equations is strongly based on the energy inequality. On the other hand, in this energy inequality we see that the nonlinear effects of the transport term ( u • ∇) • u are neglected since this term vanishes in the computations: as div( u) = 0 we formally have the identity

R 3 ( u • ∇) u • u dx = 0.
This remark yields a deeper discussion on how the nonlinear transport term intervenes in the deterministic study of the Kolmogorov dissipation law, and further, on the robustness of this theory of turbulence.

To start our discussion, we shall consider here the damped Stokes equations, which essentially writes down as the damped Navier-Stokes equation without the nonlinear transport term:

   ∂ t u = ν∆ u -∇p + f -β u, div( u) = 0, (β, ν > 0), u(0, •) = u 0 . (6.1)
As we have a linear model, the global the well-posedness of this equation is a well-known fact and we have:

Proposition 6.1 Let u 0 ∈ L 2 (R 3
) be a divergence-free initial data and let f ∈ L 2 (R 3 ) ∩ Ḣ-1 (R 3 ) be a divergence-free external force. Then, for 0 < β there exists a function

u ∈ L ∞ t L 2 x ∩ (L 2 t ) loc Ḣ1
x which is the unique strong solution of the equation (6.1). Moreover, the following energy equality holds:

u(t, •) 2 L 2 = u 0 2 L 2 -2ν t 0 u(s, •) 2 Ḣ1 ds + 2 t 0 R 3 f • u(s, •)dx ds -2β t 0 u(s, •) 2 L 2 ds. (6.
2)

The energy equality (6.2) also yields the Lemma 4.1 which is the key tool in the proof of Theorem 2. Consequently, we can follow the same estimates performed in this proof to obtain analogous lower and upper bounds on the energy dissipation rate according to the Kolmogorov dissipation law:

Theorem 3 Under the same hypothesis of the Theorem 2, let u ∈ L ∞ t L 2 x ∩(L 2 t ) loc Ḣ1
x be the solution of the equation (1.12) from which we define the quantities U and E given in (1.3) and (1.5) respectively. Then, we have the estimates 1 2

U 3 0 ≤ E ≤ 101 10 U 3 0 .
The main interest of this result is the fact that we are able to obtain a Kolmogorov type estimate even in the framework of the (damped) Stokes model. The Stokes equations are a merely a non turbulent model precisely due to the lack of the nonlinear transport term and this raises the problem of the pertinence of this theory for fluids defined in the whole space R 3 .

The fractional Navier-Stokes equations

We study now the fractional damped Navier-Stokes equations introduced in (1.10). Our first result establishes the existence of weak solutions and an energy inequality.

Theorem 4 Consider u 0 ∈ L 2 (R 3 ) be a divergence-free initial data and for some 0

< α < 4 let f ∈ L 2 (R 3 ) ∩ Ḣ-α 2 (R 3
) be a divergence free, time independent, external force. Then, for 0 < β there exists a function

u = u α,β ∈ L ∞ loc ([0, +∞[, L 2 (R 3 )) ∩ L 2 loc ([0, +∞[, Ḣ α 2 (R 3 )),
which is a weak solution of the equation (1.10) that satisfies the energy inequality:

u(t, •) 2 L 2 ≤ u 0 2 L 2 -2ν t 0 u(s, •) 2 Ḣ α 2 ds + 2 t 0 R 3 f • u(s, •)dx ds -2β t 0 u(s, •) 2 L 2 ds. (7.1)
The proof of this theorem is rather standard and it is based in the hyperviscosity method of Beirão da Vega [START_REF] Beirão | On suitable weak solutions to the Navier-Stokes equations in the whole space[END_REF] which introduces the term -∆ 2 (this fact explains the technical constraint 0 < α < 4). We thus postpone its proof to the Appendix B. We also refer to [START_REF] Jarrín | Asymptotic behavior of a generalized Navier-Stokes-alpha model and applications to related models[END_REF] for another approach to prove this theorem.

Remark 7.1 Note that, in the undamped Navier-Stokes equation, when 5 2 ≤ α we have the uniqueness of the solution and an energy equality, see [START_REF] Cholewa | Fractional Navier-Stokes Equations[END_REF]. Moreover, this uniqueness properties also holds true for the damped equation (1.10) since we have a linear damping term.

Once we have this existence result, we can see the effect of the damping term -β u in the definition of the quantity U and by following the same estimates in the proof of Proposition 2.1 we have: Proposition 7.1 Within the framework of Theorem 4 the solutions of the equation (1.10) verify:

lim sup T →+∞ 1 T T 0 u(t, •) 2 L 2 dt ≤ 1 νβ f 2 Ḣ-α 2 .
Remark that the previous estimate allows us to define rigorously the quantity U given in (1.3) in the framework of the fractional and damped Navier-Stokes equation (1.10): we can write

U = lim sup T →+∞ 1 T T 0 u(t, •) 2 L 2 dt 3 0 1 2 < +∞.
Thus, assuming U > 1, we obtain the control

1 < U ≤ 1 √ νβ f Ḣ-α 2 3 2 0 . (7.2)
Now, for 0 < α < 4 and for 0 < ν, we define the energy dissipation rate E α by the expression

E α = ν lim sup T →+∞ 1 T T 0 (-∆) α 4 u(t, •) 2 L 2 dt 3 0 . ( 7.3)
Note that by following the same arguments given in the Appendix A this quantity is well defined.

With the quantities U and E α at our disposal, we will deduce now (formally) a fractional version of the Kolmogorov dissipation law. Indeed, inspired in the relationship (1.6) we should have

E α ∼ U β 1 β 2 0 , (7.4)
where β 1 and β 2 are parameters to be fixed and for this we will use a dimensional analysis. First, following some ideas of [START_REF] Vigneron | Free turbulence on R3 and T3[END_REF] (see the formula (29b) in page 12) we will assume (only for this analysis) that the velocity u is localized at the frecuencies 1 0 < |ξ| < 2 0 . Then, by the Bernstein inequalities (see [START_REF] Grafakos | Classical Fourier Analysis[END_REF]Chapter 2]) and by the expression (7.3) we have

E α ∼ ν α 0 U 2 . (7.5)
Thus, with (7.4) and (7.5) we can write

ν α 0 U 2 ∼ U β 1 β 2 0
, which lead us to the relationship

ν ∼ U β 1 -2 β 2 -α 0
. Now, the term on the right-hand side must have the same physical dimension of the viscosity parameter ν which is length 2 time . Thus, as the characteristic velocity U has a physical dimension of length time and as 0 has a physical dimension of length, a simple dimensional analysis shows that we must have β 1 = 3 and β 2 = α -1. In this way, the Kolmogorov dissipation law in the fractional case writes down as follows:

E α ∼ U 3 α-1 0 , ( Gr 1), (7.6)
Observe that in the case when α = 2 we obtain the classical damped Navier-Stokes equations (1.7), moreover by the expression (1.5) and ( 7.3) we have E = E 2 and we recover the classical Kolmogorov dissipation law (1.6).

The aim of our next result is to prove the relationship (7.6) above in the setting of the equation (1.10). From now on, only for technical reasons we shall consider the range 3 7

< α < 3.

We will explain below these constraints of the parameter α. Thereafter, by following the previous analysis done in Section 3 we start by assuming that

f ∈ L 2 (R 3 ) ∩ Ḣ-α 2 (R 3 ) satisfies supp( f ) ⊂ |ξ| < c 0 , (7.7)
for some dimensionless constant 0 < c, and moreover, for 0 > 1 and 0 < ν < 2, we also assume that

     f L 2 2-α 2 0 and 9 8 0 f Ḣ-α 2 √ ν f 7 4 L 2 , when 3 7 < α < 1, f L 2 α+ 1 2 0 and 9 8 0 f Ḣ-α 2 √ ν f 7 4 L 2 , when 1 ≤ α ≤ 3. (7.8)
This set of conditions will allow us to justify our computations, precisely, the ones given in the proof of Lemma 7.1 below where we need to consider the cases α < 1 and 1 ≤ α separately. Moreover, these conditions can be simultaneously satisfied by considering, for example, a function f which is constant in the Fourier level over the ball B(0, c 0 ) and such that

   1 7 2 -α 2 0 ≤ f L ∞ ≤ 3+ 2α 3 0 , when 3 7 ≤ α < 1, 1 α+2 0 ≤ f L ∞ ≤ 3+ 2α 3 0 , when 1 ≤ α < 3.
Remark 7.2 Since we assume 0 > 1, the lower constraint 3 7 ≤ α is needed for the first estimates, while the upper constraint α < 3 is crucial for the second estimates.

