On the very weak solution for the Oseen and Navier-Stokes equations

In a three dimensional bounded possibly multiply-connected domain, we prove the existence and uniqueness of vector potentials in Lp theory, associated with a divergence-free function and satisfying some boundary conditions. We also present some results concerning scalar potentials and weak vector potentials. Furthermore, variousSobolev-type inequalities are given.

1. Introduction and notations. In this work, we are interested in some questions concerning the Navier-Stokes equations, defined in Ω a bounded open set of R 3 with boundary Γ: where u denotes the velocity and q the pressure and both are unknown. The external force f , the compressibility condition h and the boundary condition for the velocity g are given functions. The vector fields and matrix fields (and the corresponding spaces) defined over Ω or over R 3 are respectively denoted by boldface Roman and special Roman.

CHÉRIF AMROUCHE AND M.ÁNGELES RODRÍGUEZ-BELLIDO
proved the existence of weak solution (u, q) ∈ W 1,p (Ω) × L p (Ω) for any 3 2 < p < 2 when h = 0 and g satisfies the above conditions. More recently, Kim [17] improves Serre's existence and regularity results on weak solutions of (NS) for any 3 2 ≤ p < 2 (including the case p = 3 2 ), when the boundary of Ω is connected (I = 0) provided h is small in an appropriate norm (due to the compatibility condition between h and g , then g is also small in the corresponding appropriate norm).
The purpose of our work is to develop a unified theory of very weak solutions of the Dirichlet problem for Stokes, Oseen and Navier-Stokes equations (and also for the Laplace equation), see Theorem 4.6 and Theorem 5.3. One important question is to define rigorously the traces of the vector functions which are living in subspaces of L p (Ω) (see Lemma 2.10 and Lemma 2.11). We prove existence and regularity of very weak solutions (u, q) ∈ L p (Ω) × W −1,p (Ω) of Stokes and Oseen equations for any 1 < p < ∞ with arbitrary large data belonging to some Sobolev spaces of negative order. In the case of Navier-Stokes equations the existence of very weak solution is proved for arbitrary large external forces, but with a smallness condition for both h and g . Uniqueness of very weak solutions is also proved for small enough data.
Existence of very weak solution u ∈ L 3 (Ω), for arbitrary large external forces f ∈ H −1 (Ω), h = 0 and arbitrary large boundary condition g ∈ L 2 (Γ) and without assuming condition (1), was proved first by Marusic-Paloka in [20] (see Theorem 5) with Ω a bounded simply-connected open set of class C 1,1 . But the proof of Theorem 5 becomes correct only if either condition (1) or condition (56) holds. The same result was proved by Kim [17] for arbitrary large external forces f ∈ [W 1,3/2 0 (Ω) ∩ W 2,3 (Ω)] , for small h ∈ [W 1,3/2 (Ω)] and g ∈ W −1/3,3 (Γ) and where the boundary of Ω is supposed connected (I = 0). Remark that the space chosen for the divergence condition h is not correct, because D(Ω) is not dense in W −1/3,3 (Γ) and his dual is not a subspace of distributions. Similar argument can be done for the space chosen for the external forces f . The origin of this mistake (also present everywhere in the same paper [17]) is due to the fact that when we want to solve a boundary value problem, it is necessary to have an adequate Green formula and corresponding density lemmas.
In a close context, we also consider the case where the data, and then the solutions, belong to fractionary Sobolev spaces W s,p (Ω) with s a real number possibly not integer (see Theorem 4.7) The work is organized as follows: In the remains of this section, we recall the definitions of some spaces and their respective norms.
