Quantification of the unique continuation property for the nonstationary Stokes problem

The purpose of this work is to establish stability estimates for the unique continuation property of the nonstationary Stokes problem. These estimates hold without prescribing boundary conditions and are of logarithmic type. They are obtained thanks to Carleman estimates for parabolic and elliptic equations. Then, these estimates are applied to an inverse problem where we want to identify a Robin coeﬃcient deﬁned on some part of the boundary from measurements available on another part of the boundary


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 where u and p denote respectively the fluid velocity and the fluid pressure.Since the work made by Fabre and Lebeau in [14], 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.
In this work, we want to quantify these unique continuation properties.More precisely, we want to derive stability inequalities which assert that, if the measurements of u and p made on an interior domain ω or on a boundary part Γ are small, then u and p stay small on the whole domain Ω.In what follows, we will prove the following result: Theorem 1.
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 [9] (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 [22] and [19], local estimates are proved which only require measurements on u.In [22], 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: where A is a compact subset of Ω and 0 < θ < 1.In [19], 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 [1], [10], [23].In this case, it is well established that the best possible rate for the global stability is logarithmic.We refer to the overview [2] 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 [8] and in [9].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 Such system may be viewed as a simple model of the blood flow in the cardiovascular system (see [24] and [26]) or of the airflow in the respiratory tract (see [4]).We refer to [13] 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 [3] and [12] and with the heat equation to [6] (see also the references therein).
In [8], 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 [6] 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 [8] 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.
Then, there exists α > 0 independent of ε, C > 0 which depends on ε, ν 0 and N 0 such that We have used the following notation 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 ( [1]).This sometimes may lead to estimates of log-log type like in [5] or [7].In our case, the interior doubling inequality obtained in [19] (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 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 [9], our quantification result was based on local Carleman inequalities ( [18], [21] and [25]) obtained thanks to Gårding inequalities involving pseudodifferential computation.The same inequalities were used in [23] to quantify the unique continuation property for the Laplace equation.We refer to the survey [20] (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 [15].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 [27] and the inequality in the elliptic case can be proved with the methods presented in [15].
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 for all (u, p) solution of ( 1) in In this inequality, F is defined by If we compare estimate (8) with the equivalent estimate proved for the stationary Stokes equation in [9] (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 [9]) except that we need an additional measurement of u in H 1 (0, T ; L 2 (Γ)) for the estimate (8).For parabolic equations like heat equation, it is proved in [27] 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 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 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 (11) 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.
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 (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 where F is given by (9).Using this estimate in the right hand-side of (13), we get Let us introduce an open set Ω 0 such that Ω \ Ω ∩ Ω ⊂⊂ Ω 0 ⊂⊂ Ω.We have, according to interpolation inequalities, We apply again Proposition 2 and we get We gather this inequality with ( 14) and we get that there exists a constant C > 0 such that To conclude the proof, we notice that 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 Since Ω 0 ⊂ Ω, we can choose a sufficiently large N > 5 such that These constants δ and N only depend on the domains Γ and Ω.Let ǫ > 0 be fixed and choose We arbitrarily fix and where λ is a large enough fixed positive parameter.For 1 ≤ i ≤ 5, we define We then define For 1 ≤ i ≤ 5, we denote by For these domains, the following lemma holds: Lemma 1.The sets ( D i ) and (D i ) satisfy the following properties: (iii) For all 1 ≤ i ≤ 5, with with according to (15).This implies that ϕ(t, x) > µ 5 which shows the first inclusion.
We get a contradiction and this allows to conclude that (iv) According to the definition of D i , (22) 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 , Then, there exists C > 0 such that: 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 .