We also remark that (as 0 > 1) the quantity f L ∞ is larger as long as α → 3 -, i.e., we have a stronger external force. Finally, let us mention we think that the technical constraint 3 7 ≤ α < 3 could be removed with another particular (well-prepared) function f . Once the force is fixed, following (3.4) we define F =

f L 2 3 2 0
and we fix the damping parameter as in (3.5), i.e. β = F 3 2 . In this particular setting we have:

Theorem 5 Let u 0 ∈ L 2 (R 3
) be a divergence-free initial data; and consider a length scale 1

0 and 0 < ν < 2. Consider also f ∈ L 2 (R 3 ) ∩ Ḣ-α 2 (R 3
) a divergence-free, time independent external force which verifies (7.7) and (7.8)

. Let u ∈ L ∞ t L 2 x ∩ (L 2 t ) loc Ḣ α 2
x be a solution of the equation (1.10) associated to u 0 , f , α and β (where β is defined as in (3.5)).

If the Grashof number defined in (3.6) is large enough ( i.e. Gr 1) then for 3 7 ≤ α < 3 we have the relationship

U 3 α-1 0 ∼ E α , (7.9)
As pointed out in Remark 7.1, in the range 5 2 ≤ α < 3 we have the uniqueness of weak global solutions of the equation (1.10), but in our current study of the estimates (7.9) this property does not play any substantial role. Nevertheless, we think a deeper study of possible connections between the Kolmogorov type estimates and the uniqueness (and regularity) of weak solutions of the fractional Navier-Stokes equations could be an interesting matter of further research.

Proof of Theorem 5. First, we establish an analogous estimate to the one given in Lemma 4.1.

Lemma 7.1 Under the hypothesis of the Theorem 5 we have the controls

1 F ≤ U ≤ F 3 2 α-1 0 . (7.10)
Proof. By the energy inequality (7.1), we also have the estimate (4.2). From this estimate, by setting γ(2 -γ) = 1 U 3 < 1 and as we have β 2 = F 3 , we get F ≤ U . At this point, we emphasize that by this last inequality, by definition of the quantity F (see (3.4)) and by the estimate (7.2), we obtain the estimate

f L 2 3 2 0 ≤ 1 √ νβ f Ḣ-α 2 3 2 0
which holds true as long as the second condition in (7.8) is satisfied with 0 < ν < 2.

On the other hand, by setting now γ(2 -γ) = .

Here, it is also worth mentioning the condition • Lower estimate. By the energy inequality (7.1) and by following the same estimates performed to obtain the inequality (4.3), we can write

1 2(α-1) 0 F 4 < 1 is equivalent to 2-α 2 0 ≤ f L 2 . When
(β -1) U 2 ≤ E α .
Then, we have

β α-1 0 U - α-1 0 U U 3 α-1 0 ≤ E α ,
where we must find a lower bound for each expression in the term on the left-hand side. As

β 2 = F 3
, for the first term we have

β α-1 0 U = F 3 2 α-1 0 U
. Moreover, by the estimate (7.10) we have

1 ≤ F 3 2 α-1 0 U
. On the other hand, for the second term, as by the estimate (7.10) we have F ≤ U then we obtain -

α-1 0 F ≤ - α-1 0 U . Then, as Gr = F 3 0 ν 2 we can write - α-1 0 F = - α+2 0 ν 2 Gr .
We thus have the lower bound 1 -

α+2 0 ν 2 Gr ≤ β α-1 0 U - α-1 0
U from which we get the lower estimate

1 - α+2 0 ν 2 Gr U 3 α-1 0 ≤ E α .
Now, we shall need the following estimate

α+2 0 ν 2 Gr ≤ 1 2 , (7.11)
which follows from the condition (7.8) according to the different cases of the parameter α. Indeed, since Gr = . Thereafter, when 1 ≤ α < 3 this estimate is precisely given by the first condition in (7.8).

With the estimate (7.11) at our disposal, we obtain the lower estimate 1 2

U 3 α-1 0 ≤ E α .
• Upper estimate. Always by the energy inequality (7.1) and by following the same estimates performed to obtain the inequality (4.5), for 0 < ρ we have

2E α ≤ 1 ρ F 2 + ρU 2 .
Then, we set ρ = α-1 0 10 U to obtain

E α ≤ 10 U α-1 0 F 2 + α-1 0 10 U.
By the lower estimate in (7.10) we have F 2 ≤ U 2 and then we get

E α ≤ 10 U 3 α-1 0 + α-1 0 10 U = U 3 α-1 0 10 + 2(α-1) 0 10U 2 ≤ U 3 α-1 0 10 + 2(α-1) 0 10F 2 .
Finally, always by the estimate (7.11) we have

2(α-1) 0 F 2 = 2α+4 0 ν 4 Gr 2 ≤ 1 4
, which yields the upper estimate

E α ≤ 401 40 U 3 α-1 0 .
Theorem 5 is now proven.

A The energy dissipation rate

We prove here that que quantity E given in (1.5) is well defined. Indeed, by the energy inequality (1.2) we just write

2ν T 0 u(t, •) 2 Ḣ1 dt ≤ u 2 L 2 + 2 T 0 f , u(t, •) Ḣ-1 × Ḣ1 dt ≤ u 2 L 2 + 2 T 0 f Ḣ-1 u(t, •) Ḣ1 dt ≤ u 2 L 2 + T ν f 2 Ḣ-1 + ν T 0 u(t, •) 2 Ḣ1 dt, hence we have ν T 0 u(t, •) 2 Ḣ1 dt ≤ u 2 L 2 + T ν f 2 Ḣ-1 .
Then, we divide each term by T and by 3 0 , and moreover, by letting T → +∞ we have the control E ≤

f 2 Ḣ-1 ν 3 0 < +∞.

B Existence of weak solutions for the system (1.10)

There are several methods to obtain a weak solution of the equation

   ∂ t u = -ν(-∆) α 2 u -P(( u • ∇) u) + f -β u, div( u) = 0, α, β, ν > 0, u(0, •) = u 0 .
Perhaps the simpler way is to follow the hyperviscosity approach given by [START_REF] Beirão | On suitable weak solutions to the Navier-Stokes equations in the whole space[END_REF] and to consider the following model

∂ t u = -∆ 2 u -ν(-∆) α 2 u -P(( u • ∇) u) + f -β u, ε > 0. (B.1)
In the sequel, we shall follow the program given in [START_REF] Beirão | On suitable weak solutions to the Navier-Stokes equations in the whole space[END_REF] to construct global in time weak solutions of the equation (1.10). We shall mainly detail the estimates involving the terms -β u and -ν(-∆) α 2 u which are different from the classical case.

First step: Weak solutions for the equation (B.1). We recall that the operator e -εt∆ 2 is defined in the Fourier level as e -εt|ξ| 4 ϕ for all ϕ belonging to the Schwartz class. For T > 0 small enough, we will solve the following fixed point problem

u(t, x) = e -εt∆ 2 u 0 (x) + t 0 e -ε(t-s)∆ 2 f (x)ds - t 0 e -ε(t-s)∆ 2 (( u • ∇) u)(s, x)ds - t 0 e -ε(t-s)∆ 2 (βI d + ν(-∆) α 2 ) u(s, x)ds (B.2) in the space L ∞ ([0, T ], L 2 (R 3 )) ∩ L 2 ([0, T ], Ḣ2 (R 3 )) with the norm • T = • L ∞ t L 2 x + √ ε • L 2 t Ḣ2
x . We thus write

u T = e -εt∆ 2 u 0 T (A) + t 0 e -ε(t-s)∆ 2 f (•)ds T (B) + t 0 e -ε(t-s)∆ 2 P(( u • ∇) u)(s, •)ds T (C) + t 0 e -ε(t-s)∆ 2 (βI d + ν(-∆) α 2 ) u(s, •)ds T (D)
.

