Yavar Kian

Quang Sang Phan

Eric Soccorsi

Carleman estimate for infinite cylindrical quantum domains and application to inverse problems

Keywords: AMS 2010 Mathematics Subject Classification: 35R30 Inverse problem, Schrödinger equation, scalar potential, Carleman estimate, infinite cylindrical domain

We consider the inverse problem of determining the time independent scalar potential q of the dynamic Schrödinger equation in an infinite cylindrical domain Ω, from one Neumann boundary observation of the solution. Assuming that q is known outside some fixed compact subset of Ω, we prove that q may be Lipschitz stably retrieved by choosing the Dirichlet boundary condition of the system suitably. Since the proof is by means of a global Carleman estimate designed specifically for the Schrödinger operator acting in an unbounded cylindrical domain, the Neumann data is measured on an infinitely extended subboundary of the cylinder.

Introduction 1.What we are aiming for

In the present paper we consider the infinite cylindrical domain Ω = ω × R, where ω is a connected bounded open subset of R n-1 , n ≥ 2, with C 2 -boundary ∂ω. Given T > 0 we examine the following initial boundary value problem    -iu ′ -∆u + q(x)u = 0, in Q := (0, T ) × Ω, u(0, x) = u 0 (x), x ∈ Ω, u(t, x) = g(t, x), (t, x) ∈ Σ := (0, T ) × Γ, (1.1) where Γ := ∂ω × R and the ' stands for ∂ ∂t . Here u 0 (resp., g) is the initial (resp., boundary) condition associated to (1.1) and q is a function of x ∈ Ω only.

Since Γ is unbounded we make the boundary condition in the last line of (1.1) more precise. Writing x := (x ′ , x n ) with x ′ := (x 1 , . . . , x n-1 ) ∈ ω for every x ∈ Ω we extend the mapping

C ∞ 0 ((0, T ) × R; H 2 (ω)) -→ L 2 ((0, T ) × R; H 3/2 (∂ω))) v → [(t, x n ) ∈ (0, T ) × R → v(t, •, x n ) |∂ω ], (1.2)
to a bounded operator from L 2 ((0, T) × R; H 2 (ω)) into L 2 ((0, T) × R; H 3/2 (∂ω)), denoted by γ 0 . Then for every u ∈ C 0 ([0, T ]; H 2 (Ω)) the above mentioned boundary condition reads γ 0 u = g. The main purpose of this article is to extend [START_REF] Baudouin | Uniqueness and stability in an inverse problem for the Schrödinger equation[END_REF][Theorem 1], obtained for the dynamic Schrödinger operator in a bounded domain, to the case of the unbounded waveguide Ω. That is to say, to prove Lipschitz stability in the determination of the scalar potential q (assumed to be known outside some fixed compact subset of Ω) from one boundary measurement of the normal derivative of the solution u to (1.1).

The method of derivation of [START_REF] Baudouin | Uniqueness and stability in an inverse problem for the Schrödinger equation[END_REF][Theorem 1] is by means of a Carleman estimate for the Schrödinger equation, which was established in [START_REF] Baudouin | Uniqueness and stability in an inverse problem for the Schrödinger equation[END_REF][Proposition] for a bounded domain. Since the unknown part of q is compactly supported, then it seems at first sight quite reasonnable to guess that this question (arising from the problem of evaluating the electrostatic quantum disorder in nanotubes, see §1.2 below) could well be answered by adapting the above technique to some suitable truncation (for some bounded domain of R n ) of u. Nevertheless we shall prove that such a strategy necessarily adds unexpected "control" terms (i.e. "volume observations" of the solution) in the right hand side of the corresponding stability inequality, and is therefore inaccurate.

It turns out that this inconvenience can be avoided upon substituting some specifically designed Carleman estimate for the Schrödinger equation in a unbounded cylindrical domain to the one of [START_REF] Baudouin | Uniqueness and stability in an inverse problem for the Schrödinger equation[END_REF][ Proposition 3]. This new global Carleman inequality, established in Proposition 3.3, is the main novelty of this paper as it is the main tool for generalizing the Lipschitz stability inequality of [START_REF] Baudouin | Uniqueness and stability in an inverse problem for the Schrödinger equation[END_REF][Theorem 1] to the unbounded domain Ω under consideration.

It is worth noticing that the field of applications of Proposition 3.3 go beyond the study of inverse compactly supported quantum scalar coefficients PDE problems examined in this framework. As a matter of facts we prove with the aid of Proposition 3.3 in a companion article [START_REF] Kian | Hölder stable determination of a quantum scalar potential in unbounded cylindrical domains[END_REF], that scalar potentials which are not necessarily known outside some given compact subset of Ω, may nevertheless be Hölder stably retrieved from one boundary measurement of the normal derivative of u ′ .

Finally, let us mention that the Lipschitz stability estimate [START_REF] Baudouin | Uniqueness and stability in an inverse problem for the Schrödinger equation[END_REF][Theorem 1] was established under the additional technical assumption that both u and u ′ are time square integrable bounded functions of the space variables. As we aim to generalize this result to the case of the unbounded domain Ω, we do the same here by supposing (1.6). Nevertheless we stress out that sufficient conditions on q and the initial state u 0 (which are not detailed in this text in order to avoid the inadequate expense of the size of the paper) ensuring assumption (1.6), can be found in [START_REF] Kian | Hölder stable determination of a quantum scalar potential in unbounded cylindrical domains[END_REF][Theorem 1.1].

Physical motivation: estimating the electrostatic quantum disorder in nanotubes

The equations of (1.1) describe the evolution of the wave function of a charged particle (in a "natural" system of units where the various physical constants such as the mass and the electric charge of the particle are taken equal to one), under the influence of the "electric" potential q. Moreover the quantum motion of this particle is constrained by the waveguide Ω. Carbon nanotubes exhibit unusual physical properties, which are valuable for electronics, optics and other fields of materials science and technology. These peculiar nanostructures have a length-to-diameter ratio up to 10 8 /1. This justifies why they may be rightfully modelled by infinite cylindrical domains such as Ω. Unfortunately their physical properties are commonly affected by the inevitable presence of electrostatic quantum disorder, see e. g. [START_REF] Chang | Conductance through a single impurity in the metallic zigzag carbon nanotube[END_REF][START_REF] Kane | Coulomb Interactions and Mesoscopic Effects in Carbon Nanotubes[END_REF]. This motivates for a closer look into the inverse problem of estimating the strength of the "electric impurity potential" q from the (partial) knowledge of the wave function u.

Existing papers

There is a wide mathematical literature dealing with uniqueness and stability in inverse coefficient problems related to partial differential equations, see e. g. [START_REF] Bellassoued | Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients[END_REF][START_REF] Bellassoued | Lipschitz stability in determining density and two Lamé coefficients[END_REF][START_REF] Cristofol | Stability estimate in an inverse problem for nonautonomous magnetic Schrödinger equations[END_REF][START_REF] Isakov | Uniqueness and stability in multi-dimensional inverse problems[END_REF][START_REF] Yu | Global uniqueness and stability in determining coefficients of wave equations[END_REF][START_REF] Klibanov | Lipschitz stability of an inverse problem for an acoustic equation[END_REF]. For the stationnary (elliptic) Schrödinger equation, Bukhgeim and Uhlmann proved in [START_REF] Bukhgeim | Recovering a potential from partial Cauchy data[END_REF] that the knowledge of the Dirichlet to Neumann (DN) map measured on some part of the boundary determines uniquely the potential. In dimension n ≥ 3 this result was improved by Kenig, Sjöstrand and Uhlmann in [START_REF] Kenig | The Calderon problem with partial data[END_REF] and L. Tzou proved in [START_REF] Tataru | Carleman estimates, unique continuation and controllability for anisotropic PDEs[END_REF] that the electric potential (together with the magnetic field) depends stably on the Cauchy data even when the boundary measurement is taken only on a subset that is slightly larger than half of the boundary. Their method is essentially based on the construction of a rich set of "optics geometric solutions" to the Dirichlet problem. In these three papers, the knowledge of the DN map, taking infinitely many boundary measurements of the solution to the Schrödinger equation, is actually required.

The problem of stability in determining the time-independent electric potential in the dynamic Schrödinger equation from a single boundary measurement was treated by Baudouin and Puel in [START_REF] Baudouin | Uniqueness and stability in an inverse problem for the Schrödinger equation[END_REF]. This result was improved by Mercado, Osses and Rosier in [START_REF] Mercado | Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights[END_REF]. In these two papers, the main assumption is that the part of the boundary where the measurement is made satisfies a geometric condition related to geometric optics condition insuring observability (see Bardos, Lebeau and Rauch [START_REF] Bardos | Sharp sufficient conditions for the observation, control and stabilization from the boundary[END_REF]). This geometric condition was relaxed in [START_REF] Bellassoued | Logarithmic stability in the dynamical inverse problem for the Schrödinger equation by arbitrary boudary observation[END_REF] under the assumption that the potential is known near the boundary.

In all the above mentioned articles the Schrödinger equation is defined in a bounded domain. In the present paper we rather investigate the problem of determining the scalar potential of the Schrödinger equation in an infinite cylindrical domain. There are only a few mathematical papers dealing with inverse boundary value problems in an unbounded domain available in the mathematical literature. In [START_REF] Li | Inverse problems with partial data in a slab[END_REF] Li and Uhlmann prove uniqueness in the determination of the scalar potential in an infinite slab from partial DN map. In [START_REF] Cardoulis | Inverse problems for the Schrödinger operator in an unbounded strip[END_REF] Cardoulis, Cristofol and Gaitan obtain Lipschitz stability from a single lateral measurement performed on one side of an unbounded strip.

More specifically for an inverse boundary value problem stated in the waveguide geometry examined in this work we refer to [START_REF] Choulli | Recovering the twisting function in a twisted waveguide from the DN map[END_REF][START_REF] Choulli | Stable determination of time dependent scalar potential from boundary measurements in a periodic quantum waveguide[END_REF] where stability is claimed for various coefficients of the Schrödinger equation from the knwoledge of the DN map. Here we investigate the same type of problems, but in absence of any information given by the DN map, by a method based on an appropriate Carleman estimate. We refer to [START_REF] Albano | Carleman estimates for the Euler-Bernoulli plate operator[END_REF][START_REF] Baudouin | Uniqueness and stability in an inverse problem for the Schrödinger equation[END_REF][START_REF] Tataru | Carleman estimates, unique continuation and controllability for anisotropic PDEs[END_REF] for actual examples of this type inequalities designed for the Schrödinger equation. The original idea of using a Carleman estimate to solve inverse problems goes back to the pioneering article [START_REF] Bukhgeim | Uniqueness in the large of a class of multidimensional inverse problems[END_REF] by Bukhgeim and Klibanov. Since then this technique has then been widely and succesfully used by numerous authors, see e.g. [START_REF] Baudouin | Uniqueness and stability in an inverse problem for the Schrödinger equation[END_REF][START_REF] Bellassoued | Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients[END_REF][START_REF] Bellassoued | Inverse boundary value problem for the dynamical heterogeneous Maxwell system[END_REF][START_REF] Bellassoued | Lipschitz stability in determining density and two Lamé coefficients[END_REF][START_REF] Choulli | Recovering the twisting function in a twisted waveguide from the DN map[END_REF][START_REF] Yu | On Carleman estimates for hyperbolic equations[END_REF][START_REF] Yu | Lipschitz stability in inverse parabolic problems by Carleman estimates[END_REF][START_REF] Isakov | Carleman estimates with the Neumann boundary condition and application to the observability inequality and inverse problems[END_REF][START_REF] Klibanov | Lipschitz stability of an inverse problem for an acoustic equation[END_REF][START_REF] Mercado | Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights[END_REF], in the study of inverse wave propagation, elasticity or parabolic problems.

