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

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

Dinh Duong Nguyen
email: dinh-duong.nguyen@univ-rennes1.fr

Rotational forms of Large Eddy Simulation turbulence models: modeling and mathematical theory

Keywords: Fluid mechanics, Turbulence models, Rotational Large Eddy Simulation models, Navier-Stokes Equations. 2010 MSC: 76D05, 35Q30, 76F65, 76D03, 35Q35

In this paper we present a derivation of a back-scatter rotational Large Eddy Simulation model, which is the extension of the Baldwin and Lomax model to nonequilibrium problems. The model is particularly designed to mathematically describe a fluid filling a domain with solid walls and consequently the differential operators appearing in the smoothing terms are degenerate at the boundary. After the derivation of the model, we prove some of the mathematical properties coming from the weighted energy estimates and which allow to prove existence and uniqueness of a class of regular weak solutions.

Introduction

The aim of this paper is twofold: From one side we are deriving in a consistent way a rotational Large Eddy Simulation model, capable of taking into account of back-scatter of energy; from another side we are also showing, by using rather elaborate functional analysis tools, the existence of weak solutions for the models we propose.

Recall that, the motion of a turbulent incompressible flow in a 3D domain Ω can be simulated by using a turbulence model such as the following eddy viscosity model1 :

(1.1)

v t + div (v ⊗ v) -div (2ν + ν turb ) Dv + ∇p = f , div v = 0,
where v t = ∂ t v for simplicity, v = v(t, x) = (v 1 (t, x), v 2 (t, x), v 3 (t, x)) is the mean velocity of the fluid, p = p(t, x) the mean pressure, ν > 0 the kinematic viscosity, ν turb ≥ 0 the eddy viscosity (also known as the turbulent viscosity), f = f (t, x) = (f 1 (t, x), f 2 (t, x), f 3 (t, x)) the external source term, Dv = 1 2 (∇v + ∇v t ) the deformation stress of the mean velocity, and "div " stands for the divergence operator.

In the whole paper we will consider the problem with homogeneous Dirichlet boundary conditions, i.e., v = 0 on (0, T ) × ∂Ω, and this poses certain technical problems, which are not present in the case of homogeneous turbulence treated in the whole space or in the space-periodic setting. However, numerical simulations would require the use of wall laws (see [START_REF] Rebollo | Mathematical and numerical foundations of turbulence models and applications[END_REF]).

One basic problem in turbulence modeling is the determination of the eddy viscosity ν turb , for which there are many options (see a comprehensive presentation of this question in [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]). One of the most popular models (and one among the first introduced) is the Smagorinsky one [START_REF] Smagorinsky | On the application of numerical methods to the solution of systems of partial differential equations arising in meteorology[END_REF] for which the eddy viscosity is given by

ν turb = κ 2 |Dv|,
where κ is the von Kármán dimensionless constant (the value of which is about 0.41) and is the Prandtl mixing length (see [START_REF] Prandtl | Prandtl-Essentials of fluid mechanics[END_REF]). The peculiarity of the modeling and of the equations derived is the degeneracy of the differential operators (by means of the function (x), which is vanishing at the boundary). The models we study can be interpreted as obtained with the application of a differential filter with radius vanishing near to the boundary; hence, the model is not over-smoothing the boundary layer. The analysis of wall-laws or of other boundary conditions requires tools not developed yet for this problem.

In the case of a flow over a plate, which is identified with the plane (x, y, 0), then the domain is Ω = R 2 × {z > 0} and one finds in Obukhov [START_REF] Obuhov | Turbulence in an atmosphere with inhomogeneous temperature[END_REF] the following law = (z) = κz.

Considering a bi-layer model for a turbulent boundary layer over a plate, Baldwin & Lomax [START_REF] Baldwin | Thin-layer approximation and algebraic model for separated turbulent flows[END_REF] suggested -from heuristic arguments-to use in the inner part of boundary layer the following formula (1.2)

ν turb = κ 2 (z)|ω|,
where ω = curl v denotes the mean vorticity, while the function (not a constant now) is determined by the Van Driest formula [START_REF] Van Driest | On turbulent flow near a wall[END_REF],

(z) := κ z (1 -e -z/A );

here A depends on the oscillations of the plate and on the kinematic viscosity ν, while z ≥ 0 is again the distance from the plate. As it is well-known, the Smagorinsky model is over-diffusive, and model (1.2) looks to be a very interesting alternative, leading by (1.1) to the system

(1.3) v t + div (v ⊗ v) -ν∆v -div (κ 2 (z)|ω| Dv) + ∇p = f , div v = 0.
However, the eddy viscosity term -div (κ2 (z)|ω| Dv) in (1.3) does not follow the rotational structure of formula (1.2). In [START_REF] Rong | Extension of a simplified Baldwin-Lomax model to nonequilibrium turbulence: model, analysis and algorithms[END_REF], the authors suggest a purely rotational form curl (κ 2 (x)|ω|ω) -which is consistent with (1.2)-yielding the following system (1.4)

v t + div (v ⊗ v) -ν∆v + curl (κ 2 (x)|ω|ω) + ∇π = f , div v = 0,
for some modified pressure term π.

In addition to being over-diffusive, the Smagorinsky model (but this limitation is also shared by non adaptive eddy viscosity models) is not capable of taking into account phenomena of back-scatter of energy. Consequently, system (1.4) seems of interest limited to (statistically) stationary or equilibrium flows. A first complete existence theory for the Baldwin & Lomax model in the steady case has been recently given in [START_REF] Berselli | On the existence of weak solutions for the steady Baldwin-Lomax model and generalizations[END_REF].

In order to consider more complex physical settings, a variant has been proposed in [START_REF] Rong | Extension of a simplified Baldwin-Lomax model to nonequilibrium turbulence: model, analysis and algorithms[END_REF] including a non-smoothing dispersive term, in the same spirit as in Voigt models (also written as Voight sometimes). The mathematical theory in this case needs to handle degenerate operators and weighted estimates. For this reason, in [START_REF] Amrouche | Turbulent flows as generalized Kelvin-Voigt materials: Modeling and analysis[END_REF] we have modeled a back-scatter term of a Voigt form such as -α div ( (x) Dv t ), where α > 0 denotes the length scale, for turbulence evolving towards a statistical equilibrium, where (x) is a smooth positive function, vanishing only at the boundary of the domain and with a prescribed rate. In [START_REF] Amrouche | Turbulent flows as generalized Kelvin-Voigt materials: Modeling and analysis[END_REF] we also studied the properties of the corresponding PDE system, in conjunction with TKE (turbulent kinetic energy equations). In [START_REF] Rong | Extension of a simplified Baldwin-Lomax model to nonequilibrium turbulence: model, analysis and algorithms[END_REF], the authors suggested instead a back-scatter term under rotational form, such as curl ( 2 (x) ω t ), obtaining the following system:

(1.5)

v t + curl ( 2 (x) ω t ) + div (v ⊗ v) -ν∆v + curl (κ 2 (x)|ω|ω) + ∇π = f , div v = 0,
for some modified pressure term π.

In this paper we show:

1. How to derive systems (1.4) and (1.5) from a standard turbulence modeling procedure, 2. Existence and uniqueness results of classes of weak solutions for these systems supplemented with smooth enough initial data and Dirichlet homogeneous boundary conditions, under certain reasonable mathematical assumptions.

The main mathematical result we prove is the following.

Theorem 1.1. Assume that the domain Ω is bounded and smooth (not necessarily with a flat boundary) and that the function : Ω → R + is of class C 2 and satisfies the two following properties:

(1.6) (x) ≈ d(x, ∂Ω) for x close to ∂Ω,
where d(x, ∂Ω) denotes the distance from the boundary, and

(1.7) ∀ K ⊂⊂ Ω, ∃ K ∈ R + s.t. (x) ≥ K > 0 ∀ x ∈ K.
Assume in addition that f ∈ L 2 (0, T ; L 2 (Ω) 3 ) and 2 v 0 ∈ W 1,3 0,σ (Ω). Then, system (1.5) with v(0) = v 0 in Ω and v = 0 on (0, T ) × ∂Ω has a unique "regular weak" solution.

Theorem 1.1 is a consequence of the weighted estimate (5.1) below, which is the main mathematical result of this paper and of a proper application of monotonicity techniques, coupled with localization of the test functions.

Plan of the paper. The paper is organized as follows: In Section 2 we set the mathematical framework that we use in the whole paper. Sections 3 and 4 are devoted to modeling and to explain the motivations for the systems (1.4) and (1.5). The proof of the main weighted estimate (5.1) is provided in Section 5. Finally, in Section 6 we present the proof of Theorem 1.1.

The research of LCB that led to the present paper was partially supported by a grant of the group GNAMPA of INdAM and by the project of the University of Pisa within the grant PRA 2018 52 UNIPI Energy and regularity: New techniques for classical PDE problems. LCB thanks also Prof. Tatsien Li for the kind invitation to the conference "China-Italy Conference on PDEs and Their Applications" (Fudan Univ. Shanghai, PRC, Dec 9-13, 2019). The results of the paper has been developed also after discussion originated during the conference. RL Thanks Prof. W. Layton for pointing out to us the questions raised by the Baldwin-Lomax model.

Functional setting

In the sequel Ω ⊂ R 3 will be a smooth and bounded open set, as usual we write x = (x 1 , x 2 , x 3 ) for all x ∈ R 3 . In particular, we assume that the boundary ∂Ω is of class C 0,1 , such that the normal unit vector n at the boundary is well defined and other relevant properties hold true. We also define the distance d(x, A) of a point from a closed set For our analysis we will use the customary Lebesgue (L p (Ω), . p ) and Sobolev spaces (W k,p (Ω), . k,p ) of integer index k ∈ N and with 1 ≤ p ≤ ∞. The L 2 (Ω)-norm will be denoted by . for simplicity. We use boldface for vectors, matrices and tensors. We recall that L p 0 (Ω) denotes the subspace of L p (Ω) with zero mean value, while W 1,p 0 (Ω) is the closure of the smooth and compactly supported functions with respect to the . 1,p -norm. As usual we denote H 1 0 (Ω) = W 1,2 0 (Ω). In addition, if Ω is bounded and if 1 < p < ∞, the following two relevant inequalities hold true:

A ⊂ R
1) the Poincaré inequality (2.2) ∃ C P (p, Ω) > 0 : u p ≤ C P ∇u p ∀ u ∈ W 1,p 0 (Ω) 3 ;
2) the Korn inequality

(2.3) ∃ C K (p, Ω) > 0 : ∇u p ≤ C K Du p ∀ u ∈ W 1,p 0 (Ω) 3 .
The Korn inequality allows to control the full gradient in L p (Ω) by its symmetric part, for functions which are zero at the boundary (cf. Malek, Nečas, Rokyta, and Ružička [START_REF] Malek | Weak and Measure-valued Solutions to Evolutionary PDEs[END_REF]). Classical results (cf. Bourguignon and Brezis [START_REF] Bourguignon | Remarks on the Euler equation[END_REF]) concern controlling the full gradient with curl & divergence. The following inequality holds true: For all s ≥ 1 and 1 < p < ∞, there exists a constant C = C(s, p, Ω) such that,

u s,p ≤ C div u s-1,p + curl u s-1,p + u • n s-1/p,p,∂Ω + u s-1,p ,
for all u ∈ W s,p (Ω) 3 , where . s-1/p,p,∂Ω is the trace norm as explained below. This same result has been later improved by von Wahl [START_REF] Wahl | Estimating ∇u by div u and curl u[END_REF] obtaining, under geometric conditions on the domain, the following estimate without lower order terms: Let Ω be such that b 1 (Ω) = b 2 (Ω) = 0, where b i (Ω) denotes the i-th Betti number, that is the the dimension of the i-th homology group H i (Ω, Z). Then, there exists C = C(p, Ω) such that (2.4)

∇u p ≤ C div u p + curl u p , for all u ∈ W 1,p (Ω) 3 satisfying either (u • n) |∂Ω = 0 or (u × n) |∂Ω = 0.
As usual in fluid mechanics, when working with incompressible fluids, it is natural to incorporate the divergence-free constraint directly in the function spaces. These spaces are built upon completing the space of solenoidal smooth vector fields with compact support, denoted as φ ∈ C ∞ 0,σ (Ω) 3 , in an appropriate topology. For 1 < p < ∞ we define (2.5)

       L p σ (Ω) := φ ∈ C ∞ 0,σ (Ω) 3 . p , W 1,p 0,σ (Ω) := φ ∈ C ∞ 0,σ (Ω) 3 . 1,p
.