(B.3)
For the term (A), by [START_REF] Lemarié-Rieusset | The Navier-Stokes Problem in the 21st Century[END_REF], Theorem 18.2 (page 352), we have the estimate e -εt∆ 2 u 0

T ≤ C u 0 L 2 .
In order to study the term (B), we can adapt the estimates given in [START_REF] Lemarié-Rieusset | The Navier-Stokes Problem in the 21st Century[END_REF], Theorem 18.2, to obtain the following:

Lemma B.1 Let g ∈ L 2 t L 2
x . Then we have

t 0 e -ε(t-s)∆ 2 g(s, •)ds T ≤ C 1 g L ∞ t L 2 x , with C 1 = C T 1 2 + √ εT 1 2 + 1 √ ε . (B.4)
Proof. For the first term of the norm • T , for 0 < t ≤ T we have

t 0 e -ε(t-s)∆ 2 g(•)ds L 2 ≤ t 0 e -ε(t-s)∆ 2 g(•) L 2 ds ≤ t 0 g(s, •) L 2 ds ≤ T 1 2 g L 2 t L 2 x . (B.5)
On the other hand, for the second term of the norm • T , we have

√ ε t 0 e -ε(t-s)∆ 2 g(•)ds L 2 t Ḣ2 x ≤ √ ε t 0 e -ε(t-s)∆ 2 g(•)ds L 2 t H 2 x ≤ √ ε (I d -∆) t 0 e -ε(t-s)∆ 2 g(•)ds L 2 t L 2 x ≤ √ ε (I d -∆) 2 t 0 e -ε(t-s)∆ 2 (I d -∆) -1 g(•)ds L 2 t L 2 x ≤ C √ ε t 0 e -ε(t-s)∆ 2 (I d -∆) -1 g(•)ds L 2 t L 2 x + C √ ε ∆ 2 t 0 e -ε(t-s)∆ 2 (I d -∆) -1 g(•)ds L 2 t L 2 x .
For the first term on the right-hand side, by (B.5) we can write

C √ ε t 0 e -ε(t-s)∆ 2 (I d -∆) -1 g(•)ds L 2 t L 2 x ≤ C √ εT 1 2 (I d -∆) -1 g L 2 t L 2 x ≤ C √ εT 1 2 g L 2 t L 2 x
Moreover, for the second term on the right-hand side, by the maximal regularity of the bi-Laplacian operator we can write

C √ ε ∆ 2 t 0 e -ε(t-s)∆ 2 (I d -∆) -1 g(•)ds L 2 t L 2 x ≤ C √ ε g L 2 t L 2
x . By gathering all these estimates we get the wished estimate (B.4).

Once we have the estimate (B.4) at our disposal, be setting g = f (recall that f ∈ L 2 (R 3 ) is time-independent) for the term (B) we have

t 0 e -ε(t-s)∆ 2 f (•)ds T ≤ C 1 T 1 2 f L 2 . We thus can write e -εt∆ 2 u 0 + t 0 e -ε(t-s)∆ 2 f (•)ds T ≤ C u 0 L 2 + C 1 T 1 2 f L 2 . (B.6) T ε 1 4 + √ εT 3 4 + 1 √ ε 1 + T 1 2
by [START_REF] Lemarié-Rieusset | The Navier-Stokes Problem in the 21st Century[END_REF],

Theorem 18.2, we have

t 0 e -ε(t-s)∆ 2 P(( u • ∇) u)(s, •)ds T ≤ C B T 1 8 u 2 T . (B.7)
Finally, to study the term (D) we write

t 0 e -ε(t-s)∆ 2 (βI d + ν(-∆) α 2 ) u(s, •)ds T ≤ β t 0 e -ε(t-s)∆ 2 u(s, •)ds T + ν t 0 e -ε(t-s)∆ 2 (-∆) α 2 u(s, •)ds T
For the first term on the right-hand side, we use again the estimate (B.4) (with g = u) to write

β t 0 e -ε(t-s)∆ 2 u(s, •)ds T ≤ βC 1 u L 2 t L 2 x ≤ βC 1 T 1 2 u L ∞ t L 2 x ≤ βC 1 T 1 2 u T . (B.8)
For the second term on the right-hand side, we shall need the following:

Lemma B.2 Let 0 < α < 4 and let u ∈ L ∞ t L 2 x ∩ L 2 t Ḣ2
x . The following estimates hold:

1) When 0 < α ≤ 2 we have t 0 e -ε(t-s)∆ 2 (-∆) α 2 u(s, •)ds T ≤ C T ε 1-α 4 + √ εT 1 2 u T .
2) When 2 < α < 4 we have

t 0 e -ε(t-s)∆ 2 (-∆) α 2 u(s, •)ds T C T ε 1-α 4 + ε 3 4 -α 8 T 6-α 8 u T .
Proof. For 0 < α < 4 we estimate the first term in the norm • T . By duality and by the Cauchy-Schwarz inequality we have

t 0 e -ε(t-s)∆ 2 (-∆) α 2 u(s, •)ds L 2 = sup ϕ L 2 ≤1 R 3 t 0 e -ε(t-s)∆ 2 (-∆) α 2 u(s, x)ds • ϕ(x)dx = sup ϕ L 2 ≤1 t 0 R 3 u(s, x) • e -ε(t-s)∆ 2 (-∆) α 2 ϕ(x)dx ds ≤ u L ∞ t L 2 x sup ϕ L 2 ≤1 t 0 e -ε(t-s)∆ 2 (-∆) α 2 ϕ L 2 ,
where we must study the integral above. By the Plancherel identity we write

t 0 e -ε(t-s)∆ 2 (-∆) α 2 ϕ L 2 = t 0 R 3 e -2|(ε(t-s)) 1 4 ξ| 4 |(ε(t -s)) 1 4 ξ| 2α | ϕ(ξ)| 2 dξ 1 2 (ε(t -s)) -α 4 ds ≤ C ϕ L 2 t ε 1-α 4 ≤ C T ε 1-α 4 .
We thus obtain

t 0 e -ε(t-s)∆ 2 (-∆) α 2 u(s, •)ds L 2 ≤ C T ε 1-α 4 u T . (B.9)
For the second term in the norm • T , always by the Plancherel identity we can write

√ ε t 0 e -ε(t-s)∆ 2 (-∆) α 2 u(s, •)ds L 2 t Ḣ2 = √ ε (-∆) 1+ α 2 t 0 e -ε(t-s)∆ 2 u(s, •)ds L 2 t L 2 x ,
where we must treat the cases 0 < α ≤ 2 and 2 < α < 4 separately. For the first case, by the maximal regularity of the bi-Laplacian operator we have

√ ε (-∆) 1+ α 2 t 0 e -ε(t-s)∆ 2 u(s, •)ds L 2 t L 2 x ≤ √ εC u L 2 t L 2 x ≤ √ εC T 1 2 u L ∞ t L 2 x ≤ √ εC T 1 2 u T . (B.10)
For the second case, as 2 < α < 4 we have 1 < α 2 < 2 and for 0 < s < 1 we write α 2 = 1 + s. Then, always by the maximal regularity of the bi-Laplacian operator we can write

√ ε (-∆) 1+ α 2 t 0 e -ε(t-s)∆ 2 u(s, •)ds L 2 t L 2 x = √ ε (-∆) 2 t 0 e -ε(t-s)∆ 2 (-∆) s u(s, •)ds L 2 t L 2 x ≤ √ ε C (-∆) s u L 2 t L 2 x . Then, since u ∈ L ∞ t L 2 x ∩ L 2 t Ḣ2
x and since 0 < s < 1, for p = 8 α-2 > 2 (recall that 2 < α < 4) by the interpolation inequalities with θ = 1 -s 2 ∈ ( 1 2 , 1), we have

√ εC (-∆) s u L 2 t L 2 x ≤ √ εC u L 2 t Ḣ2s x ≤ √ εCT 1 2 -1 p u L p t Ḣ2s x ≤ √ εC T 1 2 -1 p u θ L ∞ t L 2 x u 1-θ L 2 t Ḣ2 x ≤ √ εC ( √ ε) θ-1 T 1 2 -1 p u θ L ∞ t L 2 x ( √ ε u L 2 t Ḣ2 x ) 1-θ ≤ ( √ ε) θ C T 1 2 -1 p u T = ε 3 4 -α 8 C T 6-α 8 u T .
Then we obtain

√ ε (-∆) 1+ α 2 t 0 e -ε(t-s)∆ 2 u(s, •)ds L 2 t L 2 x ≤ ε 3 4 -α 8 C T 6-α 8 u T . (B.11)
The first point stated in this lemma follows from the estimates (B.9) and (B.10), while the second point follows from the estimates (B.9) and (B.11).

