L Miguel Rodrigues
email: lmrodrig@fourier.ujf-grenoble.fr

Asymptotic stability of Oseen vortices for a density-dependent incompressible viscous fluid

Keywords: Mathematics subject classification (2000). 76D05, 35Q30, 35B35 Density-dependent incompressible Navier-Stokes equations, weak inhomogeneity, Oseen vortex, asymptotic stability, self-similar variables, vorticity equation

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Asymptotic stability of Oseen vortices for a density-dependent incompressible viscous fluid

Luis Miguel Miguel

Introduction

In this paper we consider the motion of a weakly inhomogeneous incompressible viscous fluid in the two-dimensional Euclidean space. We can describe the fluid by the pair (ρ, u), ρ = ρ(t, x) ∈ R + being the density field and u = u(t, x) ∈ R 2 the velocity field. The evolution we consider here is governed by the density-dependent incompressible Navier-Stokes equations:

   ∂ t ρ + (u • ∇) ρ = 0 ∂ t u + (u • ∇) u = 1 ρ (△u -∇p) div u = 0 (1)
where p = p(t, x) ∈ R is the pressure field, which is determined (up to a constant) by the incompressibility condition which yields the elliptic equation:

div 1 ρ ∇p = div 1 ρ △ u -(u • ∇) u . (2)
Alternatively, we can represent the fluid motion using the vorticity field ω = curl u ∈ R rather than the velocity. Note that, in the two-dimensional context, curl(f 1 , f 2 ) stands for ∂ 1 f 2 -∂ 2 f 1 . Therefore the evolution equations for (ρ, ω) become

∂ t ρ + (u • ∇) ρ = 0 ∂ t ω + (u • ∇) ω = div 1 ρ (∇ω + ∇ ⊥ p) (3)
where p is again determined by [START_REF] Danchin | Estimates in Besov spaces for transport and transportdiffusion equations with almost Lipschitz coefficients[END_REF], and u is recovered from ω via the Biot-Savart law:

u(x) = 1 2π R 2 (x -y) ⊥ |x -y| 2 ω(y) dy (4)
for x ∈ R 2 , with (z 1 , z 2 ) ⊥ = (-z 2 , z 1 ). We also denote u = K BS ⋆ ω, where K BS is the Biot-Savart kernel:

K BS (x) = 1 2π
x ⊥ |x| 2 . Without loss of generality, we assume throughout the present paper that the viscosity of the fluid is equal to one.

We refer to the monograph [START_REF] Lions | Mathematical topics in fluid mechanics[END_REF] for a general presentation of the available mathematical results on incompressible Navier-Stokes equations. We also mention the work of B.Desjardins on the global existence of weak solutions [START_REF] Desjardins | Linear transport equations with initial values in Sobolev spaces and application to the Navier-Stokes equations[END_REF][START_REF] Desjardins | Global existence results for the incompressible densitydependent Navier-Stokes equations in the whole space[END_REF], and, closer to the spirit of the present paper, the work of R.Danchin on wellposedness in Besov spaces [START_REF] Danchin | Local and global well-posedness results for flows of inhomogeneous viscous fluids[END_REF]. Let us emphasize that both Danchin and Desjardins work with the velocity formulation [START_REF] Danchin | Local and global well-posedness results for flows of inhomogeneous viscous fluids[END_REF] and do not assume the density ρ to be bounded away from zero. In more physical terms, they allow for regions of (almost complete) vacuum, which create technical difficulties.

In contrast, not only shall we not allow the density to be close to zero but we shall only consider weakly inhomogeneous fluids, namely we shall assume that the density ρ is close to a positive constant which, without loss of generality, we take equal to one. Remark that if the initial density is constant in space i.e. if the fluid is initially homogeneous, then the density remains equal to this constant for all subsequent times. Therefore, in such a case, system [START_REF] Danchin | Local and global well-posedness results for flows of inhomogeneous viscous fluids[END_REF] reduces to the usual incompressible Navier-Stokes equations. Moreover, since div(∇ ⊥ p) = 0, the pressure term disappears from system (3) which thus reduces to

∂ t ω + (u • ∇) ω = △ ω . (5)
Again, a wealth of information on the Cauchy problem for the homogeneous incompressible Navier-Stokes equations can be found in [START_REF] Lions | Mathematical topics in fluid mechanics[END_REF] or [START_REF] Gilles | Recent developments in the Navier-Stokes problem[END_REF]. Concerning the long-time behavior of the solutions of the vorticity equation [START_REF] Gallagher | Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity[END_REF], the work of Th.Gallay and C.E.Wayne has revealed the important role played by a family of explicit self-similar solutions, Oseen vortices, given by ρ ≡ 1, u = α u G and ω = α ω G , where α ∈ R is a parameter and

ω G (t, x) = 1 t G x √ t , u G (t, x) = 1 √ t v G x √ t with G(ξ) = 1 4π e -|ξ| 2 /4 , v G (ξ) = 1 2π ξ ⊥ |ξ| 2 1 -e -|ξ| 2 /4 .
For the Oseen vortex (1, α ω G ), the quantity |α| is actually its Reynolds number.

If the initial vorticity ω 0 is integrable, it is proved in [START_REF] Gallay | Global stability of vortex solutions of the two-dimensional Navier-Stokes equation[END_REF] that the corresponding solution of (5) converges to α ω G in L 1 -norm as t → ∞, where α := R 2 ω 0 . Moreover, it was shown in [START_REF] Gallagher | Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity[END_REF][START_REF] Gallagher | On the uniqueness of the solution of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity[END_REF] that α ω G is the unique solution of the vorticity equation [START_REF] Gallagher | Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity[END_REF] with initial data α δ 0 . Note also that u G is not square integrable, since |v G (ξ)| ∼ 1 |ξ| as |ξ| → ∞, which means that Oseen vortices are not finite energy solutions in the sense of Leray [START_REF] Leray | Sur le mouvement d'un liquide visqueux emplissant l'espace[END_REF]. More generally, when dealing with incompressible flows of integrable vorticity and non-zero global circulation, one needs to consider infinite energy solutions.

Even though the homogeneous incompressible Navier-Stokes equations provide a good model in many situations, all real fluids are, at least slightly, inhomogeneous and it is therefore important and relevant, both from a practical and theoretical point of view, to investigate whether the predictions of the homogeneous model are meaningful for the density-dependent model, especially in the weakly inhomogeneous regime. The goal of this paper is to address this question in the particular case of Oseen vortices. These explicit solutions persist in the density-dependent case if we assume ρ ≡ 1, and it is therefore natural to ask whether they play there the same role as in the homogeneous case. While the general answer to this question is unknown, we treat here one important aspect: are Oseen vortices stable solutions for the density-dependent incompressible Navier-Stokes equations ? In other words, does the theory predicts that these self-similar solutions may be observed ?

Before stating what we mean exactly by stability, let us recall an important property of the Navier-Stokes equations: scaling invariance. For any λ > 0, if (ρ(t, x), ω(t, x)) is a solution of (3), so is

D λ (ρ, ω) = (ρ (λ 2 t, λx), λ 2 ω (λ 2 t, λx)) .
Correspondingly, the velocity field u(t, x) and the pressure p(t, x) are rescaled into λ u(λ 2 t, λx) and λ 2 p(λ 2 t, λx). As is easily verified, Oseen vortices are selfsimilar, in the sense that D λ (1, α ω G ) = (1, α ω G ), for any α ∈ R and any λ > 0. To study these solutions, it is therefore more convenient to rewrite (3) in self-similar variables. Following [START_REF] Gallay | Invariant manifolds and the longtime asymptotics of the Navier-Stokes and vorticity equations on R 2[END_REF] , we set

(τ, ξ) := (ln t, x √ t ) . (6)
Motivated by scaling invariance, we will work with new quantities (r, w, v, Π) related to the former by r(τ, ξ) = ρ (e τ , e

Then the corresponding evolution equations for (r, w) are

∂ τ r + (v -1 2 ξ) • ∇ r = 0 ∂ τ w + (v -1 2 ξ) • ∇ w -w = div 1 r (∇ω + ∇ ⊥ Π) (8)
where again v is obtained from w by the Biot-Savart law and ∇Π by solving

div 1 r ∇Π = div 1 r △ v -(v • ∇) v . (9)
By construction, Oseen vortices correspond to stationary solutions of (8) of the form (1, α G), where α is any real number. We prescribe initial data for the original equations at t = 1 rather than at t = 0, hence at τ = 0 for the new equations.

In order to state our main result, we now write down the evolution equations for a perturbation of an Oseen vortex. Given α ∈ R, we work with new quantities (b, w) related to the former by b = 1 r -1 and w = wα G. The evolution equations for (b, w) are

∂ τ b + (v -1 2 ξ) • ∇ b = 0 ∂ τ w -(L -α Λ) w + v • ∇ w = div b (∇w + ∇ ⊥ Π) (10)
where

Lf = △f + 1 2 ξ • ∇f + f , Λf = v G • ∇f + (K BS ⋆ f ) • ∇G . ( 11
)
Here v is obtained from w by the Biot-Savart law, w and v are recovered by

w = α G + w , v = α v G + v , (12)
and ∇Π is obtained by solving

div (1 + b) ∇Π = div (1 + b) △v -(v • ∇) v . (13)
Let us point out that, thanks to the linearity of the Biot-Savart law, we both have

v = K BS ⋆ w and v = K BS ⋆ w.
Note that L is the usual Fokker-Planck type operator which generates the evolution corresponding to the heat equation in self-similar variables. On the other hand, Λ is the non-local first-order operator resulting from the linearization around w = G of the transport term of [START_REF] Gallay | Global stability of vortex solutions of the two-dimensional Navier-Stokes equation[END_REF]. More precisely, if v = K BS ⋆ w and (v, w) satisfies [START_REF] Lions | Mathematical topics in fluid mechanics[END_REF], then v

• ∇w = α Λ w + v • ∇ w, since v G ⊥ ∇G.
Before stating our result of stability, we introduce the function spaces and the norms that we will encounter throughout the present paper. For 1 ≤ p ≤ ∞, we denote by L p (R 2 ) the usual L p -space and by | f | p the usual L p -norm of a function f : R 2 → R or f : R 2 → R 2 . Similarly, for s ∈ R, we denote by H s (R 2 ) the usual Sobolev space and by | f | H s the corresponding norm. Sobolev norms will be convenient to specify the regularity of our perturbations, but we also need weighted norms to describe the localization properties. Indeed, even in the homogeneous case, we know that it is impossible to get a convergence rate in time if we do not assume that the perturbations decay sufficiently fast at infinity in space (see [START_REF] Gallay | Global stability of vortex solutions of the two-dimensional Navier-Stokes equation[END_REF]). Instead of using polynomial weights as in [START_REF] Gallay | Global stability of vortex solutions of the two-dimensional Navier-Stokes equation[END_REF], we choose here the Gaussian weight G -1 2 , which is more restrictive but naturally related to the Oseen vortices and has several technical advantages. For instance, on the Hilbert space

