Muriel Boulakia

Quantification of the unique continuation property for the nonstationary Stokes problem

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Quantification of the unique continuation property for the nonstationary Stokes problem

Introduction

Let Ω be a regular bounded connected open set of class C 2 in dimension 3. For any fixed final time T > 0, we define Q = (0, T ) × Ω. We consider the nonstationary Stokes problem

u t -∆u + ∇p = 0, in Q, div u = 0, in Q, (1)
where u and p denote respectively the fluid velocity and the fluid pressure. Since the work made by Fabre and Lebeau in [START_REF] Fabre | Prolongement unique des solutions de l'equation de Stokes[END_REF], the unique continuation property of this system is a well-known property. It is given by the following result Proposition 1. Let ω be a nonempty open subset of Ω. If (u, p) is a solution of (1) which belongs to L 2 (0, T ; H 1 loc (Ω)) × L 2 loc (Q) and if u = 0 in (0, T ) × ω , then u = 0 and p is constant in Q.

This property directly implies

Corollary 1. Let Γ ⊂ ∂Ω be a nonempty open subset of ∂Ω. If (u, p) is a solution of (1) which belongs to L 2 (0, T ; H 2 (Ω)) × L 2 (0, T ; H 1 (Ω)) and if u = 0, ∇u • npn = 0 on (0, T ) × Γ, then u = 0, p = 0 in Q.

Let us emphasize that these stability inequalities hold without prescribing any boundary conditions on the solution. The logarithmic nature of the inequalities comes from the fact that we estimate the norm of u and p on the whole domain Ω. If we are interested by interior estimates of u and p, we get inequalities of Hölder type (Propositions 2 and 3).

If we compare this result to its counterpart for the steady equation in [START_REF] Boulakia | Stability estimates for the unique continuation property of the Stokes system and for an inverse boundary coefficient problem[END_REF] (we refer to Theorem 1.4 in this paper), we notice that we need the same kind of measurements on u and p. In particular, in both cases, extra measurements are necessary compared to the unique continuation property proved by Fabre and Lebeau. This is linked to the fact that we need global measurements on the velocity and the pressure. In [START_REF] Lin | Optimal three-ball inequalities and quantitative uniqueness for the Stokes system[END_REF] and [START_REF] Ignatova | Strong unique continuation for the Navier-Stokes equation with nonanalytic forcing[END_REF], local estimates are proved which only require measurements on u. In [START_REF] Lin | Optimal three-ball inequalities and quantitative uniqueness for the Stokes system[END_REF], a three-balls inequality for the stationary Stokes problem is proved which only involves the L 2 -norm of the velocity. It leads to a quantification of the unique continuation property like in Theorem 1 of the following type:

u L 2 (A) ≤ C u 1-θ L 2 (Ω) u θ L 2 (ω)
where A is a compact subset of Ω and 0 < θ < 1. In [START_REF] Ignatova | Strong unique continuation for the Navier-Stokes equation with nonanalytic forcing[END_REF], the authors prove a local stability estimate which only involves the velocity for the solution of the Navier-Stokes equation. They assume that the data belong to a Gevrey class and enforce specific conditions on the solution which are satisfied if periodic boundary conditions are prescribed.

For the quantification of the unique continuation property of the Laplace equation, let us quote among others the works [START_REF] Alessandrini | Optimal stability for inverse elliptic boundary value problems with unknown boundaries[END_REF], [START_REF] Bourgeois | About stability and regularization of ill-posed elliptic Cauchy problems: the case of Lipschitz domains[END_REF], [START_REF] Phung | Remarques sur l'observabilité pour l'équation de laplace[END_REF]. In this case, it is well established that the best possible rate for the global stability is logarithmic. We refer to the overview [START_REF] Alessandrini | The stability for the Cauchy problem for elliptic equations[END_REF] and the references therein for works on the stability estimates for elliptic equations.

The proof of our main result relies on estimates of propagation of smallness in the interior and up to the boundary. These estimates are stated in section 2 and give simultaneous estimates on u and p. Another method could have been to prove a global estimate on the velocity alone with the help of adapted Carleman estimates. According to Stokes equation, this would allow to directly get an estimate on ∇p and, thanks to an adapted Poincaré inequality, this leads to an estimate on p if we also have measurements of p on an arbitrary sub-domain. This alternative method seems to lead to similar measurements as the ones in the inequalities given by Theorem 1.

In a second step, we will be interested in applying this quantification result to get a stability estimate for an inverse problem which has already been studied in [START_REF] Boulakia | Stability estimates for a Robin coefficient in the two-dimensional Stokes system[END_REF] and in [START_REF] Boulakia | Stability estimates for the unique continuation property of the Stokes system and for an inverse boundary coefficient problem[END_REF]. Our objective will be to identify a Robin coefficient defined on some part of the boundary from measurements available on another part of the boundary. To study this inverse problem, it is capital to have an estimate of the pressure and the velocity on the whole domain like the one given by Theorem 1.

More precisely, we assume that the boundary ∂Ω is composed of two open non-empty parts Γ 0 and Γ e such that Γ e ∪ Γ 0 = ∂Ω and Γ e ∩ Γ 0 = ∅ and we consider the following problem

           u t -∆u + ∇p = 0, in Q, div u = 0, in Q, ∇u • n -pn = g, on (0, T ) × Γ e , ∇u • n -pn + qu = 0, on (0, T ) × Γ 0 , u(0, •) = u 0 , in Ω. (6)
Such system may be viewed as a simple model of the blood flow in the cardiovascular system (see [START_REF] Quarteroni | Analysis of a geometrical multiscale model based on the coupling of ODEs and PDEs for blood flow simulations[END_REF] and [START_REF] Vignon-Clementel | Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries[END_REF]) or of the airflow in the respiratory tract (see [START_REF] Baffico | Multiscale modeling of the respiratory tract[END_REF]). We refer to [START_REF] Egloffe | Étude de quelques problèmes inverses pour le système de Stokes[END_REF] for a presentation in this last area of application. In these contexts, the real geometry is truncated and the properties of the upstream domain are condensed on the boundary conditions which are prescribed on the artificial boundary. The boundary part Γ e corresponds to the external boundary on which measurements are available and the boundary part Γ 0 corresponds to an artificial boundary on which Robin boundary conditions are prescribed. For similar studies with the identification of a Robin coefficient with the Laplace equation, we refer to [START_REF] Alessandrini | Detecting nonlinear corrosion by electrostatic measurements[END_REF] and [START_REF] Chaabane | Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems[END_REF] and with the heat equation to [START_REF] Bellassoued | Stability estimate for an inverse boundary coefficient problem in thermal imaging[END_REF] (see also the references therein).

In [START_REF] Boulakia | Stability estimates for a Robin coefficient in the two-dimensional Stokes system[END_REF], we proved a stability result (see Theorem 4.18 in this reference) which holds for a parameter q which does not depend on time and for measurements made on the interval [0, +∞[. As in [START_REF] Bellassoued | Stability estimate for an inverse boundary coefficient problem in thermal imaging[END_REF] for the Laplace equation, this result relies on properties satisfied by the semigroup generated by the operator associated to the problem and is proved by comparing the solution of the non-stationary problem with the solution of the stationary problem. The quantification of the unique continuation property given by Theorem 1 allows to generalize the result given in [START_REF] Boulakia | Stability estimates for a Robin coefficient in the two-dimensional Stokes system[END_REF] to a parameter q which depends on time and to measurements made on a finite interval. More precisely, we have the following result: Theorem 2. Let Ω be of class C 2,1 and Γ ⊆ Γ e be a nonempty open subset of the boundary of Ω. Let

ν 0 > 0 and N 0 > 0. Let u 0 ∈ H 4 (Ω) ∩ V , g ∈ H 2 (0, T ; L 2 (Γ e )) ∩ H 1 (0, T ; H 3 2 (Γ e )) be non identically zero and q 1 , q 2 ∈ H 2 (0, T ; H 2 (Γ 0 )) such that q 1 , q 2 ≥ ν 0 on (0, T ) × Γ 0 . We assume that u 0 H 4 (Ω) + g H 2 (0,T ;L 2 (Γe))∩H 1 (0,T ;H 3 2 (Γe)) + 2 j=1 q j H 2 (0,T ;H 2 (Γ0)) ≤ N 0 .
We denote by (u j , p j ) the solution of system (6) with q = q j for j = 1, 2. Let K be a compact subset of

