Chérif Amrouche
email: cherif.amrouche@univ-pau.fr

Huy Hoang

Peter D Lax

L p weighted theory for Navier-Stokes equations in exterior domains

Keywords: Navier-Stokes equations, Oseen equations, weighted Sobolev spaces, fluid mechanics AMS class: 35Q30, 76D03, 76D05, 76D07

This paper is devoted to some mathematical questions related to the stationary Navier-Stokes problem in three-dimensional exterior domains. Our approach is based on a combination of properties of Oseen problems in R 3 and in exterior domains of R 3 .

Introduction and preliminary results

This paper continues our previous studies in [START_REF] Amrouche | The stationary three-dimensional Navier-Stokes Equations with a non-zero constant velocity at infinity[END_REF] related to the three-dimensional stationary Navier-Stokes equations. Let Ω be a bounded open region of R 3 , not necessarily connected, with a Lipschitz-continuous boundary and let Ω be the complement of Ω . We suppose that Ω has a finite number of connected components and each connected component has a connected boundary, so that Ω is connected. In this paper, we study the following exterior Navier-Stokes problem:

(N S)          -ν∆u + u.∇u + ∇π = f in Ω, div u = 0 in Ω, u = 0 on Γ, u → u ∞ at infinity,
where ν > 0, f and u ∞ ∈ R 3 are respectively the viscosity of the fluid, the external force field acting on the fluid and a given constant vector of R 3 . The problem consists in looking for the velocity field u = (u 1 , u 2 , u 3 ) of the fluid and the pressure function π. We shall assume that the origin of the coordinate frame is attached to Ω . The third equation of the system states that the fluid adheres at the surface of the body, which is the common no-slip condition. Since the domain Ω is unbounded, the last equation is really necessary. In this equation, we have two different cases concerning the behavior of u at infinity. If u ∞ = 0, the flow is at rest at infinity and in the remaining case, if u ∞ = 0, the flow is past at infinity.

In this paper, we are interested in considering the case Ω being an exterior domain in R 3 and u ∞ = 0. We note that the case Ω = R 3 was considered in our previous paper [START_REF] Amrouche | The stationary three-dimensional Navier-Stokes Equations with a non-zero constant velocity at infinity[END_REF]. Our purpose is to study some regularity properties of the weak solutions to the problem (N S).

To our knowledge, in the three-dimensional situation, following Farwig [START_REF] Farwig | The stationary Navier-Stokes equations in a 3D-exterior domain, Recent topics on mathematical theory of viscous incompressible fluid[END_REF] and Galdi [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations[END_REF], they consider the problem (N S) in the case u ∞ = 0 or u ∞ = 0. In the case u ∞ = 0, they consider the external force field f belonging to the classical spaces L p (Ω), and in [START_REF] Farwig | The stationary Navier-Stokes equations in a 3D-exterior domain, Recent topics on mathematical theory of viscous incompressible fluid[END_REF] with the weight (1 + |x |) α for some p and α ∈]0, 1[. The solutions are obtained in the homogeneous Sobolev spaces with or without the weight. In this paper, we are interested in the case in which the external force field belongs to the weighted Sobolev spaces W -1,p 0 , that permits us to obtain generalized solutions in the weighted Sobolev spaces W 1,p 0 . We consider also the case in which the external force field belongs to L q or L q ∩ W -1,p 0 and some regularity properties. Our main interest is directed at L p -regularity of weak solutions, under suitable assumptions on the right-hand side f. This point is improved in this paper. We assume different levels of regularity of f, and then describe the corresponding level of smoothness of the weak solutions associated to f. We refine a regularity theory which may be found in [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations[END_REF]. Galdi assumes that f ∈ L p (Ω) for all p ∈ (1, p 0 ], with some p 0 > 3 (see Section IX.7 [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations[END_REF]). More precisely, in Theorem 5.9, we recover Galdi's regularity results. This paper is organised as follows: In this section, the problem will be introduced and we recall well-known results about weighted Sobolev spaces. In Section 2, a result about existence of weak solutions for the problem (N S) will be presented. In next sections, we shall obtain some regularity properties of the weak solution u and the associated pressure π. In Section 4, the exterior Oseen problem is considered. The identity energy will be given in the last section.

In this paper, we use bold type characters to denote vector distributions or spaces of vector distributions with 3 components and C > 0 usually denotes a generic constant the value of which may change from line to line. We shall also denote by B R the open ball of radius R > 0 centered at the origin and B R = R 3 -B R . In particular, since Ω is bounded, we can find some R 0 such that Ω ⊂ B R0 and we introduce, for any R ≥ R 0 , the set

Ω R = Ω ∩ B R and Ω R = Ω -Ω R .
We now recall the main notations and results , concerning the weighted Sobolev spaces, which we shall use later on. We define D(Ω) to be the linear space of infinite differentiable functions with compact support on Ω. Now, let D (Ω) denote the dual space of D(Ω), often called the space of distributions on Ω. We denote by ., . the duality pairing between D(Ω) and D(Ω). Remark that when f is a locally integrable function, then f can be identified with a distribution by

f, ϕ = Ω f (x) . ϕ(x) dx.
Given a Banach space B, with dual space B and a closed subspace X of B, we denote by B ⊥ X (or more simply X ⊥ , if there is no ambiguity as to the duality product) the subspace of B orthogonal to X, i.e.

B ⊥ X = X ⊥ = {f ∈ B |∀ v ∈ X, < f, v >= 0} = (B/X) .
The space X ⊥ is also called the polar space of X in B . A typical point in R 3 is denoted by x = (x 1 , x 2 , x 3 ) and its norm is given by r

= |x | = (x 2 1 + x 2 2 + x 2 3 ) 1 2
. We define the weight function ρ(x ) = 1 + r. For each p ∈ R and 1 < p < ∞, the conjugate exponent p is given by the relation

1 p + 1 p = 1.
We now define the weighted Sobolev space

W 1,p 0 (Ω) = {u ∈ D (Ω), u w 1 ∈ L p (Ω), ∇u ∈ L p (Ω)},
where

w 1 = (1 + r) if p = 3, (1 + r) ln(2 + r) if p = 3.
This space is a reflexive Banach space when endowed with the norm:

|| u|| W 1,p 0 (Ω) = (|| u w 1 || p L p (Ω) + ||∇u || p L p (Ω) ) 1/p .
We also introduce the space

W 2,p 0 (Ω) = {u ∈ D (Ω), u w 2 ∈ L p (Ω), ∇u w 1 ∈ L p (Ω), D 2 u ∈ L p (Ω)},
where

w 2 =    (1 + r) 2 if p / ∈ { 3 2 , 3}, (1 + r) 2 ln(2 + r), otherwise,
which is a Banach space equipped with its natural norm given by

|| u|| W 2,p 0 (Ω) = (|| u w 2 || p L p (Ω) + || ∇u w 1 || p L p (Ω) + ||D 2 u|| p L p (Ω) ) 1/p .
We note that the logarithmic weight only appears if p = 3 or p = 3 2 and all the local properties of W 1,p 0 (Ω) (respectively, W 2,p 0 (Ω)) coincide with those of the corresponding classical Sobolev space W 1,p (Ω) (respectively, W 2,p (Ω)). For

m = 1 or m = 2, we set • W m, p 0 (Ω) = D(Ω) W m, p 0 (Ω)
and we denote the dual space of

• W m, p 0 (Ω) by W -m,p 0 (Ω), which is the space of distributions. When Ω = R 3 , we have W m,p 0 (R 3 ) = • W m, p 0 (R 3 ). If Ω is a Lipschitz exterior domain, then • W 1, p 0 (Ω) = { v ∈ W 1,p 0 (Ω); v = 0 on Γ }. 3 If Ω is a C 1,1 exterior domain, then • W 2, p 0 (Ω) = { v ∈ W 2,p 0 (Ω); v = ∂ n v = 0 on Γ },
where ∂ n v is the normal derivative of v. For all λ ∈ N 3 where 0 ≤ |λ| ≤ 2m with m = 1 or m = 2, the mapping

u ∈ W m,p 0 (Ω) → ∂ λ u ∈ W m-|λ|,p 0 (Ω)
is continuous. Also recall the following Sobolev embeddings (see [START_REF] Adams | Sobolev Spaces[END_REF]):

W 1,p 0 (Ω) → L p * (Ω) where p * = 3p 3 -p and 1 < p < 3. (1.1)
Consequently, by duality, we have

L q (Ω) → W -1,p 0 (Ω) where q = 3p 3 + p and p > 3/2. (1.2)
Note also that if ∇u ∈ L p (Ω) with p > 3 (respectively, p = 3) and u ∈ L r (Ω) for some r ≥ 1, then we have u ∈ L ∞ (Ω) (respectively, u ∈ L q (Ω) for any real q ≥ r). Moreover,

• For all u ∈ W 1,3 0 (Ω) ∩ L r (Ω), we have

|| u || L q ≤ C ( || ∇u || L 3 + || u || L r ) for all q ≥ r; (1.3)
• For all u ∈ W 1,p 0 (Ω) ∩ L r (Ω) with p > 3, we have

|| u || L q ≤ C ( || ∇u || L p + || u || L r ) for all q ∈ [ r, ∞ ]. (1.4)
We introduce the space

X 1,p 0 (Ω) = u ∈ W 1,p 0 (Ω); ∂u ∂x 1 ∈ W -1,p 0 (Ω)
which is a Banach space equipped with the following norm

||u|| X 1,p 0 (Ω) = || u w 1 || L p (Ω) + 3 i=1 || ∂u ∂x i || L p (Ω) + || ∂u ∂x 1 || W -1,p 0 (Ω) .
We also introduce the space

• X 1, p 0 (Ω) = { u ∈ X 1,p 0 (Ω); u = 0 on Γ },
and we know that D(Ω) is dense in [START_REF] Amrouche | On the Oseen problem in threedimensional exterior domains[END_REF]). Now we introduce the following Lemma.

• X 1, p 0 (Ω) (cf.
Lemma 1.1. Let Ω ⊂ R 3 be a Lipschitz exterior domain. Assume that u ∈ • W 1, p 0 (Ω) such that ∂u ∂x 1 ∈ L q (Ω) with 1 < 2 p + 1 q . Then u ∈ L r (Ω) with 1 r = 1 3 ( 2 p + 1 q -1)
and we have the estimate as follows

||u|| L r (Ω) ≤ C(||u|| W 1,p 0 (Ω) + || ∂u ∂x 1 || L q (Ω) ). (1.5)
Proof. We extend u by zero outside Ω and denote u the extended function.

Then u ∈ W 1,p 0 (R 3 ) and

∂ u ∂x 1 ∈ L q (R 3 ). We set X p,q (R 3 ) = {v ∈ W 1,p 0 (R 3 ); ∂v ∂x 1 ∈ L q (R 3 )}.
It is easy to prove that D(R 3 ) is dense in X p,q (R 3 ), i.e, there exists ϕ k ∈ D(R 3 ) such that ϕ k → u in X p,q (R 3 ). Thanks to Babenko [START_REF] Babenko | On stationary solutions of the problem of flow past a body of a viscous incompressible fluid[END_REF], we have the following inequality

||ϕ k || L r (R 3 ) ≤ C|| ∂ϕ k ∂x 2 || 1/3 L p (R 3 ) || ∂ϕ k ∂x 3 || 1/3 L p (R 3 ) || ∂ϕ k ∂x 1 || 1/3 L q (R 3 ) ≤ C(||∇ϕ k || L p (R 3 ) + || ∂ϕ k ∂x 1 || L q (R 3 )
)

with 1 r = 1 3 ( 2 p + 1 q -1). Since (ϕ k ) is bounded in L r (R 3 ), then u ∈ L r (R 3
) and we obtain (1.5).

We introduce the Lemma 1.2. Let Ω ⊂ R 3 be a Lipschitz exterior domain and u ∈

• X 1, p 0 (Ω). i) If 1 < p < 3, then u ∈ L 4p 4-p (Ω) ∩ L
+ || u || L 3p 3-p (Ω) ≤ C || u || X 1, p 0 (Ω) . (1.6)
ii) If p = 3, then there exists a unique constant k(u) such that u + k(u) ∈ r≥12 L r (Ω) and the following estimate holds

|| u + k(u) || L r (Ω) ≤ C || u || X 1, p 0
(Ω) for any r ≥ 12.

