Manabu Tsukada

Proposition of Dynamic Virtual Link for IPv6 GeoNetworking (GN6) in ETSI

Background

Intelligent Transportation Systems (ITS) are systems deployed to optimize the road traffic and realize safe, efficient and comfortable human mobility. Cooperative ITS is a new vision of ITS where vehicles, the roadside infrastructure, traffic control centers, road users, road authorities, road operators, etc. exchange and share information based on a common communication architecture -known as the ITS station reference architecture [ISO-21217-CALM-Arch, ETSI-EN-302-665-Arch] -supporting all types of ITS use cases over a diversity of access technologies (11p, 11n, 3G/4G, infra-red, ...). The building blocks of the ITS station are specified within ISO, ETSI, IETF and IEEE.

[ETSI-TS-102-636-6-1-GN6] specifies the transmission of IPv6 packets over ETSI GeoNetworking protocol [ETSI-TS-102-636-4-1-GN] via a protocol adaptation sub-layer referred as the GN6ASL (GeoNetworking to IPv6 Adaptation Sub-layer) (See Figure 1). The definition of Geographical Virtual Link (GVL) in [ETSI-TS-102-636-6-1-GN6] is described as bellow:

A Geographical Virtual Link (GVL) is a link-local multicast-capable virtual link spanning multiple physical links with geographically scoped boundaries.

Each GVL shall be associated with one single GeoNetworking GEOBROAD-CAST/GEOANYCAST area stored in the per-GVL MIB attributes (itsGn6aslGvlAreaCenterLatitude, itsGn6aslGvlAreaCenterLongitude, itsGn6aslGvlAreaDistA itsGn6aslGvlAreaDistB and itsGn6aslAreaAngle). These attributes are referred to as the GVL Area and shall be: • derived from a received GeoNetworking header encapsulating a Router Advertisements (RA) message [11] as described in clause 10.2.1; or • assigned by the ITS station management entity only in roadside ITS stations.

Topological Virtual Link

The definition of Topological Virtual Link (TVL) in [ETSI-TS-102-636-6-1-GN6] is described as bellow:

A Topological Virtual Link (TVL) is a non-broadcast multi-access (NBMA) virtual link spanning multiple physical links with topologically scoped boundaries. IPv6 multicast traffic may not be exchanged through a TVL.

4 Limitation of Draft EN 302 636-6-1 V0.1.2 (2012-11) We describe the limitation of Draft EN 302 636-6-1 V0.1.2 (2012-11) in the Section. Note that this study is based on year November 2012 versions of the ETSI Standards and that the standards constantly evolve. As a result, parameters, primitives and virtual links on GN6ASL may have changed at the time of reading.

V2V scenario

As GVL is created upon the reception of Router Advertisements from Access Routers specified in [rfc4862], the communication using GVL cannot be enabled before the reception of RA. TVL is not used for multicast nor geographical scoped communication (i.e. Geo-Broadcast). Thus current specification will not work in the V2V scenarios where there are no infrastructure around the vehicles.

Router solicitation with GVL

As GVL is created upon the reception of Router Advertisements from Access Routers, the communication using GVL cannot be enabled before the reception of RA. Thus the mobile router cannot send the router solicitation before receiving the RA. Usually, the router solicitation should be sent in order to solicit the router advertisement from access router.

Geographical scoped communication

The GeoArea of GVL is configured in access router and shared among all the GeoAdhoc router. Thus no GeoAdhoc router can configure the GeoArea of Geographic scoped communication in GN6ASL, except for access router. In other words, the application in the vehicle ITS Station cannot configure the destination GeoArea of GeoBroadcast, but only select the existent GVLs created by RA at the moment.

INRIA technical paper (29th November 2012)

Dropped Packet before checking IPv6 routing table

Upon the reception of IPv6 GeoNetworking packet on GN6ASL, GN6ASL must select the virtual link to pass the IPv6 packet to IPv6 layer. However, in the current specification, the some IPv6 packets are dropped in GN6ASL without checking the routing table nor the multicast route table, even when the received node is the GN destination or received node is inside the destination GeoArea. This is because current rule of virtual interface selection has the case where the received IPv6 packets does not matched with any of the virtual interface selection rules in GN6ASL (See figure 2 for simplified selection rule, and see Section 8.2.2 in [ETSI-TS-102-636-6-1-GN6] for detail). Packets are dropped in GN6ASL in following cases (even the receiver is inside of the destination GeoArea, or the receiver is the GN destination of the packet):

• GeoArea of the GeoBroadcast destination does not matched with any of GVL in the receiver, and it does not bring Router advertisement.

• GeoArea of the GeoAnycast destination does not matched with any of GVL in the receiver.

• GeoUnicast brings IPv6 packet to the destination of global address and no GVL exists in the receiver.

Proposition of Dynamic Virtual Link (DVL)

To overcome the limitation arisen in Section 4,w ep r o p o s eD y n a m i cV i r t u a lL i n k( D V L ) in this section. Section 5.1 provides the definition of DVL and Section 5.2 presents the comparison of properties of TVL, DVL and GVL.

INRIA technical paper (29th November 2012)

Definition

A Dynamic Virtual Link (DVL) is multicast-capable virtual link spanning multiple physical links with geographically scoped boundaries.

A DVL is created when the physical interface is up, and exactly one DVL exists per a single physical interface. DVL shall be associated with one single GeoNetworking Geo-Broadcast /GeoAnycast area (also called GeoArea in a n d specified in [ETSI-TS-102-636-4-1-GN]) stored in the following DVL MIB attributes:

• itsGn6aslDvlAreaCenterLatitude • itsGn6aslDvlAreaCenterLongitude • itsGn6aslDvlAreaDistA • itsGn6aslDvlAreaDistB • itsGn6aslDvlAreaAngle
These attributes are referred as the DVL Area and shall be assigned by the ITS station management entity. And if the attributes are not specified, the DVL shall not used to send link-local and global multicast. In the case, DVL is only used for the reception of the packet and sending link-local and global unicast.

Symmetric access is not guaranteed in the communication via DVL, therefore IPv6 ND shall be disabled.

In sender side GN6ASL in a GeoAdhoc Router, DVL is used in V2V scenario where no infrastructure is around the vehicle nor no RA is received as described in Section 4.1. Also Router solicitation can be sent via DVL before receiving the Router advertisement in the situation where we described in Section 4.2. In addition, destination GeoArea can be specified as the sender needs by configuring DVL Area (i.e. DVL MIB attributes) stored in the management entity.

In the receiver side GN6ASL in a GeoAdhoc router, DVL act as a default interface in the virtual interface selection. Therefore, no packet is dropped in GN6ASL during the virtual selection procedure (See this problem in Section 4.4).

Property

Table 1 shows the properties of TVL, DVL and GVL.

TVL is a virtual link spanning multiple physical links with topologically scoped boundaries (but the boundary is never applied because TVL employes always GeoUnicast as an underlying GN routing mode). On the other hand, DVL and GVL are spanning multiple physical links with geographically scoped boundaries. The scope of DVL (destination GeoArea) is dynamically configured as geographically scoped boundaries in the management entity in each node and thus it is variable in each nodes connected to the link. The scope of GVL (destination GeoArea) is statically set by the scope of the received router advertisement and is invariable among the nodes connected to the link. As a result, only GVL guarantees symmetric access among the nodes on the link.

Non-broadcast multi-access (NBMA) is adapted in TVL as a virtual interface whereas Ethernet V2.0/IEEE 802.3 LAN virtual interfaces is used in DVL and GVL. In an access GeoAdhoc router, all of TVL, DVL and GVL are created when the physical communication interface is up. On the other hand, the other GeoAdhoc routers create only only TVL and DVL, but creates no GVL. GVL is setup only by the reception of router advertisement. TVL and DVL exists exactly one per the physical communication interfaces, and GVL is created upon the reception of Router advertisement (thus number of GVL depends on the number of different router advertisement received on the node).

As IPv6 address assignment, only link-local address shall be assigned to TVL and DVL whereas both Global and link-local address is assigned to GVL interfaces. The way of the allocation of the link-local address on TVL, DVL and GVL is proposed in [INRIA-report-extended-IID].

Modified Operations

The section describes the modified operation by introducing DVL. The modified operation on both sender and receiver side GeoAdhoc router is summarized in Table 2. Link-local multicast is transmitted by GeoBroadcast from the multicast-capable virtual interface specified by application (DVL and GVL). Destination GeoArea is specified in the MIB in the management entity in both cases.

IPv6 Link-local Unicast

Received Geobroadcast is received on correspondent GVL (GeoArea exact match). If no GVL matched, the packet received on the DVL.

Global Unicast

Global unicast packets are sent to GN6ASL via one of the virtual link (DVL and GVL) according to the routing table. The destination GN address is derived from MID of the next hop to the IPv6 global destination address.

Global Unicast using GeoUnicast is received on suitable GVL (prefix match, SO PV match, etc). Otherwise, received on DVL.

Global Multicast

Using GeoBroadcast via DVL or GVL. Destination GeoArea is specified in the MIB in the management entity in both cases.

Global Multicast using GeoBroadcast is received on a correspondent GVL (area exact match, SO PV match, etc). Otherwise received on the DVL. When an IPv6 link-local unicast packets are passed to the GN6ASL via TVL, DVL or GVL, GN6ASL derives MID of the destination IPv6 link-local address as the destination GN address of the packet. There are no difference in the GN header nor the MAC header depending on which virtual interface (among TVL, DVL and GVL) is used. The only difference is the source IPv6 link-local address in the IPv6 header (The way of the allocation of the link-local address on TVL, DVL and GVL is proposed in [INRIA-report-extended-IID]).

Upon the reception of the IPv6 link-local unicast using GeoUnicast, the GN6ASL selects the virtual interface to pass the packet to the IPv6 layer. The selection is done by the matching between the destination IPv6 address and IPv6 link-local address assigned to the virtual interface.

IPv6 Link-local Multicast

IPv6 link-local multicast shall not be forwarded beyond IPv6 router. The packet has linklocal multicast address in the destination and link-local unicast address allocated to the virtual interface in the source address, and the communication interface is specified by the application. TVL shall not be specified as an output interface, because the interface has not multicast capability. IPv6 link-local multicast packet is encapsulated by GeoBroadcast header.

When an IPv6 link-local multicast packets are passed to the GN6ASL via DVL or GVL, GN6ASL set the destination GeoArea from the MIB for the virtual interface (DVL MIB attributes or DVL MIB attributes).

Upon the reception of the IPv6 link-local multicast using GeoBroadcast, the GN6ASL selects the virtual interface to pass the packet to the IPv6 layer. When the destination GeoArea in the GN header and the GVL area are matched, the IPv6 packets are passed to IPv6 layer via the correspondent GVL. When none of GVLs matches, the DVL is selected as the interface to the IPv6.

IPv6 Global Unicast

IPv6 global unicast has IPv6 global address in both source and destination address. IPv6 global unicast is forwarded beyond IPv6 router, thus the source and destination address in the packets may not be the address assigned to the virtual interface of the GeoAdhoc router. The IPv6 global unicast packet is encapsulated by GeoUnicast header.

Upon passing an IPv6 global unicast packet to the GN6ASL, the IPv6 layer selects an interface to the GN6ASL based on the routing table (i.e. the longest match principle). TVL shall not be selected because no routing entry suppose to indicate the interface, because no route will be exchanged via non-multicast capable TVL1 . The destination GN address of the GeoUnicast packet is derived from MID of the next hop IPv6 address to the IPv6 global destination address.

Upon the reception of the IPv6 global unicast packet using GeoUnicast, the GN6ASL performs the virtual interface selection. The selection includes matching between IPv6 prefix of a GVL and the destination prefix, matching between source node position vector and GVL area and etc. If none of these criteria is not applied, DVL is selected as default virtual interface to pass the packet to the IPv6 layer.

IPv6 Global Multicast

IPv6 global multicast has an IPv6 multicast address in the destination address and an IPv6 global address in the source address. IPv6 global multicast is forwarded beyond IPv6 router, thus the source address in the packets may not be the address assigned to the virtual interface of the GeoAdhoc router. IPv6 global multicast packet is encapsulated by GeoBroadcast header.

Upon passing an IPv6 global unicast packet to the GN6ASL, the IPv6 layer selects zero or more interface to the GN6ASL based on the multicast route table. TVL shall not be selected because multicast listener should not exist via TVL. GN6ASL set the destination GeoArea from the MIB for the virtual interface in both cases of DVL and GVL (DVL MIB attributes and DVL MIB attributes).