Main results

In this section we state the main results of this article and briefly comment on them.

For M > 0 and p ∈ W 2,∞ (Ω) fixed, we define the set of "admissible scalar potentials" as

Q M (p) := {q ∈ W 2,∞ (Ω; R), q W 2
,∞ (Ω) ≤ M and q(x) = p(x) for a.e. x ∈ Γ}.

Here the identity q |Γ = p Γ is understood in the sense of the usual trace operator from L ∞ (R, H 2 (ω)) into L ∞ (R, H 3/2 (∂ω)). By selecting q ∈ Q M (p) we thus enjoin fixed value to q on the boundary, which is the mesurement on Γ of the scalar potential we want to determine. Similar (or stronger) "compatibility conditions" imposed on inverse problems coefficients have already been used in various contexts, see e.g. [START_REF] Bellassoued | Logarithmic stability in the dynamical inverse problem for the Schrödinger equation by arbitrary boudary observation[END_REF][START_REF] Bellassoued | Inverse boundary value problem for the dynamical heterogeneous Maxwell system[END_REF][START_REF] Cristofol | Stability estimate in an inverse problem for nonautonomous magnetic Schrödinger equations[END_REF].

We are concerned with the stability issue around any q 1 ∈ Q M (p), i.e. we want to upper bound the L 2 norm of q 1 -q 2 by some increasing function of the difference u 1 -u 2 . That is to say that q 2 ∈ Q M (p) and the solution u j , for j = 1, 2, to (1.1), where q j is substituted for q, are known, while q 1 is unknown.

Theorem 1.1. For p ∈ W 2,∞ (Ω), M > 0, ℓ > 0 and α > 0 fixed, let u 0 ∈ H 4 (Ω; R) obey u 0 (x) ≥ α > 0, x ∈ ω × (-ℓ, ℓ), (1.3)
let q j ∈ Q M (p), j = 1, 2, fulfill q 1 (x) = q 2 (x), x ∈ ω × (R \ (-ℓ, ℓ)), (1.4)
and let u j denote the

C 1 ([0, T ]; H 2 (Ω) ∩ H 1 0 (Ω)) ∩ C 2 ([0, T ]; L 2 (Ω))-solution to (1.1)
associated to u 0 , g = γ 0 G and q j , where

G(t, x) := u 0 (x) + it(∆ -p)u 0 (x), (t, x) ∈ Q. (1.5) Assume that u 2 ∈ H 1 (0, T ; L ∞ (Ω)). (1.6)
Then for every L > ℓ, there exist Γ * ⊂ ∂ω × (-L, L) and a constant C > 0 depending only on L, T , M, ω and Γ * , such that we have

q 1 -q 2 L 2 (Ω) ≤ C ∂ ν (u ′ 1 -u ′ 2 ) L 2 ((0,T )×Γ * ) + u 1 -u 2 H 1 (0,T ;H 1 (ω×S L )) , (1.7
)

with S L := (-L, -ℓ) ∪ (ℓ, L).
Moreover there is a subboundary γ * ⊂ ∂ω, such that the estimate

q 1 -q 2 L 2 (Ω) ≤ C ′ ∂ ν u ′ 1 -∂ ν u ′ 2 L 2 ((0,T )×γ * ×R) , (1.8)
holds for some positive constant C ′ depending only on ℓ, T , M, ω and γ * .

Under the prescribed conditions (1.3)-(1.5) and assumption (1.6), the first statement (1.7) of Theorem 1.1 claims Lipschitz stability in the determination of the scalar potential appearing in the dynamic Schrödinger equation in Ω from two different observations of the solution u to (1.1).

• The first one is a lateral measurement on some subboundary of ∂ω × (-L, L) of the normal derivative

∂ ν u(t, x) := ∇u(t, x) • ν(x), (t, x) ∈ Σ,
where ν is the outward unit normal to Γ. Since

ν(x) = ν(x ′ ) = ν ′ (x ′ ) 0 , x = (x ′ , x n ) ∈ Γ, (1.9)
where ν ′ (x ′ ) ∈ R n-1 denotes the outgoing normal vector to ∂ω computed at x ′ , we notice that

∂ ν u(t, x) = ∂ ν ′ u(t, x) := ∇ x ′ u(t, x) • ν ′ (x ′ ), (t, x) = (t, x ′ , x n ) ∈ Σ, (1.10)
where ∇ x ′ stands for the gradient operator w.r.t. x ′ ∈ ω.

• The second observation is an internal measurement of u which is performed in each of the two "slices" S -

L := ω × (-L, -ℓ) and S + L := ω × (ℓ, L) of Ω. Although the R n -Lebesgue measure of S ±
L can be made arbitrarily small by taking L sufficiently close to ℓ, this "volume observation" cannot be removed from the right hand side of (1.7) by taking L asymptotically close to ℓ since the prefactor (i.e. the constant C) tends to infinity as L goes to ℓ.

The occurence of this internal measurement of u in (1.7) is due to the unbounded geometry of Ω. More precisely this is a direct consequence of the technique used for the derivation of the stability inequality (1.7), which is by means of a global Carleman estimate for the Schrödinger equation in a bounded domain. Indeed, this strategy requires a cut off function with first derivative supported in (-L, -ℓ) ∪ (ℓ, L), which gives rise to the measurement of u in S ± L . Notice that the use of a Carleman estimate known to be valid in a bounded domain of R n only, was made possible here since the difference q 1q 2 is compactly supported in R n . A fact that follows from assumption (1.4) expressing that the scalar potential to be retrieved is known outside some fixed bounded set.

One way to get rid of both volume observations simultanesously is to use a global Carleman estimate specifically designed for the unbounded quantum waveguide Ω, which is stated in Proposition 3.3. This yields (1.8), implying that the electrostatic quantum potential is now Lipschitz stably retrieved in Ω from only one lateral measurement of the normal derivative ∂ ν u on some subboundary of Σ. This result is similar to the one obtained in a bounded domain (under the same assumption as (1.6)) by Baudouin and Puel, see [START_REF] Baudouin | Uniqueness and stability in an inverse problem for the Schrödinger equation[END_REF][ Theorem 1]. It should nevertheless be noticed that, contrarily to (1.7), and despite of the fact that the scalar potential under identification is assumed to be known outside a compact set, the Neumann data required in the right hand side of (1.8) is measured on an infinitely extended subboundary of Σ.

Finally, it is worth mentioning that Theorem 1.1 applies for a wide class of subboundaries Γ * . Indeed, given a

C 2 -domain O in R n obeying ω × (-ℓ, ℓ) ⊂ O ⊂ ω × (-L, L), this is the case for any Γ * ⊃ {x ∈ ∂O, (x -x 0 ) • ν O (x) ≥ 0}, where ν O is the outward unit normal to ∂O and x 0 is arbitrary in R n \ O. The same remark holds true for γ * ⊃ {x ′ ∈ ∂ω, (x ′ -x ′ 0 ) • ν ′ (x ′ ) ≥ 0}, where x ′ 0 is arbitrarily fixed in R n-1 \ ω. Theorem 1.
1 immediately entails the followig uniqueness result.

Corollary 1.2. Under the conditions of Theorem 1.1, it holds true for all q j ∈ Q M (p), j = 1, 2, that any of the two following assumptions

∂ ν u ′ 1 (t, x) = ∂ ν u ′ 2 (t, x) for a.e. (t, x) ∈ (0, T ) × Γ * u 1 (t, x) = u 2 (t, x) for a.e. (t, x) ∈ (0, T ) × ω × S L , or ∂ ν u ′ 1 (t, x) = ∂ ν u ′ 2 (t, x) for a.e. (t, x) ∈ (0, T ) × γ * × R, yields q 1 = q 2 .

Outline

The paper is organized as follows. Section 2 deals with the direct problem associated to (1.1). Namely §2.1 gathers existence and uniqueness results for the solution to the dynamic Schrödinger equation in the infinite domain Ω and §2.2 is devoted to the study of the direct problem for the linearized system associated to (1.1). In §2.3 the corresponding solution is suitably extended to a function of [-T, T ] × Ω which is continuous with respect to the time variable t. This is required by the method used in the proof of the stability inequalities (1.7) and (1.8), which is the purpose of Section 4. It is by means of the global Carleman estimate for the Schrödinger equation in a bounded domain (resp., in an unbounded waveguide) stated in Proposition 3.2 (resp., Proposition 3.3) for (1.7) (resp., for (1.8)). Finally §4.1 contains the completion of the proof of (1.7)-(1.8).

Analysis of the direct problem

In this section we establish some existence, uniqueness and regularity properties of the solution to (1.1) needed for the analysis of the inverse problem in section 3.

Existence and uniqueness results

This subsection gathers two existence and uniqueness results for (1.1). We start by recalling from [START_REF] Choulli | Stable determination of time dependent scalar potential from boundary measurements in a periodic quantum waveguide[END_REF][Proposition 2.1] the following:

Proposition 2.1. Let M > 0. Then for all q ∈ Q M , v 0 ∈ H 1 0 (Ω) ∩ H 2 (Ω) and f ∈ W 1,1 (0, T ; L 2 (Ω)), there is a unique solution v ∈ Z 0 := C([0, T ]; H 1 0 (Ω) ∩ H 2 (Ω)) ∩ C 1 ([0, T ]; L 2 (Ω)) to the boundary value problem    -iv ′ -∆v + qv = f, in Q, v(0, x) = v 0 , x ∈ Ω, v(t, x) = 0, (t, x) ∈ Σ.
Moreover we have

v Z 0 ≤ C v 0 H 2 (Ω) + f W 1,1 (0,T ;L 2 (Ω)) ,
for some constant C > 0 depending only on ω, T and M.

