Didier Bresch
email: didier.bresch@univ-savoie.fr

Pascal Noble
email: noble@math.univ-lyon1.fr

Mathematical derivation of viscous shallow-water equations with zero surface tension

Keywords: Navier-Stokes, shallow water, thin domain, free surface, asymptotic analysis, Sobolev spaces AMS subject classification: 35Q30, 35R35, 76A20, 76B45, 76D08

The purpose of this paper is to derive rigorously the so called viscous shallow water equations given for instance page 958-959 in [A. Oron, S.H

Introduction

This paper deals with a free boundary problem of nonstationary Navier-Stokes equations in thin domain justifying the derivation of the viscous shallow water equations without surface tension. For information, such a system may be used to describe long-scale evolution of thin liquid films : See Equations (4.12a)-(4.12d) written in [START_REF] Oron | Long-scale evolution of thin liquid films[END_REF]. Depending on the geometry, there are two kinds of free boundary problems for the motion of a viscous fluid. The first one describes the motion of an isolated mass of fluid bounded by a free boundary and the second one describes the motion of a fluid occupying a semi-infinite domain in R n bounded above by a free surface and below by a fixed part of the boundary. We will consider in the present paper the second problem and refer the interested reader to [START_REF] Oron | Long-scale evolution of thin liquid films[END_REF], [START_REF] Gerbeau | Derivation of shallow water equations and numerical validation DCDS[END_REF] for applications for thin liquid films and rivers. First mathematical results for free boundary problems were local existence theorems: A first local existence theorem published in 1977 by V.A. Solonnikov, in [START_REF] Solonnikov | Solvability of a problem on the motion of a viscous incompressible fluid bounded by a free surface[END_REF] ; a local existence theorem for the second problem by J.T. Beale, in 1980, in [START_REF] Beale | The Initial Value Problem for the Navier-Stokes Equations with a Free Surface[END_REF]. The following years brought other local existence theorems for equations of motion of incompressible fluids: see for instance [START_REF] Allain | Un problème de Navier-Stokes avec surface libre et tension superficielle[END_REF], [START_REF] Allain | Small time existence for the Navier-Stokes equations with a free surface[END_REF], [START_REF] Tani | Small time existence for the three dimensional incompressible Navier-Stokes equations with free surface[END_REF], [START_REF] Teramoto | The initial value problem for a viscous incompressible flow down an inclined plane[END_REF], [START_REF] Teramoto | On the Navier-Stokes flow down an inclined plane[END_REF]. The next step in investigating free boundary problems for incompressible Navier-Stokes equations was to obtain global existence theorems initiated by J.T. Beale in his paper [START_REF] Beale | Large Time Regularity of Viscous Surface Waves[END_REF] published in 1984. This paper was devoted to the motion of a fluid contained in a three dimensional infinite ocean. The first global existence result concerning the motion of a fixed mass of a fluid bounded by a free surface appeared, in the paper [START_REF] Solonnikov | On an unsteady flow of a finite mass of a liquid bounded by a free surface[END_REF], by V.A. [START_REF] Solonnikov | On an unsteady flow of a finite mass of a liquid bounded by a free surface[END_REF]. The method used to prove the global existence results in the two papers mentioned above are completely different. J.T. Beale examined the surface waves after transformating it to the equilibrium domain through an adequate change of variable (dilatation in the vertical variable) while V.A. Solonnikov applied Lagrangian coordinates and this way transformed the considered drop problem to the initial domain. Other global existence theorem can be found. All global existence results for incompressible fluids are obtained for initial data sufficiently close to an equilibrium solution. Note that A. Tani, in [START_REF] Tani | Small time existence for the three dimensional incompressible Navier-Stokes equations with free surface[END_REF] [START_REF] Tani | Large time existence of surface waves in incompressible viscous fluids with or without surface tension[END_REF], used the strategy developped by V.A. Solonnikov for free surface problem to get a local existence result through the use of Lagrangian coordinates to fix the domain and the method of successive approximations. In con-trast to J.T. Beale, the regularity of the solution obtained by A. Tani is sharp. Note that in J.T. Beale existence of arbitrary T > 0 but for sufficiently small initial data in dependence on T is also proved. The result is obtained in the same class of functions than the local existence. To the author's knowledge, in all these studies Dirichlet boundary conditions at the bottom are considered instead of Navier-Slip boundary conditions. Only recent papers with some rigid-lid assumptions consider Navier-Stokes equations in thin domains with Navier-Slip conditions, see for instance [START_REF] Chueshov | Interface boundary value problem for the Navier-Stokes equations in thin two-layer domains[END_REF], [START_REF] Iftimie | Navier-Stokes equations in thin 3D domains with Navier boundary conditions[END_REF]. In this paper, we will adopt the Lagrangian transform strategy in order to prove existence and stability of the viscous shallow water solution. To do that, we have to prove the well-posedness in thin domains with a special attention to constants in classical Sobolev inequalities and regularity in Stokes problem. We also prove the stability result in the Lagrangian form as in the paper written by D. Hoff, see [START_REF] Hoff | Uniqueness of weak solutions of the Navier-Stokes equations of multidimensional, compressible flow[END_REF], where uniqueness of weak solutions of the Navier-Stokes equations of multidimensional, compressible flow is proved using the Lagrangian formulation. We remark also that, using the Lagrangian formulation, no surface tension term is needed compared to [START_REF] Bresch | Mathematical justification of a shallow water model[END_REF]. Remark that there exists, at least, two different shallow-water type systems mainly derived from two different bottom boundary conditions choices for Navier-Stokes equations with free surface: friction boundary conditions, no-slip boundary conditions. The system based on friction boundary conditions assumption is studied in the present paper and was mentionned in [START_REF] Oron | Long-scale evolution of thin liquid films[END_REF]. The system based on no-slip assumption has been formally derived in [START_REF] Boutounet | Shallow water flows for arbitrary topography[END_REF] and recently justified in [START_REF] Bresch | Mathematical justification of a shallow water model[END_REF] assuming non zero surface tension. Reader interested by shallow-water equations and related topics introduction is refered to the recent Handbook [START_REF] Bresch | Shallow-Water Equations and Related Topics[END_REF].

The paper will be divided in four Sections. The first one presents the Navier-Stokes equations with free surface and the viscous shallow-water approximation. The second section is devoted to the linearized Navier-Stokes equations around the approximate solution built from a solution of the viscous shallow-water equations. We give anisotropic estimates and Korn's inequality in thin domains and then prove the well posedness of the linearized Navier-Stokes equations. The main theorem and its corollary are proved in Section 4 that means well posedness and convergence of the full Navier-Stokes equations in Lagrangian form. On the last section, we draw some conclusions on this paper and present some perspectives to this work.

2 Navier-Stokes equations with free surface and viscous shallow-water approximation

In this section, we write 3d Navier-Stokes equations for a thin layer of an incompressible Newtonian fluid with a free surface. We then show how to obtain an approximate solution of these equations in the shallow water scaling: for that purpose, we will use smooth solution (see W. Wang and C.J. Xu [START_REF] Wang | The Cauchy problem for viscous shallow water equations[END_REF]) of a viscous shallow water model ( 6) derived for instance in A. Oron, S.H. Davis and S.G. Bankoff [START_REF] Oron | Long-scale evolution of thin liquid films[END_REF] (see also J.-F. Gerbeau and B. Perthame [START_REF] Gerbeau | Derivation of shallow water equations and numerical validation DCDS[END_REF]). We construct formally a second order approximate solution (with respect to the aspect ratio ε which measures the relative shallowness of the fluid layer) which will help to solve the Cauchy problem for Navier-Stokes equations with free surface and provides also convergence to viscous shallow-water equations. For this purpose, we formulate the equations in Lagrangian coordinates.