Upon the reception of the IPv6 global multicast using GeoBroadcast, the GN6ASL selects the virtual interface to pass the packet to the IPv6 layer. When the destination GeoArea in the GN header and the GVL area are matched, the IPv6 packets are passed to IPv6 layer via the correspondent GVL. When none of GVLs matches, DVL is selected as default virtual interface to pass the packet to the IPv6 layer.

Consideration

About TVL

As we propose to add Dynamic Virtual Link (DVL) in GN6 specification, the importance of TVL reduces much. We consider that there are two future directions for TVL.

• TVL is no longer necessary and abolish it

• TVL remain as a virtual link that has a different role from DVL, employing To-poBroadcast as as underlaying GN routing mode Let's see the first possibility. As we have seen in Table 1, the properties of TVL and DVL is almost same, except for the scope and the access. The current GN6 specification says that TVL has topological boundary, however it is never used with GeoBroadcast, but only GeoUnicast. Therefore the boundary is never applied in the geographic routing of GN6 packets. In addition to this, the current GN6 specification also says that TVL has non-broadcast multi-access (NBMA) interface. But we don't see the advantage of bringing NBMA (that used for circuit based network concept) to the IPv6 GeoNetworking specification. Thus we consider TVL and DVL function can be merged into one virtual interface.

The other possibility is remaining TVL as a virtual interface that has different role with DVL. In current specification, IPv6 packet is never encapsulated by TopoBroadcast header. Therefore, we can consider TVL as the virtual link that employ TopoBroadcast as underlaying GN routing mode. To realize this, the restriction that TVL has non-broadcast multi-access (NBMA) must be removed. Certain IPv6 implementation sends the router advertisement with link-local unicast as a reaction of the router solicitation in order not to broadcast the message in all the node in the link.

Router Advertisement with Unicast

Both of GVL specified in [ETSI-TS-102-636-6-1-GN6] and DVL proposed in the documents cannot adapt router advertisement with unicast. Because the receiver cannot determine the GVL area upon the reception of such router advertisement.

One possible solution for the problem is disabling the router advertisement using Unicast with GVL. This must be a modification of the IPv6 layer behavior.

Extended GN6_SAP

GN6ASL looks the DVL MIB attributes up in the management entity upon receiving IPv6 packets from the IPv6 Layer on DVL in order to determine the destination GeoArea. The determination of the destination GeoArea can be upgraded by Extended GN6_SAP (EGN6_SAP) described in Appendix C 2.2.2 in [ETSI-TS-102-636-6-1-GN6].

Table 3 shows the comparison of GN6_SAP and EGN6_SAP. The IPv6 layer can use GN6_SAP without modification of the IPv6 while EGN6_SAP requires the IPv6 layer to specify a geographical area in the format specified by .

As GeoArea of DVL can be configured dynamically unlikely to GVL, the IPv6 layer can specify different destination GeoArea per packet using EGN6_SAP using DVL. When the destination GeoArea is not specified in EGN6_SAP, the GN6ASL shall refer DVL MIB attributes.

GN6_SAP (Appendix

Discussion

We consider virtual link needs to be specified as a parameter of GN6-UNITDATA.request and EGN6-UNITDATA.request.

Conclusion

In the document, we presented the limitation of Draft EN 302 636-6-1 V0.1.2 (2012-11) at this moment. In order to overcome all of these issues, Dynamic Virtual Link (DVL) is introduced as a new link in addition to GVL and TVL specified in [ETSI-TS-102-636-6-1-GN6]. A DVL is multicast-capable virtual link spanning multiple physical links with geographically scoped boundaries as well as GVL. However, unlike to GVL, GeoAdhoc router can INRIA technical paper (29th November 2012) create DVL without reception of router advertisement. This property enables GeoAdhoc router using IPv6 in V2V scenario without depending on the roadside infrastructure. In the receiver side, GN6ASL selects DVL as a default link to pass the received packets to the IPv6 layer, when no criteria matches any of GVL.

As considerations, we raise the issue about TVL. Because of the introduction of DVL, TVL is no longer used in any situation. We consider there are two potential solutions for this issue. Then we found a minor issues to both of GVL and DVL regarding to router advertisement with unicast that allowed in [rfc4861]. For the solution, the slight modification on the IPv6 layer may be necessary. Finally, we discuss about per-packet GeoArea specification using EGN6_SAP.

We consider that further discussion is necessary in ETSI ITS_WG3 (ITS_WG3@LIST.ETSI.ORG) for approval of the specification of [ETSI-TS-102-636-6-1-GN6].

FigureFigure 1 :

 1 Figure 1: GN6ASL in the ITS station architecture

  INRIA technical paper (29th November 2012)

Figure 2 :

 2 Figure 2: Dropped packet before checking IPv6 routing table

  IPv6 link-local unicast shall not be forwarded beyond IPv6 routers. The packet has linklocal unicast address in both IPv6 source and destination address, and the communication INRIA technical paper (29th November 2012) the virtual interface specified by application (TVL, DVL or GVL). The destination GN address is derived from MID of the destination link-local address Link-local unicast using GeoUnicast packets is received on the interface (TVL, DVL or GVL) that has MID of the destination link-local address. The way of link-local address allocation is proposed in [INRIA-report-extended-IID] Linklocal Multicast

[ rfc4861 ]

 rfc4861 permits for router advertisement by being sent with both link-local unicast and multicast as follows. Section 4.2 Router Advertisement Message Format Destination Address: Typical ly the Source Address of an invoking Router Solicitation or the al lnodes multicast address. INRIA technical paper (29th November 2012)

Table 1 :

 1 Properties of TVL, GVL and DVL

	INRIA technical paper (29th November 2012)

Table 2 :

 2 Sender / Receiver Operation interface is specified by the application. IPv6 link-local unicast packet is encapsulated by GeoUnicast header.

Table 3 :

 3 GN6SAP and EGN6SAP

	C.2.2.1)	EGN6_SAP (Appendix C.2.2.2)
	GN6-UNITDATA.request(	EGN6-UNITDATA.request(
	destination address,	destination,
	source address,	source address,
	gn6sdu,	scope,
	priority	gn6sdu,
	)	priority
		)

INRIA technical paper (29th November 2012)

IPv6 Unicast shall not be received on TVL neither, because no virtual interface selection criteria matches. See the description about IPv6 Global Unicast reception.INRIA technical paper (29th November

2012)


Muriel Boulakia

Anne-Claire Egloffe

Céline Grandmont

Stability estimates for a Robin coefficient in the two-dimensional Stokes system *

Keywords: Inverse boundary coefficient problem, Stokes system, Robin boundary condition, Identifiability, Carleman inequality, Logarithmic stability estimate

In this paper, we consider the Stokes equations and we are concerned with the inverse problem of identifying a Robin coefficient on some non accessible part of the boundary from available data on the other part of the boundary. We first study the identifiability of the Robin coefficient and then we establish a stability estimate of logarithm type thanks to a Carleman inequality due to A. L. Bukhgeim [12] and under the assumption that the velocity of a given reference solution stays far from 0 on a part of the boundary where Robin conditions are prescribed.

Introduction

Let us consider an open Lipschitz bounded connected domain Ω of R d , d ≥ 2. 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 = ∅ (Figure 1 gives an example of such a geometry in dimension 2).

We denote by n the exterior unit normal to Ω and let τ = (τ 1 , . . . , τ d-1 ) be d -1 vectors of R d such that (n, τ ) is an orthogonal basis of R d .

We introduce the following boundary problem:

                
∂ t u(t, x) -∆u(t, x) + ∇p(t, x) = 0, ∀x ∈ Ω, ∀ t > 0, div u(t, x) = 0, ∀x ∈ Ω, ∀ t > 0, ∂u ∂n (t, x) -p(t, x)n(x) = g(t, x), ∀x ∈ Γ e , ∀ t > 0, ∂u ∂n (t, x) -p(t, x)n(x) + q(x)u(t, x) = 0, ∀x ∈ Γ 0 , ∀ t > 0, u(0, x) = u 0 (x), ∀x ∈ Ω.

(1.1)

Notice that we assume that the Robin coefficient q defined on Γ 0 only depends on the space variable.

Our objective is to determine the coefficient q from the values of u and p on Γ e .

Such kinds of systems naturally appear in the modeling of biological problems like, for example, 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 airflow in the lungs (see [START_REF] Baffico | Multiscale modeling of the respiratory tract[END_REF]). For an introduction on the modeling of the airflow in the lungs and on different boundary conditions which may be prescribed, we refer to [START_REF] Egloffe | Étude de quelques problèmes inverses pour le système de Stokes[END_REF]. The part of the boundary Γ e represents a physical boundary on which measurements are available and Γ 0 represents an artificial boundary on which Robin boundary conditions or mixed boundary conditions involving the fluid stress tensor and its flux at the outlet are prescribed. Similar inverse problems have been widely studied for the Laplace equation [START_REF] Alessandrini | Stable determination of corrosion by a single electrostatic boundary measurement[END_REF], [START_REF] Bellassoued | Stability estimate for an inverse boundary coefficient problem in thermal imaging[END_REF], [START_REF] Chaabane | Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems[END_REF], [START_REF] Chaabane | Identification of Robin coefficients by the means of boundary measurements[END_REF], [START_REF] Cheng | Stable determination of a boundary coefficient in an elliptic equation[END_REF] and [START_REF] Sincich | Lipschitz stability for the inverse Robin problem[END_REF]. This kind of problems arises in general in corrosion detection which consists in determining a Robin coefficient on the inaccessible portion of the boundary thanks to electrostatic measurements performed on the accessible boundary. Most of these papers prove a logarithmic stability estimate ( [START_REF] Alessandrini | Stable determination of corrosion by a single electrostatic boundary measurement[END_REF], [START_REF] Bellassoued | Stability estimate for an inverse boundary coefficient problem in thermal imaging[END_REF], [START_REF] Chaabane | Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems[END_REF] and [START_REF] Cheng | Stable determination of a boundary coefficient in an elliptic equation[END_REF]). We mention that, in [START_REF] Chaabane | Identification of Robin coefficients by the means of boundary measurements[END_REF], S. Chaabane and M. Jaoua obtained both local and monotone global Lipschitz stability for regular Robin coefficient and under the assumption that the flux g is non negative. Under the a priori assumption that the Robin coefficient is piecewise constant, E. Sincich has obtained in [START_REF] Sincich | Lipschitz stability for the inverse Robin problem[END_REF] a Lipschitz stability estimate. To prove stability estimates, different approaches are developed in these papers. A first one consists in using the complex analytic function theory (see [START_REF] Alessandrini | Stable determination of corrosion by a single electrostatic boundary measurement[END_REF], [START_REF] Chaabane | Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems[END_REF]). A characteristic of this method is that it is only valid in dimension 2. Another classical approach is based on Carleman estimates (see [START_REF] Bellassoued | Stability estimate for an inverse boundary coefficient problem in thermal imaging[END_REF] and [START_REF] Cheng | Stable determination of a boundary coefficient in an elliptic equation[END_REF]). In [START_REF] Bellassoued | Stability estimate for an inverse boundary coefficient problem in thermal imaging[END_REF], the authors use a result proved by K.D. Phung in [START_REF] Phung | Remarques sur l'observabilité pour l'équation de laplace[END_REF] to obtain a logarithmic stability estimate which is valid in any dimension for an open set Ω of class C ∞ . This result has been generalized in [START_REF] Bourgeois | About stability and regularization of ill-posed elliptic Cauchy problems: the case of C 1,1 domains. M2AN Math[END_REF] and [START_REF] Bourgeois | About stability and regularization of ill-posed elliptic Cauchy problems: the case of Lipschitz domains[END_REF] to C 1,1 and Lipschitz domains. Moreover, in [START_REF] Bellassoued | Stability estimate for an inverse boundary coefficient problem in thermal imaging[END_REF], the authors use semigroup theory to obtain a stability estimate in long time for the heat equation from the stability estimate for the Laplace equation.