Let the space

X 0 := γ 0 (W 2,2 (0, T ; H 2 (Ω))), (2.11)
be equipped with the following quotient norm

g X 0 = inf{ G W 2,2 (0,T ;H 2 (Ω)) ; G ∈ W 2,2 (0, T ; H 2 (Ω)) satisfies γ 0 G = g}.
Evidently, each g ∈ X 0 admits an extension G 0 ∈ W 2,2 (0, T ; H 2 (Ω)) verifying

G 0 W 2,2 (0,T ;H 2 (Ω)) ≤ 2 g X 0 .
Further, put

L := {(u 0 , g) ∈ H 2 (Ω) × X 0 ; u 0 = g(0, •) on Γ}.
(2.12)

Then, setting u = v+G 0 , where v is defined by Proposition 2.1 for f = i∂ t G 0 +∆G 0 -qG 0 and v 0 = u 0 -G 0 (0, .), we obtain the following existence and uniqueness result, which is similar to [START_REF] Choulli | Stable determination of time dependent scalar potential from boundary measurements in a periodic quantum waveguide[END_REF][Corollary 2.1].

Theorem 2.2. Let M > 0. Then for all q ∈ Q M and all (u 0 , g) ∈ L there is a unique

solution u ∈ Z := C([0, T ]; H 2 (Ω)) ∩ C 1 ([0, T ]; L 2 (Ω))
to the boundary value problem (1.1). Moreover we have the following estimate

u Z ≤ C( u 0 H 2 (Ω) + g X 0 ), (2.13)
where C is some positive constant depending only on ω, T and M.

Linearized problem

In this subsection we establish some useful regularity properties of the solution to the linearized problem associated to (1.1) by applying the existence and uniqueness results stated in §2.1.

To this purpose, we let p, M, q j , j = 1, 2, u 0 , g and G be the same as in Theorem 1.1. Since G ∈ W 2,2 (0, T ; H 2 (Ω)) and g(0, •) = u 0 on Γ, we have (u 0 , g) ∈ L . Applying Theorem 2.2 there is thus a unique solution u j ∈ Z, j = 1, 2, to the following system

   -iu ′ j -∆u j + q j (x)u j = 0, in Q, u j (0, x) = u 0 (x), x ∈ Ω, u j (t, x) = g(t, x), (t, x) ∈ Σ. (2.14)
Further, differentiating (2.14) with respect to t for j = 2, we obtain that u ′ 2 is solution to the boundary value problem

   -iu ′′ 2 -∆u ′ 2 + q 2 (x)u ′ 2 = 0, in Q, u ′ 2 (0, x) = ũ0 (x), x ∈ Ω, u ′ 2 (t, x) = g ′ (t, x), (t, x) ∈ Σ, (2.15)
where ũ0

:= i(∆ -q 2 )u 0 . Since G ′ (t, x) = i(∆ -p)(x) for a.e. (t, x) ∈ Q by (1.5) then we have G ′ ∈ W 2,2 (0, T ; H 2 (Ω)
) and we get g ′ (0, x) = ũ0 (x) for a.e. x ∈ Γ from the identity q 2 = p on Γ. Consequently (ũ 0 , g ′ ) ∈ L . Therefore u ′ 2 ∈ Z from (2.15) and Theorem 2.2. As a consequence we have

u 2 ∈ C 1 ([0, T ]; H 2 (Ω)) ∩ C 2 ([0, T ]; L 2 (Ω)).
Moreover, it follows from (2.14) that u := u 1u 2 is solution to the linearized system

   -iu ′ -∆u + q 1 u = f, in Q, u(0, x) = 0, x ∈ Ω, u(t, x) = 0, (t, x) ∈ Σ, (2.16)
with f := (q 2q 1 )u 2 . Since u 2 ∈ Z and q 1q 2 ∈ L ∞ (Ω) then f ∈ W 1,1 (0, T ; L 2 (Ω)) so we find that u ∈ Z 0 by Proposition 2.1. Last, we deduce from (2.16

) that v := u ′ satisfies    -iv ′ -∆v + q 1 v = f ′ , in Q, v(0, x) = i(q 2 -q 1 )(x)u 0 (x), x ∈ Ω. v(t, x) = 0, (t, x) ∈ Σ, (2.17)
with f ′ = (q 2q 1 )u ′ 2 ∈ W 1,1 (0, T ; L 2 (Ω)). Since q 1 = q 2 on Γ we get that i(q 2q 1 )u 0 ∈

H 1 0 (Ω) ∩ H 2 (Ω). Therefore v = u ′ 1 -u ′ 2 ∈ Z 0 by Proposition 2.1. Bearing in mind that u ′
2 ∈ Z we deduce from this that u ′ 1 ∈ Z. Summing up, we have obtained that

u j ∈ C 1 ([0, T ]; H 2 (Ω)) ∩ C 2 ([0, T ]; L 2 (Ω)), j = 1, 2.

Time symmetrization

This subsection is devoted to showing that the solution v ∈ Z 0 to (2.17) may be extended to a solution of the same system over the time span [-T, T ].

To this end we extend v (resp., f ′ ) on (-T, 0) × Ω by setting v(t, x) := -v(-t, x) (resp., f ′ (t, x) := -f ′ (-t, x)) for (t, x) ∈ (-T, 0) × Ω. Since the initial conditions v(0, •) and f ′ (0, •) are purely complex valued according to (2.17), the mappings t → v(t, x) and t → f ′ (t, x) are thus continuous at t = 0 for a.e. x ∈ Ω. Therefore, we have

v ∈ C 0 ([-T, T ]; H 1 0 (Ω) ∩ H 2 (Ω)) ∩ C 1 ([-T, T ]; L 2 (Ω)) and f ′ ∈ W 1,1 (-T, T ; L 2 (Ω)
). Moreover v is solution to (2.17) over the whole time span (-T, T ):

   -iv ′ -∆v + q 1 v = f ′ , in Q e := (-T, T ) × Ω, v(0, x) = i(q 2 -q 1 )(x)u 0 (x), x ∈ Ω. v(t, x) = 0, (t, x) ∈ Σ e := (-T, T ) × Γ.
(2.18)

Carleman estimate for the dynamic Schrödinder operator

The main purpose of this section is to derive a global Carleman estimate for the Schrödinger operator in an unbounded cylindrical domain O × R, where O is a bounded domain of R m , m ≥ 1, from the same type of results obtained for O by Baudouin and Puel in [START_REF] Baudouin | Uniqueness and stability in an inverse problem for the Schrödinger equation[END_REF][Proposition 3]. (ii) ∂ ν β(x) := ∇ β(x).ν(x) < 0 for all x ∈ ∂O\Γ O , where ν is the outward unit normal vector to ∂O; Notice that there are actual functions β verifying Assumption 3.1, such as

What is known in a bounded domain

(iii) ∃Λ 1 > 0, ∃ǫ > 0 such that we have λ|∇ β(x) • ζ| 2 + D 2 β(x, ζ, ζ) ≥ ǫ|ζ| 2 for all ζ ∈ R m , x ∈ O and λ > Λ 1 , where D 2 β(x) := ∂ 2 β(x) ∂x i ∂x j 1≤i,j≤m
O ∋ x → |x -x 0 | 2 , where x 0 is arbitrary in R m \ O and Γ O ⊃ {x ∈ ∂O, (x -x 0 ) • ν(x) ≥ 0}.
Further we put β := β + K, where K := r β ∞ for some r > 1, (3.20) and define the two following weight functions for λ > 0:

ϕ(t, x) := e λβ(x) (T + t)(T -t)
and η(t, x) := e 2λKe λβ(x)

(T + t)(T -t) , (t, x) ∈ (-T, T ) × O. (3.21)
Finally, for all s > 0 we introduce the two following operators acting in (

C ∞ 0 ) ′ ((-T, T )× O): M 1,m := i∂ t + ∆ + s 2 |∇η| 2 and M 2,m := isη ′ + 2s∇η.∇ + s(∆η). (3.22)
It is easily seen that M 1,m (resp., M 2,m ) is the adjoint (resp., skew-adjoint) part of the operator e -sη L m e sη , where L m is given by (3.19).

Having said that we may now state the following global Carleman estimate, borrowed from [START_REF] Baudouin | Uniqueness and stability in an inverse problem for the Schrödinger equation[END_REF]

(O)) satisfying L m w ∈ L 2 ((-T, T ) × O) and ∂ ν w ∈ L 2 (-T, T ; L 2 (Γ O )).
The proof of Proposition 3.2 can be found in [START_REF] Baudouin | Uniqueness and stability in an inverse problem for the Schrödinger equation[END_REF][section 2].

Extension to the case of unbounded cylindrical domains

Let ω be Ω = ω × R be the same as in §1. We use the same notation as in section 1 and section 2, i.e. every point x ∈ Ω is written x = (x ′ , x n ) with x ′ = (x 1 , . . . , x n-1 ) ∈ ω, and we note Q e = (-T, T ) × Ω. We shall now derive from 3.2 a global Carleman estimate for the Schrödinger operator

L n := -i∂ t -∆, (3.24) acting in (C ∞ 0 ) ′ ((-T, T ) × Ω). Here ∆ := ∆ x ′ + ∂ 2 xn where ∆ x ′ := Σ n-1 j=1 ∂ 2
x j is the Laplacian in ω. where ∇ x ′ (resp., ∆ x ′ ) stands for the gradient (resp., the Laplacian) operator w.r.t. x ′ ∈ ω. (3.20). Extend β to Ω = ω × R by setting β(x) = β(x ′ ) for all x = (x ′ , x n ) ∈ Ω. Define ϕ and η (resp., the operators M j,n for j = 1, 2) by (3.21) (resp., (3.22)), upon substituting Ω for O and n for m. Then there are two constants s 0 > 0 and C > 0, depending only on T , ω and γ * , such that the estimate

s e -sη ∇ x ′ v 2 L 2 (Qe) + s 3 e -sη v 2 L 2 (Qe) + j=1,2 M j,n e -sη v 2 L 2 (Qe) ≤ C s e -sη ϕ 1/2 (∂ ν β) 1/2 ∂ ν v 2 L 2 ((-T,T )×γ * ×R) + e -sη L n v 2 L 2 (Qe) , (3.25)
holds for all s ≥ s 0 and any function v

∈ L 2 (-T, T ; H 1 0 (Ω)) verifying L n v ∈ L 2 (Q e ) and ∂ ν v ∈ L 2 (-T, T ; L 2 (γ * × R)).
Proof. Since the function β does not depend on the infinite variable x n then the same is true for ϕ and η according to (3.21). Therefore we have

M 1,n = i∂ t + ∆ + s 2 |∇ x ′ η| 2 and M 2,n = isη ′ + 2s(∇ x ′ η) • ∇ x ′ + s(∆ x ′ η), (3.26)
We denote by F xn the partial Fourier with respect to x n , i.e.

