Matthieu Hillairet Ceremade

Peter Wittwer
email: peter.wittwer@unige.ch

Asymptotic description of solutions of the exterior Navier Stokes problem in a half space

We consider the problem of a body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded domain. This situation is modeled by the incompressible Navier-Stokes equations in an exterior domain in a half space, with appropriate boundary conditions on the wall, the body, and at infinity. We focus on the case where the size of the body is small. We prove in a very general setup that the solution of this problem is unique and we compute a sharp decay rate of the solution far from the moving body and the wall.

Introduction

In the present paper we discuss solutions of the Navier-Stokes equations for the stationary flow around a body that moves with constant speed parallel to a wall in an otherwise unbounded space filled with a fluid. The mathematical formulation of the problem is as follows. Let Ω + = {x = (x, y) ∈ R 2 | y > 1}, and let B t = {(x, y) ∈ R 2 | (x, y) + te 1 ∈ B}, where e 1 = (0, 1) and where B is a bounded open connected subset of Ω + such that B ⊂ Ω + . As a function of t ≥ 0, the set B t corresponds to a body which is immersed in a fluid and moves at constant speed from right to left parallel to the wall ∂Ω + . The flow around this body is modeled by the Navier-Stokes equations

∂ t U = -(U • ∇) U + ∆U -∇P , ∇ • U = 0 , (1)
in Ω t = Ω + \ B t with the boundary conditions (the boundary ∂Ω + is at rest and we choose no slip boundary conditions at the surface of the body),

U| ∂Ω+ = 0 , lim x ∈ Ωt |x|→∞ U(x, t) = 0 , U| ∂Bt = -e 1 . (2)
We are interested in the construction of solutions of equations ( 1)-( 2) that are stationary when viewed in a reference frame attached to the moving body. We therefore set U(x, t) = u(x + te 1 ) , P (x, t) = p(x + te 1 ) , and get the following stationary problem:

-(u • ∇) u -∂ x u + ∆u -∇p = 0 , ∇ • u = 0 , (3)
in Ω + \ B, with the boundary conditions u| ∂B = -e 1 , u| ∂Ω+ = 0 , lim

x ∈ Ω+ |x|→∞ u(x) = 0 . (4)
Note that we have set without restriction of generality all the physical constants and the speed of the moving body equal to one. This can always be achieved by an appropriate scaling. With this choice of normalization the Reynolds number of the moving body corresponds to the diameter ε of B. The problem contains a second length-scale, which is the distance h of (the center of) B from the wall ∂Ω + .

In this paper, we are interested in the regime where ε is small, and in particular small with respect to h.

The system (3) with boundary conditions ( 4) is related to the so-called exterior Navier-Stokes problem:

-λ ((u -u ∞ ) • ∇) u + ∆u -∇p = 0 , ∇ • u = 0 , in R n \ B ( 5
)
u| ∂B = u * , lim |x|→∞ u(x) = 0 . ( 6
)
where B is a bounded open connected subset of R n with smooth boundary, λ ∈ R is the Reynolds number, u ∞ ∈ R n is a prescribed asymptotic velocity and u * ∈ H 1/2 (∂B) is a given boundary condition. Most of the methods for solving this problem are extensively described in the fundamental book of G.P. Galdi [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations[END_REF]. We give a brief outline of some results in the following lines.

The first methods to solve such exterior problems go back to the pioneering work of J. Leray [START_REF] Leray | Etude de diverses équations intégrales non linéaires et de quelques problèmes de l'hydrodynamique[END_REF]. In this reference, the author introduces an invading domain method yielding existence of at least one weak solution to (5)- [START_REF] Fischer | On two-dimensional slow viscous flows past obstacles in a half-plane[END_REF] whose velocity-field u satisfies ∇u ; L 2 (R n \ B) < ∞. A comparable result is obtained by H. Fujita [START_REF] Fujita | On the existence and regularity of the steady-state solutions of the Navier-Stokes theorem[END_REF]. Similar weak solutions are constructed also for exterior Navier Stokes system [START_REF] Finn | On the stationary solutions of the Navier-Stokes equations in two dimensions[END_REF] with other types of boundary conditions on ∂B (see [START_REF] Serre | Chute libre d'un solide dans un fluide visqueux incompressible[END_REF] and [START_REF] Weinberger | On the steady fall of a body in a Navier-Stokes fluid[END_REF]). The only shortcoming of these weak solutions is that insufficient information is obtained on the behavior at infinity. In the case n = 2 with u ∞ = 0, it is still not known whether the vanishing condition at infinity is satisfied by weak solutions or not (see [START_REF] Amick | On the asymptotic form of Navier-Stokes flow past a body in the plane[END_REF][START_REF] Gilbarg | Asymptotic properties of steady plane solutions of the Navier-Stokes equations with bounded Dirichlet integral[END_REF] and [START_REF] Russo | On the existence of d-solutions of the steady-state navier-stokes equations in plane exterior domains[END_REF] for recent developments in this question). This difficulty is linked to the famous Stokes Paradox which holds in two space-dimensions. For the geometry of the present paper, existence of weak solutions for (3) decaying at infinity, combined with other boundary conditions, is studied in [START_REF] Hillairet | Chute stationnaire d'un solide dans un fluide visqueux incompressible le long d'un plan incliné. partie ii[END_REF].

In the case u ∞ = 0, a more refined description of the asymptotic behavior of solutions to (5)-( 6) is given in a second series of papers. These results rely on the idea that the dominating system at infinity is the Oseen system :

λu ∞ • ∇u + ∆u -∇p = 0 , ∇ • u = 0 . (7)
A detailed comprehension of the asymptotics of solutions to this linear system enables to construct solution to ( 5)-( 6) via a standard perturbation technique and then to compute the asymptotics of the constructed solutions. Such an analysis is performed by K.I. Babenko in the 3D-setting [START_REF] Babenko | Theory of perturbations of stationary flows of a viscous incompressible fluid in the case of small Reynolds numbers[END_REF], and by R. Finn and D.R. Smith [START_REF] Finn | On the stationary solutions of the Navier-Stokes equations in two dimensions[END_REF] and L.I. Sazonov [START_REF] Sazonov | Justification of the asymptotic expansion of the solution of a two-dimensional flow problem for small Reynolds numbers[END_REF] in the 2D-setting. This method is transposed to the geometry studied in the present paper by T. Fischer, G.C. Hsiao and W.L. Wendland in [START_REF] Fischer | On two-dimensional slow viscous flows past obstacles in a half-plane[END_REF]. In this last case the difficulty linked to the Stokes paradox is less limitative. In particular, the Stokes problem :

∆u -∇p = 0 , ∇ • u = 0 , in Ω + \ B. ( 8
)
u| ∂B = u * , u| ∂Ω+ = 0 , lim |x|→∞ u(x) = 0 , (9)
is well-posed. In [START_REF] Fischer | On two-dimensional slow viscous flows past obstacles in a half-plane[END_REF] existence of solutions to (3)-( 4) is obtained via a perturbation method based on this linear Stokes problem. Nevertheless, the dominating system at infinity in our case is still the Oseen system with u ∞ = e 1 so that no precise asymptotics of the constructed solutions is given in [START_REF] Fischer | On two-dimensional slow viscous flows past obstacles in a half-plane[END_REF]. This computation requires a very careful analysis of the Oseen linear system in the half space, the analysis of which is not yet available with these former methods. For completness, we mention here that the properties of the Stokes system in the geometry of the present paper is studied in the more general framework of weighted Sobolev spaces in [START_REF] Amrouche | Exterior Stokes problem in the half-space[END_REF]. No equivalent study for the Oseen system is provided to our knowledge.

The present paper uses a dynamical-system approach for studying the asymptotics of solutions to an exterior Navier Stokes problem. In this method, the first idea is to interpret one coordinate as a time. Then, one rewrites (3) as a system of nonlinear evolution equations. Solutions are constructed via a perturbation method in function spaces enabling to compute the exact long-time behavior. In return, one obtains solutions to (3)-( 4) with detailed asymptotics. This program is applied successfully to the case of the 3D exterior Navier-Stokes system in [START_REF] Wittwer | Leading order down-stream asymptotics of stationary Navier-Stokes flows in three dimensions[END_REF] and of the 2D half-space problem, with the solid B replaced by a smooth source term with compact support, in a previous publication of the authors [START_REF] Hillairet | Existence of stationary solutions of the Navier-Stokes equations in two dimensions in the presence of a wall[END_REF]. In this last reference, the solutions to the system of nonlinear evolution equations are computed performing a Fourier transform in the transversal direction (i.e., with respect to x in our case). This is the reason why we replace the obstacle by a source term with compact support in [START_REF] Hillairet | Existence of stationary solutions of the Navier-Stokes equations in two dimensions in the presence of a wall[END_REF].

In the present paper, we prove existence of solutions to (3)-( 4) with a detailed asymptotics by combining the invading method of Leray and the dynamical-system approach. Since we apply in part perturbation methods, our results hold only for small Reynolds numbers. The role of Reynolds number is played by the diameter of the solid B in our setting. More precisely, let S be a bounded open subset of R 2 containing the origin, with a smooth boundary, and let h be a positive parameter which fixes the center of the body with respect to the boundary. Then, we set S ε := (0, 1 + h) + εS and rewrite our system as :

-

(u • ∇) u -∂ x u + ∆u -∇p = 0 , ∇ • u = 0 , (10)
in Ω \ S ε , with the boundary conditions

u| ∂Sε = -e 1 , u| ∂Ω+ = 0 , lim x ∈ Ω+ |x|→∞ u(x) = 0 . (11)
In what follows [START_REF] Gilbarg | Asymptotic properties of steady plane solutions of the Navier-Stokes equations with bounded Dirichlet integral[END_REF] together with boundary conditions [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] is referred to as Problem 1. The following theorem is our main result.

Theorem 1 For ε sufficiently small, there exists a unique weak solution u of Problem 1. Furthermore, there exists a constant C ε < ∞ such that, for all (x, y)

∈ Ω + \ S ε , |u(x, y)| ≤ C ε y 3 2 . ( 12
)
A precise definition of weak solutions for Problem 1 is given in Section 1. For the sake of simplicity we only give a bound for the decay of the weak solution in [START_REF] Hillairet | Chute stationnaire d'un solide dans un fluide visqueux incompressible le long d'un plan incliné. partie ii[END_REF]. Nevertheless, a precise first order for the asymptotics is available with our techniques. Such computations are performed in an independent paper (see [START_REF] Boeckl | Decay estimates for steady solutions of the navier-stokes equations in two dimensions in the presence of a wall[END_REF]). This bound is the critical ingredient for proving uniqueness in the frame of weak solutions to Problem 1.

Our strategy to obtain detailed information on weak solutions of Problem 1 at infinity is divided in five steps. First, we show the existence of weak solutions for Problem 1 by the invading method of Leray. Second, we use a cut-off function to obtain, from a weak solution (u, p) of Problem 1, a weak solution (ũ, p) to

-(ũ • ∇) ũ -∂ x ũ + ∆ũ -∇p = f , ∇ • ũ = 0 , (13)
with the boundary conditions ũ| ∂Ω+ = 0 , lim

x∈Ω+ x→∞ ũ(x, y) = 0 , (14)
The system (13) together with boundary conditions ( 14) is referred to as Problem 2 in what follows. Note that, in this new system we keep the divergence-free condition: the cut-off function is applied to the stream function of u. This enables to compute explicitly the source term f . So, in the third step, we show that, for ε small enough, the function f satisfies the smallness condition formulated in our previous paper [START_REF] Hillairet | Existence of stationary solutions of the Navier-Stokes equations in two dimensions in the presence of a wall[END_REF], so that there exists at least one α-solution (u α , p α ) for Problem 2 (see Section 3 for the definition of α-solutions). In the fourth step, we prove a weak-strong uniqueness result for Problem 2. Once again, this weak-strong argument applies to weak-solutions and α-solutions constructed for ε small enough. The uniqueness of solutions for Problem 2 does not directly imply the uniqueness of solutions for Problem 1, because different solutions of Problem 1 may lead to different functions f . So in a last step, we prove uniqueness of weak solutions for Problem 1 for ε small enough.

Sets and function spaces

In the whole paper, we use the standard notations for function spaces such as L p (O) for Lebesgue spaces and W m,p (O) or H m (O) for Sobolev spaces. We denote by C m (O) the spaces of continuous functions having m continuous derivative (m might be infinite). We use the subscript c to specify that function have compact support in the set O. Given a Banach space X and p ∈ X, the norm of p in X is denoted by p; X and, if X is a function space containing the constants:

p; X/R := inf{ p + c; X , c ∈ R} . (15)
This latter notation is very useful for pressures which are defined up to an additive constant in systems such as [START_REF] Babenko | Theory of perturbations of stationary flows of a viscous incompressible fluid in the case of small Reynolds numbers[END_REF].