In §2, some preliminary results are stated, including density lemmas, general trace's results, characterization of dual spaces and trace's result for very weak solutions. In §3, we present Stokes' results related to the very weak, weak and strong solution. Some of them generalized those appearing in [3] in order to be extended hal-00549173, version 1 -21 Dec 2010   ON THE VERY WEAK SOLUTION FOR OSEEN AND NAVIER-STOKES EQS   3 to the Oseen and Navier-Stokes systems. In §4, we extend the results fo §3 for the Oseen system. The first two main results in this paper are presented here: one about existence and uniqueness of very weak solution for the Oseen equations in L p (Ω) × W −1,p (Ω) with 1 < p < ∞ (see Theorem 4.6), and another one related to the regularity of solutions for the Oseen equations (see Theorem 4.7). We consider in particular the case where the external forces f and the divergence condition h are not regular, more precisely f ∈ W σ−2,p (Ω) and h ∈ W σ−1,p (Ω) with 1 p < σ ≤ 2. In §5, existence of very weak solution for the Navier-Stokes system is obtained, using a fixed point technique over the Oseen system, first for the case of small data and then for arbitrary large external forces f but sufficiently small h and g in a domain possibly multiply-connected. The results is stated in Theorem 5.3. Regularity results for this system are obtained in Theorem 5.4. The complete proofs of results can be seen in [5].
In all this work, if we do not say anything else, Ω will be considered as a Lipschitz open bounded set of R 3 . When Ω is connected, we will say Ω is a domain. We will only specify the regularity of Ω when it to be different from the regularity presented above.

Functional framework.
In what follows, for any s ∈ R, p denotes a real number such that 1 < p < ∞ and p stands for its conjugate: 1/p + 1/p = 1. We shall denote by m the integer part of s and by σ its fractional part: for all |α| = m, when s = m + σ is nonnegative and is not an integer. The space W s,p (R 3 ) is a reflexive Banach space equipped by the norm: in the first case, and by the norm In the special case of p = 2, we shall use the notation H s (R 3 ) instead of W s, 2 (R 3 ). Now, we introduce the Sobolev space It is known that H s,p (R 3 ) = W s,p (R 3 ) if s is an integer or if p = 2. Furthermore, for any real number s, we have the following embeddings: be identified to a space of distributions in Ω. For this reason, we define W s,p 0 (Ω) as the closure of D(Ω) in W s,p (Ω) and we denote by W −s, p (Ω) its dual space.
For every s > 0, we denote by W s,p (Ω) the space of all distributions in Ω which are restrictions of elements of W s,p (R 3 ) and by W s,p (Ω) the space of functions u ∈ W s,p (Ω) such that the extension u by zero outside of Ω belongs to W s,p (R 3 ).
2. Preliminary results. We present here some trace results, density results, De Rham's theorems and characterizations of some spaces, either known or designed specially for the Stokes, Oseen and Navier-Stokes problems, that will be used in the following sections.
Let Ω be a bounded open set of class C k,1 , for some integer k ≥ 0. Let s be real number such that s ≤ k + 1, s − 1/p = m + σ, where m ≥ 0 is an integer and 0 < σ < 1.
We recall also the following embeddings: where k is a non negative integer.
Then, we introduce the following spaces: Recall now two versions of De Rham's Theorem, the first one proved by G. de Rham [22] Then, there exists a distribution π in D (Ω) such that f = ∇π.