L 2 w (R 2 ) := {f | G -1 2 f ∈ L 2 (R 2
)}, the linear operators L and Λ become respectively symmetric and skew-symmetric. More generally, we shall use the weighted L p -space defined as follows, for any 1 ≤ p ≤ +∞,

L p w (R 2 ) := { f | G -1 2 f ∈ L p (R 2 ) } (14) with corresponding norms | f | w,p := | G -1 2 f | p .
We are now able to state the main result of this paper:

Theorem 1 Let α ∈ R, 0 < s < 1 and 0 < γ < 1 2 .
There exist ε 0 > 0 and K > 0 such that, for any

0 < ε < ε 0 , if 1. b 0 belongs to L 2 w (R 2 )∩L ∞ w (R 2 )∩H s+2 (R 2 ) with | b 0 | w,2 ≤ ε, | b 0 | w,∞ ≤ ε, and G -1 2 ∇b 0 belongs to L q (R 2 ) for some q > max(4, 2 s ), 2. w 0 belongs to L 2 w (R 2 ) ∩ H s (R 2 ) with | w 0 | w,2 ≤ ε and R 2 w 0 = 0,
then there exists a unique solution (b, w) of [START_REF] Gilles | Recent developments in the Navier-Stokes problem[END_REF] with initial data (b 0 , w 0 ) such that

1. b ∈ L ∞ loc (R + ; H s+2 (R 2 )), 2. G -1 2 b ∈ L ∞ (R + ; L 2 (R 2 ) ∩ L ∞ (R 2 )), G -1 2 ∇b ∈ L ∞ loc (R + ; L q (R 2 )) 3. w ∈ L ∞ loc (R + ; H s (R 2 )) ∩ L 2 loc (R + ; H s+1 (R 2 )), 4. G -1 2 w ∈ L ∞ (R + ; L 2 (R 2 )) ∩ L 2 (R + ; L 2 (R 2 )), G -1 2 ∇ w ∈ L 2 (R + ; L 2 (R 2 )), G -1 2 |ξ| w ∈ L 2 (R + ; L 2 (R 2 )) .
Moreover this solution satisfies | w(τ ) | w,2 ≤ K ε e -γ τ , for any τ > 0.

Theorem 1 shows that Oseen vortices, which are self-similar solutions of the density-dependent Navier-Stokes equations, are stable with respect to small localized perturbations of the density and the vorticity. It is very important to note that we do not make any smallness assumption on the parameter α ∈ R so that we do treat Oseen vortices with arbitrarily high Reynolds numbers |α|. However, unlike in the homogeneous case [START_REF] Gallay | Global stability of vortex solutions of the two-dimensional Navier-Stokes equation[END_REF], we are not able to consider large perturbations of the vorticity, due to the lack of appropriate Lyapunov functions for the density-dependent system, and unfortunately, in Theorem 1, the allowed size ε 0 of the perturbations is a decreasing function of the Reynolds number |α| tending to zero as |α| goes to infinity.

Without loss of generality, we assume in Theorem 1 and throughout this paper that the vorticity perturbation w has zero average. Indeed, if w = α G+ w with w small and δ := R 2 w = 0, one can always rewrite w = (α + δ) G + w with w small and R 2 w = 0. This zero-mean condition is preserved under the evolution defined by [START_REF] Gilles | Recent developments in the Navier-Stokes problem[END_REF] and is necessary to show that w(τ ) converges to zero as τ → ∞, i.e. to obtain an asymptotic stability result for the vorticity.

To make this asymptotic stability more concrete, we express it now in the original variables. Under the assumptions of Theorem 1, if (ρ, ω) is the solution of (3) defined by [START_REF] Gallay | Invariant manifolds and the longtime asymptotics of the Navier-Stokes and vorticity equations on R 2[END_REF] with w = α G + w and r = 1 1+b , then the vorticity ω(t, x) satisfies t

1 2 | e |x| 2 8t (ω(t) -α ω G (t)) | 2 ≤ C t γ , t ≥ 1 . Moreover this implies t 1-1 p | ω(t) -α ω G (t) | p ≤ C p t γ , t 1 2 -1 q | u(t) -α u G (t) | q ≤ C q t γ for any 1 ≤ p ≤ 2, 2 < q < +∞, t ≥ 1. Note that self-similarity implies | ω G (t) | p = C t -(1-1 p ) and | u G (t) | p = C t -( 1 2 -1 q )
. In contrast, since the density perturbation 1 ρ -1 is just transported by the divergence-free velocity field u, formally it remains constant in law, thus in any L p -norm.

The rest of the paper is organized as follows. In a preliminary section, we collect estimates on the Biot-Savart kernel, thus on the velocity in terms of the vorticity, and on the pressure. In the second section, we establish various estimates for the solution of a linearized density equation. Similarly, in the third one, we study a linearized vorticity equation. The final section is devoted to the proof of Theorem 1.

However there is another underlying structure that the reader may find useful to keep in mind. Indeed, to establish the existence part of Theorem 1, following [START_REF] Danchin | Local and global well-posedness results for flows of inhomogeneous viscous fluids[END_REF], we build a sequence of solutions of a linearization of [START_REF] Gilles | Recent developments in the Navier-Stokes problem[END_REF]. In order to show the convergence of this sequence, we will use local-in-time estimates in Sobolev norms for solutions of this linearized system. To be more precise, in a first step, we establish global estimates for solutions of the linearized system, which control the density in weighted L p spaces and the vorticity in H 1 . In a second step, we use the previous results to prove first global estimates of the density in H 2-ε (with a loss of regularity), then local estimates of the vorticity in H s+1 , and finally local estimates of the density in H s+2 . Then we use these results to establish the existence and uniqueness parts of the theorem via estimates on differences of two solutions. In both second section and third section, where we study the linearized density and vorticity equations, we shall clearly indicate which estimates are needed for the first step and the second step respectively.

Let us make clear that the global estimates show decay in time and enable us to keep some quantities small, leading to stability, whereas the local estimates are only used to show the existence for all time of a unique solution, allowing growth in time but precluding blow-up in a finite time. It should also be emphasized that since we consider infinite energy solutions even the existence for small time was unknown. Actually, concerning local existence, one should also note that the localization of the vorticity necessary to obtain a decay rate is also useful to get low regularity estimates of the velocity field as provided by inequality (21) below.

In what follows, the original (and physical) time will never appear again, so for notational convenience henceforth we use the letter t to denote the rescaled time τ .

Preliminaries

If f is integrable over R 2 , we define its Fourier transform to be the function f defined for any η ∈ R 2 by

f(η) = R 2 f (ζ) e iη•ζ dζ .
Concerning function spaces, we will also need the following convention. For any σ ∈ R, we denote by Ḣσ (R 2 ) the usual homogeneous Sobolev space on

R 2 equipped with | f | Ḣσ := | I σ f | 2 , where I := (-△) 1 2 .

Biot-Savart kernel

The goal of this subsection is to enable us to estimate the velocity in terms of the vorticity. For this purpose, we collect here some estimates on v in terms of w when v is obtained from w by the Biot-Savart law. Recall that, in this case, for almost every

ξ ∈ R 2 , v(ξ) = 1 2π R 2 (ξ -η) ⊥ |ξ -η| 2 w(η) dη (15)
where (

x 1 , x 2 ) ⊥ = (-x 2 , x 1 ), that is v = K BS ⋆ w, where K BS is the Biot- Savart kernel K BS (x) = 1 2π x ⊥ |x| 2 . Also note that in terms of Fourier transform, (15) becomes v(η) = i η ⊥ |η| 2 ŵ(η) . (16)
Most of these estimates are already known, but for the sake of completeness we try to give proofs rather than references whenever this is not too long.

The following proposition gathers estimates in L p -spaces.

Proposition 1

1. Let 1 < p < 2 < q < +∞ be such that 1 + 1 q = 1 2 + 1 p . There exists C > 0 such that, if w belongs to L p (R 2 ), then (15) defines a v in L q (R 2 ) and | v | q ≤ C | w | p . ( 17
) 2. Let 1 ≤ p < 2 < q ≤ +∞ and 0 < θ < 1 be such that θ p + 1-θ q = 1 2 . There exists C > 0 such that, if w belongs to L p (R 2 ) ∩ L q (R 2 ), then (15) defines a v in L ∞ (R 2 ) and | v | ∞ ≤ C | w | θ p | w | 1-θ q . ( 18
)
3. Let 1 < p < +∞. There exists C > 0 such that, if w belongs to L p (R 2 ) and v is defined from w by (15), then ∇v belongs to L p (R 2 ) and

| ∇v | p ≤ C | w | p . (19)
In addition, in all cases, we have div v = 0 and curl v = w.

Let us emphasize that we do not estimate v in L 2 (R 2 ). Indeed, one can easily derive from (16) that to make v square integrable one must assume that w has zero circulation, that is R 2 w = 0. This would exclude Oseen vortices.

Proof. Part 1 follows from a Young-like inequality called Hardy-Littlewood-Sobolev inequality (see for instance Theorem V.1 in [START_REF] Elias | Singular integrals and differentiability properties of functions[END_REF]). Indeed, K BS is weakly L 2 but not square integrable.

Part 2 is trivial when w ≡ 0. If not, we remark that from Hölder's inequalities, we obtain

|v(ξ)| ≤ 1 2π {|η|≤R} |w(ξ -η)| 1 |η| dη + 1 2π {|η|≥R} |w(ξ -η)| 1 |η| dη ≤ C | w | q R 1-2 q + C | w | p 1 R 2 p -1
, for almost every ξ ∈ R 2 and any R > 0. Aiming to optimize this inequality, we choose R = (

| w |p | w |q ) β with β = 1-θ 2 p -1 = θ 1-2 q
and derive (18). Part 3 holds since differentiating (15) yields that ∇v is obtained from w by a singular integral kernel of Calderón-Zygmund type (see Theorem II.3 in [START_REF] Elias | Singular integrals and differentiability properties of functions[END_REF]).

The following proposition gathers estimates in Sobolev spaces.