In some proofs, we shall need a smooth covering of Ω + or of Ω + \ {(0, 1 + h)}. For this purpose, we introduce here some particular subsets of Ω + . First, we denote B(λ) the open balls with center (0, 1 + h) i.e. given λ > 0, we denote by

B(λ) = (x, y) ∈ Ω + such that |(x, y) -(0, 1 + h)| < λ .
We note in particular, that, since S is bounded, there exists ε 0 > 0, such that S ε ⊂ B(h/3) for ε < ε 0 . We keep the classical convention B((x, y), r) for balls with center (x, y) ∈ R 2 and radius r > 0. We introduce (∆ n ) n∈N an increasing covering of Ω + such that, for all n ∈ N :

• ∆ n has a smooth boundary

• B( (1 + n)h ) ⊂ ∆ n ⊂ B( (2 + n)h ).
Furthermore we define, for n ∈ N, the sets A n by A n = ∆ n \ B(2 -n h). Therefore, for all n ∈ N, A n has a smooth boundary.

Weak solutions for Problem 1

In this section, we consider the theory of weak solutions for Problem 1. The main result of this section is the following theorem.

Theorem 2 There exists a family (u ε ) ε>0 which is defined for ε sufficiently small and such that:

(i) for all ε > 0, u ε is a weak solution of Problem 1 for S ε , (ii) given η > 0 there exists 0 < ε η such that, for all ε < ε η there holds u ε ; D ≤ η, (iii) there exists a pressure p ε such that (u ε , p ε ) satisfies [START_REF] Hillairet | Existence of stationary solutions of the Navier-Stokes equations in two dimensions in the presence of a wall[END_REF] in Ω + \ S ε and, given m ∈ N, there holds:

u ε ; C m+1 (A 2 ) + p ε ; C m (A 2 )/R ≤ C m u ε ; D . ( 16
)
for some universal constant C m depending only on m.

We refer the reader to the introduction for the definition of A 2 . We introduce function spaces and the definition of weak solutions for Problem 1 just below.

The proof of this result is divided in three steps. First, we recall the method of Leray for the construction of weak solutions. We obtain in this way a family of weak solutions which satisfy a particular uniform bound with respect to the (small) size of the obstacle. Eventually, we prove that this family of solutions tends to 0 in the sense of Theorem 2.

Definition of weak solutions

To begin with, the size ε of the obstacle is fixed such that S ε ⊂ B(h/3). Let (u, p) be a smooth solution of Problem 1 for S ε . We extend u from Ω + \ S ε to the whole of Ω + by setting u = -e 1 on S ε . Let w be a smooth divergence-free vector-field with compact support in Ω + which is equal to a given constant vector field W on S ε . Then, if we multiply equation [START_REF] Gilbarg | Asymptotic properties of steady plane solutions of the Navier-Stokes equations with bounded Dirichlet integral[END_REF] by w and integrate over Ω + \ S ε we get

Ω+\Sε (∆u -∇p) • w dx = Ω+\Sε [(u + e 1 ) • ∇u] • w dx . (17)
In order to unburden notations we have suppressed in [START_REF] Sazonov | Justification of the asymptotic expansion of the solution of a two-dimensional flow problem for small Reynolds numbers[END_REF], and in what follows, the arguments of functions when no confusion is possible. Applying Green's identity to the left-hand side of (17) leads to the equality

Ω+\Sε (∆u -∇p) • w dx = ∂(Ω+\Sε) T (u, p)n • w dσ - 1 2 Ω+\Sε ∇u+ [∇u] ⊤ : ∇w+ [∇w] ⊤ dx ,
where T (u, p) = (∇u+[∇u] ⊤ )-pI and where n is the outward normal on ∂(Ω + \S ε ). Using the boundary conditions for u, which imply in particular that ∇u vanishes on S ε , and using that w = W on S ε , we obtain that u satisfies

Ω+ ∇u : ∇w dx + Ω+ [(u + e 1 ) • ∇u] • w dx = -Σ • W , (18)
with the vector

Σ = - ∂Sε T (u, p)n dσ . ( 19
)
The vector Σ is the force which the fluid exerts on S ε . If we replace, on a formal level, w by u in [START_REF] Serre | Chute libre d'un solide dans un fluide visqueux incompressible[END_REF], we obtain, as W = -e 1 in this case :

Ω+ |∇u| 2 dx + Ω+ [(u + e 1 ) • ∇u] • u dx = Σ • e 1 . (20)
Integrating by parts yields, as u is divergence-free:

Ω+ [(u + e 1 ) • ∇u] • u dx = 1 2 Ω+ u • ∇|u| 2 + ∂ x |u| 2 dx = 0 , (21)
and therefore [START_REF] Weinberger | On the steady fall of a body in a Navier-Stokes fluid[END_REF] reduces to

Ω+ |∇u| 2 dx = Σ • e 1 . ( 22
)
We conclude that if (u, p) is a solution of Problem 1 which decays sufficiently rapidly at infinity, then u satisfies the integral equation ( 18) and we have the identity (22), which means that ∇u ∈ L 2 (Ω + ).

The above discussion motivates the following functional setting for weak solutions of Problem 1. Let D be the vector space of smooth divergence-free vector-fields with compact support in Ω + . We equip D with the scalar product

((w 1 , w 2 )) = Ω+ ∇w 1 : ∇w 2 dx . ( 23
)
For functions in D we have

Ω+ ∇w 1 : ∇w 2 dx = 1 2 Ω+ (∇w 1 + [∇w 1 ] ⊤ ) : (∇w 2 + [∇w 2 ] ⊤ ) dx . (24)
Let D be the Hilbert space with respect to the scalar product (23) obtained by completion of D. Let D ε ⊂ D be the vector-fields w ∈ D which are constant on S ε and let D ε be the closure of D ε in D. On D ε we define the function Γ by

Γ : D ε -→ R 2 w -→ W = 1 |S ε | Sε w(x) dx . ( 25
)
It follows from Hardy's inequality (see Proposition 18 below) that Γ is bounded. For convenience later on we define, for all W ∈ R 2 ,

D ε W = {w ∈ D ε | w |S ε = W} , D ε W = {w ∈ D ε | w |S ε = W} . ( 26
)
Such spaces have been studied extensively in [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations[END_REF]Chapter III.5]. In particular, we emphasize that with our smoothness assumptions on ∂S ε we have that

D ε W = D ε W .
Following the work of Leray, we now define weak solutions for Problem 1:

Definition 3 A vector-field u is called a weak solution of Problem 1, if (i) u ∈ D ε -e1 , (ii) there exists a vector Σ ∈ R 2 , such that for all w ∈ D ε Ω+ ∇u : ∇w dx + Ω+ [(u + e 1 ) • ∇u] • w dx = -Σ • Γ(w) , (27)
and

Ω+ |∇u| 2 dx ≤ Σ • e 1 . (28)
The following standard lemma shows that weak solutions are well defined.

Lemma 4 Let (u, v) ∈ D 2 and let w ∈ D with Supp(w) ⊂ O ⊂⊂ Ω + 1 . Then, Ω+ [(u + e 1 ) • ∇v] • w dx = - Ω+ [(u + e 1 ) • ∇w] • v dx , (29)
and

Ω+ [(u + e 1 ) • ∇v] • w dx ≤ C(O) u; L 4 (O) v; D w; L 4 (O) + v; D w; L 2 (O) . ( 30
)
Below we show that, given a weak solution u of Problem 1, one can construct a function p such that the couple (u, p) satisfies the equation [START_REF] Gilbarg | Asymptotic properties of steady plane solutions of the Navier-Stokes equations with bounded Dirichlet integral[END_REF] in the classical sense. We will call p the pressure associated with the weak solution u. Using the ellipticity of the Stokes operator together with the smoothness of the boundary of the fluid domain, it is possible to prove that (u, p) ∈ C ∞ (Ω + \ S ε ), and that the boundary conditions [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] on S ε and on ∂Ω + are satisfied in the classical sense. Therefore, weak solutions have all the requested properties of classical solutions, and the only difficulty with weak solutions is that their rate of decay at infinity remains unknown. A bound on the decay rate, like [START_REF] Hillairet | Chute stationnaire d'un solide dans un fluide visqueux incompressible le long d'un plan incliné. partie ii[END_REF], is crucial in order to prove uniqueness of solutions.

Existence of weak solutions

In this section, we prove:

Theorem 5 There exist constants K < ∞ and ε 1 > 0 such that if ε < ε 1 , there exists at least one weak solution u for Problem 1 for S ε , satisfying the further bound u; D ≤ K.

The proof is based on the exhaustion method of Leray. Namely, we consider a nested sequence of finite domains that converge to Ω + and, for any domain of this sequence, we prove existence of one approximate weak solution having support in this domain and satisfying a suitable estimate. Our result then follows by a compactness argument. Many aspects of the proof are standard, but the uniform bound is new to our knowledge.

Sketch of proof for Theorem 5

In this proof the size ε of the obstacle is again fixed such that S ε ⊂ B(h/3). We mention further assumptions on ε when needed. We consider the sequence (∆ n ) n≥1 given in the introduction. This sequence satisfies, for all n ∈ N :

• ∆ n is a bounded open set having a smooth boundary • S ε ⊂⊂ ∆ n ⊂ ∆ n+1 • n∈N ∆ n = Ω + . Given ∆ n , we define D ε,n and D ε,n W by D ε,n = {w ∈ D ε | w | Ω + \∆n = 0} , D ε,n W = {w ∈ D ε W | w | Ω + \∆n = 0} . ( 31
)
With these conventions, the definition of approximate weak solutions for Problem 1 is:

Definition 6 Let n ∈ N. A vector-field u is called an approximate weak solution on ∆ n if: (i) u ∈ D ε,n -e1 , (ii) for all w ∈ D ε,n 0 , Ω+ ∇u : ∇w dx + Ω+ [(u + e 1 ) • ∇u] • w dx = 0 . (32)
Before giving a sketch of the proof of Theorem 5, we mention that, since D ε,n 0 is a closed subspace of D ε,n of codimension two, the Lagrange multiplier theorem implies the existence of a vector Σ ∈ R 2 , such that for all w ∈ D ε,n

Ω+ ∇u : ∇w dx + Ω+ [(u + e 1 ) • ∇u] • w dx = -Σ • Γ(w) . ( 33
)
The vector Σ is the force associated with the approximate weak solution u. Since u ∈ D ε,n , we can replace w by u in (33), and an integration by parts yields

Ω+ |∇u| 2 dx = Σ • e 1 . (34)
The energy (in)equality is therefore a consequence of Definition 6, and for this reason we do not need to impose it in the definition of approximate weak solutions in contrast with the definition of weak solutions.

The proof of Theorem 5 is based on the following two lemmas:

Lemma 7 There exists a constant ε 1 > 0 such that if ε < ε 1 there exists at least one approximate weak solution on ∆ n , for all n ∈ N.

Lemma 8 Let ε be as in Lemma 7 and n ∈ N. There exists a constant K < ∞, such that u; D + |Σ| ≤ K for any approximate weak solution u on ∆ n with associated force Σ.

Proofs for these lemmas are given in Section 2.2.2. We sketch now the remaining steps of the proof of Theorem 5 assuming that ε < ε 1 .

(i) By Lemma 7, there exists a sequence (u n , Σ n ) n≥1 such that u n is an approximate weak solution on ∆ n with associated force Σ n . By Lemma 8 this sequence is bounded in D × R 2 . One can therefore extract a subsequence (u ni , Σ ni ) i≥1 , such that (u ni ) i≥1 converges in D weakly to u and such that (Σ ni ) i≥1 converges in R 2 strongly to Σ. By Hardy's inequality the sequence (u ni ) i≥1 is bounded in H 1 (S ε ) . We can therefore extract a subsequence which converges in L 2 (S ε ) strongly to u. Since u n = -e 1 , for all n ∈ N, we find that u ∈ D ε -e1 .

(ii) Given w ∈ D ε there exists n w > 0, such that w ∈ D ε,n for all n ≥ n w . Therefore, we have for i sufficiently large

Ω+ ∇u ni : ∇w dx + Ω+ [(u ni + e 1 ) • ∇u ni ] • w dx = -Σ ni • Γ(w) . (35)
Since H 1 (∆ nw ) is compactly imbedded in L 4 (∆ nw ), we find using Lemma 4, that (35) remains valid in the limit. This shows that

Ω+ ∇u : ∇w dx + Ω+ [(u + e 1 ) • ∇u] • w dx = -Σ • Γ(w) . (36)
In the weak limit we have moreover that

Ω+ |∇u| 2 dx ≤ lim inf i→∞ Ω+ |∇u ni | 2 dx = lim inf i→∞ Σ ni • e 1 = Σ • e 1 . (37)
Combining ( 36) and (37) we conclude that there exists Σ ∈ R 2 such that, for all w ∈ D ε ,

Ω+ ∇u : ∇w dx + Ω+ [(u + e 1 ) • ∇u] • w dx = -Σ • Γ(w) , (38)
and that

Ω+ |∇u| 2 dx ≤ Σ • e 1 . (39)
This completes the proof of Theorem 5.