Then, there exists π ∈ W −m+1,p (Ω) such that f = ∇π. If in addition the set Ω is connected, then π is defined uniquely, up to an additive constant, and there exists a positive constant C, independent of f, such that: The two next lemmas are density results: Sketch of the proof. Let be a linear and continuous mapping in H p (Ω) such that , v = 0 for any v ∈ D σ (Ω). We want to prove that = 0. Since H p is a subspace of L p (Ω), we can extend to L ∈ L p (Ω). We will suppose that Ω is bounded, connected but eventually multiply-connected (when Ω is not connected, we can repeat the procedure above in each connected component of Ω), being 1≤i≤I ω i its wholes, and its boundary Γ is Lipschitz-continuous. We denote by ω 0 the exterior of Ω, by Γ 0 the exterior boundary of Ω and by Γ i , 1 ≤ i ≤ I, the other components of Γ. The duality between W −1/p,p (Γ i ) and W 1/p,p (Γ i ), and W −1/p,p (Γ 0 ) and W 1/p,p (Γ 0 ), will be denoted by ·, · Γi and ·, · Γ0 , respectively. By De Rham's Lemma 2.3, there exists a unique q ∈ W 1,p (Ω) ∩ L p 0 (Ω) such that L = ∇q and We extend L by zero out of Ω and denote the extension by L. Then, for any From that, we deduce that, thanks to De Rham's Lemma 2.2, there exists h ∈ D (R 3 ) verifying ∇h ∈ L p (R 3 ) such that L = ∇h (see Lemma 2.1 in [4]). It is clear that h ∈ W 1,p loc (R 3 ). As h is unique up to an additive constant and ∇h = 0 in ω 0 , we can choose this constant in such a way that h = 0 in ω 0 . Therefore, we deduce that: . . , I} be a fixed index, choosing v j ∈ D σ (Ω) such that v j ·n, 1 Γ k = δ jk for 1 ≤ k ≤ I and v j · n, 1 Γ0 = −1, we can deduce that c j = 0. In consequence, for every v ∈ H p (Ω), we have: In the sequel, we will use the following space and we set X p,p (Ω) = X p (Ω). Observe that the space X p,p (Ω) were used in [3] in order to define very weak solution for the Stokes problem. In the case of Navier-Stokes problem, the generalization to the space X r,p (Ω) is necessary. In this sense, the proof of the next result follows from an argument appearing in [2].
Lemma 2.5. The space D(Ω) is dense in X r,p (Ω) and for all q ∈ W −1,p (Ω) and ϕ ∈ X r ,p (Ω), we have Next lemmas characterize the space (X r,p (Ω)) and give a density result.
As a consequence of Lemma 2.5, we have the following embeddings: where the second embedding holds if 1 r ≤ 1 p + 1 3 .

Lemma 2.7.
Let Ω be a Lipschitz bounded open set. Then, the space D(Ω) is dense in (X r,p (Ω)) .
One of the main difficulties for the definition of a very weak solution for Stokes, Oseen and Navier-Stokes problems is to give a meaning to the trace, because we are not in the classical variational framework. We shall use the spaces 1 : T p,r (Ω) = {v ∈ L p (Ω); ∆v ∈ (X r ,p (Ω)) }, T p,r,σ (Ω) = {v ∈ T p,r (Ω); ∇·v = 0}, endowed with the topology given by the norm:

and
H p,r (div; Ω) = {v ∈ L p (Ω); ∇ · v ∈ L r (Ω)}, which is equipped with the graph norm. Next density lemmas will be necessary: For the following two lemmas, we will need to introduce the space: that can also be described (see [3]) as: Observe that the range space of the normal derivative γ 1 : In these lemmas, we prove that the tangential trace of functions v of T p,r,σ (Ω) belongs to the dual space of Z p (Γ), which is: Recall that we can decompose v into its tangential, v τ , and normal parts, that is: 3 can be extended by continuity to a linear and continuous mapping, still denoted by γ τ , from T p,r (Ω) into W −1/p,p (Γ). The Green formula reads: for any v ∈ T p,r (Ω) and Lemma 2.11.
i) The space D(Ω) 3 is dense in H p,r (div; Ω). ii) Let 1 < p < ∞ and r > 1 such that 1 r ≤ 1 p + 1 3 . The mapping γ n : v → v · n| Γ on the space D(Ω) 3 can be extended by continuity to a linear and continuous mapping, still denoted by γ n , from H p,r (div; Ω) into W −1/p,p (Γ), and we have the Green formula: for any v ∈ H p,r (div; Ω) and ϕ ∈ W 1,p (Ω),

Lemma 2.12.
Let Ω be a Lipschitz bounded open set. Let h ∈ L r (Ω) and g ∈ W −1/p,p (Γ) be given such that the condition (11) holds. For every ε > 0, there exist and verifying In all the rest of this work , if we do not say anything else, we assume that Ω is a bounded connected open set of class C 1,1 .