Once we have the estimates (B.6), (B.7) and the estimates given in Lemma B.2 at our disposal, for a time T > 0 and for ε > 0 by the Banach contraction principle we obtain

u ε ∈ L ∞ ([0, T ], L 2 (R 3 )) ∩ L 2 ([0, T ], Ḣ2 (R 3 )) a local solution of the equations (B.2).
This solution also verifies the hyperviscosity equation (B.1) and by standard computations, for t ≥ 0 we can obtain the following energy inequality

u ε (t, •) 2 L 2 + 2ν t 0 u(s, •) 2 Ḣ α 2 ds + 2ε t 0 u ε (s, •) 2 Ḣ2 ds + 2β t 0 u ε (s, •) 2 L 2 ds ≤ u 0 2 L 2 + 2 t 0 R 3 f (x) • u ε (s, x)dxds.
(B.12)

We thus deduce the following control

u ε (t, •) 2 L 2 + ν t 0 u(s, •) 2 Ḣ α 2 ds + 2ε t 0 u ε (s, •) 2 Ḣ2 ds ≤ u 0 2 L 2 + t f 2 Ḣ-α 2 ν (B.
∞ t ) loc L 2 x ∩ (L 2 t ) loc Ḣ α 2 x . Consequently, there exists u ∈ (L ∞ t ) loc L 2 x ∩ (L 2 t ) loc Ḣ α 2 x
and there exits a subsequence (ε n ) n∈N such that for all 0 < T < +∞ we have u εn → u in the weak-* topology of the spaces L ∞ ([0, T ], L 2 (R 3 )) and L 2 ([0, T ], Ḣ α 2 (R 3 )). Moreover, by the Rellich-Lions lemma (see [START_REF] Lemarié-Rieusset | The Navier-Stokes Problem in the 21st Century[END_REF], Theorem 12.1) we also have that u εn → u in the strong topology of the space L 2 loc ([0, +∞[×R 3 ). From these convergences we can deduce that the sequence P ( u εn • ∇) u εn n∈N converges to P ( u • ∇) u in the weak-* topology of the space (L 2 t ) loc H α 2 -4 x and then the limit u is a weak solution of the equation (1.7).

Finally, from the energy inequality (B.12) by applying standard methods (see [START_REF] Lemarié-Rieusset | The Navier-Stokes Problem in the 21st Century[END_REF], Theorem 18.2) we obtain the energy inequality (7.1). Theorem 4 is proven.

C A Kolmogorov type estimate for another choice of the damping parameter

By following some ideas of [START_REF] Chamorro | On the Kolmogorov dissipation law in a damped Navier-Stokes equation[END_REF] and [11, Chapter 1], from the length 0 and the viscosity parameter ν we set now the damping parameter

β = ν 2 0 . (C.1)
We remark that this value of the damping parameter is different to the one given in (3.5) since it does not depend on the quantity f L 2 . Moreover, this choice of the damping parameter can be motivated from the space-periodic setting as follows: when we consider the classical Navier-Stokes equations with space-periodic conditions in the box Ω

= [0, 0 ] 3 , for u ∈ L ∞ t L 2 x ∩ L 2 t Ḣ1
x a generic Leray solution, by the well-known energy inequality and by the Poincaré's inequality we have the control:

lim sup T →+∞ 1 T T 0 u(t, •) 2 L 2 (Ω) dt ≤ 2 0 ν 2 f 2 Ḣ-1 ,
see [START_REF] Jarrín | Descriptions déterministes de la turbulence dans les équations de Navier-Stokes[END_REF]Section 1.2.3] for all the details. Thus, when comparing this control to the one given in Proposition 2.1 (in the setting of the whole space) we observe that the identity (C.1) naturally appears as a choice for the damping parameter β. In this sense, we remark that the damping term -β u in the equations (1.7) acts as compensation for the lack of the Poincaré inequality in the whole space R 3 .

In the following theorem we show that the value of the damping parameter given in the identity (C.1) also allows to prove an estimate from above according to the Kolmogorov dissipation law (1.11) in the deterministic setting of the damped Navier-Stokes equations (1.7).

Theorem 6 With the same hypothesis of Theorem 2, consider now the damping parameter β given in (C.1). Moreover, let the Grashof number Gr and the Reynolds number Re defined in (3.6) and (1.4) respectively.

There exists a constant 0 < c, which do not depend on any physical quantity, such that if 2Gr ≤ Re then the following estimate holds:

E ≤ c 1 Gr + 1 U 3 0 , (C.2)
where the quantities U and E are given in (1.3) and (1.5) respectively.

We emphasize that here we need to assume the (technical) control 2Gr ≤ Re which, to our knowledge, does not follow from the hypothesis of this theorem. In contrast, when we set the damping parameter β as is (3.5), this type of control can be obtained from the hypothesis of Theorem 2. See the Corollary 4.1 above.

Proof of Theorem 6. The proof of the estimate (C.2) is based on the following inequalities: Proposition C.1 There exists a numerical constant 0 < c such that if 2Gr ≤ Re then we have the

inequality F ≤ c 1 Gr + 1 U 2 0 .
Proof. We consider here the differential equation in (1.7) where we must multiply each term by f and integrate in the spatial variable. We recall that since the external force f ∈ L 2 (R 3 ) satisfies the frequency localization given in (3.1) then it belongs to all the Sobolev spaces H s (R 3 ) with s ≥ 0 and we can write

R 3 ∂ t u(t, x) • f (x) dx = ν R 3 ∆ u(t, x) • f (x) dx - R 3 P(( u • ∇) u)(t, x) • f (x) dx + f 2 L 2 -β R 3 u(t, x) • f (x) dx.
Then, we have

f 2 L 2 = R 3 ∂ t u(t, x) • f (x) dx -ν R 3 ∆ u(t, x) • f (x) dx + R 3 P(( u • ∇) u)(t, x) • f (x) dx + β R 3 u(t, x) • f (x) dx, (C.3)
where we must study each term on the right-hand side. For the first term, as f is a time-independent function we have

R 3 ∂ t u(t, x) • f (x) dx = ∂ t R 3 u(t, x) • f (x)dx.
For the second term, by the Cauchy-Schwarz inequality we obtain

-ν R 3 ∆ u(t, x) • f (x) dx = -ν R 3 u(t, x) • ∆ f (x)dx ≤ ν u(t, •) L 2 ∆ f L 2 .
For the third term, as f is divergence-free vector field, by integrating by parts and by the Hölder inequalities, we write

R 3 P(( u • ∇) u)(t, x) • f (x) dx = R 3 ( u • ∇) u(t, x) • f (x)dx = - 3 i,j=1 R 3 u i u j (t, x)∂ j f i (x)dx ≤ u(t, •) 2 L 2 ∇ ⊗ f L ∞ .
Finally, for the fourth term, by the Cauchy-Schwarz inequality we have

β R 3 u(t, x) • f (x)dx ≤ β u(t, •) L 2 f L 2 .
With these estimates at hand, we get back to the identity (C.3) to write

f 2 L 2 ≤ ∂ t R 3 u(t, x) • f (x)dx + ν u(t, •) L 2 ∆ f L 2 + u(t, •) 2 L 2 ∇ ⊗ f L ∞ + β u(t, •) L 2 f L 2 .
For T > 0, in each term of this inequality we take the time average 1 T T 0 (•)dt; and always by the fact that f does not depend on the time variable, by the Cauchy-Schwarz inequality (in the time variable) we get

f 2 L 2 ≤ 1 T R 3 u(T, x) • f (x)dx - R 3 u 0 (x) • f (x)dx + ν 1 T T 0 u(t, •) 2 L 2 dt 1 2 ∆ f L 2 + 1 T T 0 u(t, •) 2 L 2 dt ∇ ⊗ f L ∞ + β 1 T T 0 u(t, •) 2 L 2 dt 1 2 f L 2 .
To study the first term on the right-hand side, we shall need the following estimate which is given by the damping parameter -β u in the equations (1.7).