{(t, x) ∈ (ε, T -ε) × Γ 0 / u 1 = 0} for some ε > 0 and m > 0 be such that |u 1 | ≥ m on K.
Then, there exists α > 0 independent of ε, C > 0 which depends on ε, ν 0 and N 0 such that

q 1 -q 2 C(K) ≤ 1 m C log C u 1 -u 2 L 2 ((0,T )×Γ) + p 1 -p 2 L 2 ((0,T )×Γ) + ∇(p 1 -p 2 ) • n L 2 ((0,T )×Γ) α . (7)
We have used the following notation

V = {v ∈ H 1 (Ω)/ div v = 0 in Ω}.
In the hypotheses of this theorem, the existence of the constant m and of the compact K is ensured by the continuity of u 1 and the fact that u 1 can not be identically null on (0, T ) × Γ 0 . This last property is due to the unique continuation property (Corollary 1) and the hypothesis that g is non identically null. Through m and K, the estimate given in this theorem depends on u 1 . To get an estimate on the whole set (0, T ) × Γ 0 , it would be necessary to prove a lower bound on the velocity obtained thanks to a doubling inequality on the boundary ( [START_REF] Alessandrini | Optimal stability for inverse elliptic boundary value problems with unknown boundaries[END_REF]). This sometimes may lead to estimates of log-log type like in [START_REF] Ballerini | Stable determination of an immersed body in a stationary Stokes fluid[END_REF] or [START_REF] Bellassoued | Stability of the determination of the surface impedance of an obstacle from the scattering amplitude[END_REF]. In our case, the interior doubling inequality obtained in [START_REF] Ignatova | Strong unique continuation for the Navier-Stokes equation with nonanalytic forcing[END_REF] (Theorem 2.1) with an exponential rate with respect to the radius of the ball leads us to believe that we could obtain a log-log inequality.

In the next section, we present local estimates of u and p in the interior of the domain or near the boundary. We then gather these inequalities to prove Theorem 1. Section 3 is dedicated to the proof of these local estimates. At last, in Section 4, we apply our estimates to the identification problem of a Robin coefficient and prove Theorem 2.

Local estimates of the solution

In what follows, we will use the following notation: for t 1 < t 2 , we define

H 1,0 ((t 1 , t 2 ) × Ω) = {u ∈ L 2 ((t 1 , t 2 ) × Ω)/∇u ∈ L 2 ((t 1 , t 2 ) × Ω)}.
Theorem 1 will be proved with the help of three propositions that we state now. The proofs of these propositions rely on local Carleman estimates for parabolic and elliptic equations. In [START_REF] Boulakia | Stability estimates for the unique continuation property of the Stokes system and for an inverse boundary coefficient problem[END_REF], our quantification result was based on local Carleman inequalities ( [START_REF] Hörmander | The analysis of linear partial differential operators. III[END_REF], [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF] and [START_REF] Robbiano | Théorème d'unicité adapté au contrôle des solutions des problèmes hyperboliques[END_REF]) obtained thanks to Gårding inequalities involving pseudodifferential computation. The same inequalities were used in [START_REF] Phung | Remarques sur l'observabilité pour l'équation de laplace[END_REF] to quantify the unique continuation property for the Laplace equation. We refer to the survey [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF] (and the references therein) for a general presentation of these local Carleman estimates in the elliptic and parabolic cases. Here, the local Carleman estimates that we will use are derived through direct computations. Like the global Carleman inequalities, they are obtained thanks to the method of Fursikov and Imanuvilov [START_REF] Fursikov | Controllability of evolution equations[END_REF]. We call them local Carleman estimates because they are stated on a subdomain of (0, T ) × Ω where we do not prescribe boundary conditions on the solutions. Regarding the Carleman inequalities that we will use, the inequality for the parabolic case is stated in [START_REF] Yamamoto | Carleman estimates for parabolic equations and applications[END_REF] and the inequality in the elliptic case can be proved with the methods presented in [START_REF] Fursikov | Controllability of evolution equations[END_REF].

The first proposition gives an estimate of u and p in the interior of Ω with respect to measurements on a part of the boundary of Ω: Proposition 2. Let Γ ⊂ ∂Ω be a nonempty open subset of ∂Ω and let Ω 0 be a nonempty open set such that Ω 0 ⊂ Ω ∪ Γ and ∂Ω 0 ∩ ∂Ω Γ. There exists θ ∈ (0, 1) and, for any ε > 0, there exists

C > 0 such that u H 1,0 ((ε,T -ε)×Ω0) + p H 1,0 ((ε,T -ε)×Ω0) ≤ C u H 1,0 (Q) + p H 1,0 (Q) 1-θ F θ , (8)
for all (u, p) solution of (1) in H 1,0 ((0, T ) × Ω) × H 1,0 ((0, T ) × Ω). In this inequality, F is defined by

F = u H 1 ((0,T )×Γ) + ∇u • n L 2 ((0,T )×Γ) + p L 2 (0,T ;H 1 (Γ)) + ∇p • n L 2 ((0,T )×Γ) . (9)
If we compare estimate [START_REF] Boulakia | Stability estimates for a Robin coefficient in the two-dimensional Stokes system[END_REF] with the equivalent estimate proved for the stationary Stokes equation in [START_REF] Boulakia | Stability estimates for the unique continuation property of the Stokes system and for an inverse boundary coefficient problem[END_REF] (see Proposition 2.6 in this reference), we see that the norms of the measurements are similar (the norms of the measurements in (8) correspond to the L 2 -norms in time of the norms of the measurements in Proposition 2.6 in [START_REF] Boulakia | Stability estimates for the unique continuation property of the Stokes system and for an inverse boundary coefficient problem[END_REF]) except that we need an additional measurement of u in H 1 (0, T ; L 2 (Γ)) for the estimate [START_REF] Boulakia | Stability estimates for a Robin coefficient in the two-dimensional Stokes system[END_REF]. For parabolic equations like heat equation, it is proved in [START_REF] Yamamoto | Carleman estimates for parabolic equations and applications[END_REF] that this norm can not be removed, otherwise the estimate fails.

The second proposition gives an estimate of u and p in the interior of Ω with respect to measurements in the interior: Proposition 3. Let ω be a nonempty open subset of Ω and let Ω 0 ⊂ Ω be a nonempty open set relatively compact in Ω. There exists θ ∈ (0, 1) and, for any ε > 0, there exists C > 0 such that

u H 1,0 ((ε,T -ε)×Ω0) + p H 1,0 ((ε,T -ε)×Ω0) ≤ C u H 1,0 (Q) + p L 2 (Q) 1-θ ( u L 2 ((0,T )×ω) + p L 2 ((0,T )×ω) ) θ , ( 10
) for all (u, p) solution of (1) in L 2 (0, T ; H 1 (Ω)) × L 2 (Q).
In these two propositions, the exponent θ only depends on the geometry of the domain, whereas the constants C also depend on ǫ.

And the last proposition gives an estimate of u and p on the boundary with respect to measurements in the interior: Proposition 4. There exists a neighborhood Ω of ∂Ω, a nonempty open subset ω ⊂ Ω relatively compact in Ω, a constant α > 0 and, for all ε > 0, there exists a constant C > 0 such that

u C([ε,T -ε];C 1 ( Ω∩Ω)) + p C([ε,T -ε]×( Ω∩Ω)) ≤ CM log CM u L 2 ((0,T )×ω) + p L 2 ((0,T )×ω) α (11) for all (u, p) solution of (1) in H 1 (0, T ; H 3 (Ω)) ∩ H 2 (0, T ; H 1 (Ω)) × H 1 (0, T ; H 2 (Ω)), where M := u H 1 (0,T ;H 3 (Ω)) + u H 2 (0,T ;H 1 (Ω)) + p H 1 (0,T ;H 2 (Ω)) . (12)
Again, in this proposition, the exponent α only depends on the geometry of the domain, whereas the constants C also depend on ǫ.

