Stability estimates for Navier-Stokes equations and application to inverse problems

In this work, we present some new Carleman inequalities for Stokes and Oseen equations with non-homogeneous boundary conditions. These estimates lead to log type stability inequalities for the problem of recovering the solution of the Stokes and Navier-Stokes equations from both boundary and distributed observations. These inequalities fit the well-known unique continuation result of Fabre and Lebeau [18]: the distributed observation only depends on interior measurement of the velocity, and the boundary observation only depends on the trace of the velocity and of the Cauchy stress tensor measurements. Finally, we present two applications for such inequalities. First, we apply these estimates to obtain stability inequalities for the inverse problem of recovering Navier or Robin boundary coefficients from boundary measurements. Next, we use these estimates to deduce the rate of convergence of two reconstruction methods of the Stokes solution from the measurement of Cauchy data: a quasi-reversibility method and a penalized Kohn-Vogelius method.


Introduction and main results
For a nonempty bounded open subset Ω of R N (N = 2 or N = 3), we consider a pair velocity-pressure (v, p) ∈ H 2 (Ω) × H 1 (Ω) solution of the following linearized Navier-Stokes equations (also called Oseen equations): Above and in the following, ν > 0 is a constant which represents the kinematic viscosity of the fluid, f ∈ L 2 (Ω), d ∈ H 1 (Ω) and z 1 ∈ L ∞ (Ω) and z 2 ∈ W 1,r (Ω) with r > 2 if N = 2, r = 3 if N = 3. (1.2) In the following, z 1 and z 2 will be two solutions of the Navier-Stokes equations in Ω. More precisely, if z 1 and z 2 are two solutions of the Navier-Stokes equations, then their difference v = z 1 − z 2 verifies (1.1). The pair (v, p) is not completely determined by System (1.1). However, if we have some additional observation, such as the value of the velocity v in a nonempty (and arbitrary small) open subset ω ⊂ Ω, namely v = v obs in ω, (1.3) or the value of the Cauchy data (v, σ(v, p)n) on a nonempty open subset Γ obs of ∂Ω, namely v = g D on Γ obs , σ(v, p)n = g N on Γ obs , (1.4) then Fabre and Lebeau's Theorem guarantees the uniqueness of the corresponding pair (v, p) (see [18]). However, the related stability inequality expressing the (conditional) continuous dependence of (v, p) with respect to f L 2 (Ω) , d H 1 (Ω) and to some norm (v, p) Obs (corresponding to one of the above mentioned observation) are not yet proved for system (1.1). Indeed, up to our knowledge, the most recent result quantifying the Fabre and Lebeau's unique continuation theorem in the Stokes case is the following one given in [ [10]). Assume that Ω is of class C ∞ . There exists d 0 > 0 such that for all d > d 0 there exists C > 0 such that, for all solution (v, p) ∈ H 2 (Ω)×H 2 (Ω) of the Stokes equations −ν∆v + ∇p = 0 in Ω, div v = 0 in Ω, v L 2 (Γ obs ) + ∂v ∂n L 2 (Γ obs ) + p L 2 (Γ obs ) + ∂p ∂n L 2 (Γ obs ) 1/2 .
As underlined by the authors themselves, this result does not depend exclusively on the needed observations (1.3) or (1.4) and then does not fit the Fabre and Lebeau's Theorem. The first main results of the present paper are stability inequalities for the Oseen equations (1.1) which are quantified versions of Fabre and Lebeau's uniqueness Theorem (see Theorem 1.2 below) and, in this sense, improve the previous work of Boulakia et al. It allows to obtain analogous stability inequalities for the Navier-Stokes equations. Then, in a second step, we give examples of applications for some parameter identification problems as well as for some error estimates for numerical reconstruction methods.
Stability inequalities. In order to state our main theorem, we need some assumptions and notations. Here and in the following, C > 0 denotes a generic constant which, unless otherwise stated, only depends on the geometry and which may change from line to line, and K ≥ e e denotes a constant which satisfies: max 1 , z 1 L ∞ (Ω) , ∇z 2 L r (Ω) ≤ ln(ln K). (1.5) Moreover, ω denotes a nonempty open subset of Ω and Γ obs denotes a nonempty open subset of ∂Ω. In this paper, n is the outward unit normal to ∂Ω which is assumed to be of class C 2 and the stress tensor is defined by σ(u, p) def = 2νD(u) − p I, where I is the identity matrix and D(y) def = 1 2 ∇y + t ∇y is the symmetrized gradient. We prove (see Subsections 3.1 and 3.2) the following The above theorem allows us to obtain stability estimates for the Navier-Stokes equations. Let (z i , π i ) ∈ H 2 (Ω) × H 1 (Ω), i = 1, 2, satisfy (1.9) and z 1 − z 2 L 2 (Ω) ≤ CK M ln 1 + M z 1 − z 2 H 3/2 (Γ obs ) + σ(z 1 , π 1 )n − σ(z 2 , π 2 )n H 1/2 (Γ obs ) .