Proposition 2 1. Let s ∈ R. There exists C > 0 such that, if w belongs to Ḣs-1 (R 2 ), then, if v is defined by (15), v belongs to Ḣs (R 2 ) and | v | Ḣs ≤ C | w | Ḣs-1 . (20) 2. Let 0 < s < 1. There exists C > 0 such that, if (1 + |ξ|) w belongs to L 2 (R 2 ), then, if v is defined by (15), v belongs to Ḣs (R 2 ) and | v | Ḣs ≤ C | (1 + |ξ|) w | 2 . ( 21
)
Proof. Part 1 is a direct consequence of the Fourier formulation (16). Part 2 is thus reduced to estimate | w | Ḣs-1 . If 0 < s < 1, we note that

| w | 2 Ḣs-1 = C R 2 | ŵ(ξ)| 2 |ξ| 2(1-s) dξ ≤ C | ξ |≤1 | ŵ(ξ)| 2 |ξ| 2(1-s) dξ + C | ξ |≥1 | ŵ(ξ)| 2 dξ .
The second term of the right member is dominated by | w | 2 2 . Choosing p such that p > 2 s and applying first Hölder's inequalities then Sobolev's embeddings, we obtain for the first term

| ξ |≤1 | ŵ(ξ)| 2 |ξ| 2(1-s) dξ ≤ C | ŵ | 2 p ≤ C | ŵ | 2 H 1 .
Finally, gathering every piece yields

| w | 2 Ḣs-1 ≤ C | ŵ | 2 H 1 + C | w | 2 2 ≤ C | (1 + |ξ|) w | 2 2 .
This concludes the proof.

Pressure estimates

Keeping in mind equation ( 13), we gather some estimates for a solution Π of the following equation:

div (1 + b)∇Π = div F . (22)
We begin with estimates in L p -spaces.

Proposition 3

Let 1 < p < +∞.

There exist C > 0 and κ > 0 such that if

F belongs to L p (R 2 ) and b to L ∞ (R 2 ) with κ | b | ∞ < 1, then (22)
has a unique solution Π (up to a constant) such that ∇Π belongs to L p (R 2 ), and

| ∇Π | p ≤ C 1 -κ | b | ∞ | F | p . (23)
2. Let 1 < p ≤ +∞ and 1 < q, r < +∞ be such that 1 r = 1 p + 1 q . There exist C > 0 and κ > 0 such that if F belongs to L q (R 2 ) and, for

i = 1, 2, b i belongs to L ∞ (R 2 ) ∩ L p (R 2 ) with κ | b i | ∞ < 1 and Π i solves div (1 + b i )∇Π i = div F , then ∇ (Π 2 -Π 1 ) belongs to L r (R 2 ) and | ∇ (Π 2 -Π 1 ) | r ≤ C (1 -κ | b | ∞ ) 2 | b 2 -b 1 | p | F | q . ( 24
)
Proof. To prove part 1 we want to obtain ∇Π, the solution of ( 22), in terms of F as a perturbation of Leray projectors. Let P be the Leray projector, that is the projector on divergence-free vector fields along gradients, and let Q = I-P.

Remark that Q F gives the solution ∇Π of (22) when b ≡ 0. Now we can rewrite (22

) as div ∇Π = div ( F -b ∇Π ) then ∇Π = Q (F -b ∇Π) thus (I + Q b) ∇Π = Q F . Since Q is continuous on L p , there exists κ > 0 such that | Q b f | p ≤ κ | b | ∞ | f | p for f ∈ L p (R 2 ). Thus, if κ | b | ∞ < 1, I + Q b is invertible on L p , and
∇Π = (I + Q b) -1 Q F (25)
gives the unique solution, with the expected bound.

To prove part 2 reminding (25) we write

∇Π 1 = (I + Q b 2 ) -1 (I + Q b 2 ) (I + Q b 1 ) -1 Q F ∇Π 2 = (I + Q b 2 ) -1 (I + Q b 1 ) (I + Q b 1 ) -1 Q F
then subtracting and factorizing yields

∇(Π 2 -Π 1 ) = (I + Q b 2 ) -1 Q (b 1 -b 2 ) (I + Q b 1 ) -1 Q F .
Now the continuity of the operator Q on L r reduces (24) to an estimate on

| (b 1 -b 2 ) (I + Q b 1 ) -1 Q F | r .
At last applying first Hölder's inequalities then the continuity on L q concludes the proof.

In order to estimate solutions of (22) in Sobolev spaces, we first state some useful commutator estimates of Kato-Ponce type (see [START_REF] Kato | Commutator estimates and the Euler and Navier-Stokes equations[END_REF]). Let us recall that I = (-△)

1 2 .
Lemma 1 Let 0 < s ≤ 1 and σ > 1.

1. There exists C > 0 such that if

I s f belongs to L 2 (R 2 ) and g to H σ (R 2 ), then I s (f g) -f I s g belongs to L 2 (R 2 ) and | I s (f g) -f I s g | 2 ≤ C | I s f | 2 | g | H σ . ( 26
)
2. There exists C > 0 such that if

I s f belongs to H σ (R 2 ) and g to L 2 (R 2 ), then I s (f g) -f I s g belongs to L 2 (R 2 ) and | I s (f g) -f I s g | 2 ≤ C | I s f | H σ | g | 2 . ( 27
)
Proof. Let us first note that there exists C > 0 such that

| ĥ | 1 ≤ C | h | H σ ,
for any h in H σ (R 2 ). This comes applying Hölder's inequalities to

R 2 | ĥ(η)| dη = R 2 1 (1 + |η| 2 ) σ 2 (1 + |η| 2 ) σ 2 | ĥ(η)| dη .
Therefore in order to prove the lemma it is sufficient to establish

| I s (f g) -f I s g | 2 ≤ C | I s f | 2 | ĝ | 1 , (28)
| I s (f g) -f I s g | 2 ≤ C | I s f | 1 | g | 2 . ( 29
)
Now set h = I s (f g)f I s g. We have ĥ

(η) = 1 (2π) 2 R 2 (|η| s -|η -ζ| s ) f (ζ) ĝ(η -ζ) dζ
for almost every η ∈ R 2 . Thanks to the following basic fact:

||η| s -|η ′ | s | ≤ |η -η ′ | s , 0 < s ≤ 1 (30) for η, η ′ ∈ R 2 , we obtain | ĥ(η)| ≤ 1 (2π) 2 R 2 |ζ| s | f (ζ)| |ĝ(η -ζ)| dζ .
At last depending on how we apply Young's inequalities we obtain either (28) or (29).

We now state the announced estimates in Sobolev norms.

Proposition 4 Let 0 < s < 1 and σ > 1.

There exists C > 0 and κ > 0 such that if

F belongs to H s (R 2 ), b to L ∞ (R 2 ) with κ | b | ∞ < 1 and I s b belongs to H σ (R 2 ), then, if Π solves (22), I s ∇Π belongs to L 2 (R 2 ) and | I s ∇Π | 2 ≤ C 1 -κ | b | ∞ (| I s F | 2 + 1 1 -κ | b | ∞ | I s b | H σ | F | 2 ) . (31)
Proof. Applying I s to (22) and commuting b and

I s yields div (1 + b)∇I s Π = div [b, I s ]∇Π + div I s F .
Applying then (23) to this equation we obtain

| I s ∇Π | 2 ≤ C 1 -κ | b | ∞ (| I s F | 2 + | [b, I s ]∇Π | 2 ) .
Now using first (27) then applying (23) once again yields

| [b, I s ]∇Π | 2 ≤ C | I s b | H σ | ∇Π | 2 ≤ C 1 -κ | b | ∞ | I s b | H σ | F | 2 .
Gathering everything leads to (31).

Density equation

In this section, we gather information on the following linearization of the density equation:

∂ t b + (ν - 1 2 ξ) • ∇ b = 0 ( 32
)
where ν is a divergence-free vector-field, α ∈ R and ν = α v G + ν. By linearization, we mean that we do not assume that ν is obtained from a solution w of the vorticity equation in [START_REF] Gilles | Recent developments in the Navier-Stokes problem[END_REF], which involves b.

Actually in the cases we will consider ν -1 2 ξ generates a flow since ∇ ν belongs to

L 2 loc (R + ; L ∞ (R 2 )) (and ν belongs to L ∞ loc (R + ; L ∞ (R 2
)). Thus our point is not to prove the existence of a solution to (32) but to establish bounds for such a solution.

We begin with estimates in L p -spaces, weighted or not. We recall that L p w (R 2 ) for 1 ≤ p ≤ ∞ is the weighted space defined in (14). The next proposition is the density part of the announced first step of the proof.

Proposition 5 Let T > 0. Assume that ν is a divergence-free vector field belonging to L 2 (0, T ; L ∞ (R 2 )). Then b, the solution of (32) with initial data b 0 , satisfies

1. for any 1 ≤ p ≤ +∞, provided b 0 ∈ L p (R 2 ), | b(t) | p ≤ | b 0 | p e -t p , for 0 < t < T ; (33) 2. for any 1 ≤ p ≤ +∞, provided G -1 2 b 0 ∈ L p (R 2 ), | b(t) | w,p ≤ | b 0 | w,p e -t p e 1 8 t 0 | ν(s) | 2 ∞ ds , for 0 < t < T. (34)
Proof. In order to prove part 1 for 1 ≤ p < +∞, multiply (32) by sgn(b) |b| p-1 , where sgn is the usual sign function, and integrate by parts to obtain

d dt | b | p p = - R 2 (ν - 1 2 ξ) • ∇ |b| p = -| b | p p
since div ν = 0 and div ξ = 2. Integrating gives (33) in this case. The case p = +∞ follows letting p go to infinity.

To prove the weighted part of the proposition for 1 ≤ p < +∞, start as in the former and obtain

d dt | b | p w,p = - R 2 G -p 2 (ν - 1 2 ξ) • ∇ |b| p = R 2 G -p 2 p 4 ξ • (ν - 1 2 ξ) -1 |b| p since ∇G -p 2 = G -p 2 p 4 ξ. Now, since ξ ⊥ v G (ξ), we have ξ • (ν(ξ) - 1 2 ξ) = ξ • ν(ξ) - 1 2 |ξ| 2 ≤ 1 2 | ν(ξ)| 2 hence d dt | b | p w,p ≤ (-1 + p 8 | ν | 2 ∞ ) | b | p w,p .
Again integrating achieves the proof for finite p and the case p = +∞ follows letting p go to infinity.

The next proposition corresponds to the density part of the announced second step: local-in-time estimates in Sobolev norms. In order to prove some part of it we will need the following commutator lemma.