To define this function, we can take χ(t, x) = χ ϕ(t,x)−µ3 µ4−µ3 We have the following estimate: for all 1 ≤ i, j ≤ 3, for all We apply the Carleman inequality for parabolic equations on the domain D 2 (see Theorem 3.2 in [27]) 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 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 (26), we get the existence of a constant C > 0 such that: 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: where C 0 does not depend on ε.Then, since by Lemma 1 (i), we obtain: 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 We introduce a cut-off function χ in C 2 (Ω) such that 0 ≤ χ ≤ 1 and As previously for χ, this function can be defined explicitly with the help of χ: We have the following estimate: for all 1 ≤ i, j ≤ 3, for all x ∈ Ω, Let us define π = χ p.Using that D 2 ⊂ D 1 and χ = 1 on D 2 , inequality (28) becomes By taking the divergence of the first equation of (1), we obtain that ∆p = 0 in Q.Thus, π is solution of We apply to π the classical Carleman inequality for elliptic equations (which can be proved as in [15]) 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 , (s|∇π| 2 + s3 |π| 2 )e 2s φdσ (32) By using Lemma 1 (iv), there exists C > 0 such that 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 , where Let us remark that, thanks to inequality ( 16), we have for (t, x) ∈ (0, T ) × B: which implies that ϕ ≤ µ 3 on (0, T ) × B. Thus, inequality (33) implies that: We sum up inequalities ( 27) and (34).The second term in the right hand-side of inequality ( 27) with the gradient of p is absorbed by the left hand-side of inequality (34), for s large enough.Then, since by for C 2 = max(C 0 , C 1 ) which is independent of ε.This implies that, for all s ≥ s 0 : 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 : where m ∈ N is such that 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 : Thus, thanks to Lemma 2, we have proved estimate (8).
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 where ω 0 is a nonempty open subset of Ω such that ω 0 ⊂ ω.Next, we take N > 5 large enough so that where δ is now defined by δ = d 0 C(Ω) and we choose β > 0 which satisfies (16).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 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 [27] 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 Let us define χ by (24) 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 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 ( [15]) 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 , where Moreover, according to Cacciopoli inequality ( [17]), since ∆p = 0 in (0, T ) × Ω, we have 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) where B(0, R) is the open ball of center 0 and of radius R, and by Let (u, p) be a solution of (1).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 )) 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, 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: with and Let us define the operator P φ by: for a regular scalar function f , and by for a regular vector-valued function F = (F 1 , F 2 , F 3 ).We can rewrite system (38) as follows: Let ε > 0 be given.We consider and we define and 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 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 and, for all where C > 0 is a constant which only depends on R, T , ε and γ.
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 where the operator [P φ , χ η ] is defined by for all vector-valued function v.We denote by In the following, we consider that η ∈ 0, R 8 is given.We apply the Carleman estimate for parabolic equations (Theorem 3.2 in [27]) 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 , 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 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 For the first term in the right hand-side of (44), we first notice that there exists C > 0 such that |∇q| 2 e 2sϕ dx dt according to (43) and that there exists C > 0 such that 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 |∇q| 2 e 2sϕ dx dt + C η 4 e 2sα1 v 2 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 Note that, according to (41), E(η) ⊂ D 3R 2 and 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 Moreover, there exists a constant C > 0 depending only on R and γ such that, for all x ∈ D 3R 2 , we have 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 The function π η is solution of where the operator [P φ , ζ η ] is defined by 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 , 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 Thus, if we set s = se −λβ(t−t0) 2 in inequality (50) and if we integrate in time over (t 0 − ǫ, t 0 + ǫ), we obtain that there exists a constant s 0 > 0 and a constant C such that, for all s > s 0 , where J 2 is given by Thus, inequality (49) becomes 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 we obtain that, for all s ≥ s 0 , where C η = α 2 − α 1 > 0 and M is given by (12).Since Q , we can in fact estimate v and q on the whole set Q(4η): for all s ≥ s 0 Since C η = e 4λη − e 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 , Note that the previous inequality is in fact valid for all s ≥ 0 since = 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 Moreover, this inequality still holds if J 1 + J 2 = 0. Now, let us introduce the following set 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 where 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 ′ ).
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 L 2 (Q(4η)) .