(1.11) Moreover, we have curl (z 1 − z 2 ) (L 2 (Ω)) 2N−3 + π 1 − π 2 L 2 (Ω) We stress that these stability estimates respect the well known unique continuation result of Fabre and Lebeau (see [18]) since the observation in ω only concerns the velocity, and since the observation on Γ obs only concerns v Γ obs and σ(v, p)n Γ obs . Indeed, Fabre and Lebeau's Theorem states that every velocity v solution of in Ω, (1.13) which is identically zero in ω must be zero in Ω (and then p is constant, see [18, Proposition 1.1] for precise statements). In particular, no information is required on p to obtain this result. Moreover, as a direct consequence of the above mentioned uniqueness result, we can easily deduce that, if a smooth solution (v, p) of System (1.13) satisfies v = 0 and σ(v, p)n = 0 on Γ obs , then, v = 0 and p = 0 in Ω. Therefore, inequalities (1.6), (1.7) and (1.8) are quantifications of Fabre and Lebeau's uniqueness theorem. The proof of Theorem 1.2 is based on global Carleman inequalities for the Oseen system with non-homogeneous data. Quantitative results for unique continuation are classically obtained thanks to Carleman inequalities and three-spheres inequalities. We refer to the topical review of Alessandrini et al. [3] and to the references therein for elliptic cases; see also the works of Le Rousseau et al. in [25]. However, there is not so much results available on quantitative uniqueness for systems. About Stokes system we shall mention the works of Boulakia et al. in [9,10] for stability estimates and of Ballerini in [6] and Lin et al. in [26] for some other connected results.
Applications to inverse problems. We obtain stability inequalities for the problem of recovering Navier or Robin boundary coefficients. For this, we assume that Γ obs and Γ 0 are two nonempty open subsets of ∂Ω such that Γ obs ∩ Γ 0 = ∅ and we consider on Γ 0 a non penetration condition given by z · n = 0 and a friction law given by 2ν [D(z)n] τ + αz = 0 (subscript τ denotes the tangential component). The aim is to reconstruct the friction coefficient α from Cauchy data on Γ obs . Thus, we consider two solutions (z i , π i ) ∈ H 2 (Ω)× H 1 (Ω) (i = 1, 2) of the Navier-Stokes equations 14) associated to two friction coefficients α i ∈ H 1/2 (Γ 0 ) ∩ L ∞ (Γ 0 ) (i = 1, 2) in the Navier type boundary conditions on Γ 0 : We also consider the reconstruction of the Robin coefficient, still denoted α, in the case of the classical Robin boundary conditions on Γ 0 given by: Notice that the H 1/2 (Γ 0 )-regularity of α i is necessary to have a H 2 (Ω) × H 1 (Ω)-regularity of the solutions.
be two pairs solution of the Navier-Stokes equations (1.14) with the boundary conditions (1.15) or (1.16) which satisfy (1.5) for some K ≥ e e . Let N def = {x ∈ Γ 0 , z 1 (x) = 0 and z 2 (x) = 0}, assume that K is a compact subset of Γ 0 \N with a nonempty interior and let m > 0 be a constant such that max(|z 1 | , |z 2 |) ≥ m on K. Then, for any M > 0 such that z 1 − z 2 H 2 (Ω) + π 1 − π 2 H 1 (Ω) ≤ M , the following inequality holds: Here, the constant C does not depend only on the geometry but also on α i L ∞ (Γ 0 ) for i = 1, 2.