CHÉRIF AMROUCHE AND M.ÁNGELES RODRÍGUEZ-BELLIDO
3. The Stokes problem. Before starting the study of the Oseen and Navier-Stokes problems, we focus on the study of the Stokes problem in order to make an appointment about all the knowing results about this system. Recall that the Stokes problem is: with the compatibility condition: Basic results on weak and strong solutions of problem (S) in L p (Ω) Sobolev spaces may be summarized in the following theorem (see [3], [8], [12]).
, and satisfying the compatibility condition (11), the Stokes problem (S) has exactly one solution u ∈ W 1,p (Ω) and q ∈ L p (Ω)/R. Moreover, there exists a constant C > 0 depending only on p and Ω such that: q ∈ W 1,p (Ω) and there exists a constant C > 0 depending only on p and Ω such that: In the case of a bounded domain Ω which is only Lipschitz, the result of point i) is only valid for a more restricted p. In fact, if f = 0, h = 0 and g ∈ W 1−1/p,p (Γ) with Γ g · n = 0, then there exists ε > 0 such that if 2 ≤ p ≤ 3 + ε, and if f ∈ W −1,p (Ω), h = 0 and g = 0, then the result is valif for a ε such that (3 + ε)/(2 + ε) < p < 3 + ε (see [7]).
We are interested in the case of singular data satisfying the following assumptions: and r ≤ p. (14) Recall that the space (X r ,p (Ω)) is an intermediate space between W −1,r (Ω) and W −2,p (Ω) (see embeddings (5)).
We recall the definition and the existence result of very weak solution for the Stokes problem.
Definition 3.2 (Very weak solution for the Stokes problem). We say that (u, q) ∈ L p (Ω)×W −1,p (Ω) is a very weak solution of (S) if the following equalities hold: For any ϕ ∈ Y p (Ω) and π ∈ W 1,p (Ω), where the dualities on Ω and Γ are defined by: Note that W 1,p (Ω) → L r (Ω) and Y p (Ω) → X r ,p (Ω) if 1 r ≤ 1 p + 1 3 , that means that all the brackets and integrals have a sense. Proposition 1. Suppose that f, h, g satisfy (14). Then the following two statements are equivalent: is a very weak solution of (S), ii) (u, q) satisfies the system (S) in the sense of distributions.
Sketch of the proof. i) Let (u, q) very weak solution to problem (S). It is clear that −∆u + ∇q = f and ∇ · u = h in Ω and consequently u belongs to T p,r (Ω). Using Lemma 2.11 point ii), Lemma 2.10 and (3), we obtain Since for any ϕ ∈ Y p (Ω), Consequently u · n = g · n in W −1/p,p (Γ) and finally u = g on Γ.
ii) The converse is a simple consequence of Lemma 2.11 point ii), Lemma 2.10 and (3).
The following result is a variation from Proposition 4.11 in [3], which was made for f = 0 and h = 0. In the case r = p, we have and satisfying the compatibility condition (11). Then, the Stokes problem (S) has exactly one solution u ∈ L p (Ω) and q ∈ W −1,p (Ω)/R. Moreover, there exists a constant C > 0 depending only on p and Ω such that: Moreover u ∈ T p (Ω) and More generally, taking into account that now we use f ∈ (X r ,p (Ω)) instead of f ∈ (X p (Ω)) and h ∈ L r (Ω) instead of h ∈ L p (Ω), we can adapt Proposition 2 obtaining: Theorem 3.3. Let f, h, g be given satisfying (14) and (11). Then, the Stokes problem (S) has exactly one solution u ∈ L p (Ω) and q ∈ W −1,p (Ω)/R. Moreover, there exists a constant C > 0 depending only on p and Ω such that: Moreover u ∈ T p,r (Ω) and In with the corresponding estimates.