Proofs of Lemma 7 and Lemma 8

In these proofs n ∈ N is fixed. Since D ε,n and D ε,n 0 are closed subspaces of D ε , they are Hilbert spaces with respect to the scalar product (23). The space D ε,n -e1 is not empty. Indeed, let χ be a smooth cut-off function that is equal to one outside the disk B(2h/3) and equal to zero inside the disk B(h/3). As

S ε ⊂ B(h/3) the function Ũ-e1 (x, y) = -∇ ⊥ ((1 -χ) y) satisfies Ũ-e1 ∈ D ε,n -e1
. We note that D ε,n -e1 is an affine subspace of D ε,n with direction D ε,n 0 . For technical reason (see ( 41)), we also introduce U -e1 the unique minimizer of the D-norm amongst the velocity-fields in D ε,n -e1 . This velocity field satisfies :

1. D ε,n -e1 = U -e1 + D ε,n 0
2. ((U -e1 , w)) = 0 for all velocity fields w ∈ D ε,n 0 .

We now reformulate the existence of an approximate weak solution on ∆ n as a fixed point problem for a functional equation. First, we note that Lemma 4 implies that for all u ∈ D ε,n -e1 , the map

D ε,n 0 → R w → Ω+ [(u + e 1 ) • ∇u] • w dx ,
((b * n (u), w)) = Ω+ [(u + e 1 ) • ∇u] • w dx , ∀ w ∈ D ε,n 0 . ( 40
)
With these definitions we find, on the one hand, that u is an approximate weak solution on ∆ n if and

only if u = U -e1 + v, with v a solution of the functional equation v = b * n (U -e1 + v) , v ∈ D ε,n 0 . (41)
On the other hand, (30) together with (29) imply that b * n is continuous on D ε,n -e1 equipped with the L 4 (∆ n )norm. Using that H 1 0 (∆ n ) is compactly imbedded in L 4 (∆ n ) yields that b * n is completely continuous, i.e., for any given bounded sequence (v i ) i≥1 in D ε,n 0 , there exists a subsequence (v ij ) j≥1 such that the sequence (b * n (U -e1 + v ij )) j≥1 converges strongly in D ε,n 0 . Hence, the Leray-Schauder fixed point theorem (see [START_REF] Kavian | Introduction à la théorie des points critiques et applications aux problèmes elliptiques[END_REF] or [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Theorem 11.6,p. 286] for more details) guarantees the existence of a solution of (41) by proving a suitable estimate on a priori solutions to an auxiliary problem. This estimate is the content of the following proposition.

Proposition 9 There exist constants ε 1 > 0 and C < ∞, such that for all ε < ε 1 , λ ∈ [0, 1] and all

(u, Σ) ∈ D ε,n -e1 × R 2 which satisfy Ω+ ∇u : ∇w + λ Ω+ [(u + e 1 ) • ∇u] • w = -Σ • Γ(w) , ∀ w ∈ D ε,n , (42)
we have the bound u;

D + |Σ| ≤ C.
Because of the Leray-Schauder theory, this lemma implies Lemma 7. Then, Lemma 8 is proved assuming ε < ε 1 and applying this proposition to the constructed approximate solution (in this case λ = 1).

Proof of Proposition 9. First we note that given (u, Σ, n, λ) as in Proposition 9 we can set w = u in (42), and we obtain (34). Hence, it suffices to find a bound on Σ. For this purpose, we introduce an additional family of cut-off functions χ δ . This family truncates in balls around the point (0, 1 + h). Namely, let ζ : R → R be a smooth function such that

ζ(s) = 1 , ∀ s < 0 , ζ(s) = 0 , ∀ s > 1 . (43)
Then, given 0 < δ < h/3, we set for (x, y) ∈ Ω + ,

χ δ (x, y) = ζ |(x, y -1 -h)| δ -1 . (44)
With this definition, we have χ δ = 1 in B(δ) while χ δ = 0 in the exterior of B(2δ). Now, given (u, Σ, n, λ) and an obstacle S ε , we set δ(ε) = λ 0 ε, with λ 0 = sup{|(x, y)|, (x, y) ∈ S}, and define, for arbitrary W ∈ R 2 , the test-function:

w ε = -∇ ⊥ χ δ(ε) (x, y) W ⊥ • ((x, y) -(0, 1 + h)) . (45)
Since S ε tends homothetically to a point when ε → 0, we can choose ε 0 (say ε 0 = h/(3λ 0 ) for instance) such that w ε is equal to W on S ε and equal to zero outside B(2h/3) for ε < ε 0 . Thus, we can use w ε as a test-function in (42). By construction of w ε , there exists a universal constant C 1 such that

w ε ; D + w ε ; L ∞ (R 2 ) ≤ C 1 |W| , (46)
and we get from (42) the inequality

|Σ • W| ≤ u; D w ε ; D + λ u + e 1 ; L 2 (B(2δ(ε))) u; D w ε ; L ∞ (Ω + ) ≤ C 1 |W| u; D + u + e 1 ; L 2 (B(2δ(ε))) u; D .
Since u |S ε = -e 1 , Poincaré's inequality implies that there exists a constant C 2 such that

u + e 1 ; L 2 (B(2δ(ε))) ≤ C 2 u; D .
A scaling argument shows C 2 = εC 2 with a constant C 2 independent of ε and u (see [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations[END_REF]Exercise 4.10] for a construction of C 2 ). Therefore,

|Σ| ≤ C 1 u; D + C 2 ε u; D 2 . (47)
From ( 47) and (34) we find that if ε satisfies moreover ε < 1/(2C 1 C 2 ), we have a bound on |Σ| and u; D which is independent of n, λ, and ε, namely

u; D ≤ |Σ| 1/2 ≤ 2C 1 . ( 48
)

Weak solutions for obstacles of vanishing size

From now on, we choose once and for all (u ε ) ε<ε1 a bounded family of D such that u ε is a weak solution of Problem 1 for S ε for all ε < ε 1 . Such a sequence exists according to Theorem 5. In this section we complete the proof of Theorem 2 by showing that the sequence (u ε ) ε<ε1 converges to zero when the size of the obstacle tends to zero. As a by-product, we also obtain the pressure p associated with a weak solution.

The convergence is proved in the family of spaces (C m (A n )) (m,n)∈N 2 . We refer the reader to Section 1.1 for the definition of sets (A n ) n∈N . We recall here that they satisfy the following fundamental properties

• for all n ∈ N, A n ⊂ A n ⊂ A n+1 ,
• for all n ∈ N, A n has a smooth boundary,

• n∈N A n = Ω + \ (0, 1 + h).
Theorem 2 is a straightforward consequence of the following two lemmas which we prove in the following subsections.

Lemma 10 Given η > 0 there exists 0 < ε η such that u ε ; D ≤ η for all ε < ε η .

Lemma 11 Let (n, m) ∈ N 2 and ε < ε 1 such that S ε ⊂⊂ [ R 2 + \ A n+m+1 ].
Then, there exists a constant C m,n , depending only on m and n, for which any weak solution u of Problem 1 for S ε such that u; D ≤ 1 satisfies (i) there exists a pressure p such that (u, p) is solution to [START_REF] Gilbarg | Asymptotic properties of steady plane solutions of the Navier-Stokes equations with bounded Dirichlet integral[END_REF] (ii) the following estimate holds true

u; H m+1 (A n ) + p; H m (A n )/R ≤ C m,n u; D . (49)
Remark:

One might be tempted to assume that the smallness estimate of Theorem 2 is straightforward, since the fluid is moving only due to the no-slip boundary condition on ∂S ε . The smaller the body, the smaller should be the fluid flow which is induced by this boundary condition so that the flow should be zero in the limit of a body of vanishing size. The following scaling argument shows that, because of the Stokes paradox, things are not quite as simple. Let (u ε ) ε>0 a family of weak solutions for S ε and

Ω ε + = {(x, y) ∈ R 2 | (εx, εy -(h + 1)) ∈ Ω + }, and let v ε (x, y) = u ε (εx, εy -(h + 1)
) and q ε (x, y) = p ε (εx, εy -(h + 1)). This scaling does not affect the D-norm, so that ∇v ε ; L 2 (Ω ε + ) = ∇u ε ; L 2 (Ω + ) . Therefore, if the family (u ε ) ε>0 is bounded in D, the family ( ∇v ε ; L 2 (Ω ε + ) ) ε>0 is also bounded, and the functions v ε satisfy in Ω ε + the equation

-ε (v ε • ∇) v ε -ε∂ x v ε + ∆v ε -∇q ε = 0 , ∇ • v ε = 0 , (50)
with the boundary condition v ε = -e 1 on ∂S 1 . Using the same line of arguments as in the previous section we can therefore extract a subsequence converging in the topology induced by the D-norm to some function v for which ∇v; L 2 (R 2 ) is finite and which solves the Stokes equations in R 2 \ S 1 with the boundary condition v = -e 1 on ∂S 1 . By the Stokes paradox this implies that v = -e 1 (see [9, Theorem 2.2 p. 253]), and therefore the sequence v ε does not converge to zero with ε. Note that this remark does not contradict Theorem 2. It only means that, when ε goes to zero, v ε does take values close to -e 1 on a part of the domain that increases in size and covers eventually all of R 2 . The diameter of this region remains however small compared with 1/ε, and its size therefore converges to zero in the un-scaled variables.

Proof of Lemma 10

We prove Lemma 10 by contradiction. First, we assume that there exists η 0 > 0, sequences (ε i ) i∈N ∈ (0, ε 1 ) N and (u i , Σ i ) i∈N ∈ (D × R 2 ) N such that lim ε i = 0, that u i = u εi has associated force Σ i and is such that u i ; D ≥ η 0 for all i ∈ N. Then, the sequence (u i , Σ i ) i∈N is bounded. This implies the existence of a pair (u, Σ) ∈ D × R 2 and of a subsequence (u ij , Σ ij ) j∈N such that u ij ⇀ j→∞ u weakly in D and such that Σ ij → j→∞ Σ strongly in R 2 .

We now proceed as in the proof of Proposition 9. Let χ δ be the cut-off function defined in (44) and define, as in (45) for 0 < δ < h/3 and arbitrary W ∈ R 2 the test-function w δ

w δ (x, y) = -∇ ⊥ χ δ (x, y) W ⊥ • ((x, y) -(0, 1 + h)) ,
which has support in B(2h/3). As in (46) there exists a universal constant C 1 < ∞ such that

w δ ; D + w δ ; L ∞ (Ω + ) ≤ C 1 |W| . ( 51
)
Since lim ε i = 0, there exists i δ such that for i ≥ i δ the function w δ is an admissible test-function, and we have :

Ω+ ∇u i : ∇w δ dx + Ω+ [(u i + e 1 ) • ∇u i ] • w δ dx = -Σ i • W .
In the limit as i goes to infinity we therefore get

Ω+ ∇u : ∇w δ dx + Ω+ [(u + e 1 ) • ∇u] • w δ dx = -Σ • W , so that, |Σ| ≤ C 1 ∇u; L 2 (B(2δ/3)) 1 + u + e 1 ; L 2 (B(2δ/3)) .
Letting δ go to 0, yields Σ = 0, i.e., Σ i → i→∞ 0 which by the energy estimate (28) implies that lim u i ; D = 0, in contradiction with our assumption.

Proof of Lemma 11

Let (n, m) ∈ N 2 and ε, u be given as in Lemma 11. At first, we recall how to construct the pressure associated with u. We test (27) with smooth divergence free vector-fields having compact support in Ω + \ S ε . This shows that u is a generalized solution in the sense of [ Since, for all Ω ′ ⊂⊂ (Ω + \ S ε ), we have f ∈ H -1 (Ω ′ ) with the bound

f ; H -1 (Ω ′ ) ≤ C(Ω ′ ) u; H 1 (Ω ′ ) 2 + u; D (52)
we can apply [8, lemma IV.1.1,p. 186] to construct a function p ∈ L 2 loc (Ω + \ S ε ) such that, in the sense of distributions, ∆u

-∇p = f , ∇ • u = 0 , (53)
in Ω + \ S ε . Classically, this pressure p is unique up to a finite number of constants (equal to the number of connected components of Ω + \ S ε ). We call p the pressure associated with u and we indeed have that (u, p) satisfies [START_REF] Gilbarg | Asymptotic properties of steady plane solutions of the Navier-Stokes equations with bounded Dirichlet integral[END_REF].

The remainder of Lemma 11 is obtained via an induction argument (with respect to m ∈ N). Namely, we prove that, for all k ≤ m the following statement holds true:

There exist constants C m,k depending only on m and k such that :

u; H k+1 (A n+m-k ) + p; H k (A n+m-k )/R ≤ C m,k u; D . (P k )
Proof, initialization: The restriction of (u, p) to A n+m is a solution of the Stokes equations with source term f =(u + e 1 ) • ∇u and boundary data u = u | ∂A n+m . Hence, combining [8, theorem 1.1 p. 188] and (52), and using that u; D ≤ 1, we find that

u; H 1 (A n+m ) + p; L 2 (A n+m )/R ≤ C 0 u; D + u; D 2 ≤ C m,0 u; D . (54)
Our statement holds true for k = 0.