Lemma C.1 With the same hypothesis of Theorem 1, the solutions u ∈ (L ∞ t ) loc L 2 x ∩ (L 2 t ) loc Ḣ1

x of the equation (1.7) verify the following control in time for all t ≤ 0:

u(t, •) 2 L 2 ≤ e -βt u 0 2 L 2 + f 2 Ḣ-1 νβ .
Proof. Consider the functions u δn which are solutions of the regularized equation (2.7). Our starting point is then the equality (2.8):

d dt u δn (t, •) 2 L 2 = -2ν u δn (t, •) 2 Ḣ1 + 2 f , u δn (t, •) Ḣ-1 × Ḣ1 -2β u δn (t, •) 2 L 2 .
Since f ∈ Ḣ-1 (R Once we have this estimate at our disposal, in the first term on right-hand side, first we apply the Cauchy-Schwarz inequality and by this estimate we obtain

f 2 L 2 ≤ 1 T e -βT u 0 2 L 2 + 1 νβ f 2 Ḣ-1 + u 0 2 L 2 + f 2 L 2 + ν 1 T T 0 u(t, •) 2 L 2 dt 1 2 ∆ f L 2 + 1 T T 0 u(t, •) 2 L 2 dt ∇ ⊗ f L ∞ + β 1 T T 0 u(t, •) 2 L 2 dt 1 2 f L 2 .
In each term of this inequality, we divide by 3 0 and by T , we also apply the lim sup when T → +∞. Thus, by definition of the quantities F and U given in (3.4) and (1.3) respectively, we get

F 2 ≤ ν U ∆ f L 2 3/2 0 + U 2 ∇ ⊗ f L ∞ + β U F, hence, F ≤ ν U ∆ f L 2 3/2 0 F + U 2 ∇ ⊗ f L ∞ F + β U, (C.4)
where we still need to study the first term and the second term on the right-hand side. For the first term, we recall that , by recalling that Re = U 0 ν , and moreover, if we assume that 2Gr ≤ Re then we can write

F ≤ c 2ν U 2 0 + U 2 0 = c 2ν U 0 + 1 U 2 0 = c 2 Re + 1 U 2 0 ≤ c 1 Gr + 1 U 2 0 .
Proposition C.2 For the quantities E, U and F defined in (1.5), (1.3) and (3.4) respectively, we have the following inequality E ≤ F U .

Proof. By the energy inequality (2.1) and by the Cauchy-Schwarz inequality (first in the spatial variable and then in the temporal variable) we can write 2ν

T 0 u(t, •) 2 Ḣ1 dt ≤ u 0 2 L 2 + 2 T 0 R 3 f (x) • u(t, x) dx dt ≤ u 0 2 L 2 + 2 T 0 f L 2 u(t, •) L 2 dt ≤ u 0 2 L 2 + f L 2 T 1/2 T 0 u(t, •) 2 L 2 dt 1 2
.

Thereafter, in each term of this inequality we divide by 3 0 and by T ; and we also apply the lim sup when T → +∞. By definition of the quantities E, F and U we obtain the wished inequality.

3 2

 3 defined in (3.5): we thus obtain the wished dissipation law with an important damping term.

Corollary 4 . 1

 41 From the lower estimate in (4.1) and with the definition of Re and Gr given in (1.4) and (3.6) respectively, we obtain Gr ≤ Gr 1 we also have that Re 1.

3 2 0 U

 30 Now, by the definition (3.5) we have β = F and since by the upper bound of the Lemma 4.1 we have 1 ≤ F 3 2 , we can thus write 1 -

5 2 0f L 2 2 U 3 0

 52223 to obtain the claimed control. Whit this remark we have: 1 ≤ E, which is the lower estimate in (3.7).

5 2 0

 52 ≤ f L 2 , which is the case since we do have f L which is the upper estimate of (3.7).

3 2 .

 2 Moreover, by(3.6)

1 2 14. 16 .ν f 7 2 L 2

 121622 On the other hand, by the second identity in (5.1) and by the identity (5.3) we have √

F 4 <

 4 1 we obtain the estimate U ≤ F 3 2 α-1 0

3 7 ≤

 7 α < 1, this estimate is precisely the first condition in(7.8). On the other hand, when 1 ≤ α < 3, since 0 > 1 by the second condition in (7.8) we have f L 2

F 3 0ν 2 α+ 1 2 0≤ f L 2 .

 3222 the estimate above is equivalent to the estimate 2 When 3 7 ≤ α < 1 this estimate is satisfied by the first condition in (7.8) since we have 0 > 1 and f L 2

  [START_REF] Kolmogorov | The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers[END_REF] which allows us to extend the local solution u ε to the whole interval [0, +∞[. Second step: Passage to the limit and weak solutions for the equation (1.10). By the control (B.13) the family ( u ε ) ε>0 of solutions of the equation (B.1) is uniformly bounded (respect to the parameter ε) in the space (L

3 / 2 0F 2 f | L 2 ≤ c 2 f L 2 2≤ c 0 0

 322220 = f L 2 . Moreover, by the frequency localization of f given in (3.1), for the first term we have ∆ f L 2 c is the dimensionless constant in (3.1). For the second term, always by (3.1) and by the Bernstein inequalities (see [9, Chapter 2]) there exists a numerical constant 0 < c 0 such that we have∇ ⊗ f L ∞ FThus, we set a constant 0 < c such that max c 2 , c 0 , 1 ≤ c, and by these estimates we can write

1 + 1 U 3 0.

 113 Now we are able to prove the estimate (C.2). In the inequality given in Proposition C.1 we multiply by U to get F U ≤ c Gr Then, by the inequality given in Proposition C.2 we obtain the wished estimate (C.2). Theorem 6 is proven.

  for all time t ∈ [0, +∞[. Now, we will recover this control in time for the limit function u: we regularize in the time variable the quantity u δn (t, •) 2 L 2 by a convolution product with a positive functionw ∈ C ∞ 0 ([-η, η]) (for η > 0) such that R w(t)dt = 1.In this way, in the previous inequality we havew * u δn (t, •) L 2 ≤ w * u δn (t, •) 2 L 2 ≤ w * e -βt u 0 Moreover, since ( u δn ) n∈N converges weakly- * to u in (L ∞ t ) loc (L 2 x ) then w * u δn (t, •) converges weakly- * to w * u(t, •) in L 2 (R 3 ) and then we can write w * u(t, •) 2 L 2 ≤ lim inf Finally, for t a Lebesgue point of the function t → u(t, •) 2 L 2 we have the wished control in time and we can extend this inequality to all time t ∈ [0, +∞[ by the weak continuity of the function t → u(t, •) 2 L 2 .

							f 2 Ḣ-1 ν	+ ν u δn (t, •) 2 Ḣ1 and then we get
	d dt	u δn (t, •) 2 L 2 ≤	f 2 Ḣ-1 ν	-ν u δn (t, •) 2 Ḣ1 -2β u δn (t, •) 2 L 2 .
	Then, we write
		d dt	u δn (t, •) 2 L 2 ≤	f 2 Ḣ-1 ν	-β u δn (t, •) 2 L 2 ,
	and by the Grönwall inequality we have the control:
	u δn (t, •) 2 L 2 ≤ e -βt u 0	2 L 2 +	f 2 Ḣ-1 νβ	1 -e -βt ≤ e -βt u 0	2 L 2 +	f 2 Ḣ-1 νβ	,
								2 L 2 +	f 2 Ḣ-1 νβ	.
								f 2 Ḣ-1 νβ	.