Remark 1.5. We stress the fact that the previous estimate (1.17) depends on the solutions z 1 and z 2 through the choice of the compact set K and the constant m. To complete this result, it would be interesting to obtain a quantitative estimate of the vanishing rate of z, like what is done in [4] in the case of the Laplace equation. Remark 1.6. Note that the assumptions of Theorem 1.4 guarantee that z 1 , z 1 are continuous. Then if K exists, the constant m > 0 exists and depends on z 1 , z 1 on K. The existence of K is known in the case of Robin boundary conditions (1.16) if z 1 (or z 2 ) is not identically equal to zero in Ω. It is an easy consequence of Fabre and Lebeau's theorem. But in the case of Navier conditions (1.15) and if one of the z i is not trivial, the existence of a nonempty open subset of Γ 0 on which z 1 and z 2 both vanish is a difficult issue. Indeed, it reduces to study the existence of a non trivial vector field v solution to an homogeneous Oseen equation (see (4.1) below) and such that v = ∂ n v = 0 on a nonempty open subset of Γ 0 . The difficulty relies on the fact that, unlike the Robin case, no additional information on the pressure is available. Remark 1.7. We can obtain a better estimate assuming more regularity on (v, p). More precisely, for k ≥ 2 and n ∈ N suppose that (v, p) ∈ H k (Ω) × H k−1 (Ω), k ≥ 2 and α i ∈ H n (K), i = 1, 2. Then, using an interpolation argument, we can obtain for any M > 0 and N > 0 such that v H 2 (Ω) + p H 1 (Ω) ≤ M and v H k (Ω) + p H k−1 (Ω) ≤ N that for all θ ∈ [0, 1] (see Remark 4.1): . (1.18) For k = 3 and θ = n = 0, we obtain a result similar to the one presented in [9, Theorem 4.3].
Theorem 1.4, which completes the previous results given by Boulakia et al in [9,10], finds applications in the modeling of biological problems as blood flow in the cardiovascular system (see [27] and [31]) or airflow in the lungs (see [5]). For the Laplace equation, these kind of stability estimates for the Robin coefficient have been widely studied: see for example the works of Chaabane et al. in [14,13], Alessandrini et al. in [2], Sincich in [29], Bellassoued et al. in [7] and Cheng et al. in [15].
Finally, we present another application of our stability estimates in the context of numerical reconstruction methods. More precisely, we focus on the stable reconstruction of the solution of a data completion problem (also known as Cauchy problem) for the Stokes equations: for given in Ω, (1.19) and such that v = g D and σ(v, p)n = g N on Γ obs .
Estimates (1.7) and (1.8) imply the uniqueness of the solution of the data completion problem. However, there exists Cauchy data (g D , g N ) for which it does not admit any solution.
Hence, regularization methods are needed to stably reconstruct (v, p) from (g D , g N ). We study two standard regularization methods: a quasi-reversibility regularization and a penalized Kohn-Vogelius regularization.
For any (g D , g N ) ∈ H 3/2 (Γ obs )×H 1/2 (Γ obs ), both the quasi-reversibility problem (1.20) and the Kohn-Vogelius minimization problem (1.21) admits a unique solution (v ε , p ε ). Moreover, if the initial data completion problem admits a solution (v, p), then v ε converges to v strongly in H 2 (Ω) and p ε converges to p strongly in H 1 (Ω). Furthermore, the stability estimates we obtain in the present paper (proved in Section 5) provide the rate of convergence of both methods (for a survey on the connection between stability estimates and rates of convergence of regularization methods, we refer to [22]): is the exact solution of the data completion problem (1.19), we have the following error estimates for both quasi-reversibility method and penalized Kohn-Vogelius method: Notations. All along this paper, Ω is a nonempty bounded open subset of R N (N = 2 or N = 3) with a boundary ∂Ω of class C 2 , ω is a nonempty open subset of Ω, Γ obs and Γ 0 are nonempty open subsets of ∂Ω, Γ obs ∩ Γ 0 = ∅, and Γ C obs denotes the complement of Γ obs , namely Γ C obs def = ∂Ω\Γ obs . We here summarize the needed notations in the case N = 3 which can be easily adapted for N = 2. We denote by n = t (n 1 , n 2 , n 3 ) the outward unit normal to ∂Ω which has a C 1 extension to a neighborhood of ∂Ω. Above and in the following t denotes the transpose. For a scalar function w or a vector field y = t (y 1 , y 2 , y 3 ), we define ∇w on ∂Ω, we define the normal derivatives ∂w ∂n def = (∇w) · n and ∂y ∂n def = (∇y)n and the tangential gradients ∇ τ w def = ∇w − ∂w ∂n n and ∇ τ y . We also introduce the notations y n def = (y · n) n and y τ def = y − y n for the normal and the tangential components of y on ∂Ω. The divergence of y is defined by div y def = 3 j=1 ∂ x j y j and the curl of w or y is defined by We will also need to use the tangential divergence operator on ∂Ω that we denote by div τ . We recall that D(y) def = 1 2 ∇y + t ∇y denotes the symmetrized gradient and σ(y, p) def = 2νD(y) − p I the stress tensor, where I denotes the identity matrix and ν > 0 is the constant which represents the kinematic viscosity of the fluid we consider.