Before the inductive step of the proof, we need to compute an L ∞ estimate on u inside A n+m . To this end, we recall that we have by construction that A n+m ⊂ ( A n+m ∩ Ω + ) ⊂ A n+m+1 . Hence there exists a smooth truncation function χ ∈ C ∞ (Ω + ), such that χ = 1 on A n+m and χ = 0 outside A n+m+1 . We make the dependance of χ upon n implicit for legibility. Let ũ = χu and p = χp. Then (ũ, p) is a solution of the Stokes system

∆ũ -∇p = f on A n+m+1 , ∇ • ũ = g on A n+m+1 , with ũ = 0 on ∂A n+m+1 , (55)
where

f = χ(u + e 1 ) • ∇u + 2∇χ • ∇u + (∆χ) u -p∇χ , g = u • ∇χ . (56)
Using the bound (54) on (u, p), we find that, for a given q < 2 (say q = 3/2), there holds f ∈ L q (A n+m+1 ) and g ∈ W 1,q (A n+m+1 ). Furthermore we have,

f; L q (A n+m+1 ) + g; W 1,q (A n+m+1 ) ≤ C m,q u; H 1 (A n+m+1 ) + u; H 1 (A n+m+1 ) 2 + p; L 2 (A n+m+1 ) .
Applying [8, Exercise IV.6.2, p. 232] we get that ũ ∈ W 2,q (A n+m+1 ) and p ∈ W 1,q (A n+m+1 ), and that ũ; W 2,q (A n+m+1 ) + p;

W 1,q (A n+m+1 )/R ≤ C m,q u; H 1 (A n+m+1 ) + u; H 1 (A n+m+1 ) 2 + p; L 2 (A n+m+1 ) . ( 57
)
Note that we can always replace p by p + c before truncation, so that we can replace p; L 2 (A n+m+1 ) by p; L 2 (A n+m+1 )/R in the right-hand side of the last inequality, as well as in the estimates that follow. Combining (57) with (54), we get ũ; W 2,q (A n+m+1 ) + p; W 1,q (A n+m+1 )/R ≤ C m,q u; D .

Therefore, we have in particular that u

∈ W 2,q (A n+m ) ⊂ L ∞ (A n+m ) with u; L ∞ (A n+m )) ≤ K m,n u; D .

Proof, inductive step:

Assuming that for k ≤ m, there exist constants C m,k depending only on m and k such that :

u; H k+1 (A n+m-k ) + p; H k (A n+m-k )/R ≤ C m,k u; D ,
we apply again the same truncation technique as described above. Namely, we introduce χ ∈ C ∞ (Ω + ) a smooth truncation function such that χ = 1 on A n+m-k-1 and χ = 0 outside A n+m-k and we let ũ = χu and p = χp. Then (ũ, p) is a solution of the Stokes system (55), on A n+m-k with homogeneous boundary condition. Hence, we get by the ellipticity of the Stokes operator that ũ;

H k+2 (A n+m-k ) + p; H k+1 (A n+m-k )/R ≤ Cm,k f ; H k (A n+m-k )) + g; H k+1 (A n+m-k ) .
(59) We also have

f; H k (A n+m-k )) + g; H k+1 (A n+m-k ) ≤ Cm,k u; L ∞ (A n+m ) u; H k+1 (A n+m-k ) + u; H k+1 (A n+m-k ) 2 + u; H k+1 (A n+m-k ) + p; H k (A n+m-k )/R ,
and therefore there exists, by the induction assumption, a constant C m,k+1 , such that

u; H k+2 (A n+m-k-1 ) + p; H k+1 (A n+m-k-1 )/R ≤ C m,k+1 u; D .
This completes the inductive step and ends the proof.

3 Behavior of weak solutions at large distance from the obstacle.

In this section we show that the weak solutions of Problem 1 constructed above decay at infinity with the expected rate. Namely, we prove: Theorem 12 There exists ε e > 0, such that, for all ε < ε e , the weak solution u ε satisfies the decay estimate,

|u ε (x, y)| ≤ C ε y 3 2 . ∀ (x, y) ∈ Ω + \ S ε . ( 60
)
for some C ε < ∞.

This result is proved in three steps by comparing weak solutions with α-solutions. First, we show how to construct solutions for Problem 2 by truncating a weak solution for Problem 1. We prove in particular that, when the solid is sufficiently small, weak solutions to Problem 1 provided by Theorem 2 yield weak solutions to Problem 2 with a source term which is arbitrary small, so that we are able to construct α-solutions. We conclude by proving that any weak solution coincides with the α-solution when the source-term is sufficiently small.

Truncation procedure

We start this section by describing how to construct a solution for Problem 2 by truncating a weak solution for Problem 1. Let

Π : D ∩ C ∞ (Ω + \ B(h/4)) -→ C ∞ (Ω + \ B(h/4)) w -→ ψ(x, y) = - y 1 w(x, z) • e 1 dz .
The divergence-free condition satisfied by w implies that ∇ ⊥ Π[w] = w, and that

Π[w](x, y) = γ w ⊥ • dγ ,
for any path γ such that γ(0) = (0, 1) and γ(1) = (x, y). Hence, it is sufficient that w is smooth in Ω + \ B(h/3) in order for the associated stream-function Π[w] to be smooth in Ω + \ B(h/4). More precisely, for all m ∈ N, there exists a constant C m , such that,

Π[w]; C m (B(2h/3) \ B(h/3)) ≤ C m w; C m-1 (A 2 ) ∀ w ∈ D ∩ C ∞ (Ω + \ B(h/4)) . (61)
We introduce a truncation function χ ∈ C ∞ (R 2 ) which satisfies

χ(x, y) =    0 if |(x, y) -(0, 1 + h)| <h/3 ∈ [0, 1], if |(x, y) -(0, 1 + h)| ∈ (h/3, 2h/3) 1 if |(x, y) -(0, 1 + h)| > 2h/3
and define truncation operators T v and T π for the velocity and the pressure as follows

T v : D ∩ C ∞ (Ω + \ B(h/4)) -→ C ∞ (Ω + ) w -→ ∇ ⊥ [χΠ[w]] and T π : C ∞ (Ω + \ B(h/4)) -→ C ∞ (Ω + ) q -→ χq
These operators are well-defined, since the truncation function χ vanishes identically in B(h/4). For any w ∈ D ∩ C ∞ (Ω + \ B(h/4)) and q ∈ C ∞ (Ω + \ B(h/4)), we have by a straightforward application of ( 61) that

(T-i) T v [w] ∈ D ∩ C ∞ (Ω + ), and T π [q] ∈ C ∞ (Ω + ), (T-ii) T v [w] = w and T π [q] = q in Ω + \ B(2h/3), (T-iii) Given m ∈ N, there exists a constant C m such that T v [w]; C m+1 (B(2h/3) \ B(h/3)) ≤ C m w; C m+1 (A 2 ) , T π [q]; C m (B(2h/3) \ B(h/3)) ≤ C m q; C m (A 2 ) . Next for (w, q) ∈ (D ∩ C ∞ (Ω + \ B(h/4))) × C ∞ (Ω + \ B(h/4)) we define the function f ∈ C ∞ (Ω + \ B(h/4)) by f = (w + e 1 ) • ∇w -∆w + ∇q ,
and we define f = 0 inside B(h/4). Finally we define the function T N S[w, q] on Ω + by

T N S[w, q] = -χf + [( w + e 1 ) • ∇ w -∆ w + ∇q] ,
where ( w, q) = (

T v [w], T π [q]). Given (w, q) ∈ (D ∩ C ∞ (Ω + \ B(h/4))) × C ∞ (Ω + \ B(h/4
)), the above properties of the truncation operators T v and T π imply that the function T N S[w, q] satisfies:

(S-i) T N S[w, q] is smooth and has compact support in B(2h/3) \ B(h/3),

(S-ii) The truncated functions w = T v [w] and q = T π [q] satisfy:

( w + e 1 ) • ∇ w -∆ w + ∇q = χf + T N S[w, q] , in Ω + , ∇ • w = 0 , in Ω + .
with f = (w + e 1 ) • ∇w -∆w + ∇q, (S-iii) Given m ∈ N, there exists a constant C m such that

T N S[w, q]; C m (B(2h/3) \ B(h/3)) ≤ C m 1 + w; C m+2 (A 2 ) w; C m+2 (A 2 ) + q; C m+1 (A 2 )/R . ( 62
)
Applying this construction to any weak solution of Problem 1 yields a solution of Problem 2 for the source term computed with T N S. To prepare the last weak-strong uniqueness argument of this section, we show that such solutions of Problem 2 obtained by truncation satisfy a further energy property. This is the content of the next proposition.

Proposition 13 Given ε such that S ε ⊂⊂ B(h/4) and a weak solution u of Problem 1 for S ε with associated pressure p, the vector-field ũ = T v [u] satisfies (i) ũ ∈ D, (ii) for all w ∈ D, there holds:

Ω+ ∇ũ : ∇w dx + Ω+ [(ũ + e 1 ) • ∇ũ] • w dx = Ω+ f • w dx , ( 63
)
and

Ω+ |∇ũ| 2 dx ≤ Ω+ f • ũ dx , ( 64
)
with f = T N S[u, p].
As for the case of Problem 1, we emphasize that f and the test-functions w have compact support so that the integrals in ( 63) and (64) are well-defined. A velocity-field ũ satisfying (i) and (ii) for a given f ∈ C ∞ c (Ω + ) is called a weak solution for Problem 2 with source term f . Proof. First we recall that ellipticity estimates for the Stokes system imply that any weak solution u for S ε with associated pressure p satisfies

(u, p) ∈ D ∩ C ∞ (Ω + \ B(h/4)) × C ∞ (Ω + \ B(h/4)) .
Hence ũ = T v [u], p = T π [p] and T N S[u, p] are well-defined. Moreover (u, p) is a classical solution of the Navier Stokes equations outside S ε and in particular in Ω + \ B(h/4).

In order to show that ũ is a weak solution of Problem 2 we first use (T-i) to conclude that ũ ∈ D. Then, since (u, p) is a classical solution to the Navier Stokes equations in Ω + \ B(h/4), the second point (S-ii) implies that we have

(ũ + e 1 ) • ∇ũ -∆ũ + ∇p = T N S[u, p]
in Ω + . If we multiply this equality by w ∈ D and integrate by parts we obtain (63) for ũ, with

f = T N S[u, p].
The main difficulty of the proof is to obtain the energy estimate (64) for ũ. For this purpose, we multiply the Navier Stokes equations satisfied by (ũ, p) on B(5h/6) by ũ. Integrating by parts yields

B(5h/6) |∇ũ| 2 dx = B(5h/6) f • ũ dx + ∂B(5h/6) T (ũ, p)n • ũ + |ũ| 2 2 (ũ + e 1 ) • n dσ . (65)
Next, multiplying the Navier Stokes equation satisfied by (u, p) on B(5h/6) \ S ε by u and integrating by parts gives

B(5h/6) |∇u| 2 dx = Σ • e 1 + ∂B(5h/6) T (u, p)n • u + |u| 2 2 (u + e 1 ) • n dσ , (66)
with Σ the associated force applied on S ε . By definition, we have

Ω+ |∇u| 2 dx ≤ Σ • e 1 . (67)
Subtracting (66) from (67) yields

Ω+\B(5h/6) |∇u| 2 dx ≤ - ∂B(5h/6) T (u, p)n • u + |u| 2 2 (u + e 1 ) • n dσ . (68)
Since outside B(2h/3) we have by construction that u = ũ and p = p, we get by combining (65) and ( 68)

Ω+ |∇ũ| 2 dx ≤ Ω+ T N S[u, p] • ũ dx .
This completes the proof.

Existence of α-solutions

The second step of the proof of Theorem 12 is to construct an α-solution for Problem 2 with the source term f ε obtained by truncation of a weak solutions u ε . To keep this paper self-contained, we recall the definition and the main properties of α-solutions. See [START_REF] Hillairet | Existence of stationary solutions of the Navier-Stokes equations in two dimensions in the presence of a wall[END_REF], for details.

Definition 14

We define for fixed α, r ≥ 0 the function µ α,r :

R×[1, ∞) → (0, ∞) by µ α,r (k, t) = 1 1 + (|k| t r ) α . (69)
We define, for fixed α ≥ 0, and p, q ≥ 0, B α,p,q to be the Banach space of functions

f ∈ C(R 0 × [1, ∞), C), R 0 = R \ {0}, for which the norm f ; B α,p,q = sup t≥1 sup k∈R0 f (k, t) 1 t p µ α,1 (k, t) + 1 t q µ α,2 (k, t) is finite. Furthermore, we set U α = B α, 5 2 ,1 × B α, 1 2 ,0 × B α, 1 2 ,1 .
Formally, it is possible to compute the velocity-field u = (u, v), of a solution (u, p) to Problem 2 with source term f := (F 1 , F 2 ), as the inverse fourier transform, with respect to x, of a pair (û, v) :

u(x, y) = R e ikx û(k, y)dk, v(x, y) = R e ikx v(k, y)dk, ∀ (x, y) ∈ Ω + ,
the pair (û, v) satisfying:

û(k, y) = -η(k, y) + φ(k, y) v(k, y) = ω(k, y) + ψ(k, y), ∀ (k, y) ∈ R × (1, ∞),
with (ω, η, φ, ψ) a solution to

∂ y ω = -ik η + Q1 , (70)
∂ y η = (ik + 1)ω + Q0 , (71)
∂ y ψ = ik φ -Q1 , (72)
∂ y φ = -ik ψ + Q0 . (73)
The source terms ( Q0 , Q1 ) is computed as follows :

Q0 = 1 2π (û * ω) + F2 , (74)
Q1 = 1 2π (v * ω) -F1 . (75)
Here F1 and F2 stand for the fourier transform, with respect to x, of F 1 and F 2 respectively. When the solution (ω, û, v) given by the solution of (70)-( 75) satisfies (ω, û, v) ∈ U α with α > 3, the velocity-field u = (u, v) constructed this way is a weak solution to Problem 2 in the sense of Proposition 13.