The equations and an existence theorem for the associated viscous shallow-water system

In this paper, we consider the motion of a thin layer of Newtonian fluid with a constant density, set to 1. The differential operators ∇ x , div x will concern the horizontal coordinates x = (x 1 , x 2 ). In the following, the fluid velocity u is denoted (u H , u V ) where u H ∈ R 2 (resp u V ∈ R) is the horizontal (resp. vertical) fluid velocity. Navier-Stokes equations are then written, in their non dimensional form as

∂ t u + u.∇u + 1 F 2 0 ∇p = - e 3 F 2 0 + 1 Re div D(u) , div u = 0, (1)
for all t > 0 and (x, z) ∈ Ω t the fluid domain

Ω t = {(x, z) / x ∈ X, z ∈ (0, h(x, t))} , X = T n , R n , n = 1, 2.
The nondimensional numbers F 0 , Re are respectively the Froude and Reynolds numbers defined as

F 2 0 = U 2 g L , Re = L U ν ,
with U a characteristic velocity of the fluid and L a characteristic wavelength. Navier-Stokes equations are completed by boundary conditions. The impermeability condition at the free surface reads

∂ t h + u H | z=h .∇ x h = u V | z=h , (2)
which means that the fluid velocity is parallel to the free surface. We also assume a Navier slip condition at the bottom

u V | z=0 = 0, ∂ z u H | z=0 = γu H | z=0 , (3)
and at the free surface, we assume the continuity of the fluid stress

1 Re D(u) - p F 2 0 Id | z=h n = - p atm F 2 0 n. (4)
Here D(u) = ∇u + ∇u T is the total deformation tensor of the fluid and can be written as

D(u) = D x (u H ) ∂ z u H + ∇ x u V (∂ z u H + ∇ x u V ) T 2∂ z u V , D x (u H ) = ∇ x u H +(∇ x u H ) T .
The vector n is the normal to the free surface:

1 + |∇ x h| 2 n = -∇ x h 1 .
Since the fluid pressure is defined up to a constant, we suppose that p atm = 0. Note that equation ( 4) can be equivalently written as

p| z=h F 2 0 = 2∂ z u V -(∇ x h) T D x (u H )(∇ x h) | z=h Re 1 -|∇ x h| 2 , (5)
∂ z u H + ∇ x u V | z=h = D x (u H ) - 2∂ z u V -(∇ x h) T D x (u H )∇ x h 1 -|∇ x h| 2 | z=h ∇ x h.
In this paper, we will assume that the fluid domain is relatively thin and set h = εh , F 2 0 = εF 2 with ε ≪ 1. As a result, the Froude number F = U √ gH with H = εL, the characteristic fluid height, is the physical Froude number as the ratio between fluid velocity and wavespeed of perturbations at the free surface. Moreover, following the method of derivation by Gerbeau and Perthame [START_REF] Gerbeau | Derivation of shallow water equations and numerical validation DCDS[END_REF] to obtain viscous shallow water equations, we suppose that the friction at the bottom is small: γ = εγ. Let us define an approximate solution of Navier-Stokes system [START_REF] Allain | Un problème de Navier-Stokes avec surface libre et tension superficielle[END_REF][START_REF] Allain | Small time existence for the Navier-Stokes equations with a free surface[END_REF][START_REF] Beale | The Initial Value Problem for the Navier-Stokes Equations with a Free Surface[END_REF][START_REF] Beale | Large Time Regularity of Viscous Surface Waves[END_REF]. For that purpose, we introduce (h 0 , u 0 ) a smooth solution of the shallow water system mentionned by A. Oron, S.H. Davis, S.G. Bankoff [START_REF] Oron | Long-scale evolution of thin liquid films[END_REF], see page 958-959 with C → ∞:

∂ t h 0 + div x (h 0 u 0 ) = 0, h 0 ∂ t u 0 + u 0 .∇ x u 0 ) + ∇ x h 2 0 2F 2 = 1 Re div x h 0 D x (u 0 ) (6) + 2 Re ∇ x h 0 div x (u 0 ) - γu 0 Re .
For the moment, we carry out a formal computation: we will make more precise assumptions on this solution in the sequel. We present a second order apporximate solutions, that means we search for an approximate solution of [START_REF] Allain | Un problème de Navier-Stokes avec surface libre et tension superficielle[END_REF][START_REF] Allain | Small time existence for the Navier-Stokes equations with a free surface[END_REF][START_REF] Beale | The Initial Value Problem for the Navier-Stokes Equations with a Free Surface[END_REF][START_REF] Beale | Large Time Regularity of Viscous Surface Waves[END_REF] in the form h a = εh 0 and

u a,H = u 0 +u 1 z+u 2 z 2 2 , u a,V = w 0 +w 1 z+w 2 z 2 2 +w 3 z 3 6 , p a = p 0 +p 1 z+p 2 z 2 2 .
In the main Theorem we need higher order approximate solution that can be obtained in the same way, see [START_REF] Bresch | Mathematical justification of a shallow water model[END_REF] for construction of high order approximation of Navier Stokes systems with a free surface. Inserting this ansatz in the divergence free condition, one has

w 1 = -div x u 0 , w 2 = -div x u 1 , w 3 = -div x u 2 .
Moreover, the boundary conditions at the bottom imposes

w 0 = 0, u 1 = εγu 0 .
Next, using the second equation of (1) and boundary condition [START_REF] Boutounet | Shallow water flows for arbitrary topography[END_REF], one obtains the approximation of p as

p a = εh 0 -z - 2εF 2 Re div x u 0 - 2ε 3 γF 2 Re div x u 0 .
Next, we insert u a and p a into the second equation of (5): identifying order O(ε) terms, one proves that

u 2 + ∇ x w 1 = D x (u 0 ) + 2div x (u 0 )Id ∇ x h 0 h 0 - γu 0 h 0 .
Remark. For the approximate solution, we can calculate an ansatz up to any order on derivatives of (h 0 , u 0 ) and polynomial dependence with respect to z. We refer to [START_REF] Bresch | Mathematical justification of a shallow water model[END_REF] for more details. We end up at order which gives a remaining term of order (z 3 ) for interior equations and (ε 3 ) at free surface.

A Navier-Stokes type system is satisfied by the high order shallow water approximation (u a , p a ). The adequate ansatz satisfies

∂ t u a + u a .∇u a + 1 εF 2 ∇p a = - e 3 εF 2 + 1 Re div D(u a ) + O(z 3 ), div u a = O(z 3 ). (7)
Moreover, the boundary conditions at the bottom are exactly satisfied and the boundary conditions at the free surface read

∂ t h a + u a,H | z=ha ∇ H h a -u a,V | z=ha = O(ε 3 ), D(u a ) Re - p a εF 2 | z=ha n a = O(ε 3
). For the sake of completeness, we recall the existence and uniqueness result related to the viscous shallow water system (6) that can be obtained using the same procedure than in [START_REF] Wang | The Cauchy problem for viscous shallow water equations[END_REF]

Theorem 1 Let s > 0, u 0 (0), h 0 (0)-1 ∈ H s+2 (R 2 ), h 0 (t = 0)-1 H s+2 (R 2 ) ≪ 1.
Then there exists a positive time T , a unique solution (u 0 , h 0 ) of ( 6) such that

u 0 , h 0 -1 ∈ L ∞ (0, T ; H 2+s (R 2 )), ∇u 0 ∈ L 2 (0, T ; H 2+s (R 2 )).
Furthermore, there exists a constant c such that if

h 0 (t = 0) -1 H s+2 (R 2 ) + u 0 (t = 0) H s+2 (R 2
) ≤ c then we can choose T = +∞ with the same control on (h 0 , u 0 ).

Remark: the smallness assumption on h 0 (t = 0) -1 can be replaced by h 0 (t, x) > α > 0 to prove local well posedness provided that we work in Besov spaces: see the paper of R. Danchin [START_REF] Danchin | Well-Posedness in Critical Spaces for Barotropic Viscous Fluids with Truly Not Constant Density[END_REF] for more details. However, we will need here this particular assumption to prove the well posedness of free surface Navier-Stokes equations in dimension 3 and convergence to shallow water equations. In dimension 2 for free surface Navier-Stokes equations, we just need an assumption similar to the one of [START_REF] Danchin | Well-Posedness in Critical Spaces for Barotropic Viscous Fluids with Truly Not Constant Density[END_REF], namely that there is no vacuum. Of course the results are only local in time: for arbitrary time interval, we need a smallness assumption on h 0 (t = 0) -1 and u 0 (t = 0).

For global existence of weak solutions to a related viscous shallow water system, a result has been proved in [START_REF] Bresch | Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasigeostrophic model[END_REF] assuming an extra turbulent drag term.

Remark: In this paper, we will choose s > 4. Indeed, a careful analysis of the remainder shows that the contained fourth order derivatives of (h 0 , u 0 ) have to be Lipschitz.