A basic tool in mathematical fluid mechanics is the construction of a continuous right inverse of the divergence operator with zero Dirichlet boundary conditions. An explicit construction is due to the Bogovskiȋ and it is reviewed in Galdi [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems[END_REF]Ch. 3]. The following results holds true.

Proposition 2.1. Let ω ⊂ R 3 be a bounded smooth domain and let f ∈ L p 0 (ω). Then, there exists at least one u = Bog ω (f ) ∈ W 1,p 0 (ω) 3 which solves the boundary value problem

div u = f in ω, u = 0 on ∂ω.
Among other spaces, the operator Bog ω is linear and continuous from L p 0 (ω) to W 1,p 0 (ω) 3 , for all p ∈ (1, ∞).

Part I Modeling

In this part we perform the modeling leading to the model (1.4) in Section 3 and the rotational back-scatter model (1.5) in Section 4.

On the Baldwin & Lomax model

We start by recalling some facts about the Baldwin & Lomax model which will be used later on. Let Ω ⊂ R 3 denote the flow domain. We decompose any field ψ = ψ(t, x) with (t, x) ∈ R + × Ω, as the sum of its mean (denoted by a bar) and its fluctuation,

ψ = ψ + ψ ,
as suggested by Reynolds [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]. The bar operator denotes any linear statistical filter that does not need to be specified, beside that we assume it verifies at least the Reynolds rules:

     v t + ω × v -ν∆v + ∇ p + |v| 2 2 = f , div v = 0, (3.2)
where (v, p) denotes the pair of the velocity and the pressure, and the alternative form of the convective term follows by using the well-known identity

(v • ∇) v = 1 2 ∇|v| 2 + ω × v,
where ω = curl v. We apply the bar operator to (3.2). By using the Reynolds rules (3.1), one obtains to the following system

(3.3) v t + ω × v + ω × v -ν∆v + ∇q = f , div v = 0,
where the force is chosen such that f = f for simplicity. The Bernoulli pressure and the fluctuation of the vorticity are given, respectively, by

q = p + |v| 2 2 and ω = curl v .
This leads to the issue of modeling the turbulent flux term ω × v only by mean (averaged) quantities. According to the Helmholtz-Hodge theorem, under reasonable regularity and decay assumptions, there exists a unique vector field

A (R) such that (3.4) curl A (R) = ω × v , div A (R) = 0,
and in what follows we call A (R) the "rotational Reynolds stress." As usual in turbulent modeling, the fundamental question is how to express A (R) in terms of averaged quantities.

It is natural to assume that A (R) is a function of the mean vorticity ω. Following the standard Reynolds-stress modeling-procedure and respecting the divergence free constraint div A (R) = 0, we are led to set (3.5)

A (R) = ν turb ω + ∇ψ,
for some scalar function ψ which will be specified later on. Notice that from the Reynolds rules combined with the Schwarz theorem, we have div ω = 0. Therefore, taking the divergence of (3.5) and using div A (R) = 0 yields a Poisson equation for ψ:

-∆ψ = div (ν turb ω), hence, since ω is divergence-free, ψ = (-∆) -1 (∇ν turb • ω).
In conclusion, the closure assumption for the rotational Reynolds stress can be expressed as follows

(3.6) A (R) = ν turb ω + ∇(-∆) -1 (∇ν turb • ω).
Taking the curl of (3.6) gives curl

A (R) = curl (ν turb ω).
Therefore, according to the Baldwin & Lomax model if ν turb = κ 2 (x)|ω|, and by noting that

       1 2 |v| 2 = 1 2 |v| 2 + k, div (v ⊗ v) = ω × v + ∇ |v| 2 2 ,
(where k = 1 2 |v | 2 denotes the turbulent kinetic energy) we get as closure equations from (3.3) the following system:

v t + div (v ⊗ v) -ν∆v + curl (κ 2 (x)|ω|ω) + ∇(p + k) = f , div v = 0,
which yields to the system (1.4) by setting the modified pressure π = p + k, and where we recall that ω = curl v. In a vorticity/velocity formulation it is also relevant to consider the rotational form of the convective term, hence

v t + ω × v -ν∆v + curl (κ 2 (x)|ω|ω) + ∇q = f , div v = 0.

Introduction of the rotational back-scatter term

Following a modeling similar to that already employed in [START_REF] Amrouche | Turbulent flows as generalized Kelvin-Voigt materials: Modeling and analysis[END_REF], we show in this section how to derive the following model

(4.1) v t + curl ( 2 ω t ) + ω × v -ν∆v + curl (κ 2 |ω| ω) + ∇q = f , div v = 0,
for a turbulent flow evolving towards a statistical equilibrium.

Equation (3.3) combined with (3.4) becomes:

(4.2) v t + ω × v -ν∆v + curl A (R) + ∇q = f , div v = 0.
According to Leray's result [START_REF] Leray | Sur le mouvement d'un liquide visqueux emplissant l'espace[END_REF], we know any turbulent solution (smooth enough to carry on all the calculations) to (4.2) satisfies the energy inequality

(4.3) 1 2 d dt v(t) 2 + ν ∇v(t) 2 + curl A (R) , v(t) ≤ f (t), v(t) ,
in the sense of distributions over (0, T ), provided that the boundary conditions do not bring additional terms (such as occurs i) with the no-slip boundary condition; ii) when Ω = R 3 , or iii) in the space periodic case, for instance). Let us set

I (t) := curl A (R) , v(t) .
The aim of what follows is to study the contribution of this term in the energy inequality (4.3). To do so, we use the well-known formula (4.4)

ν turb = C k √ k,
relating eddy viscosity ν turb and turbulent kinetic energy k, see [START_REF] Rebollo | Mathematical and numerical foundations of turbulence models and applications[END_REF]. Then, we combine (4.4) with ν turb = κ 2 |ω|, leading to the closure equation for k

(4.5) k = 2 2 |ω| 2 = 2 2 |curl v| 2 .
We assume now that the production of turbulent kinetic energy is mainly due to small scales eddies, which are in a statistical equilibrium and that no-stratification occurs. By a straightforward generalization of what is done in [START_REF] Rebollo | Mathematical and numerical foundations of turbulence models and applications[END_REF]Sec. 4.4.1], we get the following equation for k:

(4.6) k t + v • ∇k + div e v = A (R) • ω -ε + f • v ,
where the rotational turbulent dissipation is given in this case by ε = ν|ω | 2 , and e denotes the fluctuation of the kinetic energy of the fluctuation e = 1 2 |v | 2 . The combination of (4.5) and (4.6) gives the formal following energy equality:

(4.7) d dt Ω k(t) = Ω 2 ω t • ω = I (t) - Ω ε(t) + f , v .
From (4.3) and (4.7) it follows the following inequality

(4.8) f rac12 d dt v(t) 2 + ω(t) 2 + ν ∇v(t) 2 + √ ε(t) 2 ≤ f (t), v(t) + f , v .
The energy inequality (4.8) suggests to add the term 2 ω t to the rotational Reynolds stress in formula (3.6), leading to the following expression for the no-equilibrium rotational Reynolds stress (4.9)

A (R) = 2 ω t + ν turb ω + ∇(-∆) -1 (∇ν turb • ω).
When we plug (4.9) into (4.2) to get the following energy inequality

(4.10) 1 2 d dt v(t) 2 + ω(t) 2 + ν ∇v(t) 2 + √ ν turb ω(t) 2 ≤ f (t), v(t) .
We compare (4.8) and (4.10), which is consistent when the following compatibility condition is satisfied: Part II

(4.11) √ ν turb ω(t) 2 + f , v ≤ √ ε(t)

Analysis of the model

In this part of the paper we perform the mathematical analysis of the back-scatter rotational model, by using established methods of analysis for non-Newtonian fluids. We first present the main weighted estimate in Section 5 and then proving the existence and uniqueness results in Section 6, as stated in Theorem 1.1.

Main estimate

In this section we show a bound involving the weighted-curl and weighted-gradient, which does not follow directly from the classical tools combining weighted estimates and harmonic analysis. As employed in [START_REF] Berselli | On the existence of weak solutions for the steady Baldwin-Lomax model and generalizations[END_REF] it can be shown that for fields in W 1,p 0,σ (Ω) one can prove the weighted estimate

Ω |∇v(x)| p w(x) dx ≤ C(w, Ω, p) Ω |curl v(x)| p w(x) dx,
provided that the weight function w ∈ L 1 loc (R 3 ), which is s.t. w ≥ 0 a.e., belongs to the Muckenhoupt class A p , for 1 < p < ∞, that is there exists C such that sup

Q⊂R n - Q w(x) dx - Q w(x) 1/(1-p) dx p-1 ≤ C,
where Q denotes a cube in R 3 . It is well-known that the powers of the distance function

w(x) = d(x, ∂Ω) α are Muckenhoupt weights of class A p if and only if -1 < α < p -1,
hence in the relevant cases we could not infer the required estimates if

(x) = d(x, ∂Ω) α , for α ≥ p -1.
In our case, we can prove a crucial estimate in a different and direct way, by using the special Hilbert structure when p = 2. Despite being based on elementary direct computations the following lemma plays a fundamental role on the analysis of the rotational back-scatter system (1.5).

Lemma 5.1. Assume that the function is such that 2 ∈ W 2,∞ (Ω) and let v ∈ W 1,2 0,σ (Ω). Then, there exists a positive constant C( ) = C( D 2 2 ∞ ) such that

(5.1) Ω 2 |∇v| 2 dx ≤ Ω 2 |curl v| 2 dx + C( ) Ω |v| 2 dx.
Proof. We start from the well-known vector-calculus identity

(5.2) -∆v = curl (curl v) -∇(div v) = curl (curl v),
that holds for any divergence free vector field v. Then multiplying (5.2) by 2 v and integrating by parts on Ω we obtain3 :

(5. 3)

Ω ∇v : ∇( 2 v) dx = Ω (curl v) • (curl ( 2 v)) dx,
where the fact that v = 0 on ∂Ω has been used. We will explicitly rewrite both sides of (5.3). The left-hand side (l.h.s) of (5.3) can be rewritten as follows

Ω ∇v : ∇( 2 v) dx = Ω 3 i,j=1 ∂ j v i ∂ j ( 2 v i ) dx = 1 2 Ω 3 i,j=1 ∂ j v 2 i ∂ j 2 dx + Ω 3 i,j=1 (∂ j v i ) 2 2 dx = 1 2 Ω 3 j=1 ∂ j 3 i=1 v 2 i ∂ j 2 dx + Ω 2 |∇v| 2 dx = 1 2 Ω 3 j=1 ∂ j |v| 2 ∂ j 2 dx + Ω 2 |∇v| 2 dx = 1 2 Ω ∇|v| 2 • ∇ 2 dx + Ω 2 |∇v| 2 dx = - 1 2 Ω |v| 2 ∆ 2 dx + Ω 2 |∇v| 2 dx, (5.4)
where v = (v 1 , v 2 , v 3 ) and in the last equality in (5.4) we used integration by parts possible again since v = 0 on ∂Ω. The right-hand side (r.h.s) of (5.3) can be rewritten as

Ω (curl v) • (curl ( 2 v)) dx = Ω 2 |curl v| 2 dx + Ω (curl v) • (∇ 2 ) × v dx, (5.5)
where the identity curl ( 2 v) = 2 curl v + (∇ 2 ) × v has been used. Combining (5.4) and (5.5) yields (5.6)

Ω 2 |∇v| 2 dx = Ω 2 |curl v| 2 dx + Ω (curl v) • (∇ 2 ) × v dx + 1 2 Ω |v| 2 ∆ 2 dx.
To conclude we are going to estimate the two last integrals on the r.h.s of (5.6). To handle the vector product we use the properties of the Ricci totally anti-symmetric tensor ijk (such that (a × b) i = 3 j=1 3 k=1 ijk a j b k ) and we rewrite in components the integrand as follows

(curl v) • (∇ 2 ) × v = 3 i,j,k,m,p=1 ijk (∂ j v k ) imp (∂ m 2 )v p = 3 j,k,m,p=1 (δ jm δ kp -δ jp δ km )(∂ j v k ) (∂ m 2 )v p = 3 j,k,m=1 [(∂ j v k ) (∂ m 2 )v k -(∂ j v k ) (∂ k 2 )v j ] = 1 2 3 j,m=1 (∂ j |v| 2 ) (∂ m 2 ) - 3 j,k=1 (∂ j v k ) (∂ k 2 )v j .
Hence, passing to the integral, and recalling that v vanishes at the boundary and is divergence-free, we can integrate by parts to obtain

Ω (curl v) • (∇ 2 ) × v dx = 1 2 3 j,m=1 Ω (∂ j |v| 2 ) (∂ m 2 ) dx - 3 j,k=1 Ω (∂ j v k ) (∂ k 2 )v j dx = - 1 2 3 j,m=1 Ω |v| 2 ∂ jm 2 dx + 3 j,k=1 Ω v j v k ∂ jk 2 dx, which leads to Ω (curl v) • (∇ 2 ) × v dx ≤ C D 2 2 ∞ v 2 .
In addition, the other integral on the r.h.s of (5.6) is bounded by

1 2 Ω |v| 2 ∆ 2 dx ≤ C D 2 2 ∞ v 2 ,
which combines to the previous estimate ending the proof.