Lemma 2 Let s ≥ 1 and σ > 1.

There exists C > 0 such that if

I s f belongs to L 2 (R 2 ) and g to H σ (R 2 ), then I s (f g) -f I s g belongs to L 2 (R 2 ) and | I s (f g) -f I s g | 2 ≤ C | I s f | 2 | g | H σ + C | ∇f | H σ | I s-1 g | 2 .
(35)

Proof. The proof is essentially the same as Lemma 1's except that here (30) is replaced by

||ζ| s -|ζ ′ | s | ≤ C |ζ -ζ ′ | (|ζ -ζ ′ | s-1 + |ζ ′ | s-1 ), for ζ, ζ ′ ∈ R 2 . ( 36
)
Note that we could have obtained, as in Lemma 1, various estimates depending on the way we apply Young's inequalities.

Proposition 6 Let T > 0.

1. Let 0 < s ≤ 2 and 0 < ε < s. Assume ν is a divergence-free vector field with ∇ ν ∈ L 1 (0, T ; H 1 (R 2 )).

Then there exists C T > 0 independent of ν such that, for any initial data

b 0 ∈ H s (R 2 ), any solution b ∈ L ∞ (0, T ; H s-ǫ (R 2 )) of (32) satisfies | b(t) | H s-ε ≤ C T | b 0 | H s exp (C T t 0 | ∇ν(τ ) | H 1 dτ ) 2 , for 0 ≤ t ≤ T. (37) 2. Let s > 2.
Assume ν is a divergence-free vector field with ∇ ν ∈ L 1 (0, T ; H s-1 (R 2 )). Then (32) has a unique solution b ∈ L ∞ (0, T ; H s (R 2 )), for any initial data b 0 ∈ H s (R 2 ). Moreover there exists C > 0 independent of b 0 and ν such that b satisfies

| b(t) | H s ≤ C | b 0 | H s e s-1 2 t exp C t 0 | ∇ν(τ ) | H s-1 dτ , for 0 ≤ t ≤ T. (38)
Proof. We refer to Theorem 0.1 in [START_REF] Danchin | Estimates in Besov spaces for transport and transportdiffusion equations with almost Lipschitz coefficients[END_REF] for the first part of the proposition.

Let us only add that here we apply the former theorem with σ = s, ǫ = ε and p = r = p 2 = r 2 = 2.

To prove the second part, we start computing [I s , ξ 2 ] • f = -s 2 I s-2 div f , for any vector field f . Thus applying I s to equation (32) and commuting yield

∂ t I s b + (ν - 1 2 ξ) • ∇ I s b = s 2 I s b -[I s , ν] • ∇b .
Then multiplying by I s b and integrating lead to

1 2 d dt | I s b | 2 2 - s -1 2 | I s b | 2 2 = - R 2 I s b [I s , ν] • ∇b
since div ν = 0. Now use Cauchy-Schwarz' inequality and apply Lemma 2 (with σ = s -1) to get 1 2

d dt | I s b | 2 2 - s -1 2 | I s b | 2 2 ≤ C | ∇ν | H s-1 | b | 2 H s .
At last combine the former with

1 2 d dt | b | 2 2 + 1 2 | b | 2 2 ≤ 0 to obtain 1 2 d dt | b | 2 H s - s -1 2 | b | 2 H s ≤ C | ∇ν | H s-1 | b | 2 H s
which yields (38) by a mere integration.

The last estimate we state for the linearized transport equation ( 32) is intended to be used for the proofs of the convergence of our iterative scheme and of the uniqueness of our solutions. Indeed we estimate the difference of two solutions of equations of type (32).

Proposition 7 Let T > 0. Assume that, for i = 1, 2, ν i is a divergence-free vector field belonging to

L 2 (0, T ; W 1,∞ (R 2 )). If, for i = 1, 2, b i is a solution of ∂ t b i + (ν i - 1 2 ξ) • ∇ b i = 0 ,
where ν i = α v G + ν i , with initial data b 0 , then b 1 and b 2 satisfy 1. provided that G -1 2 ∇b 0 belongs to L p (R 2 ), for some 1 ≤ p ≤ +∞,

| ∇b i (t) | w,p ≤ | ∇b 0 | w,p e -t( 1 p - 1
2 ) e

1 8 t 0 | νi(s) | 2 ∞ ds e t 0 | ∇νi (s) |∞ ds (39)
for i = 1, 2 and 0 ≤ t ≤ T ;

2. provided that G -1 2 b 0 belongs to L p (R 2 ) and G -1 2 ∇b 0 belongs to L q (R 2 ), for some 1 ≤ p < q ≤ +∞,

| (b 2 -b 1 )(t) | w,p ≤ e 1 8 t 0 | ν2(s) | 2 ∞ ds sup 0≤s≤t | ∇b 1 (s) | w,q t 0 | ( ν 2 -ν 1 )(s) | r ds (40) for 0 ≤ t ≤ T , where r is such that 1 p = 1 q + 1 r .
Proof. In order to prove the first part of the proposition, we start differentiating the equation for b 1 to get for j = 1, 2,

∂ t ∂ j b 1 + (ν 1 - 1 2 ξ) • ∇ ∂ j b 1 = -∂ j ν 1 • ∇b 1 + 1 2 ∂ j b 1 .
From this, following the proof of (34), we obtain for j = 1, 2,

d dt | ∂ j b 1 | p w,p ≤ (-1 + p 2 + p 8 | ν 1 | 2 ∞ ) | ∂ j b 1 | p w,p + p | ∇ν 1 | ∞ | ∇b 1 | p w,p .
Now combining the inequalities for j = 1, 2 and integrating lead to (39). Then to prove part 2 observe that b 2b 1 satisfies

∂ t (b 2 -b 1 ) + (ν 2 - 1 2 ξ) • ∇ (b 2 -b 1 ) = -( ν 2 -ν 1 ) • ∇b 1 .
Now following again the proof of (34) yields

d dt | δ b | p w,p ≤ (-1 + p 8 | ν 2 | 2 ∞ ) | δ b | p w,p + p | δ b | p-1 w,p | ∇b 1 | w,q | ν 2 -ν 1 | r where δ b = b 2 -b 1 .
It is now straightforward to derive (40).

Vorticity equation

As announced, we now study a linearization of the vorticity equation:

∂ t w -(L -α Λ) w + ν • ∇ w = div b (∇w + ∇ ⊥ Π) ( 41
)
where L and Λ are as in [START_REF] Leray | Sur le mouvement d'un liquide visqueux emplissant l'espace[END_REF], b is a real function, ν is a divergence-free vector field,

α ∈ R, v = K BS ⋆ w , v = α v G + v , w = α G + w , ν = α v G + ν ,
and ∇Π is obtained by solving

div (1 + b) ∇Π = div (1 + b) △v -(ν • ∇) v . ( 42
)
Remind that we always assume R 2 w 0 = 0.

Weighted estimate

In this subsection we establish a global-in-time estimate in weighted L 2 -spaces.

Proposition 8 Let α ∈ R, K 0 > 0.
There exist ε 0 > 0 and C > 0 such that if b is a real function and ν a divergence-free vector field such that 1. for 0 < t < T , for any

1 ≤ p ≤ +∞, 2 ≤ q ≤ +∞, | b(t) | p ≤ | b 0 | p e -t p , | b(t) | w,q ≤ | b 0 | w,q e -t q e K0 2. for 0 < t < T , | ν(t) | 8 ≤ K 0 , t 0 | ν | 2 ∞ ≤ 1 24 3. and | b 0 | w,4 ≤ ε 0 , | b 0 | w,∞ ≤ ε 0 then any solution w ∈ L ∞ (0, T ; L 2 w (R 2 )) of (41), with initial data w 0 ∈ L 2 w (R 2 ), satisfies, for any 0 < t < T , | w(t) | 2 w,2 + C t 0 | w | 2 w,2 + | ∇ w | 2 w,2 + | |ξ| w | 2 w,2 ≤ 2 | w 0 | 2 w,2 + C |α| | b 0 | w,4 . (43)
Note that the assumptions on b corresponds to the first proposition of the previous section. Note also that once α and K 0 are fixed, since L 2 w is embedded in any L p , 1 ≤ p ≤ 2, and H 1 is embedded in any L q , 2 ≤ q < ∞, inequalities (18) and (43) enable us to make

t 0 | v | 2
∞ as small as we want provided we take w 0 and b 0 small enough. At last note that the proposition enables us to bound

t 0 | ∇ v | 2
H 1 , which can be used in (37).

Proof. Our strategy is to multiply (41) by G -1 w and integrate to bound

d dt | w | 2 w,2 .
In what follows, we examine each term arising once multiplied by G -1 w and integrated on R 2 .

• Let us emphasize first that (41) preserves R 2 w. Hence R 2 w = 0.

Let L := G -1 2 (-L) G 1 2 . A direct calculation shows that L = -△ + |ξ| 2 16 -1 2
is a harmonic oscillator with spectrum {0, 1 2 , 1, 3 2 , . . .}. Moreover 0 is a simple eigenvalue with eigenvector G

1 2 . In particular, if f belongs to the domain of L with R 2 G 1 2 f = 0, then R 2 f Lf ≥ 1 2 | f | 2 2 .
Coming back to L, we obtain: if G -1 2 w belongs to the domain of L with R 2 w = 0, then, for any 0

< γ < 1 2 , R 2 G -1 w L w ≤ - 1 2 (1 -γ) | w | 2 w,2 + γ R 2
G -1 w L w thus integrating by part from the formula for L

R 2 G -1 w L w ≤ - 1 2 (1 -2γ) | w | 2 w,2 -γ | ∇(G -1 2 w) | 2 2 + | |ξ| 4 w | 2 w,2
and expanding

R 2 G -1 w L w ≤ - 1 2 (1 -2γ) | w | 2 w,2 -γ | ∇ w | 2 w,2 + 2 | |ξ| 4 w | 2 w,2 + 2 R 2 G -1 ∇ w • ξ 4 w hence R 2 G -1 w L w ≤ - 1 2 (1 -2γ) | w | 2 w,2 -γ 1 3 | ∇ w | 2 w,2 + 1 2 | |ξ| 4 w | 2 w,2 . (44) • Recalling that Λ w = v G • ∇ w + v • ∇G, we obtain R 2 G -1 w Λ w = 0 . (45) Indeed , from v G (ξ) ⊥ ξ and ∇G -1 = -ξ 2 G -1 , we derive R 2 G -1 w v G • ∇ w = - 1 2 R 2 G -1 ξ 2 • v G w 2 = 0 .
And, on the other hand, using the identity η ⊥ • ξ = -ξ ⊥ • η and the explicit formula (15) for the Biot-Savart law, we derive