Reduction of Navier-Stokes equations to a fixed domain

In what follows, we write the Navier-Stokes equations in a fixed domain: this is done using a Lagrangian formulation of the equations. We first search the fluid velocity and pressure in the form

u = u a + u, p = p a + εF 2 Re p
Navier-Stokes equations (1) then reads

∂ t u + (u a + u).∇ u + u.∇ u a + ∇ p Re = 1 Re div D( u) + f a , div u = 0. ( 8
)
in the fluid domain Ω ε,t = {(x, z)/ x ∈ X, 0 ≤ z ≤ εh(x, t)}. The boundary conditions are then written as

u V | z=0 = 0, ∂ z u H | z=0 = εγ u H | z=0 . (9)
At the free surface, one has

D( u) -p z=εh n ε = g a , ∂ t (εh) + (u a,H + u H )| z=εh .∇ x (εh) = u a,V + u V | z=εh . (10)
Next, we introduce the Lagrangian change of variable

dX dt = u a,H (t, X, Z) + u H (t, X, Z), dZ dt = u a,V (t, X, Z) + u V (t, X, Z).
For the sake of simplicity, we assume that the initial fluid domain is X×(0, ε): the rigorous justification will remain true for | < h 0 (t = 0) > -1| ≪ 1 and γ sufficienty large. Denote x 0 ∈ X, z 0 ∈ (0, ε), the coordinates in this domain and A the Jacobian matrix of the Lagrangian change of variable

A =    ∂X ∂x 0 ∂X ∂z 0 (∇ x 0 Z) T ∂Z ∂z 0    .
The chain rules are given by

∇ x 0 ∂ z 0 = A T ∇ x ∂ z , ∇ x 0 u ∂ z 0 u ∇ x 0 w T ∂ z 0 w = ∇ x u ∂ z u ∇ x w T ∂ z w A.
The fluid height is defined implicitely as εh t, X(t, x 0 , ε) = Z(t, x 0 , ε) so that the impermeability condition is satisfied. Moreover, one has

ε∇ x h(t, X(t, , x 0 , ε)) = ( ∂X ∂x 0 ) -T ∇ x 0 Z(t, x 0 , ε).
The Lagrangian velocity is defined as u = u(t, X) and u defined on the fixed thin domain X × (0, ε). We further introduce the Lagrangian pressure p = p(t, X). In that setting, Navier-Stokes equations are written as

∂ t u + (A -1 u).∇u a + A -T ∇p Re = A -T Re div A T P -∇A : P + f a , (11) div(A -1 u) = σ a , (12)
where f a = f a (t, X), u a = u a (t, X), (∇A :

P) i = j,k ∂ k A j,i P j,k
. The matrix P is the transformed deformation tensor:

P = (∇u)A -1 A -T + A -T (∇u) T A -T .
The boundary conditions at the free surface z 0 = ε reads

P| z 0 =ε -p| z 0 =ε Id n = 0, n = -( ∂X ∂x 0 ) -T ∇ x 0 Z(t, x 0 , ε) 1 , (13)
whereas the boundary conditions at the bottom z = 0 are written as

u V | z 0 = 0, det( ∂X ∂x 0 )|∂ z 0 u H | z 0 =0 = εγ u H | z 0 =0 . (14)
The main result of the paper is Theorem 2 There exists C, K > 0 so that if we assume u 0 2 ≤ C 2 ε 3/2 , then there is a unique strong solution of ( 11)-( 14) such that sup (0,T )

u 2 2 + T 0 u 2 3 + ∂ t u 1 + p 2 2 + ∂ t p 3 0 < Kε 3/2 .
This gives the expected existence result of Navier-Stokes equations with free surface namely the system (1), ( 2), (3), [START_REF] Boutounet | Shallow water flows for arbitrary topography[END_REF] and its convergence to the solution of the shallow-water system in a Lagrangian form namely a control of the solution of ( 8)- [START_REF] Coutand | Unique solvability of the free-boundary Navier-Stokes equations with surface tension[END_REF] with respect to ε.

Corollary 1 Let (h 0 , u 0 ) be the solution of the viscous shallow-water system [START_REF] Bresch | Shallow-Water Equations and Related Topics[END_REF]

such that (h 0 -1, u 0 ) small enough in C(0, T ; H 2+s (R 2 ) with s > 4. Let (h ǫ 0 , u ǫ 0 ) such that (u ǫ 0 -u a )| t=0 2 ≤ Cε 3/2
and h ε 0 = h 0 and u ǫ 0 satisfying the compatibility conditions. Then System (1), ( 2), (3), ( 5) is well posed and u ǫ -u a satisfy in its Lagrangian form the estimates given by Theorem 2.

Note that there is no assumption on the time interval: the solution of Navier-Stokes system is defined on the same existence time as the solution of the viscous shallow-water system [START_REF] Bresch | Shallow-Water Equations and Related Topics[END_REF]. The associated linear system. Next we introduce a linear problem that will be usefull to prove the well-posedness of the full Navier-Stokes system. Indeed, this will be done using a fixed point argument. First, let us introduce the following Lagrangian system of coordinates

dX 0 dt = u 0 (X 0 ), dZ 0 dt = -Z 0 div u 0 (X 0 ),
with the initial conditions X 0 (0, .) = x 0 , Z 0 (0, .) = z 0 . It is easily proved that X 0 , Z 0 satisfy

∂X 0 ∂z 0 = 0, Z 0 (t, x 0 , z 0 ) = z 0 h 0 (t, X 0 (t, x 0 )), det ∂X 0 ∂x 0 = h -1 0 (t, X 0 (t, x 0 )).
In what follows, we will consider a special linear problem that is obtained if we substitute X = X 0 and Z = Z 0 in Navier-Stokes equations written in Lagrangian coordinates and drop order one differential operators. More precisely, this system is written as

∂ t u + A -T 0 ∇p Re = A -T 0 Re div(A T 0 P 0 ) + f, (15)
with

P 0 = ∇(u)A -1 0 A -T 0 + A -T 0 (∇u) T A -T 0 . The divergence condition reads div(A -1 0 u) = σ. ( 16
)
The boundary conditions at the bottom are

u V | z 0 =0 = 0, h -1 0 ∂ z 0 u H | z 0 =0 = εγ u H | z 0 =0 + g 1 , (17)
whereas the boundary conditions at the free surface read

A T 0 P 0 -p Id | z 0 =ε e 3 = g 2 . ( 18
)
3 Linearized Navier-Stokes around the approximate solution of viscous shallow-water equations

This section is devoted to study the linearized Navier-Stokes equations around the approximate solution of viscous shallow-water [START_REF] Merkin | An infinite period bifurcation arising in roll waves down an open inclined channel[END_REF][START_REF] Nishida | Navier-Stokes flows down an inclined plane: downward periodic motion[END_REF][START_REF] Nishida | Hopf bifurcation in viscous incompressible flow down an inclined plane[END_REF][START_REF] Oron | Long-scale evolution of thin liquid films[END_REF] that was introduced in the previous section. We begin by a subsection dealing with Sobolev inequalities, Korn inequality and regularity result for Stokes problem in thin domain. In the second subsection, we give well posedness result for the full linearized system.

Anisotropic estimates in thin domains

In this part, we recall Sobolev inequalities that arises in thin domains and pay a special attention to the constants arising in such inequalities. We also consider a linear Stokes problem in a thin domain that will be useful to obtain estimates on the Sobolev norms of (u, p).

Sobolev inequalities

In what follows, we use that mean of u 1 in the vertical direction is 0 to write Poincaré, Agmon and Ladyzhenskaia anisotropic inequalities in thin domains. Here the domain Ω is Ω = X n × (0, ε). In this paper, we will use the following notations:

u k = u H k (Ω) , for any k ∈ N with the convention H 0 (Ω) = L 2 (Ω). Similarly, one denotes |u| k = |u| H k (X) .
Proposition 1 There exists a constant C independent of ε such that the following Ladyzhenskaia and Agmon estimates hold true:

u L 6 ≤ Cε -1 3 u 1 , u L ∞ ≤ Cε -1 2 u 2 .
The proof of this proposition is found in [START_REF] Temam | Navier-Stokes equations in three-dimensional thin domains with various boundary conditions[END_REF]. From this estimate and the inequality uv

L 2 ≤ ε 1 6 u L 6 v L 6
, one easily proves the following tame estimates for the products of functions and composition of functions.

Lemma 1 We will use the following estimates

: if t > n 2 |u v| H s (X) ≤ C |u| H t (X) |v| H s (X) + |u| H s (X) |v| H t (X) , u v H s (Ω) ≤ C √ ε u H t (Ω) v H s (Ω) + u H s (Ω) v H t (Ω) ,
We have also the following property: For all F : R p → R smooth enough with

F (0) = 0 then F (u) s ≤ C( u L ∞ ) u s .
We will also need the following trace theorems:

Lemma 2 Let u ∈ H 1 is so that u| z=0 = 0 then one has |u| 1 2 ≤ C u 1 , with C independent of ε. In the general case, one has |u| 1 2 ≤ C √ ε u 1 .
This result is easily proved with Fourier series, that reduces the problem to a 1d problem. The trace of a product is estimated using the following result.

Lemma 3 Let u ∈ H -1 2 and v ∈ H 2 then, one has |uv| -1 2 ≤ C √ ε |u| -1

Korn inequality in thin domain

In what follows, we prove the following result.