Remark 1. In Theorem 1, the hypotheses of regularity on the solution come from the hypotheses of regularity made in Proposition 4. In Propositions 2 and 3, the regularity of the solutions is much weaker (even if, we have to give a sense to the norms which appear in the measurements on the boundary given by ( 9)). In Proposition 4, if we do not assume that u belongs to H 2 (0, T ; H 1 (Ω)) and remove the norm of u in H 2 (0, T ; H 1 (Ω)) in M , we can prove inequality [START_REF] Boyer | Éléments d'analyse pour l'étude de quelques modèles d'écoulements de fluides visqueux incompressibles[END_REF] with the norm of u in H 1 (0, T ; L 2 (ω)) instead of L 2 ((0, T ) × ω) in the right hand-side.

Remark 2. In Proposition 3, we only assume that p belongs to L 2 (Q) and we get an estimate of p in H 1,0 ((ε, Tε) × Ω 0 ). Since, for all t ∈ (0, T ), p(t, •) is a solution of the Laplace equation in Ω, results on the interior regularity for elliptic problem ( [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]) directly implies that p belongs to H 1,0 ((0, T ) × Ω 0 ).

These three propositions will allow to prove the quantification of the unique continuation property given in Theorem 1:

Proof of Theorem 1.

1. We first apply Proposition 4 and we obtain the existence of a neighborhood Ω of ∂Ω, an open subset ω ⊂ Ω relatively compact in Ω and a constant α > 0 such that, for all ε > 0

u C([2ε,T -2ε];C 1 ( Ω∩Ω)) + p C([2ε,T -2ε]×( Ω∩Ω)) ≤ CM log CM u L 2 ((ǫ,T -ǫ)×ω) + p L 2 ((ǫ,T -ǫ)×ω) α , (13)
for some C > 0. Let us now apply Proposition 2 on ω. We get the existence of constants C > 0 and 0 < θ < 1 such that

u L 2 ((ε,T -ε)×ω) + p L 2 ((ε,T -ε)×ω) ≤ C p H 1,0 (Q) + u H 1,0 (Q) 1-θ F θ ≤ CM 1-θ F θ
where F is given by [START_REF] Boulakia | Stability estimates for the unique continuation property of the Stokes system and for an inverse boundary coefficient problem[END_REF]. Using this estimate in the right hand-side of ( 13), we get

u C([2ε,T -2ε];C 1 ( Ω∩Ω)) + p C([2ε,T -2ε]×( Ω∩Ω)) ≤ CM log CM F α . (14)
Let us introduce an open set Ω 0 such that Ω \ Ω ∩ Ω ⊂⊂ Ω 0 ⊂⊂ Ω. We have, according to interpolation inequalities,

u C([2ε,T -2ε];C 1 (Ω0)) + p C([2ε,T -2ε]×Ω0) ≤ CM 7/8 ( u L 2 (2ε,T -2ε;H 1 (Ω0)) + p L 2 (2ε,T -2ε;H 1 (Ω0)) ) 1/8 .
We apply again Proposition 2 and we get

u C([2ε,T -2ε];C 1 (Ω0)) + p C([2ε,T -2ε]×Ω0) ≤ CM 7/8 (M 1-θ F θ ) 1/8 = CM F M θ/8
We gather this inequality with ( 14) and we get that there exists a constant C > 0 such that

u C([2ε,T -2ε];C 1 (Ω)) + p C([2ε,T -2ε]×Ω) ≤ CM log CM F α .
To conclude the proof, we notice that

u L 2 (0,T ;H 1 (Γ)) + p L 2 (0,T ;H 1 (Γ)) ≤ C u 1/3 L 2 ((0,T )×Γ) u 2/3 L 2 (0,T ;H 2 (Ω)) + C p 1/3 L 2 ((0,T )×Γ) p 2/3 L 2 (0,T ;H 2 (Ω)) ≤ CG 1/3 M 2/3 and u H 1 (0,T ;L 2 (Γ)) ≤ C u 1/2 L 2 ((0,T )×Γ) u 1/2 H 2 (0,T ;H 1 (Ω)) ≤ CG 1/2 M 1/2 . Thus, F ≤ CG 1/3 M 2/3 which leads to (2).
2. We proceed in the same way as in the first step except that we apply Proposition 3 instead of Proposition 2.

3 Proof of the local estimates

Estimates in the interior of the domain: proof of Propositions 2 and 3

Let us first define some well-chosen weight functions which will be useful in the proof of Proposition 2.

To do so, we take again the setting of Proposition 2: we introduce Γ ⊂ ∂Ω a nonempty open subset of ∂Ω and Ω 0 a nonempty open set such that Ω 0 ⊂ Ω ∪ Γ and ∂Ω 0 ∩ ∂Ω Γ. Then, we consider Ω a domain

of class C 2 such that Ω ⊂ Ω, Γ = ∂Ω ∩ Ω and ∂Ω\Γ ⊂ ∂ Ω. Let d ∈ C 2 ( Ω) be such that d(x) > 0 for all x ∈ Ω, d(x) = 0 for x ∈ ∂ Ω and |∇d(x)| > 0 for x ∈ Ω. We define δ = d C( Ω) .
Since Ω 0 ⊂ Ω, we can choose a sufficiently large N > 5 such that

Ω 0 ⊂ x ∈ Ω/d(x) > 5 N δ . (15)
These constants δ and N only depend on the domains Γ and Ω. Let ǫ > 0 be fixed and choose

β > 0 such that 3β 2 ε 2 < δ < 2βε 2 . ( 16
)
We arbitrarily fix

t 0 ∈ ( √ 2ε, T - √ 2ε) and set, for all (t, x) ∈ R × R 3 ψ(t, x) = d(x) -β(t -t 0 ) 2 (17)
and

ϕ(t, x) = e λψ(t,x) , ( 18
)
where λ is a large enough fixed positive parameter. For 1 ≤ i ≤ 5, we define

µ i = e λ i N δ-βε 2 N . ( 19
)
We then define

Ω 1 a sub-domain of Ω such that Ω 0 ⊂ Ω 1 , ∂Ω 1 ∩ ∂Ω = ∅, ∂Ω 1 ∩ ∂Ω Γ and d C(Ω\Ω1) ≤ 1 3N δ. (20)
For 1 ≤ i ≤ 5, we denote by

D i = x ∈ Ω 1 /d(x) > i N δ D i = (t, x) ∈ R × Ω 1 /ϕ(t, x) > µ i .
For these domains, the following lemma holds:

Lemma 1. The sets ( D i ) and (D i ) satisfy the following properties:

(i) t 0 -ε √ N , t 0 + ε √ N × Ω 0 ⊂ D 5 ⊂ . . . ⊂ D 1 . (ii) For all 2 ≤ i ≤ 5, D i ⊂ t 0 - √ 2ε, t 0 + √ 2ε × D i-1 . (iii) For all 1 ≤ i ≤ 5, ∂D i ⊂ Σ 1,i ∪ Σ 2,i (21)
with

Σ 1,i ⊂ t 0 - √ 2ε, t 0 + √ 2ε × Γ and Σ 2,i = (t, x) ∈ t 0 - √ 2ε, t 0 + √ 2ε × Ω 1 /ϕ(t, x) = µ i . (iv) For all 1 ≤ i ≤ 5, ∂ D i ⊂ Σ 1,i ∪ Σ 2,i (22)
with

Σ 1,i ⊂ Γ ∩ ∂Ω 1 and Σ 2,i = x ∈ Ω 1 /d(x) = i N δ . Proof of Lemma 1. (i) Let (t, x) ∈ t 0 - ε √ N , t 0 + ε √ N × Ω 0 . We notice that ψ(t, x) = d(x) -β(t -t 0 ) 2 > d(x) - βǫ 2 N > 5 N δ - βǫ 2 N
according to [START_REF] Fursikov | Controllability of evolution equations[END_REF]. This implies that ϕ(t, x) > µ 5 which shows the first inclusion.