In this article, we prove an identifiability result and a logarithmic stability estimate for the Stokes equations with Robin boundary conditions 1.1 under the assumption that the velocity of a given reference solution stays far from 0 on a part of the boundary where Robin conditions are prescribed. We would like to highlight why this assumption appears for the inverse problem of recovering a Robin coefficient. Let us consider (u i , p i ) be solutions of system 1.1 associated to q = q i , for i = 1, 2. Using the boundary conditions on Γ 0 , we obtain

(q 2 -q 1 )u 1 = q 2 (u 1 -u 2 ) + ∂u 1 ∂n - ∂u 2 ∂n -(p 1 -p 2 )n.
When u 1 vanishes, difficulties occur to estimate the difference between the Robin coefficients q 2 -q 1 . In the case of the scalar Laplace equation, it is possible to determine the sign of u solution of

         ∆u = 0, in Ω, ∂u ∂n
= g, on Γ e , ∂u ∂n + qu = 0, on Γ 0 , on any compact subset K ⊂ Γ 0 under some positivity assumption on the flux g (see [START_REF] Chaabane | Identification of Robin coefficients by the means of boundary measurements[END_REF]). Such a result comes from properties specific to harmonic functions, like for instance the maximum principle. When the flux g has a variable sign, G. Alessandrini, L. Del Piero and L. Rondi provide in [START_REF] Alessandrini | Stable determination of corrosion by a single electrostatic boundary measurement[END_REF] a quantitative control of the vanishing rate of u that allows to estimate the difference between the Robin coefficients on {x ∈ Γ 0 /d(x, ∂Ω\Γ 0 ) > d)}, for any d > 0, by using methods of complex analytic function theory. Moreover, G. Alessandrini and E. Sincich proved in [START_REF] Alessandrini | Detecting nonlinear corrosion by electrostatic measurements[END_REF] that the oscillation of u on Γ 0 is bounded from below by a constant depending on the a priori data only. To do so, they use unique continuation estimates for the Laplace equation. Due to the methods employed, it does not seem that we can extend these results to the Stokes system. This is why we estimate the Robin coefficient on a compact subset K ⊂ Γ 0 on which u 1 does not vanish. This estimate and the set K depend on u 1 , and knowing whether one can control our solution, for well chosen data, on the whole set Γ 0 or on any compact subset K ⊂ Γ 0 remains an open problem. Note however that in a really particular case (detailed in Remark 4.9), one can obtain a logarithmic estimate on the whole set Γ 0 .

The paper is organized as follows. The second section contains preliminary results on the regularity of the solution. In the third section, we are interested in the identifiability of the Robin coefficient q. Under some regularity assumptions and using the theorem of unique continuation for the Stokes equations proved in [START_REF] Fabre | Prolongement unique des solutions de l'equation de Stokes[END_REF], we prove that if two measurements of the velocity are equal on (0, T ) × Γ, where Γ ⊆ Γ e is a non-empty open subset of the boundary, then the two corresponding Robin coefficients are also equal on Γ 0 . Section 4 corresponds to the main part of our article. The results of this section are only valid in dimension 2. We prove a stability estimate, first for the stationary problem and then for the evolution problem. To do this, we use a global Carleman inequality due to A. L. Bukhgeim which is only valid in dimension 2 (see [START_REF] Bukhgeȋm | Extension of solutions of elliptic equations from discrete sets[END_REF]). The stability estimate for the unsteady problem is deduced from the stability estimate for the stationary problem thanks to the semigroup theory. We end Section 4 by concluding remarks and perspectives to this work.

When we are not more specific, C is a generic constant, whose value may change and which only depends on the geometry of the open set Ω and of the boundaries Γ e and Γ 0 . Moreover, we denote indifferently by | | a norm on R n , for any n ≥ 1.

We are going to start with some preliminary results which will be useful in the subsequent sections.

Preliminary results

In this section we study the well-posedness of the system and the regularity of the solution.

Regularity of the stationary problem

Let us first consider the stationary case:

             -∆u + ∇p = f, in Ω, div u = 0, in Ω, ∂u ∂n -pn = g, on Γ e , ∂u ∂n
-pn + qu = 0, on Γ 0 .

(2.1)

For g ∈ H -1 2 (Γ e ) d and v ∈ H Proposition 2.1. Let α > 0, f ∈ L 2 (Ω) d , g ∈ H -1 2 (Γ e ) d and q ∈ L ∞ (Γ 0 ) be such that q ≥ α on Γ 0 . System 2.1 admits a unique solution (u, p) ∈ V × L 2 (Ω). Moreover, there exists a constant C(α) > 0 such that u H 1 (Ω) d ≤ C(α)( g H -1 2 (Γe) d + f L 2 (Ω) d ). (2.2)
Proof of Proposition 2.1. The variational formulation of the problem is: find u ∈ V such that for every v ∈ V ,

Ω ∇u : ∇v + Γ0 qu • v =< g, v |Γe > -1 2 , 1 2 ,Γe + Ω f • v.
For all (u, v) ∈ V × V , we denote by

a q (u, v) = Ω ∇u : ∇v + Γ0 qu • v, (2.3)
and for all v ∈ V ,

L 1 (v) =< g, v |Γe > -1 2 , 1 2 ,Γe + Ω f • v.
We easily verify that a q is a continuous symmetric bilinear form. Since q ≥ α > 0, according to the generalized Poincaré inequality, the bilinear form a q is coercive on V . On the other hand, L 1 is a continuous linear form on V . Thus we prove the existence and uniqueness of u ∈ V solution of 2.1 by using Lax-Milgram Theorem. We obtain simultaneously estimate 2.2. We prove the existence and uniqueness of p ∈ L 2 0 (Ω) in a classical way, using De Rham Theorem. The fact that p is unique in L 2 (Ω) comes from the boundary conditions. We refer to [START_REF] Boyer | Éléments d'analyse pour l'étude de quelques modèles d'écoulements de fluides visqueux incompressibles[END_REF] for a complete proof in the case of Neumann boundary condition.

Next we want to derive regularity properties of the solution. Let us first recall existence and regularity results for the Stokes problem with Neumann boundary condition proved in [START_REF] Boyer | Éléments d'analyse pour l'étude de quelques modèles d'écoulements de fluides visqueux incompressibles[END_REF]. 1 . We assume that:

Proposition 2.2. Let k ∈ N. Assume that Ω is of class C k+1,
(f, h) ∈ H k (Ω) d × H k+ 1 2 (∂Ω) d .
Then the solution (u, p) of

     -∆u + ∇p = f, in Ω, div u = 0, in Ω, ∂u ∂n -pn = h, on ∂Ω, belongs to H k+2 (Ω) d × H k+1 (Ω)
and there exists a constant C > 0 such that:

u H k+2 (Ω) d + p H k+1 (Ω) ≤ C( h H k+ 1 2 (∂Ω) d + f H k (Ω) d ).
In order to study the Stokes system with Robin boundary conditions, one needs to specify to which space the Robin coefficient q belongs. As stated in Proposition 2.4, we will assume that q belongs to some Sobolev space H s (Γ 0 ) where s is large enough so that qu |Γ0 belongs to H r (Γ 0 ) if u |Γ0 belongs to H r (Γ 0 ). This stability in the Sobolev spaces will allow to apply the previous proposition (Proposition 2.2). Before stating the regularity result, let us state the following lemma:

Lemma 2.3. Let r, s ∈ R, with s > d-1
2 and 0 ≤ r ≤ s. Let q ∈ H s (Γ 0 ). The linear operator

T : H r (Γ 0 ) → H r (Γ 0 ) u → qu
is continuous. Furthermore, the following estimate holds true

qu H r (Γ0) ≤ C q H s (Γ0) u H r (Γ0) .
Proof of Lemma 2.3. Since s > d-1 2 , H s (Γ 0 ) is a Banach algebra (see [START_REF] Adams | Pure and Applied Mathematics[END_REF]) and thus T ∈ L(H s (Γ 0 ), H s (Γ 0 )) and T s = sup u∈H s (Γ0),u =0

T u H s (Γ 0 ) u H s (Γ 0 ) ≤ q H s (Γ0) . Moreover, since H s (Γ 0 ) → L ∞ (Γ 0 ), T ∈ L(L 2 (Γ 0 ), L 2 (Γ 0 )) and T 0 = sup u∈L 2 (Γ0),u =0 T u L 2 (Γ0) u L 2 (Γ0) ≤ q L ∞ (Γ0) ≤ C q H s (Γ0) .
Thus, the result follows by interpolation (see [START_REF] Bergh | Interpolation spaces. An introduction[END_REF] or [START_REF] Lunardi | Interpolation theory[END_REF]).

From Proposition 2.2 and Lemma 2.3 we deduce the following result:

Proposition 2.4. Let k ∈ N and s ∈ R with s > d-1 2 and s ≥ k + 1 2 . Assume that Ω is of class C k+1,1 . Let α > 0, M > 0, f ∈ H k (Ω) d , g ∈ H k+ 1
2 (Γ e ) d and q ∈ H s (Γ 0 ) such that q ≥ α on Γ 0 . Then the solution (u, p) of system 2.1 belongs to H k+2 (Ω) d × H k+1 (Ω). Moreover, there exists a constant C(α, M ) > 0 such that for every q ∈ H s (Γ 0 ) satisfying q H s (Γ0) ≤ M ,

u H k+2 (Ω) d + p H k+1 (Ω) ≤ C(α, M )( g H k+ 1 2 (Γe) d + f H k (Ω) d ).
Proof of Proposition 2.4. Let us prove the result for k = 0. Let h = -qu |Γ0 + g. According to Proposition 2.1, u belongs to H 1 (Ω) d . We obtain from Lemma 2.

3 for r = 1/2 that qu |Γ0 ∈ H 1 2 (Γ 0 ) d , which implies, since g ∈ H 1 2 (Γ e ) d and Γ e ∩ Γ 0 = ∅, that h ∈ H 1 2 (∂Ω) d . Using Proposition 2.2 with k = 0 we obtain that (u, p) ∈ H 2 (Ω) d × H 1 (Ω) and: u H 2 (Ω) d + p H 1 (Ω) ≤ C( h H 1 2 (∂Ω) d + f L 2 (Ω) d ).
But, since by assumption, q H s (Γ0) ≤ M , we have from Lemma 2.3 with r = 1/2, that:

h H 1 2 (∂Ω) d ≤ C(M )( u H 1 2 (∂Ω) d + g H 1 2 (Γe) d ).
We obtain:

u H 2 (Ω) d + p H 1 (Ω) ≤ C(M )( g H 1 2 (Γe) d + u H 1 (Ω) d + f L 2 (Ω) d ).
Thus we obtain the result for k = 0 using the inequality of Proposition 2.1. We then proceed by induction to prove the result for any k ∈ N.

Remark 2.5. Note that the space to which the Robin coefficient q belongs is not optimal. One could surely obtain similar regularity result for a less regular Robin coefficient. In fact, the key argument to proceed by induction in the proof of Proposition 2.4 is that qu |Γ0 ∈ H k+ 1 2 (Γ 0 ) d , for u ∈ H k+ 1 2 (Γ 0 ) d (this property allows to apply the regularity result given by Proposition 2.2).

Regularity of the evolution problem.

Concerning the initial problem 1.1, we can prove, using the Galerkin method, the following regularity results. For the sake of completeness, the proof of Theorem 2.6 is given in the appendix.

Theorem 2.6. Let s ∈ R be such that s > d-1 2 , T > 0, α > 0 and u 0 ∈ V . We assume that

Ω is of class C 1,1 , g ∈ H 1 (0, T ; H 1 2 (Γ e ) d ) and q ∈ H s (Γ 0 ) is such that q ≥ α on Γ 0 . Then problem 1.1 admits a unique solution (u, p) ∈ L 2 (0, T ; H 2 (Ω) d ) ∩ H 1 (0, T ; L 2 (Ω) d ) ∩ L ∞ (0, T ; V ) × L 2 (0, T ; H 1 (Ω)).
The following corollary will be useful when we will prove stability estimates for the evolution problem 1.1.