(1.7)

iii) If 3 < p < 4, then there exists a unique constant k(u) such that u + k(u) ∈ L 4p/(4-p) (Ω) ∩ L ∞ (Ω) and the following estimate holds

|| u + k(u) || L 4p 4-p (Ω) + || u + k(u) || L ∞ (Ω) ≤ C || u || X 1, p 0 (Ω) . (1.8) Proof. Let u ∈ • X 1, p 0 (Ω) with 1 < p < 4.
Extend u by zero outside Ω and denote u by the extended function. It is clear that u belongs to W 1,p 0 (R 3 ). It remains to prove that

∂ u ∂x 1 ∈ W -1,p 0 (R 3 ). Let R 0 > 0 be a real and sufficient large such that Ω is contained in B R0 and R 1 , R 2 be reals such that R 2 > R 1 > R 0 .
Choose now some functions ψ 1 and ψ 2 satisfying

ψ 1 ∈ C ∞ (R 3 ), ψ 1 (x ) = 0 if |x | ≤ R 1 , ψ 1 (x ) = 1 if |x | ≥ R 2 , ∀x ∈ R 3 , ψ 1 (x ) + ψ 2 (x ) = 1.
We then can write

u = uψ 1 + uψ 2 = u 1 + u 2 . It is easy to prove that ∂ u 1 ∂x 1 and ∂ u 2 ∂x 1 belong to W -1,p 0 (R 3 ), then ∂ u ∂x 1 ∈ W -1,p 0 (R 3 ) and we can deduce u ∈ X 1,p 0 (R 3 ). Moreover, || u || X 1,p 0 (R 3 ) ≤ C || u || X 1,p 0 (Ω) . Since -∆ u + ∂ u ∂x 1 ∈ W -1,p 0 (R 3 ) and < -∆ u + ∂ u ∂x 1 , 1 > W -1,p 0 (R 3 )×W 1,p 0 (R 3 ) = 0 if p < 3/2,
we know from Theorem 4.4 [START_REF] Amrouche | Weighted Sobolev spaces for a scalar model of the stationary Oseen equations in R 3[END_REF] there exists a unique v ∈ X 1,p 0 (R 3 )∩L 4p/(4-p) (R 3 ) such that

-∆v + ∂v ∂x 1 = -∆ u + ∂ u ∂x 1
and satisfying the following estimate

|| v || X 1,p 0 (R 3 ) + || v || L 4p/(4-p) (R 3 ) ≤ C || -∆ u + ∂ u ∂x 1 || W -1,p 0 (R 3 ) ≤ C || u || X 1,p 0 (Ω) .
(1.9)

The function z = u -v ∈ X 1,p 0 (R 3 ) verifying the equation

-∆z + ∂z ∂x 1 = 0 in R 3 ,
then z is a polynomial that belongs to W 1,p 0 (R 3 ). Therefore, there exists a constant k such that z = k, with k = 0 if 1 < p < 3. It means that u -k = v in Ω. The estimate (1.6) is immediately deduced from (1.9). The estimates (1.7) and (1.8) are consequences of (1.3) and (1.4).

Remark 1.3. The above result is available for all u ∈ X 1,p 0 (Ω) because we know that u can be extended by P u ∈ X 1,p 0 (R 3 ).

Defining now

X 2,p 0 (Ω) = { v ∈ W 2,p 0 (Ω); ∂v ∂x 1 ∈ L p (Ω) }.
Note that W 2,p 0 (Ω) → L p * (Ω) where p * = 3p 3 -2p and 1 < p < 3/2.

By duality, we have

L q → W -2,p 0
(Ω) where q = 3p 2p + 3 and p > 3.

Note also that if v ∈ W 2,p 0 (Ω) with 3 2 ≤ p < 3 and ∇v ∈ L r (Ω) for some r, then ∇v ∈ L q (Ω) for all q ≥ r if p = 3/2 and ∇v ∈

L r (Ω) ∩ L ∞ (Ω) if 3/2 < p < 3.
We now introduce a lemma concerning the extension of X 2,p 0 (Ω) in R 3 .

Lemma 1.4. Assume v ∈ X 2,p 0 (Ω). Then there exists v ∈ X 2,p 0 (R 3 ) such that v = v in Ω and || v || X 2,p 0 (R 3 ) ≤ C || v || X 2,p 0 (Ω) .
(1.10)

Proof. We know that there exists an linear and continuous extended operator P of W 2,p 0 (Ω) in W 2,p 0 (R 3 ). Setting v = P v and using again the partition of unity

v = vψ 1 + vψ 2 ,
then it is easy to prove that v ∈ W 2,p 0 (R 3 ) and v satisfies the estimate (1.10). Proposition 1.5. Let Ω ⊂ R 3 be a Lipschitz exterior domain and u ∈ X 2, p 0 (Ω).

i) If 1 < p < 3/2, then u ∈ L 2p/(2-p) (Ω) ∩ L 3p/(3-2p) (Ω). ii) If 3/2 ≤ p < 2,
then there exists a unique constant k such that u+k ∈ L q (Ω) for all q ≥ 2p/(2 -p).

Proof. The proof is similar as in the one of Lemma 1.2 by using once again the partition of unity and Proposition 4.3 [START_REF] Amrouche | Weighted Sobolev spaces for a scalar model of the stationary Oseen equations in R 3[END_REF].

Proposition 1.6. Let Ω ⊂ R 3 be a Lipschitz exterior domain and u ∈ X 2, p 0 (Ω). i) If 1 < p < 3, then ∇u ∈ L 4p/(4-p) (Ω) ∩ L 3p/(3-p) (Ω).
ii) If p = 3, then there exists a unique k ∈ P 1 , independent on x 1 , such that

∇(u + k) ∈ r≥12 L r (Ω).
iii) If 3 < p < 4, then there exists a unique k ∈ P 1 , independent on

x 1 , such that ∇(u + k) ∈ L 4p/(4-p) (Ω) ∩ L ∞ (Ω).
Proof. This proposition is a consequence of Lemma 1.2 and Remark 1.3.

Existence of weak solutions in weighted Sobolev spaces

First of all, we shall study the existence of weak solutions of Navier-Stokes problem in weighted Sobolev spaces in this chapter. Without loss of generality, we can set u ∞ = λe 1 with e 1 = (1, 0, 0) and λ ≥ 0. From now on, we consider the case of a fixed λ > 0.

In 1933, Jean Leray [START_REF] Leray | Sur le mouvement d'un liquide visqueux emplissant l'espace[END_REF] who introduced the concept of the weak solution:

Definition 2.1. A weak solution to the problem (N S) is a field u ∈ H 1 loc (Ω) vanishing on ∂Ω, with ∇u ∈ L 2 (Ω), div u = 0 in Ω and lim |x |→∞ S2 |u(σ|x |) - u ∞ |dσ = 0 where S 2 is the unit sphere of R 3 such that for all ϕ ∈ V(Ω) = {v ∈ D(Ω), div v = 0}: ν Ω ∇u : ∇ϕ dx + Ω (u. ∇u) . ϕ dx = f, ϕ .
As in [START_REF] Alliot | On the regularity and decay of the weak solutions to the steady-state Navier-Stokes equations in exterior domains[END_REF], it is easy to prove the following theorem.

Theorem 2.2. Let Ω ⊂ R 3 be a Lipschitz exterior domain. Given a force f ∈ W -1,2 0 (Ω), the problem (N S) has a weak solution u satisfying u -u ∞ ∈ W 1,2
0 (Ω) and there exists a function π ∈ L 2 loc (Ω), unique up to a constant, such that (u, π) solves the problem (N S) in the sense of distributions and we have the following estimation

||u -u ∞ || W 1,2 0 (Ω) ≤ C ν || f || W -1,2 0 (Ω) + C(ν)|u ∞ |(1 + |u ∞ |). (2.1)
In Theorem 2.2, we see that a pressure π locally belongs to L 2 (Ω). At the beginning, we shall establish, without additional assumption, of the properties of integrability at infinity of the pressure. Proposition 2.3. Let Ω ⊂ R 3 be an exterior domain and let f ∈ W -1,2 0 (Ω). The pressure π obtained in Theorem 2.2 has a representative such that

π = τ 1 + τ 2 with τ 1 ∈ L 2 (Ω) and τ 2 ∈ W 1,3/2 0 (Ω).
Proof. Let R 1 and R 2 be reals such that R 2 > R 1 > R 0 and choose some functions ψ 1 and ψ 2 such that

ψ 1 ∈ C ∞ (R 3 ), ψ 1 (x ) = 0 if |x | ≤ R 1 , ψ 1 (x ) = 1 if |x | ≥ R 2 , ∀x ∈ R 3 , ψ 1 (x ) + ψ 2 (x ) = 1. Let v = u -u ∞
where u is a solution given by Theorem 2.2 and let π ∈ L 2 loc (Ω) be the associated pressure. We define (v 1 , π 1 ) as follows

(v 1 , π 1 ) = (vψ 1 , πψ 1 ) in Ω, (v 1 , π 1 ) = (0, 0) in Ω , and set (v 2 , π 2 ) = (vψ 2 , πψ 2 ) in Ω. It is easy to check that (v 1 , π 1 ) ∈ W 1,2 0 (R 3 )× L 2 loc (R 3 ) and (v 2 , π 2 ) ∈ H 1 (Ω 2 )×L 2 (Ω 2 )
. Moreover, we can establish the equalities in the sense of distributions (respectively in

D (R 3 ) if i = 1 and in D (Ω 2 ) if i = 2): -ν∆v i + λ ∂v i ∂x 1 + ∇π i = f i and div v i = g i , (2.2)
where

f i = [fψ i -νv∆ψ i -2ν∇v∇ψ i + π∇ψ i ] + [λv ∂ψ i ∂x 1 -(v.∇v)ψ i ] := k i + h i , g i = -v.∇ψ i . (2.3) Since ψ 1 is C ∞ on R 3 with supp ψ 1 ⊂ Ω,
we have naturally denoted by f ψ 1 the distributions on R 3 given by:

∀ϕ ∈ D(R 3 ), < f ψ 1 , ϕ > R 3 = < f, ϕψ 1 > Ω .
This notation also applies to each other term in the definition (2.3) with i = 1. Considering now with i = 2, the regularity of v and π near the boundary depends on the regularity of (f 2 , g 2 ) and on the properties of the Oseen problem in the bounded domain Ω 2 . Similarly, the regularity of v and π near the infinity depends on the regularity of (f 1 , g 1 ) and on the properties of the Oseen problem in the bounded domain R 3 . We have π = π 1 + π 2 and from Theorem 2.2, we obtain π 2 ∈ L 2 (Ω). Thus, the main of the proof deals with the properties of π 1 and therefore of (f 1 , g 1 ). We consider

-ν∆a 1 + λ ∂a 1 ∂x 1 + ∇b 1 = k 1 and div a 1 = -v∇ψ 1 in R 3 . (2.4)
Since ψ 1 is bounded and has bounded derivatives with compact support, it is easy to check that the term fψ 1 , v∆ψ 1 , ∇v∇ψ 1 and π∇ψ

1 belong to W -1,2 0 (R 3 )
and because W 1,2 0 (R 3 ) ⊂ L 6 (R 3 ) then we have v.

∂ψ 1 ∂x 1 ∈ L q (R 3 ) for all q ∈ [1, 6].
Even simple is to prove that

g 1 = -v.∇ψ 1 ∈ L 2 (R 3 ) ∩ W -1,2 0 (R 3 ) and therefore ∂g 1 ∂x 1 ∈ W -2,2 0 (R 3
) satisfying the following compatibility condition

∂g 1 ∂x 1 , 1
W -2,2 0 (R 3 )×W 2,2 0 (R 3 ) = 0.
Applying Theorem 1.10 [START_REF] Amrouche | The stationary three-dimensional Navier-Stokes Equations with a non-zero constant velocity at infinity[END_REF], there exists a unique solution

(a 1 , b 1 ) ∈ (X 1,2 0 (R 3 )× L 2 (R 3 )) of (2.4) such that a 1 ∈ L r1 (R 3
) where 4 ≤ r 1 ≤ 6. Thanks to Hölder inequality, we deduce that (v.∇v)ψ 1 ∈ L 3/2 (R 3 ) and, in particular, we have

v. ∂ψ 1 ∂x 1 ∈ L 3/2 (R 3
). Therefore, from Theorem 1.9 (see [START_REF] Amrouche | The stationary three-dimensional Navier-Stokes Equations with a non-zero constant velocity at infinity[END_REF]), the system as follows

-ν∆a 2 + λ ∂a 2 ∂x 1 + ∇b 2 = h 1 and div a 2 = 0 in R 3 , (2.5) has a unique solution (a 2 , b 2 ) ∈ L s1 (R 3 )×W 1,3/2 0 (R 3 ) such that ∇a 2 ∈ L r2 (R 3 ), ∇ 2 a 2 ∈ L 3/2 (R 3 ) and ∂a 2 ∂x 1 ∈ L 3/2 (R 3 ) for all s 1 ∈ [6, ∞) and r 2 ∈ [12/5, 3]. We set z = v 1 -a 1 -a 2 and θ = π 1 -b 1 -b 2 . Subtracting (2.
2) to (2.4) and (2.5), we get

-ν∆z + λ ∂z ∂x 1 + ∇θ = 0 and div z = 0 in R 3 . (2.6)
Therefore, we have

-ν∆curl z + λ ∂(curl z) ∂x 1 = 0 in R 3 ,
and we get Ψ = curl z, then for i = 1, 2, 3,

-ν∆Ψ i + λ ∂ψ 1 ∂x 1 = 0 in R 3 ,
where

Ψ i ∈ L 2 (R 3 ) + L r2 (R 3 ) → S (R 3 ). Then, from Lemma 4.1 [9], Ψ is a polynomial which belongs to L 2 (R 3 ) + L r2 (R 3 ). Consequently, Ψ = 0 = curl z and div z = 0. Therefore, -∆z = curl curl z + ∇ div z = 0 in R 3 .
Similarly, it is easy to prove that z is a constant, then we can deduce from (2.6) that ∇θ = 0 and by the way the existence of a constant c such that

π 1 = b 1 + b 2 + c. Therefore, the proposition is proved setting τ 1 = π 2 + b 1 , τ 2 = b 2 .