The fact that

D i+1 ⊂ D i for 1 ≤ i ≤ 4 is obvious. (ii) Now, let (t, x) belong to D i , for 2 ≤ i ≤ 5. Since ϕ(t, x) > µ i , d(x) -β(t -t 0 ) 2 > i N δ - βǫ 2 N . (23)
Thus

β(t -t 0 ) 2 < 1 - i N δ + βǫ 2 N < 2βǫ 2 1 - i N + βǫ 2 N < 2βǫ 2
thanks to [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]. This implies that |tt 0 | < √ 2ǫ. On the other hand, we deduce from (23) that

d(x) > i N δ - βǫ 2 N > i N δ - 2 3N δ > i -1 N δ
using again [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]. Thus,

D i ⊂ t 0 - √ 2ε, t 0 + √ 2ε × D i-1 . (iii) Let (t, x) belong to ∂D i . If x belongs to Ω 1 , then ϕ(t, x) = µ i (otherwise, ϕ(t, x) > µ i and thus (t, x) ∈ Di ). Thus, ( 21
) holds with Σ 1,i ⊂ t 0 - √ 2ε, t 0 + √ 2ε × ∂Ω 1 and Σ 2,i = (t, x) ∈ t 0 - √ 2ε, t 0 + √ 2ε × Ω/ϕ(t, x) = µ i . Let (t, x) ∈ Σ 1,i .
We have two cases. Either x belongs to ∂Ω 1 ∩ ∂Ω ⊂ Γ or x belongs to ∂Ω 1 \ (∂Ω 1 ∩ ∂Ω). In the second case, according to [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF],

d(x) ≤ 1 3N δ.
Moreover, since ϕ(t, x) ≥ µ i ≥ µ 1 we have that

β(t -t 0 ) 2 ≤ d(x) + βǫ 2 N - 1 N δ ≤ - 2 3N δ + βǫ 2 N < 0
according to [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF].

We get a contradiction and this allows to conclude that

Σ 1,i ⊂ t 0 - √ 2ε, t 0 + √ 2ε × Γ.
(iv) According to the definition of D i , [START_REF] Lin | Optimal three-ball inequalities and quantitative uniqueness for the Stokes system[END_REF]

holds for Σ 1,i ⊂ ∂Ω 1 and Σ 2,i = x ∈ Ω 1 /d(x) = i N δ . If x ∈ ∂Ω 1 \ (Γ ∩ ∂Ω 1 ), then according to (20), d(x) ≤ 1 3N δ. Thus, x / ∈ D i . This implies that Σ 1,i ⊂ Γ ∩ ∂Ω 1 .
Before starting the proof of Proposition 2, we give the following classical lemma Lemma 2. Let A > 0, B > 0, C 1 > 0, C 2 > 0 and D > 0. We assume that there exists c 0 > 0 and γ 1 > 0 such that D ≤ c 0 B and for all γ ≥ γ 1 ,

D ≤ Ae C1γ + Be -C2γ .
Then, there exists C > 0 such that:

D ≤ CA C 2 C 1 +C 2 B C 1 C 1 +C 2 .
Proof of Proposition 2. In this proof, C > 0 stands for a generic constant which may depend on Ω, Γ, T , λ and ǫ but which is independent of s and t 0 .

Let χ ∈ C 2 (R × Ω) be a cut-off function such that 0 ≤ χ ≤ 1 and χ(t, x) = 0, if ϕ(t, x) ≤ µ 3 , 1, if ϕ(t, x) ≥ µ 4 . (24)
To define this function, we can take χ(t, x) = χ ϕ(t,x)-µ3 µ4-µ3

where χ ∈ C 2 (R) is such that 0 ≤ χ(t, x) ≤ 1 and χ(ξ) = 0, if ξ ≤ 0, 1, if ξ ≥ 1. (25)
We have the following estimate: for all 1 ≤ i, j ≤ 3, for all

(t, x) ∈ R × R 3 , |∂ t χ(t, x)| + |∂ i χ(t, x)| + |∂ 2 ij χ(t, x)| ≤ C. ( 26
) Then v = χu satisfies ∂ t v -∆v = ∂ t χu -∆χu -2∇χ • ∇u -χ∇p in Q.
We apply the Carleman inequality for parabolic equations on the domain D 2 (see Theorem 3.2 in [START_REF] Yamamoto | Carleman estimates for parabolic equations and applications[END_REF]) with the weight ϕ: for all fixed λ large enough, there exist a constant s 0 > 0 and a constant C such that, for all s > s 0

D2 s|∇v| 2 + s 3 |v| 2 e 2sϕ dxdt ≤ C D2 |∂ t χu-∆χu-2∇χ•∇u| 2 e 2sϕ dxdt+C D3 |∇p| 2 e 2sϕ dxdt+C ∂D2 e 2sϕ (s|∇ t,x v| 2 +s 3 |v| 2 )dσdt
Let us mention that the constants C and s 0 do not depend on t 0 since, if we look at the dependence of the domain D 2 with respect to t 0 , we see that the domains D 2 are in translation with each other with respect to t 0 . By the definition of χ (24), the first term in the right hand-side of this inequality is in fact an integral on {(t, x) ∈ Q/µ 3 ≤ ϕ(t, x) ≤ µ 4 }. Thus, using [START_REF] Vignon-Clementel | Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries[END_REF], we get the existence of a constant C > 0 such that:

D2 |∂ t χu -∆χu -2∇χ • ∇u| 2 e 2sϕ dxdt ≤ Ce 2sµ4 u 2 H 1,0 (Q) .
Moreover, for the boundary integral in the right hand-side, we use Lemma 1 (iii) for i = 2. We obtain that there exists a constant C > 0 such that:

∂D2 e 2sϕ (s|∇ t,x v| 2 + s 3 |v| 2 )dσdt ≤ Ce C0s T 0 Γ |∂ t u| 2 + |∇u| 2 + |u| 2 dσdt
where C 0 does not depend on ε. Then, since by Lemma

1 (i), t 0 - ε √ N , t 0 + ε √ N × Ω 0 ⊂ D 5 ⊂ D 2 ,
we obtain:

e 2sµ5 t0+ ε √ N t0-ε √ N Ω0 s|∇u| 2 + s 3 |u| 2 dxdt ≤ Ce 2sµ4 u 2 H 1,0 (Q) + C D3 |∇p| 2 e 2sϕ dxdt + Ce C0s T 0 Γ |∂ t u| 2 + |∇u| 2 + |u| 2 dσdt. ( 27
)
Let us now obtain estimates on the pressure p to estimate the second term in the right hand-side.

According to Lemma 1 (ii), we have

D3 (s|∇p| 2 + s 3 |p| 2 )e 2sϕ dxdt ≤ t0+ √ 2ε t0- √ 2ε D2 (s|∇p| 2 + s 3 |p| 2 )e 2sϕ dxdt. (28)
We introduce a cut-off function χ in C 2 (Ω) such that 0 ≤ χ ≤ 1 and

χ(x) = 1, if d(x) ≥ 2 N δ, 0, if d(x) ≤ 3 2N δ. (29)
As previously for χ, this function can be defined explicitly with the help of χ:

χ(x) = χ 2N δ d(x) - 3 2N δ .
We have the following estimate: for all 1 ≤ i, j ≤ 3, for all x ∈ Ω,

|∂ i χ(x)| + |∂ ij χ(x)| ≤ C. (30)
Let us define π = χ p. Using that D 2 ⊂ D 1 and χ = 1 on D 2 , inequality (28) becomes

D3 (s|∇p| 2 + s 3 |p| 2 )e 2sϕ dxdt ≤ t0+ √ 2ε t0- √ 2ε D1 (s|∇π| 2 + s 3 |π| 2 )e 2sϕ dxdt. (31)
By taking the divergence of the first equation of (1), we obtain that ∆p = 0 in Q. Thus, π is solution of

∆π = p∆ χ + 2∇p • ∇ χ in Q.
We apply to π the classical Carleman inequality for elliptic equations (which can be proved as in [START_REF] Fursikov | Controllability of evolution equations[END_REF]) on D 1 with φ = e λd : for all fixed λ large enough, there exist constants s0 , C and C 1 such that, for all s > s0 ,

D1 (s|∇π| 2 + s3 |π| 2 )e 2s φdx ≤ C D1 |p∆ χ + 2∇p • ∇ χ| 2 e 2s φdx + C ∂ D1
(s|∇π| 2 + s3 |π| 2 )e 2s φdσ (32)