For r ≥ 0 we denote by L 2 (Ω), L 2 (∂Ω), H r (Ω), H r (∂Ω), H r 0 (Ω), the usual Lebesgue and Sobolev spaces of scalar functions in Ω or in ∂Ω, and we write in bold the spaces of vector-valued functions: We recall that z 1 , z 2 are vector fields satisfying (1.2). Moreover, we use the following particular constant: ( 1.22) We also recall that C > 0 denotes a generic constant only depending on the geometry. In particular, it is independent on z 1 , z 2 and on the parameters s, λ appearing in Carleman inequalities of sections 2 and 3.
Organization of the paper. The paper is organized as follows. The Section 2 is dedicated to the proof of Carleman inequalities for the non-homogeneous Oseen equations (see Theorem 2.3). It is obtained by combining a domain extension argument with Carleman inequalities for compactly supported solutions of the Stokes equations. Then in Section 3, we deduce a Hölder type interior estimates for a distributed observation as well as log type stability inequalities for both distributed and boundary observations. In particular, Theorem 1.2 is proved in subsections 3.1 and 3.2 and Theorem 1.3 is proved in subsection 3.3. Finally, we present some applications in the last sections. The Section 4 concerns the proof of stability inequalities for the inverse problem of recovering Navier and Robin coefficients (proof of Theorem 1.4) and Section 5 is dedicated to the proof of error estimates for some numerical reconstruction methods (proof of Theorem 1.8).

Carleman Inequality for Stokes and Oseen equations
for some positive constant c 0 > 0. For the existence of such a function see for instance [19] or [30,Appendix III]. Here, the set O plays the role of Ω or of an extension Ω of Ω which is used in Section 3 below.
The main aim of this section is to prove a Carleman inequality for the non homogeneous Oseen equations. For that, we first prove a Carleman inequality for a pair velocity-pressure in H 2 0 (O) × H 1 0 (O) and then we use a domain extension argument to recover the nonhomogeneous case.

Carleman Inequality in the case of homogeneous boundary data
Let us first recall a standard Carleman inequality for the Laplace equation: satisfies the following inequality: Proof. Inequality (2.2) for k = 1 is given for instance in [ We deduce the following Carleman inequality for Stokes equations: Theorem 2.2. There exist C > 0, λ > 1 and s > 1 such that for all λ ≥ λ and s ≥ s, and for all (v, p) ∈ H 2 0 (O) × H 1 0 (O) the following inequalities hold: (2.7) Then, by applying (2.2) for k = 0 to (2.6) we obtain (2.4).
Next, we introduce another open subset ω 0 ⋐ ω and apply (2.2) for k = 0 to (2.5) to obtain: Let us replace the local term in curl v by a local term in v. For that, we introduce a function ρ ∈ C ∞ c (ω) such that 0 ≤ ρ ≤ 1 and ρ = 1 in ω 0 . Using an integration by parts in ω, we get and with Cauchy-Schwarz inequality: By combining (2.8) with the above inequality for ǫ > 0 small enough, we obtain Finally, (2.3) is obtained by first applying (2.2) for k = 1 to (2.7) and next using the estimate of curl v given by (2.9).

Carleman Inequality in the case of non-homogeneous boundary data
In this section, we prove a Carleman inequality for the Oseen equations: Above and in the following, and we use the following notation for the particular constant: We recall that C > 0 denote a generic constant only depending on the geometry and independent on s, λ, z 1 , z 2 .