Regularity of weak solutions

Let v = uu ∞ where u is the weak solution of the Navier-Stokes problem (N S) given by Theorem 2.2. Then we rewrite the Navier-Stokes problem (N S)

as follows:

(N S)              -ν∆v + λ ∂v ∂x 1 + ∇π = f -v.∇v in Ω, div v = 0 in Ω, v = -u ∞ on Γ, v -→ 0 if |x| → ∞. (3.1)
We start our studies by adding assumptions on the force field f. First, we assume additionally that f ∈ W -1,3 0 (Ω), and then, we will consider the case more generally f ∈ W -1,2 0 (Ω) ∩ W -1,p 0 (Ω) with p ≥ 3. Following this idea, we state and prove the Theorem 3.1. Let Ω ⊂ R 3 be an exterior domain with a C 1,1 boundary. Given p ≥ 3 and f ∈ W -1,2 0 (Ω)∩W -1,p 0 (Ω). Then, each weak solution u to the problem

(N S) satisfies v ∈ W 1,2 0 (Ω) ∩ W 1,p 0 (Ω) ∩ L r1 (Ω) and ∂v ∂x 1 ∈ W -1,r2 0 (Ω) (3.2)
for any r 1 ≥ 6 and any r 2 ≥ 3. Besides, the associated pressure has a representative

π ∈ L 3 (Ω) ∩ L p (Ω), (3.3)
and if p > 3, then we have v ∈ L ∞ (Ω).
Proof. We use once again the partition of unit introduced in Proposition 2.3. We first prove the case p = 3 and then consider the case p > 3.

a) The case p = 3:

f ∈ W -1,2 0 (Ω) ∩ W -1,3 0 (Ω).
Let u be a weak solution of (N S) given by Theorem 2.2 and

v = u -u ∞ . Since v ∈ L 6 (Ω) and v.∇v = div (v ⊗ v), we have that v.∇v ∈ W -1,3 0 (Ω), v ∂ψ 1 ∂x 1 ∈ L 3/2 (R 3 ) → W -1,3 0 (R 3 ) and f ψ 1 -(v.∇v)ψ 1 ∈ W -1,3 0 (R 3 ). Moreover, since v ∈ H 1 loc (Ω) and π ∈ L 2 (Ω 2 )
, we deduce easily from Sobolev imbedding theorem that

-2ν∇v∇ψ 1 -νv∆ψ 1 + π∇ψ 1 ∈ W -1,3 0 (R 3 ), -v.∇ψ 1 ∈ L 3 (R 3 ).
Hence, the pair (f 1 , g 1 ) belongs to W -1,3

0 (R 3 ) × L 3 (R 3 ). Otherwise, we can easily see that g 1 ∈ L 3/2 (R 3 ) ⊂ W -1,3 0 (R 3 ) and therefore ∂g 1 ∂x 1 ∈ W -2,3 0 (R 3 )
satisfying the following compatibility condition

∂g 1 ∂x 1 , 1
W -2,3 0 (R 3 )×W 2,3/2 0 (R 3 ) = 0.
Then, applying Theorem 1.10 [START_REF] Amrouche | The stationary three-dimensional Navier-Stokes Equations with a non-zero constant velocity at infinity[END_REF], the following Oseen system

-ν∆w + λ ∂w ∂x 1 + ∇q = f 1 and div w = g 1 in R 3 (3.4) has a unique solution (w, q) ∈ (X 1,3 0 (R 3 ) × L 3 (R 3 )) such that w ∈ L r (R 3
) for any r ≥ 12. We set z = v 1 -w and θ = π 1 -q. Subtracting (2.2) to (3.4), we get

-ν∆z + λ ∂z ∂x 1 + ∇θ = 0 and div z = 0 in R 3 .
Proceeding analogously as in the proof of Proposition 2.3, we can deduce that ∇z = 0 in R 3 . Since z belongs to W 1,2 0 (R 3 ) + W 1,3 0 (R 3 ), then z must be a constant c and ∇v

1 = ∇w. As z ∈ L 6 (R 3 ) + L 12 (R 3 ), then c = 0, i.e. v 1 = w and v 1 ∈ W 1,2 0 (R 3 ) ∩ W 1,3 0 (R 3 ). Moreover, we have v 1 ∈ L r1 (R 3 ) and ∂v 1 ∂x 1 ∈ W -1,r2 0 (R 3
) for any r 1 ≥ 6 and any r 2 ≥ 3. Since z = 0, we deduce that ∇θ = 0, then θ must be a constant, i.e, there exists a constant a such that π 1 = q + a with q ∈ L 3 (R 3 ). Let us now come to the regularity near the boundary. Recall that

(v 2 , π 2 ) ∈ H 1 (Ω 2 ) × L 2 (Ω 2 ) satisfies (2.2) with i = 2.
Moreover, we can prove-like we proved-that

(f 2 , g 2 ) ∈ W -1,3 (Ω 2 ) × L 3 (Ω 2 ).
Thanks to Green's formula and div v = 0, we have

Ω2 g 2 (x)dx = - Γ ψ 2 u ∞ .ndσ. (3.5)
With such data, and since Ω 2 has C 1,1 boundary, we can deduce from Proposition 4.2 [START_REF] Amrouche | On the Oseen problem in threedimensional exterior domains[END_REF] that

(v 2 , π 2 ) ∈ W 1,3 (Ω 2 ) × L 3 (Ω 2 ) which immediately imply that (v 2 , π 2 ) ∈ W 1,3 0 (Ω) × L 3 (Ω).
This ends the proof of the case p = 3.

b) The case p > 3:

Let f ∈ W -1,2 0 (Ω) ∩ W -1,p 0 (Ω). It is clear that f ∈ W -1,3 0
(Ω) and since we have proved the theorem for p = 3, we know that

v ∈ W 1,2 0 (Ω) ∩ W 1,3 0 (Ω) ∩ L r1 (Ω) for any r 1 ≥ 6 and π ∈ L 3 (Ω). Then (f 1 , g 1 ) ∈ W -1,p 0 (R 3 ) × L p (R 3 ) and (f 2 , g 2 ) ∈ W -1,p (Ω 2 ) × L p (Ω 2 ).
As in the case a), we prove that

(v 1 , π 1 ) ∈ W 1,p 0 (R 3 ) × L p (R 3 ) and (v 2 , π 2 ) ∈ W 1,p 0 (Ω) × L p (Ω), i.e, v ∈ W 1,p 0 (Ω) and π ∈ L p (Ω). Moreover v ∈ L ∞ (Ω). The proof is complete.
From Sobolev embedding theorem and the properties of the duality, we know that L 3/2 (Ω) → W -1,3 0 (Ω). If we now reinforce the assumptions of Theorem 3.1, f belongs to L 3/2 (Ω) instead of W -1,3 0 (Ω), we can prove the following.

Theorem 3.2. Let Ω ⊂ R 3 be an exterior domain with C 1,1 boundary. i) Assume that f ∈ W -1,2 0 (Ω) ∩ L 3/2 (Ω). Then each weak solution u to the problem (N S) satisfies v ∈ W 1,2 0 (Ω) ∩ W 1,3 0 (Ω) ∩ L r1 (Ω), (3.6)
∂v ∂x 1 ∈ L 3/2 (Ω) ∩ L 3 (Ω) ∩ W -1,r2 0 (Ω) and ∇ 2 v ∈ L 3/2 (Ω) (3.7)
for any r 1 ≥ 9 2 , r 2 ≥ 3. Besides, the associated pressure π belongs to W

1,3/2 0 (Ω). ii) Let 3 2 < p < 3. Assume that f ∈ W -1,2 0 (Ω) ∩ L p (Ω). Then each solution u to the problem (N S) satisfies v ∈ W 1,2 0 (Ω) ∩ W 1,p * 0 (Ω) ∩ L r1 (Ω) and ∂v ∂x 1 ∈ W -1,r2 0 (Ω) (3.8) for any r 1 ∈ [3p, ∞] if 3 2 < p < 2, for any r 1 ∈ [6, ∞] if 2 ≤ p < 3
and for any r 2 ≥ 3. Besides, the associated pressure satisfies

π ∈ L 3 (Ω) ∩ L p * (Ω) (3.9)
where p * = 3p 3 -p

. Moreover, we have

∇ 2 v ∈ L p (Ω), ∂v ∂x 1 ∈ L p (Ω) and π ∈ W 1,p 0 (Ω). (3.10)
Proof. i) Let u be a weak solution of (N S). Since L 3/2 (Ω) → W -1,3 0 (Ω), from Theorem 3.1, we know that u and π satisfy (3.2) and (3.3)with p = 3. Now it remains to prove that v belongs to L 9/2 (Ω) and ∂v ∂x 1 , ∇ 2 v, ∇π belong to

L 3/2 (Ω). It is then clear that f 1 ∈ L 3/2 (R 3 )and g 1 ∈ X 1,3/2 0 (R 3
). Then, by applying Theorem 1.9 [START_REF] Amrouche | The stationary three-dimensional Navier-Stokes Equations with a non-zero constant velocity at infinity[END_REF], the following Oseen system

-ν∆w + λ ∂w ∂x 1 + ∇µ = f 1 and div w = g 1 in R 3 , (3.11)
has a unique solution (w, µ)

such that w ∈ L s (R 3 ), ∇w ∈ L r (R 3 ), ∇ 2 w ∈ L 3/2 (R 3 ), ∂w ∂x 1 ∈ L 3/2 (R 3 ) and the pressure µ ∈ W 1,3/2 0 (R 3 ) for all s ∈ [6, ∞)
and r ∈ [12/5, 3]. We set z = v 1 -w and θ = π 1 -µ. Subtracting (3.1) to (3.11), we get

-ν∆z + λ ∂z ∂x 1 + ∇θ = 0 and div z = 0 in R 3 .
By the analogous techniques as in the proof of Theorem 3.1, we conclude v 1 = w,

π 1 = µ ∈ W 1,3/2 0 (R 3 ), ∂v 1 ∂x 1 ∈ L 3/2 (R 3 ) and ∇ 2 v 1 ∈ L 3/2 (R 3
). Thanks to Lemma 1.4 [START_REF] Amrouche | The stationary three-dimensional Navier-Stokes Equations with a non-zero constant velocity at infinity[END_REF] with q = 3 2 , we can deduce v 1 ∈ L 9/2 (R 3 ). Let us now come to the regularity near the boundary. First, we verify easily that (f 2 , g 2 ) ∈ L 3/2 (Ω 2 ) × W 1,3/2 (Ω 2 ). With such data, and since Ω 2 has C 1,1 boundary, we can deduce from Proposition 4.3 [START_REF] Amrouche | On the Oseen problem in threedimensional exterior domains[END_REF] that Hölder's inequality, we obtain v.∇v ∈ L q (Ω) for all 3 2 ≤ q 2 < 3 and then

(v 2 , π 2 ) ∈ W 2,3/2 (Ω 2 ) × W 1,3/2 (Ω 2 ) which immediately imply that (v 2 , π 2 ) ∈ W 2,3/2 0 (Ω) × W 1,3/2 (Ω). Finally, since v = v 1 + v 2 and π = π 1 + π 2 ,
f -v.∇v ∈ L p (Ω).
Proceeding similarly as in the previous case, we prove (3.10). By applying Lemma 1.1, we have v ∈ L 3p (Ω) and we deduce (3.8). Finally, we obtain π ∈ L p * (Ω) from π ∈ W 1,p 0 (Ω). The theorem is completely proved.

In Theorem 3.2 (i), we proved v ∈ L r1 (Ω) for any r 1 ≥ 9/2. To obtain v ∈ L r1 (Ω) with r 1 < 9/2, we have to assume additionally a condition for f. We can state the Theorem 3.3. Let Ω ⊂ R 3 be an exterior domain with C 1,1 boundary. Assume that f ∈ W -1,2 0 (Ω) ∩ L 3/2 (Ω) ∩ L 4/3 (Ω). Then each weak solution u and the associate pressure π to the problem (N S) satisfy the results in Theorem 3.2 i). Moreover,

∇ 2 v ∈ L 4/3 (Ω), ∂v ∂x 1 ∈ L 4/3 (Ω), π ∈ W 1,4/3 0
(Ω) and v ∈ L r1 (Ω) (3.12)

for any r 1 ≥ 4.