By using Lemma 1 (iv), there exists C > 0 such that

∂ D1 |∇π| 2 + |π| 2 dσ ≤ C Γ |p| 2 + |∇p| 2 dσ.
Thus, if we take s = se -λβ(t-t0) 2 and if we integrate inequality (32) over (t 0 -√ 2ǫ, t 0 + √ 2ǫ), thanks to the properties (29) and (30) satisfied by χ, we deduce from inequality (31) that there exist a constant s 0 and a constant C such that, for all s > s 0 ,

D3 (s|∇p| 2 + s 3 |p| 2 )e 2sϕ dxdt ≤ C T 0 B (|p| 2 + |∇p| 2 )e 2sϕ dxdt + Ce C1s T 0 Γ |p| 2 + |∇p| 2 dσdt (33)
where

B := x ∈ Ω/ 3 2N δ < d(x) < 2 N δ .
Let us remark that, thanks to inequality (16), we have for (t, x) ∈ (0, T ) × B:

ψ(t, x) = d(x) -β(t -t 0 ) 2 < 2 N δ ≤ 3 N δ - βǫ 2 N ,
which implies that ϕ ≤ µ 3 on (0, T ) × B. Thus, inequality (33) implies that:

D3 s|∇p| 2 + s 3 |p| 2 e 2sϕ dxdt ≤ Ce 2sµ3 p 2 H 1,0 ((0,T )×B) + Ce C1s T 0 Γ |p| 2 + |∇p| 2 dσdt. (34)
We sum up inequalities ( 27) and (34). The second term in the right hand-side of inequality [START_REF] Yamamoto | Carleman estimates for parabolic equations and applications[END_REF] with the gradient of p is absorbed by the left hand-side of inequality (34), for s large enough. Then, since by

Lemma 1 (i), t 0 - ε √ N , t 0 + ε √ N × Ω 0 ⊂ D 5 ⊂ D 3 , we obtain that, for all s ≥ s 0 , e 2sµ5 t0+ ε √ N t0-ε √ N Ω0 s|∇p| 2 + s 3 |p| 2 + s|∇u| 2 + s 3 |u| 2 dxdt ≤ Ce 2sµ4 p 2 H 1,0 (Q) + u 2 H 1,0 (Q) + Ce C2s T 0 Γ |∂ t u| 2 + |∇u| 2 + |u| 2 + |p| 2 + |∇p| 2 dσdt,
for C 2 = max(C 0 , C 1 ) which is independent of ε. This implies that, for all s ≥ s 0 :

u 2 H 1,0 ((t0-ε √ N ,t0+ ε √ N )×Ω0) + p 2 H 1,0 ((t0-ε √ N ,t0+ ε √ N )×Ω0) ≤ Ce -2s(µ5-µ4) p 2 H 1,0 (Q) + u 2 H 1,0 (Q) + Ce C2s F 2 ≤ Ce -C3s p 2 H 1,0 (Q) + u 2 H 1,0 (Q) + Ce C2s F 2 (35)
where C 3 only depends on δ, N and λ and where F is given by ( 9).

As already noticed, the constants in the right hand-side of (35) are independent of t 0 . Let us take the following values for t 0 :

t 0,j = √ 2ǫ + jǫ √ N , with j = 0, . . . , m
where m ∈ N is such that

√ 2ǫ + mǫ √ N ≤ T - √ 2ǫ ≤ √ 2ǫ + (m + 1)ǫ √ N ≤ T.
If we sum up over j the estimates (35) obtained with t 0 = t 0,j for 0 ≤ j ≤ m, we obtain estimates of u and p in ( √ 2ε, T -√ 2ε) × Ω 0 . Thus, replacing √ 2ǫ by ǫ, we obtain, for all s ≥ s 0 :

u H 1,0 ((ε,T -ε)×Ω0) + p H 1,0 ((ε,T -ε)×Ω0) ≤ Ce -sC3/2 p H 1,0 (Q) + u H 1,0 (Q) + Ce sC2/2 F. (36)
Thus, thanks to Lemma 2, we have proved estimate [START_REF] Boulakia | Stability estimates for a Robin coefficient in the two-dimensional Stokes system[END_REF].

The proof of Proposition 3 follows exactly the same steps as the proof of Proposition 2, so we will only explain the main arguments and stress the main differences with the previous proof. We introduce a function d 0 which belongs to C 2 (Ω) and which satisfies

d 0 > 0 in Ω, d 0 = 0 on ∂Ω, |∇d 0 | > 0 in Ω \ ω 0
where ω 0 is a nonempty open subset of Ω such that ω 0 ⊂ ω. Next, we take N > 5 large enough so that

Ω 0 ⊂ x ∈ Ω/d 0 (x) > 5 N δ .
where δ is now defined by δ = d 0 C(Ω) and we choose β > 0 which satisfies [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]. We keep the same definitions ( 17), ( 18) and ( 19) for, respectively, ψ, ϕ and µ i with d 0 instead of d. Moreover, we define

D i = x ∈ Ω/d 0 (x) > i N δ and D i = (t, x) ∈ R × Ω/ϕ(t, x) > µ i .
Points (i) and (ii) of Lemma 1 still hold with these new definitions.

Proof of Proposition 3. If we adapt the proof of Theorem 3.1 in [START_REF] Yamamoto | Carleman estimates for parabolic equations and applications[END_REF] to our new weight ϕ, we get the following local Carleman estimate: Let D ⊂ (0, T )×Ω be a domain of class C 2 such that, for all t ∈ [0, T ], the boundary of the domain D∩{t} is composed of a finite number of smooth surfaces. For all fixed λ large enough, there exist a constant s 0 > 0 and a constant C such that, for all s > s 0 , for all v in

H 1,0 (Q) satisfying ∂ t v -∆v ∈ L 2 (Q) and supp v ⊂⊂ D D s|∇v| 2 + s 3 |v| 2 e 2sϕ dxdt ≤ C D |∂ t v -∆v| 2 e 2sϕ dxdt + C ((0,T )×ω)∩D s 3 |v| 2 e 2sϕ dxdt.
Let us define χ by [START_REF] Quarteroni | Analysis of a geometrical multiscale model based on the coupling of ODEs and PDEs for blood flow simulations[END_REF] with the new definition of ϕ and take v = χu. Since the support of v is relatively compact in D 2 , we can apply this inequality to v in D 2 . This implies that

e 2sµ5 t0+ ε √ N t0-ε √ N Ω0 s|∇u| 2 + s 3 |u| 2 dxdt ≤ Ce 2sµ4 u 2 H 1,0 (Q) + C D3 |∇p| 2 e 2sϕ dxdt + C T 0 ω s 3 |u| 2 e 2sϕ dxdt.
To estimate the second term in the right hand-side, we notice that (31) still holds with π = χp where χ is defined by (29) with d 0 instead of d. Then, we apply to π the standard elliptic Carleman estimate ( [START_REF] Fursikov | Controllability of evolution equations[END_REF]) in D 1 ⊂⊂ Ω with our new weight φ = e λd0 for λ fixed large enough. Arguing in a similar way as in the proof of Proposition 2, we get that there exist a constant s0 > 0 and a constant C such that, for all s > s0 ,

D3 (s|∇p| 2 + s 3 |p| 2 )e 2sϕ dxdt ≤ Ce 2sµ3 p 2 H 1,0 ((0,T )×B) + Cs 3 T 0 ω |p| 2 e 2sϕ dxdt
where

B := x ∈ Ω/ 3 2N δ < d 0 (x) < 2 N δ ⊂⊂ Ω.
Moreover, according to Cacciopoli inequality ( [START_REF] Gilbarg | Elliptic partial differential equations of second order, volume 224 of Grundlehren der Mathematischen Wissenschaften[END_REF]), since ∆p = 0 in (0, T ) × Ω, we have

p 2 H 1,0 ((0,T )×B) ≤ C p 2 L 2 ((0,T )×Ω) .
We then proceed exactly as in the proof of Proposition 2 to conclude the proof.