In [START_REF] Hillairet | Existence of stationary solutions of the Navier-Stokes equations in two dimensions in the presence of a wall[END_REF] the following existence theorem is proved:

Theorem 15 Let α > 3, f ∈ C ∞ c (Ω + )
, and let f be the Fourier transform with respect to x of f . If f; W α is sufficiently small, then there exists an α-solution ū being the inverse Fourier transform (with respect to x) of û ∈ U α , with û satisfying û; U α ≤ C α f ; W α , for some constant C α depending only on the choice of α.

The α-solution ū satisfies:

1. ū ∈ H 1 0 (Ω + ),
2. there exists a constant C such that:

ū; H 1 0 (Ω + ) ≤ C û; U α , and |ū(x, y)| ≤ C û; U α y 3/2 , ∀ (x, y) ∈ Ω + .
We now show that when the obstacle size is small, the function f ε = T N S[u ε , p ε ] satisfies the condition of Theorem 15. This reads: Lemma 16 Given α > 3, there exists ε α > 0 such that, for all ε < ε α the weak solution u ε with associated pressure p ε is such that Problem 2 with source term f ε = T N S [u ε , p ε ] admits an α-solution ūε . Moreover, there exists C α < ∞ depending only on α such that the α-solution satisfies ûε ; U α ≤ C α u ε ; D , where ûε is the Fourier transform of ūε with respect to x.

Proof. First, let η 0 be a sufficiently small parameter to be fixed later on and denote by m the integer part of α + 1. Applying Theorem 2, there exists ε α,η such that for all ε < ε α,η the weak solution u ε with associated pressure p ε satisfy :

u ε ; C m+2 (A 2 ) + p ε ; C m+1 (A 2 )/R ≤ C m u ε ; D ≤ C α η 0 .
As a consequence, the source-term f ε := T N S[u ε , p ε ] obtained after truncation satisfies (see (S-i) and (S-iii)):

• f ε has compact support in B(3h/4) \ B(h/3) • f ε ; C m+2 (A 2 ) ≤ K α u ε ; D ≤ K α η 0
Denoting by f any component of f ε we apply then the following classical computation. The function f ∈ C ∞ c (Ω + ) has support in B(2h/3). Hence, the Fourier transform f of f is well-defined and continuous on Ω + . Moreover we have, for y ≥ 1 and k ∈ R,

f (k, y) = 2h/3 -2h/3 e ikx f (x, y) dx .
Integration by parts implies the existence of a constant C such that, for y ≥ 1 and

k ∈ R 0 , f (k, y) ≤ C f ; C 0 (Ω + ) , and f (k, y) ≤ C f ; C m (Ω + ) |k| m .
Using that f has compact support in y, we obtain that

f (k, y) ≤ C f ; C m (Ω + ) y p (1 + (|k| y) m ) + f ; C m (Ω + ) y q (1 + (|k| y 2 ) m ) ,
for arbitrary m ∈ N. In particular, there holds:

f ; B α,p,q ≤ K α p,q f ; C α (Ω + ) . (76)
Keeping the previous notations for the Fourier transform, we have

f ε ; W α ≤ K α u ε ; D ≤ K α η 0 .
Finally, for η 0 sufficiently small we apply Theorem 15. This yields an α-solution ūε for Problem 2 with source term f ε . Furthermore, this solution satisfies:

ûε ; U α ≤ Cα f ε ; W α ≤ Cα K α u ε ; D .
This completes the proof.

Weak-strong uniqueness of solution for Problem 2

So far, we have shown that a weak solution u of Problem 1 for S ε with associated pressure p provides a weak solution ũ of Problem 2 for source term f = T N S [u, p] by truncation. We have also shown that, for small obstacles, we can construct an α-solution ūε for source terms f ε obtained after truncation of u ε . In this section, we prove:

Theorem 17 Given α > 3, there exists η α > 0 such that, given an α-solution ū for source-term f ∈ C ∞ c (Ω + ) such that û; U α < η α , any weak solution ũ of Problem 2 with source term f coincides with ū.

Consequently, choosing α = 4, for instance, and a sufficiently small obstacle, we have, by Lemma 16 that ûε ; U α ≤ η α . Hence, we can apply this theorem to ũε := T v [u ε ]. This yields that ũε coincides with ūε . Since by construction the weak solution ũε coincides with u ε outside a compact set, u ε also coincides with the α-solution ūε outside a compact set and inherits its asymptotic properties. Thus, this weak-strong uniqueness result ends the proof of Theorem 12.

Theorem 17 is a generalization of [START_REF] Hillairet | Existence of stationary solutions of the Navier-Stokes equations in two dimensions in the presence of a wall[END_REF]Theorem 8], where it was shown that for small f any weak solution ũ ∈ H 1 0 (Ω + ) of Problem 1, is an α-solution. This theorem was not general enough for the present purposes because weak solutions that are obtained by truncation merely satisfy ũ ∈ D. The remainder of this section is devoted to this new uniqueness proof.

Sketch of proof for Theorem 17

We set α > 3 and fix ū an α-solution with a source-term f . It has been shown in [START_REF] Hillairet | Existence of stationary solutions of the Navier-Stokes equations in two dimensions in the presence of a wall[END_REF]Section 3], that such an α-solution is also a weak solution of Problem 2 for f . Hence, we have ( 63 We now assume that Next we assume that for any α-solution ū we have:

Ω+ ∇ũ : ∇ū dx + Ω+ (ũ + e 1 ) • ∇ũ • ū dx = Ω+ f • ū dx , (H1) and that
Ω+ (v + e 1 ) • ∇w • ū dx = - Ω+ (v + e 1 ) • ∇ū • w dx, ∀ (v, w) ∈ D 2 . ( H3
)
This assumption is also proved below. Together with the previous inequality we get ũ -ū; D 2 ≤ -

Ω+ (ũ + e 1 ) • ∇ū • ũ dx + Ω+ (ū + e 1 ) • ∇ū • ũ dx ≤ Ω+ (ū -ũ) • ∇ū • ũ dx ≤ Ω+ (ū -ũ) • ∇ū • (ũ -ū) dx ,
as it yields from (H3) :

Ω+ (ū -ũ) • ∇ū • ū dx = 0.
Finally, we assume that for any α-solution ū we have:

Ω+ v • ∇ū • w dx ≤ K û; U α v; D w; D , ∀(v, w) ∈ D 2 , ( H4
)
and we get ũ -ū;

D 2 ≤ C û; U α ũ -ū; D 2 .
For η α sufficiently small we have C û ; U α < 1/2, so that ũ -ū; D = 0. This completes the proof up to the technical points (H1)-(H4) which are proved in the following sections.

Proof of (H2) and (H4)

We first establish some additional conditions for the trilinear form

Ω+ (u + e 1 ) • ∇v • w dx (77)
to be well defined. The main tool is the Hardy inequality for functions in D:

Proposition 18 For all w ∈ D,

Ω+ |w(x, y)| 2 (y -1) 2 dx dy ≤ 4 w; D 2 .
Therefore, we have the following continuity result for the trilinear form:

Proposition 19 There exists a constant C such that for all (v, w) ∈ D and all

ū ∈ H 1 loc (Ω + , dx) ∩ L 2 (Ω + , dx) such that yu ; L ∞ (Ω + ) < ∞ , and ∇ū ∈ H 1 loc (Ω + , dx) ∩ L 2 (Ω + , y 2 dx) such that y 2 ∇ū ; L ∞ (Ω + < ∞ , respectively, we have Ω+ (v + e 1 ) • ∇w • ū dx ≤ C (1 + v; D ) w; D ū; L 2 (Ω + ) + yū; L ∞ (Ω + ) , (78)
and

Ω+ (v + e 1 ) • ∇ū • w dx ≤ C (1 + v; D ) w/y; L 2 (Ω + ) y∇ū; L 2 (Ω + ) + y 2 ∇ū; L ∞ (Ω + ) . (79)
Proof. We denote by I 1 and I 2 the integrals in (78) and (79), respectively. Let (ū, v, w) ∈ H

1 loc (Ω + ) × D 2 . If ū; L 2 (Ω + ) + yū; L ∞ (Ω + ) < ∞, we can split I 1 into two integrals I 1 = Ω+ v • ∇w • ū dx - Ω+ ∂ x w • ū dx . (80)
The second integral on the right hand side of (80) can be bounded by the Cauchy Schwarz inequality.

For the first integral we have

Ω+ v • ∇w • ū dx = Ω+ v y • ∇w • yū dx (81) ≤ v/y; L 2 (Ω + ) ∇w; L 2 (Ω + ) yū; L ∞ (Ω + ) ,
and we obtain the bound on I 1 by applying the Hardy inequality (note that y ≥ y -1). The integral I 2 is bounded similarly.

This proposition is suitable for α-solutions. Indeed, if ū = (u, v), is an α-solution, for α > 3, denoting by û = (û, v) (respectively ûx = (û x , vx ) and ûy = (û y , vy ) ) the Fourier transform with respect to x of ū (respectively ∂ x ū and ∂ y ū) there holds :

• (û, v) ∈ B α,1/2,0 × B α,1/2,1 , • (û x , vx ) ∈ B α-1,3/2,2 × B α-1,3/2,3 • (û y , vy ) ∈ B α-1,3/2,1 × B α-1,3/2,2 with : ûx ; B α-1,3/2,2 × B α-1,3/2,3 + ûy ; B α-1,3/2,1 × B α-1,3/2,2 ≤ C α û; U α .
(see [13, pp. 685-686] for more details). Moreover, we have: Proposition 20 Let p, q > 0, α > 1, s ∈ [2, ∞] and let f be the inverse Fourier transform of f ∈ B α,p,q .

Then there exists a constant C depending only on α and s such that, for any y > 1, f (•, y) ∈ L s (R), and

f (•, y); L s (R) ≤ C y e f ; B α,p,q .
where e = min(1 -1/s + p, 2(1 -1/s) + q).

Proof. Given f ∈ B α,p,q , and y > 1, the function f (x, y) is the inverse Fourier transform with respect to k of the function f (k, y), which is continuous on R 0 and satisfies for k

∈ R 0 | f (k, y)| ≤ 1 y p (1 + (|k|y) α ) + 1 y q (1 + (|k|y 2 ) α f ; B α,p,q . Therefore f (•, y) ∈ L r (R) for all r ∈ [1, 2] so that f ∈ L s (R) for all s ∈ [2, ∞]. Moreover, given s ≥ 2 and r ≤ 2 the conjugate exponent, i.e. 1 r + 1 s = 1, there exists a constant C s , such that f (•, y); L s (R) ≤ C s f (•, y); L r (R) . (82)
By a scaling argument, we have

∞ -∞ 1 (1 + |k|y) rα dk ≤ C 1 r,α y , and
∞ -∞ 1 (1 + |k|y 2 ) rα dk ≤ C 2 r,α y 2 ,
which, together with (82) gives

f (•, y); L s (R) ≤ C α,r 1
y p+ 1 r + 1 y q+ 2 r f ; B α,p,q ,
as required. Proposition 20 implies that, if f ∈ B α,p,q for p > 0, q > 0 and α > 1, then f ∈ L 2 (Ω + ) and we have, for all (x, y)

∈ Ω + , |f (x, y)| ≤ C y min(p+1,q+2) f ; B α,p,q .
In particular, for an α-solution ū = (u, v), we have ū ∈ L 2 (Ω + ), y∇ū ∈L 2 (Ω + ), and according to the remark before the proposition, there holds :

|u(x, y)| + |v(x, y)| ≤ C y 3/2 û; U α , ( 83
) |∇u(x, y)| + |∇v(x, y)| ≤ C y 5/2 û; U α , (84)
for all (x, y) ∈ Ω + . Consequently, we can use the bound (79) for w ∈ D, and we find that

Ω+ (ū + e 1 ) • ∇ū • w dx ≤ C ( û; U α + 1) û; U α w; D .
This implies that for the α-solution ū the linear form L ū,

L ū[w] = Ω+ (ū + e 1 ) • ∇ū • w dx
is continuous on D, so that the weak formulation (63) for ū can be extended to D. This completes the proof of assumption (H2).

With similar arguments we obtain, that for arbitrary (v, w)

∈ D 2 Ω+ v • ∇ū • w dx ≤ K y 2 ∇ū; L ∞ (Ω + ) v; D w; D ≤ K û; U α v; D w; D .
This completes the proof of assumption (H4).