Proof. From Theorem 3.2 and Theorem 5.26 [START_REF] Amrouche | On the Oseen problem in threedimensional exterior domains[END_REF], we have the following estimate

λ|| ∂v ∂x 1 || L 3/2 (Ω) ≤ C(|| f -v.∇v || L 3/2 (Ω) + ||u ∞ || W 4/3,3/2 (Γ) ) ≤ C(|| f || L 3/2(Ω) + || v || L 6 (Ω) ||∇v || L 2 (Ω) + ||u ∞ || W 4/3,3/2 (Γ) ) ≤ C(|| f || L 3/2 (Ω) + || v || L 6 (Ω) ||f || W -1,2 0 (Ω) + ||u ∞ || W 4/3,3/2 (Γ) ). (3.13)
Applying Lemma 1.4 [START_REF] Amrouche | The stationary three-dimensional Navier-Stokes Equations with a non-zero constant velocity at infinity[END_REF] with q = 3/2, we can deduce

||v|| L 9/2 (Ω) + λ|| ∂v ∂x 1 || L 3/2 (Ω) ≤ C(|| f || L 3/2 (Ω) + || v || L 6 (Ω) + 1).
We define the sequence {q k } as follows

2q k+1 2 -q k+1 = 3q k , k ∈ N (3.14)
with q 0 = 2. Clearly, the sequence {q k } is strictly decreasing and converges to 4/3. By induction, we can deduce for 4/3 ≤ q k ≤ 2 with k ∈ N that

||v|| L 3q k+1 (Ω) + λ|| ∂v ∂x 1 || L q k+1 (Ω) ≤ C(|| f || L q k+1 (Ω) + || v || L 2q k+1 /(2-q k+1 ) (Ω) + 1) ≤ C(|| f || L q k+1 (Ω) + || v || L 3q k (Ω) + 1).
(3.15) Thanks to Babenko [START_REF] Babenko | On stationary solutions of the problem of flow past a body of a viscous incompressible fluid[END_REF] and (3.15), we have the following estimate

|| v || L 3q k+1 (Ω) ≤ C || ∂v ∂x 1 || 1/3 L q k+1 (Ω) || ∂v ∂x 2 || 1/3 L 2 (Ω) || ∂v ∂x 3 || 1/3 L 2 (Ω) ≤ C || ∂v ∂x 1 || 1/3 L q k+1 (Ω) ≤ C (1 + || v || L 3q k (Ω) ) 1/3 . (3.16)
Then we deduce

1 + || v || L 3q k+1 (Ω) ≤ C (1 + || v || L 3q k (Ω) ) 1/3 ≤ C 1+ 1 3 +...+ 1 3 k (1 + || v || L 3q 0 (Ω) ) 1 3 k .
When k → +∞, then q k → 4/3 and we can deduce v ∈ L 4 (Ω) with the following estimate

1 + || v || L 4 (Ω) ≤ C (1 + || v || L 6 (Ω) ) ≤ C.
Since v ∈ L 4 (Ω) and ∇v ∈ L 2 (Ω), we obtain fv.∇v ∈ L 4/3 (Ω) and we deduce (3.12). The Theorem is completely proved.

The exterior Oseen problem

For our studies, we shall introduce the following problem. Let

a fixed z ∈ L 3 (Ω) such that div z = 0 in Ω, (4.1)
we search a solution (w, θ) to the following Oseen problem

-ν∆w + λ ∂w ∂x 1 + z.∇w + ∇θ = f in Ω, div w = 0 in Ω, w = u * on Γ. (4.2)
We introduce the space

V p (Ω) = {v ∈ • W 1, p 0 (Ω), div v = 0}.
As in [START_REF] Alliot | On the regularity and decay of the weak solutions to the steady-state Navier-Stokes equations in exterior domains[END_REF], we can prove the following lemma Proof. We use once again the partition of unit with the role of w (θ, respectively) as v (π, respectively) introduced in Proposition 2.3. Proceeding analogously as in the proof of Theorem 3.1, we can deduce that θ belongs to L 2 (Ω). Moreover,

we have ∂w ∂x 1 ∈ W -1,2 0
(Ω) because ∆w, z.∇w, ∇θ and f belong to W -1,2 0 (Ω).

Thanks to Lemma 1.2, we deduce w ∈ L 4 (Ω). (Ω). Then Problem (4.2) has a unique solution (w, θ) ∈ (

• W 1, 2 0 (Ω) ∩ L 4 (Ω)) × L 2 (Ω) with ∂w ∂x 1 ∈ W -1,2 0
(Ω) and w satisfies the energy equality

ν Ω |∇w | 2 dx = < f , w > W -1,2 0 (Ω)× • W 1, 2 0 (Ω) . (4.3)
Moreover, we have the estimate

||w || L 4 (Ω) + ||∇w|| L 2 (Ω) + || ∂w ∂x 1 || W -1,2 0 (Ω) + ||∇θ|| L 2 (Ω) ≤ C(||f || W -1,2 0 (Ω) + ||z.∇w || W -1,2 0 (Ω) ). (4.4)
Proof. The existence of (w, θ) ∈ (

• W 1, 2 0 (Ω) ∩ L 4 (Ω)) × L 2 (Ω) such that ∂w ∂x 1 ∈ W -1,2 0
(Ω) is given by Lemma 4.1 and Corollary 4.2. Since the space V(Ω) is dense in V 2 (Ω), for any ϕ ∈ V 2 (Ω), we have [START_REF] Amrouche | The stationary Oseen equations in R 3 . An approach in weighted Sobolev spaces[END_REF]), for all ψ, ϕ ∈

ν Ω ∇w.∇ϕdx + λ < ∂w ∂x 1 , ϕ > W -1,2 0 (Ω)× • W 1, 2 0 (Ω) + < z.∇w, ϕ > W -1,2 0 (Ω)× • W 1, 2 0 (Ω) = < f, ϕ > W -1,2 0 (Ω)× • W 1, 2 0 (Ω) . (4.5) Since D(Ω) is dense in • X 1, 2 0 (Ω) (see
• X 1, 2 0 (Ω), we obtain ∂ψ ∂x 1 , ϕ W -1,2 0 (Ω)× • W 1, 2 0 (Ω) = -ψ, ∂ϕ ∂x 1 • W 1, 2 0 (Ω)×W -1,2 0 (Ω)
.

Then, we deduce ∂ψ ∂x 1 , ψ

W -1,2 0 (Ω)× • W 1, 2 0 (Ω) = 0 (4.6)
and we have for any ϕ ∈ V 2 (Ω), div (z ⊗ ϕ), ϕ

W -1,2 0 (Ω)× • W 1, 2 0 (Ω) = - Ω z i ϕ j ∂ϕ j ∂x i dx = - 1 2 Ω z i ∂ϕ 2 j ∂x i dx = 0. (4.7)
From (4.6) and (4.7) and (4.5), we have (4.3). The uniqueness of (w, θ) is a immediate consequence of (4.3).

We consider the following nonhomogeneous problem. (Ω). Then Problem (4.2) has a unique solution (w, θ) ∈ (X 1,2 0 (Ω) ∩ L 4 (Ω)) × L 2 (Ω) and we have the estimate

||w || L 4 (Ω) + ||∇w|| L 2 (Ω) + || ∂w ∂x 1 || W -1,2 0 (Ω) + ||∇θ|| L 2 (Ω) ≤ C(||f || W -1,2 0 (Ω) + ||z.∇w || W -1,2 0 (Ω) + ||u * || H 1/2 (Γ) ). (4.8)
Proof. It is easily to show Lemma 4.4 by applying Lemma 4.3 and Lemma 5.8 [START_REF] Amrouche | On the Oseen problem in threedimensional exterior domains[END_REF] with the case p = 2.

Our objective is to consider the Navier-Stokes equations by using the properties of the Oseen equations. We now consider some properties of the Oseen equations. Beforehand, we introduce

Y 1,p 0 (Ω) = X 1,p 0 (Ω) ∩ L 4p/(4-p) (Ω) if 1 < p < 4, X 1,p 0 (Ω) if p ≥ 4, and
• Y 1, p 0 (Ω) = { v ∈ Y 1,p 0 (Ω); v = 0 on Γ }, with the same definition when Ω = R 3 . Now defining N + p (Ω) = {(u, π) ∈ • Y 1, p 0 (Ω) × L p (Ω), T(u, π) = (0, 0) in Ω}, N - p (Ω) = {(u, π) ∈ • Y 1, p 0 (Ω) × L p (Ω), T * (u, π) = (0, 0) in Ω}, with T(u, π) = (-∆u + ∂u ∂x 1 + ∇π, -div u),
and its adjoint

T * (u, π) = (-∆u - ∂u ∂x 1 + ∇π, -div u).
Moreover, if 1 < p < 4, u satisfies the properties i)-iii) of Lemma 1.2. We introduce the characterization of the kernel N + p (Ω). (see [START_REF] Amrouche | On the Oseen problem in threedimensional exterior domains[END_REF]).

Lemma 4.5. Let Ω be an exterior domain with a C 1,1 boundary.

1) If 1 ≤ p < 4, then N + p (Ω) = {(0, 0)}. 2) If p ≥ 4, then N + p (Ω) = {(λ c -c, µ c ); c ∈ R 3 } where (λ c , µ c ) ∈ r>4/3 Y 1,r 0 (Ω) × s>3/2 L s (Ω)
is the unique solution of the following system

-∆λ c + ∂λ c ∂x 1 + ∇µ c = 0, div λ c = 0 in Ω, λ c = c on Γ. (4.9)
Remark that we have the similar results for N - p (Ω). We now introduce the Theorem 4.6. [START_REF] Amrouche | On the Oseen problem in threedimensional exterior domains[END_REF] Let Ω be an exterior domain with a C 1,1 boundary. Assume that f ∈ W -1,p 0 (Ω) and u * ∈ W 1/p ,p (Γ). Moreover, if 1 < p ≤ 4/3, assume that we have the compatibility condition

∀(v, η) ∈ N - p (Ω), < f, v > Ω + < (∇v -ηI).n, u * > Γ = 0. (4.10) i) If 1 < p < 4, then the following problem -∆u + ∂u ∂x 1 + ∇π = f, div u = 0 in Ω, u = u * on Γ (4.11) has a unique solution (u, π) ∈ Y 1,p 0 (Ω) × L p (Ω) satisfying the estimate || u || Y 1,p 0 (Ω) + || π || L p (Ω) ≤ C ( || f || W -1,p 0 (Ω) + || u * || W 1/p ,p (Γ) ). ii) If p ≥ 4, then problem (4.11) has a solution (u, π) ∈ Y 1,p 0 (Ω) × L p (Ω), unique up to an element of N + p (Ω), satisfying the estimate inf (v,η)∈N + p (Ω) ( || u+v || Y 1,p 0 (Ω) +|| π+η || L p (Ω) ) ≤ C ( || f || W -1,p 0 (Ω) +|| u * || W 1/p ,p (Γ) ).
The next Lemma characterizes the kernel N p,q (Ω) of the exterior Oseen system:

N p,q (Ω) = {(u, π) ∈ [Y 1,p 0 (Ω) + Y 1,q 0 (Ω)] × [L p (Ω) + L q (Ω)], T(u, π) = (0, 0) in Ω , u = 0 on Γ} with 1 < p < q < ∞. Lemma 4.7.
Let Ω be an exterior domain with a C 1,1 boundary and

1 < p < q < ∞. i) If q < 4, then N p,q (Ω) = {(0, 0)}. ii) If q ≥ 4, then N p,q (Ω) = {(λ c -c, µ c ); c ∈ R 3 } where (λ c , µ c ) ∈ r>4/3 Y 1,r 0 (Ω) × s>3/2 L s (Ω)
is the unique solution of the system (4.9). Moreover, we have

λ c ∈ L s (Ω) ∩ L ∞ (Ω) for all s > 2.
Proof.

Let (z , θ) ∈ N p,q (Ω), then z = u -v with u ∈ Y 1,p 0 (Ω), v ∈ Y 1,q 0 (Ω) and u = v on Γ. Let now v ∈ Y 1,q 0 (R 3 ) be an extended function of v outside Ω. We set u = u in Ω, u = v outside Ω and z = u -v. Then u ∈ Y 1,p 0 (R 3
), z = 0 outside Ω and we can prove that div z = 0 in R 3 . We now extend θ by 0 outside Ω and denote θ its extended function. It is easy to see that θ ∈ L p (R 3 )+L q (R 3 ). Now setting

h = -∆ z + ∂ z ∂x 1 + ∇ θ, then supp h ⊂ Γ and h ∈ W -1,p 0 (R 3 ).
1) The case p > We now set that y = wz and k = α -θ. Hence, we have -∆y + ∂y ∂x 1 + ∇k = 0 and div y = 0 in R 3 , and we deduce

-∆curl y + ∂(curl y ) ∂x 1 = 0 in R 3 .
We take Φ = curl y . Then, for i = 1, 2, 3, we have

-∆Φ i + ∂Φ i ∂x 1 = 0 where Φ i ∈ L p (R 3 ) + L q (R 3 ) → S (R 3 ). It is deduced that Φ is a polynomial which belongs to L p (R 3 ) + L q (R 3 ). Consequently, Φ = 0 = curl y . Therefore,
-∆y = curl curl y + ∇div y = 0.