Although that in [14] Theorem 3 the authors obtain a similar result, observe that the domain is considered of class C 2,1 instead of class C 1,1 , and the divergence term h ∈ L p (Ω) instead of h ∈ L r (Ω). Moreover, our solution is obtained in the space T p,r (Ω) which has been clearly characterized contrary to the space W 1,p (Ω) appearing in [14] which is not charaterized, is completely abstract and is obtained as closure of W 1,p (Ω) for the norm where A r is the Stokes operator with domain equal to W 2,p (Ω) ∩ W 1,p 0 (Ω) ∩ L p σ (Ω) and P r is the Helmholtz projection operator from L r (Ω) onto L r σ (Ω).
Corollary 1. Let f, h, g be given satisfying (11) and Then the solution u given by Theorem 3.3 belongs to W 1,r (Ω). Moreover, if f 1 belongs to L r (Ω), then the solution q given by Theorem 3.3 belongs to L r (Ω). In the both cases, we have the corresponding estimates.
Corollary 2. Let us consider h and g satisfying: Moreover, there exists a constant C = C(Ω, p, r) such that: The following corollary gives the existence of a unique Stokes solution (u, q) in fractionary Sobolev spaces of type W σ,p (Ω) × W σ−1,p (Ω), with 0 < σ < 2 by using an interpolation argument. Corollary 3. Let s be a real number such that 0 ≤ s ≤ 1.
4. The Oseen problem. We want to study the existence of a generalized, strong and very weak solutions for the problem (O), given by:  verifying the compatibility condition (11) for p = 2. Then, the problem (O) has a unique solution (u, q) ∈ H 1 (Ω) × L 2 (Ω)/R. Moreover, there exist some constants C 1 > 0 and C 2 > 0 such that: where C 1 = C(Ω) and Proof. In order to prove the existence of solution, first (using Lemma 3.3 in [3], for instance) we lift the boundary and the divergence data. Then, there exists u 0 ∈ H 1 (Ω) such that ∇ · u 0 = h in Ω, u 0 = g on Γ and: Therefore, it remains to find (z , q) = (u − u 0 , q) in H 1 0 (Ω) × L 2 (Ω) such that: −∆z − v · ∇z + ∇q = f and ∇ · z = 0 in Ω, z = 0 on Γ.

ON THE VERY WEAK SOLUTION FOR OSEEN AND NAVIER-STOKES EQS 13
where b is a trilinear antisymmetric form with respect to the last two variables, well-defined for v ∈ L 3 (Ω), z , ϕ ∈ H 1 0 (Ω). (We can recover the pressure π thanks to the De Rham's Lemma 2.3). By Lax-Milgram's Theorem we can deduce the existence of a unique z ∈ H 1 0 (Ω) verifying: which added to estimate (22) makes (20).
Thanks to De Rham's Lemma 2.3, there exists a unique q ∈ L 2 (Ω)/R such that: . Finally, estimate (21) follows from the previous equation and estimate for z .
Thus, we focus on the getting of a strong estimate for (u λ , q λ ). Let ε > 0 with 0 < λ < ε/2. We consider where v is the extension of v by zero to R 3 and ρ ε is the classical mollifier. By regularity estimates for the Stokes problem, we have Now, we use the decomposition (27) in order to bound the term v λ · ∇u λ L p (Ω) . We observe first that v ε λ,2 L s (Ω) Recall that W 2,p (Ω) → W 1,k (Ω) (29) for any k ∈ [1, p * ], with 1 p * = 1 p − 1 3 , if p < 3, for any k ≥ 1 if p = 3 and for any k ∈ [1, ∞] if p > 3. Moreover the embedding is compact for any q ∈ [1, p * [ if p < 3, for any q ∈ [1, ∞[ if p = 3 and for q ∈ [1, ∞] if p > 3. Then, using the Hlder inequality and the Sobolev embedding, we obtain v ε λ,2 · ∇u λ L p (Ω) ≤ v ε λ,2 L s (Ω) ∇u λ L k (Ω) ≤ C ε u λ W 2,p (Ω) where 1 k = 1 p − 1 s , which is well defined (see the defintion of the real number s). For the second estimate, we consider two cases. i) Case p ≤ 2. Let r ∈ ]3, ∞] be such that 1 p = 1 r + 1 2 and t ≥ 1 such that 1+ 1 Using the estimate (25) From (32) and (31), we deduce that ii) Case p > 2. First, we choose the exponent q given in (30) such that q > 2. For any ε , we known that there exists C ε > 0 such that Let first consider p < 3 and choose q < p * and close of p * . Then, there exist r > 3 such that 1 p = 1 r + 1 q and t > 1 such that 1 + 1 where we choose q = ∞ if p > 3 and q large enough if p = 3. In the both cases, in order to control the first term on the right hand side of (28) with the term on the left hand side, we fix ε and ε small enough to obtain Thus, we deduce that (u λ , q λ ) satisfies (33), where we replace v L 3 by v L s .