Proof of (H1)

The following proposition shows that α-solutions are approximated by velocity-fields of compact support:

Proposition 21 Let α > 3 and let ū := (u, v) be an α-solution. Then there exists a sequence

(ū n ) n∈N ∈ D N , such that for any n ∈ N i) ūn ∈ C ∞ (Ω + ) ii) ūn = ū in B((0, 0), n) ∩ Ω + and ūn = 0 outside B((0, 0), 2n) ∩ Ω + .
iii) There exists a constant C(ū) such that, for all n ∈ N,

ū -ūn ; L ∞ (Ω + ) + y∇(ū -ūn ); L 2 (Ω + ) + y(ū -ūn ); L ∞ (Ω + ) + y 2 ∇(ū -ūn ); L ∞ (Ω + ) ≤ C(ū) .
Proof. Let α > 3 and ū = (u, v) be an α-solution and let ψ = Π[ū] be the corresponding streamfunction. Proposition 20 implies that ū ∈ H 1 0 (Ω + ) and we have the bounds (83), (84). Therefore we have not only that ψ ∈ C 1 (Ω + ) and that ∇ ⊥ ψ(x, y) = ū(x, y), but also that ψ ∈ L ∞ (Ω + ), and that, for all (x, y)

∈ Ω + , |ψ(x, y)| ≤ C û; B α,1/2,0 , (85)
with the previous notations, and, since ū is smooth in Ω + , the stream-function ψ is also smooth in Ω + . Let

ζ n (x, y) = ζ |(x, y)| n -1 , and ūn = ∇ ⊥ [ζ n ū] .
Then, ūn ∈ D and it satisfies i) and ii) for any n ∈ N, and

ūn -ū = (ζ n -1)ū + ψ∇ ⊥ ζ n .
Using a scaling argument, one shows that ∇ζ n ; L 2 (Ω + ) is uniformly bounded for n ∈ N, and therefore we have the uniform bound, Therefore, the previous pointwise bound (84) on ∇ū implies that y 2 (∇ū -∇ū n ); L ∞ (Ω + ) is finite and remains uniformly bounded for n ∈ N. This completes the proof of Proposition 21.

ūn -ū; L 2 (Ω + ) ≤ ū; L 2 (Ω + ) + C ζ ψ; L ∞ (Ω + ) .
Combining Proposition 19 and Proposition 21 we are now able to prove (H1). Indeed, let ũ ∈ D be a weak solution for f and let ū be the corresponding α-solution. From Proposition 21 we get that there exists a sequence ūn ∈ D N which approximates ū, and, since ūn has bounded support, equation ( 63) is satisfied by ūn . Using the bounds satisfied by ũ and ū we get Finally, using that f has compact support, we get, for n sufficiently large,

Ω+ f • ūn dx = Ω+ f • ū dx .
Passing to the limit in (63) with ūn we obtain (63) with ū. This completes the proof of (H1).

Proof of (H3)

With arguments similar to the ones in the previous subsection we show that we have for any (v, w) ∈ D The Hardy inequality implies that w/y ∈ L 2 (Ω + ). Consequently, we have the following limit, Therefore, the same identity is true for w. This proves (H3).

Uniqueness of solutions for Problem 1

To conclude the paper, we sketch the proof that weak solutions for small obstacles are also unique. We note that this result is not included Theorem 17. First, this previous theorem applies to ũε . Hence, it gives information on u ε only far from S ε where ũε coincides with u ε . Second, the "unique" α-solution ūε , to which ũε is compared, depends itself on the source term obtained from u ε in the truncation procedure. However another weak solution to Problem 1 could create another source term. Our final result is Theorem 22 There exists ε u > 0, such that, for all ε < ε u , if u is a weak solution to Problem 1 for S ε then u = u ε .

Proof. The following proof is very close to the proof of Theorem 17. Hence, we only sketch the main ideas. First, we fix α > 3 and choose ε u 0 such that, for all ε < ε u 0 , any weak solution u ε is equal to the α-solution ūε outside B(2h/3). Furthermore, there holds (see Lemma 16): ûε ; U α ≤ C α u ε ; D .

for some constant C α depending only on α. Here, ûε stands once again for the Fourier transform of ūε with respect to x. Now, let ε < ε u 0 and let u be a weak solution of Problem 1 for S ε . Following the sketch of proof of wTheorem 17, we obtain that Therefore, one proves, as in sections 3.3.2 to 3.3.4, suitable continuity and antisymmetric properties of the trilinear form I, when applied to u ε , and using the fact that in I ext the weak solution u ε coincides with the α-solution ūε . This yields

u -u ε ; D 2 ≤ Ω+ (u -u ε ) • ∇u ε • (u -u ε ) dx .
u -u ε ; D 2 ≤ C ûε ; U α + u ε ; D u -u ε ; D 2 ≤ C α u ε ; D u -u ε ; D 2 .

  is a continuous linear form. By the Riesz-Fréchet theorem we can therefore define a continuous map b * n from D ε,n -e1 to D ε,n 0 by the formula

8 ,

 8 Definition IV.1.1, p. 185] of the Stokes equation in Ω + \ S ε with source term f = (u + e 1 ) • ∇u .

  ) and (64) for ũ and ū.To estimate ũ -ū, we use the D-normũ -ū; D 2 = ũ; D 2 + ū; D 2 -2 Ω+ ∇ũ : ∇ū dx ,where, applying (64) :ũ; D 2 ≤ Ω+ f • ũ dx , and ū; D 2 ≤ Ω+ f • ū dx .

Ω+∇ū:

  ∇ũ dx + Ω+ (ū + e 1 ) • ∇ū • ũ dx = Ω+ f • ũ dx . (H2) These assumptions are proved below. Combining (H1) and (H2) yields ũ -ū; D 2 ≤ Ω+ (u + e 1 ) • ∇ũ • ū + Ω+ (ū + e 1 ) • ∇ū • ũ .

Similarly,

  y∇ζ n ; L ∞ (Ω + ) is uniformly bounded for n ∈ N, and as a consequence we have for all (x, y)∈ Ω + |y(ū n -ū)(x, y)| ≤ |yū(x, y)| + C ζ |ψ(x, y)| .(86)Using (83) and (85), we find that the right-hand side in (86) is uniformly bounded for (x, y, n) ∈ Ω + × N.For the derivatives of ūn and ū, we have|∇ū n (x, y) -∇ū(x, y)| ≤ |∇ū(x, y)| + C |∇ζ n (x, y)| |ū(x, y)| + |∇ 2 ζ n (x, y)| |ψ(x, y)| .(87)Applying scaling techniques as above, one obtains for (x, y, n) ∈ Ω + × N,y∇ 2 ζ n ; L 2 (Ω + ) ≤ C ζ , |y 2 ∇ 2 ζ n (x, y)| ≤ C ζ .(88)From (88) and (87) and (83) we gety∇(ū n -ū); L 2 (Ω + ) ≤ y∇ū; L 2 (Ω + ) + C ∇ζ n ; L 2 (Ω + ) yū; L ∞ (Ω + ) + y∇ 2 ζ n ; L 2 (Ω + ) ψ; L ∞ (Ω + ) ,which yields a uniform bound with respect to n. Finally, we have, y 2 |∇ū n (x, y) -∇ū(x, y)| ≤ |y 2 ∇ū(x, y)| + C |y∇ζ n (x, y)| |yū(x, y)| + |y 2 ∇ 2 ζ n (x, y)| |ψ(x, y)| .

  Ω+ ∇ũ : (∇ū -∇ū n ) dx ≤ C(ū) Ω+\B((0,0),n) ∇ũ : ∇ū n dx = Ω+ ∇ũ : ∇ū dx .To bound the trilinear form, we now apply (78) and getΩ+ (v + e 1 ) • ∇w • (ū -ūn ) dx ≤ C (1 + v; D ) ∇w; L 2 (Ω + \ B((0, 0), n)) ū -ūn ; L 2 (Ω + ) + y (ū -ūn ) ; L ∞ (Ω + ) ≤ C(ū) (1 + v; D ) ∇w; L 2 (Ω + \ B((0, 0), n)) .Passing to the limit in n, we get lim n→∞ Ω+ (v + e 1 ) • ∇w • ūn dx = Ω+ (v + e 1 ) • ∇w • ū dx .

2

 2 and ū an α-solution, lim n→∞ Ω+ (v + e 1 ) • ∇w • ūn dx = Ω+ (v + e 1 ) • ∇w • ū dx .

From ( 79

 79 ) we getΩ+ (v + e 1 ) • ∇(ū -ūn ) • w dx ≤ C (1 + v; D ) w y ; L 2 (Ω + \ B((0, 0), n)) y 2 ∇(ū n -ū); L ∞ (Ω + ) + y∇(ū n -ū); L 2 (Ω + ) ≤ C(ū) (1 + v; D ) w/y; L 2 (Ω + \ B((0, 0), n)) .

  lim n→∞ Ω+ (v + e 1 ) • ∇ū n • w dx = Ω+ (v + e 1 ) • ∇ū • w dx .Since the approximation ūn has compact support, for any fixed n ∈ N, we haveΩ+ (v + e 1 ) • ∇w • ūn dx = -Ω+ (v + e 1 ) • ∇ū n • w dx .

  The technicalities which arise here are analogous to (H1)-(H4), and are justified by splitting integrals as follows:I(v, w, z) := Ω+ (v + e 1 ) • ∇w • z dx = I int (v, w, z) + I ext (v, w, z) ,whereI ext (v, w, z) := Ω+\B(2h/3) (v + e 1 ) • ∇w • z dx , I int (v, w, z) := B(2h/3) (v + e 1 )• ∇w • z dx .

We use the standard notation A ⊂⊂ B to mean that the closure Ā is a compact subset of B.

Acknowledgements. The authors would like to thank G. P. Galdi for suggesting the technique of proof of Theorem 17, and J. Guillod for the careful reading of preliminary versions of the paper.

* Supported in part by the Swiss National Science Foundation.

1

According to Theorem 5, there exists ε u 1 such that, if ε < ε u 1 , we have u ε ; D < 1/(2C α ). This completes the proof.


J Kiszka

P J Ersts

V Ridoux

Structure of a toothed cetacean community around a tropical island (Mayotte Island, Mozambique Channel)

Keywords: barrier reef slope, cetaceans, community composition, distribution, encounter rates, Indian Ocean, Mayotte, odontocetes

We describe the structure of a toothed cetacean community around the island of Mayotte (South-West Indian Ocean, 45°10'E, 12°50'S), using data collected from small boat-based surveys conducted between July 2004 and June 2006. In all, 16 odontocete species were recorded. Diversity (Shannon-Weaver index) was particularly high along the outer slope of the barrier reef. Patterns of spatial distribution underscore the existence of three main cetacean habitat types: the inner lagoon (Indo Pacific bottlenose dolphin Tursiops aduncus, and humpback dolphin Sousa chinensis), the outer reef slope (Spinner dolphin Stenella longirostris, Pantropical spotted dolphin S. Attenuate and melon-headed whale Peponocephala electra) and oceanic waters deeper than 500 m (e.g. Blainville's beaked whale Mesoplodon densirostris). Group characteristics were highly variable among species, with oceanic small delphinids characterised by larger group sizes than strictly coastal and non-delphinid oceanic species. The outer slope of the barrier reef appears to be of primary importance in terms of density and diversity of odontocetes around Mayotte. Results support the hypothesis that a number of cetacean species, particularly several delphinid species, are dependant on coral reef complexes.

Introduction

A biological community can be defined as the populations of organisms that co-exist in an ecosystem (or habitat). Descriptions of biological communities may consider all taxonomic groups in an ecosystem or be limited to a single functional or taxonomic group (e.g. marine top predators or cetaceans respectively). Characterising biological communities is necessary not only for understanding ecosystem structure and functioning (including trophodynamics), but for providing baseline information against which effects of ecosystem changes can be gauged and to identify critical areas for conservation management. Cetaceans perform a role as top predator in various marine ecosystems (e.g. coastal, slope-associated and oceanic, etc.). The distribution, diversity and group characteristics of cetacean communities have been described for marine ecosystems from polar to tropical waters, including Antarctic waters [START_REF] Thiele | Cetacean distribution in eastern Antarctica (80-150°E) during austral summer of 1995/1996[END_REF], the Mediterranean Sea [START_REF] Gannier | Summer distribution and relative abundance of delphinids in the Mediterranean Sea[END_REF], off the Bahamas [START_REF] Macleod | Diversity, relative density and structure of the cetacean community in summer months east of Great Abaco, Bahamas[END_REF]), the Gulf of Mexico [START_REF] Maze-Foley | Cetaceans in the oceanic northern gulf of Mexico: distributions, group sizes and inter-specific associations[END_REF], the South-West Atlantic [START_REF] Moreno | Habitat characteristics of dolphins of the genus Stenella (Cetacea: Delphinidae) in the southwest Atlantic Ocean[END_REF] and in French Polynesia [START_REF] Gannier | Distribution of cetaceans off the Society Islands (French Polynesia) as obtained from dedicated surveys[END_REF][START_REF] Gannier | Cetaceans of the Marquesas Islands (French Polynesia): distribution and relative abundance as obtained from a small boat dedicated survey[END_REF]. These studies, conducted at the scale of oceanic basins, regions or archipelagos, have shown that cetacean species partition their habitat according to a number of abiotic and biotic environmental variables, such as physiography and primary production. Most cetacean habitat studies have found that depth was one of the primary environmental features explaining cetacean distribution (e.g. [START_REF] Cañadas | Cetacean distribution related to depth and slope in the Mediterranean waters of southern Spain[END_REF].