R 2 G -1 w v • ∇G = - R 2 w(ξ) v(ξ) • ξ 2 dξ = - 1 4π R 2 ×R 2 w(ξ) (ξ -η) ⊥ • ξ |ξ -η| 2 w(η) dη dξ = 1 4π R 2 ×R 2 w(ξ) η ⊥ • ξ |ξ -η| 2 w(η) dη dξ = - 1 4π R 2 ×R 2 w(ξ) ξ ⊥ • η |ξ -η| 2 w(η) dη dξ = - R 2 G -1 w v • ∇G . Thus R 2 G -1 w v • ∇G = 0.
• Using Hölder's inequalities, we also obtain

| R 2 G -1 w ν • ∇ w| ≤ 6 | ν | 2 ∞ | w | 2 w,2 + 1 24 | ∇ w | 2 w,2 . (46)
• Integrating by part, we obtain

R 2 G -1 w div (b ∇ w) = - R 2 G -1 b |∇ w| 2 - 1 2 R 2 G -1 b w ξ • ∇ w
then, using Hölder's inequalities and the fact that

| b(t) | ∞ ≤ | b 0 | ∞ , | R 2 G -1 w div (b ∇ w)| ≤ 5 4 | b 0 | ∞ | ∇ w | 2 w,2 + 1 4 | b 0 | ∞ | |ξ| w | 2 w,2 . (47)
• In the same way, since

| b(t) | 2 ≤ | b 0 | 2 e -t 2 , we have | R 2 G -1 w div (b ∇G)| = | 1 2 R 2 b ξ • ∇ w + 1 4 R 2 b w |ξ| 2 | ≤ C | b 0 | 2 e -t 2 (| ∇ w | w,2 + | w | w,2 ) thus | R 2 G -1 w div (b ∇G)| ≤ C | b 0 | 2 ( e -t + | ∇ w | 2 w,2 + | w | 2 w,2 ) . (48)
• Finally, integrating by part, using Hölder's inequalities and applying inequality (23), we obtain, for b 0 small enough in

L ∞ , | R 2 G -1 w div (b ∇ ⊥ Π) | = | 1 2 R 2 G -1 b w ξ • ∇ ⊥ Π + R 2 G -1 b ∇ w • ∇ ⊥ Π | ≤ C 1 -κ | b 0 | ∞ (| |ξ| w | w,2 + | ∇ w | w,2 ) × ( | b 0 | w,∞ e K0 | (1 + b) △ v | 2 + | b 0 | w,4 e K0 e -t 4 | α (1 + b) △v G -(ν • ∇) v | 4 ). Now, on one hand, since | b(t) | ∞ ≤ | b 0 | ∞ , estimate (20) yields
| (1 + b) △ v | 2 ≤ C (1 + | b 0 | ∞ ) | ∇ w | 2 .
On the other hand, similarly, we have

| (1 + b) △v G | 4 ≤ C (1 + | b 0 | ∞ ) .
At last, Hölder's inequalities, estimate (19) and Sobolev's embeddings yields

| (ν • ∇) v | 4 ≤ C (|α| + | ν | 8 ) (|α| + | w | 8 ) ≤ C (|α| + | ν | 8 ) (|α| + | w | H 1 )
.

Taking this into account yields when

κ | b 0 | ∞ ≤ 1 2 | R 2 G -1 w × div (b ∇ ⊥ Π)| ≤ e -t 2 C |α| e K0 | b 0 | w,4 (1 + |α| + | ν | 8 ) 2 + | w | 2 w,2 C e K0 | b 0 | w,4 (1 + |α| + | ν | 8 ) 2 + | ∇ w | 2 w,2 C e K0 (| b 0 | w,∞ + | b 0 | w,4 (1 + |α| + | ν | 8 )) + | |ξ| w | 2 w,2 C e K0 (| b 0 | w,∞ + | b 0 | w,4 (1 + |α|)) . (49)
It only remains to us to gather everything after setting γ = 1 4 in (44) and integrate in time in order to obtain, when

κ | b 0 | ∞ ≤ 1 2 , | w(t) | 2 w,2 × (1 -12 t 0 | ν | 2 ∞ ) + C t 0 | w | 2 w,2 × (1 -e K0 | b 0 | w,4 (1 + |α| + | ν | 8 ) 2 ) + C t 0 | ∇ w | 2 w,2 × (1 -e K0 ( | b 0 | w,∞ +| b 0 | w,4 (1 + |α| + | ν | 8 )) + C t 0 | |ξ| w | 2 w,2 × (1 -e K0 (| b 0 | w,∞ + | b 0 | w,4 (1 + |α|))) ≤ | w 0 | 2 w,2 + C |α| × | b 0 | w,4 e K0 (1 + |α| + | ν | 8 ) 2
which yields the proposition since

t 0 | ν | 2 ∞ ≤ 1 24 .

Sobolev estimate

In this subsection we prove a local-in-time estimate in Sobolev norms for solutions of equation ( 41). Remind that I = (-△)

1 2 . Proposition 9 Let 0 < s < 1, 1 + s < s ′ < 2, 1 < s ′′ < 2 -s and α ∈ R.
There exists ε 0 > 0 and, for K > 0, there exists C > 0 such that if b is a real function and ν a divergence-free vector field such that

sup [0,T ] | b | ∞ ≤ | b 0 | ∞ ≤ ε 0 , sup [0,T ] | b | H s ′ ≤ K , sup [0,T ] | w | 2 ≤ K , T 0 | w | 2 w,2 ≤ K , T 0 | ∇ w | 2 2 ≤ K , T 0 | ν | 2 ∞ ≤ K , sup [0,T ] | I s ν | 2 ≤ ε 0 , T 0 | I s ν | 2 H s ′′ ≤ K , then any solution w ∈ L ∞ (0, T ; Ḣs (R 2
)) of ( 41), with initial data w 0 ∈ Ḣs (R 2 ), satisfies for any 0 < t < T ,

| I s w (t) | 2 2 + C t 0 | I s ∇ w | 2 2 ≤ C e Ct ( | I s w 0 | 2 2 + K ) . (50)
Note that (33) and (37) can provide us the validity of the assumptions on b, and (43) both the validity of the assumptions on w and, thanks to estimates (18), ( 20) and ( 21), the validity of estimates on ν when ν = v. Conversely (50) can be used in (38), again when ν = v.

Proof. We choose σ such that 1 < σ < 1 + s, s + σ < s ′ and σ < s ′′ . The only role of σ is to make clearer our use of commutator estimates (26) and (27).

Our strategy is to apply I s to (41), then multiply by I s w and estimate each term arising, in order to bound

d dt | I s w | 2 2 . • First we compute the commutator [I s , L] = s 2 I s and obtain R 2 I s w I s L w = - R 2 |∇I s w| 2 + 1 + s 2 R 2 |I s w| 2 (51)
since integrating by parts yields

R 2 f Lf = -R 2 |∇f | 2 + 1 2 R 2 |f | 2 . • On one hand, we have R 2 I s w I s ((v G • ∇) w) = R 2 I s w (v G • ∇) I s w + R 2 I s w ([I s , v G ] • ∇) w with, integrating by parts, R 2 I s w (v G • ∇) I s w = 1 2 R 2 (v G • ∇) |I s w| 2 = 0
and, using Hölder's inequalities and inequality (27),

| R 2 I s w ([I s , v G ] • ∇) w| ≤ C | I s w | 2 | I s v G | H σ | ∇ w | 2 .
On the other hand, we have

R 2 I s w I s (( v • ∇) G) = R 2 I s w ( v • ∇) I s G + R 2 I s w ([I s , v] • ∇) G
with, using Hölder's inequalities,

| R 2 I s w ( v • ∇) I s G| ≤ C | I s w | 2 | v | ∞ | ∇I s G | 2
and, using both Hölder's inequalities and inequality (26),

| R 2 I s w ([I s , v] • ∇) G| ≤ C | I s w | 2 | I s v | 2 | ∇G | H σ .
Using estimate (21) and estimate (18) combined with Sobolev's embeddings, we derive |

R 2 I s w I s Λ w| ≤ C (| w | 2 w,2 + | ∇ w | 2 2 ) . (52)
• In the same way, we have

R 2 I s w ( ν • ∇) I s w = 0
and, applying Hölder's inequalities and inequality (26),

| R 2 I s ∇ w • ([I s , ν] w)| ≤ C | I s ∇ w | 2 | I s ν | 2 | w | H σ thus, since 0 ≤ σ ≤ 1 + s, | R 2 I s w I s (( ν • ∇) w)| ≤ C | I s ν | 2 | I s ∇ w | 2 2 + C | I s ν | 2 | w | 2 2 . (53)
• Integrating by parts yields

R 2 I s w I s div( b∇ w) = - R 2 b |I s ∇ w| 2 - R 2 I s ∇ w [I s , b] ∇ w .
Applying (27), we derive

| R 2 I s w I s div( b∇ w)| ≤ (| b 0 | ∞ + ε) | I s ∇ w | 2 2 + C ε | b | 2 H s+σ | ∇ w | 2 2 (54)
where ε > 0 is intended to be chosen small enough.