Corollary 2.7. Let s ∈ R be such that s > d-1

2 and s ≥ 3 2 , T > 0, α > 0 and u 0 ∈ H 3 (Ω) d ∩ H. We assume that Ω is of class C 2,1 , g ∈ H 2 (0, T ; H 3 2 (Γ e ) d ) and that q ∈ H s (Γ 0 ) is such that q ≥ α on Γ 0 . Then, problem 1.1 admits a unique solution (u, p) ∈ L ∞ (0, T ;

H 3 (Ω) d )∩H 1 (0, T ; H 2 (Ω) d )∩ H 2 (0, T ; L 2 (Ω) d ) × L ∞ (0, T ; H 2 (Ω)) ∩ H 1 (0, T ; H 1 (Ω)).
Proof of Corollary 2.7. Let (u, p) be the solution of 1.1. Let us consider the following system:

                 ∂ t v -∆v + ∇ζ = 0, in (0, T ) × Ω, div v = 0, in (0, T ) × Ω, ∂v ∂n -ζn = ∂ t g, on (0, T ) × Γ e , ∂v ∂n -ζn + qv = 0, on (0, T ) × Γ 0 , v(0) = ∆u 0 -∇p 0 , in Ω, (2.4)
where p 0 ∈ H 2 (Ω) is defined as the solution of the following elliptic boundary problem:

         ∆p 0 = 0, in Ω, p 0 = ∂u 0 ∂n • n -g |t=0 • n, on Γ e , p 0 = ∂u 0 ∂n • n + qu 0 • n, on Γ 0 .
According to Theorem 2.6, we obtain that (v, ζ) belongs to L 2 (0, T ;

H 2 (Ω) d ) ∩ H 1 (0, T ; L 2 (Ω) d ) ∩ L ∞ (0, T ; V ) × L 2 (0, T ; H 1 (Ω)). Remark that (∂ t u, ∂ t p
) is solution of system 2.4 in the distribution sense on (0, T ). Thus, by uniqueness, (v, ζ) = (∂ t u, ∂ t p). Then, since q ∈ H s (Γ 0 ) and

(∂ t u, g) ∈ L ∞ (0, T ; V )×L ∞ (0, T ; H 3 2 (Γ e ) d ) we deduce from Proposition 2.4 that (u, p) ∈ L ∞ (0, T ; H 3 (Ω) d )× L ∞ (0, T ; H 2 (Ω)).

Identifiability

Unique continuation

We start by recalling a unique continuation result for the Stokes equations proved in [START_REF] Fabre | Prolongement unique des solutions de l'equation de Stokes[END_REF].

Theorem 3.1. We denote by Q = (0, T ) × Ω and let O be an open subset in Q. The horizontal component of O is C(O) = {(t, x) ∈ Q/ ∃ x 0 ∈ Ω, (t, x 0 ) ∈ O}. Let (u, p) ∈ L 2 (0, T ; H 1 loc (Ω)) d × L 2 loc (Q) be a weak solution of ∂ t u -∆u + ∇p = 0, in (0, T ) × Ω, div u = 0, in (0, T ) × Ω, satisfying u = 0 in O then u = 0 and p is constant in C(O).
From this theorem, we easily deduce the following result which will be useful in the next subsection.

Corollary 3.2. Let δ > 0, x 0 ∈ ∂Ω, t 0 ∈ (0, T ) and r > 0 be such that γ = (t 0 -δ, t 0 + δ) × (B(x 0 , r) ∩ ∂Ω) is an open set in (0, T ) × ∂Ω. Let (u, p) ∈ L 2 (0, T ; H 2 (Ω) d ) × L 2 (0, T ; H 1 (Ω)) be solution of: ∂ t u -∆u + ∇p = 0, in (0, T ) × Ω, div u = 0, in (0, T ) × Ω,
satisfying u = 0 and ∂u ∂n -pn = 0 on γ. Then u = 0 and p = 0 in (t 0 -δ, t 0 + δ) × Ω.

Proof of Corollary 3.2. We extend u and p by 0 on (t 0 -δ,

t 0 + δ) × (B(x 0 , r) ∩ Ω c ): ũ (resp p) = u (resp p), in (t 0 -δ, t 0 + δ) × Ω, 0, in (t 0 -δ, t 0 + δ) × (B(x 0 , r) ∩ Ω c ),
and we denote Ω = Ω ∪ B(x 0 , r). Let us verify that (ũ, p) ∈ L 2 (0, T ;

H 1 (Ω) d ) × L 2 (0, T ; L 2 (Ω)) is still a solution of the Stokes equations in Ω. Let v ∈ D( Ω) d .
We check by integration by parts in space that almost everywhere in t ∈ (t 0 -δ, t 0 + δ):

Ω ∂ t ũ • v + Ω ∇ũ : ∇v - Ω p div v = 0.
Moreover div ũ = 0 in (t 0 -δ, t 0 + δ) × Ω. Therefore, we can apply Theorem 3.1 to (ũ, p): (ũ, p) = (0, 0) in (t 0 -δ, t 0 + δ) × Ω which implies that u = 0 and p is constant in (t 0 -δ, t 0 + δ) × Ω.

At last, the fact that ∂u ∂n -pn = 0 on γ implies that p = 0 in (t 0 -δ, t 0 + δ) × Ω.

Application

Proposition 3.3. Let s > d-1 2 , T > 0, α > 0, x e ∈ Γ e , r > 0, g ∈ H 1 (0, T ; H 1 2 (Γ e ) d
) be non identically zero, u 0 ∈ V and q j ∈ H s (Γ 0 ) be such that q j ≥ α on Γ 0 for j = 1, 2. Let (u j , p j ) be the weak solutions of 1.1 with q = q j for j = 1, 2. We assume that u 1 = u 2 on (0, T ) × (B(x e , r) ∩ Γ e ). Then q 1 = q 2 on Γ 0 .

Proof of Proposition 3.3. We are going to prove Proposition 3.3 by contradiction: we assume that q 1 is not identically equal to q 2 on Γ 0 . Thanks to Theorem 2.6, we have (u j , p j ) ∈ L 2 (0, T ; H 2 (Ω) d ) × L 2 (0, T ; H 1 (Ω)) for j = 1, 2. We define by u = u 1 -u 2 and p = p 1 -p 2 . Let us notice that (u, p) is the solution of the following problem:

             ∂ t u -∆u + ∇p = 0, in (0, T ) × Ω, div u = 0, in (0, T ) × Ω, ∂u ∂n -pn = 0, on (0, T ) × Γ e , ∂u ∂n -pn + q 1 u 1 -q 2 u 2 = 0, on (0, T ) × Γ 0 .
By assumption, u = 0 and ∂u ∂n -pn = 0 on (0, T )×(B(x e , r)∩Γ e ). Thus, according to Corollary 3.2, u 1 = u 2 and p 1 = p 2 in (0, T ) × Ω. Consequently, we deduce from

∂u 1 ∂n -p 1 n + q 1 u 1 = 0, on (0, T ) × Γ 0 , ∂u 1 ∂n -p 1 n + q 2 u 1 = 0, on (0, T ) × Γ 0 , that u 1 (q 1 -q 2 ) = 0 on (0, T ) × Γ 0 . (3.1)
By assumption, q 1 is not identically equal to q 2 . Since s > d -1 2 , q 1 and q 2 are continuous on Γ 0 . Thus, we can find an open set κ ⊂ Γ 0 with a positive measure such that:

(q 1 -q 2 )(x) = 0, ∀x ∈ κ. Equation 3.1 implies that u 1 ≡ 0 on (0, T ) × κ and then u 1 is the solution of          ∂ t u 1 -∆u 1 + ∇p 1 = 0, in (0, T ) × Ω, div u 1 = 0, in (0, T ) × Ω, u 1 = 0, on (0, T ) × κ, ∂u 1 ∂n -p 1 n = 0, on (0, T ) × κ.
Applying again Corollary 3.2, we obtain that u 1 = 0 and p 1 = 0 in (0, T ) × Ω. This is in contradiction with the fact that g is non identically zero.

Stability estimates

In this section, we assume that d = 2 and that the open set Ω ⊂ R 2 is of class C 3,1 .

We are going to prove stability estimates for the inverse problem we are interested in by using a global Carleman inequality which is stated in Lemma 4.1.

First, in Theorem 4.3, we state a stability estimate for the stationary problem. Then we deduce from this theorem two stability estimates for the evolution problem 1.1 by using an inequality coming from the analytic semigroup theory. To be more precise, we treat separately the case where g does not depend on time (see Theorem 4.18) and the case where g depends on time (see Theorem 4.21).

Carleman inequality

Let us state a global Carleman inequality proved by A. L. Bukhgeim in [START_REF] Bukhgeȋm | Extension of solutions of elliptic equations from discrete sets[END_REF]:

Lemma 4.1. Let Ψ ∈ C 2 (Ω).
We have:

Ω (∆Ψ|u| 2 + (∆Ψ -1)|∇u| 2 )e Ψ ≤ Ω |∆u| 2 e Ψ + ∂Ω ∂Ψ ∂n |u| 2 + |∇u| 2 + 2 ∂|∇u| 2 ∂τ e Ψ (4.1) for all u ∈ C 2 (Ω).
The proof of this result, which is only valid in dimension 2, uses computational properties of function defined on C (in particular, the fact that 4∂ z ∂ z = ∆).

Remark 4.2. The result is still true for u ∈ H 3 (Ω). Indeed, for all u ∈ H 3 (Ω), there exists

(u n ) n∈N ∈ C 2 (Ω) N such that u n → u in H 3 (Ω). ( 4
∂Ω ∂Ψ ∂n ∂|∇u| 2 ∂τ e Ψ ≤ 2 Ψ C 1 (Ω) e Ψ C 0 (Ω) 2 i=1 ∂Ω |∇u| • |∇∂ i u| < ∞.
We have: .

∂Ω ∂Ψ ∂n ∂|∇u n | 2 ∂τ - ∂|∇u| 2 ∂τ e Ψ ≤ C Ψ C 1 (Ω) e Ψ C 0 (Ω) 2 i,j=1 ∂Ω |∂ j u| 2 1 2 ∂Ω |∂ ij u n -∂ ij u| 2 1 2 + C Ψ C 1 (Ω) e Ψ C 0 (Ω) 2 i,j=1 ∂Ω |∂ ij u n | 2 1 2 ∂Ω |∂ j u n -∂ j u|
According to 4.2, the sequence

(∂ ij u n ) n∈N converges in L 2 (∂Ω) towards ∂ ij u and ∂ ij u n L 2 (∂Ω)
is bounded by a constant independent of n. Then, equality 4.3 follows from 4.2.

The stationary case

For the stationary problem:

             -∆u + ∇p = 0, in Ω, div u = 0, in Ω, ∂u ∂n -pn = g, on Γ e , ∂u ∂n -pn + qu = 0, on Γ 0 , (4.4)
we have the following stability estimate.

Theorem 4.3. Let α > 0, M 1 > 0, M 2 > 0, (g, q j ) ∈ H 5 2 (Γ e ) 2 × H 5 2 (Γ 0 ) for j = 1, 2 be such that g is not identically zero, g H 5 2 (Γe) ≤ M 1 , q j ≥ α on Γ 0 and q j H 5 2 (Γ0)
≤ M 2 . We denote by (u j , p j ) the solution of sustem 4.4 associated to q = q j for j = 1, 2. Let K be a compact subset of {x ∈ Γ 0 /u 1 (x) = 0} and m > 0 be a constant such that |u 1 | ≥ m on K.

Then there exist positive constants

C(M 1 , M 2 , α) and C 1 (M 1 , M 2 , α) such that q 1 -q 2 L 2 (K) ≤ 1 m C(M 1 , M 2 , α) ln C1(M1,M2,α) u1-u2 L 2 (Γe ) 2 + p1-p2 L 2 (Γe ) + ∂p 1 ∂n - ∂p 2 ∂n L 2 (Γe) 1 2
.

(4.5)

Remark 4.4. Since g is not identically zero, Corollary 3.2 ensures that {x ∈ Γ 0 /u 1 (x) = 0} is not empty. Moreover, according to Proposition 2.4, u 1 is continuous on Ω, thus we obtain the existence of a compact K and a constant m as in Theorem 4.3. We notice however that the constants involved in the estimate 4.5 and the set K depend on u 1 . Finding a uniform lower bound for any solution u of system 4.4 remains an open question. We refer to [START_REF] Chaabane | Identification of Robin coefficients by the means of boundary measurements[END_REF], [START_REF] Alessandrini | Stable determination of corrosion by a single electrostatic boundary measurement[END_REF] and [START_REF] Alessandrini | Detecting nonlinear corrosion by electrostatic measurements[END_REF] for the case of the scalar Laplace equation.

Remark 4.5. In [START_REF] Cheng | Stable determination of a boundary coefficient in an elliptic equation[END_REF], the same kind of inequality is proved for the Laplacian problem with Robin boundary conditions under the hypothesis that the measurements are small enough. Here, we free ourselves from this smallness assumption on the measurements.

Remark 4.6. If we compare this result with the identifiability property (Proposition 3.3), we notice that we need additional measurements on the solution. In Proposition 3.