3

) we have 2 f , u δn (t,

•) Ḣ-1 × Ḣ1 ≤ n-→+∞ w * u δn (t, •) 2 L 2 ≤ w * e -βt u 0 2 L 2 +


Dorothée Arzounian

Mathilde De Kerangal

Alain De Cheveigné
email: alain.de.cheveigne@ens.fr

A sliding two-alternative forced choice paradigm for pitch discrimination

Keywords: pitch, frequency discrimination, adaptive procedure, context dependency

) Thresholds did differ between conditions for the new task, showing a deleterious effect of inserting relatively large steps in the frequency sequence. (3) Thresholds also differed between conditions for the classic task, showing an advantage for a fixed frequency standard. There was no indication that results were more variable with either task, and we found no reason not to use the new sliding 2AFC task in lieu of the classic 2-tone-per-trial 2AFC task.

I. INTRODUCTION

Pitch is a prominent quality of sound important for music and speech, the object of keen interest since antiquity [START_REF] De Cheveigné | Continuous versus discrete frequency changes: different detection mechanisms[END_REF][START_REF] Plack | Pitch Perception[END_REF][START_REF] Mcdermott | Music perception, pitch, and the auditory system[END_REF][START_REF] Oxenham | Pitch Perception[END_REF]. Pitch discrimination has been explored in many studies, to characterize perceptual and sensory limits [START_REF] Moore | Frequency difference limens for short-duration tones[END_REF][START_REF] Bernstein | Pitch discrimination of diotic and dichotic tone complexes: Harmonic resolvability or harmonic number?[END_REF], Demany et al., 2009[START_REF] Oxenham | Pitch perception beyond the traditional existence region of pitch[END_REF][START_REF] Micheyl | Characterizing the dependence of puretone frequency difference limens on frequency, duration, and level[END_REF], Micheyl et al., 2010a) in the psychophysical tradition, to explore effects of context or memory [START_REF] Matthews | The effect of stimulus range on two-interval frequency discrimination[END_REF][START_REF] Ries | Effects of recurrent tonal information on auditory working memory for pitch[END_REF][START_REF] Nahum | From Comparison to Classification: A Cortical Tool for Boosting Perception[END_REF][START_REF] Raviv | How Recent History Affects Perception: The Normative Approach and Its Heuristic Approximation[END_REF][START_REF] Mathias | Stimulus uncertainty and insensitivity to pitch-change direction[END_REF], Micheyl et al., 2010b), to characterize interindividual differences (Semal andDemany, 2006, Mathias et al., 2010) and to relate them to factors such as musicianship [START_REF] Kishon-Rabin | Pitch discrimination: Are professional musicians better than non-musicians[END_REF][START_REF] Nikjeh | Hearing of note: an electrophysiologic and psychoacoustic comparison of pitch discrimination between vocal and instrumental musicians[END_REF][START_REF] Peretz | Congenital amusia: a disorder of fine-grained pitch discrimination[END_REF][START_REF] Micheyl | Influence of musical and psychoacoustical training on pitch discrimination[END_REF][START_REF] Tervaniemi | Pitch discrimination accuracy in musicians vs nonmusicians: an event-related potential and behavioral study[END_REF][START_REF] Bianchi | Pitch Discrimination in Musicians and Non-Musicians: Effects of Harmonic Resolvability and Processing Effort[END_REF], intellectual abilities or impairment [START_REF] Acton | Sensory discrimination as related to general intelligence[END_REF]Schroeder, 2001, Parsons et al., 2009), effects of training and plasticity [START_REF] Demany | Learning to perceive pitch differences[END_REF][START_REF] Carcagno | Pitch Discrimination Learning: Specificity for Pitch and Harmonic Resolvability, and Electrophysiological Correlates[END_REF][START_REF] Micheyl | Influence of musical and psychoacoustical training on pitch discrimination[END_REF][START_REF] Goldsworthy | Training improves cochlear implant rate discrimination on a psychophysical task[END_REF][START_REF] Carcagno | Subcortical Plasticity Following Perceptual Learning in a Pitch Discrimination Task[END_REF], or electrophysiological responses [START_REF] Nikjeh | Hearing of note: an electrophysiologic and psychoacoustic comparison of pitch discrimination between vocal and instrumental musicians[END_REF][START_REF] Barker | Human auditory cortical responses to pitch and to pitch strength[END_REF][START_REF] Carcagno | Subcortical Plasticity Following Perceptual Learning in a Pitch Discrimination Task[END_REF][START_REF] Petacchi | Increased activation of the human cerebellum during pitch discrimination: A positron emission tomography (PET) study[END_REF][START_REF] Clinard | Aging alters the perception and physiological representation of frequency: Evidence from human frequency-following response recordings[END_REF][START_REF] Tervaniemi | Pitch discrimination accuracy in musicians vs nonmusicians: an event-related potential and behavioral study[END_REF]. A common observation is large interindividual differences, with thresholds ranging from about 0.2% for trained subjects and musicians to greater than 10% for naive subjects. Marked training effects are also reported.

Discrimination thresholds can be measured using various methods. Adaptive procedures are usually preferred over constant stimuli as they ensure a dense sampling of the psychometric curve in the region of the threshold. That region is hard to situate prior to the experiment because of interindividual differences and training effects. In one common procedure (two-interval two-answer forced choice, 2I-2AFC) two tones are presented on each trial with slightly different frequencies, and the subject is required to answer which one has a higher pitch. In a variant of this procedure, the tones have either the same or a different frequency, and the subject is required to judge whether the pitch of the two tones is the same or different. The frequency difference is varied adaptively to converge on the threshold (usually corresponding to 70.7% or 79% correct). Other studies have used 3 tones [START_REF] Bernstein | Pitch discrimination of diotic and dichotic tone complexes: Harmonic resolvability or harmonic number?[END_REF][START_REF] Kishon-Rabin | Pitch discrimination: Are professional musicians better than non-musicians[END_REF][START_REF] Nikjeh | Hearing of note: an electrophysiologic and psychoacoustic comparison of pitch discrimination between vocal and instrumental musicians[END_REF][START_REF] Bianchi | Pitch Discrimination in Musicians and Non-Musicians: Effects of Harmonic Resolvability and Processing Effort[END_REF] or 4 tones [START_REF] Semal | Individual differences in the sensitivity to pitch direction[END_REF][START_REF] Mathias | Stimulus uncertainty and insensitivity to pitch-change direction[END_REF][START_REF] Amitay | A comparison of adaptive procedures for rapid and reliable threshold assessment and training in naive listeners[END_REF] per trial.

In this paper, we investigate a one-tone-per-trial "sliding 2AFC" paradigm in which the subject is presented with a sequence of tones and asked, after each tone, to judge whether its pitch is higher or lower than that of the preceding tone. A first motivation is to improve the yield (number of useful judgments per unit time) relative to classic paradigms, so as to reduce experimental time or increase statistical power for threshold estimates. This is particularly useful in experiments that require a large number of trials (e.g. [START_REF] Demany | Learning to perceive pitch differences[END_REF]. A second motivation is to facilitate the study of the effect of prior context on pitch judgments. It is well known that trial-to-trial roving of the frequency of the first tone of each pair in tasks involving 2 tones per trial increases thresholds [START_REF] Bull | Recognition Memory for Pitch of Fixed and Roving Stimulus Tones[END_REF][START_REF] Ahissar | Dyslexia and the failure to form a perceptual anchor[END_REF][START_REF] Matthews | The effect of stimulus range on two-interval frequency discrimination[END_REF], reflecting a more general sensitivity of pitch judgments to the history of prior tone frequencies [START_REF] Ruusuvirta | Proactive interference of a sequence of tones in a two-tone pitch comparison task[END_REF][START_REF] Raviv | How Recent History Affects Perception: The Normative Approach and Its Heuristic Approximation[END_REF]. Studying such effects should be easier with a continuous paradigm, in which all tones play the same role, and all are followed by a judgment, than in 2-, 3-or 4interval tasks in which successive tones play different roles.