• Similarly,

| R 2 I s w I s div( b∇G)| ≤ (| b 0 | ∞ +ε) | I s ∇ w | 2 2 + C (| b 0 | ∞ + 1 ε | b | 2 H s+σ ) . ( 55
)
• First, integrating by parts yields

R 2 I s w I s div(b∇ ⊥ Π) = - R 2 I s ∇ w • b I s ∇ ⊥ Π - R 2 I s ∇ w • ([I s , b]∇ ⊥ Π)
with, using Hölder's inequalities,

| R 2 I s ∇ w • b I s ∇ ⊥ Π| ≤ | I s ∇ w | 2 | b 0 | ∞ | I s ∇Π | 2
and, using Hölder's inequalities and inequality (27),

| R 2 I s ∇ w • ([I s , b]∇ ⊥ Π)| ≤ C | I s ∇ w | 2 | b | H s+σ | ∇Π | 2 .
Besides, on one hand, estimate (23) applied to equation (42) and Hölder's inequalities imply

| ∇Π | 2 ≤ C (|α| + |α| 2 + | △ v | 2 + |α|| ∇ v | 2 + | ν | ∞ | ∇ v | 2 + |α|| ν | ∞ )
thus with estimates on Biot-Savart kernel

| ∇Π | 2 ≤ C (|α| + |α|| ν | ∞ + |α| 2 + (|α| + | ν | ∞ ) | w | 2 + | ∇ w | 2 ) .
On the other hand, estimate (31) applied to (42) and Hölder's inequalities imply

| I s ∇Π | 2 ≤ C (| b | H s+σ | ∇Π | 2 + | I s ((1 + b) △v) | 2 + | I s (ν • ∇) v | 2 )
with, commuting I s and b thanks to (27), after some calculation,

| I s ((1 + b) △v) | 2 ≤ C | b | H s+σ (|α| + | ∇ w | 2 ) + C (1 + | b 0 | ∞ ) (|α| + | I s ∇ w | 2 )
and, in the same way, commuting I s and ν,

| I s ((ν •∇) v) | 2 ≤ C (|α|+| I s ν | H σ ) (|α|+| w | 2 )+C (|α|+| ν | ∞ ) (|α|+| I s w | 2 ) .
Therefore, gathering these inequalities,

| R 2 I s w × I s div( b ∇ ⊥ Π) | ≤ C | I s w | 2 2 | b 0 | ∞ (|α| 2 + | ν | 2 ∞ ) + C | I s ∇ w | 2 2 (| b 0 | ∞ + ε) + C | b | 2 H s+σ 1 ε + | b 0 | ∞ × |α| 2 + |α| 2 | ν | 2 ∞ + |α| 4 + (|α| 2 + | ν | 2 ∞ ) | w | 2 2 + | ∇ w | 2 2 + C | b 0 | ∞ |α| 2 (1 + | b 0 | 2 ∞ + |α| 2 + | ν | 2 ∞ ) +(|α| 2 + | I s ν | 2 H σ ) (|α| 2 + | w | 2 2 ) + | b | 2 H s+σ (|α| 2 + | ∇ w | 2 2 ) . (56)
Putting all these points together and integrating yields

| I s w | 2 2 + C t 0 | I s ∇ w | 2 2 1 -| b 0 | ∞ (1 + |α|) -sup [0,t] | I s ν | 2 ≤ | I s w 0 | 2 2 + C t 0 | I s w | 2 2 1 + | b 0 | ∞ | ν | 2 ∞ + |α| (1 + | b 0 | ∞ t 0 | ν | 2 ∞ |α| | b 0 | ∞ ) + C t 0 | ν | 2 ∞ | b 0 | ∞ |α| 2 + (|α| 2 + sup [0,t] | w | 2 2 ) (1 + | b 0 | ∞ ) sup [0,t] | b | 2 H s+σ + C t 0 | ∇ w | 2 2 |α| + (1 + | b 0 | ∞ ) sup [0,t] | b | 2 H s+σ + C t 0 | w | 2 w,2 |α| + |α| 2 | b 0 | ∞ + sup [0,t] | I s ν | 2 + |α| 2 (1 + | b 0 | ∞ ) sup [0,t] | b | 2 H s+σ + C t 0 | I s ν | 2 H σ | b 0 | ∞ (|α| 2 + sup [0,t] | w | 2 2 ) + C t |α| (1 + |α| 3 ) (1 + | b 0 | ∞ ) (1 + sup [0,t] | b | 2 H s+σ ) .
A Gronwall-type argument achieves the proof.

Estimate for convergence

We now establish an estimate on the difference of two solutions of equations of type (41), intending to prove convergence of our iterative scheme and uniqueness of solutions of [START_REF] Gilles | Recent developments in the Navier-Stokes problem[END_REF]. For i = 1, 2, consider

∂ t w i -(L -α Λ) w i + ( ν i • ∇ ) w i = div b i (∇w i + ∇ ⊥ Π i ) (57)
where L and Λ are as in ( 11), b i , Ω i are real functions, α ∈ R,

v i = K BS ⋆ w i , ν i = K BS ⋆ Ω i , v i = α v G + v i , ν i = α v G + ν i , w i = α G + w i , Ω i = α G + Ω i ,
and ∇Π i is obtained by solving

div (1 + b i ) ∇Π i = div (1 + b i ) △v i -(ν i • ∇) v i . (58)
Note that we choose to write ν = K BS ⋆ Ω to stress the symmetry of the hypotheses on Ω and w.

In what follows for concision's sake we denote δf = f 2f 1 for any functions f 1 , f 2 .

Proposition 10

Let α ∈ R, K > 0, σ > 2, 0 < η < s < 1 and max( 2 η , 4) < p < +∞. There exists ε 0 > 0 and, for K ′ , T > 0, there exists C > 0 such that if w 1 , w 2 satisfy (57) with

1. | b 0 | w,4 ≤ ε 0 , | b 0 | w,∞ ≤ ε 0 2. for 0 < t < T , for i = 1, 2, for any 1 ≤ r ≤ +∞, | b i (t) | r ≤ | b 0 | r e -t r , | b i (t) | w,r ≤ K | b 0 | w,r e -t r 3. for 0 < t < T , for i = 1, 2, | b i (t) | H σ ≤ K ′ 4. for 0 < t < T , for i = 1, 2, | Ω i (t) | 2 w,2 + t 0 | ∇ Ω i | 2 w,2 ≤ ε 0 | Ω i (t) | H s + t 0 | ∇ Ω i | 2 H s ≤ K ′ 5. for 0 < t < T , for i = 1, 2, | w i (t) | 2 w,2 + t 0 | ∇ w i | 2 w,2 ≤ ε 0 | w i (t) | H s + t 0 | ∇ w i | 2 H s ≤ K ′ then for 0 < t < T , | δ w(t) | 2 w,2 + C t 0 | δ w | 2 w,2 + | ∇(δ w) | 2 w,2 + | |ξ| (δ w) | 2 w,2 ≤ C t 0 (1 + | w 1 | 2 w,p + | ∇ w 1 | 2 H η ) (| δb | 2 w,p + | δ Ω | 2 w,2 ) . ( 59
)
Proof. Combining (60) for i = 1, 2, we derive

∂ t (δ w) -(L -α Λ) (δ w) + ν 2 • ∇ (δ w) -div b 2 (∇(δ w) + ∇ ⊥ Π) = -((δ ν) • ∇) w 1 + div (δb)∇w 1 + div (b 2 ∇ ⊥ R) + div (b 2 ∇ ⊥ (δS)) + div (δb)∇ ⊥ Π 1 (60)
with Π, R, S 1 and S 2 obtained by solving

   div (1 + b 2 )∇Π = div (1 + b 2 )△(δ v) -( ν 2 • ∇) (δ ν) div (1 + b 2 )∇R = div -α(v G • ∇) (δ v) + (δb)△ v 1 -((δ ν) • ∇) v 1 div (1 + b i )∇S i = div (1 + b 1 )△v 1 -(ν 1 • ∇) v 1 , for i = 1, 2 . Note that Π 2 = Π + R + S 2 and Π 1 = S 1 .
Our strategy is again to multiply (60) by G -1 (δ w), integrate on R 2 and estimate each term arising to bound d dt | δ w | 2 w,2 . We deal with each term coming from the left member of equality (60) as we did in the linearized vorticity equation (41). Let us only show how to deal with the other terms.

First of all, let us emphasize that

| G r ∇(G r ′ f ) | 2 2 is controlled by | ∇f | 2 w,2 + | |ξ| f | 2 w,2 provided that r + r ′ = -1 2 .
• First integrating by parts and applying Hölder's inequalities, we obtain

| R 2 G -1 (δ w) div( w 1 (δ ν))| ≤ | G 1 2 ∇(G -1 (δ w)) | 2 | G -1 2 w 1 | p | δ ν | q
where 2 < q < +∞ is such that 1 p + 1 q = 1 2 . Then using (17)

R 2 G -1 (δ w) div( w 1 (δ ν))| ≤ ε | G 1 2 ∇(G -1 (δ w)) | 2 2 + C ε | w 1 | 2 w,p | δ Ω | 2 w,2 (61)
where ε is intended to be chosen small enough.

• Similarly, with the same q, we also have

| R 2 G -1 (δ w) div((δb)∇w 1 )| ≤ | G 1 2 ∇(G -1 (δ w)) | 2 | G -1 2 (δb) | p | ∇w 1 | q thus using Sobolev's embeddings | R 2 G -1 (δ w) div((δb)∇w 1 )| ≤ ε | G 1 2 ∇(G -1 (δ w)) | 2 2 + C ε | ∇w 1 | 2 H η | δb | 2 w,p (62)
where ε is again intended to be chosen small enough.

• In quite the same way, we obtain

| R 2 G -1 (δ w) div(b 2 ∇ ⊥ R)| ≤ C | G 1 2 ∇(G -1 (δ w)) | 2 | b 0 | w,∞ | ∇R | 2 .
With pressure estimate (23), estimate on Biot-Savart law (17), Hölder's inequalities and Sobolev's embeddings we can derive

| R 2 G -1 (δ w) div(b 2 ∇ ⊥ R)| ≤ ε | G 1 2 ∇(G -1 (δ w)) | 2 2 + C ε | b 0 | 2 w,∞ | ∇ w 1 | 2 H η | δb | 2 p + C ε |α|| b 0 | 2 w,∞ | δ w | 2 2 + C ε | b 0 | 2 w,∞ | w 1 | 2 w,2 | δ Ω | 2 w,2 (63)
where ε is once again intended to be chosen small enough.

• Using estimate (24) instead of estimate (23) and inequalities ( 17) and ( 19), since

1 p < 1 2 -1 p , we can obtain | R 2 G -1 (δ w) div(b 2 ∇ ⊥ (δS))| ≤ ε | G 1 2 ∇(G -1 (δ w)) | 2 2 + C ε | b 0 | 2 w,∞ | δb | 2 p × | ∇w 1 | 2 H η + | Ω 1 | 2 w,2 | w 1 | 2 w,p (64)
where ε is still intended to be chosen small enough.

• At last, integrating by parts and applying Hölder's inequalities, we have

| R 2 G -1 (δ w) div((δb)∇ ⊥ Π 1 | ≤ | G 1 2 ∇(G -1 (δ w)) | 2 | δb | w,p | ∇ ⊥ Π 1 | q .
where 2 < q < +∞ is again such that 1 p + 1 q = 1 2 . Again with pressure estimate (23), estimates on Biot-Savart law (17) and (19), Hölder's inequalities and Sobolev's embeddings we can derive

| R 2 G -1 (δ w) div((δb)∇ ⊥ Π 1 | ≤ ε | G 1 2 ∇(G -1 (δ w)) | 2 2 + C ε | δb | 2 w,p | ∇w 1 | 2 H η + | Ω | 2 w,2 | w 1 | 2 w,p (65)
where ε is once again intended to be chosen small enough. Gathering everything and integrating yield (59).