Since y ∈ Y 1,p 0 (R 3 ) + Y 1,q 0 (R 3 ), then y must be a constant c and ∇w = ∇ z . Moreover, we obtain ∇k = 0 in R 3 . Then k is a constant belonging to L p (R 3 ) + L q (R 3 ), it means α = θ in R 3 . a) The case q < 4: As y ∈ L 4p/(4-p) (R 3 )+L 4q/(4-q) (R 3 ), then c = 0. Therefore, w = z in R 3 and w = 0 on Γ. Since p < 4, from Theorem 4.6, then w = 0 in Ω, i.e., z = 0 in Ω. Therefore, ∇θ = 0 in Ω and we can deduce that θ is a constant which belongs to L p (Ω) + L q (Ω). Hence, θ = 0 in Ω. b) The case q ≥ 4: There exists a constant c = (c 1 , c 2 , c 3 ) such that wz = c and w = c on Γ. Consider now the following problem

-∆λ i + ∂λ i ∂x 1 + ∇µ i = 0, div λ i = 0 in Ω, λ i = e i on Γ, (4.12)
where (e 1 , e 2 , e 3 ) is the canonical basis of R 3 . We know that the system (4.12) has a unique solution

(λ i , µ i ) such that λ i ∈ r>4/3 Y 1,r 0 (Ω) and µ i ∈ r>3/2 L r (Ω).
If p < 4, from Theorem 4.6, w is unique and then w i = c.e i = c.λ i on Γ, therefore w i = c.λ i in Ω. Now we set λ c = (c.λ 1 , c.λ 2 , c.λ 3 ) and µ c = c.µ with µ = (µ 1 , µ 2 , µ 3 ). By construction of λ c and µ c , we deduce that (λ c , µ c ) is the unique solution of the following system

-∆λ c + ∂λ c ∂x 1 + ∇µ c = 0, div λ c = 0 in Ω, λ c = c on Γ, such that λ c ∈ r>4/3 Y 1,r 0 (Ω) and µ c ∈ r>3/2 L r (Ω). It is easy to see that (w , α) = (w , θ) = (λ c , µ c ) in Ω. Then we obtain (z , θ) = (λ c -c, µ c ) in Ω. If p ≥ 4, we obtain again (z , θ) = (λ c -c, µ c )
in Ω by proceeding similarly as in the case p < 4.

2) The case

1 < p < 4 3 : We set m = (m j ) j with m j =< h j , 1 > W -1,p 0 (R 3 )×W 1,p 0 (R 3 )
and H i = h i -δδ ij m j where δ is the Dirac distribution and δ ij denotes the Kronecker symbol. From Theorem 1.10 [START_REF] Amrouche | The stationary three-dimensional Navier-Stokes Equations with a non-zero constant velocity at infinity[END_REF], there exists a unique solution

(w 0 , α 0 ) ∈ Y 1,p 0 (R 3 ) × L p (R 3 ) such that -∆w 0 + ∂w 0 ∂x 1 + ∇α 0 = H , div w 0 = 0 in R 3 .
We now set that w = w 0 -O.m and α = α 0 -P.m

where (O, P) is the fundamental solution of Oseen equations. Then we have

-∆w + ∂w ∂x 1 + ∇α = h, div w = 0 in R 3 .
Moreover, proceeding as in the case 1 of this Lemma, we obtain ∇w = ∇ z and α = θ in R 3 . Note now that the pair (a, b) ∈ N - p (Ω) satisfies the Green's formula: for all (ψ, ξ)

∈ D(Ω) × D(Ω), Ω {(-∆ψ + ∂ψ ∂x 1 + ∇ξ)a -b div ψ}dx = < (∇a -bI).n, ψ > Γ , (4.13)
where <, > Γ denotes the duality pairing between W -1/p ,p (Γ) and W 1/p ,p (Γ).

Thanks to the density of D(Ω) in X 1,p 0 (Ω) and D(Ω) in L p (Ω), applying (4.13) with (ψ, ξ)

= (w 0 , α 0 ) ∈ X 1,p 0 (Ω) × L p (Ω) and (a, b) = (v β -β, θ β ) ∈ N - p (Ω) (β ∈ R 3 ), we obtain < (∇v β -θ β I).n, w 0 > W -1/p ,p (Γ)×W 1/p ,p (Γ) = 0. (4.14)
a) The case q < 4: Then we have wz = 0 in R 3 and w = 0 on Γ. Therefore, we deduce w 0 = O.m on Γ. From (4.14), we have

m < (∇v β -θ β I).n, O > Γ = 0. (4.15)
By some calculs, we can obtain

0 = Ω {(-∆O + ∂O ∂x 1 + ∇P)v β -θ β div O}dx = < (∇v β -θ β I).n, O > Γ -β Γ ∂O ∂n ,
then we deduce < (∇v β -θ β I).n, O > Γ = 0 and from (4.15), m = 0. Then (w 0 , α 0 ) = (0, 0) in Ω and we can deduce that (w , α) = (0, 0) and (z , θ) = (0, 0) in Ω.

b) The case q ≥ 4: There exists a constant c such that wz = c in R 3 and w = c on Γ. Then we have w 0 = c + O.m on Γ. Applying (4.14), we deduce that

< (∇v β -θ β I).n, c + O.m > Γ = 0.
We set that µ = c + O.m. It is easy to prove that µ ∈ W 1/r ,r (Γ) for all 4 3 < r < 4. Thanks to Theorem 4.6, the following system

-∆y 0 + ∂y 0 ∂x 1 + ∇κ = 0, div y 0 = 0 in Ω, y 0 = µ on Γ,
has a unique solution (y 0 , κ) ∈ Y 1,r 0 (Ω) × L r (Ω). Then (y 0 -w 0 , κ -α 0 ) ∈ N p,r (Ω) and we deduce that (y 0 , κ) = (w 0 , α 0 ). Moreover, we can see that µ ∈ W 1/p ,p (Γ). Then there exists (s, ω) ∈ Y 1,q 0 (Ω) × L q (Ω) such that -∆s + ∂s ∂x 1 + ∇ω = 0, div s = 0 in Ω, s = µ on Γ.

Then (s -w 0 , ω -α 0 ) ∈ N r,q (Ω) and (w 0 , α 0 ) ∈ Y 1,q 0 (Ω) × L q (Ω). Therefore, we deduce that (w , α) ∈ Y 1,q 0 (Ω) × L q (Ω). Since w = c on Γ and thanks to the characterization of N q,q (Ω), we obtain that (z , θ) = (λ c -c, µ c ).

Finally, in the case

p = 4 3 , let ϕ ∈ D(R 3 ) satisfying R 3 ϕ i = 1. We set U = O * ϕ and K = P * ϕ.
The reasonning can be applied by remplacing δδ ij by ϕ i , U by O and K by P.

Thanks to the above lemma, we immediately deduce the following corollary.

Corollary 4.8. Let Ω be an exterior domain with a C 1,1 boundary. Assume

f ∈ W -1,p 0 (Ω), u * ∈ W 1/p ,p (
Γ) with 1 < p < 4 satisfying the compatibility condition (4.10) and (u, π) ∈ Y 1,p 0 (Ω) × L p (Ω) be the unique solution of the system (4.11). If in addition, f ∈ W -1,q 0 (Ω) and u * ∈ W 1/q ,q (Γ) with 1 < q < 4 satisfying the compatibility condition (4.10) by remplacing p by q, then we also have (u, π) ∈ Y 1,q 0 (Ω) × L q (Ω).

We denote by [q] the integer part of q. For any k ∈ N, P k (respectively, P ∆ k ) stands for the space of polynomials (respectively, harmonic polynomials)

Proof. The proof can be obtained by proceeding similarly as in the one of Lemma 4.7.

The following corollary is immediately deduced from the previous lemma.

Corollary 4.11. Let Ω be an exterior domain with a C 1,1 . Let f ∈ L p (Ω), u * ∈ W 1+1/p ,p (Γ) with 1 < p < 2 and (u, π) ∈ Z p (Ω) × W 1,p 0 (Ω) be the unique solution of the system (4.11). If in addition, f ∈ L q (Ω), u * ∈ W 1+1/q ,q (Γ) with 1 < q < 2, then we also have (u, π) ∈ Z q (Ω) × W 1,q 0 (Ω).

5 More regularity for the velocity field u and the pressure π of the Navier-Stokes system

We now introduce the following result which we shall need in this part. The proof of this lemma is similar as the one of Lemma 4.2 [START_REF] Amrouche | The stationary three-dimensional Navier-Stokes Equations with a non-zero constant velocity at infinity[END_REF].

Lemma 5.1. Let Ω ⊂ R 3 be a Lipschitz exterior domain and z ∈ L 4 (Ω) such that div z = 0. Then, for all ε > 0, there exist ρ = ρ(ε, z) > 0 and a sequence

(z k ) k∈N ∈ L 3 (Ω) ∩ L 4 (Ω), such that div z k = 0, satisfying z k → z in L 4 (Ω). (5.1)
Moreover, there exist sequences (a k ) and

(b k ) in L 3 (Ω) ∩ L 4 (Ω) satisfying for each k ∈ N z k = a k + b k with ||a k || L 4 (Ω) ≤ ε and supp b k ⊂ Ω ρ . (5.2)
From now on, Ω is an exterior domain with C 1,1 boundary in R 3 . Note that L 6/5 (Ω) → W -1,2 0 (Ω) and L 3/2 (Ω) → W -1,3 0 (Ω), and with the previous results in hand, we can now prove the following theorem.

Theorem 5.2. Assume that f ∈ L 6/5 (Ω) ∩ L 3/2 (Ω). Then each weak solution (u, π) to the problem (N S), satisfies

v ∈ L q (Ω) for all q ∈ [3, ∞), π ∈ W 1,6/5 0 (Ω) ∩ W 1,3/2 0 (Ω), ∇v ∈ L 12/7 (Ω) ∩ L 3 (Ω), ∇ 2 v ∈ L 6/5 (Ω) ∩ L 3/2 (Ω), ∂v ∂x 1 ∈ L 6/5 (Ω) ∩ L 3 (Ω). (5.3)
Proof. Let u be a weak solution of (N S). As f satisfies the hypothesis of Theorem 3. (Ω). Moreover, since W 2,6/5 (Ω ρ ) → W 1,3/2 (Ω ρ ) with compact imbedding, we have w km → w in W 1,3/2 (Ω ρ ) with

(w k , θ k ) ∈ X 1,2 0 (Ω) × L 2 (Ω) satisfying -ν∆w k + λ ∂w k ∂x 1 + v k .∇w k + ∇θ k = f and div w k = 0 in Ω (5.4) with w k = -u ∞ on Γ.
||∇w || L 3/2 (Ωρ) = 1, (5.9)
and -ν∆w + λ ∂w ∂x 1 + v.∇w + ∇θ = 0 in Ω.

(5.10)

Since w ∈ W 1,2 0 (Ω) and θ ∈ L 2 (Ω), then we have ∆w and ∇θ belonging to W -1,2 0 (Ω). On the other hand, we deduce that v.∇w = div (v⊗w) ∈ W -1,2 0 (Ω) because v and w belong to L 4 (Ω). Since L 6/5 (Ω) → W -1,2 0 (Ω) we also have

∂w ∂x 1 ∈ W -1,2 0
(Ω). Moreover, w is divergence free and, because of (5.8), it has also zero trace at the boundary. Then, we deduce w = 0 in Ω which contradicts (5.9). Thanks to (5.5), (5.6) and (5.7), we have the following estimation

|| w k || L 3 (Ω) + || ∇w k || L 12/7 (Ω) || ∂w k ∂x 1 || L 6/5 (Ω) + || ∇ 2 w k || L 6/5 (Ω) + || θ k || W 1,6/5 0 (Ω) ≤ C(|| f || L 6/5 (Ω) + || v || L 6 (Ω) || f || L 6/5 (Ω) + || u ∞ || W 1+1/6,6/5 (Γ) ).
We can show that there exist a subsequence of (w k ) k which converges weakly towards w in W (Ω) ∩ L 3 (Ω) and a subsequence of (θ k ) k which converges weakly towards θ in W 1,6/5 0

(Ω) being a solution of the system as follows -ν∆w + λ ∂w ∂x 1 + v.∇w + ∇θ = f and div w = 0 in Ω.

We set y = vw and χ = π -θ. Then we deduce that (y, χ) is a solution of the following system -ν∆y + λ ∂y ∂x 1 + v.∇y + ∇χ = 0 and div y = 0 in Ω.

Since y satisfies the energy equality (4.3) with f = 0, we deduce that y = 0 then χ = 0. Thanks to uniqueness arguments, we show that w = v and θ = π.

Theorem is completely proved.

Thanks to Theorem 3.2 (part ii), Theorem 5.2, Sobolev embedding theorem and by duality arguments, we can prove the following.