Existence and uniqueness results

Throughout this section, we assume that assumptions (1.6) and (1.7) in Theorem 1.1 hold, that is (x) = O( d(x, ∂Ω)) near the boundary and is strictly positive inside the domain. Moreover, we also assume that f ∈ L 2 (0, T ; L 2 (Ω) 3 ) and v 0 ∈ W 1,3 0,σ (Ω). Finally, recall that ω = curl v. As in (2.1) we write d(x, ∂Ω) = d(x). Without loss of generality, and according to the modeling introduced in

v t + curl (d 0 d ω t ) + ω × v -ν∆v + curl (d 0 d |ω| ω) + ∇p = f , div v = 0,
where p is some modified pressure and, again for simplicity and without loss of generality, we suppose from now on that d 0 = 1. Recall that the above system is supplemented by the Dirichlet boundary conditions v = 0 on (0, T ) × ∂Ω and the initial datum v(0) = v 0 in Ω.

In order to prove existence of weak solutions we observe that the basic a-priori estimate is obtained by testing with v itself and obtaining (after integration by parts, if solutions are smooth to perform all computations) the following energy inequality for all s ∈ (0, T )

v(s) 2 + √ d ω(s) 2 + ν s 0 ∇v 2 dt + 2 s 0 Ω d |ω| 3 dxdt ≤ v 0 2 + √ d ω 0 2 + C ν s 0 Ω f 2 dxdt.
Here, the vanishing contribution of the rotational convection term has been used (after modifying the pressure) and the dimensionless constant C comes from applying the Poincaré and Young inequalities. It follows by using (2.2) and a natural f ∈ L 2 (0, T ; L 2 (Ω) 3 ) assumption, that

v ∈ L ∞ (0, T ; L 2 (Ω) 3 ) ∩ L 2 (0, T ; H 1 0 (Ω) 3 ), d 1/2 ω ∈ L ∞ (0, T ; L 2 (Ω) 3 ), d 1/3 ω ∈ L 3 (0, T ; L 3 (Ω) 3 ).
Hence, from one side we have for the mean velocity v the same estimates valid for the Leray-Hopf weak solutions of the Navier-Stokes equations; On the other side we have further estimates on the mean vorticity which are weighted by the distance from the boundary, hence not enough to directly apply standard methods. We observe that both (dispersive/back-scatter and dissipative/eddy viscosity) the additional degenerate terms pose some mathematical difficulties: If in the system (6.2) one would have been given the following smoothing term curl (|ω|ω), then the a-priori estimate, and the divergence-free constraint with (2.4) will imply directly that v ∈ L 3 (0, T ; W 1,3 0,σ (Ω)), allowing us to apply the same tools valid for the Smagorinsky model as in [START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF]. Here the estimates degenerate at the boundary (being the mean vorticity weighted by the distance function d) and this prevents from using the solution itself as a legitimate test function.

Next, if the dispersive term would have been given by curl (ω t ) = -∆v t , (where the equality is valid for divergence-free functions) the same well-known tools valid for the Voigt model can be used as in [START_REF] Cao | Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models[END_REF]. We note in particular that in problem (6.2) the presence of this dispersive term does not allow us to prove by comparison the classical regularity in negative spaces for the time derivative v t (as needed by Aubin-Lions type compactness results); In addition it is also not easy to prove from the weak formulation that the solution is weakly continuous in L 2 (Ω) as required by the compactness results à la Hopf (or in the refined form of Landes and Mustonen [START_REF] Landes | A strongly nonlinear parabolic initial-boundary value problem[END_REF]). Each term poses some questions which can be separately handled, but the combination of the effects of both weighted terms requires to have a precise interplay between some local (in space) estimates on a double approximated system.

For these reasons we first -regularize the system by a hyper-dissipative term and we then approximate it by a Galerkin procedure. We first pass to the limit in the Galerkin system and then pass to the limit in the smoothed system, by using further regularity on the time derivative which is obtained in a way similar to [START_REF] Amrouche | Turbulent flows as generalized Kelvin-Voigt materials: Modeling and analysis[END_REF].

The approximate system: existence and further regularity

For simplicity we assume from now on that f = 0, but the introduction of an external force f ∈ L 2 (0, T ; L 2 (Ω) 3 ) can be done with minor changes. Moreover, throughout the section, we assume = √ d, but observe that assumptions (1.6) and (1.7) would be enough.

In order to apply the standard Galerkin method and monotonicity, we approximate the system (6.2) by the following one (6.3)

v t + curl (d ω t ) + ω × v -ν∆v + curl (d |ω |ω ) -div (|Dv |Dv ) + ∇p = 0, div v = 0,
and we study it with homogeneous Dirichlet boundary conditions.

The above system falls within the standard class of monotone problems as those considered in Lions [START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF] and Ladyžhenskaya [START_REF] Ladyžhenskaya | The mathematical theory of viscous incompressible flow[END_REF] in the analysis of the Smagorinsky model. Here, in addition to the standard Smagorinsky model, we have two perturbation terms, which can be easily handled.

We have the following result.

Theorem 6.1. Let be given v 0 ∈ W 1,3 0,σ (Ω), then there exists a unique weak solution v to system (6.3) with v 0 as initial datum and with homogeneous Dirichlet boundary conditions. This means that v ∈ L ∞ (0, T ;

L 2 σ (Ω) 3 ) ∩ L 3 (0, T ; W 1,3 0,σ (Ω)), is such that (6.4) T 0 Ω (ω × v ) • φ + ν∇ω : ∇φ + d |ω |ω • curlφ + |Dv |Dv : Dφ dxdt = T 0 Ω v • φ t + d ω • curl φ t dxdt + Ω v 0 • φ(0) + d ω 0 • curl φ(0) dx, for all φ ∈ C ∞ 0,σ ([0, T ) × Ω) 3 . Proof.
Testing by divergence free test vector fields as is custom, we do not consider the pressure term that can be recovered through the usual ways. The proof is based on an application of the Galerkin method to prove existence of approximate solutions. Denoting by v ,m ∈ V m for all t ∈ (0, T ) a finite dimensional approximation to v one has the following energy estimate for all s ∈ [0, T ]

v ,m (s) 2 + √ d ω ,m (s) 2 + 2ν s 0 ∇v ,m 2 dt + 2 s 0 Dv ,m 3 3 dt + 2 s 0 Ω d |ω ,m | 3 dxdt ≤ v 0 2 + √ d ω 0 2 , which shows, by using (2.2)-(2.3) that v ,m ∈ L ∞ (0, T ; L 2 σ (Ω) 3 ) ∩ L 3 (0, T ; W 1,3 0,σ (Ω)),
with estimates depending on > 0, but independent of m ∈ N.

Next, testing with v t , we can see that the contribution of the rotational convective term can be estimated as follows:

Ω (ω × v) • v t dx ≤ v t v 6 ω 3 ≤ v t v 6 ∇v 3 ≤ C v t ∇v Dv 3 ≤ 1 2 v t 2 + C ∇v 2 Dv 2 3 ,
for smooth enough v, where we used the Korn inequality (2.3).

By using v ,m t as test function and the previous estimates (where v is replaced by v ,m ) we then obtain the following differential inequality

1 2 v ,m t (s) 2 + √ d ω ,m t (s) 2 + d dt ν 2 ∇v ,m 2 + d dt 3
Dv ,m 3 3 + d 3dt Ω d |ω ,m | 3 dx ≤ C ∇v ,m 2 Dv ,m 2 3 .
An application of the Gronwall lemma -possible since v ,m ∈ L 3 (0, T ;

W 1,3 0,σ (Ω))-shows that v ,m ∈ L ∞ (0, T ; W 1,3 0,σ (Ω)) and v ,m t ∈ L 2 (0, T ; L 2 (Ω) 3 ),
again uniformly in m ∈ N. The above estimates with Aubin-Lions compactness lemma (cf. [START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF]) are enough to infer that, for each fixed > 0, there exists 9 ),

v ∈ L ∞ (0, T ; W 1,3 0,σ (Ω)) ∩ H 1 (0, T ; L 2 σ (Ω)), such that when m → +∞ v ,m * v in L ∞ (0, T ; W 1,3 0,σ (Ω)), v ,m v in L 3 (0, T ; W 1,3 0,σ (Ω)), v ,m t v t in L 2 (0, T ; L 2 σ (Ω) 3 ), √ d ω ,m t √ d ω t in L 2 (0, T ; L 2 σ (Ω) 3 ), |Dv ,m |Dv ,m χ 1 in L 3/2 (0, T ; L 3/2 (Ω) 9 ), |Dv ,m |Dv ,m * χ 1 in L ∞ (0, T ; L 3/2 (Ω)
d 2/3 |ω ,m |ω ,m χ 2 in L 3/2 (0, T ; L 3/2 (Ω) 3 ), d 2/3 |ω ,m |ω ,m * χ 2 in L ∞ (0, T ; L 3/2 (Ω) 3 ), v ,m → v in L 2 (0, T ; L q σ (Ω) 3 ) ∀ q < ∞.
The above convergences are enough to pass to the limit in the approximate equations, except in the monotone terms.

In particular, for the Baldwin & Lomax term, it follows that

T 0 Ω d |ω ,m |ω ,m • curl φ dxdt = T 0 Ω d 2/3 |ω ,m |ω ,m • d 1/3 curl φ dxdt m→+∞ → T 0 Ω χ 2 • d 1/3 curl φ dxdt = T 0 Ω d 1/3 χ 2 • curl φ dxdt,
for all smooth functions φ with compact support. Hence, one gets (the trick of distributing powers of the distance function on the integrands will be used several times in the sequel)

T 0 Ω (ω × v) • φ + ν∇ω : ∇φ + χ 1 : Dφ + d 1/3 χ 2 • curlφ dxdt = T 0 Ω v • φ t + d ω • curl φ t dxdt + Ω v 0 • φ(0) + d ω 0 • curl φ(0) dx,
and for almost all 0 ≤ s 0 ≤ s ≤ T it holds

1 2 v (s) 2 + √ d ω (s) 2 + s s 0 ν ∇v (t) 2 + Ω χ 1 : Dv + d 1/3 χ 2 • ω dx dt = 1 2 v (s 0 ) 2 + √ d ω (s 0 ) 2 .
Hence, to show that v is a solution to (6.3) one needs to prove that (6.5)

χ 1 = |Dv |Dv and χ 2 = d 2/3 |ω |ω , at least almost everywhere in (0, T ) × Ω.
This can be proved by using the standard monotonicity argument (Minty-Browder trick) as developed in the time evolution problem in [START_REF] Ladyžhenskaya | The mathematical theory of viscous incompressible flow[END_REF][START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF]. The only thing to be verified is that the function v ,mv , is a legitimate test function. This follows by the regularity of the time derivative we proved. Hence, the classical argument proceeds as in (cf. [14, p. 207]) showing that the approximate solution v satisfies

s 0 Ω (χ 1 -|Dφ|Dφ) : (Dv -Dφ) dxdt ≥ 0, s 0 Ω (χ 2 -d 2/3 |curl φ|curl φ) • (d 1/3 ω -d 1/3 curl φ) dxdt s 0 Ω (d 1/3 χ 2 |curl φ|curl φ) • (ω -curl φ) dxdt ≥ 0,
for a.e. s ∈ [0, T ] and for arbitrary φ ∈ L 3 (0, T ; W 1,3 0,σ (Ω)), since they are both coming from monotone terms. This is enough to imply by monotonicity of the functions

B → |B|B and b → d α |b|b,
(which is valid for all matrices B, vectors b, α ∈ R + , and smooth functions d such that d > 0 for all x ∈ Ω, cf. [START_REF] Berselli | On the existence of weak solutions for the steady Baldwin-Lomax model and generalizations[END_REF]Lem. 3.2]) that the equalities in (6.5) hold true. We finally proved that there exists v such that (6.6)

T 0 Ω v t • φ + d ω t (s) • curl φ + (ω × v ) • φ dx + T 0 Ω ν ∇v : ∇φ + |Dv |Dv : Dφ + d |ω | ω • curl φ dxdt = 0,
at least for all φ ∈ L 3 (0, T ; W 1,3 0,σ (Ω)). Well-known estimates can be also applied to show that the solution v is unique. Remark 6.1. Due to the regularity of the solution of the approximated system we can use the function v ,mv as test function. In the case of the non-regularized system (6.2) we will see that localization in the space variable is needed and this is not compatible with the finite dimensional Galerkin approximation.