The diversity and density of marine top predators including cetaceans at local (insular or archipelago) scales appears to be high relative to ocean-basin or regional scales [START_REF] Gannier | Distribution of cetaceans off the Society Islands (French Polynesia) as obtained from dedicated surveys[END_REF], 2002[START_REF] Baird | Studies of odontocete population structure in Hawaiian waters: results of a survey through the main Hawaiian Islands in May and June 2003[END_REF]. Similar to continental margins, where the land plunges to the deep oceanic waters, insular slopes potentially provide more abundant resources and perform essential functions such as nutrient cycling (Levin and Dayton 2009). Turbulence and vertical mixing in island channels are believed to create nutrient-rich conditions around archipelagos (Gilmartin and Revelante 1974). The formation of these isolated, nutrient-rich regions, especially in the oligotrophic regimes of the tropics where ocean productivity is generally low, is the primary reason why islands and archipelagos can serve as 'oases' of biodiversity. These oases are of critical importance for conservation and management actions that require examination over a range of spatial and temporal scales. Despite this, relatively few studies of cetacean community structure around tropical islands and atolls have been conducted thus far [START_REF] Macleod | Diversity, relative density and structure of the cetacean community in summer months east of Great Abaco, Bahamas[END_REF][START_REF] Anderson | Observations of cetaceans in the Maldives, 1990-2002[END_REF][START_REF] Dulau-Drouot | Cetacean diversity off La Réunion (France)[END_REF][START_REF] Hermans | Marine mammal diversity in the remote waters of Aldabra atoll, southern Seychelles[END_REF].

Mayotte is characterised by diverse ecosystems that are in close proximity to each other, i.e. mangroves, fringing reefs, a large semi-closed lagoon, barrier and double-barrier reef systems, and deep oceanic waters within a few kilometres from shore [START_REF] Quod | The Comoros archipelago[END_REF].

The cetacean community is mostly composed of delphinids but it also includes large odontocetes (e.g. ziphiids, kogiids, physeterids), blue whales Balaenoptera musculus and humpback whales Megaptera novaeagliae. Although some of the species recorded are rare, all are present year-round, with the exception of humpback whales.

In view of the growing need to identify critical areas for marine biodiversity conservation, both locally and regionally, this paper describes the general structure of the odontocete community encountered around the island of Mayotte. We present the diversity of species occurring in the island's surrounding waters in relation to the main habitat types and provide details on the spatial distribution and encounter rates of the most common species.

Materials and methods

Study area

Mayotte (45 o 10'E, 12 o 50'S) is situated in the northern Mozambique Channel and is part of the Comoros archipelago (Figure 1). The island is almost entirely surrounded by a 197 km long barrier reef, with a second double-barrier in the south-west and the immerged reef complex of Iris Bank in the north-west. There are a series of deep passes through the reefs, some of which are the sites of ancient rivers [START_REF] Quod | The Comoros archipelago[END_REF]. The area of the lagoon and surrounding reef complexes is about 1 500 km² with an average of 20 m and a maximum depth of 80 m in the western, older region of the lagoon. There are some 20 small islets in the lagoon, ranging from 1 ha to 242 ha, each of which is surrounded by fringing reefs. There are approximately 670 ha of mangrove forests around the main island, especially in protected bays [START_REF] Quod | The Comoros archipelago[END_REF]. The insular slope on the exterior of the barrier reef is very steep and contains many submarine canyons. Broad canyons, with numerous volcanoes and landslides, deeply incise the slope [START_REF] Audru | Bathymay: la structure sous-marine de Mayotte révélée par l'imagerie multifaisceaux[END_REF].

Data collection

Between July 2004 and June 2006, small boat-based surveys were undertaken in the waters surrounding Mayotte. Several types of boats were used: a 7-m catamaran equipped with two four-stroke, 60-hp outboard engines; a 7-m boat equipped with two two-stroke, 40-hp outboard engines; and a 6.4 m cabin boat equipped with one four-stroke, 150-hp outboard engine. Surveys were conducted throughout the study period during daylight hours between 07:00h and 18:00h, in sea conditions not exceeding Beaufort 3. Survey vessels did not follow pre-defined transects but every attempt was made to sample each habitat type within the surrounding waters of Mayotte, i.e. coastal areas (mangrove fronts, fringing reef), lagoon waters, barrier reef-associated areas (inner and outer slopes) and oceanic/slope waters (>500 m). Constant GPS logging was used to collect geographic positions every five seconds between departure from and return to the harbour, using a hand-GPS Garmin Gecko®.

When cetaceans were encountered, standard sighting data were recorded; i.e. species, group size (maximum, minimum, best estimate) and geographic position. For small aggregations of cetaceans, group size was defined as the number of animals at the surface within five body lengths of each other [START_REF] Smolker | Sex differences in patterns of association among Indian Ocean bottlenose dolphins[END_REF]. Large aggregations of small delphinids often consisted of a super group, comprised of several smaller animal units or aggregations (typically 2-20), spaced several dozen-hundreds meters apart (typically 50-200 m), moving in the same direction and exhibiting similar patterns of behaviour. For these large aggregations, group size reflects the size of the super group not the individual aggregations.

Data analysis

Only data for odontocetes were used in this study. Ten geographic zones were defined around the island, based on their general location and environmental characteristics (see Figure 1, Table 2). These geographic zones were grouped into three broad habitat categories to assess cetacean diversity for each habitat type: inner lagoon, outer reef slope (depth <500 m) and oceanic waters (depth >500 m). Encounter rate was defined as the number of sightings per unit of effort (N sightings/effort), expressed in hours. Species richness (S), which is the number of species present in an area, was considered to be an inappropriate measure of diversity on its own because it fails to take into account whether each species is rare or common. Therefore the Shannon-Weaver index (H) was also employed. The Shannon-Weaver index is one of several diversity indices used to measure diversity in categorical data. This diversity measure is derived from information theory and measures the order (or disorder) observed within a particular system. In ecological studies, this order is characterised by the number of individuals observed for each species in the sample plot:

1 - ln S i = H Pi Pi = ∑
where Pi is the relative abundance of each species. The Shannon-Weaver index was calculated for the whole study area and for each of the three broad habitat categories.

Values of the median, minimum and inter-quartile ranges of depth are provided to describe bathymetric preferences for each species. Depth data provided by Service Hydrographique et Océanographique de la Marine were associated with each sighting location using an overlay technique in a GIS. After GPS track data were downloaded a track point for each second was estimated using interpolation. The tracking data were then post-processed to isolate portions of track spent 'on effort', which were subsequently used to calculate the effort within each 2 km 2 and geographic zone (Table 2, Figure 2).

Results

General

Between July 2004 and June 2006, more than 441 hours were spent in 'search mode', actively searching for marine mammals around Mayotte. Search effort did not vary across months and years (Kruskal-Wallis test: H = 4.167; df = 3: p = 0.244). Because the main harbour is located on the north-east coast, observation effort was greater off the east coast, in the south and the north. The western portion of the lagoon and deep oceanic waters were surveyed less (Table 1, Figure 1). Melon-headed whales Peponephala electra, pantropical spotted dolphins Stenella attenuata and spinner dolphins S. Longirostris had the largest group sizes (mean = 287.8, 70.9 and 72.8 respectively; Table 1), and were frequently encountered on the outer reef slope. More coastal species, such as Indo-Pacific humpback whales Sousa chinensis (mean = 2.4) and bottlenose dolphins Tursiops aduncus (mean = 6.5), had the smallest group size (Table 2).

Diversity and distribution

During this study, 16 odontocete species were recorded (species Richness), including 11

Delphinidae belonging to nine genera, two Ziphiidae, two Kogidae and one Physeteridae (Table 1). The Shannon-Weaver index of diversity for the entire region was 1.76, but the index varied between geographic zones: H = 0.57 for the inner lagoon (four species recorded), H = 1.31 for the outer reef slope (five species) and H = 0.62 for the oceanic waters (12 species). The higher index for the outer-reef slope area was due to equitability in abundance between the species present in this area (reflected in their group sizes) -a community with an equitable distribution of abundances between species is more diverse than a community with variable species abundances. Conversely, in oceanic waters, there was greater variability in group sizes between species (high group size in delphinids vs. low in the largest toothed whales), lowered the index. Spatial distribution of cetaceans encountered around Mayotte was highly variable. Spinner and pantropical spotted dolphins had similar distributions along the outer-reef slope and on the Iris Bank and were rarely observed inside the lagoon (Figure 2). Indo-Pacific bottlenose and humpback dolphins Sousa chinensis were observed mainly inside the lagoon and the former were also regularly observed on the Iris Bank, in waters <40 m (Figure 3a, Table 3). Melon-headed whales were found on the outer reef slope area and in the shallower waters of the Iris Bank, but were never sighted inside the lagoon (Table 1 and3, Figure 3b).

The other delphinids were oceanic species and were observed farther offshore, including common bottlenose dolphins (mean depth at encounter, MDE=509 m), Risso's dolphins

Encounter rates

Encounter rates were derived for the coastal species (Indo-Pacific bottlenose and humpback dolphins), pantropical spotted and spinner dolphins, melon-headed whales, oceanic delphinids, and the large toothed whales (beaked and sperm whales) in each geographic zone (Table 2). Although search effort varied between geographic zones, certain trends were apparent, especially for the most common species, once sightings were standardised for this.

For coastal species, the highest encounter rate was in the western portion of the lagoon (0.36 group h -1 ), in the south-eastern lagoon (0.25 group h -1 ) and on the Iris Bank (0.22 group h -1 ). Spinner dolphins were encountered regularly on the eastern outer slope (1.04 group h -1 ), but more frequently along the southern slope (1.94 group h -1 ). A similar trend was found for pantropical spotted dolphins in the latter area (0.93 group h -1 ). Beaked whales were rarely encountered and only in waters deeper than 500 m.

Discussion

To date, at least 31 cetacean species have been recorded in the south-west Indian Ocean, including 23 odontocetes (Kiszka et al. 2009). The odontocete community around Mayotte has a number of notable characteristics. Species richness within the area is high, especially in waters deeper than 500 m (12 species recorded vs. five in the outer-reef slope area and four inside the lagoon). The Shannon-Weaver index was significantly lower in oceanic waters (>500 m) and inside the lagoon, because of the high variability of abundance among species in these regions. For example, in oceanic waters, delphinids have a significantly higher abundance (defined by group size) than beaked whales and sperm whales. Conversely, on the outer-reef slope, species richness is lower (five species) but abundance there is more similar among species, making the outer-reef slope community more diverse.] A previous description of cetacean diversity around Mayotte by [START_REF] Kiszka | Cetacean diversity around the Mozambique Channel island of Mayotte (Comoros archipelago)[END_REF] However, certain oceanic species that have been recorded in the South-West Indian Ocean

have not yet been recorded around Mayotte. These include Cuvier's beaked whale Ziphius cavirostris, the rough-toothed dolphin Steno bredanensis and the striped dolphin Stenella coeruleoalba. Their absence in this study may be an artefact of lower observation effort undertaken in offshore waters. Overall, the high diversity of odontocetes recorded around the island may be linked to the diversity of habitat types encountered there, especially in comparison to other oceanic islands that do not have lagoon or/and extended coral complexes. However, it remains difficult to compare study areas directly, because the number of species recorded is also linked to the spatial and temporal distribution of effort.

Detailed descriptions of the distributions of four delphinid around Mayotte are provided by [START_REF] Gross | A preliminary study of habitat and resource partitioning among co-occurring tropical dolphins around Mayotte, southwest Indian Ocean[END_REF]. Our study confirms that around the island, the Indo-Pacific bottlenose dolphin has a coastal and shallow-water distribution. The coastal affinity of this species has been documented in other areas of the south-west Indian Ocean, such as at La Réunion [START_REF] Dulau-Drouot | Cetacean diversity off La Réunion (France)[END_REF]) and off the south coast of Zanzibar [START_REF] Stensland | Abundance, distribution and behavioral ecology of Indo-Pacific bottlenose (Tursiops aduncus) and humpback (Sousa chinensis) dolphins off the south coast of Zanzibar[END_REF].