(F xn ψ)(x ′ , k) = ψ(x ′ , k) := 1 (2π) 1/2 R e -ikxn ψ(x ′ , x n )dx n , ψ ∈ L 2 (Ω), x ′ ∈ ω, k ∈ R,
and consider the unitary group U t = e -it∂ 2 xn in L 2 (Ω), defined by

U t := F -1 xn e itk 2 F xn , t ∈ R,
where F -1 xn is the inverse transform to F xn and e itk 2 stands for the multiplier by the corresponding complex number. Therefore we have

(U t ψ)(x) = e -it∂ 2 xn ψ(x) := F -1 xn e itk 2 ψ(•, k) (x), ψ ∈ L 2 (Ω), x ∈ Ω, t ∈ R. (3.27)
We notice from (3.27) that

[i∂ t , U t ] = ∂ 2 xn U t = U t ∂ 2 xn , t ∈ R, (3.28)
since ∂ 2 xn commutes with U t , and that

[∂ x j , U t ] = [e -sη , U t ] = 0, j = 1, . . . , n -1 and s, t ∈ R, (3.29)
since η does not depend on x n . Let us now introduce the function

w(t, x) := U t v(t, x), (t, x) ∈ Q e . (3.30)
Then we have w(

•, x n ) ∈ L 2 (-T, T ; H 1 0 (ω)) and ∂ ν ′ w(•, x n ) ∈ L 2 (-T, T ; L 2 (γ *
)) for a.e. x n ∈ R. Further, upon differentiating (3.30) w.r.t. t, we deduce from (3.28) that

i∂ t w = U t (i∂ t v + ∂ 2 xn v). (3.31)
This and the first commutator identity in (3.29) yields

(-i∂ t -∆ x ′ )w(t, x) = U t L n v(t, x), (t, x) ∈ Q e , which entails L n-1 w(•, x n ) = U t L n v(•, x n ), x n ∈ R. (3.32)
As a consequence we have

L n-1 w(•, x n ) ∈ L 2 ((-T, T ) × ω) for a.e. x n ∈ R, whence s e -sη ∇ x ′ w(•, x n ) 2 L 2 ((-T,T )×ω) + s 3 e -sη w(•, x n ) 2 L 2 ((-T,T )×ω) + j=1,2 M j,n-1 e -sη w(•, x n ) 2 L 2 ((-T,T )×ω) (3.33) ≤ C s e -sη ϕ 1/2 (∂ ν β) 1/2 ∂ ν ′ w(•, x n ) 2 L 2 ((-T,T )×γ * ) + e -sη L n-1 w(•, x n ) 2 L 2 ((-T,T )×ω) ,
for every s ≥ s 0 , by Proposition 3.2. The next step of the proof involves noticing from (3.29)-(3.31) that

e -sη w(•, x n ) = U t e -sη v(•, x n ), e -sη ∇ x ′ w(•, x n ) = U t e -sη ∇ x ′ v(•, x n ), (3.34)
and, with reference to (3.26), that

M j,n-1 e -sη w(•, x n ) = U t M j,n e -sη v(•, x n ), j = 1, 2, (3.35)
for each t ∈ (-T, T ) and a.e. x n ∈ R. Similarly, we get

e -sη L n-1 w(•, x n ) = U t e -sη L n v(•, x n ), t ∈ (-T, T ), x n ∈ R, (3.36)
by combining the second identity of (3.29) with (3.32). Finally, bearing in mind that ϕ and β are independendent of x n , we deduce from (1.9)-(1.10) and the second part of (3.29) that

e -sη ϕ 1/2 (∂ ν β) 1/2 ∂ ν ′ w(•, x n ) = U t e -sη ϕ 1/2 (∂ ν β) 1/2 ∂ ν v(•, x n ), t ∈ (-T, T ), x n ∈ R.
(3.37) Therefore, putting (3.33)-(3.37) together, we obtain for s ≥ s 0 and a.e.

x n ∈ R that s U t e -sη ∇ x ′ v(•, x n ) 2 L 2 ((-T,T )×ω) + s 3 U t e -sη v(•, x n ) 2 L 2 ((-T,T )×ω) + j=1,2 U t M j,n e -sη v(•, x n ) 2 L 2 ((-T,T )×ω) ≤ C s U t e -sη ϕ 1/2 (∂ ν β) 1/2 ∂ ν v(•, x n ) 2 L 2 ((-T,T )×γ * ) + U t e -sη L n v(•, x n ) 2 L 2 ((-T,T )×ω) .
Integrating the above inequality w.r.t. x n over R and taking into account that the operator U t is isometric in L 2 (Ω) for each t ∈ (-T, T ) thus yields

s e -sη ∇ x ′ v 2 L 2 (Qe) + s 3 e -sη v 2 L 2 (Qe) + j=1,2 M j,n e -sη v 2 L 2 (Qe) ≤ C s U t e -sη ϕ 1/2 (∂ ν β) 1/2 ∂ ν v 2 L 2 ((-T,T )×γ * ×R) + e -sη L n v 2 L 2 (Qe) , (3.38)
provided s ≥ s 0 . Last, setting Φ := e -sη ϕ 1/2 (∂ ν β) 1/2 ∂ ν v and using the unitarity of the operator F xn in L 2 (R), we derive from (3.27) that 4 Proof of Theorem 1.1

U t Φ 2 L 2 ((-T,T )×γ * ×R) = T -T γ * F -1 xn e itk Φ(t, x ′ , •) 2 L 2 (R) dx ′ dt = T -T γ * e itk Φ(t, x ′ , •) 2 L 2 (R) dx ′ dt = T -T γ * Φ(t, x ′ , •) 2 L 2 (R) dx ′ dt = Φ 2 L 2 ((-
In view of section 3 we are now in position to solve the inverse problem under study. More precisely we apply the Bugkhgeim-Klibanov method presented in [START_REF] Bukhgeim | Uniqueness in the large of a class of multidimensional inverse problems[END_REF] to prove the stability inequality (1.7) (resp., the estimate (1.8)) with the aid of Proposition 3.2 (resp., Proposition 3.3).

Proof of the stability inequality (1.7)

We first introduce the following notation used throughout the entire section: for all d > 0 we write Ω d := ω × (-d, d).

Fix L > ℓ, set r := (L + ℓ)/2 and consider a domain

O in R n , with C 2 boundary ∂O, obeying Ω ℓ ⊂ Ω r ⊂ O ⊂ Ω L . (4.39) We next choose χ ∈ C ∞ 0 (R xn , [0, 1]) such that χ(x n ) := 1 if |x n | ≤ ℓ 0 if |x n | ≥ r.
w |O ∈ C 0 ([-T, T ]; H 1 0 (O) ∩ H 2 (O)) ∩ C 1 ([-T, T ]; L 2 (O)) (4.40)
and is solution to the following system

   -iw ′ -∆w + q 1 w = χf ′ -Kv, in (-T, T ) × O, w(0, x) = iχ(q 2 -q 1 )(x)u 0 (x), x ∈ O. w(t, x) = 0, (t, x) ∈ (-T, T ) × ∂O, (4.41)
where K := [∆, χ] = χ + 2 χ∂ xn . Here χ (resp., χ) is a shorthand for the first (resp., second) derivative of χ.

As a preamble to the proof of the stability inequality (1.7) we first establish two elementary technical results.

(M 1 φ)φdtdx = Re (-T,0)×O ∂ t φφdtdx + Im (-T,0)×O ∆φφdtdx + s 2 ∇ηφ 2 L 2 (-T,0)×O = Re (-T,0)×O ∂ t φφdtdx + Im ∇φ 2 L 2 ((-T,0)×O) = Re (-T,0)×O ∂ t φφdtdx ,
so the result follows readily from this and (4.42). M j e -sη w 2 L 2 ((-T,T )×O) , (4.43) the notation M j , for j = 1, 2, being a shorthand for M j,n .

Proof. In view of Lemma 4.1 and the Cauchy-Schwarz inequality, we have

J ≤ 2 M 1 φ L 2 ((-T,T )×O) φ L 2 ((-T,T )×O) ≤ s -3/2 M 1 (e -sη w) 2 L 2 ((-T,T )×O) + s 3 e -sη w 2 L 2 ((-T,T )×O) .
This and (4.43) yields the desired result.

Completion of the proof

The next step of the proof involves majorizing I(w) with the aid of the Carleman inequality of Proposition 3.2. In view of (4.40)-(4.41) and (4.43) the estimate

I(w) ≤ C s e -sη ϕ 1/2 (∂ ν β) 1/2 ∂ ν w 2 L 2 ((-T,T )× Γ) + e -sη (χf ′ -Kv) 2 L 2 ((-T,T )×O) ≤ C s e -sη ϕ 1/2 (∂ ν β) 1/2 ∂ ν w 2 L 2 ((-T,T )× Γ) + j=0,1 e -sη ∇ j v 2 L 2 ((-T,T )×(Ωr \Ω ℓ )) + e -sη χf ′ 2 L 2 ((-T,T )×Ωr) , (4.44)
holds with Γ := Γ O for s ≥ s 0 . Here and henceforth C denotes some generic positive constant. Bearing in mind that η(0, x) = inf t∈(-T,T ) η(t, x) for all x ∈ Ω and that f ′ = (q 2q 1 )u ′ 2 , we have e -sη χf ′ L 2 ((-T,T )×Ωr) ≤ C e -sη(0,•) (q 1q 2 ) L 2 (Ω ℓ ) . (4.45)

Here we used the fact that u ′ 2 L ∞ ((-T,T )×Ω ℓ ) < ∞. On the other hand, the function ϕ∂ ν β being bounded in (-T, T ) × Γ as well, we deduce from (4.44)-(4.45) that

I(w) ≤ C s e -sη(0,•) ∂ ν w 2 L 2 ((-T,T )× Γ) + j=0,1 e -sη(0,•) ∇ j v 2 L 2 ((-T,T )×(Ωr\Ω ℓ )) + e -sη(0,•) (q 1 -q 2 ) 2 L 2 (Ω ℓ ) ≤ C obs + e -sη(0,•) (q 1 -q 2 ) 2 L 2 (Ω ℓ ) , s ≥ s 0 , (4.46)
where we have set

obs := s e -sη(0,•) ∂ ν w 2 L 2 ((-T,T )× Γ) + j=0,1 e -sη(0,•) ∇ j v 2 L 2 ((-T,T )×(Ωr \Ω ℓ )) . ( 4
J = e -sη(0,•) χ(q 1 -q 2 )u 0 2 L 2 (O) ≤ Cs -3/2 obs + e -sη(0,•) (q 1 -q 2 ) 2 L 2 (Ω ℓ ) , (4.48)
for all s ≥ s 0 . Further since J ≥ α 2 e -sη(0,•) (q 1q 2 ) 2 L 2 (Ω ℓ ) from the assumption (1.3) we get that (α 2 -Cs -3/2 ) e -sη(0,•) (q 1q 2 ) 2 L 2 (Ω ℓ ) ≤ Cs -3/2 obs, s ≥ s 0 , (4.49) from (4.48). Thus chosing s ≥ s 0 so large that α 2 -Cs -3/2 ≥ α 2 /2 and taking into account that inf x∈Ω ℓ e -sη(0,x) > 0, we derive from (4.47) and (4.49) that

q 1 -q 2 2 L 2 (Ω ℓ ) ≤ C ∂ ν w 2 L 2 ((-T,T )× Γ) + j=0,1 ∇ j v 2 L 2 ((-T,T )×(Ωr\Ω ℓ )) . (4.50)
Now, recalling from §2.3 that ∇ j v L 2 ((-T,T )×(Ωr \Ω ℓ )) = 2 ∇ j v L 2 ((0,T )×(Ωr \Ω ℓ )) for j = 0, 1, and from the identity w = χv that ∂ ν w L 2 ((-T,T )× Γ) = 2 ∂ ν w L 2 ((0,T )× Γ) , we derive from (4.50) that

q 1 -q 2 2 L 2 (Ω) ≤ C ∂ ν w 2 L 2 ((0,T )× Γ) + j=0,1 ∇ j v 2 L 2 ((0,T )×(Ωr \Ω ℓ )) . (4.51)
Here we used the identity q 1q 2 L 2 (Ω ℓ ) = q 1q 2 L 2 (Ω) arising from (1. Putting this together with (4.51) and the identity v = (u 1u 2 ) ′ , we end up getting (1.7).