We extend ψ to O (while keeping the same name) in a such a way that: be a linear continuous map (given for example by Stein's theorem, see [1]), also continuous from We also denote by z 1 , z 2 some continuous extensions of z 1 , z 2 for the L ∞ and W 1,r norms in O respectively. The From the continuity of the extension operator E we have: Moreover, since z 2 ∈ W 1,r ( O), we use the Hölder's inequality and the continuous embedding Thus, gathering (2.17), (2.18) and (2.19) and choosing λ ≥ m(z 1 , z 2 ) c for c large enough (and depending only on the geometry), the terms in z 1 , z 2 at the right hand side of inequality (2.17) can be absorbed and we obtain In above calculations, we have used the fact that ψ ≤ c 0 in O\O and (2.16). Then (2.12) follows by combining the above inequality with (2.20). Finally, to prove (2.13), we first apply (2.4) to ( v, p) which gives: Then, using (2.18) and (2.19) to estimate the last above integral, we obtain for c large enough and λ ≥ c m(z 1 , Hence, we use (2.20) and the rest of the proof is the same as for (2.12).

Stability estimates for Oseen and Navier-Stokes Equations
In this section we use the Carleman inequalities given in Theorem 2.3 to obtain several stability estimates for both distributed and boundary observation. In particular, we prove Theorems 1.2 and 1.3. We first prove a Hölder type interior estimates and a global log type estimates for a distributed observation. Then, we use an extension of the domain procedure to obtain a global log type estimates for a boundary observation. We recall that Ω is a nonempty bounded open subset of R N (N = 2 or N = 3) with a boundary ∂Ω of class C 2 , that Γ obs is a nonempty open subset of ∂Ω and that ω is a nonempty open subset of Ω. Moreover, z 1 , z 2 are vector fields satisfying (1.2) and we use the following notation for the particular constant: (3.1) We also recall that C > 0 denotes a generic constant only depending on the geometry and in particular independent on s, λ, z 1 , z 2 .
Then, we deduce the following Theorem 3.5. There exist c > 0 and c * > 0 such that for all z 1 , Proof. We apply Theorem 3.4 and, for s >ŝ, we introduce A such that we rewrite (3.6) as v L 2 (Ω) ≤ e sC * A + c * s M where C * def = e c * λ . First, if A = 0, since the previous inequality is true for all s, we obtain v L 2 (Ω) = 0 and then (3.9) holds. In the following, we assume A = 0.
Next, we suppose that 1 2C * ln(1 + M A ) ≥ s and we choose s = 1 and next, using the fact that 1

Stability estimates with boundary observation
We now prove the following theorem from which we deduce the logarithm estimates stated in Theorem 1.2 as in the proof of Theorem 3.5. Notice that the first estimate (1.6) is given in the previous Theorem 3.5 (see (3.9)).  Let Ω be an extension of Ω of class C 2 through Γ obs (see Figure 1), namely Ω is of class C 2 , ∂Ω ∩ Ω = Γ obs .

There exists an extension
In particular, v 2 with (w, ∂w ∂n , q) = (g obs , h obs , k obs ) on Γ obs . Then, let us denote by S the linear continuous extension operator given by Stein's theorem (see [1]): .
We also denote by T the linear continuous operator of restriction to Ω: . Finally, by denoting ( w, q) It is easily checked that ( v, p) ∈ H 2 ( Ω) × H 1 ( Ω).
Proof of Theorem 3. 6. In what follows, z 1 , z 2 denote some continuous extensions to R N of z 1 , z 2 , for the L ∞ and the W 1,r norm respectively. Let us consider the extensions Ω and ( v, q) ∈ H 2 ( Ω) × H 1 ( Ω) given by Lemma 3.7. Let us consider ω ⋐ Ω\Ω a non empty bounded open subset. We summarize these notations in Figure 1.