Proposition 2 There is a constant C > 0 independent of ε such that

2 D(u) 2 L 2 (Ω) + εγ|u H | z=0 | 2 L 2 (X) ≥ C u 2 H 1 (Ω) , for all u ∈ H 1 (Ω) so that divu = 0 and u V | z=0 = 0.
Proof. This is a consequence of Poincaré inequality and trace estimates that the L 2 norm of u H and u V are estimated as

u V L 2 (Ω) ≤ Cε ∂ z u V L 2 (Ω) , u H L 2 (Ω) ≤ C ε|u H | z=0 | L 2 (X) +ε ∂ z u H L 2 (Ω) .
We will prove a refined inequality on the deformation tensor when

X = T n 2 D(u) 2 L 2 (Ω) ≥ C ∇u 2 L 2 (Ω) .
For that purpose, we decompose the functions into Fourier series in the horizontal variables, (u

H , u V ) = k∈Z 2 (u k (z), w k (z)) exp(ik.x).
Note that a similar result is obtained when X = R n , using Fourier transform. We find, using the divergence free condition

2 D(u) 2 L 2 (Ω) = 2 k∈Z 2 ε 0 k 2 1 |u k,1 | 2 + k 2 2 |u k,2 | 2 + |k.u k | 2 + k∈Z 2 ε 0 |k 2 u k,1 + k 1 u k,2 | 2 + |u ′ k,1 + z 0 k 1 k.u k | 2 + |u ′ k,2 + k 2 z 0 k.u k |, ∇u 2 L 2 (Ω) = k∈Z 2 ε 0 |k| 2 |u k | 2 + |k.u k | 2 + | z 0 k.u k | 2 + |u ′ k | 2 .
where u

′ (z) = du dz . Setting U k = z 0 u k , there exists C k ∈ (0, 2) so that ε 0 2k 2 1 |U ′ 1 | 2 + 2k 2 2 |U ′ 2 | 2 + 2|k.U ′ | + |k 2 U ′ 1 + k 1 U ′ 2 | 2 + |U ′′ 1 + k 1 k.U| 2 + ε 0 |U ′′ 2 + k 2 k.U| 2 ≥ C k ε 0 |k| 2 |U ′ | 2 + |k.U ′ | 2 + |k.U| 2 + |U ′′ | 2 , ( 19
)
for all U ∈ H 1 (0, ε) 2 so that U| z=0 = 0. We prove the inequality for each modes. We must prove that there is C > 0 independent of k so that C k ≥ C, forall k ∈ Z 2 . If k = 0, the two quantities are equal and one can choose

C 0 = 1.
In what follows, we assume that k = 0. Denote C k ∈ (0, 2) the optimal value for the mode k. Using test function in D(0, ε), one can prove that, if C k = 1, an optimal solution necessarily satisfies the differential system

D 2 U 1 +k 2 2 DU 1 = k 1 k 2 DU 2 , D 2 U 2 +k 2 1 DU 2 = k 1 k 2 DU 1 , ∀z ∈ (0, ε). ( 20
)
where D is the differential operator D = d 2 dz 2 -|k| 2 Id. This space of solutions is 8-dimensional and reduced to a 6-dimensional one, using the condition U| z=0 = 0. Moreover, the eigenfunction U can be written U(z) = Ũ (|k|z), where Ũ ∈ S, the vector space defined as

Ũ (z) = -c s a 1 z + b 1 sinh(z) + c 1 (cosh(z) -1) + s c a 2 sinh(z) + b 2 z sinh(z) + c 2 z cosh(z) . ( 21
)
where we have denoted k as k = |k|(c, s) and a i , b i , c i ∈ R, i = 1, 2. Then we have to show that there is

C independent of k, ε so that Q 2 ( Ũ , k) ≥ C Q 1 ( Ũ, k)
, for all Ũ ∈ S and Q i the quadratic forms defined as

Q 2 (U, k) = M 0 2c 2 (U ′ 1 ) 2 + 2s 2 (U ′ 2 ) 2 + 2(cU ′ 1 + sU ′ 2 ) 2 + M 0 (sU ′ 1 + cU ′ 2 ) 2 + U ′′ 1 + c(cU 1 + sU 2 ) 2 + U ′′ 2 + s(cU 1 + sU 2 ) 2 , Q 1 (U, k) = M 0 |U ′ | 2 + (cU ′ 1 + sU ′ 2 ) 2 + |U ′′ | 2 + (cU 1 + sU 2 ) 2 ,
with M = ε|k|. Let us denote q i (k), i = 1, 2 the symmetric matrix associated to the quadratic forms

Q i (., k), i = 1, 2. If C k is the minimum value for which the inequality Q 2 ( Ũ , k) ≥ C k Q 1 ( Ũ , k
) for all Ũ is satisfied, then det(q 2 (k) -C k q 1 (k)) = 0 and one is left with an eigenvalue problem. Let us study the roots of P (X) = det(q 2 (k) -X q 1 (k)). First, an integration by parts in the quadratic form

Q 2 (., k) yields Q 2 (U, k) = Q2 (U, k) + Q 1 (U, k) Q2 (U, k) = 2 cU 1 + sU 2 (M) cU ′ 1 + sU ′ 2 (M). The quadratic form Q2 (., k) is a product of two linear forms Q2 (U, k) = (v 1 (k), a)(v 2 (k), a) with (v 1 (k), v 2 (k)) ∈ R 6
and a ∈ R 6 represents the coordinates of U in the basis of S. Then the matrix q2 (k) associated to Q2 (., k) is given by q2 (k

)a = (v 1 (k), a)v 2 (k) + (v 2 (k), a)v 1 (k). As a consequence, for any a ∈ (v 1 (k), v 2 (k)) ⊥ , one has q 2 (k)a = q 1 (k)a. One can prove that v 1 (k), v 2 (k)
are not colinear so that X = 1 is a root of P with multiplicity 4. There remains two extremal values which correspond to minimal and maximal value of C k . We know that C k ∈ (0, 2): let us show that X = 2 is exactly the maximum value for C k . It is sufficient to show that there is a divergence free vector fields (u, w) so that J(u, w) = J(u, w) T and w(0) = 0, J(u, w) being the Jacobian matrix of u, w. This is done choosing a potential flow (u, w) = (∇ψ, ∂ z ψ) so that ψ z (0) = 0 and ∆ψ = 0. Split the functions into their Fourier modes, one shows that it suffices to choose ψ k (z) = α k cosh(|k|z) with α ∈ l 2 (Z 2 ) chosen so that the H 1 regularity of u, w is satisfied. As a result, the polynom P is factorized in the form

P (X) = Γ k (X -1) 4 (X -2)(X -Λ(k))
and there is a unique minimal value Λ(k) which depends continuously of M = ε|k| and σ = c + is ∈ S 1 . We analyse Λ(k) when M ∼ ε and M → ∞.

We have calculated with a formal computation software the expansion of P with respect to M at M = 0:

P (X) = M 4 8 9 (Z -1) 5 (Z -2) + O(M 6 ),
so that Λ(k) ∼ 1 as ε → 0 and k lies in a fixed compact set. Next we deal with the limit M → ∞: Assume that there is U M so that MQ 1 (U M , k) = 1 and MQ 2 (U M , k) → 0. This is equivalent to the existence of ŨM so that

1 0 M 2 (c Ũ1,M + s Ũ2,M ) 2 + | Ũ′′ M | 2 M 2 + | Ũ′ M | 2 + (c Ũ′ 1,M + s Ũ′ 2,M ) 2 = 1, lim M →∞ c Ũ1,M (1) + s Ũ2,M (1) c Ũ′ 1,M + s Ũ′ 2,M M = -1.
Denote f M = c Ũ1,M + s Ũ2,M : it is easily proved that f M (1) and f ′ M (1) are estimated as

f M (1) 2 = 2 1 0 f M (z)f ′ M (z)dz ≤ 2 M , 1 M 2 f ′ M (1) 2 ≤ 1 M 2 1 0 (f ′ M (z)) 2 dz + 2 M 2 1 0 1 z |f ′ M (s)f ′′ M (s)|dsdz ≤ 3 M .
This yields the contradiction with lim M →∞ M -1 f M (1)f ′ M (1) = -1 and concludes the proof of the Korn's inequality. 2

Well posedness of linearized Navier-Stokes equations 3.2.1 A special divergence free linear problem

In this section, we consider the special linear problem obtained from [START_REF] Merkin | An infinite period bifurcation arising in roll waves down an open inclined channel[END_REF][START_REF] Nishida | Navier-Stokes flows down an inclined plane: downward periodic motion[END_REF][START_REF] Nishida | Hopf bifurcation in viscous incompressible flow down an inclined plane[END_REF][START_REF] Oron | Long-scale evolution of thin liquid films[END_REF], assuming h 0 = 1, u 0 = 0: the more general case |h 0 -1| ≪ 1 and |u 0 | ≪ 1 will be treated in the next section through a perturbation argument. In the case considered here, A 0 = Id and the linear problem is