Another species documented in our study, the Indo-Pacific humpback dolphin, was encountered infrequently, which precluded detailed analysis of its distribution and habitat characteristics. Along the outer-reef slope, spinner and pantropical spotted dolphins were encountered regularly; these were the most abundant cetacean species found around Mayotte. A comparative habitat analysis by [START_REF] Gross | A preliminary study of habitat and resource partitioning among co-occurring tropical dolphins around Mayotte, southwest Indian Ocean[END_REF] confirmed that these two species overlap in their distributions, as well as in their isotopic niches, which could indicate possible competition between these sibling species. Habitat features of spinner dolphin around Mayotte are slightly different from those in other areas. In French Polynesia, Hawaii, and the Maldivian atolls, these dolphins enter atolls, sheltered bays and lagoons through reef channels in the morning and leave in the afternoon to feed overnight [START_REF] Würsig | A spinner dolphin's day[END_REF][START_REF] Anderson | Observations of cetaceans in the Maldives, 1990-2002[END_REF][START_REF] Gannier | Environmental variables affecting the residence of spinner dolphins (Stenella longirostris) in the Bay of Tahiti (French Polynesia)[END_REF]. Around Mayotte, the spinner dolphins under study generally inhabited the outer-reef slope, within a greater depth range than has been previously reported (Norris et al 1994, Gannier and[START_REF] Gannier | Environmental variables affecting the residence of spinner dolphins (Stenella longirostris) in the Bay of Tahiti (French Polynesia)[END_REF]. In addition, their mean school size of 72.8 animals was slightly higher than in other areas, such as La Réunion Melon-headed whales have been reported at a number of island groups, including the Hawaii archipelago, the Philippines, French Polynesia and in the Indian Ocean [START_REF] Gannier | Distribution of cetaceans off the Society Islands (French Polynesia) as obtained from dedicated surveys[END_REF], 2002[START_REF] Baird | Studies of odontocete population structure in Hawaiian waters: results of a survey through the main Hawaiian Islands in May and June 2003[END_REF][START_REF] Anderson | Observations of cetaceans in the Maldives, 1990-2002[END_REF][START_REF] Dolar | Abundance and distributional ecology of cetaceans in the central Philippines[END_REF][START_REF] Kiszka | Cetacean diversity around the Mozambique Channel island of Mayotte (Comoros archipelago)[END_REF][START_REF] Kiszka | Preliminary account of cetacean diversity and humpback whale (Megaptera novaeangliae) group characteristics around the Union of the Comoros (Mozambique Channel)[END_REF][START_REF] Dulau-Drouot | Cetacean diversity off La Réunion (France)[END_REF][START_REF] Brownell | Behavior of melon-headed whales, Peponocephala electra, near oceanic islands[END_REF]. They are generally accepted as having a global distribution, preferring deep tropical and warm-temperate waters [START_REF] Perryman | Melon-headed whale Peponocephala electra Gray, 1846[END_REF]. In contrast to areas such as Hawaii, La Réunion and the Gulf of Mexico, melon-headed whales around Mayotte were encountered in shallower waters, in the vicinity of the barrier reef where they appeared to engage in resting/socialising behaviour. This daylight behaviour, which has been observed in other areas throughout the species' range [START_REF] Brownell | Behavior of melon-headed whales, Peponocephala electra, near oceanic islands[END_REF], suggests that melon-headed whales use shallower waters to rest and socialise, but feed in deeper waters (probably on the slope).

Several large oceanic delphinids (e.g. Risso's dolphin, short-finned pilot whale, false killer whale), beaked whales (e.g. Blainville's beaked whale) and sperm whales (sperm whale, dwarf and pygmy sperm whales) were encountered during our study, but relatively infrequently. These species are found throughout the south-west Indian Ocean [START_REF] Leatherwood | Cetaceans and cetacean research in the Indian Ocean Sanctuary In[END_REF]Donovan 1991, Kiszka et al. 2009), preferring the slope and oceanic waters [START_REF] Baird | Studies of odontocete population structure in Hawaiian waters: results of a survey through the main Hawaiian Islands in May and June 2003[END_REF], Whitehead 2003[START_REF] Macleod | Habitat utilization by Blainville's beaked whales of Great Abaco, northern Bahamas, in relation to seabed topography[END_REF]. However, the encounter rate for Blainville's beaked whales was particularly high around Mayotte (0.09 groups h -1 in waters >500 m), similar to the rate as observed off Little Bahama Bank in the Caribbean (0.07 groups h -1 , MacLeod and Zuur 2005) but higher than in the main Hawaiian Islands (0.012 groups h -1 ; RW Baird, pers. comm. Cascadia Research Collective). The abundance of beaked whales encountered around Mayotte could be attributed to the number of broad submarine canyons that deeply incise the outer slope of the island, which may concentrate the main prey of these teuthophageous predators [START_REF] Macleod | Review of data on diets of beaked whales: evidence of niche separation and geographic segregation[END_REF][START_REF] Audru | Bathymay: la structure sous-marine de Mayotte révélée par l'imagerie multifaisceaux[END_REF].

It is evident that the outer-reef slope is of primary importance in terms of density and diversity of odontocetes around Mayotte. This particular habitat (or collection of habitats) provides resting and foraging areas for several species, such as spinner dolphins, pantropical spotted dolphins and melon-headed whales [START_REF] Norris | Behavior of the Hawaiian spinner dolphin, Stenella longirostris[END_REF][START_REF] Würsig | A spinner dolphin's day[END_REF][START_REF] Brownell | Behavior of melon-headed whales, Peponocephala electra, near oceanic islands[END_REF]. Many oceanic species also make regular incursions into these habitats, including the short-finned pilot whale, which have been observed in close proximity to the barrier reef around Mayotte while resting. Shallow waters that provide protected areas with few predators, in close proximity to oceanic foraging habitats, apparently provides an attractive environment for cetaceans. The affinity of cetaceans for the outer reef slope suggests probable dependence of the cetaceans on coral reef systems as major feeding and resting areas. The current decline of coral reefs, both at the global and regional scale in the western Indian Ocean [START_REF] Macclanahan | Western Indian Ocean coral communities: bleaching responses and susceptibility to extinction[END_REF], should be considered as a possible long-term loss of toothed cetacean habitat.

Conclusion

The lagoon and adjacent outer-slope waters of Mayotte support a high diversity of toothed cetaceans, particularly delphinids. This community includes coastal, semi-pelagic/oceanic and oceanic species. The high diversity of species combined with the sizes of aggregations 1 underline the importance of Mayotte to cetaceans. It is noteworthy that there is a large overlap in the distribution of several delphinid, especially in species living along the outerreef slope, as shown by [START_REF] Gross | A preliminary study of habitat and resource partitioning among co-occurring tropical dolphins around Mayotte, southwest Indian Ocean[END_REF]. Because species should occupy their own niche, some fine-scale segregation processes should occur, which need to be assessed through indepth habitat analyses. Our results provide important, previously unavailable, descriptive information that is critical for conservation and management efforts. Human activities, especially maritime traffic fishing pressure and disturbances from commercial whale and dolphin watching activities, are escalating in the coastal and lagoon waters of Mayotte.

Further effort is needed to assess the spatial and temporal interactions between maritime human activities and cetaceans around this rapidly developing island.

1

  Figure 3b). Although rarely encountered, larger toothed whales such as Blainville's beaked whale Mesoplodon densirostris(Table3), pygmy sperm whales Kogia breviceps (MDE=705 m), dwarf sperm whales (K. sima) (MDE=919 m) and Longman's beaked whales Indopacetus pacificus (MDE=1 945 m), were also observed in deep waters off the barrier reef and over the slope (Figure3c).

  documented the presence of 17 species, of which 15 were odontocetes. There was also an unsubstantiated sighting (photographic evidence lacking) of a ginkgo-toothed beaked whale Mesoplodon ginkgodens. Our study did not confirm the presence of M. ginkgodens but added the pygmy sperm whale Kogia breviceps to the species list for Mayotte waters. In addition, the killer whale Orcinus orca has been observed on several occasions by whale-watching operators in the recent years (N Bertrand, Sea Blue Safari, pers. comm.). The species richness of the odontocete community around Mayotte is high relative to other tropical islands and archipelagos such as La Réunion in the Mascarenes (eight odontocete species; Dulau-Drouot et al. 2008), Great Abaco in northern Bahamas (seven odontocete species; MacLeod et al. 2004), Aldabra in southern Seychelles (12 odontocete species; Hermans and Pistorius 2008), and the whole Hawaiian archipelago (14 odontocete species; Baird et al. 2003).

(

  mean = 51.2) and in the Maldives (mean = 58.2)[START_REF] Anderson | Observations of cetaceans in the Maldives, 1990-2002[END_REF]. The reasons why spinner dolphins do not regularly use lagoonal waters around Mayotte, even though the habitat conditions appear to be ideal for this species, remain uncertain. One possibility is that they are excluded from the lagoon by the presence of Indo-Pacific bottlenose dolphins, a larger, and possibly more dominant, territorial species. Pantropical spotted dolphins demonstrate a wide range of distribution and habitat characteristics around Mayotte, utilising both shallow and oceanic waters along the outer-reef slope. This species was most frequently observed close to the reef on the outer-reef slope. In Golfo Dulce, along the Pacific coast of Costa Rica, pantropical spotted dolphin occur in shallow waters (mean = 92.7 m; Cubero-Pardo 2007), whereas at La Réunion, the species is only encountered in relatively deep waters (mean = 881 m; Dulau-Drouot et al. 2008).

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: Mayotte Island, showing (a) the subareas defined for encounter rate calculations and (b) spatial representation of search effort during July 2004-June 2006

  Figure 1

Table 1 :

 1 The number of sightings, cumulative number of individuals and group size characteristics of cetaceans encountered around Mayotte from July 2004 to June 2006

	566
	Species	Common name	Number of sightings	Frequency sightings (%)	Number of individuals (cumulative)	Frequency of cumulated (%) individuals	Mean size group	Range	SD
	Stenella longirostris	Spinner dolphin	177	48.5	9 242	59.7	72.8	3-500	87.1
	Stenella attenuata	Pantropical spotted dolphin	85	23.3	2 553	16.5	70.9	3-300	71.9
		Indo-Pacific bottlenose			414	2.7
	Tursiops aduncus	dolphin	64	17.5			6.5	1-15	3.5
	Peponephala electra	Melon-headed whale	9	2.5	2 590	16.7	287.8	140-450	84.2
		Indo-Pacific Humpback			17	0.1
	Sousa chinensis	dolphin	7	1.9			2.4	1-3	0.8
	Mesoplodon densirostris	Blainville's beaked whale	6	1.6	14	0.1	2.3	1-5	1.6
	Grampus griseus	Risso's dolphin	5	1.4	44	0.3	8.8	2-20	6.8
		Common bottlenose				1
	Tursiops truncatus	dolphin	2	0.5	160		80	40-120	-
	Pseudorca crassidens	False killer whale	2	0.5	250	1.6	125	100-150	-
	Kogia sima	Dwarf sperm whale	2	0.5	3	0.01	1.5	1-3	-
	Kogia breviceps	Pygmy sperm whale	1	0.3	6	0.03	6	-	-
	Mesoplodon pacificus	Longman's beaked whale	1	0.3	1	0.006	1	-	-
	Physeter macrocephalus	Sperm whale	1	0.3	11	0.07	11	-	-
	Globicephala macrorhynchus	Long-finned pilot whale	1	0.3	60	0.4	60	-	-
	Lagenodelphis hosei	Fraser's dolphin	1	0.3	120	0.8	120	-	-
	Feresa attenuata	Pygmy killer whale	1	0.3	4	0.03	4	-	-
	TOTAL		365	100	15 489	100	53.8	1-500
	567 568 569 570

Table 2 :

 2 Search effort, number of sightings and encounter rates (in parentheses) of cetaceans encountered aroundMayotte, July 2004-June 2006

				Number of sightings
	Geographic zone	Effort (h)	Bottlenose and humpback dolphins	Spinner dolphin	Pantropical spotted dolphin	Melon-headed whale	Large toothed whales	Other delphinids
	Eastern slope	48.2	0	50 (1.04)	19 (0.39)	2 (0.04)	0	0
	Iris Bank	46	10 (0.22)	40 (0.87)	20 (0.43)	1 (0.02)	0	0
	North-eastern lagoon	75.7	12/3 (0.2)	0	1 (0.01)	0	0	0
	Northern slope	14.4	0	6 (0.42)	2 (0.14)	0	0	1 (0.07)
	Northern lagoon	32.5	6 (0.18)	13 (0.4)	3 (0.09)	0	0	0
	South-eastern lagoon	52.6	12/2 (0.25)	2 (0.04)	2 (0.04)	0	0	0
	Southern slope	23.7	0/1 (0.04)	46 (1.94)	22 (0.93)	0	0	0
	Southern lagoon	56.8	16/1 (0.28)	6 (0.11)	2 (0.04)	0	0	0
	Western slope	4.8	0	1 (0.21)	1 (0.21)	1 (0.21)	0	1 (0.21)
	Western lagoon	22	8 (0.36)	2 (0.09)	1 (0.05)	0	0	0
	Total	376.8	64/7 (0.18)	166 (0.44)	73 (0.19)	4 (0.01)	0	2 (0.01)
	Entire survey area	441.9	64/7 (0.16)	177 (0.40)	85 (0.19)	9 (0.02)	11 (0.02)	12 (0.03)

Table 3 :

 3 Depth preferences of the most frequently sighted cetacean species around Mayotte from July 2004-June 2006

	Species	Common name

Acknowledgements -Funding for field work was provided by the Ministère de l'Energie, l'Ecologie, le Développement Durable et de l'Aménagement du Territoire (MEEDDAT) and the Collectivité Départementale de Mayotte (CDM). Data were collected during a programme conducted by the Office National de la Chasse et de la Faune Sauvage (ONCFS) and the Agriculture and Forestry Office (Direction de l'Agriculture et de la Forêt). We thank Robert L Brownell and Colin MacLeod for their helpful comments on the early version of the manuscript. We also thank Robin Rolland, Alban Jamon, Julien Wickel, Wilfrid Fousse, Ismaël Ousseni (DAF), Sarah Caceres, Franck Charlier, Denis Girou (ONCFS), Didier Fray (CDM) and the personnel of Brigade Nature (CDM/ONCFS) for assistance in the field. Colin MacLeod, Robert L. Brownell and Robin W. Baird are thanked for providing helpful comments on the earlier versions of the manuscript. The valuable comments of Sal Cerchio and one anonymous reviewer are appreciated.