The primary aim of the present study is to check whether the new procedure introduces any unexpected artifacts, such as increased bias or variability. For this we run both procedures with comparable parameters on the same subjects, in interleaved blocks to control for effects of time and practice on performance. A desirable outcome, relative to our goals, would be a lack of difference between the thresholds measured with the new and classic procedures, however a lack of effect is hard to demonstrate statistically [START_REF] Kriegeskorte | Circular analysis in systems neuroscience: the dangers of double dipping[END_REF]. For that reason, we also included conditions for which we do expect different thresholds, to establish whether the experiment is sufficiently sensitive to reveal the hypothesized artifacts, should they exist. Thus, we ran five different conditions, three with a classic two-interval 2AFC procedure, and two with the new sliding 2AFC procedure.

II. METHODS

The principal aim of these experiments is to validate a continuous procedure to measure pitch discrimination performance, and compare it to a standard procedure using 2 tones per trial. In order to achieve this goal, five conditions were presented, distributed over 3 sessions.

A. Subjects

15 subjects took part in the experiments, 10 women and 5 men aged from 21 to 27 years. 7 of them were musicians. Subjects who had played an instrument for at least 5 years and still had a regular practice were qualified as musicians. All musician subjects were members or former members of a music school or conservatory. One non-musician woman was excluded from the analysis because of her abnormally poor performance on the tasks.

Participants had no history of audiological or neurological disorders, and reported normal hearing which was confirmed by an audiogram in the frequency band used in the experiments. The procedures were approved by the CERES ethics committee (IRB 20131100001072), the participants were informed of the procedures and a written consent was obtained from each of them. All subjects performed all three sessions.

B. Stimuli

Stimuli consisted of 100 ms pure tones with 10 ms raised cosine onset and offset ramps.

For the sliding one-tone-per-trial procedure (conditions O1 and O2) a response was requested after each tone. For the two-tone procedure (conditions T1, T2, T3), a response was requested after each pair of tones presented with a 500 ms stimulus onset asynchrony.

In both procedures, the response triggered the onset of a new tone starting the next trial after a 500 ms interval. In the one-tone procedure, the frequency varied from one tone to the next with random direction, starting from 𝑓 " = 1000 Hz. An unbiased random walk can produce extreme frequency values; to avoid this situation the probability of an up transition (0.5 at 1000Hz) decreased linearly in frequency by a factor 0.0003 / Hz, so that frequency remained near 1000 Hz. In this region, up and down transitions were approximately equiprobable (see Results). In both procedures, the magnitude of the frequency step between the tones to be compared was determined according to a weighted one-down algorithm to track the 75 % correct point on the psychometric function [START_REF] Kaernbach | Simple adaptive testing with the weighted up-down method[END_REF]. After every wrong answer the relative frequency difference Δ𝑓/𝑓 was multiplied by 2, after every correct answer it was divided by 2 -/. . For each experimental block the step size was initially set to 10% and adapted with a single adaptive track (for rapid convergence) until the first wrong answer after at least 12 trials, after which the procedure switched to 4 interleaved independent tracks [START_REF] Leek | An interleaved tracking procedure to monitor unstable psychometric functions[END_REF]. The size of step 𝑗 was adjusted based on the response to trial 𝑗 -4. The interleaved tracks were used to reduce serial correlation between successive frequency step sizes, for the purpose of analysis in a future study.

Stimuli were synthesized at a sampling rate of 44.1 kHz and outputted via a Meridian Explorer2 sound card with 16 bit resolution. Sounds were delivered diotically through Beyerdynamic DT 770 pro 250 ohm headphones, at a listening level of 70 dB SPL.

Listeners were tested individually in a double-walled sound attenuating booth.

Instructions were provided verbally, and displayed again on a graphic interface. Subjects used the keyboard to answer, and received visual feedback (green for correct vs red for incorrect) after each response.

C. Procedures and conditions

For the sliding one-tone procedure (conditions O1, O2), the subject had to report after each tone whether it was higher or lower than the previous tone (upper right panel of Fig. 1). In condition O1 the tone-to-tone frequency step size was determined by the adaptive procedure. In condition O2 one step out of two was determined by the adaptive procedure and the other was set to 6% (one semitone). The response after that step was not taken into account for the adaptive procedure.

For the two-tone-per-trial procedure (conditions T1, T2, T3) the subject had to report after each tone pair whether the second tone was higher or lower than the first. The within-pair step size was determined by the adaptive procedure, but conditions T1, T2, and T3 differed in how the between-pair step size was determined. In condition T1 it was determined by the random procedure, leading to a similar stimulus sequence pattern as in O1. In condition T2 the between-pair step size was zero (the first tone of a pair had the same frequency as the second tone of the preceding pair). In condition T3 the frequency of the first tone was always 1 kHz (fixed standard; lower right panel of Fig. 1).

Conditions O1 and T1 share very similar tone sequences, and are expected to yield the same thresholds in the absence of an artifact specific to the one-or two-tone procedure.

Condition O2 is expected to yield larger thresholds than O1 because small near-threshold steps are interspersed with relatively large steps (6%, a semitone). Condition T2 is expected to yield smaller thresholds than T1 because the first tone of a pair is, in effect, repeated twice. Condition T3 is also expected to yield smaller thresholds than other conditions because the first tone of each pair (standard) is the same. Subject had to give an answer (pitch went up or down) after each tone. Bottom left: two-tone-per-trial procedure. Subjects had to give an answer after each pair of tones. Top right: two conditions that used the one-tone procedure. For O1 all intervals were determined by the adaptive procedure, for O2 one interval out of two was set to ±1 semitone. Bottom right: three conditions that used the two-tone procedure. For T1 all intervals (within and between pairs) were determined by the adaptive procedure (as in O1). For T2 the first tone of each pair had the same frequency as the second tone of the preceding pair.

For T3 the first tone of each pair was 1 kHz.

The experiment was carried out in 3 one-hour sessions (including instructions and training) happening on different days. Two conditions were presented in each session, interleaved into 6 blocks to control for effects of time and practice (for example due to learning or fatigue).

In each block, the subject performed one condition. In the first session, the subjects performed the main conditions O1 and T1, in the second session they performed conditions T2 and T3, and in the third and last session, they performed conditions O1 and O2. 120 trials were presented per block, except for condition O2 where 240 trials were presented per block (responses to 6% steps were ignored in the analysis). Each session started with two short training blocks (60 trials each, 120 for O2), one for each condition in that session. This was to familiarize the subjects with the procedures and ensure that they understood what feature of the sound they had to listen to.

D. Analysis

A psychometric function of the form: 𝑃 = 0.5 + 0.5 1 1 + 𝑒 56 757 8 where 𝑃 is the probability of a correct answer and 𝑥 = log Δ𝑓/𝑓 , was fit to the data by adjusting the two parameters 𝑠 and 𝑥 " , using a least-squares error method. Δ𝑓 refers to the difference between the frequencies of the two tones to be compared, and 𝑓 is (arbitrarily) the lowest of these two frequencies.

III. RESULTS

Fourteen paid subjects performed 3 sessions of 6 blocks each. In each session, two conditions were interleaved for a total of 360 responses per condition (720 for condition O2 of which 360 were retained for analysis). Condition O1 was presented twice over two sessions.

Individual results are shown in Fig. 3. The thresholds in condition O1 are presented separately for session 1 (O1a) and session 3 (O1b). The range of threshold values is consistent with the literature [START_REF] Micheyl | Influence of musical and psychoacoustical training on pitch discrimination[END_REF], although one study reported considerably larger values [START_REF] Raviv | How Recent History Affects Perception: The Normative Approach and Its Heuristic Approximation[END_REF]. The lowest thresholds found here (~0.2 %) are comparable to those reported for trained subjects in classic papers on pitch discrimination [START_REF] Moore | Frequency difference limens for short-duration tones[END_REF][START_REF] Micheyl | Influence of musical and psychoacoustical training on pitch discrimination[END_REF][START_REF] Demany | Learning to perceive pitch differences[END_REF].