Main results

We now use our various estimates to derive our main results.

Existence and uniqueness

Before proving a result of existence and uniqueness of solution of equations ( 10), we state a lemma that will make a link between norm estimates and convergence of the iterative scheme.

Lemma 3 Let T > 0 and 1 < p ≤ +∞. Let (f k ) be a sequence in L ∞ (0, T ; R + ), and (g k ) a bounded sequence in L p (0, T ; R + ) be such that for 0 < t < T and k ∈ N,

f k+1 (t) ≤ t 0 f k g k .
Then (f k ) is uniformly summable, namely, for 0 < t < T ,

k≥0 f k (t) ≤ C T .
Proof. Using Hölder inequalities and iterating yield f k (t) ≤ K C k ( t k k! ) 1-1 p , where C is a bound for (g k ) in L p (0, T ; R + ).

We can now prove the existence and uniqueness parts of Theorem 1.

Proof.

• Existence. We build a sequence ((b k , w k )) k∈N * such that, for any k ∈ N * ,

∂ t b k+1 + (v k -1 2 ξ) • ∇ b k+1 = 0 ∂ t w k+1 -(L -α Λ) w k+1 + v k • ∇ w k+1 = div b k (∇w k+1 + ∇ ⊥ Π k+1 )
where L and Λ are as in [START_REF] Leray | Sur le mouvement d'un liquide visqueux emplissant l'espace[END_REF] with initial data (b 0 , w 0 ). For k = 0, we solve the system with v k (t) ≡ 0 and b k (t) ≡ 0.

Let us show how we propagate bounds on (b k , w k ).

Step 1 Fix K 0 > 0 and choose ε 0 > 0 small enough. We can propagate which provides us, thanks to Proposition 1 and Sobolev's embeddings,

| v k | 8 ≤ C | w k | 8 5 ≤ C | w k | w,2 ≤ K 0 t 0 | v k | 2 ∞ ≤ C t 0 | w k | 2 w,2 + | ∇ w k | 2 2 ≤ min( 1 24 , K 0 ) .
Step 2 Again choosing ε 0 small enough independently of t and using Proposition 2 we can obtain, when 0 < s < 1 and 1 < s ′′ < 2s,

t 0 | ∇v k | H 1 ≤ C(t + t 1 2 ( t 0 | ∇ v k | 2 H 1 ) 1
2 ) ≤ C(t

+ t 0 | w k | 2 H 1 ) | I s v k (t) | 2 ≤ C | w k (t) | w,2 ≤ C K0 ε 0 t 0 | I s v k | 2
for some 0 < η < s such that 2 η < p < +∞. Now in order to apply Lemma 3 with

f k = | (δb) k | 2 w,p + | (δ w) k | 2 w,2 , remark that -since (G -1 2 w k ) is bounded in L ∞ (R + ; L 2 (R 2
)) and (∇(G -1 2 w k )) in L 2 (R + ; L 2 (R 2 )), (G -1 2 w k ) is bounded in L r (R + ; L p (R 2 )), for some 2 < r < +∞, by interpolation and Sobolev embeddings

  , ξ) = e τ ω (e τ , e τ 2 ξ) , Π(τ, ξ) = e τ p (e τ , e τ 2 ξ) .

  , ( v k ) is obtained from ( w k ) by the Biot-Savart law, v k = α v G + v k , w k = α G + w k , for k ∈ N * ,and (∇Π k ) is obtained by solving, for anyk ∈ N * , div (1 + b k ) ∇Π k+1 = div (1 + b k ) △v k+1 -(v k • ∇) v k+1

1. for any 1 ≤ 0 | w k | 2 w, 2 + | ∇ w k | 2 w, 2 + | |ξ| w k | 2 w, 2 ≤

 10222222 p ≤ +∞, 2 ≤ q ≤ +∞, thanks to Proposition 5,| b k (t) | p ≤ | b 0 | p e -t p , | b k (t) | w,q ≤ | b 0 | w,q e -t q e K0and thanks to Proposition 8,| w k (t) | 2 w,2 + C K0 t C K0 (| w 0 | 2 w,2 + | b 0 | w,4 )

+ C t 0 | 0 | 0 (| w k | 2 2 +Step 3 2 w, 2 ≤ C T t 0 ( 1 +

 000232201 w k | 2 w,2 + | ∇ w k | 22 ) ≤ K 0 and propagate, for 0 < t < T , 1. when 1 + s < s ′ < 2, thanks to Proposition 6,| b k (t) | H s ′ ≤ C K0,T2. and thanks to Proposition 9,| I s w k (t) | 2 2 I s ∇ w k | 2 2 ≤ C K0,Twhich provides us, for 0 < t < T ,t ∇v k | H s+1 ≤ C (t + t | I s ∇ w k | 2 2 )) ≤ C K0,Tthus, thanks to Proposition 6, for 0 < t < T ,| b k (t) | H s+2 ≤ C K0,T .We now prove the convergence of the scheme. Set (δb) k = b k+1 -b k and (δ w) k = w k+1w k . Choose max(4, 2 s ) < p < q. Propositions 7 & 10 give us, for T > 0, for any 0 < t < T and any k ∈ N * ,| (δb) k+1 (t) | 2 w,p + | (δ w) k+1 (t) | | w k | 2 w,p + | ∇ w k | 2 H η ) (| (δb) k | 2 w,p + | (δ w) k | 2 w,2 )

Acknowledgments. I warmly thank Thierry Gallay for having proposed to me and supervised this work.

since ( w k ) is bounded in L ∞ (R + ; L 2 (R 2 )) ∩ L 2 (0, T ; H s+1 (R 2 )), (∇ w k ) is bounded in L r ′ (0, T ; H η (R 2 )) for some 2 < r ′ < +∞, by interpolation.

Thus (b k ) converges in L p w (R 2 ) and ( w k ) in L 2 w (R 2 ), locally uniformly in time. This implies at once that (b k ) and ( w k ) also converges in L 2 (R 2 ). Now by interpolation

, for any 0 < η ′′ < s.

These properties enable us to take the limit in the sequence of equations.

Note that we recover the regularity on the limit by a mere application of Fatou's lemma.

• Uniqueness. We obtain a bound similar to (66) for the difference of two solutions. Then Gronwall lemma gives the result.

Asymptotic behavior

We now state the asymptotic part of Theorem 1. Note that under the hypotheses of Theorem 1, the following assumptions are fulfilled.

Theorem 2 Let α ∈ R. For any 0 < γ < 1 2 , there exist ε 0 > 0 and K, K ′ > 0 such that if (b, w) is a solution of [START_REF] Gilles | Recent developments in the Navier-Stokes problem[END_REF] with initial data (b 0 , w 0 ), such that

Then we deal with the other terms of the vorticity equation as we did to obtain estimate (43), except for the pressure term and

obtained thanks to estimate (18) and Sobolev's embeddings. We treat the pressure term as follows:

This yields


Pierre-Alain Muller
email: pa.muller@uha.fr

The TopModL Initiative

We believe that there is a very strong need for an environment to support research and experiments on model-driven engineering. Therefore we have started the TopModL project, an open-source initiative, with the goal of building a development community to provide:

• an executable environment for quick and easy experimentation, • a set of source files and a compilation tool chain, • a web portal to share artefacts developed by the community.

The aim of TopModL is to help the model-engineering research community by providing the quickest path between a research idea and a running prototype. In addition, we also want to identify all the possible contributions, understand how to make it easy to integrate existing components, while maintaining architectural integrity. At the time of writing we have almost completed the bootstrap phase (known as Blackhole), which means that we can model TopModL and generate TopModL with TopModL.

Beyond this first phase, it is now of paramount importance to gather the best possible description of the requirements of the community involved in model-driven engineering to further develop TopModL, and also to make sure that we are able to reuse or federate existing efforts or goodwill. This paper is more intended to set up a basis for a constructive discussion than to offer definitive answers and closed solutions.

Introduction -About model-driven engineering

At the end of the year 2000, the OMG proposed a radical move from object composition to model transformation [START_REF] Bézivin | From Object Composition to Model Transformation with the MDA[END_REF] , and started to promote MDA 2 (Model-Driven Architecture) a modeldriven engineering framework to manipulate both PIMs (Platform Independent Models) and PSMs (Platform Specific Models). The OMG also defined a four level meta-modeling architecture, and UML was elected to play a key role in this architecture, being both a general purpose modeling language, and (for its core part) a language to define metamodels. As MDA will become mainstream, more and more specific metamodels will have to be defined, to address domain specific modeling requirements. Examples of such metamodels are CWM (Common Warehouse Metamodel) and SPEM (Software Process Engineering Metamodel). It is likely that MDA will be applied to a wide range of different domains.

While preeminent in the current days, MDA is only a specific case, and we suggest considering model-driven engineering as a wider research field, which includes the study of the following issues:

• What are the essential entities for model-driven engineering?

• How to classify these entities?

• How to translate models into executable code?

• What are the essential operations of model-driven engineering?

• How to classify these operations?

• How to separate and merge the business and platform aspects?

• How to build transformation systems?

• How to maintain a model-driven application?

• How to migrate a legacy application to a model-driven application?

• How to integrate conventional application with model-driven applications?

• Which abstractions and notations should be used to support the previous points?

• What kind of supporting environment should be defined?

Obviously the scope of model-driven engineering is wide and a lot of work is still ahead of us. We believe that a common research platform which would provide the fundamentals services required by model-driven engineering would significantly contribute to the advance of research in this field.

Basic principles of model-driven engineering

The point is to identify the fundamental characteristics upon which to build model-driven engineering. In our specific case, this means identify the requirements for a supporting environment dedicated to model-driven experiments.

The fundamental principles identified so far by the TopModL initiative are:

• The fact that everything is a model. For TopModL, models are first-class entities; everything is expressed explicitly in terms of models, including business models, platform models, executable models, debugging models, trace models, transformation models, process models…

• The notions of languages, models and roles. A model is expressed in a language; this language is a model which plays the role of meta-model for the models expressed in that language.

• The fact that TopModL itself is a model. We want TopModL to be model-driven; we want everything in TopModL to be explicit and customizable.

• The fact that everything is explicit, including the meta-modeling framework. For instance TopModL does not require the M3 to be MOF, it does not even require a 4 layer meta-modeling architecture.