Proof of Theorem 1.1

We now consider the original problem (without the -regularization) and give the proof of the main result of the paper.

Proof of Theorem 1.1. The proof is divided into two steps. Let us start with the existence part.

Step 1: Existence part. To construct weak solutions to (6.2) we consider the limit → 0 of solutions to (6.3). By the estimate coming from the energy inequality we also have by using (5.1) the following inequality, for all s ∈ (0, T )

1 2 v (s) 2 + min 1, 1 2C( ) √ d ∇v (s) 2 + 2ν s 0 ∇v 2 dt + 2 s 0 Dv 3 3 dt + 2 s 0 Ω d |ω | 3 dxdt ≤ v 0 2 + √ d ω 0 2 , which shows that v ∈ L ∞ (0, T ; L 2 σ (Ω) 3 ) ∩ L 2 (0, T ; W 1,2 0,σ (Ω)) and d 1/3 ω ∈ L 3 ((0, T ) × Ω),
with estimates independent of > 0. We now extract further information from the other bound which is independent of , namely

√ d ∇v ∈ L 2 (0, T ; L 2 (Ω) 9 ),
coming from the other term on the left-hand side. We use now the inequality

v H 1/2 (Ω) 3 ≤ C √ d ∇v ∀ v ∈ W 1,2 0 (Ω) 3 ,
which is a simplification of that proved in [1, Thm. 3.1] and H 1/2 (Ω) denotes the famous critical fractional Sobolev space.

Here, we have the full gradient instead of the deformation tensor on the right-hand side and since we are working with the Galerkin approximations we need to verify it at least for functions in W 1,2 0 (Ω), instead that for general distributions: This is why the estimate is less technical than that in [START_REF] Amrouche | Turbulent flows as generalized Kelvin-Voigt materials: Modeling and analysis[END_REF]. By using the Sobolev embedding H 1/2 (Ω) ⊂ L 3 (Ω), valid in three space dimensions, we finally have the following version of a classical Lions and Magenes result (6.7)

v 3 ≤ C v 1/2,2 ≤ C √ d ∇v ∀ v ∈ W 1,2 0 (Ω) 3 .
This is still not enough for our purposes, but we pass at the estimate obtained testing with v t . We can also write the following estimate, which follows as in [1, Sec. 4] (6.8)

Ω (ω × v ) • v t dx ≤ v t 3 v 6 ω ≤ C v t 3 ∇v 2 ≤ 1 2 min 1, 1 2C( ) √ d ∇v t 2 + C 1 ( ) ∇v 4 ,
valid for smooth functions for some C 1 ( ). At the level of the Galerkin approximation we can use the above estimate and then the bound is inherited by the limit in m → +∞. Hence, by testing by v ,m t the Galerkin system and by using Lemma 5.1, with the estimation on the convective term (6.8), we get (after passing to the limit m → +∞) the following differential inequality

1 2 v t 2 + 1 2 min 1, 1 2C( ) √ d ∇v t 2 + d dt ν 2 ∇v 2 + d dt 3 Dv 3 3 + d 3dt Ω d |ω | 3 dx ≤ C 1 ( ) ∇v 4 .
In particular, for all s ∈ (0, T ) it holds

ν 2 ∇v (s) 2 ≤ ν 2 ∇v 0 2 + 3 Dv 0 3 3 + 1 3 Ω d |ω 0 | 3 dx + C 1 ( ) s 0 ∇v 4 dt,
which shows that, by using the Gronwall lemma (see for example [1, Lemma 4.1])

ν 2 ∇v (s) 2 ≤ ν 2 ∇v 0 2 + 3 Dv 0 3 3 + 1 3 Ω d |ω 0 | 3 dx exp C 1 ( ) s 0 ∇v 2 dt ≤ ν 2 ∇v 0 2 + 1 3 Ω d |ω 0 | 3 dx exp C 1 ( ) 2ν v 0 2 + √ d ω 0 2 =: F ( , v 0 ),
where we have used the uniform estimate for v in L 2 (0, T ; W 1,2 0 (Ω) 3 , previously proved. Therefore, from the above differential inequality we get, for all s ∈ (0, T )

1 2 s 0 v t 2 + min 1, 1 2C( ) √ d ∇v t 2 dt + ν 2 ∇v (s) 2 + 1 3 Ω d |ω (s)| 3 dx + 3 Dv (s) 3 3 ≤ ν 2 ∇v 0 2 + 1 3 Ω d |ω 0 | 3 dx + C 1 ( )F ( , v 0 ) v 0 2 + √ d ω 0 2 ,
for all > 0. The latter implies in particular that

v t ∈ L 2 (0, T ; L 3 (Ω) 3 ) ∩ H 1/2 (Ω) 3 ) and v ∈ L ∞ (0, T ; W 1,2 0,σ (Ω)),
with bounds uniform in > 0. (The validity of the estimates can be justified working again with the Galerkin approximation showing estimates not depending on m in a standard way.) We can now use this information to pass to the limit as → 0.

In particular, by the a priori estimates, and since d > 0 for all x ∈ Ω observe that we can infer

d K T 0 K |ω | 3 dxdt ≤ T 0 Ω d |ω | 3 dxdt, with 0 < d K := min x∈K d(x).
Next, being the right-hand side bounded independently of > 0 this shows that we have (up to a sub-sequence) L 3 -weak convergence in (0, T ) × K.

Considering a family of closed balls B q,rq ⊂ Ω with rational center q ∈ Q 3 and rational radius r q ∈ Q + which form a covering of Ω, and using a diagonal argument we can show that we can find a sub-sequence {ω } converging in L 3 in any compact set of (0, T ) × Ω.

Moreover, one has also the weak- * convergence in L ∞ (0, T ; L 3 (K)). By collecting all information coming from the above a priori estimates, we can infer that there exists v ∈ W 1,2 (0, T ; 3 ), (6.10) 3 ), (6.12) |Dv |Dv 0 in L 3/2 (0, T ; L 3/2 (Ω) 9 ), (6.13) 3 ), (6.14) (6.17) and by Aubin-Lions lemma

L 3 σ (Ω) ∩ H 1/2 (Ω) 3 ) ∩ L ∞ (0, T ; W 1,2 0,σ (Ω)), with ω ∈ L ∞ (0, T ; L 3 loc (Ω) 3 ), such that v * v in L ∞ (0, T ; W 1,2 0,σ (Ω)), (6.9) √ d ω * √ d ω in L ∞ (0, T ; L 2 (Ω)
v t v t in L 2 (0, T ; H 1/2 (Ω) 3 ∩ L 3 σ (Ω) 3 ), (6.11) √ d ω t √ d ω t in L 2 (0, T ; L 2 (Ω)
d |ω |ω χ in L 3/2 (0, T ; L 3/2 (Ω)
d |ω |ω * χ in L ∞ (0, T ; L 3/2 (Ω) 3 ), (6.15) ω ω in L 3 (0, T ; L 3 (K) 3 ), ∀ K ⊂⊂ Ω, (6.16) ω * ω in L ∞ (0, T ; L 3 (K) 3 ), ∀ K ⊂⊂ Ω,
v → v in L 2 (0, T ; W 3/4,2 0,σ (Ω) 3 ) ⊂ L 2 (0, T ; L 4 (Ω) 3
), (6.18) All terms in the equation with the weak formulation (6.4) for v pass to the limit, except the nonlinear one concerning the Baldwin & Lomax stress tensor. We obtain then (6.19)

T 0 Ω v t • φ + d ω t • curl φ + (ω × v) • φ + ν∇v : ∇φ + χ • curl φ dxdt = 0,
for all smooth test functions φ with compact support in (0, T ) × Ω.

The last step is to show that the limit v (and it curl ω) satisfies the system (6.2) in a weak sense. To this end it would be classical to take the difference between the equation satisfied by v and that satisfied by v, test by the difference and show that the limit vanishes. This is needed to show that d |ω |ω → d |ω|ω, at least a.e. in (0, T ) × Ω. All the other terms work fine, the only problem is then to make sure that the integral below is well-defined (6.20)

T 0 Ω (d |ω |ω -d |ω|ω) • (ω -ω) dxdt → 0,
and to show that it vanishes. The a priori estimates we have on the solution are not enough for this results: the integral in (6.20) can be well-defined if taken over a compact subset of K ⊂ Ω, being ω ∈ L 3 loc (Ω) 3 for a.e. t ∈ [0, T ], but not over the whole domain Ω. In order to overcome this problem we have to localize. So let us fix an open ball B := B(x, R) ⊂ Ω and take a cut-off function 0

≤ η ∈ C ∞ 0 (Ω)) such that η(x) = 1 if x ∈ B/2 := B(x, R/2), η(x) = 0 if x ∈ Ω\B.
In this way, since for a.e. t ∈ (0, T ) it follows that v(t) ∈ L 3 (Ω) 3 , and ω(t) ∈ L 3 (B) 3 we have that

η (v -v) |∂B = 0, div (η (v -v)) = ∇η • (v -v) ∈ L 3 (B), curl(η (v -v)) = ∇η × (v -v) + η (ω -ω) ∈ L 3 (B) 3 ,
it follows then by (2.4) that η (v -v) ∈ L 3 (0, T ; W 1,3 0 (B) 3 ). Concerning the regularity, for all > 0 the vector η (v -v) will be suitable as test function, but it still not allowed since η (v -v) is not divergence-free. So in order to be able to use it we need to subtract its divergence. This can be done by means of the Bogovskiȋ operator Bog B ( . ) associated to the ball B. Note that we are using it for all fixed t ∈ [0, T ] and this does not create problems since the functions are smooth enough to consider the time as a parameter. Hence, a legitimate test function is the following one

Φ := η (v -v) -Bog B (∇η • (v -v)) in B, 0 in Ω\B.
From the continuity of the Bogovskiȋ operator as in Proposition 2.1 we can infer that supp Φ ⊂ B for all t ∈ [0, T ] and

Φ ∈ L ∞ (0, T ; W 1,2 0,σ (Ω)) ∩ L 3 (0, T ; W 1,3 0,σ (Ω)).
Moreover, from the convergence of the approximated sequence we also have, by interpolation, that v

-v → 0 in L 3 (0, T ; L 3 (Ω) 3 ), hence Φ → 0 in L 3 (0, T ; L 3 (Ω) 3 ), (6.21) Φ 0 in L 3 (0, T ; W 1,3 0 (B) 3 ), (6.22) Bog B (∇η • (v -v)) → 0 in L 3 (0, T ; W 1,3 0 (B) 3 ). (6.23)
We then obtain from the weak formulation of the regularized problem (6.6) the following equality

T 0 Ω η d |ω |ω -d |ω|ω • ω -ω dxdt = - T 0 Ω d |ω |ω -d |ω|ω • ∇η × v m -v dxdt + T 0 Ω d |ω |ω -d |ω|ω • curl Bog B (∇η • (v m -v)) dxdt -ν T 0 Ω D(v m -v) : DΦ dxdt + T 0 Ω ω × v -ω × v • Φ dxdt + T 0 Ω d 1/3 χ -d |ω|ω • curl Φ dxdt - T 0 Ω |Dv |Dv : DΦ dxdt - T 0 Ω (v t -v t ) • Φ dxdt - T 0 Ω d (ω t -ω t ) • curl Φ dxdt =: (I) + (II) + (III) + (IV ) + (V ) + (V I) + (V II) + (V III).
The strong L 3 (0, T ; L 3 (Ω) 3 ) convergence of v m and the continuity of the Bogovskiȋ operator, with (6.21) imply that (I) and (II) vanish as → 0 (we also used that the function d is uniformly bounded). We write then the following equality

(III) = -ν T 0 Ω η|D(v m -v)| 2 dxdt -ν T 0 Ω D(v m -v) : ∇η ⊗ (v m -v) dxdt + ν T 0 Ω D(v m -v) : D Bog B (∇η • (v m -v)) dxdt,
where the first term is non-positive and the second and third one vanish on account of (6.21)-(6.23). The convergence of (IV ) follows from uniform bounds in L 2 (0, T ; W 1,2 (Ω) 3 ) and (6.21). The term (V ) → 0 due to (6.23) and the bound in L 3/2 ((0, T ) × B) of χ and |ω|ω. Next, (V I) → 0, due the L 3 (0, T ; W 1,3 (B) 3 ) bound of v mv and (6.13).