∂ t u + ∇p Re = 1 Re div D(u) + f, divu = 0, (22)
with the boundary conditions at the bottom

u V | z 0 =0 = 0, ∂ z 0 u H | z 0 =0 = εγ u H | z 0 =0 + g 1 , (23)
and at the free surface

(∂ z 0 u H + ∇ x 0 u V )| z 0 =ε = g 2,1 , (2∂ z 0 u V -p)| z 0 =ε = g 2,2 . (24)
We prove the existence of a weak solution. The weak formulation of [START_REF] Tani | Small time existence for the three dimensional incompressible Navier-Stokes equations with free surface[END_REF][START_REF] Tani | Large time existence of surface waves in incompressible viscous fluids with or without surface tension[END_REF][START_REF] Temam | Navier-Stokes equations in three-dimensional thin domains with various boundary conditions[END_REF] is written as

∂ t Ω (u, φ) + 1 2 Re Ω D(u) : D(φ) + εγ Re X (u H | z 0 =0 , φ H | z 0 =0 ) = Ω (f, φ) + 1 Re X (g 2 , φ| z 0 =ε ) -(g 1 , φ H | z 0 =0 ), (25)
for all φ ∈ (H 1 (Ω)) 3 so that divφ = 0 and φ V | z 0 =0 = 0. In what follows, we prove the existence (and uniqueness) of a solution of [START_REF] Teramoto | The initial value problem for a viscous incompressible flow down an inclined plane[END_REF].

Proposition 3 Let f ∈ L 2 (0, T ), (H 1 (Ω) ′ ) and g ∈ L 2 (0, T ), H -1 2 (T) . For any u 0 ∈ L 2 div (Ω), there is a unique weak solution u ∈ L 2 (0, T ), H 1 (Ω) of ( 25) and ∂ t u ∈ L 2 (0, T ), (H 1 (Ω)) ′ . Moreover, it satisfies the energy estimates

max (0, t) u 2 0 + C t 0 u 2 1 ≤ u 0 2 0 + C t 0 f 2 (H 1 (Ω)) ′ + |g 2,2 | 2 -1 2 + 1 ε |g 2,1 | 2 -1 2 + |g 1 | 2 -1 2 . ( 26
)
Proof. The proof follows from a classical Galerkin approximation process:

we only prove here the energy estimate. Substituting φ = u in the weak formulation [START_REF] Teramoto | The initial value problem for a viscous incompressible flow down an inclined plane[END_REF], one obtains

∂ u 2 0 ∂t + 1 2 Ω D(u) : D(u) + εγ T |u H | z=0 | 2 0 = f, u (H 1 ) ′ ,H 1 + (g 2 , u| z=ε ) H -1 2 ,H 1 2 -(g 1 , u H | z=0 ) H -1 2 ,H 1 2 . ( 27
)
Using the Korn's inequality on the functional space H 1 div (Ω) 3 so that u V | z=0 = 0, one proves that there exists a constant C > 0 independent of ε so that

u 2 1 ≤ C Ω D(u) : D(u) + εγ T |u H | z=0 | 2 .
Moreover, using lemma 2, one has the trace estimates

|u V | z=ε | 1 2 ≤ C u V 1 , √ ε|u H | 1 2 ≤ C u H 1 .
Using a Young's inequality, one can prove that for any η > 0, there exists c(η) > 0 so that

|(g 2,1 , u H | z=ε ) -(g 1 , u H | z=0 ) H -1 2 ,H 1 2 | ≤ c(η) ε |g 2,1 | 2 -1 2 + |g 1 | 2 -1 2 + η u H 2 1 , |(g 2,2 , u V )| H -1 2 ,H 1 2 | ≤ c(η)|g 2,2 | 2 -1 2 + η u V 2 1 |(f, u)| (H 1 ) ′ ,H 1 ≤ c(η) f 2 (H 1 ) ′ + η u 2 1 .
Inserting this inequality into [START_REF] Wang | The Cauchy problem for viscous shallow water equations[END_REF] gives the estimate

∂ ∂t u 2 0 + C u 2 1 ≤ C f 2 (H 1 ) ′ + |g 2,1 | 2 -1 2 + 1 ε |g 2,1 | 2 + |g 1 | 2 .
Then integrating this equation on the interval (0, t) yields [START_REF] Teramoto | On the Navier-Stokes flow down an inclined plane[END_REF]. From the weak formulation, one easily proves

t 0 ∂ t u 2 (H 1 ) ′ ≤ u 0 2 2 + t 0 f 2 (H 1 ) ′ + |g 2,1 | 2 -1 2 + 1 ε |g 2,1 | 2 -1 2 + |g 1 | 2 -1 2
Using an interpolation argument, one has u ∈ C (0, t), L 2 div and the uniqueness follows, letting f = g i = 0 and u 0 = 0 into [START_REF] Teramoto | On the Navier-Stokes flow down an inclined plane[END_REF]. This completes the proof of the proposition.2 This is a standard argument that there exists a pressure p ∈ L 2 ((0, t), L 2 ) so that Ω p(., t) = 0 and

Ω (u t , φ) + D(u) : D(φ) - Ω p div φ + εγ X u H | z=0 φ H | z=0 = Ω (f, φ) + X (g 2 , φ| z=ε ) -(g 1 , φ H | z=0 ) ∀φ ∈ H 1 , φ V | z=0 = 0. (28)
Moreover, the pressure term p satisfies

t 0 p 2 0 ≤ C u 0 0 + t 0 f 2 (H 1 ) ′ + |g 2,2 | 2 -1 2 + 1 ε |g 2,1 | 2 -1 2 + |g 1 | 2 -1 2 ,
where C is a constant independent of ε. Next, we prove a result on the regularity of the weak solution.

Proposition 4 Let f ∈ L 2 (0, T ), H 1 such that ∂ t f ∈ L 2 (0, T ), (H 1 ) ′ and g i ∈ L 2 (0, T ), H 3 2 , ∂ t g i ∈ L 2 (0, T ), H -1 2
which satisfies the compatibility conditions

(∂ z 0 u H + ∇ x 0 u V ) z 0 =ε (t = 0, .) = g 2,1 (0, .), ∂ z 0 u H | z 0 =0 -εγu H | z 0 =0 (t = 0, .) = g 1 (0, .).
and div u 0 = 0, u 0 ∈ H 2 . Then the weak solution u verifies

max (0, t) u 2 2 + u t 2 0 + t 0 u(s, .) 2 3 + u t (s, .) 2 1 ds ≤ u 0 2 2 + f (0, .) 2 0 + g i (0, .) 2 1 2 + t 0 f 2 1 + f t 2 (H 1 ) ′ + |g 2,2 | 2 3 2 + |g 2,2,t | 2 -1 2 + 1 ε |g i,1 | 2 3 2 + |g i,1,t | 2 -1 2
Proof. Let us first prove that there exists an "initial" pressure p 0 ∈ H 1 (Ω), satisfying the mixed Dirichlet/Neuman problem

-∆p 0 = -div ∆ u 0 + f (0) , p 0 | z 0 =ε = 2∂ z 0 u 0,V | z 0 =ε -g 2,2 (0), ∂ z 0 p 0 | z 0 =0 = εγdiv x 0 u 0,H | z=0 + div x 0 g 1 (0) + f V (0).
This problem is well posed since f (0) ∈ L 2 and g i (0) ∈ H 1 2 with a compactness argument respectively on f ∈ L 2 (0, t; H 1 ) ∩ H 1 ((0, t); (H 1 ) ′ ) and g ∈ L 2 ((0, t); H 3 2 ) ∩ H 1 (0, t; H -1 2 ). We split this problem into three parts: first, it is easily proved using a Lax Milgram argument that there exists a unique solution p 0,1 ∈ H 1 (Ω) so that

-∆p 0,1 = -div ∆u 0 + f (0) , p 0,1 | z 0 =ε = ∂ z p 0,1 | z 0 =0 = 0.
This is a straighforward computation (with Fourier series) to prove that there exists C, independent of ε, such that

p 0,1 1 ≤ C f (0) 0 + u 0 2 .