Let (u, p) ∈ H 4 (Ω) 2 × H 3 (Ω) be the solution in Ω of -∆u + ∇p = 0, div u = 0.
Then, there exist C > 0, C 1 > 0 and d 0 > 0 such that for all d > d 0 :

u L 2 (Γ0) 2 + ∇u L 2 (Γ0) 4 + p L 2 (Γ0) + ∇p L 2 (Γ0) 2 ≤ dC u H 3 (Ω) 2 + p H 3 (Ω) ln C 1 d2 u H 3 (Ω) 2 + p H 3 (Ω) u L 2 (Γe ) 2 + ∂u ∂n L 2 (Γe ) 2 + p L 2 (Γe ) + ∂p ∂n L 2 (Γe) 1 2 . (4.6)
Proof of Lemma 4.7. The proof is based on the Carleman inequality of Lemma 4.1 for an appropriate choice of Ψ. Note that we will apply 4.1 twice: one time for the velocity u and one time for the pressure p. The weight function Ψ is chosen in order to estimate the traces on Γ 0 with respect to the ones on Γ e .

Step 1: choice of Ψ. We choose Ψ as in [START_REF] Cheng | Stable determination of a boundary coefficient in an elliptic equation[END_REF]. There exists Ψ 0 ∈ C 2 (Ω) non identically zero such that:

∆Ψ 0 = 0 in Ω, Ψ 0 = 0 on Γ 0 , Ψ 0 ≥ 0 on Γ e , ∂Ψ 0 ∂n < 0 on Γ 0 . Indeed, let χ ∈ C 2 (∂Ω) such that χ = 0 on Γ 0 , χ ≥ 0 on Γ e ,
and χ non identically zero on Γ e . The boundary value problem :

∆Ψ 0 = 0, in Ω, Ψ 0 = χ, on ∂Ω, (4.7)
has a unique solution Ψ 0 ∈ C 2 (Ω). Note that Ψ 0 is not constant because χ is non identically zero. So, from the strong maximum principle, Ψ 0 > 0 in Ω. According to Hopf Lemma, we have ∂Ψ 0 ∂n < 0 on Γ 0 .

Let λ > 0. We denote by Ψ 1 ∈ C 2 (Ω) the unique solution of the boundary value problem:

∆Ψ 1 = λ, in Ω, Ψ 1 = 0, on ∂Ω.
From the comparison principle and the strong maximum principle, we have Ψ 1 < 0 in Ω.

Moreover, according to the Hopf Lemma, we have ∂Ψ 1 ∂n > 0 on ∂Ω.

Let us consider Ψ = Ψ 1 + sΨ 0 , for s > 0. To summarize, the function Ψ has the following properties: ∆Ψ = λ in Ω, Ψ = 0 on Γ 0 , Ψ ≥ 0 on Γ e , and s ∂Ψ 0 ∂n ≤ ∂Ψ ∂n ≤ ∂Ψ 1 ∂n on Γ 0 .

Step 2: We first apply Lemma 4.1 to u. Using the fact that ∆u = ∇p, we have:

Ω (∆Ψ|u| 2 +(∆Ψ -1)|∇u| 2 )e Ψ ≤ Ω |∇p| 2 e Ψ + ∂Ω ∂Ψ ∂n |u| 2 + |∇u| 2 + 2 ∂|∇u| 2 ∂τ e Ψ . (4.8)
Then, we apply once again Lemma 4.1 to p:

Ω (∆Ψ|p| 2 +(∆Ψ -1)|∇p| 2 )e Ψ ≤ Ω |∆p| 2 e Ψ + ∂Ω ∂Ψ ∂n |p| 2 + |∇p| 2 + 2 ∂|∇p| 2 ∂τ e Ψ .
(4.9)

We have ∆p = div(∆u) = 0 hence Let us study each of the terms. We have:

Ω |∆p| 2 e Ψ =
Γ0 ∂Ψ 1 ∂n (|u| 2 + |∇u| 2 + |p| 2 + |∇p| 2 ) ≤ C( u 2 H 3 (Ω) 2 + p 2 H 3 (Ω) ).
Moreover, since ∂Ψ ∂n ≤ ∂Ψ 1 ∂n on Γ 0 , we obtain:

2 Γ0 ∂Ψ ∂n ∂|∇p| 2 ∂τ + ∂|∇u| 2 ∂τ ≤ C( u 2 H 3 (Ω) 2 + p 2 H 3 (Ω) ).
Since, on Γ e , ∂Ψ ∂n ≤ sC for s ≥ 1, we have:

Γe ∂Ψ ∂n (|u| 2 + |∇u| 2 + |p| 2 + |∇p| 2 )e Ψ ≤ Cs Γe (|u| 2 + |∇u| 2 + |p| 2 + |∇p| 2 )e Ψ .
Using Cauchy-Schwarz inequality, we obtain:

2 Γe ∂Ψ ∂n ∂|∇p| 2 ∂τ + ∂|∇u| 2 ∂τ e Ψ ≤ sC( u H 3 (Ω) 2 + p H 3 (Ω) ) Γe (|∇p| 2 + |∇u| 2 )e 2Ψ 1 2
.

Note that e Ψ depends on s on Γ e . Hence, reassembling these inequalities, inequality 4.10 becomes:

θ Γ0 (|u| 2 + |∇u| 2 + |p| 2 + |∇p| 2 ) ≤ C K s + 1 s ( u 2 H 3 (Ω) 2 + p 2 H 3 (Ω) ) , (4.11)
where In order to study the dependence with respect to s of K s , we define B by:

K s = ( u H 3 (Ω) 2 + p H 3 (Ω) ) Γe (|∇p| 2 + |∇u| 2 )e 2Ψ
B = u L 2 (Γe) 2 + p L 2 (Γe) + ∂u ∂n L 2 (Γe) 2 + ∂p ∂n L 2 (Γe) . (4.13)
Let us estimate the first term in the expression of K s . Remark that, thanks to classical interpolation inequalities (see [START_REF] Adams | Pure and Applied Mathematics[END_REF]), there exists C > 0 such that for all f ∈ H 2 (Γ e ):

∂f ∂τ L 2 (Γe) ≤ f H 1 (Γe) ≤ C f 1 2 L 2 (Γe) f 1 2
H 2 (Γe) .

Applying the previous inequality, there exists C > 0 such that:

Γe ∂u ∂τ 2 ≤ C u H 3 (Ω) 2 u L 2 (Γe) 2 and Γe ∂p ∂τ 2 ≤ C p H 3 (Ω) p L 2 (Γe) . (4.14)
We obtain, using the fact that ∇v = ∂v ∂n n + ∂v ∂τ τ on ∂Ω for all v ∈ H 2 (Ω), and thanks to inequality 4.14, that there exists C > 0 such that:

Γe (|∇p| 2 + |∇u| 2 )e 2Ψ ≤e 2ks ∂p ∂n 2 L 2 (Γe) + ∂u ∂n 2 L 2 (Γe) 2 + e 2ks C u H 3 (Ω) 2 u L 2 (Γe) 2 + C p H 3 (Ω) p L 2 (Γe) ≤Ce 2ks u H 3 (Ω) 2 ∂u ∂n L 2 (Γe) 2 + u L 2 (Γe) 2 + Ce 2ks p H 3 (Ω) ∂p ∂n L 2 (Γe) p L 2 (Γe) ≤Ce ks u H 3 (Ω) 2 + p H 3 (Ω) B,
where k = max Γ Ψ 0 . Similarly, for the second term in the expression of K s we prove that

Γe (|u| 2 + |∇u| 2 + |p| 2 + |∇p| 2 )e Ψ ≤Ce ks u H 3 (Ω) 2 + p H 3 (Ω) B ≤Ce ks u H 3 (Ω) 2 + p H 3 (Ω) 3 2 B 1 2 .
Thus, using the two previous inequalities, according to the definition 4.12 of K s , we obtain,

K s ≤ Ce ks u H 3 (Ω) 2 + p H 3 (Ω) 3 2 B 1 2 .

Let us denote by

A = u H 3 (Ω) 2 + p H 3 (Ω) .
Hence we get from 4.11:

Γ0 (|u| 2 + |∇u| 2 + |p| 2 + |∇p| 2 ) ≤ C A 3 2 e ks B 1 2 + A 2 s ,
for all s ≥ 1. Remark that this inequality is trivially verified for 0 < s ≤ 1 by continuity of the trace mapping. Let d ≥ 1. To summarize, we have proved that:

Γ0 (|u| 2 + |∇u| 2 + |p| 2 + |∇p| 2 ) ≤ CA 3 2 e ks B 1 2 + dA 1 2 s , ∀s > 0.
We now optimize the upper bound with respect to s. We denote by

f (s) = e ks B 1 2 + dA 1 2
s .

Let us study the function f in R * + . We have:

lim s→0 f (s) = +∞, lim s→∞ f (s) = +∞.
So since f is continuous on R + * , f reaches its minimum at a point s 0 > 0. At this point,

f (s 0 ) = 0 ⇔ B 1 2 = e -ks0 dA 1 2 ks 0 2 , thus f (s 0 ) = dA 1 2 ks 2 0 + dA 1 2
s 0 .

Hence:

Γ0 (|u| 2 + |∇u| 2 + |p| 2 + |∇p| 2 ) ≤ C dA 2 s β 0 1 k + 1 ,
where β = 1 if s 0 ≥ 1 and β = 2 otherwise. But, we notice that 1

B 1 2 = ks 0 2 e ks0 dA 1 2 ≤ ke (k+2)s0 dA 1 2
, that is to say:

1 s 0 ≤ k + 2 ln dA 1 2 kB 1 2
, if d is larger than a constant which only depends on k and on the continuity constants of the trace mapping. In the same way, when s 0 < 1 , we obtain:

1 s 2 0 ≤ 1 ln dA 1 2 ke k B 1 2
, if d is larger than a constant which only depends on k and on the continuity constants of the trace mapping. Using the fact that ln x

1 2
= 1 2 ln(x) for all x > 0 and according to the definition 4.13 of B, the desired result follows. (Γ e ) 2 and q j ∈ H 5 2 (Γ 0 ) for j = 1, 2, thanks to Proposition 2.4 applied for k = 2, there exists C(α, M 1 , M 2 ) > 0 such that:

u j H 4 (Ω) 2 + p j H 3 (Ω) ≤ C(α, M 1 , M 2 ), for j = 1, 2.
(4.15)

In the following, we denote by u = u 1 -u 2 and p = p 1 -p 2 . We have:

(q 2 -q 1 )u 1 = q 2 u + ∂u ∂n -pn, on Γ 0 . (4.16)
Consequently, since |u 1 | ≥ m > 0 on K:

q 1 -q 2 L 2 (K) ≤ 1 m C(M 2 ) u L 2 (Γ0) 2 + ∂u ∂n L 2 (Γ0) 2 + p L 2 (Γ0) . (4.17)
Let us denote by

A = u H 3 (Ω) 2 + p H 3 (Ω)
and

B = u L 2 (Γe) 2 + p L 2 (Γe) + ∂u ∂n L 2 (Γe) 2 + ∂p ∂n L 2 (Γe) .
By applying Lemma 4.7, we obtain that there exists there exists C(M 2 ) > 0, C 1 > 0 and d 0 > 0 such that for all d > d 0 :

q 1 -q 2 L 2 (K) ≤ dC(M 2 ) m A ln C 1 d2 A B 1 2 . (4.18)
We are going to concude the proof by studying the variation of the function defined by

f y (x) = x ln x y 1 2
on (y, +∞), for y = B C 1 d2 . We have

f y (x) = ln x y -1 2 ln x y 3 2
.

Let us denote by x 0 = ye 1 2 . The function f y is decreasing on (y, x 0 ] and is increasing on [x 0 , +∞). For d large enough, we have A ≥ x 0 by continuity of the trace mapping. Using 4.15 and since f y is increasing on [x 0 , +∞), we directly deduce that f y (A) ≤ f y (C(α, M 1 , M 2 )).

Using this result in inequality 4.18, we get that there exist constants C(α, M 1 , M 2 ) > 0 and C 1 (α, M 1 , M 2 ) > 0 such that:

q 1 -q 2 L 2 (K) ≤ 1 m C(α, M 1 , M 2 ) ln C1(α,M1,M2) u L 2 (Γe) 2 + p L 2 (Γe) + ∂u ∂n L 2 (Γe ) 2 + ∂p ∂n L 2 (Γe ) 1 2
.

Since ∂u ∂n = pn on Γ e , we obtain the desired inequality.

Remark 4.9. Note that the assumption that |u 1 | ≥ m > 0 on K is essential to pass from 4.16 to 4.17. Outside the set K, an estimate of q 1 -q 2 may be undetermined or highly unstable.