Concerning the terms involving the time derivative, which are the new ones with respect to the steady problem treated in [START_REF] Berselli | On the existence of weak solutions for the steady Baldwin-Lomax model and generalizations[END_REF], it follows that they both vanish as → 0. In fact, in (V II) the term v t -v t is bounded in L 2 (0, T ; L 2 (Ω) 3 ), by (6.19), while Φ vanishes strongly in L 2 (0, T ; L 2 (Ω) 3 ). Moreover, regarding (V III), we rewrite it as

T 0 Ω ( √ d ω t - √ d ω t ) • √ d Φ dxdt,
and that the quantity √ d ω t -√ d ω t is bounded in L 2 (0, T ; L 2 (Ω) 3 ) by (6.12), while √ d Φ converges strongly to zero in L 2 (0, T ; L 2 (Ω) 3 ) by (6.18). In this way we proved that min This proves, by the identification of weak and almost everywhere limits, the validity of the limit d |ω |ω d |ω|ω, at least in L 3/2 (0, T ; L 3/2 (Ω) 3 ) ending the proof of the existence part, since v satisfies for all φ ∈ C ∞ 0,σ ((0, T ) × Ω) 3 . Observe that the hypotheses on the initial datum v 0 ∈ W 1,3 0,σ (Ω) are enough to make the integrals well-defined. In the limit only the weighted estimate Ω d |ω 0 | 3 dx < ∞ is needed. So at the price of further technical questions related to approximation by smooth functions in weighted space as in Kufner [START_REF] Kufner | Weighted Sobolev spaces[END_REF], one can relax the hypotheses on the initial datum as follows:

v 0 ∈ W 1,2 0,σ (Ω) with

Ω d |ω 0 | 3 dx < ∞,
such that there exists a sequence v 0 ∈ W 1,3 0,σ (Ω) satisfying We continue now with the uniqueness part.

v 0 → v 0 in W 1,
Step 2: Uniqueness part. Since we proved existence of rather regular weak solutions, we can now prove their uniqueness. As usual we suppose that there exists two solutions v 1 , v 2 corresponding to the same initial datum. We take the difference and it follows that all estimates satisfied by the velocity are inherited by the difference and hence δ v := v 1 -v 2 and δ ω := ω 1 -ω 2 satisfy in particular the following δ v ∈ L ∞ (0, T ; W 1,2 0,σ (Ω)), √ d (δ ω) ∈ L ∞ (0, T ; L 2 (Ω) 3 ),

δ v t ∈ L 2 (0, T ; H 1/2 (Ω) 3 ∩ L 3 σ (Ω)), √ d (δ ω t ) ∈ L 2 (0, T ; L 2 (Ω) 3 ), δ ω ∈ L ∞ (0, T ; L 3 (K) 3 ), ∀ K ⊂⊂ Ω.
It follows that if we write the equation satisfied by the difference δ v, we can rigorously test by the difference itself. All terms work directly, the only one that needs to be checked is the monotone one. In fact, if we write (6.24)

T 0 Ω d |ω 1 |ω 1 -d |ω 2 |ω 2 • ω 1 -ω 2 dxdt,
this would be surely finite if ω i ∈ L 3 (0, T ; L 3 (Ω) 3 ), which we do not know. Nevertheless we can observe that, for all i, j = 1, 2 The bound ∇v 2 ∈ L ∞ (0, T ; L 2 (Ω) 9 ) and the Gronwall lemma implies that δ v(s) ≡ 0 for all s ∈ [0, T ], hence the uniqueness follows.

3

 3 as follows d(x, A) := inf y∈A |x -y|, and we denote by d(x) the distance of x from the boundary of the domain Ω (2.1) d(x) := d(x, ∂Ω) ∀ x ∈ Ω.

(3. 1 )

 1 ∂ψ = ∂ψ and ψ = ψ, for any linear differential operator ∂. Let us start by considering the following rotational form of the Navier-Stokes equations (NSE in the sequel),

  [START_REF] Amrouche | Turbulent flows as generalized Kelvin-Voigt materials: Modeling and analysis[END_REF] motivated by dimensional analysis, we consider now the back-scatter Baldwin & Lomax model with the following explicit expression for the length(6.1) (x) = d 0 d(x)for some length d 0 > 0, which is consistent with assumptions (1.6) and (1.7) and 2 (x) = d 0 d(x) ∈ C 2 (Ω), if the boundary of the domain ∂Ω is at least of class C 2 (cf. the assumptions in Lemma 5.1 and see alsoGilbarg and Trudinger [9, Ch. 14]). Consequently, we now study the existence and uniqueness problems for the following model (6.2)

2 (η0 B/ 2 d

 22 |ω |ω -|ω|ω) • (ω -ω) (|ω |ω -|ω|ω) • (ω -ω) dxdt ≤ T (x) η (|ω |ω -|ω|ω) • (ω -ω) dx ≤ T 0 B d(x) η (|ω |ω -|ω|ω) • (ω -ω) dxdt → 0,which is enough to prove that |ω |ω → |ω|ω a.e. in (0, T ) × B/2. The arbitrariness of the ball B ⊂ Ω implies that |ω |ω → |ω|ω a.e. in (0, T ) × Ω.

  • φ + d ω t • curl φ + (ω × v) • φ] dx dt + T 0 Ω [ν ∇v : ∇φ + d |ω| ω • curl φ] dxdt = 0,

2 Ωd

 2 2 0,σ(Ω) andΩ d |ω 0 | 3 dx ≤ |ω 0 | 3 dx.

dδ v 2

 2 |ω i |ω i • ω j dxdt ≤ in (6.24) is well defined, and then by monotonicity it follows thatT 0 Ω d |ω 1 |ω 1 -d|ω 2 |ω 2 • ω 1 -ω 2 dxdt ≥ 0. dt,by using the standard inequalities for the nonlinear term (as in[START_REF] Amrouche | Turbulent flows as generalized Kelvin-Voigt materials: Modeling and analysis[END_REF] Sec. 4]), since δ v(0) ≡ 0.

  Remark 4.1. The assumption in condition (4.11) can be justified as in[START_REF] Amrouche | Turbulent flows as generalized Kelvin-Voigt materials: Modeling and analysis[END_REF] Remark 2.2]. More precisely, (for a time averaging filter) this condition holds true when source term is constant f (t, x) = f (x), without turbulent fluctuation, i.e., f = 0. It implies a decrease of TKE, which means a decrease of the turbulence, towards a laminar state, or a stable statistical equilibrium, such as a grid turbulence.

2

, which we assume to be held near statistical equilibium. Hence, (4.1) follows by combining (4.2) and (4.9). Finally, (4.1) yields the model (1.5) by setting the modified pressure π = p + k. An example which satisfies (4.11) is given by the following remark.

Thanks to the divergence free constraint div v = 0, we have div (v ⊗v) = (v •∇) v where v ⊗v = (vivj) for 1 ≤ i, j ≤ 3. Therefore, these both forms are used throughout this report without any confusion, but also the "rotational form" will be used.

The divergence-free spaces W 1,p 0,σ (Ω) are defined below by (2.5).

We denote A : B = i,j aijbij for any two matrices A = (aij) and B = (bij) for 1 ≤ i, j ≤ 3.

Acknowledgments


F Mottez

J Heyvaerts ⋆2

Erratum: Magnetic coupling of planets and small bodies with a pulsar wind

Keywords:

Aims. This paper investigates the electromagnetic interaction of a relativistic stellar wind with a planet or a smaller body in orbit around the neutron star. The interaction is based on the theory of Alfvén wings adapted to the context of relativistic winds.The 2011 paper comprises a short section with numerical applications. This requires an estimate of the magnetic field in the pulsar wind, that depend on the magnetic flux Ψ in the wind asymptotic regime. This flux is badly approximated. Methods. The present erratum corrects the consequences of this wrong estimate of Ψ in three papers where it has consequences: about the magnetic drag acting on pulsar companions, and a model of fast radio bursts (FRB). Results. In those three papers, the physics is unchanged, but the numerical results must be re-scaled. The theory of FRB published in 2014 is briefly discussed.

Magnetic coupling of planets and small bodies with a pulsar wind

In Mottez & Heyvaerts (2011b), the magnetic field in the pulsar wind is expressed as a function of the magnetic flux in the asymptotic regime of the wind Ψ = B r 0 r 2 , where B r 0 is the local magnetic field, and r is the distance to the neutron star. This is an integral of motion along stream lines. In the end of section 2, in view of numerical applications, it is evaluated as Ψ = B r * R 2 * where quantities with a star refer to the neutron star (NS) surface. This approximation corresponds to the monopole solution, which overestimates the wind magnetic field by orders of magnitude. Because this is an asymptotic approximation, its value must be evaluated in the domain where the field lines are wind-like, not on the NS surface. The most admitted approximation is its value at the light cylinder Goldreich & Julian (1969), where between the NS surface and the light cylinder (LC), the magnetic field is supposed dipolar. Then,

Ψ = B r LC r 2 LC = B * R 3 * Ω * /c
where Ω is the NS angular velocity, and c is the speed of light. Then the toroidal component of the pulsar wind magnetic field is approximated by

B φ 0 = B * R 3 * Ω 2 * /rc 2 where r >> r LC . ⋆ Deceased.
The numerical estimates of the potential drop U associated with PSR 1257+21 and PSR 1620-26 in the end of section 2 are not correct. Their right values as well as the corrected values of Table 3 in Mottez & Heyvaerts (2011b) are given in four first columns of the present Table 1.

A magnetic thrust action on small bodies orbiting a pulsar

In Mottez & Heyvaerts (2011a), the same wrong evaluation of Ψ leads to an overestimate of the magnetic thrust acting on bodies orbiting a pulsar described in the joint paper .

Table 1 in Mottez & Heyvaerts (2011a) must be replaced by Table 1 of the present erratum. The conclusion is that even for small bodies orbiting the pulsars given in example, the influence of this effect on the orbit is totally negligible. The figures 2, 3, and 4 must be replaced by flat functions showing no significant evolution of the eccentricity and of the semi-major axis.

F. Mottez and J. Heyvaerts: Erratum: Magnetic coupling of planets and small bodies with a pulsar wind.

Table 1. Electric potential drop, total electric current associated to the Alfvén wing. Electrical energy ĖJmax dissipated in the Alfvén wing. Variation per (terrestrial) year of the semi-major axis, variation of the eccentricity, per year, ∆e/year), and the coefficient D/ √ GM that scales these computations.