Corollary 5.3. i) Assume that f ∈ L p (Ω) for all p ∈ [6/5, 2). Then the Navier- Stokes problem (N S) has a solution (u, π) satisfying v ∈ L q (Ω), ∇v ∈ L s1 (Ω), π ∈ W 1,s2 0 (Ω), ∇ 2 v ∈ L s2 (Ω), ∂v ∂x 1 ∈ L s3 (Ω), (5.11)
for any q ∈ [3, ∞], any s 1 ∈ [12/7, 6), any s 2 ∈ [6/5, 2) and any s 3 ∈ [6/5, 6).

ii) Assume that f ∈ L p (Ω) for all p ∈ [6/5, 3). Then we have (5.11) for any q ∈ [3, ∞], s 1 ∈ [12/7, ∞), s 2 ∈ [6/5, 3) and s 3 ∈ [6/5, ∞).

We now prove the following Theorem.

Theorem 5.4. Assume that f ∈ L p (Ω) for all p ∈ (1, 3/2]. Then each weak solution (u, π) to the problem (N S) satisfies

v ∈ L q (Ω), ∇v ∈ L s1 (Ω), π ∈ W 1,s2 0 (Ω), ∇ 2 v ∈ L s2 (Ω), ∂v ∂x 1 ∈ L s3 (Ω), (5.12)
for any q ∈ (2, ∞), any s 1 ∈ (4/3, 3], any s 2 ∈ (1, 3/2] and any s 3 ∈ (1, 3].

Proof. Remark that from Theorem 5.2, as v ∈ L 3 (Ω) and ∇v ∈ L 12/7 (Ω), we have fv.∇v ∈ L 12/11 (Ω). By applying Theorem 5.26 [START_REF] Amrouche | On the Oseen problem in threedimensional exterior domains[END_REF] with p = 12 11 , the following system (Ω). Combining with the results in Theorem 5.2, we have v ∈ L q (Ω) for all q ∈ [12/5, ∞) and ∇v ∈ L 4/3 (Ω)∩L 3 (Ω).

-ν∆w + λ ∂w ∂x 1 + ∇θ = f -v .∇v , div w = 0 in Ω; w = -u ∞ on Γ, has a unique solution (w , θ) satisfying w ∈ L 2p/(2-p) (Ω) ∩ L 3p/(3-2p) (Ω), ∇w ∈ L 4p/(4-p) (Ω) ∩ L 3p/(3-p) (Ω), ∇ 2 w ∈ L p (Ω), ∂w ∂x 1 ∈ L p (Ω) and θ ∈ W 1,p 0 (Ω),
Hence, it is easy to prove that fv.∇v belongs to L p (Ω) for all p ∈ ]1, 3/2] and we can deduce that

v ∈ L 2p 2-p (Ω) ∩ L 3p 3-2p (Ω), ∇v ∈ L 4p 4-p (Ω) ∩ L 3p 3-p (Ω), ∇ 2 v ∈ L p (Ω), ∂v ∂x 1 ∈ L p (Ω) and π ∈ W 1,p 0 (Ω).
Clearly, we have (5.12) by combining with (5.3).

Thanks to Corollary 5.3 and Theorem 5.4, we immediately obtain the following results. Corollary 5.5. i) Assume that f ∈ L p (Ω) for all 1 < p < 2. Then each weak solution (u, π) to (N S) satisfies

v ∈ L q (Ω), ∇v ∈ L s1 (Ω), π ∈ W 1,s2 0 (Ω), ∇ 2 v ∈ L s2 (Ω), ∂v ∂x 1 ∈ L s3 (Ω), (5.13
)

for any q ∈ (2, ∞], any s 1 ∈ [4/3, 6), any s 2 ∈ [1, 2) and any s 3 ∈ [1, 6).
ii) Assume that f ∈ L p (Ω) for all 1 < p < 3. Then we have (5.13) for any q

∈ (2, ∞], s 1 ∈ (4/3, ∞), s 2 ∈ (1, 3) and s 3 ∈ (1, ∞).
We now search weak solutions of Navier-Stokes system (N S) such that v ∈ L q (Ω) and π ∈ L q (Ω) for small values of q (q ≤ 2) with similar properties for ∇v. The following theorem allow us to improve the results in Theorem 5.2 by taking an additional assumption for f. In particular, if

4 3 < p < 12 7
, we obtain additionally v ∈ L q (Ω) for any q ≥ 4p 4 -p and ∇v ∈ L p (Ω).

(5.15)

Proof. From Theorem 5.2, if u is a solution of (N S), we have v satisfies (5.3). In particular,

v ∈ L 3 (Ω) ∩ L 4 (Ω) and div (v ⊗ v) ∈ W -1,3/2 0 (Ω) ∩ W -1,2 0 (Ω).
1) The case 3/2 ≤ p < 2: We have fv.∇v ∈ W -1,p 0 (Ω). As p>4/3, then the compatibility condition (4.10) is automatically satisfied. Thanks to Theorem 4.6, the following system

-ν∆w + λ ∂w ∂x 1 + ∇θ = f -v .∇v , div w = 0 in Ω; w = -u ∞ on Γ, has a unique solution (w , θ) satisfying w ∈ L 4p/(4-p) (Ω) ∩ L 3p/(3-p) (Ω), θ ∈ L p (Ω), ∇w ∈ L p (Ω) and ∂w ∂x 1 ∈ W -1,p 0 (Ω). It is easy to see that (w , θ) ∈ Y 1,p 0 (Ω) × L p (Ω) and (v , π) ∈ Y 1,3/2 0
(Ω) × L 3/2 (Ω). Applying Corollary 4.8, we have (w , θ) = (v , π), then we obtain (5.14).

2) The case 4/3 < p < 3/2: The proof is complete by combining the case 1) with the case 2). In Theorem 5.4, we know that if f ∈ L p (Ω) for all p ∈ (1, 3/2], then v satisfies (5.12). With additional assumption for f, we shall prove that the weak solutions given in Theorem 5.4 satisfy better properties. Proof. We know that (u, π) satisfies (5.12). In addition, thanks to Theorem 5.4, we have v ⊗ v ∈ L q (R 3 ) for all q > 1. Then we deduce fv.∇v ∈ W -1,r 0 (Ω). Proceeding as in Theorem 5.6, it is easy to prove that v ∈ L (Ω) and π ∈ L r (Ω). As v ∈ L q (Ω) for any q ≥ 2, we have ∂v ∂x 1 ∈ W -1,s 0 (Ω) for any s ≥ r. For the pressure, we note that thanks to (5.12), π ∈ L t (Ω) for all 3/2 < t ≤ 3 and then, we have (5.16). The Theorem is completely proved.

Since f ∈ L 6/5 (Ω) → W -1,2 0 (Ω), then in particular f ∈ W -1,3/2 0 (Ω). From the case 1) of this theorem, we have v ∈ L 4p/(4-p) (Ω) ∩ L 3p/(3-p) (Ω). Applying with p = 3/2, we have v ∈ L 12/5 (Ω) ∩ L 3 (Ω). Hence, we can show that v.∇v = div (v ⊗ v) ∈ W -1,4/3 0 (Ω) ∩ W -1,3/2 0 (Ω) and f -v.∇v ∈ W -1,p 0 (Ω).
We now prove the following theorem.

Theorem 5.9. Let 4/3 < p < ∞ and q 0 ≥ 3. Assume that f ∈ L q (Ω) ∩ W -1,p 0 (Ω) for all q ∈ (1, q 0 ]. Then the problem (N S) has a solution (u, π) satisfying the properties of Corollary 5.5 part ii). Moreover, we have π ∈ W 1,s2 0 (Ω) and ∇ 2 v ∈ L s2 (Ω) for all s 2 ∈ (1, q 0 ]. In particular, if 4/3 < p ≤ 3/2, we have additionally π ∈ L k1 (Ω) for any k 1 ≥ p.

Proof. In particular, we have f ∈ L q (Ω) for all 1 < q < 3. From Corollary 5.5 part ii), we have v ∈ L s0 (Ω), ∇v ∈ L s1 (Ω), π ∈ W 1,s2 0 (Ω),

∇ 2 v ∈ L s2 (Ω), ∂v ∂x 1 ∈ L s3 (Ω), (5.17)
for any s 0 ∈ (2, ∞], any s 1 ∈ (4/3, ∞), any s 2 ∈ (1, 3) and any s 3 ∈ (1, ∞).

Using the partition of unity, we can deduce (f 1 , g 1 ) ∈ L q (R 3 ) × X 1,q 0 (R 3 ) and (f 2 , g 2 ) ∈ L q (Ω 2 ) × W 1,q 0 (Ω 2 ) for all q ∈ (1, q 0 ] satisfying (3.5). Applying Theorem 1.9 [START_REF] Amrouche | The stationary three-dimensional Navier-Stokes Equations with a non-zero constant velocity at infinity[END_REF], Proposition 4.3 [START_REF] Amrouche | On the Oseen problem in threedimensional exterior domains[END_REF] and proceeding as in Theorem 3.1, we can obtain that π ∈ W 1,q 0 (Ω), ∇ 2 v ∈ L q (Ω), ∂v ∂x 1 ∈ L q (Ω). Combining with the previous results, we have (5.17) for all s 2 ∈ (1, q 0 ], s 3 ∈ (1, ∞). As v⊗v ∈ L r (R 3 ) for any r > 1, then fv.∇v ∈ W -1,p 0 (R 3 ). We use the same technique as in the proof of Theorem 5.6, we have (v, π) ∈ (X 1,p 0 (Ω) × L p (Ω)) such that w ∈ L s (Ω) for all 4p 4-p ≤ s ≤ 3p 3-p . Note that π ∈ L k1 (Ω) for any k 1 ≥ p if 4/3 < p ≤ 3/2. The Theorem is completely proved. Now we introduce the stress tensor T and the related stretching tensor D, T (u, π) = -πI + 2νD(u), where I is the identity matrix and D(u) = {D ij }(u) with D ij (u) = 1 2 ( ∂u i ∂x j + ∂u j ∂x i ).

We now consider the energy identity. The key idea to find the conditions to obtain the energy identity (5.18), is to test the Navier-Stokes problem with v. (5.18)

Proof. Let (u, π) be a weak solution of (N S). From Theorem 5.2, we know that (5.3) takes place. Let a 0 ∈ H 1 (Ω 2R ) where R > R 0 such that a 0 = 0 on ∂Ω, a 0 = u ∞ on ∂B 2R , div a 0 = 0 in Ω 2R . We set that a = u ∞ in B 2R and a = a 0 in Ω 2R . Then, we have au ∞ ∈ W 1,2 0 (Ω) with compact support and div a = 0. As V(Ω) is dense in V 2 (Ω) (cf. [START_REF] Alliot | On the regularity and decay of the weak solutions to the steady-state Navier-Stokes equations in exterior domains[END_REF]), there exists a sequence (ψ i ) ∈ V(Ω) with (ψ i ) va + u ∞ in V 2 (Ω) with compact support. Since v ∈ L 3 (Ω) then we deduce va + u ∞ ∈ L 3 (Ω). Testing (3.1) with (ψ i ), we obtain

ν Ω ∇v.∇ψ i dx + λ < ∂v ∂x 1 , ψ i > W -1,2 0 (Ω)× • W 1, 2 0 (Ω) + < v.∇v, ψ i > W -1,2 0 (Ω)× • W 1, 2 0 (Ω) = < f, ψ i > W -1,2 0 (Ω)× • W 1, 2 0 (Ω)
.

When i → ∞, we deduce that

ν Ω |∇v | 2 dx -ν Ω ∇v.∇a dx + λ < ∂v ∂x 1 , v + u ∞ -a > W -1,2 0 × • W 1, 2 0 + < v.∇v, v + u ∞ -a > W -1,2 0 × • W 1, 2 0 = < f, v -a + u ∞ > W -1,2 0 × • W 1, 2 0 . ( 5
+ Ω v.∇v.(v + u ∞ -a) dx = Ω f.(v -a + u ∞ ) dx.
(5.20)

2 < p < 3 ,

 23 we obtain (3.6) and (3.7). ii) Thanks to the Sobolev embedding theorem, since f ∈ L p (Ω) where 3 we can deduce that f ∈ W -1,p * 0 (Ω) with p * = 3p 3-p and p * > 3. From Theorem 3.1, we have (3.2) and (3.3) but p * plays a role as p in Theorem 3.1. From

Lemma 4 . 1 .W 1 , 2 0

 4112 Let Ω ⊂ R 3 be a Lipschitz exterior domain. Assume that z satisfies (4.1), u * = 0 and let f ∈ W -1,2 0 (Ω). Then Problem (4.2) has a solution(w, θ) ∈ • (Ω) × L 2loc (Ω). We have the following corollary.

Corollary 4 . 2 .

 42 With the same hypothesis as in Lemma 4.1, we can deduce that θ ∈ L 2 (Ω). Moreover, we have w ∈ L 4 (Ω),

Lemma 4 . 3 .

 43 Let Ω ⊂ R 3 be a Lipschitz exterior domain. Assume that z satisfies (4.1), u * = 0 and let f ∈ W -1,2 0

Lemma 4 . 4 .