The estimate (33) is uniform with respect to λ, and therefore we can extract subsequences, that we still call {u λ } λ and {q λ } λ , such that if λ → 0, u λ −→ u weakly in W 2,p (Ω), and for the pressure, there exists a sequence of real numbers k λ such that q λ + k λ → q weakly in W 1,p (Ω).
It is easy to verify that (u, q) is solution of (O) satisfying estimate (24) and this solution is unique.
Thanks to the strong regularity, we can deduce the following regularity: be given verifying the compatibility condition (11). Then, the problem (O) has a unique solution (u, q) ∈ W 1,p (Ω) × L p (Ω)/R. Moreover, there exists some constant holds.
Using quickly the reasoning given in Theorem 4.2, we can improve estimates (35) and (36) for some values of p: Proposition 3. Under the assumptions of Theorem 4.3 and supposing that 6 5 ≤ p ≤ 6, the solution (u, q) satisfies the estimate: Moreover assuming v · n = 0 on Γ, then the estimate (39) holds for any 1 < p < ∞.

ON THE VERY WEAK SOLUTION FOR OSEEN AND NAVIER-STOKES EQS
Remark 4. If we suppose that v ∈ H p (Ω), then estimate (39), where we replace the norm v L 3 (Ω) by v L p (Ω) , holds when p > 6 (and then also, by duality argument, when p < 6/5 and v ∈ H p (Ω)).
The concepts of weak and strong solutions are known for the Oseen equations. Now, we define and prove the existence of a very weak solution for the Oseen equations.
It suffices to consider the case where g · n| Γ = 0 and Ω h(x ) dx = 0, the general case is similar to the proof given in the end of Proposition 2. The result can be deduced (see [5]) applying the Riesz's Lemma.
Similarly to Corollary 3, we can prove: i) Let σ be a real number such that 0 < σ < 1. Let f = ∇ · F 0 + ∇f 1 , h and g satisfy the compatibility condition (11) with Then, the Oseen problem (O) has a unique solution (u, q) belonging to W σ,p (Ω)× W σ−1,p (Ω)/R and satisfying the estimate
Let v ∈ H s (Ω) satisfy (42). Then, the Oseen problem (O) has exactly one solution (u, q) ∈ W σ,p (Ω) × W σ−1,p (Ω)/R satisfying the estimate Proof. The proof is similar to proof of Theorem 3.4. It suffices to study the new term containing the function v .

5.
The Navier-Stokes problem. First of all, we give the definition of a very weak solution for the Navier-Stokes equations.

hal-00549173, version 1 -21 Dec 2010
In the stationary Navier-Stokes equations, the data h and g play an special role, making possible or not the existence of a very weak solution. If h and g are small enough, then the result is true. Until we now, we think that it is not possible to eliminate this latest condition.