4.2 Proof of the stability estimate (1.8)

The proof of (1.8) is very similar to the one of (1.7). Therefore, in order to avoid the inadequate expense of the size of this article, we shall skip some technical details.

The function v still denoting the solution to (2.18), we put I(v) := s e -sη ∇ x ′ v 2 L 2 (Qe) + s 3 e -sη v 2 L 2 (Qe) + j=1,2

M j e -sη v 2 L 2 (Qe) , s > 0,

where we write M j for j = 1, 2, instead of M j,n . Then arguing in the exact same way as in the derivation of Lemmae 4.1-4.2, we get that e -sη(0,•) v(0, •) 2 L 2 (Ω) = e -sη(0,•) (q 1q 2 )u 0 2 L 2 (Ω) ≤ s -3/2 I(v), (4.52) for all s > 0.

On the other hand, applying Proposition 3.3 to (2.18) we obtain

I(v) ≤ C s e -sη ϕ 1/2 (∂ ν β) 1/2 ∂ ν v 2 L 2 ((-T,T )×γ * ×R) + e -sη f ′ 2 L 2 (Qe) , (4.53)
for every s ≥ s 0 . Further, taking into account that η(0, x ′ ) = inf t∈(-T,T ) η(t, x ′ ) for all x ′ ∈ ω and f ′ = (q 2q 1 )u ′ 2 , we derive from (1.6) that e -sη f ′ L 2 (Qe) ≤ C ′ e -sη(0,•) (q 1q 2 ) L 2 (Ω) ,

for some generic positive constant C ′ . Since ϕ∂ ν β is bounded in (-T, T ) × γ * × R, (4.53)-(4.54) then yield I(v) ≤ C ′ s e -sη(0,•) ∂ ν v 2 L 2 ((-T,T )×γ * ×R) + e -sη(0,•) (q 1q 2 ) 2 L 2 (Ω) , s ≥ s 0 . (4.55) Putting (4.52) and (4.55) together we find that e -sη(0,•) (q 1q 2 )u 0 2 L 2 (Ω) ≤ C ′ s -3/2 s e -sη(0,•) ∂ ν v 2 L 2 ((-T,T )×γ * ×R) + e -sη(0,•) (q 1q 2 ) 2 L 2 (Ω) , s ≥ s 0 .

From this and (1.3) then follows for any s ≥ s 0 that (α 2 -C ′ s -3/2 ) e -sη(0,•) (q 1q 2 ) 2 L 2 (Ω) ≤ Cs -1/2 e -sη(0,•) ∂ ν v 2 L 2 ((-T,T )×γ * ×R) . (4.56) Choosing s ≥ s 0 so large that α 2 -C ′ s -3/2 ≥ α 2 /2 and using that inf x∈ω e -sη(0,x) > 0, we deduce from (4.56) that

q 1 -q 2 2 L 2 (Ω) ≤ C ′ ∂ ν v 2

  Given an arbitrary bounded domain O ⊃ R m with C 2 boundary, we consider the Schrödinger operator acting in (C ∞ 0 ) ′ ((-T, T ) × O), L m := -i∂ t -∆, (3.19) where ∆ denotes the Laplacian in O. Here we use the subscript m to emphasize the fact that the operator L m acts in a subdomain of (-T, T ) × R m . Next we introduce a function β ∈ C 4 (O; R + ) and an open subset Γ O of ∂O, satisfying the following conditions: Assumption 3.1. (i) ∃C 0 > 0 such that the estimate |∇ β(x)| ≥ C 0 holds for all x ∈ O;

and D 2

 2 β(x, ζ, ζ) denotes the R m -scalar product of D 2 β(x)ζ with ζ.

Proposition 3 . 2 . 2 M j,m e -sη w 2 L 2 (

 32222 [Proposition 3], which is the main tool for the proof of Theorem 1.1. Let β be given by (3.20), where β ∈ C 4 (O; R + ) fulfills Assumption 3.1, let ϕ and η be as in (3.21), and let L m , M 1,m and M 2,m be defined by (3.19)-(3.22). Then we may find s 0 > 0 and constant C > 0, depending only on T , O and Γ O , such that the estimate s e -sη ∇w 2 L 2 ((-T,T )×O) + s 3 e -sη w 2 L 2 ((-T,T )×O) + j=1,(-T,T )×O) ≤ C s e -sη ϕ 1/2 (∂ ν β) 1/2 ∂ ν w 2 L 2 ((-T,T )×Γ O ) + e -sη L m w 2 L 2 ((-T,T )×O) , (3.23) holds for any real number s ≥ s 0 and any function w ∈ L 2 (-T, T ; H 1 0

Proposition 3 . 3 .

 33 For n ≥ 2 fixed, let β and γ * := Γ O obey Assumption 3.1 with O = ω and m = n -1, and let β be given by

  T,T )×γ * ×R) , so (3.25) follows immediately from this and (3.38).

  Put w := χv, where v is theC 0 ([-T, T ]; H 1 0 (Ω) ∩ H 2 (Ω)) ∩ C 1 ([-T, T ]; L 2 (Ω))-solution to (2.18); Then the restriction w |O of w to O satisfies

4. 1 . 1

 11 Two auxiliary resultsLemma 4.1. For s > 0, let φ := e -sη w, where w is defined by (4.40)-(4.41). Then we haveJ := e -sη(0,•) w(0, •) 2 L 2 (O) = 2Im (-T,0)×O M 1 φφdtdx ,where M 1 stands for M 1,n .Proof. In light of (3.20)-(3.21) it holds true that lim t↓(-T ) η(t, x) = +∞ for all x ∈ O, hence lim t↓(-T ) φ(t, x) = 0. Therefore we have J = φ(0, •) 2 L 2 (O) = (-T,0)×O ∂ t |φ| 2 dtdx, from where we get that J = 2Re (-T,0)×O ∂ t φφdtdx . (4.42) On the other hand, (3.22) and the Green formula yield Im (-T,0)×O

Lemma 4 . 2 .

 42 Let w and J be the same as in Lemma 4.1. Then we have J ≤ s -3/2 I(w), s > 0, where I(w) := s e -sη ∇w 2 L 2 ((-T,T )×O) + s 3 e -sη w 2 L 2 ((-T,T )×O) + j=1,2

  4). Last we set Γ * := Γ ∩ (∂ω × [-r, r]) and Γ e := Γ ∩ (Ω L \ Ω r ) in such a way that Γ = Γ * ∪ Γ e , Γ * ∩ Γ e = ∅, by (4.39). Since χ(x n ) = 0 for every |x n | ≥ r then the function w = χv is necessarily uniformly zero in (0, T ) × (Ω L \ Ω r ). This entails∂ ν w(t, σ) = 0, (t, σ) ∈ (0, T ) × Γ e .As a consequence we have∂ ν w L 2 ((0,T )× Γ) = ∂ ν w L 2 ((0,T )×Γ * ) ≤ C ∂ ν v L 2 ((0,T )×Γ * ) .

L 2 (

 2 (-T,T )×γ * ×R) . Now (1.8) follows readily from this upon recalling from §2.2 that v = (u 1u 2 ) ′ and from §2.3 that ∂ ν v L 2 ((-T,T )×γ * ×R) = 2 ∂ ν v L 2 ((0,T )×γ * ×R) .


Myriam Graba
email: mgraba@yahoo.fr

Sabine Sauuage

Frédéric Y Moulin

Gemma Urrea

Sergi Sabaterf

Miguel Sanchez-Pérez

Interaction between local hydrodynamics and algal community in epilithic bioftlm

Keywords:

Interactions between epilithic biofilm and local hydrodynamics were investigated in an experimental fiume. Epilithic biofilm from a natural river was grown over a 41-day period in three sections with different flow velocities (0.10, 0.25 and 0.40 rn s-1 noted LV, IV and HV respectively). Friction velocities U• and boundary layer parameters were inferred from PlV measurement in the three sections and related ta the biofilm structure. The results show that there were no significant differences in Dry Mass and Ash-Free Dry Mass (g m-2 ) at the end of experiment, but velocity is a selective factor in algal composition and the biofilms' morphology differed according ta differences in water velocity. A hierarchical agglomerative cluster analysis (Bray-Curtis distances) and an Indicator Species Analysis

(IndVal) showed that the indicator taxa were Fragilaria capucina var. mesolepta in the lowvelocity (u•. = 0.010-0.012 rn s-1 ), Navicula atomus, Navicula capitatoradiata and Nitzschia frustulum in the interrnediate-velocity (u•. = 0.023-0.030 rn s-1 ) and Amphora pediculus, Cymbella proxima, Fragilaria capucina var. vaucheriae and Surirella angusta in the high-velocity (u•. = 0.033-0.050 rn s-1 ) sections. A sloughing test was performed on 40-day-old biofilms in arder ta study the resistance of epilithic biofilms ta higher hydrodynamic regimes. The results showed an inverse relationship between the proportion of detached biomass and the average value of friction velocity during growth. Therefore, water velocity during epilithic biofilm growth conditioned the structure and al gal composition of biofilm, as well as its response (ability ta resist) ta higher shear stresses. This result should be considered in modelling epilithic biofilm dynamics in streams subject ta a variable hydrodynamics regime.

1.