Here, we have used the notation ψ max def = max x∈ Ω ψ(x). Thus, by dividing the above inequality by e 2se λc 0 and using that and that λ ≥ m(z 1 , z 2 ) c, we obtain for some c * > 0 large enough (independent on λ), With a similar argument, Now, to conclude, it remains to replace the term ∂v The above inequality with the following computations

yields 4 Application: stability estimates for boundary coefficients inverse problems
In the present section, we focus on the proof of Theorem 1.4. We begin by considering the Navier boundary conditions. One can first notice that the pair (v , p) Without loss of generality, we can assume that |z 1 | ≥ m on K. Then, since ( To estimate the above right hand side, we use the following inequalities: Note that the above inequalities are an immediate consequence of the interpolation inequality · 2 L 2 (∂Ω) ≤ C · H 1 (Ω) · L 2 (Ω) which can be obtained for instance by first applying [20, Theorem 1.5.1.10] to get C > 0 such that for all u ∈ H 1 (∂Ω) and all 0 < ε < 1, and next by taking ε = u 2 Hence, from (4.2) we obtain: and we conclude using the estimate on v L 2 (Ω) given by Theorem 1.3. For the Robin boundary conditions, we proceed in exactly the same way to obtain and conclude using the estimate on v L 2 (Ω) and on p L 2 (Ω) given by Theorem 1.3 Remark 4.1. We can obtain a better estimate assuming more regularity on (v, p). For example, if (v, p) ∈ H k (Ω)×H k−1 (Ω), k ≥ 2, we can use an interpolation inequality in (4.4) to obtain Then (1.18) follows from (4.5) with the interpolation inequality · H θn (K) ≤ C · 1−θ L 2 (K) · θ H n (K) . Remark 4.2. Concerning the Navier boundary conditions, we can obtain the same result in a different way, by writing [D(v)n] τ in terms of curl v on Γ 0 . Indeed, since v · n = 0 on Γ 0 , ∇(v · n) = ∂(v · n) ∂n n and then, On the other hand, using the same kind of computations, we have Hence, we obtain that Thus, in the previous proof, we can write and conclude using the estimates (1.11) and (1.12) on v L 2 (Ω) and curl v (L 2 (Ω)) 2N−3 given by Theorem 1.3.

Application to error estimates
In this section, we consider the reconstruction of (v, p), solution of the Stokes system in Ω, knowing v and σ(v, p)n on Γ obs . In other words, we consider the data completion problem for the Stokes system, that is: from given data g D ∈ H 3/2 (Γ obs ) and As the problem is ill-posed, it is mandatory to use a stabilization method to stably reconstruct (v, p) from the data f , g D and g N . Such a stabilization method usually depends on a parameter of regularization ε > 0, and it must fulfill the two following requirements: it must have a solution for any data f , g D and g N , regardless of the existence of a solution to the corresponding Stokes problem (5.1). And its solution should converge to the solution of (5.1) when the parameter ε goes to zero, when such a solution exists.
We study below two standard methods of regularization: a quasi-reversibility method and a penalized Kohn-Vogelius method. In particular, we obtained the convergence rates of these methods directly from the estimates obtained previously.
Indeed, since σ(V , P )n · n = 2ν ∂V ∂n · n − P and it suffices to choose P = 0 and a continuous lifting V which satisfies V = g D and ν ∂V ∂n = 1 2 (g N · n)n + g N τ − ν(∇ τ g D )n on Γ obs .
Suppose now that the initial data completion problem admits a (necessarily unique) solution (v, p) ∈ H 2 (Ω) × H 1 (Ω). Then, we have the following Theorem 5.2. The solution (v ε , p ε ) ∈ H 2 (Ω) × H 1 (Ω) of the quasi-reversibility problem (5.2) converges to (v, p) ∈ H 2 (Ω) × H 1 (Ω) solution of the data completion problem for the Stokes problem (5.1) when ε tends to zero, strongly in H 2 (Ω) × H 1 (Ω). We furthermore have the estimate as test functions in the quasi-reversibility problem (5.2), which is admissible as they verify the boundary conditions, we directly obtain to equation (5.4), we obtain Going back to equation (5.4), we finally obtain which, using (5.5) and (5.7), directly leads to the estimate (5.3). Now, suppose that v ε and p ε do not converge to v and p. Then there exist ρ > 0 and ε n , sequence of strictly positive real numbers verifying ε n −−−→ n→∞ 0, such that the By equation (5.5), we know that (v n , p n ) is a bounded sequence in H 2 (Ω) × H 1 (Ω). Hence, up to a subsequence (that we still denote (v n , p n )) the sequence converges to some (w, q) weakly in H 2 (Ω) × H 1 (Ω). Then equation (5.3) and the boundary conditions verified by (v n , p n ) directly imply that (w, q) verifies the Stokes data completion problem (5.1), which in turn implies by uniqueness of such solution that w = v and q = p. Therefore, v n weakly converges to v in H 2 (Ω) and p n weakly converges to p in H 1 (Ω). But Equation (5.5) implies then that (v n , p n ) strongly converges to (v, p), which is a direct contradiction with the definition of the sequence, and therefore ends the proof.
Remark 5.3. It is not difficult to obtain the following complement to the theorem: if the initial data completion problem for the Stokes system does not admit a solution, then Otherwise, we would have a sequence of strictly positive real numbers (ε n ) n∈N verifying ε n −−−→ n→∞ 0 and (v εn , p εn ) H 2 (Ω)×H 1 (Ω) ≤ C. But using the same arguments as in the last paragraph of the proof of theorem 5.2, extracting a subsequence and passing to the limit, we would obtain a solution to the data completion problem for the Stokes system, in obvious contradiction with the assumption.