In particular, an estimate of the Robin coefficients on the whole set Γ 0 might be worst than of logarithmic type (see [START_REF] Bellassoued | Stability of the determination of the surface impedance of an obstacle from the scattering amplitude[END_REF]).

Note however that for a simplified problem, it is in fact possible to obtain a logarithmic stability estimate on the whole set Γ 0 which does not depend on a given reference solution. Assume that g = g e n and q ∈ R are such that

(A) g e ∈ R satisfies β ≤ g e ≤ M 1 , (B) α ≤ q ≤ M 2 ,
for some α > 0, β > 0, M 1 > 0 and M 2 > 0. We denote by (u ge,q , p ge,q ) the solution of system 4.4 associated to q and g = g e n. Thanks to the weak formulation of the problem, Γe u g,q • n > 0. Moreover, one can prove by contradiction and thanks to the continuity of the solution with respect to the data that there exists m 1 > 0 which depends on M 1 , M 2 , α and β such that for all (g e , q) ∈ R 2 which satisfies (A) and (B), Γe u ge,q • n ≥ m 1 .

For i = 1, 2, let q i ∈ R satisfy the assumption (B) above. We define by (u i , p i ) = (u ge,qi , p ge,qi ) the solution of system 4.4 associated with g = g e n and q = q i for i=1,2. If we multiply 4.16 by the unit normal n and we integrate on Γ 0 , we obtain:

(q 2 -q 1 ) Γ0 u 1 • n = q 2 Γ0 (u 1 -u 2 ) • n + Γ0 ∂u 1 ∂n - ∂u 2 ∂n • n - Γ0 (p 1 -p 2 ). Since u 1 is divergence free, Γ0 u 1 • n = Γe u 1 • n ≥ m 1 . Thus, we get |q 1 -q 2 | ≤C(M 1 , M 2 , α, β) u 1 -u 2 L 2 (Γ0) 2 + ∂u 1 ∂n - ∂u 2 ∂n L 2 (Γ0) 2 + p 1 -p 2 L 2 (Γ0) .
We conclude as in the proof of Theorem 4.3 and obtain that positive constants C(M 1 , M 2 , α) and

C 1 (M 1 , M 2 , α, β) such that |q 1 -q 2 | ≤ C(M 1 , M 2 , α, β) ln C1(M1,M2,α) u1-u2 L 2 (Γe) 2 + p1-p2 L 2 (Γe ) + ∂p 1 ∂n - ∂p 2 ∂n L 2 (Γe ) 1 2
.

Evolution problem

In order to use semigroup properties, we begin by introducing the Stokes operator associated with the Robin boundary conditions on Γ 0 .

Properties of the Stokes operator

We recall that the bilinear form a q is defined by 2.3.

Definition 4.10. We define the set D(A q ) as follows:

D(A q ) = {u ∈ V /∃C > 0, ∀v ∈ V, |a q (u, v)| ≤ C v L 2 (Ω) 2 },
and the operator A q : D(A q ) ⊂ H → H by: ∀u ∈ D(A q ), a q (u, v) = (A q u, v) L 2 (Ω) 2 , ∀v ∈ V.

Proposition 4.11. Let α > 0 and q ∈ L ∞ (Γ 0 ) such that q ≥ α almost everywhere on Γ 0 . The operator A q has the following properties:

1. A q ∈ L(D(A q ), H) is invertible and its inverse is compact on H.

2.

A q is selfadjoint.

As a consequence, A q admits a family of eigenvalues φ l q A q φ l q = λ l q φ l q with 0 < λ 1 q ≤ λ 2 q ≤ ... ≤ λ j q and lim j→∞ λ j q = +∞, which is complete and orthogonal both in H and V .

Proof of Proposition 4.11. It relies on classical arguments for which we refer to [START_REF] Brezis | Analyse fonctionnelle[END_REF] or [START_REF] Raviart | Introduction à l'analyse numérique des équations aux dérivées partielles[END_REF].

Remark 4.12. Let α > 0. There exists a constants µ > 0 such that for all q ∈ L ∞ (Γ 0 ) such that q ≥ α, for l ∈ N: q : (V, a q (., .)

λ l q ≥ µ. (4.19) Indeed, λ l q ≥ λ 1 q = (A q φ 1 q , φ 1 q ) L 2 (Ω) 2 = a q (φ 1 q , φ 1 q ) ≥ a α (φ 1 q , φ 1 q ) ≥ µ φ 1 q 2 L 2 (Ω) 2 = µ,
1 2 ) → (H, L 2 (Ω) 2
) is an isometry.

Proposition 4.14. Let α > 0 and q ∈ L ∞ (Γ 0 ) be such that q ≥ α almost everywhere on Γ 0 . The operator -A q generates an analytic semigroup on H. This analytic semigroup is explicitly given by: e -tAq f = l≥1 e -tλ l q (φ l q , f ) L 2 (Ω) 2 φ l q , (4.20)

for all f ∈ H.

Proof of Proposition 4.14. It follows from the construction of the operator A q . We refer to [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] and [START_REF] Dautray | Analyse mathématique et calcul numérique pour les sciences et les techniques[END_REF] for details.

Proposition 4.15. Let α > 0, M > 0, k ∈ N and s ∈ R be such that s > d-1 2 and s ≥ 1 2 + k. We assume that Ω is of class C k+1,1 and that q ∈ H s (Γ 0 ) is such that q ≥ α on Γ 0 .

Then for each f ∈ H ∩ H k (Ω) 2 , there exists u ∈ H k+2 (Ω) 2 solution of A q u = f if and only if there exists p ∈ H k+1 (Ω) such that (u, p) is solution of the following problem: Moreover, there exists a constant C(α, M ) > 0 such that for every q ∈ H s (Γ 0 ) satisfying q H s (Γ0) ≤ M :

       -∆u + ∇p = f, in Ω, div u = 0, in Ω,
u H 2+k (Ω) 2 ≤ C(α, M ) f H k (Ω) 2 .
Proof of Proposition 4.15. This result follows from the construction of the operator A q and from Proposition 2.4.

Corollary 4.16. Let α > 0, k ∈ N * and s ∈ R be such that s > d-1 2 and s ≥ 1 2 + 2(k -1). We assume that Ω is of class C 2k-1,1 and that q ∈ H s (Γ 0 ) is such that q ≥ α on Γ 0 . Then D(A k q ) → H 2k (Ω) 2 ∩ H.
Proof of Corollary 4.16. For k = 1, it is clear. Take now k = 2. Let u ∈ D(A 2 q ). We have 2 ∩H by assumption, so u ∈ H 4 (Ω) 2 ∩H thanks to the regularity properties of the solution of the Stokes problem summarize in Proposition 2.4. We conclude by induction on k.

A 2 q u = f ⇔ A q u = v A q v = f But v ∈ D(A q ) ⊂ H 2 (Ω)
Remark 4.17. Let us remark that, due to the prescribed boundary conditions, D(A q ) is not equal to H 2 (Ω) 2 ∩ H.

The flux g does not depend on t

In this paragraph, we consider the evolution problem 1.1 given in the introduction. We assume in this part that g does not depend on time. Let α > 0, M 1 > 0 and M 2 > 0. In the following, we assume that g ∈ H 5 2 (Γ e ) 2 is non identically zero and g

H 5 2 (Γe) 2 ≤ M 1 , (4.22) q ∈ H 5 2 (Γ 0 ) is such that q H 5 2 (Γ0)
≤ M 2 and q ≥ α on Γ 0 . (4.23)

Let us prove the following theorem:

Theorem 4.18. Let α > 0, M 1 > 0, M 2 > 0 and u 0 ∈ H∩H 3 (Ω) 2 . We assume that g satisfies 4.22 and that q j satisfies 4.23 for j = 1, 2. We denote by (u j , p j ) the solution of system 1.1 associated to q = q j , for j = 1, 2. Let K be a compact subset of {x ∈ Γ 0 /v 1 (x) = 0}, where (v 1 , ζ 1 ) is the solution of system 4.4 with q = q 1 and let m > 0 be a constant such that |v 1 | ≥ m on K. Then, there exist C(α, M 1 , M 2 ) > 0 and C 1 (M 1 , M 2 , α) > 0 such that

q 1 -q 2 L 2 (K) ≤ 1 m C(α, M 1 , M 2 ) ln C1(M1,M2,α) u1-u2 L ∞ (0,+∞;L 2 (Γe ) 2 ) + p1-p2 L ∞ (0,+∞;L 2 (Γe )) + ∂p 1 ∂n - ∂p 2 ∂n L ∞ (0,+∞;L 2 (Γe )) 1 2
.

Remark 4.19. Due to the method which relies on semigroup theory, we need to take measurements during an infinite time.

Proof of Theorem 4.18. For j = 1, 2, let (v j , ζ j ) be the solution of the stationary problem 4.4 with q = q j . According to Proposition 2.4, (v j , ζ j ) belongs to H 4 (Ω) 2 × H 3 (Ω) and moreover, thanks to assumptions 4.22 and 4.23, there exists a constant C(α, M 1 , M 2 ) > 0 such that

v j H 4 (Ω) 2 + ζ j H 3 (Ω) ≤ C(α, M 1 , M 2 ). (4.24)
We denote (w j , π j ) = (u j -v j , p j -ζ j ). Thanks to Theorem 4.3, we are able to estimate q 1q 2 L 2 (K) with respect to an increasing function of (v

1 -v 2 ) |Γe , (ζ 1 -ζ 2 ) |Γe and ∂ζ1 ∂n -∂ζ2 ∂n |Γe .
Our objective is now to compare the asymptotic behavior of u 1 -u 2 and p 1 -p 2 to the solution of the stationary problem v 1 -v 2 and ζ 1 -ζ 2 . More precisely, we are going to prove that:

w j (t, .) H 3 (Ω) 2 + π j (t, .) H 2 (Ω) ≤ G(t),
where G is a function which tends to 0 when t goes to +∞. This inequality, combined with Theorem 4.3, will allow us to conclude the proof of Theorem 4.18.

We have that (w j , π j ) is the solution of the following problem: for t > 0,

             ∂ t w -∆w + ∇π = 0, in Ω, div w = 0, in Ω, ∂w ∂n
-πn = 0, on Γ e , ∂w ∂n -πn + q j w = 0, on Γ 0 , completed with the initial condition w(0) = u 0 -v j . Let t > 0. We have from the theory of analytic semigroup that: w j (t, .) = e -tAq j w j (0, .).

Let η > 0. There exists a constant C > 0 independent of q j such that:

A η qj e -tAq j ≤ C e -µt t η , t > 0, η > 0, (4.26)
where µ is given by 4.19 and where is the norm operator. Using regularity result for the stationary problem given in Proposition 2.4, we have that:

w j (t, .) H 3 (Ω) 2 + π j (t, .) H 2 (Ω) ≤ C(α, M 2 ) ∂ t w j (t, .) H 1 (Ω) 2 .
Note that, thanks to Proposition 4.15 we have:

∂ t w j (t, .) H 1 (Ω) 2 = A qj w j (t, .) H 1 (Ω) 2 .
Then, since w j (t, .) is given by 4.25, and using Proposition 4.13 and estimates 4.24 and 4.26 with η = 3 2 , it follows:

w j (t, .) H 3 (Ω) 2 + π j (t, .) H 2 (Ω) ≤ C(α, M 2 ) A 3 2
q e -tAq j w j (0, .)

L 2 (Ω) 2 ≤ C(α, M 2 ) e -µt t 3 2 u 0 L 2 (Ω) 2 + v j L 2 (Ω) 2 ≤ C(α, u 0 , M 1 , M 2 ) e -µt t 3 2
.

(4.27)

We have from 4.27:

v 1 -v 2 L 2 (Γe) 2 ≤ C(α, u 0 , M 1 , M 2 ) e -µt t 3 2 + u 1 -u 2 L ∞ (0,+∞;L 2 (Γe) 2 ) .
Then, passing to the limit when t goes to infinity, we get:

v 1 -v 2 L 2 (Γe) 2 ≤ u 1 -u 2 L ∞ (0,+∞;L 2 (Γe) 2 ) .
We prove similarly:

ζ 1 -ζ 2 L 2 (Γe) ≤ p 1 -p 2 L ∞ (0,+∞;L 2 (Γe)) ,
and

∂ζ 1 ∂n - ∂ζ 2 ∂n L 2 (Γe) ≤ ∂p 1 ∂n - ∂p 2 ∂n L ∞ (0,+∞;L 2 (Γe)) .
To summarize, we have obtained:

v 1 -v 2 L 2 (Γe) 2 + ζ 1 -ζ 2 L 2 (Γe) + ∂ζ 1 ∂n - ∂ζ 2 ∂n L 2 (Γe) ≤ u 1 -u 2 L ∞ (0,+∞;L 2 (Γe) 2 ) + p 1 -p 2 L ∞ (0,+∞;L 2 (Γe)) + ∂p 1 ∂n - ∂p 2 ∂n L ∞ (0,+∞;L 2 (Γe))
.