Notations

AFDM DM

Introduction

"Epilithic biofilm" that grows on gravel, cobbles, and rocks in river beds, is a collective term for a complex microorganism community and includes algae, bacteria, and microfauna, with algae usually the dominant component. This community is the source of most primary production [START_REF] Minshall | Autotrophy in stream ecosystems[END_REF][START_REF] Lock | River epilithon: towards a structuralfunctional model[END_REF], and constitutes a food source for a number of invertebrates and fish [START_REF] Fuller | The importance of algae to stream invertebrates[END_REF][START_REF] Mayer | The importance of algae in a shaded head water stream as a food of an abundant caddisfly (Trichoptera)[END_REF][START_REF] Winterboum | Interactions among nutrients, algae and invertebrates in New Zealand mountain stream[END_REF]. It plays a major role in the metabolic conversion and partial removal of biodegradable material in rivers and streams (Mcintire, 1973;[START_REF] Saravia | Modeling periphyton dynamics in runningwater[END_REF][START_REF] Hondzo | Effects of turbulence on growth and metabolism of periphyton in a laboratory fiume[END_REF], and serves as a functional indicator of river health [START_REF] Wehr | Freshwater Algae of North America[END_REF][START_REF] Cardinale | Biodiversity improves water quality through niche partitioning[END_REF]. However, it still one of the least-studied communities despite the significant increase in the examination of aquatic microbial communities in recent years [START_REF] Stoodley | Influence of flow on the structure of bacterial biofilms[END_REF][START_REF] Battin | Effects of current velocity on the nascent architecture of stream microbial biofilms[END_REF][START_REF] Besemer | Biophysical contrais on community succession in stream biofilms[END_REF]Besemer et al., , 2009aBesemer et al., , 2009b)).

Hydrodynamics is one of the most important environmental factors (nutrient, light, temperature etc.) driving stream biofilm dynamics and structure and is generally considered the major agent of physical forcing on the biofilm [START_REF] Reiter | Interactions between the hydrodynamics of flowing water and development of a benthic algal community[END_REF][START_REF] Power | Disturbance and recovery of an algal assemblage following flooding in an Oklahoma stream[END_REF][START_REF] Biggs | Linking scales of flow variability to lotie ecosystem structure and function[END_REF]. Indeed, metabolic rates for the biofilm are controlled by the thickness of the diffusive boundary layer that develops along filaments driving the transfer of metabolites to and from cells, and they are then related to the flow water velocity (Whitford and Schumacher, 1961;[START_REF] Lock | The effect of flow patterns on uptake of phosphorus on river periphyton[END_REF][START_REF] Riber | Boundary layer and internai diffusion effects on phosphorus fluxes in lake periphyton[END_REF]. Besides, as water velocity increases, the drag forces and skin friction exerted on the community also increase, and this affects their attachment ability [START_REF] Biggs | Periphyton responses ta a hydraulic gradient in a regulated river in New Zealand[END_REF].

The effect of water velocity on epilithic biofilms biomass has been analysed in a number of studies, both by observations in natural streams (e.g. [START_REF] Biggs | Periphyton responses ta a hydraulic gradient in a regulated river in New Zealand[END_REF][START_REF] Uehlinger | Periphyton dynamics in a flood-prone pre-alpine river: evaluation of significant processes by modelling[END_REF][START_REF] Uehlinger | Effects of experimental floods on periphyton and stream metabolism below a high dam in the Swiss Alps (River Spël)[END_REF][START_REF] Boulêtreau | Assessing the importance of self-generated detachment process in river biofilm models[END_REF][START_REF] Boulêtreau | Identification of a minimal adequate madel to describe the biomass dynamics of river epilithon[END_REF][START_REF] Boulêtreau | Temporal dynamics of river biofilm in constant flows: a case study in a riverside laboratory fiume[END_REF] as well as in flumes (e.g. [START_REF] Homer | Stream periphyton development in relation to current velocity and nutrients[END_REF][START_REF] Ghosh | Current velocity and the establishment of stream algal periphyton communities[END_REF][START_REF] Hondzo | Effects of turbulence on growth and metabolism of periphyton in a laboratory fiume[END_REF][START_REF] Cardinale | Biodiversity improves water quality through niche partitioning[END_REF]. However, only local flow conditions are ultimately relevant for describing the forcing at biofilm scale, and are generally not easily inferred from mean bulk velocities, except in the case of hydraulically smooth turbulent boundary layers where only the fluid viscosity v [START_REF] Nezu | Turbulence in Open-channel Flows[END_REF] and the metabolites diffusivities need to be known. For rough turbulent boundary layers, a better description of the local flow conditions are inferred by a log law description (see [START_REF] Labiod | The relationship between stream periphyton dynamics and near-bed turbulence in rough open-channel flow[END_REF][START_REF] Graba | Effect of nearbed turbulence on chronic detachment of epilithic biofilm in artificial rough, open channel flow: experimental and modeling approaches[END_REF] that requires knowledge of the roughness length u* v Roughness Reynolds number (= U• k.lv) Flow discharge (m 3 s-1 ) Standard errer in measured values (g m-2 ) Friction velocity (rn ç 1 ) Water kinetic viscosity (10-6 m 2 ç 1 ) (or equivalently Nikuradse's equivalent sand roughness k.) and reads:

U(z) = !log(z-d) + 85 U* K ks (1)
Or from the exponential profile of [START_REF] Nezu | Turbulence in Open-channel Flows[END_REF] :

U(z) ( z ) --u;-=Duexp -C•H-d (2)
where U(z) is the mean (above the roughness sublayer) or double-average longitudinal velocity (inside the roughness sublayer), U• is the friction velocity, k. the Nikuradse's equivalent sand roughness, d the displacement height, K the Karman constant (K = 0.41), H the flow height and c, and Du are empirical constants (C, = 1 and Du = 2.3).

Observations on the interaction between water flow and biofilms (Reiter, 1989a(Reiter, , 1989b;;[START_REF] Nikora | On stream periphytonturbulence interactions[END_REF][START_REF] Nikora | A simple model of stream periphyton-flow interactions[END_REF][START_REF] Labiod | The relationship between stream periphyton dynamics and near-bed turbulence in rough open-channel flow[END_REF] have shown that friction velocity U• (which measures the drag of the flow at the bottom layer) could increase with the growth of epilithic biofilm, leading to the conclusion that stream biofilm increased bed roughness. However, [START_REF] Biggs | Periphyton responses ta a hydraulic gradient in a regulated river in New Zealand[END_REF] , [START_REF] Moulin | Experimental study of the interaction between a turbulent flow and a river biofilm growing on macrorugosities[END_REF] and [START_REF] Graba | Effect of nearbed turbulence on chronic detachment of epilithic biofilm in artificial rough, open channel flow: experimental and modeling approaches[END_REF] have found that stream biofilm decreased the drag forces and the roughness. As explained in [START_REF] Moulin | Experimental study of the interaction between a turbulent flow and a river biofilm growing on macrorugosities[END_REF] and [START_REF] Graba | Effect of nearbed turbulence on chronic detachment of epilithic biofilm in artificial rough, open channel flow: experimental and modeling approaches[END_REF], these apparently contradictory results are essentially due to a graduai transition from a completely nude bed to a biofilm-covered bed, an increase or decrease of the roughness length being observed depending on the value of the roughness length for the initial nude bed compared to a typical value for a biofilm-covered bed. Yet, beyond their apparent contradictions, these studies provide an idea of the complex interaction between local hydrodynamics and the successional stage (age, thickness and composition of the community) and physiognomy of the algal biofilm community (Reiter, 1989b).

Tools such as the hydraulic habitat preference curves Qowett et al., 1991) have been used to predict the effects of flow regulation on stream habitats (Davis and Barmuta, 1989;[START_REF] Young | Clarification of the criteria used to identify near-bed flow regimes[END_REF]. However, there is still a poor knowledge on the relationship between the near-substratum hydrodynamics and the structure and species composition of epilithic algal assemblages. It is generally admitted that Rhodophytes prefer current velocities exceeding 0.030 rn s-1 [START_REF] Seath | Mechanical adaptations to flow in freshwater red algae[END_REF], and that sorne species (Gomphonema paroulum and Gomphonema lanceolatum) prefer pools as the main habitats [START_REF] Ghosh | Current velocity and the establishment of stream algal periphyton communities[END_REF]. It is also known that while mean values and fluctuations of u and w, as defined in the Reynolds decomposition u = U + u' and w =ID+ w'. Each vertical plane yields 120 vertical profiles along around 8 cm, i.e., two hemisphere diameters in the streamwise direction.

Following the methodology of [START_REF] Nikora | Sorne observations of the effects of micro-organisms growing on the bed of an open channel on the turbulence properties[END_REF]Nikora et al. ( , 2007aNikora et al. ( , 2007b)), double-averaged quantities, i.e. quantities averaged in the two horizontal directions (noted with brackets< >) were estimated from PlV measurements in the two vertical planes by spatial averaging along the streamwise direction and between the two vertical planes. As shawn by [START_REF] Castro | Turbulence over urbantype roughness: deductions from wind-tunnel measurements[END_REF] , such double-averaged quantities extend the validity range of the log law towards the top of the roughness, deep inside the roughness sublayer, leading ta more robust estimations of the boundary layer parameters U•, ks and d. The 1000 independent measurements of u and w yield ta an estimation of {U) and< u'w' > with time convergence relative errors below 5% and 15%, respectively (using convergence errer estimates of [START_REF] Bendat | Random Data: Analysis and Measurement Procedures[END_REF] for confidence intervals of 95%). Ta fit the data with the log-law equation (1) , we followed [START_REF] Castro | Turbulence over urbantype roughness: deductions from wind-tunnel measurements[END_REF] and inferred the friction velocity U• from vertical profiles of the turbulent shear stress < u'w' > : we took the square root of the averaged value of the turbulent shear stress in bath the roughness and inertial sublayers. For nude cobbles before the inoculum, it corresponded ta the region between the top of the hemispheres at z = H and the top of the inertial sublayer, taken as z = 0.1(D-H) where Dis the water depth in arder ta remain far below the defect law region. For biofilmcovered cobbles, algal filaments moved in the camera field, sa that the top for the roughness could not be de arly identified like for nude cobbles. Therefore, we used the maximal height reached by the filaments, noted Ztop• as the lower limit of the fittingrange of the log-law equation (1) . Naturally, with biofilm accrual, this lower limit gradually raised up from 2 cm for the nude cobbles ta 3.5 cm for the 28 days old biofilm in the law velocity section. The upper limit of the fitting range was chosen equal ta the top of the inertial sublayer at z = Ztop+0.1(D-Ztop)• Further details on this method and on PlV measurements are presented in [START_REF] Moulin | Experimental study of the interaction between a turbulent flow and a river biofilm growing on macrorugosities[END_REF] and [START_REF] Graba | Effect of nearbed turbulence on chronic detachment of epilithic biofilm in artificial rough, open channel flow: experimental and modeling approaches[END_REF] . Relative errors on < u'w' > below 15% yield errors on friction velocity U• lower than 7.5%. With fitting ranges defined above, relative errors on ks and d induced by errors on (U) and< u'w' > were found ta be lower than 10%.

The roughness Reynolds number k+ (=U• ksfv, where v is water kinetic viscosity), a descriptor of the hydraulic roughness of the flow was also calculated. This number depends on the hydraulics and turbulent conditions in the near bed region (turbulent energy) but also on the dimensions and the shape of the roughness in this region (that drives the shape of the mean velocity profile). Sa the change of the values of this term gives an idea of the changes induced by the growth of the epilithic biofilm on the turbulent conditions and the flow regime in the near bed region. More specifically, for vegetal canopies, vertical exchanges of matter between the canopy and the flow above can be expressed using exchange velocity or equivalently, power functions of k., as discussed in [START_REF] Graba | Effect of nearbed turbulence on chronic detachment of epilithic biofilm in artificial rough, open channel flow: experimental and modeling approaches[END_REF] .