Name U I ĖJmax ∆a/year ∆e/year D/ √ GM Unit (V) (A) (W) (m.yr -1 ) (yr -1 ) (m 5/2
•s -1 ) PSR 1257+12 a 4.0 10 10 1.1 10 8 3.4 10 18 1.5 10 -7 0 2.1 10 3 PSR 1257+12 b 1.3 10 11 3.4 10 8 3.4 10 19 1.3 10 -8 3.3 10 -21 3.4 10 2 PSR 1257+12 c 9.7 10 10 2.6 10 8 1.9 10 19 1.0 10 -8 2.8 10 -21 3.5 10 2 PSR 1620-26 a 1.9 10 10 3.0 10 7 2.8 10 17 6.7 10 -11 0 1.1 10 2 PSR 1s b 10,000 km 8.4 10 9 2.2 10 7 1.5 10 17 6.3 10 -9 0 9.9 10 1 PSR 10ms 100 km 1.6 10 8 4.2 10 5 5.1 10 13 2.6 10 -8 3.1 10 -19 1.7 10 2 PSR 10ms 1 km 1.6 10 6 4.2 10 3 5.1 10 9 2.6 10 -6 3.1 10 -17 1.7 10 4 PSR 1s b 100 km 1.6 10 8 4.2 10 5 5.1 10 13 2.6 10 -6 3.1 10 -17 1.7 10 4 PSR 1s b 1 km 1.6 10 6 4.2 10 3 5.1 10 9 2.6 10 -4 3.1 10 -15 1.7 10 6

3. Radio emissions from pulsar companions: a refutable explanation for galactic transients and fast radio bursts

In (Mottez & Zarka 2014), because of the new evaluation of Ψ, equation ( 1) must be replaced by

B φ 0 = B r 0 v φ 0 -Ω * r v r 0 ∼ - B r 0 Ω * r c = - B r * Ω 2 * R 3 * rc 2 , (1)
and in the same section, the current in the Alfvén wing for an Earth-like orbiting body at 0.2 AU for a 1s standard pulsar is not 10 11 A but I A = 0.23 10 8 A. For the same body orbiting a recycled pulsar, I A = 10 9 A must be replaced by I A = 0.23 10 8 A. In section 4.3 (frequencies), the correct value of Ψ is Ψ = B * R 3 * Ω * /c, and Eq. ( 15) becomes

f c,s = qB * Ω * 2πmc R 3 * r 2 1 + r γr LC 2 , ( 2
)
where γ is the wind Lorentz factor, and R LC is the light cylinder radius. Equation ( 16) becomes

f c,o = 0.52 γ B * 10 5 T 1AU r 2 R * 10 4 m 3 10ms T * × 1 + π 10 5 γ 10ms T * r 1AU 2 1/2 . (3)
In section 4.4 (brightness), Eq. ( 18) becomes

ĖJ = π µ 0 c 3 R b 2 r -2 R 6 * B 2 * Ω 4 * , (4)
where R b is the radius of the body orbiting the NS. Equation ( 19) becomes

P radio = ǫ ĖJ = ǫ π µ 0 c 3 R b 2 r -2 R 6 * B 2 * Ω 4 * , (5)
and Eq. ( 20) becomes

< S > Jy = 2.7 10 -3 γ 10 5 2 ǫ 10 -3 R b 10 7 m 2 × 1AU r 2 R * 10 4 m 6 B * 10 5 T 2 × 10ms T * 4 Mpc D 2 1 GHz ∆f . (6)
On fig. 6, the frequencies (vertical axis) must be reduced by a factor 2.1 10 -2 . In fig. 7, they must be rescaled by a factor 2.1 10 -4 . In fig. 8, the distances from where a 1 Jy signal can be observed must be rescaled by a factor 2.1 10 -2 .

In the second paragraph of section 5.1, it is said that a recycled millisecond pulsar and a planet corresponding to the reference values of Eq. ( 20) except γ = 10 6 could provide a FRB signal seen from a distance of 1 Gpc. This is wrong. Actually, a young millisecond pulsar, characterized by the reference values of Eq. ( 20), except γ = 10 6 and B * = 10 7 T could provide the required FRB characteristics.

In section 5.2, third paragraph, it is said erroneously that a kilometer-sized asteroid orbiting a standard pulsar (P = 1 s, B * = 10 8 T) could explain the PSR J1928+15 event. Actually, the same asteroid orbiting a highly magnetized pulsar (P = 1 ms, B * = 10 9 T) could explain it.


Bassem Barkia
email: bassem.barkia@polytechnique.edu

Pascal Aubry

Paul Haghi-Ashtiani

Thierry Auger

Lionel Gosmain

Frédéric Schuster

Hicham Maskrot

On the origin of the high tensile strength and ductility of additively manufactured 316L stainless steel: Multiscale investigation

Keywords:

We report that 316L austenitic stainless steel fabricated by direct laser deposition (DLD), an additive manufacturing (AM) process, have a higher yield strength than that of conventional 316L while keeping high ductility. More interestingly, no clear anisotropy in tensile properties was observed between the building and the scanning direction of the 3D printed steel. Metallographic examination of the as-built parts shows a heterogeneous solidification cellular microstructure. Transmission electron microscopy observations coupled with Energy Dispersive X-ray Spectrometry (EDS) reveal the presence of chemical micro-segregation correlated with high dislocation density at cell boundaries as well as the in-situ formation of well-dispersed oxides and transition-metal-rich precipitates. The hierarchical heterogeneous microstructure in the AM parts induces excellent strength of the 316L stainless steel while the low staking fault energy of the as-built 316L promotes the occurrence of abundant deformation twinning, in the origin of the high ductility of the AM steel. Without additional post-process treatments, the AM 316L proves that it can be used as a structural material or component for repair in mechanical construction.

Introduction

Metal additive manufacturing (AM) has gained significant academic and industry interest as it provides the opportunity to made complex-shaped parts that can be used in several domains [1][2][3]. AM could revolutionize many industrial sectors by decreasing material waste, component lead-time and carbon footprint [START_REF] Herderick | Additive Manufacturing of Metals: A Review[END_REF]. AM processes can be classified into two principal classes: Powder bed fusion based technologies (PBF), such as selective laser melting (SLM) [START_REF] Tapia | [END_REF] and direct laser deposition (DLD) [6], such as laser engineered net shaping (LENS), a promising additive technology that is currently being developed [7]. The DLD/LENS process can be used to produce near-net shaped components or treat damaged or worn areas of a component [8]. The DLD/LENS technology has proved to be effective to fabricate a wide range of materials including stainless steels, titanium, nickel-base super alloys and cobalt-base alloys, etc [9][10][11][12].

316L austenitic stainless steel, in particular, is considered as one of the most widely used materials today owing to its high corrosion and oxidation resistance and its excellent strength and ductility [START_REF] Novak | Handbook of Stainless Steels[END_REF][START_REF]Handbook for Stainless Steels[END_REF]. Its area of application is increasing continuously. Currently, austenitic steels are mainly manufactured using forging, casting and extrusion [START_REF] Honeycombe | Bhadeshia HKDH Steels -Microstructure and Properties[END_REF]. The final product obtained from these conventional techniques need to welded or machined into the final shape, which makes these technologies time consuming. In addition, the machining of austenitic steels is relatively expensive and welding may induce corrosion sensitive grain boundaries [START_REF] Honeycombe | Bhadeshia HKDH Steels -Microstructure and Properties[END_REF][START_REF] Xin | [END_REF]. Thus, the austenitic stainless steel is particularly interesting for laser additive manufacturing.

The currently available literature show that a large number of previous studies were dedicated to the 3D printing of 316L, especially using SLM technique [17][18][19][20][21]. However, limited research has been carried out on 316L using LENS technology compared to SLM. Most of these studies were focused on the effect of process parameters or heat treatment on the microstructure and/or the mechanical properties of the LENS 316L [22][23][24][25][26][27]. The presence of build defects (e.g., porosity, lack of powder melting. . .) may contribute to poor mechanical properties (low elongation and/or low Young's modu- lus) of the AM 316L parts [22][23][24][25]. Furthermore, the effect of the building direction on the mechanical performances of the LENS 316L was poorly studied and rarely related to its microstructure and/or the crystallographic texture [23,26,27]. Finally, very few studies have strived to quantify the as-built microstructure at fine scale [26] and the investigation on the deformation modes of the LENS-processed 316L is still lacking. Therefore, in order to fill this gap, the aim of this study is to use the LENS technology to fabricate high-density parts of 316L with a combination of high strength and ductility (a particular interest will be granted to the effect of the building direction on microstructure and tensile properties). Fine scale TEM investigations of the microstructure before and after tensile testing have been performed to develop a better understanding on the nonequilibrium as-built microstructure and the deformation modes of the additively manufactured 316L. Based on these observations, a correlation between the microstructural features and the resulting tensile properties is proposed. Finally, mechanical results from the 3D printed 316L will be compared with that of conventionally processed 316L.

Experimental

Material & additive manufacturing processing

Gas atomized spherical 316L powder (supplied by ERASTEEL, Sweden) with particle size of 45-106 m was used as starting material. The chemical composition of the as-received powder respects the recommendation limit of the ASTM standard values for 316L [START_REF]ASTM-A276-06[END_REF] (see Table 1).

The laser metal depositions were performed using a LENS MR-7 System (OPTOMEC, Inc. Albuquerque, NM, USA). The LENS machine consists of a 3 kW Yb-fiber laser; a coaxial powder feed system, a controlled environment glove box and a motion control system. To limit the material oxidation during deposition at high temperature, the chamber was purged with high purity argon in order to maintain the oxygen concentration below 10 ppm during processing. The LENS control software lets the operator set the process parameters such as laser power, powder flow rate or layer thickness.

Two rectangular bars of 70 mm × 30 mm × 30 mm and 30 mm × 30 mm × 70 mm dimensions were fabricated from the 316L powder as shown schematically in Fig. 1. The two parts were built on a 150 mm × 50 mm × 10 mm 316L stainless steel plates with process parameters reported in Table 2. Note that a bidirectional scanning strategy was adopted, i.e., during each layer, the main direction of the hatch pattern is tilted by 90 • .

Microstructural & mechanical characterization

For microstructural analyzes, the lateral surface and the top surface refer to the surface parallel and perpendicular to the building direction, respectively, as shown in Fig. 1.

Optical microscopy (OM; ZEISS Axio Imager 2), scanning electron microscopy (SEM; JEOL JMS 7000 F; 15 kV) equipped with an electron backscatter diffraction (EBSD) detector were used to examine the microstructure and the crystallographic texture of the as-built 316L samples. At microscale, transmission electron microscopy observations and Energy Dispersive X-Ray Spectroscopy (EDS) characterization were carried out using a FEI TITAN 3 transmission electron microscope operating at 300 kV.

For phase identification, X-ray diffraction (XRD) analyzes of the as-received powder and the as-built parts were performed on a diffractometer (XRD, Brucker D8 ADVANCE) with copper radiation source under 40 kV voltage and 40 mA current. The XRD data were collected over an angular range of 40-120 • using a step size of 0.02 • .

For OM, SEM and XRD analyzes, the samples were first mechanically and then electrolytically polished in a 10% oxalic acid aqueous solution at 15 V for 20 s. For TEM observations, 3-mm disk were machined and mechanically ground to a thickness of 100 m. Final thinning of the disk to electron transparency was performed using a Tenupol-5 twin jet polisher, in a solution of 10% perchloric acid and 90% acetic acid at room temperature and an operating voltage of 15 V.

To investigate the tensile properties of the as-built bars, longitudinal (tensile axis parallel to the building direction) and transverse (tensile axis perpendicular to the building direction) cylindrical specimens of 4 mm diameter and 15 mm gage length were machined from the AM parts (see Fig. 1). Uniaxial tensile tests were conducted at room temperature on an electromechanical testing frame (MTS 20/M, 100 kN load cell) at a cross-head speed of 2.5 m/s. The yield strength y,0.2% was evaluated by measuring the proof stress at 0.2% plastic strain.

Results and discussion

Density measurement and phase constitution

The density measurement results conducted by Archimedes method showed that a high density of >99.7% was achieved for the two fabricated parts (the density of conventional 316L steel was taken as a reference value). Moreover, the SEM examination of high-resolution micrographs does not show the presence of any pores or voids. These observations suggest that the difference in the measured density between the AM 316L and the conventional 316L cannot be attributed to the presence of voids but rather to the formation of inclusions, whose density is lower than that of the matrix, as will be shown later.

A comparison of the XRD patterns for the powder and the asbuilt 316 L samples is shown in Fig. 2. Both the as-milled powder and the deposited samples consist of pure face-centered-cubic (FCC) austenite phase. Indeed, the strongest lines of ferrite should appear at 44.7 • and 65 • [29] and, their absence in Fig. 2, indicates the fully austenitic structures of the analyzed samples. The theoretical line intensities for the austenite are 100%, 44% and 24% for the lines (111), ( 200) and (220), respectively [START_REF] Bechtoldt | An X-Ray Diffraction Method for Determining the Amount of Austenite in an Austenite-Ferrite Mixture[END_REF]. These ratios were checked for the powder analysis but were not completely obeyed for the as-built 316L because of the crystallographic texture generated by the LENS process. Furthermore, compared to the as-received powder, the austenite peaks of the additively manufactured samples are slightly broader because of the presence of residual stresses and initial dislocations induced by the LENS process as will be discussed below.