A repeated measure ANOVA with factors repetition, conditions and groups demonstrated significant main effects of condition (Greenhouse-Geisser correction 𝐹(2.43, 29.21) = 33.7, 𝑝 < 0.001) and group (𝐹 1,12 = 8.35, 𝑝 = 0.014), but no main effect of repetition (𝐹(2,24) = 2.08, 𝑝 = 0.15). Interactions between repetition and condition (𝐹(10,120) = 0.85, 𝑝 = 0.58) and between condition and musicianship (𝐹 (5,60) = 1.78, 𝑝 = 0.13) were not significant. The interaction between repetition and musicianship was marginally significant (𝐹 2,24 = 3.63, 𝑝 = 0.042), reflecting a slight tendency for performance to improve over sessions for non-musicians (Fig. 3). The three-way interaction was not significant. As effects involving repetition were either non-significant or small, data were merged over blocks to produce one threshold per condition and per subject. Group-mean thresholds per condition are shown in Fig. 4, error bars indicating standard error of the mean. Focusing on pairwise comparisons, of main interest to our study we found no threshold difference between O1a and T1 (session 1), even in the absence of Bonferroni correction for multiple comparisons (𝑝 > 0.05, sign test). The lack of a significant effect carries little weight in itself, as it could result for example from a lack of power [START_REF] Kriegeskorte | Circular analysis in systems neuroscience: the dangers of double dipping[END_REF]. In this respect, it is of interest that other differences between conditions were found to be significant and of considerable magnitude.

After Bonferroni correction for 6 comparisons, the difference between conditions O1b and O2 in session 3 was significant (𝑝 = 0.011, corrected), thresholds for O2 being larger than for O1b, suggesting a deleterious effect of inserting a relatively large frequency step (6%) prior to an interval to be judged. There was a significant threshold difference between T1 and T3 (𝑝 < 0.001, corrected), thresholds for T3 being lower than for T1, indicating better performance when the first tone of all pairs was set to the same frequency (fixed standard) rather than roving. Thresholds were also lower for T3 than T2 (𝑝 = 0.011, corrected). The difference between T1 and T2 was significant before Bonferroni correction (𝑝 = 0.013) but not after correction. We found no significant difference between O1a and O1b (𝑝 > 0.05, uncorrected), again failing to demonstrate a change of performance over time. There were no differences in psychometric function slopes between any conditions.

To summarize, we found no major difference between procedures (sliding one-tone-pertrial and classic two-tone-per-trial), whereas for both procedures we found significant effects of manipulating the prior context (O1 vs O2, T1 vs T3 and T2 vs T3). There was no evidence of greater variability (over trials or subjects) or learning effects for one procedure relative to the other.

IV. DISCUSSION

The primary aim of this study was to compare a one-tone-per-trial sliding 2AFC discrimination threshold measurement procedure to a standard two-tone-per-trial 2AFC procedure. We found that both procedures were sensitive to relatively subtle manipulations of stimulus context, but yielded equivalent thresholds when applied to comparable stimulus conditions. Subjects reported no difficulty with either task and, overall, we found no indication of a problem specific to the new procedure relative to the old. Nonetheless, the lowest thresholds were found with the two-tone-per trial procedure with fixed reference (T3).

A secondary aim of this study was to compare conditions that differ in the pattern of prior context, using the same subjects and controlling for within-session and between-session serial effects so as to maximize sensitivity to potentially small effects. A common observation in the literature is that thresholds are larger if the frequency of the first tone of each pair is roved rather than constant (fixed standard) [START_REF] Bull | Recognition Memory for Pitch of Fixed and Roving Stimulus Tones[END_REF][START_REF] Ahissar | Dyslexia and the failure to form a perceptual anchor[END_REF][START_REF] Matthews | The effect of stimulus range on two-interval frequency discrimination[END_REF]. In our study, every condition involved some form of "roving" with the exception of T3. That condition yielded the lowest thresholds, consistent with prior studies. Those studies considered roving over wide frequency ranges, our study extends their conclusions to a roving range on the order of a few times the discrimination threshold. Thresholds in our study increased with the amplitude of roving (quantified as the average interval from one "standard" to the next), the highest being found for ~6% (O2) and the lowest for 0% (T3).

The deleterious effect of roving has been interpreted as reflecting an inability to form a reliable perceptual anchor [START_REF] Berliner | A perceptual anchor model for context coding in intensity perception[END_REF], or greater requirements placed on working memory [START_REF] Zhang | Auditory Discrimination Learning: Role of Working Memory[END_REF], or more general informational masking or "confusion" effects [START_REF] Mathias | Stimulus uncertainty and insensitivity to pitch-change direction[END_REF]. Those accounts invoke properties of the stimulus set as a whole; an alternative is that the internal representation of the standard on any trial is influenced by the immediately preceding tone or tones. In a companion study we found that context effects were well predicted by a linear model including (in its simplest form) a term proportional to the frequency of the immediately preceding tone roles. The influence of an irrelevant stimulus feature on decisions must lead to suboptimal performance, and a stronger influence in the case of roving might indeed lead to a higher threshold.

Subjects did not report any particular difficulty with either procedure. Subjectively, the sliding 2AFC procedure is straightforward and refreshingly brisk, as each stimulus event calls for a response. The task is analogous to perception of music, where each tone of a melody anchors both the intervals that precede and follow it. Thresholds did not differ from a standard 2-tone-per-trial 2AFC procedure with similar stimuli. This is a useful outcome, as the sliding procedure is considerably more efficient in terms of number of judgments per unit time. The greater density of judgments allows fluctuations in performance (for example related to learning, fatigue or brain state changes) to be followed with finer temporal resolution. All tones play the same role (each tone serves both as a test tone relative to the previous tone, and as a reference relative to the following tone) which is useful when studying the effects of the extended tonal history preceding each judgment. As described the procedure required subjects to give up-vs-down judgments, but it could presumably be adapted as a same-different task, for example to cater to listeners who have difficulties identifying the direction of pitch changes (Semal andDemany, 2006, Mathias et al., 2010). The sliding 2AFC procedure may also be of use to measure discrimination thresholds for other stimulus features such as intensity. give an answer (pitch went up or down) after each tone. Bottom left: two-tone-per-trial procedure. Subjects had to give an answer after each pair of tones. Top right: two conditions that used the one-tone procedure. For O1 all intervals were determined by the adaptive procedure, for O2 one interval out of two was set to ±1 semitone. Bottom right: three conditions that used the two-tone procedure. For T1 all intervals (within and between pairs) were determined by the adaptive procedure (as in O1). For T2 the first tone of each pair had the same frequency as the second tone of the preceding pair. For T3 the first tone of each pair was 1 kHz.

Figure 1 .

 1 Figure 1. Top left: illustration of the sliding one-tone-per-trial procedure.

  Figure 2 (top) shows response data of an experimental run together with the fit. The histograms in Fig. 2 (bottom) indicate the distribution of values sampled by the adaptive procedure.

Figure 2 .

 2 Figure 2. Top: psychometric function fit (line) to experimental data (circles)

Figure 3 .

 3 Figure 3. Individual frequency discrimination thresholds for each condition

Figure 4 .

 4 Figure 4. Frequency discrimination thresholds geometrically averaged over

Figure 1 .

 1 Figure captions:

Figure 2 .

 2 Figure 2. Top: psychometric function fit (line) to experimental data (circles) for a typical

Figure 3 .

 3 Figure 3. Individual frequency discrimination thresholds for each condition and each

Figure 4 .

 4 Figure 4. Frequency discrimination thresholds geometrically averaged over subjects and

ACKNOWLEDGEMENTS

This work has received support under the program ANR-10-LABX-0087 IEC and ANR-10-IDEX-0001-02 PSL*.

multiple comparisons (* for 𝒑 < 0.05, *** for 𝒑 < 0.001), 'n.s.' indicates the difference is not significant even before Bonferroni correction.