In the sloughing test fiume, PlV measurements were also performed at different values of the volumic discharge Q, and the boundary layer parameters (friction velocity u•, Nikuradse's equivalent sand roughness ks, and The roughness Reynolds number k+) were then inferred at the same way than in the main channel. As expected for rough turbulent boundary layers over rigid bottom, log law parameters d and k. were independent of Q, with d "" 1.54 ± 0.02 cm and ks :::: 1.03 ± 0.07 cm for all values of Q. Measured friction velocities U• were found ta be linearly dependent of Q, with values of u* = 1.0, 1.8, 3.0, 4.1, 5.1 and 6.4 cm s-1 for respectively Q = 5, 10, 15, 20, 25 and 30 Ls-1 •

3.

Results and discussions

Biojilm biomass

The temporal evolution of DM and AFDM (g m-2 ) in the three flow sections (Fig. 3 and Fig. 4) reveals that flow velocities have a significant influence on the values of DM at days 9 (ANOVA, P < 0.001), 15 (ANOVA, P < 0.001) and 23 (ANOVA, P < 0.05). Later (on days 29 and 35) these differences became less significant (0.05 < P < 0.2). AFDM was less sensitive ta water velocity patterns. The influence of flow velocity on AFDM was significant only up ta day 15 (ANOVA, P < 0.05) and became insignificant at days 23, 29 and 35 (0.05 < P < 0.2). These results show that the biofilm colonisation was significantly delayed by the highest flow velocity until the third week after inoculum, but the values reached at the end of the experiment approached a mean of 93.95 ± 15.74 (g m-2 , DM) and 23.10 ± 4.03 (g m-2 , AFDM) for the three sections. This can be explained by the conflicting effects of current regime and turbulence intensities. In fact, in the initial colonisation phase the highest drag forces and friction velocities slowed dawn the deposit and attachment of microbial and algal cells, resulting in a more significant colonisation in the law flow regime than in the higher ones. From the third week of the experiment, the highest diffusion and exchange in the intermediate and high-flow region accelerated the productivity of the attached cells and counterbalanced the delay registered during the colonisation phase.

Biojilm patterns and algal composition

Colonisation patterns during the first week were regular in the LV section, i.e. exhibiting the same spatial distribution for comparison with water depth), the evolution is very different. Very confined flows are generated initially (A/h = 4 and 3), and a very quick decrease in the Nikuradse's equivalent sand roughness ks and friction velocity U• is observed {Fig. 7) at the beginning of experiment when the biofilm matter covered firstly the troughs between the cobbles spaces (see Fig. Se) and brought about a change in the roughness topography, leading to a less rough boundary associated with less strong drag, so a decrease in friction velocity U• and the turbulent roughness k+ were observed (see Fig. 7).

As discussed in Moulin et al. {2008) , competing contributions from the wake and skin frictions behind cobbles and along algal filaments necessarily drive a complex evolution of the roughness length since this quantity integrates all the processes occurring in the canopy (see Nikora et al., 2007a, b). The drop of k. in deep flows is observed when filaments become longer than the initial horizontal scale prescribed by the substrates, the biofilm structure then controlling most of the friction. Indeed, the values of k 5 found at the end of the growth experiment in the present study, equal to 0.318 x 10-3 rn for the HV section and 0.360 x 10-3 rn for the LV section at day 28, compare very well with the values found by [START_REF] Labiod | The relationship between stream periphyton dynamics and near-bed turbulence in rough open-channel flow[END_REF] for same age biofilms grown on smaller substrates (values found range between 0.468 and 0.800 x 10-3 rn for 26-day-old biofilm). The main difference between the two studies cornes from the difference in the initial value of ks that depends only on the substrate length scale {rods or marbles in Labiod et al. {2007) and around 2 x 10-3 rn high hemispheres in our study). These flow measurements confirm an evolution of k. that simply expresses a transition from a bed covered with nude substrates towards a bed covered completely with a matt of biofilm.

The evolution of the near bed turbulence (evaluated by the roughness Reynolds number k+) as we can see in Fig. 7, agree with the result of Besemer et al. {2007, 2009aBesemer et al. {2007, , 2009b) and Tomés and Saba ter {2010) that the algal mats as the bacterial community modify the local architectural conditions in a way to slow down the near-substratum velocities, and thereby lessening the current effects on algal and bacterial detachment.

Sloughing test

During the sloughing test, increasing friction velocities were exerted on the sampled cobbles by increasing the tl ume flow discharge Q. For Q ranging from 0.005 m 3 ç 1 to 0.030 m 3 çl, PIV measurements yield values of friction velocity that range between 0.010 and 0.064 rn s-1 • Filming during the sloughing test (see Fig. 8) shows that the detachment of filaments begins after the friction velocity value exerted in the sloughing fiume exceeds the time-averaged value exerted during the growth experiment in the section being considered (noted u• gr)• However, the biomass loss is graduai as sloughing friction velocity increases.

For the three sections, the sloughing test eventually led to the detachment of sorne proportion of the biomass covering the hemispheres. Indeed, sorne of the biofilm strongly attached to the artificial cob bles remained after the sloughing test {Table 2), while sorne ofit, composed mostly of filaments, was taken away by the flow. The proportion of detached biomass is inversely proportional to the time-averaged value of friction velocity U• exerted during biofilm growth (see Table 2). This is in accordance with the results ofWaesche et al. {2002); [START_REF] Stoodley | Biofilm material properties as related to shear-induced deformation and detachment phenomena[END_REF]Mohle et al., 2007, on the effect of growth phase hydrodynamics on the mechanical properties and the resistance to detachment of microbial biofilms. In fact these last studies concluded also that biofilms grown under higher shear were more strongly attached and were cohesively stronger than those grown under lower shears.

4.

Conclusion

The impacts of different flow regimes on the dynamics of epilithic biofilm, its structure, algal composition and feedback on the local hydrodynamics have been evidenced by changes in the biomass and algal composition analysis. Actually, the biofilm composition and structures are expressed as different growth forms in relation to hydrodynamic descriptors. Their prevalence and the biofilm thickness is tightly related to the hydrodynamic conditions: Melosira moniliformis (O.F.Muller) Agardh was the dominant species in the three sections, while the Indicator Species Analysis (IndVal) shows that the indicator taxa were Fragilaria capucina var. mesolepta in the low-velocity (u•. = 0.010-0.012 rn s-1 ), Nauicula atomus, N. capitatoradiata and Nitzschia frustulum in the intermediate-velocity (u•. = 0.023-0.030 rn s-1 ) and A. pediculus, Cymbella proxima, Fragilaria capucina var. uaucheriae and Surirella angusta in the high-velocity group (u•. = 0.033-0.050 rn ç 1 ). An inverse relationship was found between the proportion of detached biomass and the averaged value of friction velocity during biofilm growth. Thus, the differences in biofilm structure and composition, their influence on the flow and their resistance to higher hydrodynamical regimes seem to be a function of the friction velocity U• on the boundary layer. This result supports the improvement of Labiod et al. {2007) and [START_REF] Graba | Effect of nearbed turbulence on chronic detachment of epilithic biofilm in artificial rough, open channel flow: experimental and modeling approaches[END_REF] in modelling epilithic biomass dynamics with the equation from [START_REF] Uehlinger | Periphyton dynamics in a flood-prone pre-alpine river: evaluation of significant processes by modelling[END_REF]. This, by substituting the flow discharge with friction velocity or roughness Reynolds number k+, as an external physical variable forcing the chronic detachment process. This result also sheds new light on the role of local hydrodynamics in the catastrophic detachment process associated with floods. Firstly, it suggests improving the term describing this process in the same way by considering local hydrodynamic rather than flow and mean velocity, as external physical variables for forcing the detachment process, in epilithic biofilms biomass dynamics models (e.g. [START_REF] Uehlinger | Periphyton dynamics in a flood-prone pre-alpine river: evaluation of significant processes by modelling[END_REF][START_REF] Saravia | Modeling periphyton dynamics in runningwater[END_REF]Asaeda andHong Son, 2000, 2001;[START_REF] Flipo | Biogeochemical Modelling at the River Scale: Plankton and Periphyton Dynamics[END_REF][START_REF] Boulêtreau | Assessing the importance of self-generated detachment process in river biofilm models[END_REF][START_REF] Boulêtreau | Identification of a minimal adequate madel to describe the biomass dynamics of river epilithon[END_REF]. Secondly, the detached biofilm biomass driven by a strong hydraulic perturbation is almost entirely associated with the biofilm filaments, and the results presented here support a separate description of the biomass of these filaments in biofilm dynamics modelling.

Fig. 3 -

 3 Fig. 3 -Evolution of the DM ± SE (g m M セ @ in the three flow sections (LV, IV and HV) at different days after inoculum.

Table 2 -Measurements ofbiomass (DM) in the sloughing test fiume with U• up to 0.064 ms-1 ).

 2

	Fiume	u* gn average values	DM±	DM± SE after	Detached
	section	of friction velocity during	SE before	sloughing (gm-2 )	proportion
		biofilm growth (mç 1 )	sloughing (gm-2 )
	LV	0.010	121.39 ± 2.94	57.9 ± 18.56	52%
	IV	0.025	99.11 ± 11.50	70.7 ± 10.85	29%
	HV	0.040	104.49 ± 1.25	93.5 ± 3.27	11%

Ac:knowledgements

This work has been supported by the national research project « EC2CO Ecosphère Continentale et Côtière» as part of a project entitled «Couplage et flux entre un biofilm de rivière et un écoulement turbulent : expérimentations en conditions naturelles contrôlées et modélisation numérique dans l'écosystème de la Garonne Moyenne». We wish ta thank S. Font, Y. Peltier, C. Pen and D. Baque for technical support, data acquisition and analysis.


The Effect of microstructural heterogeneities on the fatigue behaviour of 22MnB5 sheet steel Abstract This work deals with the effect of heat treatment on the fatigue strength of a rear axial beam and aims to propose a methodology suitable and reliable for fatigue design. The rear axial beam is made of sheet metal (22MnB5); the initial microstructure is ferrito-pearlitic and martensitic after the treatment. A vast experimental campaign has been undertaken to investigate the behaviour and more specially the fatigue damage mechanisms observed (with material treated and no treated) under different loading conditions: tension and shear test with different load ratios. To test a sheet on shearing an original test is also used.

SEM observations of fatigue failure surfaces, for both heat treated and nontreated specimens, show that the fatigue cracks initiate from inclusions for the specimens loaded in tension. The experiments show that the damage mechanism depends on the applied loading condition: for shear loadings no inclusions are observed at the crack initiation site.