Next we consider the problem ∆p

0,2 = 0, p 0,2 | z=ε = h, ∂ z p 0,2 | z=0 = 0, with h = 2∂ z u 0,V | z 0 =ε -g 2,2 (0) ∈ H 1 2
. Using Fourier series, this is equivalent to find p k 0,2 so that

- d 2 dz 2 p k 0,2 + |k| 2 p k 0,2 = 0, p k 0,2 | z=ε = h k , d dz p k 0,2 | z=0 = 0.
The solution is easily calculated and satisfies

p k (z) = cosh|k|z cosh(|k|ε) h k , d dz p k 0,2 2 + |k| 2 p k 0,2 ≤ C(|k|ε)|k|h 2 k , with C(y) = 1 cosh 2 y y 0 cosh(2x)dx ≤ C 1 , ∀y ≥ 0 so that p 0,2 1 ≤ C 1 |h| 1 2 .
Similarly, the solution p 0,3 of

∆p 0,3 = 0, p 0,3 | z 0 =ε = 0, ∂ z p 0,3 | z=0 = εγdiv x 0 u 0,H | z 0 =0 +div x 0 g 1 (0)+f V (0)| z 0 =0 , satisfies p 0,3 1 ≤ C 1 u 0 1 + |g 1 (0)| 1 2 + f 2 (0) 0 . Let us define an initial condition u 0 ∈ L 2 (Ω) ( that plays the role of ∂ t u(0)) as u 0 = -∇p 0 + ∆u 0 + f (0).
This satisfies the following estimate

u 0 0 + p 0 1 ≤ C 1 u 0 2 + f (0) 0 + |g 1 (0), g 2 (0)| 1 2 ,
with C 1 independent of ε. We apply the previous proposition with this initial condition that lies in the space of initial conditions of weak solutions {u ∈ L 2 / divu = 0, u V | z 0 =0 = 0} and f, g are replaced by (f t , g t ). There exists a unique weak solution u ∈ L 2 (0, t; H 1 (Ω)) associated to this problem and that satisfies the energy estimate

max (0, t) u 0 + t 0 u 1 ≤ C 1 ( u 0 2 2 + f (0) 2 0 + |(g 1 , g 2 )(0)| 2 1 2 + t 0 f t 2 (H 1 ) ′ + |g 2,2,t | 2 -1 2 + 1 ε |g 1,t | -1 2 + |g 2,1,t | 2 -1 2 . ( 29
)
Associated to this weak solution is the pressure pressure term p ∈ L 2 (which plays the role of p t ) that satisfies the estimate L 2 (0, t; L 2 )

t 0 p 2 L 2 ≤ C 1 ( u 0 2 2 + f (0) 2 0 + |g 1 (0), g 2 (0)| 2 1 2 + t 0 f t 2 (H 1 ) ′ + |g 2,2,t | 2 -1 2 + 1 ε |g 1,t | -1 2 + |g 2,1,t | 2 -1 2 . ( 30
)
Then, let us consider

u 1 = u 0 + t 0 u, p 1 = p 0 + t 0 p.
Integrating on the time intervals (0, t) the weak formulation on u and using the definition of p 0 , one can prove that u 1 is a weak solution of the initial problem with u 1 (t = 0) = u 0 . Then one finds that u = u 1 , p = p 1 and u t = u, p t = p satisfies the estimate (29). This concludes the proof of the proposition.2

Finally, we check the space regularity of the weak solution. One can prove the proposition

Proposition 5 Let f ∈ L 2 (0, T ; H 1 ) ∩ H 1 (0, T ; (H 1 ) ′ ), g i ∈ L 2 (0, T ; H 3 2 ) ∩ H 1 (0, T ; H -1 2 )
and u 0 ∈ H 2 satisfying the compatibility conditions

∂ z 0 u H + ∇ x 0 u V | z 0 =ε (t = 0) = g 2,1 (0), (∂ z 0 u H -εγu H )| z 0 =0 (t = 0) = g 1 (0).
Then there exists a constant C independent of ε such that the weak solution

u ∈ L 2 (0, T ; H 3 (Ω))∩H 1 (0, T ; H 1 (Ω)), p ∈ L 2 (0, T ; H 2 (Ω))∩H 1 (0, T ; L 2 (Ω)) satisfies max (0, T ) u 2 2 + T 0 u 2 3 + p 2 2 + u t 2 1 + p t 2 0 ≤ C 1 u 0 2 2 + f (0) 2 0 + |(g 1 (0), g 2 (0))| 2 1 2 + T 0 f 2 1 + f t 2 (H 1 ) ′ + |g 2,2 | 2 3 2 + |g 2,2,t | 2 -1 2 + 1 ε T 0 |g 1 , g 2,1 | 2 3 2 + |g 1,t , g 2,1,t | 2 -1 2 . (31)

The full linear problem

In this part, we consider the full problem with a divergence term and h 0 = 1. Let us first deal with the case h 0 = 1, u 0 = 0 (steady state) and consider the linear problem [START_REF] Tani | Small time existence for the three dimensional incompressible Navier-Stokes equations with free surface[END_REF][START_REF] Tani | Large time existence of surface waves in incompressible viscous fluids with or without surface tension[END_REF][START_REF] Temam | Navier-Stokes equations in three-dimensional thin domains with various boundary conditions[END_REF] with the divergence free condition replaced by

divu = h, (32)
with h ∈ L 2 (0, T ; H 2 ) ∩ H 1 (0, T ; L 2 ). In order to prove the well posedness of [START_REF] Tani | Small time existence for the three dimensional incompressible Navier-Stokes equations with free surface[END_REF][START_REF] Tani | Large time existence of surface waves in incompressible viscous fluids with or without surface tension[END_REF][START_REF] Temam | Navier-Stokes equations in three-dimensional thin domains with various boundary conditions[END_REF] with the divergence condition (32), we will apply proposition 4: for that purpose, we introduce u h = ∇φ so that ∆φ = h, ∂φ ∂z

| z=0 = ∂φ ∂z | z=ε = 0.
One can prove, with Fourier series, that there exists a constant C > 0 independent of ε such that

φ 4 + φ t 2 ≤ C h 2 + h t 0 .
Next, we consider f , g i such that f = f + ∆u h -u h,t and

g 1 = g 1 + εγu h | z=0 , g 1,2 = g 1,2 -2∂ xz φ| z=ε , g 2,2 = g 2,2 + 2∂ zz φ.
This is a straightfoward computation to prove that there exists a constant C independent of ε such that

f 1 ≤ C f 1 + h 2 + h t 0 , | g 1 , g 1,2 | 3 2 + | g 1,t , g 1,2,t | -1 2 ≤ C |g 1 , g 1,2 | 3 2 + |g 1,t , g 1,2,t | 2 -1 2 + h 2 + h t 0 , | g 2,2 | 3 2 + | g 2,2,t | -1 2 ≤ C |g 2,2 | 3 2 + |g 2,2,t | -1 2 + 1 √ ε h 2 + h t 0 .
The estimate on f t (H 1 ) ′ is obtained using the linear Navier-Stokes equations: more precisely, one has

f t = f t + ∆u h,t -u h,tt and divu h,tt = h tt = divu tt = div f t + 1 Re div D(u t ) - 1 Re ∇p t ) .
Then, one obtains the estimate of u h,tt (and thus f t ) in L 2 ((0, T ), (H 1 ) ′ ) norm, using elliptic regularity. Next, we apply proposition 4 on the divergence free problem for the fluid velocity u -u h and pressure p. Finally, we have proved the well-posedness of the linear problem [START_REF] Tani | Small time existence for the three dimensional incompressible Navier-Stokes equations with free surface[END_REF][START_REF] Tani | Large time existence of surface waves in incompressible viscous fluids with or without surface tension[END_REF][START_REF] Temam | Navier-Stokes equations in three-dimensional thin domains with various boundary conditions[END_REF] with (32) and the solution (u, p) satisfies the energy estimates

max (0, T ) u 2 2 + u t 2 0 + T 0 u 2 3 + u t 2 1 + p 2 2 + p t 2 0 ≤ C u 0 2 + f (0) 2 0 + |(g 1 , g 2 )(0)| 2 1 2 + T 0 f 2 1 + f t 2 (H 1 ) ′ + |g 2,2 | 2 3 2 + |g 2,2,t | 2 -1 2 + 1 ε T 0 h 2 2 + h t 2 0 + |g 1 , g 2,1 | 2 3 2 + |g 1,t , g 2,1,t | 2 -1 2 . (33)
Let us now deal with the full problem [START_REF] Merkin | An infinite period bifurcation arising in roll waves down an open inclined channel[END_REF][START_REF] Nishida | Navier-Stokes flows down an inclined plane: downward periodic motion[END_REF][START_REF] Nishida | Hopf bifurcation in viscous incompressible flow down an inclined plane[END_REF][START_REF] Oron | Long-scale evolution of thin liquid films[END_REF]. For the sake of simplicity, we assume σ = 0, the inhomogeneous case σ = 0 is treated as previously. In order to deal with the more classical divergence free condition, we introduce the Jacobian decomposition of the fluid velocity u = A 0 u. The linear problem then reads (we also drop a linear term in the form L 0 (u, ∇u, x, t))