Proposition 5.1 and Theorem 5.2 clearly show that the proposed quasi-reversibility method (5.2) is a regularization method for problem (5.1). However, if Theorem 5.2 assures the convergence of the approximated solution to the exact one, it does not give any rate of convergence. Actually, it is known (see [23, section 2.5] and the references therein) that Carleman estimates are the key argument to derive convergence rates for the quasireversibility method. This is the case for the quasi-reversibility method proposed above and we now prove Theorem 1.8 for this method: Combining this result with the previous estimates, we therefore obtain where M > 0 is such that v H 2 (Ω) + p H 1 (Ω) ≤ M . Such estimate highlights the competition between regularization and noise, which leads to the question of the optimal choice of the regularization parameter ε with respect to the amplitude of the noise δ. On this subject of the optimal choice of the regularization parameter for the quasi-reversibility method for elliptic equations, see [11,12] and the references therein.

Error estimates for the Kohn-Vogelius method
The quasi-reversibility method proposed in the previous section regularizes the data completion problem for the Stokes system by solving approximately the first two equations of (5.1) (see the estimate in Theorem 5.2) while verifying exactly the boundary conditions. The Kohn-Vogelius method we study now is somehow a symmetric method, in the sense that it solves exactly the equations in Ω with approximated boundary conditions. And again, we obtain the rate of convergence of the method using the same estimates (1.7) and (1.8).
Remark 5.5. For this Kohn-Vogelius method, we have to impose Γ obs ∩ Γ C obs = ∅ in order to guarantee that the functional is well-defined and more precisely that the pairs In the case Γ obs ∩ Γ C obs = ∅, one cannot guarantee that the solutions belong to H 2 (Ω) × H 1 (Ω) (see for example [28]).
It is not difficult to verify that the two following propositions are equivalent: Let us briefly explain why. Due to the denseness of the admissible data (see [8] or [16, section 2]), for all ε > 0, there exists f ε , g Dε , g Nε ∈ L 2 (Ω) × H 3/2 (Γ obs ) × H 1/2 (Γ obs ) such that the corresponding Stokes problem (5.1) has a solution (v ε , p ε ) and Therefore, choosing ϕ Nε = σ(v ε , p ε )n and ψ Dε = v ε on Γ C obs , it is not difficult to see that Hence, the infimum of F is 0. Furthermore, if the above sequence (ϕ Nε , ψ Dε ) is bounded in H 1/2 (Γ C obs ) × H 3/2 (Γ C obs ), one can extract a weakly convergent subsequence which leads to the existence of a solution of the Cauchy problem (5.1) using Problems (5.8) and Inequality (5.10).
Thus, to regularize the problem, we add a penalization term: for ε > 0, we introduce the functional F ε : We have the following result: Proof. Obviously, the functional F ε is continuous and strictly convex. Furthermore, it is coercive. Indeed, suppose it is not. Then there exists a sequence (ϕ m N , ψ m D ) and a constant C > 0 such that This directly implies (v ϕ m N , p ϕ m N ) H 2 (Ω)×H 1 (Ω) < C and (v ψ m D , p ψ m D ) H 2 (Ω)×H 1 (Ω) < C, which directly implies (ϕ m N , ψ m D ) H 1/2 (Γ C obs )×H 3/2 (Γ C obs ) < C by continuity of trace and normal derivative operators, which is a contradiction with the initial assumptions.
Therefore F ε is continuous, strictly convex and coercive, which implies the result (see [17]).
Now, we see that F (ϕ ex N , ψ ex D ) = 0 = F (ϕ ex N + c n, ψ ex D + c) for any c ∈ R and c ∈ R N . Therefore, similarly has previously, we have F ε (ϕ m N , ψ m D ) ≤ F ε (ϕ ex N + c n, ψ ex D + c) which implies directly implying that the weak convergences are actually strong convergences. Finally, a standard argument ad absurdum ends the proof as in the end of the proof of Theorem 5.2.
Remark 5.8. The Kohn-Vogelius functional is classically defined by F(ϕ N , . Notice that Proposition 5.6 is also valid for the associated functional F ε . The only point where the H 2 -seminorm is needed is Inequality (5.13).