General microstructure

Representative 3D optical view of the as-built microstructure is shown in Fig. 3(a). The material has a layered microstructure char-

Table 2

Processing parameters used in directed laser deposition of the two additively manufactured parts. showing grains in random colors were superimposed. The micrograph reveals that many grains include the melt-pool boundaries. This observation suggests an epitaxial growth of the grains from re-melted zones, following the maximum temperature gradient directions. At higher magnification, high-resolution SEM micrographs revealed that the as-built parts are characterized by a typical cellular structure (Fig. 4). The cells are about 3-4 m in diameter and a few tens of microns in length. The cellular structure is generally attributed to the high cooling speed that could reach 10 3 -10 4 K/s during the LENS process as reported by Wang et al [START_REF] Wang | [END_REF]. It is suggested that, with faster cooling rate; the solidification microstructure can evolve from planar to cellular to dendritic [START_REF] Dupont | Fundamentals of Weld Solidification[END_REF][START_REF] Lancaster | Metallurgy of Welding[END_REF]. Cells are equiaxed or elongated, depending on their growth direction being strongly influenced by the direction of the thermal gradient (Fig. 4(c) and (d)). Moreover, elongated cells are especially observed along the melt-pool boundaries compared to more equiaxed ones in the center of the melt-pool. Furthermore, the high-resolution SEM image reported in Fig. 4(b) revealed the presence of spherical nano-inclusions with size ≤200 nm. Because of their small size, the SEM resolution was not enough to determine the chemical composition of the nano-inclusions. TEM observations are then conducted for further understanding of the fine microstructure induced by the LENS Process.

Fig. 5(a) and (b) shows inverse pole figure (IPF) maps relative to the building direction of the top surface (S 1 D-S 2 D) and the lat-eral surface (BD-S 1 D) of the as-built 316L. The EBSD scans obtained from 1.3 mm × 1 mm areas were post-processed to evaluate the grain size by a standard metallographic technique using the intercept line method [START_REF] Underwood | [END_REF]. Grain boundaries were detected using grain detection angle of 5 • . Although the grain size varied from about 20 to 350 m, grains are quasi-equiaxed and the measured average grain size in the BD-S 2 D as well as in the S 1 D-S 2 D plane is around 60 m. Note that the achievement of equiaxed grains is desirable because high aspect ratio (grain morphology), often obtained in AM, can lead to an anisotropic plastic behavior of metallic materials: materials deform differently when being tested in different directions [START_REF] Kocks | Texture and Anisotropy: Preferred Orientations in Polycrystals and Their Effect on Materials Properties[END_REF]. The plastic anisotropy could constitute a barrier to the application of advanced alloys [START_REF] Pham | [END_REF]. It is obvious that the nearly equiaxed microstructure obtained in this study can be beneficial to the mechanical response and failure behavior of the AM steel, especially in the case of multi-axial deformation. For several set of parameters, Wang et al. [37] reported that the aspect ratio of the 304L SS made by DLD varied between 2 and 3.2. For the 316L obtained by LENS process, Yadollahi et al. [22] also observed elongated grains parallel to the building direction without quantifying the aspect ratio.

More than 90% of grain boundaries were of high angle types (Â ≥ 15 • ) with an average misorientation angle of about 32 • in section along BD and 37 • in the section perpendicular to BD. The misorientation across boundaries is an important microstructural parameter since it indicates the density of high angle grain boundaries (HAGB). HAGB provide obstacles to cleavage crack propagation [38,39] and thus enhance the fracture toughness. Few data in the literature are available concerning the grain boundary misorientation in austenitic steels fabricated by DLD/LENS technique. Yadollahi et al. [22] reported with their processing parameters that the majority of grain boundaries has very low misorientation angles (1-4 • ) and that the average misorientation angles were about 14 • in the as-built 316L and doesn't exceed 30 • after heat treatment for 2 h at 1150 • C.

{100} and {110} pole figures calculated from the EBSD maps of the lateral surface (BD-S 2 D plane) and the top surface (S 1 D-S 2 D plane) are shown in Fig. 5(c) and (d), respectively. In both cases, the pole figures show similar textures with the presence of a strong common (100) orientation aligned with the building direction BD and with the two scanning directions S 1 D and S 2 D (Maximum texture intensity of 3.6 times random). To our knowledge, such a kind of crystallographic texture has never been reported for austenitic steel made by DLD/LENS process. However, Kurzynowski et al. [20] observed the same kind of texture (i.e. strong {001} texture with respect to the building and scanning directions) in 316L made by selective laser melting (SLM). The samples were made at 200 W using the älternate-hatchscanning strategy (i.e., changing the scan direction by 90 • between two successive layers). It is important to note that, in AM process, the texture evolution is affected by both the growth directions of the grains and the maximum heat flow directions. Thijs et al. [40] reported that the directional solidification and thus the resulting crystallographic texture would depend on the adopted scanning strategy and building direction. The bidirectional scanning strategy used in this study is believed to be responsible for a reduction of the texture intensity along the building and scanning directions, which consequently reduces the anisotropy in the resulting mechanical properties.

Deep microstructure features

The TEM analyzes were performed on thin foils sectioned normal to the scanning direction S 1 D. Fig. 6(a) shows representative bright field TEM micrographs of the initial microstructure of the as-built 316L. The non-deformed material represent extensive network of dislocations at the cell boundaries. A closer view in the cell interiors revealed the presence of dislocation tangles and extended band of stacking faults as evidenced in Fig. 6(b). The dislocation cell structure formed in the AM steel is similar to the microstructure obtained under severe plastic deformation process. The thermal contraction induced by the rapid solidification is thought to be responsible for the high dislocation density in the AM parts [41].

Furthermore, the white contrast at the cell boundaries on the HAADF-STEM (Z contrast) image presented in Fig. 7(a) compared to the interior of the cells reveals an enrichment in heavier elements at the cell boundaries. This observation is supported by the corresponding STEM-EDS maps where segregation of Mo, Ni and Cr correlated with the dislocation structures can be observed (Fig. 7(a)). There is also an anti-correlation with Fe and Mn. It is notable that the dislocation substructures developed over the same length scale (3-4 m) that the cell size revealed by the SEM micrographs. It indicates that the dislocation substructures are correlated with chemical micro-segregation.

The STEM-EDS maps in Fig. 7(b) revealed the presence of Si-containing oxides (from 50 to 200 nm in size) as well as transition-metal-rich precipitates, in which Mo, Mn and Cr content varies along the cell boundaries. These same particles (i.e. Si-oxides and Cr-Mo-rich precipitates) can also be observed in the TEM bright field micrographs shown in Fig. 6. Using the "Microprobe STEM" technique on the transmission electron microscope, the electron diffraction patterns (EDPs) corresponding to the two types of inclusions were examined. The EDP analysis proved that the Si-oxides Table 3 Tensile properties of the as-built 316L in building and scanning directions.

Yield strength (MPa)

Elongation to failure (%)

Building direction 378 ± 3 54.5 ± 1 Scanning direction 440 ± 5 51.5 ± 0.8 depleted of Fe are amorphous while the transition-metal-rich precipitates were crystalline. One can expect that the crystallographic structure (i.e., amorphous or crystalline) would have an impact on the deformability of the particles and on the dislocation-inclusions interaction when the material is deformed. These oxide inclusions are generated by in-situ oxidation during the LENS process due to the presence of residual oxygen in the build chamber. It entails the possibility of using the build atmosphere for controlling the interstitial content of the 3D printed steels and for the in-situ formation of oxide-dispersed steels (ODS), as reported by Springer et al [42]. It is well established that introducing oxide particles inside a steel matrix will improve the high-temperature mechanical properties [43,44]. A fine dispersion of oxide particles can also improve the radiation tolerance of stainless steels [45,46]. The formation of nano-inclusions in the 316L during AM process was reported in previous work especially during sintering/melting process [26,47]. The inclusions were suggested to be silicate-containing chromium and manganese, which is confirmed by our TEM observations.

Mechanical properties and deformation mechanisms

Tensile tests were carried out on the as-built 316L at room temperature along the building and scanning directions. The engineering stress vs. engineering strain curves of the specimens are shown in Fig. 8. The results showed that the loading direction has slight effects on mechanical properties: it makes the 0.2% yield strength change from 378 ± 3 to 440 ± 5 MPa and total elongation from 54.5% ± 1% to 51. 5% ± 0.8% (Table 3). That is to say, regardless the loading direction, the LENS 316L produced here has good combination of strength and elongation compared to the conventional 316L (see Fig. 9). Indeed, tensile tests revealed that the AM 316 L stainless steel has higher tensile strength compared to the 316L stainless steel processed by the other conventional manufacturing process without sacrificing its ductility (Fig. 9) [22,25,[START_REF]Metals Handbook[END_REF][START_REF] Rawers | [END_REF][50][51][52]. For comparison, the 0.2% yield stress of the as-built 316L is 440 ± 5 MPa compared with 170 MPa for the wrought 316L [START_REF]Metals Handbook[END_REF]. Moreover, the difference in yield strength between the building and scanning directions is only 62 MPa, which represents a small fraction of the observed difference between the strengths of the as-built 316L and the wrought 316L (more than 200 MPa). Previous studies on AM 316L showed that the DLD/LENS process improves the yield strength but reduces the ductility especially when the material is stressed along the building direction [25,26]. Compared to the results reported by Zhang et al. [25], the elongation to failure in the building direction obtained in the present study were far higher (54.5% in the present study against 21% in Ref. [25]). The poor elongation along the building direction reported by Zhang and Zi ętala [25,26] is attributed to the presence of build defects (e.g. porosity and lack of fusion) among layer boundaries. It is evident that pre-existing dislocation structure contributes significantly to the high strength of the LENS 316L. The dislocation network as well as the chemical micro-segregation at cell boundaries formed during manufacturing hinder the dislocation motion resulting in higher yield strength compared to the conventional 316 L. Moreover, the nano-inclusions generated during the process will contribute to the increased strength of the AM 316L compared to as-cast or wrought material. The fracture surface morphologies of the as-built 316L tested in building and scanning directions were analyzed. In both cases, typical ductile fracture features were observed although some evidence of brittle fracture was also visible. As shown in Fig. 10, ductile fracture via the void growth and coalescence mechanism was the dominant fracture mode. The dimples are a few microns in size, close to the typical cell dimensions of the as-built parts. Small dimples tend to increase the area of fracture surface so they absorb more deformation energy, thus improving the ductility by delaying the fracture process. Note that some secondary cracks, seemingly transgranular, were also observed. The formation and the propagation of secondary cracking is likely favored by the presence of nano-inclusions revealed in the as-built microstructure.

Despite the weakening effect (loss of ductility) that nanoinclusions could have on the 3D printed steel, the elongation to failure of the LENS 316L remains excellent. In order to investigate the origin of such behavior, EBSD analysis and TEM observations were performed on the AM 316L strained to fracture to identify the activated deformation modes during plastic deformation.

Based on the IPF map analyzes in Fig. 11(c), the high lattice misorientation inside individual grains (10 • to 15 • ) reflect the strong dislocation slip activity (slip-induced lattice reorientation mechanism). Furthermore, the examination of the SEM micrograph of the analyzed area (Fig. 11(b)) revealed distortion of the melt-pool boundaries, reflecting the intense strain at the melt-pool interfaces. More interestingly, compared to the initial microstructure of the non-deformed material, several deformation twins are observed in the IPF map confirming that twinning is mainly induced by deformation and suggesting that the contribution of mechanical twinning to plastic deformation is substantial. Twins were oriented parallel to one another and their thickness varies from few microns to several tens of microns. The misorientation plot across the matrix/twin interfaces showed that the mechanical twins were rotated by 30-40 • relative to the matrix (Fig. 11(d)). Moreover, the phase map (not shown here) indicates that the deformed AM 316L has retained its austenitic phase against martensitic phase transformation. Conventional 316L undergoes strain-induced martensite transformation during plastic deformation as observed by Zhang et al. [53]. However, the EBSD analysis showed that the AM 316L did not transform during tensile test, suggesting that the mechanical stability of austenite is higher than that of conventional 316L. In a recent study conducted on a medium Mn austenitic steel, He et al. [54] showed that introducing initial dislocations by using a warm rolling process tend to stabilize the austenitic phase during plastic deformation at room temperature. One can assume that the dislocation cell structure formed in the as-built 316L tend to stabilize the austenitic phase of the AM steel and thus inhibit the strain-induced martensite transformation. Moreover, It is well established that the austenite stability is strongly related to the chemical composition (especially carbon and manganese) [55] but also to grain size [56]. Naghizadeh et al. [56] showed that below average grain size of ≈50 m, the tendency to martensitic transformation diminishes by decreasing grain size. Thus, in addition to the initial dense dislocations, the fine cellular microstructure of the as-built 316L (cell size of 3-4 m) would also contribute to the mechanical stability of austenite during plastic deformation.