Estimates on the boundary of the domain: proof of Proposition 4

For any x ∈ R 3 , we use the following notation x = (x 1 , x ′ ) where x 1 ∈ R and x ′ ∈ R 2 . Moreover, for all R > 0, we denote by B(0, R)

+ = {x = (x 1 , x ′ ) ∈ B(0, R)/x 1 > 0},
where B(0, R) is the open ball of center 0 and of radius R, and by

D R = {x = (x 1 , x ′ ) ∈ R 3 /0 < x 1 < R and |x ′ | < R}.
Let (u, p) be a solution of [START_REF] Alessandrini | Optimal stability for inverse elliptic boundary value problems with unknown boundaries[END_REF]. Thanks to a change of coordinates, we can straighten locally the boundary of Ω and go back to the upper half-plane. For all P ∈ ∂Ω, let φ P be such a change of variables in a neighborhood of P . The function φ P is a C 2 -diffeomorphism on B(0, r P ) for some r P > 0 and satisfies φ P (0) = P, φ P (B(0, r P ) + ) = Ω ∩ φ P (B(0, r P ))

and φ P (x 1 , x ′ ) ∈ B(0, r P ), x 1 = 0 = ∂Ω ∩ φ P (B(0, r P )).
Moreover, due to the regularity and compactness of Ω, there exists R > 0 such that ∀P ∈ ∂Ω, r P ≥ 3R and we can always assume that R < 1. Next, since ∂Ω ⊂ P ∈∂Ω φ P (B(0, R/2)), by compactness of ∂Ω,

there exist N points (P i ) 1≤i≤N of ∂Ω such that ∂Ω ⊂ N i=1 φ Pi (B(0, R/2)).
In the following, we fix 1 ≤ i ≤ N and, to simplify the notations, we set φ = φ Pi . Let us define, for all (t, x) ∈ (0, T ) × B(0, 3R) + v(t, x) = u(t, φ(x)), q(t, x) = p(t, φ(x)).

(37)

These functions satisfy the following problem:

∂ t v -div(∇vA φ ) + B φ ∇q = 0, in (0, T ) × B(0, 3R) + , div(A φ ∇q) = 0, in (0, T ) × B(0, 3R) + , (38)
with

A φ = | det(∇φ)|(∇φ) -1 (∇φ) -t ,
and

B φ = | det(∇φ)|(∇φ) -t .
Let us define the operator P φ by:

P φ f = -div(A φ ∇f ) (39)
for a regular scalar function f , and by

P φ F = (P φ F 1 , P φ F 2 , P φ F 3 ) ,
for a regular vector-valued function F = (F 1 , F 2 , F 3 ). We can rewrite system (38) as follows:

∂ t v + P φ v + B φ ∇q = 0, in (0, T ) × B(0, 3R) + , P φ q = 0, in (0, T ) × B(0, 3R) + . (40)
Let ε > 0 be given. We consider

t 0 ∈ (ε, T -ε) and x ′ 0 ∈ R 2 such that |x ′ 0 | ≤ R. We choose β > 0 and γ > 0 such that R < βε 2 < 2R and γR 4 > 1 (41)
and we define

d(x) = x 1 -γ|x ′ -x ′ 0 | 2 , ψ(t, x) = d(x) -β(t -t 0 ) 2 (42)
and

Q(η) = {(t, x) ∈ R × R 3 /x 1 < R + η and ψ(t, x) > η}. Lemma 3. For 0 < η < R 2 , we have Q(η) ⊂ (t 0 -ε, t 0 + ε) × D 3R 2 ⊂ (t 0 -ε, t 0 + ε) × B(0, 3R) + .
Proof of Lemma 3. Let (t, x) ∈ Q(η). First, we have that

x 1 < R + η < 3R 2 . Moreover, since ψ(t, x) > η x 1 > η + γ|x ′ -x ′ 0 | 2 + β(t -t 0 ) 2
This implies that x 1 > 0 and that γ|x ′ -

x ′ 0 | 2 + β(t -t 0 ) 2 < R.
According to the conditions (41) satisfied by β and γ, we obtain the first inclusion. The second inclusion is readily proved.

In order to apply local Carleman inequality, we need to introduce a cut-off function

Lemma 4. Let 0 < η < R 2 be given. We can define a function χ η ∈ C ∞ (R × R 3 ) such that 0 ≤ χ η ≤ 1, χ η (t, x) = 1, if ψ(t, x) ≥ 3η, 0, if ψ(t, x) ≤ 2η, (43)
and, for all

(t, x) ∈ (0, T ) × D 3R 2 , |∂ i χ η (t, x)| + |∂ t χ η (t, x)| ≤ C η , ∂ 2 ij χ η (t, x) ≤ C η 2 , for all 1 ≤ i, j ≤ n,
where C > 0 is a constant which only depends on R, T , ε and γ.

To prove this lemma, we can define χ thanks to the definition (25) of χ by χ η (t, x) = χ ψ(t, x) -2η η .

Proof of Proposition 4. Let us define (v η , q η ) = (χ η v, χ η q). The function v η satisfies the following equation

∂ t v η + P φ v η = -χ η B φ ∇q + ∂ t χ η v + [P φ , χ η ]v, in (0, T ) × B(0, 2R) + ,
where the operator [P φ , χ η ] is defined by

[P φ , χ η ]v = -div(A φ v∇χ t η ) -A φ ∇v∇χ η
for all vector-valued function v. We denote by

D R = x ∈ R 3 /R < x 1 < 3R 2 , |x ′ | < 3R 2 .
In the following, we consider that η ∈ 0, R 8 is given. We apply the Carleman estimate for parabolic equations (Theorem 3.2 in [START_REF] Yamamoto | Carleman estimates for parabolic equations and applications[END_REF]) in Q(η) with the weight ϕ = e λψ where ψ is given by ( 42): for all fixed λ large enough, there exists a constant s 0 > 0 and a constant C such that, for all s > s 0 ,

Q(η) 1 s |∂ t v η | 2 + s|∇v η | 2 + s 3 |v η | 2 e 2sϕ dxdt ≤ C Q(η) | -χ η B φ ∇q + ∂ t χ η v + [P φ , χ η ]v| 2 e 2sϕ dxdt + C ∂Q(η) (s|∇ t,x v η | 2 + s 3 |v η | 2 )e 2sϕ dσdt. (44)
Notice that, since the domain Q(η) is a translation of Q(0) in the direction x 1 for any η, the constants s 0 and C are independent of η.

Let us first estimate the last term in this inequality. We remark that

∂Q(η) = (t, x)/x 1 = R + η, ψ(t, x) ≥ η ∪ (t, x)/x 1 ≤ R + η, ψ(t, x) = η .
Since χ η satisfies (43), v η = |∇v η | = 0 on (t, x)/x 1 ≤ R + η, ψ(t, x) = η . Moreover, according to Lemma 3, (t, x)/x 1 = R + η, ψ(t, x) ≥ η ⊂ (t, x) ∈ (0, T ) × D 3R 2 /x 1 = R + η . Hence, there exists a constant C > 0 which does not depend on η such that

∂Q(η) (s|∇ t,x v η | 2 + s 3 |v η | 2 )e 2sϕ dσdt ≤ Ce Cs J 2 1 where J 1 = v H 1 (0,T ;H 2 ( D R )) . (45)
For the first term in the right hand-side of (44), we first notice that there exists C > 0 such that

Q(η) |χ η B φ ∇q| 2 e 2sϕ dxdt ≤ C Q(η)∩Q(2η)
|∇q| 2 e 2sϕ dx dt according to (43) and that there exists C > 0 such that

Q(η) |∂ t χ η v + [P φ , χ η ]v| 2 e 2sϕ dx dt ≤ C η 4 e 2sα1 v 2
H 1,0 ((0,T )×D 3R

2

)
where we have denoted α 1 = e 3λη and used Lemmas 3 and 4. By this way, if we denote α 2 = e 4λη , inequality (44) becomes

e 2sα2 Q(η)∩Q(4η) s|∇v| 2 + s 3 |v| 2 dxdt ≤ C Q(η)∩Q(2η)
|∇q| 2 e 2sϕ dx dt + C η 4 e 2sα1 v 2