Applying Theorem 4.3 to (v j , ζ j ) for j = 1, 2, we obtain the existence of positive constants

C(M 1 , M 2 , α) and C 1 (M 1 , M 2 , α) such that q 1 -q 2 L 2 (K) ≤ 1 m C(M 1 , M 2 , α) ln C1(M1,M2,α) v1-v2 L 2 (Γe) 2 + ζ1-ζ2 L 2 (Γe ) + ∂ζ 1 ∂n - ∂ζ 2 ∂n L 2 (Γe ) 1 2
.

We conclude by using the fact that the function

x → 1 ln 1 x increases on R * + .
Remark 4.20. Remark that

(u j , p j ) ∈ L ∞ (0, +∞; H 3 (Ω) 2 ) × L ∞ (0, +∞; H 2 (Ω)). ( 4

.28)

Let us prove 4.28. Let ν > 0. In fact, thanks to equation 4.27, we obtain that

(w j , π j ) ∈ L ∞ (ν, +∞; H 3 (Ω) 2 ) × L ∞ (ν, +∞; H 2 (Ω)),
and since u j = w j + v j and p j = π j + ζ j , we deduce that

(u j , p j ) ∈ L ∞ (ν, +∞; H 3 (Ω) 2 ) × L ∞ (ν, +∞; H 2 (Ω)).
Moreover, thanks to Corollary 2.7, we have

(u j , p j ) ∈ L ∞ (0, ν; H 3 (Ω) 2 ) × L ∞ (0, ν; H 2 (Ω)).
Thus, 4.28 follows.

The flux g depends on t

We restrict our study to the case where g is colinear to the exterior unit normal n: g = κ n. Let α > 0, M 1 > 0 and M 2 > 0. We assume that:

κ ∈ H 2 loc (0, +∞; H 3 2 (Γ e )), (4.29)
and

q ∈ H 5 2 (Γ 0 ) is such that q H 5 2 (Γ0)
≤ M 2 and q ≥ α on Γ 0 . (4.30)

Let us introduce h such that:

h ∈ H 5 2
(Γ e ) is non identically zero and h

H 5 2 (Γ0) ≤ M 1 . (4.31)
We assume that:

lim t→∞ ( κ(t, .) -h H 3 2 (Γe) + ∂ t κ(t, .) H 3 2 (Γe) + t 0 e -µ(t-s) ∂ t κ(s, .) 2 H 3 2 (Γe) ds 1 2 ) = 0, (4.32)
where µ is given by equation 4.19.

Theorem 4.21. Let α > 0, M 1 > 0, M 2 > 0 and u 0 ∈ H 3 (Ω) 2 ∩ H. We assume that h and κ satisfy respectively 4.31 and 4.29 and for j = 1, 2, q j satisfies 4.30. We denote by (u j , p j ) the solution of system 1.1 associated to q = q j . Let K be a compact subset of {x ∈

Γ 0 /v 1 (x) = 0}, where (v 1 , ζ 1 ) is the solution of              -∆v + ∇ζ = 0, in Ω, div v = 0, in Ω, ∂v ∂n -ζn = hn, on Γ e , ∂v ∂n -ζn + q 1 v = 0, on Γ 0 ,
and let m > 0 be a constant such that |v 1 | > m on K. We assume that 4.32 is verified. Then there exist C(α, M 1 , M 2 ) > 0 and C 1 (α, M 1 , M 2 ) > 0 such that

q 1 -q 2 L 2 (K) ≤ 1 m C(α, M 1 , M 2 ) ln C1(α,M1,M2) u1-u2 L ∞ (0,+∞;L 2 (Γe ) 2 ) + p1-p2 L ∞ (0,+∞;L 2 (Γe )) + ∂p 1 ∂n - ∂p 2 ∂n L ∞ (0,+∞;L 2 (Γe )) 1
+ ∂ t l(t, .) H 3 2 (Γe)
< +∞, Then κ = h + l satisfies 4.32. We note that a particular case of function satisfying 4.32 is given by l(t, x) = ω(t)ρ(x) where ω ∈ H 2 loc (0, +∞) , ρ ∈ H 3 2 (Γ e ) and lim t→∞ e tθ ω(t) = lim t→∞ e tθ ω (t) = 0.

Proof of Theorem 4.21. For j = 1, 2, we decompose u j into u j = v j + w j where (v j , ζ j ) ∈ H 4 (Ω) 2 × H 3 (Ω) is the solution of the stationary problem:

             -∆v + ∇ζ = 0, in Ω, div v = 0, in Ω, ∂v ∂n
-ζn = hn, on Γ e , ∂v ∂n -ζn + q j v = 0, on Γ 0 , and (w j , π j ) is solution of the following problem:

                 ∂ t w -∆w + ∇π = 0, in (0, +∞) × Ω, div w = 0, in (0, +∞) × Ω, ∂w ∂n -πn = (κ -h)n, on (0, +∞) × Γ e , ∂w ∂n
-πn + q j w = 0, on (0, +∞) × Γ 0 , w(0, x) = u 0 (x) -v j (x), in Ω.

We would like to perform the same reasoning as in Theorem 4.18. More precisely, we are going to prove that:

w j (t, .) H 3 (Ω) 2 + π j (t, .) H 2 (Ω) ≤ G(t),
where G is a function which tends to 0 when t goes to +∞. Since the function κ depends on t, there will be one more step than in Theorem 4.18 and that is why we assume 4.32.

We divide (w j , π j ) into two terms: w j = u 0 j + wj and π j = p 0 j + πj , where (u 0 j , p 0 j ) is solution of

                 ∂ t u 0 -∆u 0 + ∇p 0 = 0, in (0, +∞) × Ω, div u 0 = 0, in (0, +∞) × Ω, ∂u 0 ∂n -p 0 n = (κ -h)n, on (0, +∞) × Γ e ,
∂u 0 ∂n -p 0 n + q j u 0 = 0, on (0, +∞) × Γ 0 , u 0 (0, x) = 0, in Ω, and ( wj , πj ) is solution of

                 ∂ t w -∆ w + ∇π = 0, in (0, +∞) × Ω, div w = 0, in (0, +∞) × Ω, ∂ w ∂n
-πn = 0, on (0, +∞) × Γ e , ∂ w ∂n -πn + q j w = 0, on (0, +∞) × Γ 0 , w(0, x) = u 0 (x) -v j (x), in Ω.

Let t > 0. Using the same arguments as in the previous subsection, we prove that there exists C(α, u 0 , M 1 , M 2 ) > 0 such that: It remains for us to bound u 0 j (t, .) H 3 (Ω) 2 and p 0 j (t, .) H 2 (Ω) . We are going to prove that there exists a constant C(α, M 2 ) > 0 such that:

wj (t, .) H 3 (Ω) 2 + πj (t, .) H 2 (Ω) ≤ C(α, u 0 , M 1 , M 2 ) e -µt
u 0 j (t, .) H 3 (Ω) 2 + p 0 j (t, .) H 2 (Ω) ≤C(α, M 2 ) t 0 e -µ(t-s) ∂ t κ(s, .) 2 H 3 2 (Γe) ds 1 2 + C(α, M 2 ) ∂ t κ(t, .) H 3 2 (Γe) + C(α, M 2 ) κ(t, .) -h H 3 2 (Γe)
+ e -µt κ(0, .) -h , and in the following two estimates, the right hand side tends to 0 when t goes to infinity thanks to inequalities 4.33 and assumption 4.32.

H 3 2 (Γe) . ( 4
We introduce (y j , ρ j ) the solution of

             -∆y + ∇ρ = 0, in Ω, div y = 0,
in Ω, ∂y ∂n -ρn = (κ -h)n, on Γ e , ∂y ∂n -ρn + q j y = 0, on Γ 0 , for all t > 0. We know that (y j (t, .), ρ j (t, .)) ∈ H 3 (Ω) Then (y j , ρ j + p) satisfies

             -∆y j + ∇(ρ j + p) = ∇p, in Ω, div y j = 0, in Ω, ∂y j ∂n -(ρ j + p)n = 0,
on Γ e , ∂y j ∂n -(ρ j + p)n + q j y = 0, on Γ 0 , for all t > 0. Remark that, since ∇p ∈ L 2 (Ω), we have that y j (t) ∈ D(A qj ) by definition of D(A qj ).

Notice that the fact that g is colinear to n is important here to do the change of variable in the pressure. We deduce from A qj y j (t) = ∇p(t) ∈ V = D(A 1 2 qj ) that y j (t) ∈ D(A 3 2 qj ). Moreover, using Proposition 4.13 and inequality 4.37, there exists a constant C(M 2 ) > 0 such that:

A 3 2 qj y j (t, .) L 2 (Ω) 2 ≤C(M 2 ) A qj y j (t, .) H 1 (Ω) 2 = C(M 2 ) ∇p(t) H 1 (Ω) 2 ≤C(M 2 ) κ(t, .) -h H 3 2 (Γe) , (4.38)
that is to say: y j (t, .)

D(A 3 2 q j ) ≤ C(α, M 2 ) κ(t, .) -h H 3 2 (Γe) . ( 4

.39)

We can use the same argument, replacing κ -h by ∂ t κ, to prove that ∂ t y j (t, .) ∈ D(A 3 2 qj ) together with the estimate ∂ t y j (t, .)

D(A 3 2 q j ) ≤ C(α, M 2 ) ∂ t κ(t, .) H 3 2 (Γe) . ( 4

.40)

Let us consider w j = u 0 j -y j and p j = p 0 j -ρ j . The couple (w j , p j ) is solution of

                 ∂ t w -∆w + ∇p = -∂ t y j , in (0, +∞) × Ω, div w = 0, in (0, +∞) × Ω, ∂w ∂n
-pn = 0, on (0, +∞) × Γ e , ∂w ∂n -pn + q j w = 0, on (0, +∞) × Γ 0 , w(0, x) = -y j (0, x), in Ω. We know that w j is given by: w j (t, .) = -e -tAq j y j (0, .) -t 0 e -(t-s)Aq j ∂ t y j (s, .)ds.

Using the family (φ l qj ) l≥1 defined by Proposition 4.11, we have: w j (t, .) = l≥1 C l (t)φ l qj , with

C l (t) = -e -tλ l q j (y j (0, .), φ l qj ) L 2 (Ω) 2 - t 0 e -(t-s)λ l q j (∂ t y j (s, .), φ l qj ) L 2 (Ω) 2 ds.
Thus, recalling that (λ l qj ) l≥1 satisfies 4.19 and using Cauchy-Schwarz inequality, there exists C > 0 such that:

C l (t) 2 ≤ 2e -2tµ (y j (0, .), φ l qj ) 2 L 2 (Ω) 2 + C t 0 e -(t-s)µ (∂ t y j (s, .), φ l qj ) 2 L 2 (Ω) 2 ds.
We obtain from estimates 4.39 and 4.40:

w j (t, .) D(A 3 2 q j ) ≤C(α, M 2 )e -µt κ(0, .) -h H 3 2 (Γe) + C(α, M 2 ) t 0 e -µ(t-s) ∂ t κ(s, .) 2 H 3 2 (Γe) ds 1 2
.

(4.42)

Remark that, thanks to Proposition 4.15 and Proposition 4.13, we have:

w j (t, .) H 3 (Ω) 2 ≤ C(α, M 2 ) Aw j (t, .) H 1 (Ω) 2 ≤ C(α, M 2 ) A 3 2 w j (t, .) L 2 (Ω) 2 ≤ C(α, M 2 ) w j (t, .) D(A 3 2 q j ) . (4.43)
To summarize, using 4.43 and 4.42, we obtain the estimate:

w j (t, .) H 3 (Ω) 2 ≤C(α, M 2 )e -µt κ(0, .) -h H 3 2 (Γe) + C(α, M 2 ) t 0 e -µ(t-s) ∂ t κ(s, .) 2 H 3 2 (Γe) ds 1 2
.