Therefore, we present first three results related to the existence of very weak solution: the two first for the small external forces case (following the scheme used by Marusič-Paloka [20]) and the third one for the general Navier-Stokes case, always supposing that h and g are small enough in their respective norms. Last result involves the regularity for the Navier-Stokes equations.
ii) Moreover there exists a constant α 2 ∈ ]0, α 1 ] such that this solution is unique, up to an additive constant for q, if Proof. i) Existence. The existence of a very weak solution is made through the application of the Banach's fixed point theorem. We do this fixed point over the Oseen equations, written in an adequate manner. We are searching for a fixed point for the application T , where given v ∈ H 3 (Ω), T v = u is the unique solution of (O) given by Theorem 4.6. We also need to define a neighborhood B r , in the form: In order to prove the contraction of the operator, we must prove that: there exists θ ∈ ]0, 1[ such that Searching for an estimate of u 1 − u 2 L 3 (Ω) , we observe that for each i = 1, 2, we have with the estimates where C > 0 is the constant given by (43). Moreover, for estimating the difference u 1 − u 2 , we look for the problem verified by (u, q) = (u 1 − u 2 , q 1 − q 2 ), which is: Using the very weak estimates (43) made for the Oseen problem successively for u and for u 2 , we obtain that: . Thus, we obtain estimate (52) considering C 2 β (1 + r) 2 < 1 which is verified, for example, taking: Therefore, if (54) is verified, using again estimate (43) we conclude that the fixed pointū ∈ L 3 (Ω) verifies: If we also choose β such that β < (2 C) −1 , then: Setting α 1 = min (2C) −1 , (2C 2 ) −1 , then estimate (47) is satisfied. For the estimate of the associated pressure, we deduce from the equations ∇q = ∆ū −ū ·∇ū +f and (47) that: where C 1 is the continuity constant of the Sobolev embedding [X 3,3/2 (Ω)] → W −2,3 (Ω) and C 2 is the continuity constant of the Sobolev embedding W 1,3/2 0 (Ω) → L 3 (Ω), which is (48) and the proof of existence is completed.
ii) Uniqueness. We shall next prove uniqueness. Let us denote by (u 1 , q 1 ) the solution obtained in step i) and by (u 2 , q 2 ) any other very weak solution corresponding to the same data. Setting u = u 1 − u 2 and q = q 1 − q 2 . We find that −∆u + u 2 · ∇u + ∇q = −u · ∇u 1 and div u = 0 in Ω, u = 0 on Γ. As u · ∇u 1 belongs to W −1,3/2 (Ω), using uniqueness argument and Proposition 3, the function u belongs to W 1,3/2 (Ω) and we have the estimate where C 1 > 0 is given by (39). Thanks to Theorem 4.6, we have also: where C > 0 is the constant given in (43). We deduce then provided that β ≤ α 1 . Using finally the embedding W 1,3/2 (Ω) → L 3 (Ω), we obtain the estimate where C 2 is the continuity constant of the above embedding. Consequently We deduce that u = 0 and the proof of uniqueness is completed.
Sketch of the proof. We decompose the problem into two parts. First, we are looking to find a pair (v ε , q 1 ε ) solution of the problem: and then to find (z ε , q 2 ε ) solution of the problem: (see Lemma 2.7 and Lemma 2.12). The pair (u, q) = (v ε + z ε , q 1 ε + q 2 ε ) is then solution to problem (NS).
The existence of solution for (N S 1 ) follows from Theorem 5.2 and solution of (N S 2 ) is based on the classical theory and the use of Hopf's Lemma (see [13], Remark VIII.4.4 for instance).
v) Galdi et al. in [14] prove Theorem 5.3 and Theorem 5.4 point i) with f = div F 0 , where F 0 ∈ L r (Ω), h ∈ L p (Ω) and g ∈ W −1/p,p (Γ) with 1 r ≤ 1 p + 1 3 and max{2r, 3} ≤ p. They assume the domain Ω is of class C 2,1 . Moreover they suppose f , h and g sufficiently small with respect to their norms. The small condition on the external forces is in fact unnecessary.