• The independence versus the model repository. We want to have a uniform access to several repositories including EMF [START_REF]Eclipse EMF[END_REF] , MDR 4 , or XDE.

These principles shape the requirements for the services to be provided by TopModL.

Toward a research platform for Model-Driven Engineering

The goal of TopModL is to provide an infrastructure for model-driven engineering, including a reference implementation of MDA as promoted by the OMG. TopModL wants to act as a facilitator for researchers in model-driven engineering in connection with other fields including generative programming, graph transformations, domain specific languages (DSLs), or aspects.

The basic services offered by TopModL are:

• Model (meta-model) persistence The TopModL artefacts include:

• A set of source files (Java, XML, text…) and model-driven compilation tool chain to bootstrap TopModL.

• An executable environment. The first release (known as Blackhole) allows the visual edition of meta-model which conforms to the UML Infrastructure (TopModL by default uses the UML Infrastructure as M3).

• A model-driven web portal to share artefacts developed by the community (libraries of meta-models, profiles, models and transformations…).

The first release of TopModL (Blackhole) contains all the artefacts required to bootstrap TopModL. Bootstrapping means being able to use TopModL to model and generate TopModL itself.

Blackhole -Technical architecture of the bootstrap

The technical infrastructure of TopModL has been a recurring concern since the inception of the TopModL initiative (November 2003). The basic question is: What should be the existing technology (if any) onto which to layer the new developments to be done in the initiative?

The initial partners acknowledged that the answer to that question would not be obvious, and considered that trying to determine upfront the best technology would be counterproductive, and would significantly delay the timeliness delivery of the TopModL artefacts.

The decision was then taken to use the technology that the partners felt the most comfortable with at the time of the launch of the project, and not to wait for the next wonderful technology yet to come. The direct advantage of this approach was to accumulate practical experience and then to be able to make decisions based on explicit knowledge -rather than informal feelings -if a new technical element had to be incorporated.

The technologies and standards that were chosen for the bootstrap phase include:

• Java for the programming language • JMI for the metadata API

• MDR for the model repository • GEF/SWT for the graphical interface • The Eclipse framework and plugging mechanism • The UML Infrastructure as M3

It is the intent of the TopModL partners to take any appropriate action to simplify the potential transition from one technology to another one. Technology independence is achieved as much as possible by:

• Sticking to established standards when these standards are available (JMI, XMI, MOF, UML…) • Defining neutral pivot on top of alternative technical solutions, for instance to be independent of the repositories.

The following picture describes the bootstrap process of TopModL. One of the concerns of TopModL is to make explicit the meta-modeling framework, and not to constrain the usage of a meta-modeling language at the M3 modeling level. There are mainly two reasons for that. The first reason is that TopModL, as a research-oriented tool, should be able to evaluate new possible meta-modeling languages and new meta-modeling constructs. The second reason is that there are already many different meta-modeling languages available today, and that there will be even more in the future (MOF 1.3, MOF 1.4, MOF 2.0, EMF ECore, …), and so TopModL should be able to deal with all of them. As an example, the first M3 language that we want Blackhole to support is the UML 2.0 Infrastructure. In addition, there is also an architectural problem, as model repositories are relying on a given meta-modeling language for both meta-model definition and persistence (actually to shape the schema of the repository). For instance, MDR requires metamodels to be MOF 1.4 described. Again, to promote research one of the most important issue is to abstract away this MOF platform. This was possible by introducing model transformations.

The first step is to model the meta-modeling language using UML class diagram in a UML CASE tool such as the Community Edition of Poseidon 5 . Then, this model is exported in an XMI file as a UML 1.4 model, which can be taken as input by the MDR UML2MOF model transformation, which promotes UML models to MOF models. A problem is that the Infrastructure is making an extensive use of constructs that do not have their equivalent in MOF 1.4 (like package merge or refinement/subsetting of association-ends) such that the MDR UML2MOF had to be enhanced to interpret such constructs. Actually, this was done directly on the XMI resulting from the transformation, on which we have reported occurrences of these constructs, thus extending MOF 1.4 a little bit to what we called MOF 1.4++. As these constructs cannot be understood by MOF 1.4 implementation such as MDR, they are interpreted by the transf1 model transformation, for the meta-modeling language (Infrastructure in this case) to be represented as MOF 1.4. This makes possible to load the Infrastructure as a metamodel in the MDR tool, which means to be able to generate the JMI interfaces specific to the Infrastructure, to store Infrastructure models, and to be able to serialize/de-serialize Infrastructure models to XMI 1.2.

We have also developed a visual editor which connects to this repository, and we can this way edit metamodels (which conform to the UML Infrastructure). This editor is currently handcoded, but it is our intent to generate it from models, in a similar way of what is done in Netsilon [START_REF] Muller | Platform Independent Web Application Modeling[END_REF] .

At this point, TopModL is able to edit and maintain Infrastructure models, thus it is possible to model the Infrastructure in TopModL. The result is the Infrastructure modeled as an Infrastructure model. The transf2 model transformation was developed to transform Infrastructure models to MOF 1.4++ models, so that the Infrastructure modeled in TopModL can be automatically translated to a MOF 1.4++ model. By reusing this MOF 1.4++ model of the Infrastrucure, it is possible to perform again the process, this time using TopModL instead of Poseidon: TopModL is at this phase bootstrapped, i.e. TopModL is modeled and designed using TopModL. Obviously, the bootstrapping process may be repeated for any modeling language supposed to play the role of a meta-modeling language.

The second part of the picture shows how the approach can then be used to generate a tool for another modeling language such as the UML 2.0 Superstructure. Basically, the process is the same: modeling the new modeling language in TopModL as an Infrastructure model, and using the transf2 to regenerate a TopModL intended to edit and maintain models in the new modeling language. Again, the issue of modeling the concrete syntax of the new modeling language, and the behavior of the editor dedicated to it is not yet solved and needs further research. The following paragraph further motivates such CASE tool generation.

Toward a model-driven CASE tool for the UML

TopModL is a meta-environment which can be used to realize model-driven CASE tools. The TopModL development community has considered that realizing a model-driven UML CASE tool would be an excellent use case for TopModL, and would also allow bypassing the current limitations of the available tools, most notably:

• The fact that no commercial tool fully implements the OMG standards, mainly because the first generation of UML CASE tools do not rely on an explicit description of the meta-models, but on hand-coded implementations, often themselves derived from earlier modeling tools (designed before the advent of the UML). Our experience also shows that CASE tools vendors do not have the resources to realize quick update of their tools, or to satisfy specific demands (to validate a research advance or to support a specific extension, or even simply to implement a standard).

• • The hard-wired nature of existing tools which embed a lot of decisions in their code and are therefore very difficult to customize. When buried in the code, decisions are implicit, as opposed to explicit when they are expressed in models and meta-models

In this context, TopModL will develop a new kind of UML CASE tool, entirely model-driven, with all the decisions made explicit by means of models. It is more accurate to talk about a model compilation tool chain, as the resulting CASE tool will always be conforming to a user-defined set of meta-models which will define its context. These capacities of customization will concern:

• The internal repository, generated in conformance to the meta-models referenced by the context.

• The model serialization (import-export via XMI)

• The visual editors, derived from a generic graph editor by explicit customization (this feature will be of special interest to realize domain specific languages).

• The textual editors (with syntactic and semantic completion) generated from metamodels (the abstract syntax) and mapped on a concrete syntax (also expressed via models).

• The overall behavior of the tool which will be described in a process model conforming to SPEM (Software Process Engineering Model).

Moreover, all the generated (UML or other modeling language) CASE tools will present the ability to interoperate, thus allowing for instance model transformations. One possible way for generating code is to generate two TopModL CASE tools for a modeling language (e.g. UML) and for the desired target language (e.g. Java) by using its abstract syntax as a metamodel. A model transformation from the modeling language to the target language would actually transform a user-specific model to a target-language model. By taking advantage of the textual concrete syntax of the target language, the latter model may be expressed as a set of files using the target language syntax. This is another reason for defining precisely concrete syntaxes of modeling language explicitly.

Related works

There are many related works, which share a common vision with the TopModL initiative. We summarize some of these approaches below:

• Meta CASE tools including Metaedit+ 7 , Dome 8 or GME [START_REF] Ledeczi | The Generic Modeling Environment[END_REF] , provide customizable CASE tools, however they are fairly closed in the sense that they are neither opensource, nor themselves model-driven.

• Dedicated model-driven tools, which generate specific applications, like Netsilon 6 for Web information systems, Accord/UML 10 for embedded distributed real-time systems.

• OCL based tools, including KMF [START_REF]Kent Metamodeling Framework[END_REF] which generates modeling tools from the definition of modeling languages expressed as meta-models, or Octopus 12 an Eclipse plug-in OCLE 13 or the Dresden OCL toolkit 14 , which are able to check the syntax of OCL expressions, as well as the types and correct use of model elements like association roles and attributes.

• Meta-modeling frameworks like Eclipse EMF 3 , Netbeans MDR 4 or Coral 15 which offer model persistence, model serialization and programmatic access to models via an API, or integration technologies like ModelBus [START_REF] Blanc | Model Bus : Towards the interoperability of modelling tools[END_REF] . These frameworks provide part of the functionalities required by TopModL.

• Open-source modeling tools, including ArgoUML 17 of Fujuba 18 which offer significant feature at the M1 level, but lack customization at the M2 level.

One of the goals of TopModL is to understand how to reuse or leverage these related works, and to find how to integrate them as much as possible in a research platform for model-driven engineering.

Conclusion

The TopModL open-source initiative has been launched with the goal of providing tool support to the model-driven engineering research community.

The TopModL initiative groups a development community, a web portal to share the artefacts developed by the community, a set of source files and an executable program for metamodeling.

TopModL is itself a model-driven application, and a first phase known as Blackhole delivers the bootstrap of TopModL, which means that TopModL is modeled and generated with TopModL.

TopModL makes all decisions explicit, including the meta-modeling framework, by using models. However, some modeling language will still have to be defined, for instance to express concrete syntax or model transformation. Having such a variable meta-modeling framework may raise questions, for instance about the universality of the 4 layers metamodeling architecture. TopModL will provide the opportunity to experience a step further the meta-modeling activity, and will point out problems that may have not been already reported, like the necessity of modeling concrete syntaxes, or edition facilities.

Figure 1 :

 1 Figure 1 : Bootstrapping TopModL

  The lack of model-driven open-source tools, mainly because all the open-source efforts are based on programming and not modeling. TopModL is an open-model project before being an open-source project.

The TopModL Initiative

www.topmodl.org