H 1,0 ((0,T )×D 3R 2 ) + Ce Cs J 2 1 ( 46
)
where the constants C do not depend on η. We now want to estimate the term with the pressure (first term in the right hand-side). To do so, we will use the fact that q satisfies an elliptic equation and apply a Carleman estimate. First, we denote by

E(η) = {x ∈ R 3 /d(x) > η and x 1 < R + η}.
Note that, according to (41), E(η) ⊂ D 3R 2 and

Q(η) ⊆ (t 0 -ǫ, t 0 + ǫ) × E(η). (47)
In a similar way as in Lemma 4, let us introduce a cut-off function which satisfies the following properties: For a given η > 0, there exists

ζ η ∈ C 2 (R 3 ) such that 0 ≤ ζ η ≤ 1 and ζ η (x) = 1, if d(x) ≥ 2η, 0, if d(x) ≤ 3η 2 . ( 48
)
Moreover, there exists a constant C > 0 depending only on R and γ such that, for all x ∈ D 3R 2 , we have

|∂ i ζ η (x)| ≤ C η and ∂ 2 ij ζ η (x) ≤ C η 2 , for all 1 ≤ i, j ≤ 3.
We denote by π η = ζ η q. According to (47) and ( 48), we will thus be able to estimate the first term in the right hand-side of (46) thanks to the following inequality

Q(η)∩Q(2η) (s|∇q| 2 + s 3 |q| 2 )e 2sϕ dx dt ≤ t0+ǫ t0-ǫ E(η) (s|∇π η | 2 + s 3 |π η | 2 )e 2sϕ dx dt. (49)
The function π η is solution of

P φ π η = [P φ , ζ η ]q in (0, T ) × B(0, 2R) + ,
where the operator [P φ , ζ η ] is defined by

[P φ , ζ η ]q = -∇ζ t η A φ ∇q -div(qA φ ∇ζ η )
for all scalar function q. We then apply the elliptic Carleman estimate to π η in E(η) with the weight φ = e λd : for all fixed λ > 0 large enough, there exists a constant s0 > 0 and a constant C such that, for all s > s0 ,

E(η) s|∇π η | 2 + s3 |π η | 2 e 2s φ dx ≤ C E(η) |[P φ , ζ η ]q| 2 e 2s φ dx + C ∂E(η) (s|∇π η | 2 + s3 |π η | 2 )e 2s φdσ. (50)
The domain E(η) is a translation of E(0) in the direction x 1 , thus the constants s0 and C do not depend on η. Moreover, we notice that

∂E(η) = x ∈ R 3 /d(x) = η and 0 < x 1 ≤ R + η ∪ x ∈ R 3 /d(x) ≥ η and x 1 = R + η
and, by construction, |π η | = |∇π η | = 0 on x ∈ R 3 /d(x) = η and 0 < x 1 ≤ R + η . Thus, if we set s = se -λβ(t-t0) 2 in inequality (50) and if we integrate in time over (t 0ǫ, t 0 + ǫ), we obtain that there exists a constant s 0 > 0 and a constant C such that, for all s > s 0 , t0+ǫ t0-ǫ E(η)

s|∇π η | 2 + s 3 |π η | 2 e 2sϕ dx dt ≤ C η 4 e 2sα1 q 2 H 1,0 ((0,T )×D 3R 2 ) + Ce Cs J 2 2
where J 2 is given by

J 2 = q L 2 (0,T ;H 7/4 ( D R )) . (51)
Thus, inequality (49) becomes

Q(η)∩Q(2η) (s|∇q| 2 + s 3 |q| 2 )e 2sϕ dx dt ≤ C η 4 e 2sα1 q 2 H 1,0 ((0,T )×D 3R 2 ) + Ce Cs J 2 2 .
If we add up this inequality with inequality (46), the left hand-side of this inequality allows to absorb the first term in the right hand-side of (46) if we take s large enough. Since Q

(η) ∩ Q(4η) ⊂ Q(η) ∩ Q(2η),
we obtain that, for all s ≥ s 0 ,

v 2 H 1,0 (Q(4η)∩Q(η)) + q 2 H 1,0 (Q(4η)∩Q(η)) ≤ C η 4 e -2sCη M 2 + Ce Cs (J 2 1 + J 2 2 )
where C η = α 2α 1 > 0 and M is given by [START_REF] Chaabane | Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems[END_REF]. Since Q

(4η) = Q(4η) ∩ Q(η) ∪ Q(4η) ∩ {R + η ≤ x 1 < R + 4η} and since v 2 H 1,0 (Q(4η)∩{R+η<x1<R+4η}) + q 2 H 1,0 (Q(4η)∩{R+η<x1<R+4η}) is bounded by J 2 1 + J 2 2
, we can in fact estimate v and q on the whole set Q(4η): for all s ≥ s 0

v 2 H 1,0 (Q(4η)) + q 2 H 1,0 (Q(4η)) ≤ C η 4 e -2sCη M 2 + Ce Cs (J 2 1 + J 2 2 ).
Since C η = e 4ληe 3λη = e 3λη (e λη -1) ≥ e λη -1 ≥ λη, we get the existence of s 0 , C > 0 and C 1 > 0 which are independent of η such that, for all s ≥ s 0 ,

v H 1,0 (Q(4η)) + q H 1,0 (Q(4η)) ≤ C η 2 e -sλη M + Ce C1s (J 1 + J 2 ).
Note that the previous inequality is in fact valid for all s ≥ 0 since

v H 1,0 (Q(4η)) + q H 1,0 (Q(4η)) ≤ M . If J 2 1 + J 2 2
= 0, letting s → +∞, we see that v = 0 and q = 0 in Q(4η). Assume now that J 2 1 + J 2 2 = 0 and choose s such that the first term in the right hand-side has the same value as the second term in the right hand-side: we take s = 1 C 1 + λη log M (J 1 + J 2 )η 2 . Then we obtain that there exists a constant C > 0 such that, for all η ∈ 0, R

8 v H 1,0 (Q(4η)) + q H 1,0 (Q(4η)) ≤ C M η 2 C 1 C 1 +λη (J 1 + J 2 ) λη C 1 +λη ≤ C η 2 M C 1 C 1 +λη (J 1 + J 2 ) λη C 1 +λη . (52)
Moreover, this inequality still holds if J 1 + J 2 = 0. Now, let us introduce the following set

Q 1 (4η) = (t, x) ∈ R × R 3 |t -t 0 | < ǫ 2 , 4η + γ|x ′ -x ′ 0 | 2 < x 1 -β(t -t 0 ) 2 < 4η + R 2 .
Thanks to the property (41), for all η ∈ 0, R

8 , Q 1 (4η) ⊂ Q(4η). For all (t, x) ∈ Q 1 (4η), let us set x1 = x 1 -β(t -t 0 ) 2 . Then (t, x) ∈ Q 1 (4η) ⇔ t ∈ I ǫ , (x 1 , x ′ ) ∈ B( 4η
)
where

I ǫ = t 0 - ǫ 2 , t 0 + ǫ 2 , B(4η) = (x 1 , x ′ ) 4η + γ|x ′ -x ′ 0 | 2 < x1 < 4η + R 2 .
Let us set, for all t ∈ I ǫ and (

x 1 , x ′ ) ∈ B(4η), w(t, x1 , x ′ ) = v(t, x1 + β(t -t 0 ) 2 , x ′ ). Then, v C Q1(4η) = w C (Iǫ×B(4η) ≤ C w 7 8 H 1 (Iǫ;H 2 (B(4η))) w 1 8
L 2 ((Iǫ×B(4η)) .