(4.44)

Using now the regularity result for the stationary problem given in Proposition 2.4, we have:

p j (t, .) H 2 (Ω) ≤ C(α, M 2 ) ∂ t y j (t, .) H 1 (Ω) 2 + ∂ t w j (t, .) H 1 (Ω) 2 .
Since A qj w j = -∂ t y j -∂ t w j , we obtain:

p j (t, .) H 2 (Ω) ≤ C(α, M 2 ) ∂ t y j (t, .) H 1 (Ω) 2 + A qj w j (t, .) H 1 (Ω) 2 .
Thanks to Proposition 4.13, we know that

A qj w j (t, .) H 1 (Ω) 2 ≤ C(α) A 3 2 qj w j (t, .) L 2 (Ω) 2 .
Therefore, using 4.40 and 4.42, we obtain:

p j (t, .) H 2 (Ω) ≤C(α, M 2 ) e -µt κ(0, .) -h H 3 2 (Γe) + ∂ t κ(t, .) H 3 2 (Γe) + C(α, M 2 ) t 0 e -µ(t-s) ∂ t κ(s, .) 2 H 3 2 (Γe) ds 1 2
.

(4.45)

The estimate 4.34 follows from u 0 j = w j + y j , p 0 j = w j + ρ j and inequalities 4.35, 4.44 and 4.45.

Conclusion

To conclude, we have proved, under some regularity assumptions on the open set Ω and on the solution (u, p) of system 1.1, logarithmic stability estimates for the Stokes system with mixed Neumann and Robin boundary conditions. Due to the method which relies on a global Carleman inequality proved in [START_REF] Bukhgeȋm | Extension of solutions of elliptic equations from discrete sets[END_REF], these estimates are valid in dimension 2.

Our result which, as far as we know, is the first result of this type for Stokes system, could be improved in different ways. A first concern could be to prove a logarithmic stability estimate which is valid in any dimension. This will be the subject of a forthcoming work. Next, as mentioned in Remark 4.9, Robin coefficients are estimated on a compact subset K ⊂ Γ 0 which is not a fixed inner portion of Γ 0 but is unknown and depends on a given reference solution. To obtain an estimate of Robin coefficients on the whole set Γ 0 or on any compact subset K ⊂ Γ 0 is still an open question. Finally, in our stability estimates, we need measurements on Γ e of u, p and ∂p ∂n , while the identifiability result given by Proposition 3.3 only requires information on u and ∂u ∂n -pn on Γ, where Γ ⊆ Γ e is a non-empty open subset of the boundary. Therefore, it might be interesting to know whether it is possible to obtain a stability inequality with less measurement terms.

B = Ω ∇w i (x) : ∇w j (x) + Γ0 q(x)w i (x) • w j (x)dx 1≤i,j≤n ξ(t) = (ξ i (t)) 1≤i≤n and L(t) = Γe g(t, x) • w i (x)dx 1≤i≤n .
We can rewrite system A.4 in the form:

Aξ (t) + Bξ(t) = L(t), ξ(0) = ((u 0 , w i ) L 2 (Ω) d ) 1≤i≤n .
Since the matrix A is invertible, the system has a unique global solution ξ ∈ H 1 (0, T ) d . We are now going to prove that there exists a constant C > 0 independent of n ∈ N * such that: Multiplying the first equation of A.4 by ξ j , summing over j for j = 1, . . . , n and then integrating on (0, t), we obtain:

t 0 Ω u n,t • u n + t 0 Ω |∇u n | 2 + t 0 Γ0 q|u n | 2 = t 0 Γe g • u n (A.6)
Let > 0. We have, thanks to Cauchy-Schwartz and Young inequalities: Choosing small enough and using the fact that q ≥ α on Γ 0 , we obtain: u n (t, x) • w j (x)ψ (t)dxdt -Ω u n (0, x) • w j (x)ψ(0)dx, we easily pass to the limit when n goes to infinity in A.8. Remark that this inequality is still valid if we replace w j by any v ∈ V by continuity. This ends the proof of the existence of u ∈ L 2 (0, T ; V ) which satisfies A.1 in the distribution sense on (0, T ).

Let us finish the proof of Proposition A.1 by proving that the initial condition A.2 is satisfied. Let v ∈ V . We deduce from equality A.3 that d dt (u, v) L 2 (Ω) d ∈ L 2 (0, T ). Consequently, the function t → (u(t), v) L 2 (Ω) d is continuous. This gives a sense to (u(0), v) L 2 (Ω) d . Let ψ ∈ C 1 (0, T ) such that ψ(T ) = 0. Multiplying A.1 by ψ and then integrating on (0, T ), we obtain: where we recall that a q is defined by 2.3 and with l(v) = Γe g • v, for v ∈ V . Comparing equality A.9 with equality A.3, we obtain ψ(0)(u(0, .) -u 0 , v) L 2 (Ω) d = 0, for all v ∈ V . By choosing ψ such that ψ(0) = 0, equality A.2 follows.

We are now able to prove Theorem 2.6.

Proof of Theorem 2.6. We will begin by proving that ∂ t u ∈ L 2 (0, T ; H), then we will conclude by using the regularity result for the stationary problem from Proposition 2.4. Let t ∈ (0, T ). Multiplying the first equation of A.4 by ξ j , summing over j for j = 1, . . . , n and then integrating on (0, t), we obtain: We deduce that (u n ) n∈N * is bounded in H 1 (0, T ; H) ∩ L ∞ (0, T ; V ) and therefore u ∈ H 1 (0, T ; H) ∩ L ∞ (0, T ; V ).

To get regularity in space, we use the regularity result stated in Proposition 2.4 for the stationary problem. To do so, we notice that, for all t ∈ (0, T ), (u(t), p(t)) is solution of system 2.1 with f and g replaced by ∂ t u(t) and g(t). So, by Proposition 2.4 applied with k = 0, since (∂ t u, g) belongs to L 2 (0, T ; L 2 (Ω) d )×L 2 (0, T ; H 1 2 (Γ e ) d ), we deduce that (u, p) ∈ L 2 (0, T ; H 2 (Ω) d )× L 2 (0, T ; H 1 (Ω)).

Let us now prove the uniqueness of solution. Assume that u 1 and u 2 are two solutions and let w = u 1 -u 2 . Then we have for all v ∈ V : So u 1 = u 2 on (0, T ) × Ω. To conclude, thanks to system 1.1, we obtain p 1 = p 2 .

  0

Figure 1 :

 1 Figure 1: Example of an open set Ω such that Γ e ∪ Γ 0 = ∂Ω and Γ e ∩ Γ 0 = ∅ in dimension 2.

2

 2

1 2 +(|u| 2 +

 122 Γe |∇u| 2 + |p| 2 + |∇p| 2 )e Ψ . (4.12)

Remark 4 . 8 .

 48 Let Γ ⊆ Γ e be a non-empty open part of Γ e . Inequality 4.6 of Lemma 4.7 still holds if we replace Γ e by Γ in the right-hand side. To prove this, it is sufficient in the definition 4.7 of Ψ 0 to define χ ∈ C 2 (∂Ω) such that χ = 0 on Γ 0 ∪ (Γ e \ Γ) and χ ≥ 0 on Γ. Let us now prove Theorem 4.3. Proof of Theorem 4.3. Since g ∈ H 5 2

  where µ is the coercivity constant associated with the bilinear form a α . Proposition 4.13. The operator A 1 2

  ∂u ∂n -pn = 0, on Γ e , ∂u ∂n -pn + qu = 0, on Γ 0 . (4.21)

2 H 1 (

 21 Ω) d .

5 . 0 Ω

 50 According to inequality A.5, there exists u ∈ L 2 (0, T ; V ) such that, up to a subsequence, u n u in L 2 (0, T ; V ).Let j ∈ N * . Multiplying the first equation of A.4 by ψ ∈ C 1 (0, T ) such that ψ(T ) = 0 then integrating on (0, T ), we get, ∀n ≥ j:T 0 Ω u n,t (t, x) • w j (x)ψ(t)dxdt + T 0 Γ0 q(x)u n (t, x) • w j (x)ψ(t)dxdt + T ∇u n (t, x) : ∇w j (x)ψ(t)dxdt = T 0 Γe g(t, x) • w j (x)ψ(t)dxdt. t (t, x) • w j (x)ψ(t)dxdt = -

  v) L 2 (Ω) d ψ (t)dt + T 0 a q (u, v)ψ(t)dt = (u(0, .), v) L 2 (Ω) d ψ(0) + T 0 l(v)ψ(t)dt, (A.9)

Ω

  ∂ t w(t) • v + Ω ∇w(t) : ∇v + Γ0 qw(t) • v = 0, w(0) = 0. (A.11)Taking v = w(t) in A.11, we find: | 2 = 0, for all t ∈ (0, T ).

  of the boundary, in order to get the identifiability of the Robin coefficient q on Γ 0 . Here, besides a measurement on u 1 -u 2 , we need measurements on Let us begin by proving this intermediate result which gives us a logarithmic estimate of the traces of u, ∇u, p, ∇p on Γ 0 with respect to the ones on Γ e .

	∂u 1 ∂n	-	∂u 2 ∂n	(or p 1 -p 2 ) and	∂p 1 ∂n	-	∂p 2 ∂n	.
	Lemma 4.7.

3, we only have to assume that u 1 = u 2 and ∂u 1 ∂n -p 1 n = ∂u 2 ∂n -p 2 n on Γ ⊆ Γ e , where Γ is a non-empty open part

  + |∇u| 2 + |p| 2 + |∇p| 2 ) + + |∇u| 2 + |p| 2 + |∇p| 2 )e Ψ ≥ 0.

	∂Ω	∂Ψ ∂n		|u| 2 + |∇u| 2 + 2	∂|∇u| 2 ∂τ	e Ψ
							+	∂Ω	∂Ψ ∂n	|p| 2 + |∇p| 2 + 2	∂|∇p| 2 ∂τ	e Ψ ≥ 0.
	We now specify the dependence with respect to s. We denote by θ = min Γ0	∂Ψ 0 ∂n	. We note that on
	Γ 0 , e Ψ = 1. Consequently:
		-sθ	Γ0	(|u| 2 Γ0	∂Ψ 1 ∂n	|u| 2 + |∇u| 2 + |p| 2 + |∇p| 2
		+2 +	Γ0 Γe ∂Ψ ∂Ψ ∂n ∂n (|u| 2 (4.10) ∂|∇p| 2 ∂τ + ∂|∇u| 2 ∂τ + 2 Γe ∂Ψ ∂n ∂|∇p| 2 ∂τ + ∂|∇u| 2 ∂τ e Ψ

0. We now choose λ ≥ 2. By summing up inequalities 4.8 and 4.9 and by eliminating the integrals on Ω in the left hand side which are positive terms, we obtain:

  2 × H 2 (Ω) and satisfies, thanks to Proposition 2.4:y j (t, .) H 3 (Ω) 2 + ρ j (t, .) H 2 (Ω) ≤ C(α, M 2 ) κ(t, .) -h

						H	3 2 (Γe)	.	(4.35)
	Remark that y j (t, .) belongs to D(A	3 2 qj ). Indeed, there exists a unique p(t, .) ∈ H 3 (Ω) solution of
	 	∆p = 0, p = κ -h, on Γ e , in Ω,	(4.36)
		p = 0,	on Γ 0 ,
	for all t > 0 and there exists a constant C > 0 such that
	p(t, .) H 3 (Ω) ≤ C κ(t, .) -h	H	3 2 (Γe)	.	(4.37)

* This work was partially funded by the ANR-08-JCJC-013-01 (M3RS) project headed by C. Grandmont and the ANR-BLAN-0213-02 (CISIFS) project headed by L. Rosier.

A Existence and uniqueness for the unsteady problem

We study the regularity of the solution of the unsteady problem:

where q only depends on the space variable. We are going to prove Theorem 2.6. First of all, as a preliminary result, we prove the following existence result:

There exists u ∈ L 2 (0, T ; V ) such that for all v ∈ V , we have in the distribution sense on (0, T ):

and for all v ∈ V ,

Proof of Proposition A.1. We begin by proving, using a Galerkin method, that there exists u ∈ L 2 (0, T ; V ) such that

Let (w i ) i∈N * be a Hilbert basis of V which is also an orthogonal basis of H. For each n ∈ N * , we define an approximate solution as follows: we search u n ∈ V n = Span (w i ) 1≤i≤n which satisfies

where u n,t denotes ∂ t u n .

Let t ∈ (0, T ). We decompose u n (t, .) in the Hilbert basis:

We denote by