A T 0 ∂ t A 0 u + 1 Re ∇p = 1 Re div( P 0 ) + f, (34)
with

P 0 = A T 0 A 0 ∇( u)A -1 0 A -T 0 + (∇ u) T . The divergence condition reads div u = 0. ( 35
)
The boundary conditions at the bottom are written as

u V | z 0 =0 = 0, h -1 0 ∂ z 0 u H | z 0 =0 = εγ u H | z 0 =0 + g 1 , (36)
whereas the boundary conditions at the free surface read

P 0 -p Id | z 0 =ε e 3 = g 2 . (37)
Similarly to the special linear problem treated in the previous section, one can obtain a weak formulation of (34-37):

∂ t Ω A 0 u, A 0 φ + 1 Re Ω ∇φ : P 0 + εγ Re X h -1 0 ( ∂X 0 ∂x 0 u H | z 0 =0 , ∂X 0 ∂x 0 φ H | z 0 =0 ) = Ω (A 0,t φ, A 0 u) + (f, φ) + 1 Re X (g 2 , φ H | z 0 =ε ) -h -1 0 ( ∂X 0 ∂x 0 g 1 , ∂X 0 ∂x 0 φ H | z=0 ).
for all φ ∈ H 1 (Ω) so that divφ = 0 and φ V | z 0 =0 . One prove the following theorem Theorem 3 There is η 1 > 0, η 2 > 0 such that for any (h 0 , u 0 ) ∈ W 2,∞ be a smooth solution of ( 6) which first satisfies |h 0 (t = 0) -1| ∞ < η 1 and either

• (small data, large time) ∀t ∈ (0, T ), h 0 (t, .) -1, u 0 (t, .) ∞ ≤ η 2 or, • (large data, small time) T sup t∈(0,T ) h 0 (t, .) -1, u 0 (t, .) W 2,∞ ≤ η 2
Then for all source terms f, g i , i = 1, 2 and h so that

f ∈ L 2 ((0, T ); H 1 (Ω)) ∩ H 1 ((0, T ); (H 1 ) ′ (Ω)), g i ∈ L 2 ((0, T ); H 3 2 (X)) ∩ H 1 ((0, T ); H -1 2 (X)), h ∈ L 2 ((0, T ); H 2 (Ω)) ∩ H 1 ((0, T ); L 2 (Ω)).
There exists a unique weak solution (u, p) of [START_REF] Tani | Small time existence for the three dimensional incompressible Navier-Stokes equations with free surface[END_REF][START_REF] Tani | Large time existence of surface waves in incompressible viscous fluids with or without surface tension[END_REF][START_REF] Temam | Navier-Stokes equations in three-dimensional thin domains with various boundary conditions[END_REF]32) and this solution satisfies energy estimates (33) .

Proof. The proof of this theorem follows the one of proposition 4: one first constructs an approximate solution using a Galerkin method and then prove uniform estimates on this solution. The only difference here is that we have to prove the Korn's inequality when (h 0 , u 0 ) ≈ (1, 0). Let us first assume that h 0 (t = 0) = 1. In the case of small data, one can write that

Ω ∇ũ : P0 = 1 2 Ω D( u) : D( u) + O(η 2 ) ∇ u 2 , X h -1 0 | ∂X 0 ∂x 0 u H | z 0 =0 | 2 = 1 + O(η 2 ) W | u H | z 0 =0 | 2
Then, one can use the Korn's inequality proved in the case h 0 = 1 to show that for η 2 sufficiently small, there is C > 0 independent of ε so that

Ω ∇ũ : P0 + εγ X h -1 0 | ∂X 0 ∂x 0 u H | z 0 =0 | 2 ≥ C ∇ u 2 .
From the Korn inequality, we deduce energy estimates on the approximate solutions: this proves the existence of a weak solution. The proof of regularity is similar to the case of h 0 = 1, u 0 = 0: indeed the main task was to prove a Korn's inequality in this setting, which is done mainly using the fact that the Lagrangian change of variable is close to identity uniformly with respect to ε. The case of large data and small time is simply treated by considering that the Lagrangian change of variable is close to identity for T small enough and one can prove a similar Korn's inequality in this case. The more general case h 0 (t = 0) = 1 is treated by perturbation, simply through the change of variable (x, z) → x, zh 0 (t = 0, x) . 2

4 Well posedness and convergence of the full Navier-Stokes equations in Lagrangian form.

We write an iterative scheme to prove the well posedness of free surface Navier Stokes equations with Navier slip condition at the bottom together with the convergence of this set of equations to a viscous shallow water model. We will use Tychonov fixed point theorem, as in [START_REF] Coutand | Unique solvability of the free-boundary Navier-Stokes equations with surface tension[END_REF], with now the functional space

X = (u, p) ∈ L 2 (0, T ); H 3 × H 2 / (u, p) X ≤ C( u 0 2 + ε 3/2 ) , (u, p) 2 X = sup (0,T ) u 2 2 + T 0 u 2 3 + ∂ t u 2 1 + p 2 2 + ∂ t p 2 0 .
The iterative scheme is written as

∂ t u k+1 + A -1 0 u k+1 .∇u a (X 0 ) + A -T 0 ∇p k+1 R e = A -T 0 R e div A T 0 P k+1 0 -∇A 0 : P k+1 0 + F k , div A -1 0 u k+1 = H k , with H k = σ a (X 0 , Z 0 ) + (σ a (X k , Z k ) -σ a (X 0 , Z 0 )) + div (A -1 0 -A -1 k )u k
and

F k = (A -1 0 -A -1 k )u k .∇u a (X k , Z k ) + A -1 0 u k .∇ u a (X 0 , Z 0 ) -u a (X k , Z k ) + 1 R e (A -T 0 -A -T k )∇p k + (A -T k -A -T 0 ) div(A T k P k ) -∇A k : P k + 1 R e A -T 0 ∇A 0 : P k 0 -∇A k : P k + div A k T P k -A T 0 P k 0 +(f a (X k , Z k ) -f a (X 0 , Z 0 )) + f a (X 0 , Z 0 ).
This set of equation is completed with boundary conditions at the bottom:

u k+1 V | z=0 = 0, 1 h 0 ∂ z u k+1 H | z=0 = εγu k+1 H | z=0 + G 1 k , (38)
with G 1 k = εγ det(∂xX 0 ) det(∂xX k ) -1 u k H | z=0 and boundary conditions at the free surface

A T 0 P k+1 0 -p k+1 Id | z=ε e 3 = G 2 k , (39)
where

G 2 k = A T 0 (g a (X k , Z k )-g a (X 0 , Z 0 ))+g a (X 0 , Z 0 )+(P k 0 -P k )n+(P k 0 -p k Id)(n 0 -n) | z=ε with P k = (∇u k )A -1 k A -T k + A -T k (∇u k ) T A -T k , P k 0 = (∇u k )A -1 0 A -T 0 + A -T 0 (∇u k ) T A -T 0 .
The function u k+1 has the initial condition u k+1 | t=0 = u 0 . Using the well posedness results obtained in the previous section, this set of equation has a unique solution and the weak continuity of the map is done. We first prove that (u k , p k ) is bounded, uniformly on the time interval (0, T ) and uniformly with respect to ε, in a suitable norm provided that the initial condition is sufficiently small. The proof is done in 5 steps: estimates of the Lagrangian coordinates (more precisely the deviation from the "shallow water" coordinates), the boundary terms, the divergence term and nonlinear terms in momentum equations.