 44 Let Ω ⊂ R 3 be a Lipschitz exterior domain. Assume that z satisfies (4.1), u * ∈ H 1/2 (Γ) and let f ∈ W -1,2 0

4 3 :

 3 Thanks to Theorem 1.10[START_REF] Amrouche | The stationary three-dimensional Navier-Stokes Equations with a non-zero constant velocity at infinity[END_REF], there exists w ∈ Y 1,p 0 (R 3 ) and α ∈ L p (R 3 ) such that -∆w + ∂w ∂x 1 + ∇α = h and div w = 0 in R 3 .

L 4 (Ω) and ∂v ∂x 1 ∈

 41 3, then (v, π) verify (3.2), (3.3), (3.6), (3.7) and in particular, v ∈ L 4/3 (Ω). Let ε > 0, ρ > 0 and v k = a k + b k be a sequence as z k in Lemma 5.1. Since v k ∈ L 3 (Ω) and div v k = 0, from Lemma 4.4, there exists a unique solution

  i.e., (w , θ) ∈ Z 12/11 (Ω) × W 1,12/11 0 (Ω). On the other hand, from (5.3), we can show (v , π) ∈ Z 6/5 (Ω) × W 1,6/5 0 (Ω). Thanks to Corollary 4.11, we have (w , θ) = (v , π). Then, we obtain v ∈ L 12/5 (Ω)∩L12/7 (Ω), ∇v ∈ L 4/3 (Ω)∩L12/7 (Ω), ∇ 2 v, and ∂v ∂x 1 belong to L 12/11 (Ω), π ∈ W 1,12/11 0

Theorem 5 . 6 .

 56 Let 4 3 < p < 2 and f ∈ L 6/5 (Ω) ∩ L 3/2 (Ω) ∩ W -1,p 0 (Ω). Then each weak solution (u, π) to the problem (N S) satisfies (5.3). Besides, we have π ∈ L p (Ω) and ∂v ∂x 1 ∈ W -1,s 0 (Ω) for any s ≥ p. (5.14)

  By applying Theorem 4.6 and Corollary 4.8, we have (5.14) and (5.15).

Remark 5 . 7 .

 57 Because of the compatibility condition (4.10), the above problem is open for the case 1 < p ≤ 4 3 .

Proposition 5 . 8 . 3 .

 583 Given r > 4 Assume that f ∈ L p (Ω) ∩ W -1,r 0 (Ω) for all p ∈ (1, 3/2].Then each weak solution (u, π) to (N S) satisfies (5.12) and ∂v ∂x 1 ∈ W -1,s 0 (Ω) for any s ≥ r. L t (Ω) for all r ≤ t ≤ 3.(5.16)

4r 4 -

 4 r (Ω) ∩ L 3r 3-r (Ω), ∇v ∈ L r (Ω), ∂v ∂x 1 ∈ W -1,r0

Theorem 5 . 10 .

 510 Let f ∈ L 6/5 (Ω) ∩ L 3/2 (Ω) and (u, π) be a weak solution of (N S). Then we have the energy identityν Ω | ∇v | 2 dx -λ Γ T.n dσ = Ω f.v dx.

  Thanks to Theorem 5.26[START_REF] Amrouche | On the Oseen problem in threedimensional exterior domains[END_REF], we have|| w k || L 3 (Ω) + || ∇w k || L 12/7 (Ω) + || L 6/5 (Ω) + || ∇ 2 w k || L 6/5 (Ω) + || θ k || W 1,6/5 || L 6/5(Ω) + || v k .∇w k || L 6/5 (Ω) + || u ∞ || W 7/6,5/6 (Γ)), ≥ 0 depends only on λ, ν and Ω. Note now that by construction of the sequence b k , we have |b k | ≤ v k almost everywhere in Ω, we have|| v k .∇w k || L 6/5 (Ω) ≤ || a k || L 4 (Ω) || ∇w k || L 12/7 (Ω) + || b k || L 6 (Ωρ) || ∇w k || L 3/2 (Ωρ) ≤ ε|| ∇w k || L 12/7 (Ω) + || v || L 6 (Ω) || ∇w k || L 3/2 (Ωρ)(5.6)But there existsC 1 ∈ R such that ∀k ∈ N * , || ∇w k || L 3/2 (Ωρ) ≤ C 1 (|| f || L 6/5 (Ω) + || u ∞ || W 7/6,5/6 (Γ) ). (5.7) Contradicting (5.7) means that there exists a sequence (k m ) m∈N * such that, for all m ∈ N * , ||∇w km || L 3/2 (Ωρ) = 1, || -ν∆w km + λ ∂w km ∂x 1 + v km .∇w km + ∇θ km || L 6/5 (Ω) + ||w km || W 7/6,5/6 (Γ) ≤ || w km || L 3 (Ω) + || ∇w km || L 12/7 (Ω) + || ∇ 2 w km || L 6/5 (Ω) (Ω), (w km ) m is bounded in L 3 (Ω) and (θ km ) m is bounded in W Thus, there exist subsequences, again denoted by (w km ) m and (θ km ) m , such that

	where C 1 m	.
									(5.8)
	Then we deduce from (5.5), (5.6) and (5.8) that
			+||	∂w km ∂x 1	|| L 6/5 (Ω) + || θ km || W 1,6/5 0	(Ω) ≤ C.
	Therefore (w km ) m is bounded in W 0 2,6/5	(Ω) ∩ W 0 1,12/7	(Ω),	∂w km ∂x 1 m	is bounded
	in L 6/5 1,6/5 0	(Ω).
	w km	w in W 0 2,6/5	(Ω) ∩ W 0 1,12/7	(Ω),	∂w km ∂x 1	∂w ∂x 1	in L 6/5 (Ω), w km	w in
	L 3 (Ω), and θ km	θ in W	1,6/5 0
		+ ||	∂w k ∂x 1						0	(Ω)	(5.5)

≤ C(|| f

  and v.∇v are in L 3/2 (Ω). Then, we can rewrite (5.19) as followsν Ω |∇v | 2 dx -ν Ω ∇v.∇a dx + λ Ω ∂v ∂x 1 .(v + u ∞ -a) dx

		.19)
	From (5.3),	∂v ∂x 1

of degree ≤ k. If k is strictly negative integer, we set by convention P k = {0}. We introduce the following space of polynomials:

Observe that N 0 = R 3 × {0} and N 1 = P 1 × R 3 where P 1 is the space of polynomials of degree less than or equal to one not depending on x 1 .

We now introduce the space Z p (Ω) as follows:

Define that

We can characterize the kernel A + p (Ω) (see [START_REF] Amrouche | On the Oseen problem in threedimensional exterior domains[END_REF]), as follows: Lemma 4.9. Let Ω be an exterior domain with a C 1,1 boundary.

is the unique solution of the problem (4.9). iii) If p ≥ 4, then

is the unique solution of the problem (4.9).

The next lemma characterizes the kernel A p,q (Ω) of the exterior Oseen system:

Lemma 4.10. Let Ω be an exterior domain with a C 1,1 boundary and

is the unique solution of the problem (4.9). iii) If p ≥ 4, then A p,q (Ω) = { (λ c -c, µ c -η); (c, η) ∈ N 1 }, where

is the unique solution of the problem (4.9).

Next, we multiply (3.1) with u ∞ -a ∈ W 1,2 0 (Ω) having compact support. Integrating on Ω and using integration by parts, we get (5.21)

It is easy to see that Ω v.∇v.v = 0 and Ω ∂v ∂x 1 .v = 0 ( cf. [START_REF] Farwig | The stationary Navier-Stokes equations in a 3D-exterior domain, Recent topics on mathematical theory of viscous incompressible fluid[END_REF]). From (5.20) and (5.21), we have (5.18).


Alain Simac-Lejeune

Sophie Marat

Denis Pellerin

Patrick Lambert

Michèle Rombaut

Nathalie Guyader

Relevance of interest points for eye position prediction on videos

Keywords:

This papers tests the relevance of interest points to predict eye movements of subjects when viewing video sequences freely. Moreover the papers compares the eye positions of subjects with interest maps obtained using two classical interest point detectors: one spatial and one space-time. We fund that in function of the video sequence, and more especially in function of the motion inside the sequence, the spatial or the space-time interest point detector is more or less relevant to predict eye movements.

Introduction

Images contain a very large amount of data, and image analysis often begins by the selection of some "interest points" which are supposed to carry more information than the rest of the pixels. The denition of these interest points is quite subjective, and generally depends on the aim of the analysis.

In the case of still images, many interest point detectors had been proposed. They are based on the detection of points having a signicant local variation of image intensities in dierent directions (corners, line endings, isolated points of maximum -or minimum -local intensity, etc.). The most popular one is probably the Harris detector [START_REF] Harris | A combined corner and edge detector[END_REF], with scale adaptive versions ( [START_REF] Lindeberg | Feature detection with automatic scale selection[END_REF], [START_REF] Mikolajczyk | Scale and ane invariant interest point detectors[END_REF]). Dierent Gaussian based detectors have also been proposed -LoG (Laplacian of Gaussian), DoG (Dierence of Gaussians), DoH (Determinant of the Hessian). It can be noted that DoG are used in the denition of the well known SIFT (Scale Invariant Feature Transform) approach [START_REF] Lowe | Object recognition from local scale-invariant features[END_REF] [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF]. Successful applications of interest points have been proposed in image indexing [START_REF] Mikolajczyk | Indexing based on scale invariant interest points[END_REF], stereo matching [START_REF] Tuytelaars | Wide baseline stereo matching based on local, anely invariant regions[END_REF], object recognition [START_REF] Lowe | Object recognition from local scale-invariant features[END_REF], etc.

Only a few interest point detectors had been dened in the case of moving images. Laptev [START_REF] Laptev | On space-time interest points[END_REF] proposed a space-time interest point detector which is a temporal extension of the Harris detector. He used this detector for the recognition of human actions (walking, running, drinking, etc.) in movies. In [START_REF] Dollar | Behavior recognition via sparse spatio-temporal features[END_REF], Dollar proposed a space-time detector based on a 2D spatial gaussian lter jointly used with a quadrature pair of 1D temporal Gabor lters. However, this approach is limited to the detection of periodic motions, such as a bird apping its wings. In [START_REF] Scovanner | A 3-dimensional sift descriptor and its application to action recognition[END_REF], Scovanner et al. proposed a temporal extension of SIFT descriptor.

Parallel to the research about interest points, other researches have proposed models to predict where people look at when freely viewing static or dynamic images. Given an image or video sequence, these bottom-up saliency models compute saliency maps, which topographically encodes for conspicuity (or "saliency") at every location in the visual input [START_REF] Itti | A model of salincy-based visual attention for rapid scene analysis[END_REF]. The saliency is computed in two steps: rst, the visual signal is split into dierent basic saliency maps that emphasize basic visual features as intensity, color, orientation, and second, the basic saliency maps are fuzzed together to create the master saliency map. This map emphasizes which elements of a visual scene are likely to attract the attention of human observers, and by consequence their gaze. The model saliency map is then evaluated using dierent metrics. These metrics are used to compare the model saliency maps with human eye movements when looking at the corresponding scenes ( [START_REF] Tatler | Visual correlates of xation selection : eects of scale and time[END_REF], [START_REF] Torralba | Contextual guidance of eye movements and attention in real-world scenes : The role of global features on object search[END_REF]).

In the same way the visual saliency maps are compared with subject eye movements, in this papers, we test whether the interest points are related to human eye movements. The goal of this papers is to measure the similarity between the interest maps obtained with two successful interest point detectors, one static and one dynamic, and the human eye position density maps. More precisely, we focus on the specicity of these two detectors (spatial/space-time). The papers is organized as follows. Section 2 presents the two interest point detectors which are chosen in this work. The eye movement experiment and the evaluation method are detailed in section 3. A relevance analysis is described in section 4. Conclusions are given in section 5.

Selection and description of interest point detectors

In the case of still images, several works have compared the performances of different interest point detectors. In [START_REF] Schmid | Evaluation of interest point detectors[END_REF], Schmid et al. introduced two evaluation criteria: the repeatability rate and the information content. The repeatability rate evaluates the detector stability for a scene under dierent viewing conditions (ve dierent changes were tested: viewpoint changes, scale changes, image blur, JPEG compression and illumination changes). The information content measures the distinctiveness of features. Those two criteria directly measure the quality of the feature for tasks such as image matching, object recognition and 3D reconstruction. Using these two criteria the Harris detector appears to be the best interest point detector. For this reason, this detector will be chosen in the following, either in its spatial and space-time forms.

Spatial Interest Points: Harris detector

In an image, Spatial Interest Points (denoted SIP in the following) can be dened as points with signicant gradients in more than one direction. In [START_REF] Harris | A combined corner and edge detector[END_REF], Harris et al. proposed to nd such points using a second moment matrix H dened, for a pixel (x, y) having intensity I(x, y), by:

H(x, y) = ∂ 2 I ∂x 2 ∂ 2 I ∂x∂y ∂ 2 I ∂x∂y ∂ 2 I ∂y 2 (1)
In practice, the image I is rst smoothed using a Gaussian kernel g(x, y, σ) where σ controls the spatial scale at which corners are detected.