The strong twinning activity was confirmed by TEM analyzes performed on the specimen strained to fracture. The TEM micrographs shown in Fig. 12 revealed the extensive activity of mechanical twinning. Dark field (DF) imaging has been used to unambiguously identify the observed micro-twins by selecting the appropriate diffraction spot. This strong tendency towards twinning is often observed in TWIP (Twinning Induced Plasticity) steels containing high content of Mn typically higher than 15 at.%, depending on the carbon content [57].

The propensity for mechanical twinning of the as-built 316L can be related to the value of the stacking fault energy (SFE). As the SFE decreases, the stacking faults become wider and the dislocation cross-slip mechanism more difficult thus promoting the deformation twinning [START_REF] Hull | Introduction to Dislocations[END_REF]. From TEM observations of the AM 316L (see Fig. 6), extended band of stacking faults were observed suggesting that the non-deformed material has relatively low value of SFE, since the SFE is a measure the tendency for forming local stacking faults. Moreover, it is well known that a low value of SFE promotes mechanical twinning at the expense of dislocation glide [START_REF] Lu | [END_REF] and that twinning induces large plasticity during deformation [60]. Thus, the pre-existing stacking faults in the non-deformed 316L favor mechanical twinning and improve the ductility of the AM 316L. Together with high density of initial dislocations, the mechanical twins contribute also to the high strength of the AM 316L steel since twin boundaries can act as obstacles to the dislocation glide by reducing the mean free path of dislocations during plastic deformation [61,62]. The spacing between deformation micro-twins can be less than a few hundred of nanometers (Fig. 12) that is narrower than the cell size of the as-built 316L (3-4 m). It means that the influence of twinning on restricting the movement of dislocations is not negligible and would contribute to further improving the tensile strength of the AM 316L.

Conclusion

High strength and ductility 316L stainless steel samples were successfully printed in controlled atmosphere using LENS technique. Microstructural observations up to nanoscale revealed micro-cellular structure enriched with oxide particles and transition-metal-rich precipitates. The as-built components showed promising results regarding their structural integrity and resulting mechanical performances. In contrast to conventional counterparts where microstructure is homogenous, the hierarchical heterogeneous microstructures in the LENS parts induced excellent strength of 316L stainless steel. The strong deformation twinning induces large plasticity leading to high tensile ductility despite the formation of nano-inclusions in the AM steel. Furthermore, the build orientation seemed to have little to no effect on the mechanical properties of the as-built 316L. This study contributes to the understanding of the relationships between the LENS processing, microstructures, and resulting mechanical performances of 316 L stainless steel, key to satisfy the demands for new applications.

Fig. 1 .

 1 Fig. 1. Schematic of the horizontally and vertically built parts showing the positions from which tensile specimens were extracted and the areas on which microstructural analyzes were performed.

  pools generated by the laser beam. The depth and the width of the melt pools are about 450 and 900 m respectively. No evidence of cracks, pores or un-melted powder was observed. A closer 3D view of the microstructure is presented in Fig.3(b)where EBSD image quality map (IQ) and grain color map

Fig. 2 .

 2 Fig. 2. The XRD patterns taken on the powder, the lateral and the top surface of the as-built 316L.

Fig. 3 .

 3 Fig. 3. 3D view of the as-built microstructure (a) optical micrograph; (b) Image quality (IQ) + color grain map.

Fig. 4 .

 4 Fig. 4. SEM images illustrating the typical cellular microstructure at (a) low and (b-d) high magnification corresponding to the yellow areas indicated in Fig. 4(a); (b) equiaxed cells; (c,d) evidence of elongated cells at the melt pool boundaries.

Fig. 5 .

 5 Fig. 5. Electron back-scattered diffraction (EBSD) inverse pole figure (IPF) relative to the building direction + IQ in (a) the lateral surface; (b) the top surface and (c,d) the corresponding pole figures.

Fig. 6 .

 6 Fig. 6. TEM bright field micrographs of the typical microstructure of the as-built 316L: (a) cellular structure with arrays of dislocations pilling up at cell boundaries; (b) evidence of stacking faults and inclusions.

Fig. 7 .

 7 Fig. 7. (a) STEM-HAADF micrograph of the as-built 316L and the corresponding STEM-EDS chemical maps revealing segregation at cell boundaries; (b) evidence of Si-oxides and transition-metal-rich precipitates.

Fig. 8 .

 8 Fig. 8. Stress-strain curves of the as-built 316L strained at 2.5 m s -1 at room temperature.

Fig. 9 .

 9 Fig. 9. The yield strength and ductility data of the DLD-LENS and conventional 316L SS from literature (the elongation to failure (elongation at break) was used).

Fig. 10 .

 10 Fig. 10. SEM fractographs of the LENS 316L after tensile test along building direction (a) low magnification view; (b) typical ductile dimple fracture; (c) evidence of secondary cracking.

Fig. 11 .

 11 Fig. 11. (a) SEM micrograph of the cross section of the broken specimen; (b,c) SEM image of the area where EBSD analyzes were carried out with the corresponding IPF along the building direction + IQ; (d) misorientation profile along the black line in (c).

Fig. 12 .

 12 Fig. 12. (a) TEM bright-field micrograph revealing a high density of twins in the as-built sample strained to fracture; (b) the dark-field micrograph imaged using the diffraction spot outlined by the dashed circle in (c); (c) the corresponding electron diffraction pattern EDP (subscripts "T" and "M" refer to "Twin" and "Matrix" respectively).

Table 1

 1 Chemical compositions of the as-received 316 L powder compared to the ASTM specifications for 316L (wt.%).

	Element	C	Cr	Co	Mn	P	Ni	Si	S	Mo	V	Cu	Fe
	Powder	0.016	17.7	0.15	0.29	0.010	12.6	0.58	0.026	2.33	0.026	0.024	Bal.
	ASTM	0.03Max	16-18	-	2.0Max	0.045Max	10-14	0.75Max	0.03Max	2-3	-	-	Bal.

Acknowledgements

This work was supported financially by the French Alternative Energies and Atomic Energy Commission. The TEM observations were carried out using the equipment of the MATMECA consortium partially funded by the ANR under contract number (No. ANR-10-EQUIPEX-37).


Dr Anne Marchais-Roubelat

Dr Fabrice Roubelat
email: anne.roubelat@lecnam.net

HISTORY, TIME AND FUTURES STUDIES TENSIONS FROM GEOSTRATEGY ANTICIPATORY PRACTICES

Since seminal futures publications, geostrategy anticipatory practices have infused the futures field, as well as questioned the place of time and history in futures thinking. Exploring the temporal tensions of futures thinking, this paper discusses lessons from two geostrategy futures cases. Bridging history and futures studies, section 1 stresses the wicked nature of future history and discusses research directions to introduce time in futures thinking. Section 2 exhibits the context, the design and the methodological backgrounds of the two cases. Section 3 discusses the temporal tensions between factors, actors and events on the one hand and within action processes on the other hand and suggests developing a streaming approach of anticipation.

Bridging history and futures studies: tensions and directions

The wicked nature of future history

While history is viewed to supplement futures methods such as scenarios [START_REF] Bradfield | The critical role of history in scenario thinking: Augmenting causal analysis within the intuitive logic scenario development methodology[END_REF], it first served as a cornerstone to shape anticipation practices either to stress the difference between futures thinking and history or to point out that futures thinking deals with a wicked future history. In seminal futures thinking, the sense of history differs from the purpose of futures practices designed as a support to decision-making that cannot be scientific, science being timeless [START_REF] Berger | Sciences humaines et prévision[END_REF]. Although "related to history", Flechtheim's futurology cannot elicit a "chronological sequence of detailed facts" but "avail itself of interpretation, generalization, and speculation to a considerably higher degree" [START_REF] Flechtheim | History and futurology[END_REF]. However, Kahn's scenarios are about "future history" and "attempt to describe in more and less detail some hypothetical sequence of events" [START_REF] Kahn | Thinking about the unthinkable[END_REF]. But the design of hypothetical "future history" is based on a gap with history, as the very nature of historical facts (facta) are not to be confused with future events (futura) that are nothing but subjective [START_REF] Jouvenel | The art of conjecture[END_REF].

Timing futures thinking

If futures studies are "futureless" [START_REF] Sardar | The namesake: futures; futures studies; futurology; futuristic; foresight -what's in a name?[END_REF], futures practices are not timeless, and history remains a reference to futures studies. Within macro-history, "timing the future" is one of the pillars of futures thinking [START_REF] Inayatullah | Six pillars: futures thinking for transforming[END_REF]. Developmental scenarios [START_REF] Van Notten | An updated scenario typology[END_REF] and backcasting studies [START_REF] Vergragt | Backcasting for sustainability: Introduction to the special issue[END_REF] introduce time-scales in scenario design while the concept of end-state [START_REF] Burt | Why are we surprised at surprises? Integrating disruption theory and system analysis with the scenario methodology to help identify disruptions and discontinuities[END_REF] offers a time frame for scenarios. Stakeholder behavior [START_REF] Cairns | A decision-analysis-based framework for analysing stakeholder behaviour in scenario planning[END_REF] and empowerment [START_REF] Bourgeois | Foresight for all: Co-elaborative scenario building and empowerment[END_REF], as well as leadership shifts (Marchais- [START_REF] Marchais-Roubelat | Dominance, stakeholders' moves and leadership shifts: new directions for transforming futures[END_REF] question the timing of action in futures thinking, while governance issues point out the role of actors and stakeholders in change over time (Wangel, 2012). Whereas Kahn's scenarios explore decision processes like escalation [START_REF] Kahn | On escalation. Metaphors and scenarios[END_REF]) and examine "the branching points dependent of critical choices" [START_REF] Kahn | Thinking about the unthinkable[END_REF], "branching scenarios" (Cairns, Wright, Fairbrother and Philips, 2017) one [START_REF] Saritas | Mapping issues and envisaging futures: An evolutionary scenario approach[END_REF], as well as the one of emergence (Fuller, 2018). Addressing everchanging changes and the related temporal gaps [START_REF] Berger | Sciences humaines et prévision[END_REF], futures thinking also stresses the tension between long-range slow transformations over time (Botta, 2016) and ephemeral futures (Roubelat, Brassett, MacAllum, Hoffmann and Kera, 2015).

Research context and methodological background

Within the context of futures studies applied to defense issues (Saritas andBurmaoglu, 2016, Nemeth, Dew and[START_REF] Nemetha | Understanding some pitfalls in the strategic foresight processes: The case of the Hungarian Ministry of Defense[END_REF], we discuss two geostrategy anticipatory practices from two cases commissioned by the French ministry of Defense. The first case is based on the seminal archive report on security global governance frameworks to 2030 and on the book 2030, the end of globalization? issued from the report (Coutau-Bégarie, 2008). The case is supplemented by the results of a research meeting organized with Paris war college research institute (Irsem) [START_REF] Roubelat | Hervé Coutau-Bégarie et la prospective de la mondialisation[END_REF]. The second case is based on full outcomes of an action research on strategic action scenarios to 2030 designed by the authors, which supplement the action-based scenarios methodology (Marchais- Roubelat andRoubelat, 2008, 2016).

From framing to streaming. Timing the tensions of geostrategy futures

Timing the tensions between factors, actors and events

2030, the end of globalization? offers an opportunity to discuss the tensions between factors, actors and events, within the two perspectives of Coutau-Bégarie works on "new history" and on strategy [START_REF] Coutau-Bégarie | Le phénomène nouvelle histoire[END_REF], 2011). In Coutau-Bégarie's perspective, futures thinking may uncover only a part of the causal clues at work in history, as the interweaving of factors, actors and events reveals moving horizons as well as different temporalities. While the long-range evolution of factors may be discussed, actors and events can hardly be anticipated and future governance results from moving combinations of scenario archetypes that question the resilience of governance over time and the horizons of future global transformations. The role of individuals, which may be of importance in history, as well as the one of communities and of emerging actors, also appear as a tricky issue for the timing of futures thinking.

Timing the tensions from action processes

Strategic action scenarios offer a methodological framework to play factors, stakeholders and events in action processes from action rules. In such scenarios, stakeholders' acts and organizing processes move and change over time to challenge the action rules, or not. Through an iterative process, paradigm shifts, stalemates, oscillations and phase lags are explored to design new branching rules for scenarios moving over time. Such scenarios stress three tensions for stakeholders: the sustainability of their acts with the question of their reversibility over time, the transgressions from their acts with the issue of the conflicts between stakeholders, their capacity or incapacity to act with the problem of the transformations of stakeholders over time. These three issues suggest designing streaming futures, to be challenged over time from emerging moves from stakeholders.

  break lock-ins to incorporate new decision issues. Scenarios add the "what next" question to the "what if"