A probabilistic approach using the weakest link concept is used to model the fatigue. This approach leads naturally to a probabilistic Kitagawa type diagram, which in this case explains the relationship between the influence of the heat treatment and the microstructural heterogeneities.

INTRODUCTION

The 22MnB5 steel, commercially referred to as Usibor 1500 by Arcelor Mittal, has been developed with the aim of reducing the mass of structural components used in the automotive industry. This material is obtained by hot rolling and is characterised by excellent quenchability.

The good quenchability of Boron steels can be taken advantage of by using the Hot Forming Die Quenching process (HFDQ), whereby the sheet metal is austenized and subsequently stamped in a cooled die. [START_REF] Karbasian | A review on hot stamping[END_REF] To control this process, much research has been devoted to the characterization of the thermomechanical behaviour of boron steels [START_REF] Liu | Hot formation quality of high strength steel BR1500HS for hot stamping without cooling system[END_REF][START_REF] Bardelcik | Effect of cooling rate on the high strain rate properties of boron steel[END_REF].

Considerable progress has been made concerning the HFDQ process and it is now possible to vary the final microstructure of a component by optimizing the way in which the tooling is cooled. Some structural components, such as a Bpillar, may benefit from regions that have a lower strength and greater ductility for improved crash performance [START_REF] Merklein | Investigation of the thermo-mechanical properties of hot stamping steels[END_REF].

Another method of obtaining a heterogeneous microstructure is to manufacture the product by conventional sheet metal forming processes and then to locally quench zones of the component by induction heating. The rear axial beam used in Renault vehicles is a good example of the usefulness of this method. As the axial beam is principally subject to torsional loads, highly loaded zones can be identified. It is therefore, a priori, not necessary to fully heat treat the entire component as a local heat treatment in certain zones could significantly reduce the manufacturing time. The objective of this work is to propose a fatigue criterion that is capable of explicitly taking into account the effect of the heat treatment. The results of an experimental campaign to characterize the fatigue behaviour of heat treated and non-treated 22MnB5 steel under different cyclic stress states are presented. After an analysis of the failure surfaces, a probabilistic approach to take into account the combined effect of alumina inclusions and the heat treatment on the fatigue behaviour is developed.

THE MATERIAL, 22MNB5 AND EXPERIMENTAL PROCEDURE THE MATERIAL

The material studied in this work is a ferrito-pearlitic steel commercially referred to as Usibor 1500 and is used to reduce the mass of structural components and reinforcements used in the automotive industry. Its chemical composition is given in Table 1. The material is produced in the form of rolled sheets. The material is isotropic in terms of its mechanical behaviour and its mechanical properties are presented in Table 2. The heat treatment consists of an austenitization at 950 °C for 6 minutes, and then quenching in oil. SEM (Scanning Electron Microscope) observations show a ferrito-pearlitic microstructure in its initial state and martensitic after the heat treatment.

FATIGUE TEST

For the untreated material, Push-pull (R=-1 and R=0,1) fatigue tests were performed using a vibrophore Rumul testing machine at a frequency of approximately 72 Hz. The fatigue limits were evaluated using the staircase method, at 2x10 6 cycles using ten specimens per condition.

To define the shearing fatigue limit of the sheet material, an original fatigue test set-up was developed. It is based on the work of [START_REF] Galtier | Fatigue tests on thin sheet materials, fatigue design[END_REF]. Using a servo-hydraulic fatigue testing machine and the specimen geometry shown in the figure below, it is possible to generate a cyclic pure shear stress state in an area of about 10mm in diameter (see Figure 3). With this system, the test frequency is limited. Hence, in order to estimate the high cycle fatigue strength, an accelerated method has been used, that is the selfheating method. These shearing tests were conducted at 10 Hz for load ratios R = -1 and R = 0.1, a single type T thermocouple is fixed to the specimen surface via adhesive tape to measure the temperature. Blocks of 5000 cycles were applied. To estimate the average fatigue limit, the iteration empirical procedure developed by Cura et al. [START_REF] Koutiri | A probabilistic model for the high cycle fatigue behaviour of cast aluminium alloys subject to complex loads[END_REF] is used.

FATIGUE TEST RESULTS AND FRACTURE SURFACES

Table 3 summarised the results of the fatigue test campaign. For the heat treated and non-treated materials, when subject to uniaxial tensile cyclic loads (R = 0. Given the hardness of our material and the average defect size, it is possible to use the Murakami [START_REF] Murakami | Metal fatigue effects of small defects and non-metallic inclusions[END_REF] criterion to predict fatigue limits for each loading condition. These predictions are plotted in Figure 7. The Murakami [START_REF] Murakami | Metal fatigue effects of small defects and non-metallic inclusions[END_REF] criterion uses the parameter area (where area is the projected area of a defect) and the hardness to define the fatigue strength via the following equation: It can be seen that the Murakami criterion is able to accurately predict the fatigue limit of the untreated material in tension and sheari with R=0.1 and R=-1. This result is surprising because despite the fact that the damage mechanism observed in shear is not associated with the presence of a defect, the Murakami criterion results in good predictions. Concerning the heat treated material, the Murakami criterion is however less accurate for tensile loads with a R-ratio of 0.1. This hypothesis, of the existence of two fatigue damage mechanisms, depending on the type of loading condition, has been previously verified [START_REF] Pessard | Modelling the role of non-mettalic inclusions on the anisotropic fatigue behaviour of forged steel[END_REF][START_REF] Koutiri | A probabilistic model for the high cycle fatigue behaviour of cast aluminium alloys subject to complex loads[END_REF] and will be the basis of the proposed modelling approach.

( ) ( ) α σ       - + = 2 1 120 6 / 1 R area H A v w (1)
The proposed modelling framework is a flexible way of taking into account multiple, coexisting, fatigue damage mechanisms via the combination of two appropriate high cycle fatigue criteria. One criterion to model fatigue damage associated with crack initiation and a second for crack propagation (or crack arrest).

A PROBABILISTIC MULTIAXIAL FATIGUE CRITERION THAT CAN TAKE INTO ACCOUNT THE HEAT TREATMENT EFFECT

In order to model both fatigue crack initiation and propagation, the Crossland [START_REF] Crossland | Int. Conf. on Fatigue of Metals[END_REF] and Murakami [START_REF] Crossland | Int. Conf. on Fatigue of Metals[END_REF] models have been chosen. The threshold defined by each one of these criteria is then re-defined in terms of a Weibull distribution, giving a failure probability caused by each of these mechanisms.

Maximum hydrosta c stress, ∑ h,max

Experiments

Murakami criterion

The total probability of survival of the component is defined by applying the weakest link hypothesis. It is obtained by multiplying the two survival probabilities from the two different observed mechanisms:

For example, for traction-compression loads, the Crossland [START_REF] Crossland | Int. Conf. on Fatigue of Metals[END_REF] criterion can be written as:

( ) a I H a eq R k k J , max , , 2 1 3 2 3 1 σ σ σ         - + = + =
(3) and the Murakami criterion can be written as :

area K a I π σ , 65 . 0 = ∆ and ( )( ) α       - + × = ∆ - 2 1 120 10 3 . 3 3 / 1 3 R area H K v th (4)
By assuming that the scatter is the same for both mechanisms,

                                              - ′ +             ′ - +         - = α σ σ (5)
Where 01 th σ ′ and 02 th C ′ are the material parameters for the thresholds associated with the two mechanism. k is the Crossland parameter that takes into account the hydrostatic stress. Plotting the fatigue limit as a function of the defect size it is possible to obtain a Kitagawa type diagram (Figure 7)

In order to take into account the effect of the heat treatment, it is necessary to make both thresholds 01 th σ ′ and 02 th C ′ a function of the heat treatment. As well, for the Murakami criterion, it is proposed to make both thresholds depend on the hardness. This is a strong assumption that is proposed in order to demonstrate the possibilities of the proposed model. The aim is to develop an approach that is both phenomenological and empirical, which can be applied to the industrial problem of applying a local heat, the results of which are currently measured in terms of hardness. It is necessary to undertake a larger testing program in order to identify each one of these relationships. The predictions from the proposed model are shown in a Kitagawa's diagram (Figure 7).

m f         -         = - σ σ σ σ σ σ ( ) 2
K K K m f         ∆ ∆ -         ∆ ∆ ∆ = - σ Failure probability                 - - = ∫ Ω1 1 1 01 01 1 exp 1 S m th eq F dS S P σ σ                 ∆ ∆ - - = ∫ Ω 2 2 2 02 02 1 exp 1 S m th F dS K K S P Total failure probability                           ∆ ∆ +         - - = Ω Ω

CONCLUSION

The fatigue behaviour of the 22MnB5 steel has been characterized for different loading conditions and heat treatments. For this, an original shearing test setup has been employed to determine the fatigue behaviour of the 22MnB5 steel in shear. An analysis of the fatigue failure surfaces show that the crack initiation mechanism depends on the type of applied load. An original modelling approach has been developed to take into account the evolution of the fatigue behaviour depending on the applied heat treatment.

Figure 1 :

 1 Figure 1: a) rear axial beam, b) Von Mises equivalent stress for torsion loads (from an elastic simulation)

Figure 2 :

 2 Figure 2: Microstructure of the 22MnB5 steel, a) in it initial state, b) after the heat treatment

Figure 3

 3 Figure 3: a) Picture of the tool b) Evolution of the two principal deformations during a static loading of tension-compression c) Von Mises Criterion from an elastic simulation (MPa)

Figure 4 :Figure 6 :Figure 6

 466 Figure 4: a),b),c) Typical failure surfaces, d) Chemical composition of an inclusion analysed by energy dispersive X-ray spectroscopy

  56 for surface and internal defects, respectively for torsional loads and surface defects, the fatigue strength can be expressed as:

Figure 5 :

 5 Figure 5: Dang Van diagram, showing the Murakami predictions and the corresponding data for the 22MnB5 steel

  have a spherical shape, the traction-compression fatigue limit can be written as:

Figure 6 :

 6 Figure 6: Effect of the heat treatment on the Kitagawa diagram on the 22MnB5

  a-b B. ABRIVARD, a E. PESSARD, a F. MOREL, b P. DELHAYE, a B. GERIN

a LAMPA, Arts et Métiers ParisTech, Angers 49035, France b Renault SAS, Le Mans 72086, France

Table 1 :

 1 Chemical composition of the 22MnB5

	Element	C	Mn	Si	P	S	Al	Cr	B	Ti N Mo
	Weight (%)	0.206 1.191 0.264 0,019 0,007 0.02 0.199 0,002 0.02 -0,003

Table 2 :

 2 Mechanical Properties of the 22MnB5 steel

		Ultimate Tensile stress (MPa)	Yield stress (MPa)	Tensile elongation (%)	Superficial hardness HV20
	initial state	480	320 -420	20	180
	After heat treatment	1300 -1650	1000 -1250	4,5	530

Table 3 :

 3 Fatigue results obtained on the 22MnB5 using staircase method and Self Heating Method (SH test)

	Uniaxial tensile tests	Shear Test