Since the domain B(4η) is the translation of B(0) in the direction x 1 , the constants in these inequalities are independent of η. Coming back to the function v in the right hand-side, we get that

v C Q1(4η) ≤ C v 7 8 H 1 (0,T ;H 2 (B(0,3R) + )) v 1 8 L 2 (Q1(4η)) ≤ CM 7 8 v 1 8 L 2 (Q(4η)) .
The same inequalities hold for ∇v and q. Consequently, thanks to (52), we have

v C Q1(4η) + ∇v C Q1(4η) + q C Q1(4η) ≤ CM 7 8 C η 2 M C 1 C 1 +λη (J 1 + J 2 ) λη C 1 +λη 1 8 ≤ C η 1 4 M J 1 + J 2 M λη 8(C 1 +λη) . (53) Since η ∈ 0, R 8 , we have that J1+J2 M λη 8(C 1 +λη) < J1+J2 M Cη
, where C > 0 is a constant which does not depend on η. Thus,

v C Q1(4η) + ∇v C Q1(4η) + q C Q1(4η) ≤ C η 1 4 M J 1 + J 2 M Cη . Let ξ ′ = (ξ 2 , ξ 3 ) ∈ R 2 . In particular, for all |ξ ′ -x ′ 0 | < R 2γ , 0 < η < R 8 , we have: v t 0 , 4η + γ|ξ ′ -x ′ 0 | 2 , ξ ′ + ∇v t 0 , 4η + γ|ξ ′ -x ′ 0 | 2 , ξ ′ + q t 0 , 4η + γ|ξ ′ -x ′ 0 | 2 , ξ ′ ≤ C η 1 4 M J 1 + J 2 M Cη . (54)
Let us define

Q x ′ 0 = x ∈ R 3 / γ|x ′ -x ′ 0 | 2 < x 1 < R 2 , |x ′ -x ′ 0 | < R 2γ .
Thanks to the following change of variables:

(η, ξ ′ ) → (4η + γ|ξ ′x ′ 0 | 2 , ξ ′ ), we obtain:

Q x ′ 0 |v(t 0 , x)| 2 dx = 4 |ξ ′ -x ′ 0 |< R 2γ R 8 -γ 4 |ξ ′ -x ′ 0 | 2 0 v(t 0 , 4η + γ|ξ ′ -x ′ 0 | 2 , ξ ′ ) 2 dηdξ ′ .
By performing the same calculation for ∇v and q, we obtain thanks to (54),

Q x ′ 0 |v(t 0 , x)| 2 + |∇v(t 0 , x)| 2 + |q(t 0 , x)| 2 dx ≤ CM 2 R 8 0 η -1 2 J 1 + J 2 M Cη dη. ( 55
)
According to the definitions (45), (51) and (12) of J 1 , J 2 and M , there exists a constant C 2 > 0 such that J 1 + J 2 ≤ C 2 M . We notice that

R 8 0 η -1 2 J 1 + J 2 C 2 M Cη dη ≤ +∞ 0 η -1 2 e -C log( C 2 M J 1 +J 2 )η dη ≤ 1 C log( C2M J1+J2 ) -1 2 +∞ 0 C log C 2 M J 1 + J 2 η -1 2 e -C log( C 2 M J 1 +J 2
)η dη.

By performing the change of variables η 1 = C log C2M J1+J2 η, we finally obtain that

R 8 0 η -1 2 J 1 + J 2 C 2 M Cη dη ≤ C Γ 1 2 log C2M J1+J2 1 2
, where Γ is the gamma function. Thus, if we come back to (55), we obtain that, for all x ′ 0 ∈ R 2 such that |x ′ 0 | ≤ R v(t 0 , .) H 1 (Q x ′ 0 ) + q(t 0 , .)

L 2 (Q x ′ 0 ) ≤ CM log C2M J1+J2 1 4
.

Then, by applying an interpolation inequality, we obtain

sup x∈Q x ′ 0 |v(t 0 , x)| ≤ C v(t 0 , .) 7 8
H 2 (Q x ′ 0

) v(t 0 , .)

1 8 L 2 (Q x ′ 0 ) ≤ CM 7
8 v(t 0 , .)

1 8 L 2 (Q x ′ 0 )
where the constant C > 0 does not depend on x ′ 0 since the domains Q x ′ 0 are in translation with each other in the direction x ′ . Similar inequalities hold for ∇v and q. Thus, for all t 0 ∈ (ǫ, Tǫ), for all x ′ 0 such that |x ′ 0 | ≤ R .

Since B(0, R/2) .

+ ⊂ {x ∈ R 3 /0 ≤ x 1 ≤ R/2, |x ′ | ≤ R} ⊂ {x ′ 0 /|x ′ 0 |≤R} Q x ′ 0 ,
We notice that . The last step consists of coming back to an estimate on (ε, Tε) × φ Pi (B(0, R/2) + ) of u and p related to v and q by the change of variables (37). Then, if we sum up the obtained inequality for 1 ≤ i ≤ N , we get estimate [START_REF] Boyer | Éléments d'analyse pour l'étude de quelques modèles d'écoulements de fluides visqueux incompressibles[END_REF].

J 1 + J 2 ≤ v

Stability estimate for the identification of the Robin coefficient

In this section, we come back to the inverse problem presented in the introduction. In particular, we consider system (6) and we assume that ∂Ω is the union of two disjoint pars Γ 0 and Γ e . We will prove the stability result on the Robin coefficient q given by Theorem 2. Let us first state regularity results for the following problem            u t -∆u + ∇p = 0, in (0, T ) × Ω, div u = 0, in (0, T ) × Ω, ∇u • npn = g, on (0, T ) × Γ e , ∇u • npn + qu = h, on (0, T ) × Γ 0 , u(0, •) = u 0 , in Ω.

(

) 56
where M is defined by (3) and G is defined by (4). According to Proposition 6, applied with u 0 = 0, g = 0 and h = u 1 (q 2q 1 ), we have M ≤ C( u 1 (q 2q 1 ) H 2 (0,T ;L 2 (Γ0)) + u 1 (q 2q 1 ) H 1 (0,T ;H ) ≤ C(ν 0 , N 0 ), according to (57). This proves Theorem 2.

sup x∈Q x ′ 0

 0 |v(t 0 , x)| + |∇v(t 0 , x)| + |q(t 0 , x)| ≤ CM log C2M J1+J2 1 32

  this implies that sup(t0,x)∈(ǫ,T -ǫ)×B(0,R/2) + |v(t 0 , x)| + |∇v(t 0 , x)| + |q(t 0 , x)| ≤ CM

L 2 ( 8 M 7 / 8 .

 2878 (0,T )× D R ) + q L 2 ((0,T )× D R ) 1/Thus, v C ε,T -ε;C 1 B(0,R/2) + + q C (ε,T -ε)×B(0,R/2) + ≤ CM log CM v L 2 ((0,T )× D R ) + q L 2 ((0,T )× D R ) 1 32

3 2 (

 2 Γ0))

Acknowledgments: The author wants to thank Masahiro Yamamoto who suggested to work on this problem, Anne-Claire Egloffe who collaborated on this subject at the start and Céline Grandmont. The author also thanks them for fruitful discussions on this subject.

Using regularity results for the stationary Stokes problem with Neumann boundary conditions [START_REF] Boyer | Éléments d'analyse pour l'étude de quelques modèles d'écoulements de fluides visqueux incompressibles[END_REF] and the same kind of arguments as in the proof of Theorem 2.6 in [START_REF] Boulakia | Stability estimates for a Robin coefficient in the two-dimensional Stokes system[END_REF], we can prove that the solution of system (56) satisfies the two following propositions: Proposition 5. Let Ω be a bounded and connected open set in R 3 of class C 1,1 and let ν 0 > 0, N 0 > 0 be given. We assume that

Then problem (56) admits a unique solution

Moreover, this solution satisfies:

)

where C depends on ν 0 and N 0 .

Let Ω be a bounded and connected open set in R 3 of class C 2,1 and let ν 0 > 0, N 0 > 0 be given. We assume that

Then problem (56) admits a unique solution (u, p) ∈ H 2 (0, T ;

. Moreover, this solution satisfies:

)

where C depends on ν 0 and N 0 .

We are now able to prove Theorem 2. Under the hypotheses made in the statement of this theorem, according to Proposition 6, (u 1 , p 1 ) and (u 2 , p 2 ) belong to H 2 (0, T ; H 1 (Ω)) ∩ H 1 (0, T ; H 3 (Ω)) × H 1 (0, T ; H 2 (Ω)) and we have, for i = 1, 2,

The functions u = u 1u 2 and p = p 1p 2 are solutions of

Then, we can apply the first part of Theorem 1 and we get that there exists α > 0 and C > 0