To obtain SIP, Harris et al. proposed to use a feature extraction function entitled "salience function", dened by:

R(x, y) = det(H(x, y)) -k × trace(H(x, y)) 2 (2)
The parameter k is empirically adjusted between 0.04 and 0.15 (0.04 is chosen in the following). SIP correspond to high values of the salience function extracted using a thresholding step (the salience function being normalized between 0 and 255, typical threshold value is 150).

Space-Time Interest Points: Laptev detector

Laptev et al. [START_REF] Laptev | Space-time interest points[END_REF] proposed a spatio-temporal extension of the Harris detector to detect what they call "Space-Time Interest Points", denoted STIP in the following. STIP are points which are both relevant in space and time. These points are specially interesting because they focus information initially contained in thousands of pixels on a few specic points which can be related to spatiotemporal events in an image. STIP detection is performed by using the Hessian-Laplace matrix H dened, for a pixel (x, y) at time t having intensity I(x, y, t), by:

H(x, y, t) =    ∂ 2 I ∂x 2 ∂ 2 I ∂x∂y ∂ 2 I ∂x∂t ∂ 2 I ∂x∂y ∂ 2 I ∂y 2 ∂ 2 I ∂y∂t ∂ 2 I ∂x∂t ∂ 2 I ∂y∂t ∂ 2 I ∂t 2    (3)
As with the Harris detector, a gaussian smoothing is applied both in spatial and temporal domain. Two parameters σ s and σ t , one for each domain, control the spatial and temporal scale at which corners are detected. Typical values of σ s and σ t are respectively 1.5 and 1.2. In order to highlight STIP, dierent criteria have been proposed. As in [START_REF] Laptev | On space-time interest points[END_REF], we have chose the spatio-temporal extension of the Harris corner function, entitled "salience function", dened by:

R(x, y, t) = det(H(x, y, t)) -k × trace(H(x, y, t)) 3 (4)
where k is a parameter empirically adjusted at 0.04 as for SIP detection. STIP also correspond to high values of the salience function R and are obtained using a thresholding step.

Eye position experiment and comparison metric

Eye positions are usually used to evaluate saliency models. Most of these models are inspired by the biology of the human visual system especially the processing of the retina and the primary visual cortex, ( [START_REF] Itti | A model of salincy-based visual attention for rapid scene analysis[END_REF], [START_REF] Le Meur | Predicting visual xations on video based on low-level visual features[END_REF], [START_REF] Marat | Modelling spatio-temporal saliency to predict gaze direction for short videos[END_REF]). As the interest point models, these saliency models are based on low level properties of the stimuli. As the aim of this papers is to test whether the interest points are related to human eye movements, an experiment had been carried out in order to get the eye positions of subjects on particular video databases.

Experiment

We the eye positions of fteen subjects when they were viewing a video freely. This experiment was inspired by an experiment of Carmi and Itti [START_REF] Carmi | Visual causes versus correlates of attentional selection in dynamic scenes[END_REF] and is explained in more detail in [START_REF] Marat | Modelling spatio-temporal saliency to predict gaze direction for short videos[END_REF]. Fifty three videos (25 fps, 720x576 pixels per frame) are selected from heterogeneous sources. The 53 videos are cut in small snippets of 1-3 seconds (1.86 ± 0.61), that are strung together to make up 20 clips of 30 seconds. Each clip contains at most one clip snippet from each continuous source. The total amount of snippets over the 20 clips is 305. The eye data are recorded using an Eyelink II eye tracker (SR Research), recording eye positions at 500Hz. As the frame rate is 25Hz, for each frame we compute the median of the 20 values related to this frame in order to get an eye position for each subject and for each frame. To relax the constraint on the exact positions, a 2-dimensional gaussian ltering is applied on each eye position point to obtain the human eye position density map M h .

Comparison metric

The human eye position density map has to be compared to the interest maps provided by the interest point detectors. By looking what is done for saliency model evaluation, we can used several metrics ( [START_REF] Tatler | Visual correlates of xation selection : eects of scale and time[END_REF], [START_REF] Torralba | Contextual guidance of eye movements and attention in real-world scenes : The role of global features on object search[END_REF]). In this papers we use the Normalized Scanpath Saliency (NSS) [START_REF] Peters | Components of bottom up gaze allocation in natural images[END_REF], [START_REF] Peters | Applying computational tools to predict gaze direction in interactive visual environments[END_REF] that was especially designed to compare eye positions and the salient locations emphasized by a saliency map M m , and so it can be easily interpreted. The NSS is dened as follows:

N SS(t) = M h (x, y, t) × M m (x, y, t) -M m (x, y, t) σ Mm(x,y,t) (5)
where M h (x, y, t) is the human eye position density map normalized to unit mean and M m (x, y, t) a saliency map for a frame t. M m (x, y, t) and σ Mm(x,y,t) are respectively the average and the standard deviation of the model map M m (x, y, t).

In this papers the NSS is chosen to compare subjects' eye position and an interest map obtained with the interest-points detector as described below. The NSS is null if there is no link between eye positions and interest regions, negative if eye positions tend to be on non-interest regions and positive if eye positions tend to be on interest regions. To sum up, a interest point model would be a good predictor of human eye xations if the corresponding NSS value would be positive and high. For this map, we directly use a normalized version of the salience function R(x, y, t) on which a 2D Gaussian lter is applied. Figure 1 gives an example of these dierent maps. Human eye position density map ((M h ), g. 1.b) does not look like the two dierent interest maps ((M SIP ),(M ST IP ), g. 1.c/d). There are very few highlighted regions on (M h ) compared to the interest maps. The more the highlighted regions of (M h ) will be also highlighted on the dierent interest maps, the more the NSS will be high. In order to determine the relevance of M SIP and M ST IP maps according to the human eye position density map M h , the NSS is calculated for each interest map. In the following, N SS SIP (resp. N SS ST IP ), denotes the NSS values obtained with M SIP (resp. M ST IP ).

Analysis on the global database

A temporal analysis of the NSS criteria is realized. Figure 2 shows the evolution of average NSS over time (or frames) of each snippet (see section 3.1). First of all, both for SIP and STIP, we can note that the NSS values are positive which means that the interest points are relevant for eye position prediction. The second observation is related to the beginning of the evolution. The two curves present similar aspects after the rst ten frames but are quite dierent at the beginning. That can be explained by the fact that after a shot cut between two snippets, humans' gaze stay at the same position, corresponding to the previous shot, for a short period before going to an interesting region of the new shot. As SIP interest map highlights interest points in a static way, after a shot cut the interest points change immediately and consequently are dierent from the regions gazed at. Thus the NSS is low. After a small delay, the subjects gazed at regions highlighted on the SIP map, and then the NSS increases. On the contrary, as the STIP interest map is built using a sliding window considering several frames before and after the current frame, during the rst frames of a new snippet, STIP saliency map highlights interest points of the previous shot which are still gazed at by subjects.

It is particularly interesting to note that the N SS SIP values are higher than the N SS ST IP values for approximately 65% of snippets. However, the N SS ST IP average (0.54) is higher than N SS SIP average (0.50). Thus, when N SS ST IP is higher than N SS SIP , it is signicantly higher.

Analysis per semantical categories

We want to see if the interest points are more relevant for dierent semantical categories of snippets. An analysis shows that performances are dierent ac-cording to the snippet content. Among the 305 snippets, we extract a number of classes of similar content. We have chosen to present four examples of interesting behavior: Trac (18 snippets, 6% of the snippets), Team Sports (44 snippets, 14% of the snippets), Faces and/or Hands (47 snippets, 15% of the snippets) and Demonstration (30 snippets, 10% of the snippets). Figure 3 gives some images of snippets corresponding to these four classes. Table 1 summarizes the NSS values obtained for these dierent classes. This table shows the Trac and Team Sports classes have got average and maximum values of NSS greater with STIP than with SIP. Furthermore, for Trac class, the maximum value is very high. On the contrary, for the Faces/Hands class, SIP gives better average and maximum than STIP. Finally, NSS values for the demonstration class are close to 0, which means that there is a weak link between the eye positions and the interest points. To better understand these results, we give (Fig. 4) the NSS evolution for an example of snippet of each class. On this gure, the N SS ST IP evolution exhibits a local maximum (value 4) near the thirtieth image. This peak corresponds to the abrupt change in direction of one of the vehicles.

Team sport class (using a ball): The second example concerns the category of team sports (g3.b) using a ball: basketball, hand ball, rugby. These sports are characterized by rapid movements, rather erratic and with rapid changes. Furthermore, the more a player is close to the ball, or to the action, the more movements are rapid and disorderly. This context is favorable to STIP which tend to detect points with irregular motion. Figure 4.b shows the evolution of NSS over images for SIP and STIP. Clearly, link with eye positions is better for STIP than for SIP. The local maxima of N SS ST IP generally correspond to sudden changes in the action. These changes attract eyes while providing a lot of STIP. So, for this class, the contribution of the temporal component to the interest point detection seems to be relevant. However, this result is not always true for football sequences. This counterexample is probably due to the fact that football images generally gives a wide view, which induces smoother motion.

Face/hand class: The third class is composed of close-up sequences of faces and hands (for instance a music concert -g 3.c). This class represents the typical situations where the N SS SIP is higher than the N SS ST IP (65% of global database). In these sequences, the areas attracting the most attention are faces [START_REF] Cerf | Predicting gaze using low-level saliency combined with face detection[END_REF]. However, motion in these sequences is weak whereas they contain many spatial points of interest, generally located on faces or hands which are areas of visual attention. This explains the good results for SIP while adding the temporal component to interest point detection decreases the performance. Figure 4.c shows the evolution of NSS. The problem is that in these sequences all people move which induces a lot of interest points uniformly distributed within images. But in the same way, visual attention is not captured by a particular area. Thus, the correspondence between eye positions and interest points is rare.

Conclusions and perspectives

The work presented in this papers has shown that interest points provided by specic detectors are globally relevant for eye position prediction. More precisely, we have studied the dierence between spatial and space-time interest points, searching in which conditions interest points could be regarded as predictions of eye positions.

In order to analyse the relevance, the reference was built by recording eye position of subject which were compared to interest maps using the NSS metric. Experiment was run on a set of 305 snippets with various contents.

From the obtained results, we can get three main conclusions: globally, there is relevant link between eye positions and interest points (SIP and STIP). Hence interest points can be used as a prediction of gaze. The computational cost is very low regarding to other more dedicated methods. STIP provide a very good detection of eye positions when the sequence contains specials events, for examples: a car crash, somebody running and suddenly changing the direction of his run, On the contrary, when the semantic content is static (for faces and hands for example), the STIP do not work and SIP provide a very good detection of eye positions A future extension of this work could be a collaborative use of SIP and STIP according to the video content. If information about the class or type of content is a priori known, the type of detector to use (SIP or STIP) can be easily chosen.

If there is no additional information, intrinsic evaluation of STIP could help making this choice for optimum performance.

4 Relevance analysis 4 . 1

 41 Eye position density map and interest mapsWe chose to transform the sets of points (given by the eye position experiment and the interest point detectors) into maps by applying a 2D spatial gaussian lter on each point. This ltering allows to take the imprecision and the density of measures into account. For each frame of the dierent snippets presented in the previous section, three maps are worked out :Human eye position density map (M h ): which is obtained by applying a 2D Gaussian ltering on each eye position point. In this papers, it corresponds to the reference map. SIP interest map (M SIP ): this map corresponds to the SIP detector. As for the previous map, it is obtained by applying the same 2D Gaussian ltering on the SIP points. STIP interest map (M ST IP ): this map corresponds to the STIP detector.

Fig. 1 .

 1 Fig. 1. Example of maps extracted from a snippet

Fig. 2 .

 2 Fig. 2. NSS variations averaged over the 305 snippets over time plotted for the 55 rst frames of each snippets. (NSS Average on the global database: N SSSIP 0.50 -N SSST IP 0.54)

Fig. 3 .

 3 Fig. 3. Image examples of the 4 classes

Fig. 4 .

 4 Fig. 4. NSS over time for a snippet of each class

Table 1 .

 1 Average and maximum NSS values for dierent classes

			Trac Team Sports Faces/Hands Demonstration
	N SSSIP	Average Maximum	0.86 2.10	0.17 0.72	1.85 4.78	0.19 0.78
	N SSST IP	Average Maximum	1.26 4.24	0.77 1.98	0.39 3.06	0.23 0.95
	Number of snippets 18 (6%)	44 (14%)	47 (15%)	30 (10%)
	Note that the minimum values of all the interest map is 0

  N SS SIP has a good level (average level approximately 1.5) and is almost all the time over N SS ST IP .Demonstration/crowd class: The last class contains demonstrations or crowds (g 3.d). The characteristics of this class is that movements are performed by a multitude of people covering almost all of the image. Figure4.d) shows that this class is characterized by a very low NSS relatively constant, around 0.2, as well for SIP as for STIP, even if in some cases STIP seems a little better than SIP. The result indicates that the eye position do not match with the interest point.

Acknowledgements

We thank the Rhône-Alpes region for its support with LIMA project.