Step 1: Estimates of Lagrangian coordinates. Let us recall that Lagrangian coordinates satisfy the set of equations

d dt (X k -X 0 ) = u a,H (X k , Z k ) -u a,H (X 0 , Z 0 ) + u k H , d dt (Z k -Z 0 ) = u a,V (X k , Z k ) -u a,V (X 0 , Z 0 ) + u k V .
Let us first compute the basic L 2 estimate

d dt X k -X 0 2 0 + Z k -Z 0 2 0 ≤ C( u a (t, .) W 1,∞ ) X k -X 0 2 0 + Z k -Z 0 2 0 + u k 0 X k -X 0 0 + Z k -Z 0 0 . This is a straightforward consequence of Gronwall inequality that sup t∈(0,T ) X k -X 0 2 0 + Z k -Z 0 2 0 ≤ T e C(T +1) T 0 u k (t, .) 2 0 dt, with C = C sup t∈(0,T ) ( u a W 1,∞
) . There is no difficulty to extend this estimate to higher order space derivatives:

sup t∈(0,T ) X k -X 0 2 s + Z k -Z 0 2 s ≤ T e C(T +1) T 0 u k (t, .) 2 s dt, with C = C sup t∈(0,T ) ( u a W 1,∞
) . Next, we compute boundary terms at the bottom:

Step 2: Estimates of nonlinear terms at the bottom. In order to estimate the boundary term at the bottom, we will use lemma 2 and 3. The nonlinear term at the bottom reads

G k 1 = εγu k H | z=0 det(∂ x X 0 ) det(∂ x X k ) -1 | z=0 = εγu k H | z=0 F (∂ x X k -∂ x X 0 ).
We first compute an estimate for |g k 1 | 3

2

. Let us fix t 0 > 1 so as H t 0 (T 2 ) ⊂ L ∞ (T 2 ). We then have

|G k 1 | 3 2 ≤ εγ |u k H | z=0 | t 0 |F (∂ x X k -∂ x X 0 )| 3 2 + |u k H | z=0 | 3 2 |F (∂ x X k -∂ x X 0 )| t 0 ≤ εγC(|∂ x X k -∂ x X 0 | ∞ ) |u k H | z=0 | t 0 |∂ x X k -∂ x X 0 | t 0 +|u k H |z = 0| 3 2 |∂ x X k -∂ x X 0 | t 0 ≤ εγC(|X k -X 0 | t 0 +1 ) u k H t 0 + 1 2 √ ε X k -X 0 3 √ ε + u k H 2 √ ε X k -X 0 t 0 + 3 2 √ ε ≤ γC( X k -X 0 t 0 + 3 2 √ ε ) u k t 0 + 1 2 X k -X 0 3 + u k 2 X k -X 0 t 0 + 3 2 . ( 40
)
Then it is easily seen, choosing

t 0 = 3 2 , that T 0 |G k 1 (t, .)| 2 3 2 dt ≤ γC T ε u k L 2 ((0,T ),H 3 ) T sup t∈(0,T ) ( u k (t, .) 2 2 )
T 0 u k (t, .) 2 3 dt.

Using lemma 3, we next compute an estimate of

|∂ t g k 1 | -1 2
. One has

∂ t G k 1 = εγ ∂ t u k H | z=0 F (∂ x X k -∂ x X 0 )+u H | z=0 ∂ 1 F (∂ x X k -∂ x X 0 ) ∂ 2 xt (X k -X 0 )
Then, the following inequality holds:

|∂ t G k 1 | -1 2 ≤ ε |∂ t u k | -1 2 √ ε F (∂ x X k -∂ x X 0 ) 2 +|u H | z=0 G(∂ x X k -∂ x X 0 )(∂ x u k H + F 1 (∂ x X k -∂ x X 0 )| -1 2 ≤ C T ε u k L 2 ((0,T ),H 3 ) ( u k 1 + ∂ t u k 1 ) X k -X 0 3 + √ ε|∂ x u k | -1 2 F (∂ x X k -∂ x X 0 )u H 2 (41) ≤ C T ε u k L 2 ((0,T ),H 3 ) ( u k 1 + ∂ t u k 1 ) X k -X 0 3 + u k 2 X k -X 0 3 .
So that we obtain the estimate

T 0 |∂ t G k 1 | 2 -1 2 ≤ C T ε u k L 2 ((0,T ),H 3 ) T T 0 ∂ t u k 2 1 + u k 2 3 2 .
Next we compute interior estimates of "divergence" terms: this is done using proposition 1 and tame estimates of lemma 1.

Step 3: Estimate of divergence terms. One part of the divergence term

H k 1 = Tr (A -1 k -A -1 0 )∇u k satisfies H k 1 2 ≤ 1 √ ε C( ∂ x X k -∂ x X 0 ∞ ) X k -X 0 3 ∇u k 2 , H k 1 2 ≤ 1 √ ε C( T ε u k L 2 ((0,T ),H 3 ) ) X k -X 0 3 u k 3 .
As a consequence, one has

T 0 H k 1 (t, .) 2 2 dt ≤ C( T ε u k L 2 ((0,T ),H 3 ) ) ε T 0 u k 2 3 2 .
Next, we compute an estimate of ∂ t H k 1 0 . On the one hand, one has

Tr (A -1 k -A -1 0 )∇∂ t u k 0 ≤ 1 √ ε C(|∂ x X k -∂ x X 0 | ∞ ) X k -X 0 3 ∂ t u k 1 , and
Tr (∂ t A k -∂ t A 0 )∇u k 0 ≤ ∇u k ∞ C(∂ x,t X k -∂ x,t X 0 ) 0 , Tr (∂ t A k -∂ t A 0 )∇u k 0 ≤ 1 √ ε u k 3 √ ε C(|∂ x X k -∂ x X 0 | ∞ +|∂ x u k | ∞ ) X k -X 0 1 + u k 1 .
So that we obtain

T 0 ∂ t H k 1 2 0 ≤ T ε C T ε u k L 2 ((0,T ),H 3 ) T 0 ∂ t u k 2 1 dt T 0 u k (t, .) 2 3 dt.
This concludes the estimate of nonlinear divergence terms.

Step 4: Estimates of "interior" terms. The interior terms are estimated with proposition 1 and lemma 1. We first compute an estimate of (A -T 0 -

A -T k )∇p k 1 . One has (A -T 0 -A -T k )∇p k 1 ≤ A -T 0 -A -T k ∞ p k 2 + ∇(A -T 0 -A -T k )∇p k 0 ≤ A -T 0 -A -T k ∞ p k 2 + ε 1 6 ∇(A -T 0 -A -T k ) 1 √ ε p k 2 √ ε ≤ C(|∂ x X k -∂ x X 0 | ∞ ) √ ε X k -X 0 3 p k 2 . (42)
As a consequence, we obtain the estimate

T 0 (A -T 0 -A -T k )∇p k 2 1 ≤ T ε C( T ε u k L 2 ((0,T ),H 3 ) ) T 0 u k (t, .) 2 3 dt T 0 p k 2 2 .
Similarly, the time derivative of this term satisfies

T 0 ∂ t (A -T 0 -A -T k )∇p k 2 (H 1 ) ′ ≤ T ε C T ε u k L 2 ((0,T ),H 3 ) T 0 u k (t, .) 2 3 dt T 0 ∂ t p k (t, .) 2 0 dt + T ε C T ε u k L 2 ((0,T ),H 3 ) T 0 ∂ t u k (t, .) 2 1 dt T 0 p k (t, .) 2 2 dt.
The other nonlinear term with high derivatives that we consider is

f k = A -T k div(A T k P k ) -A -T 0 div(A T 0 P k 0 ).
This is a lengthly but straightforward computation to show that

T 0 f k 2 1 + ∂ t f k 2 (H 1 ) ′ ≤ T ε C T ε u k L 2 ((0,T ),H 3 ) T 0 u k 2 3 T 0 u k 2 3 + ∂ t u k 2 1 .
Step 5: estimates of boundary term at the free surface. We have to deal with the nonlinear term

G k 2 = A T 0 (P k -P k 0 )n + (P k -p k )(n 0 -n) | z=ε .
On the one hand, one has

|(P k -P k 0 )n| z=ε | 3 2 ≤ C T ε u k L 2 ((0,T ),H 3 ) |P k -P k 0 | 3 2 .
Moreover, this is a straightforward computation to show that

|P k -P k 0 | 3 2 ≤ C X k -X 0 3 √ ε X k -X 0 3 u k 3 ε .
As a consequence, one obtains For the term (P k -p k )(nn 0 )| z=ε , we first remark that (nn 0 )| z=0 = 0. As a consequence, we obtain the estimate

T 0 |(P k -p k )(n -n 0 )| z=ε | 2 3 2 ≤ 1 ε C T ε u k L 2 ((0,T ),H 3 ) T 0 u k 2 3 T 0 u k 2 3 + p k 2 2 .
Hence, using the estimates on the linear problem and using that remaining terms in the Navier-Stokes type system [START_REF] Bresch | Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasigeostrophic model[END_REF] satisfied by the high order shallow water approximation (u a , p a ) satisfy

T 0 f a (X 0 , Z 0 ) 2 1 + ∂ t f a (X 0 , Z 0 ) 2 (H 1 ) ′ +|g a 2,2 (X 0 , Z 0 )| 2 3/2 +|∂ t g a 2,2 (X 0 , Z 0 )| 2 -1/2 + 1 ε T 0
H a (X 0 , Z 0 ) 2 2 + ∂ t H a (X 0 , Z 0 ) 2 0 + |g a 1 (X 0 , Z 0 ), g a 2,1 (X 0 , Z 0 )| 2 3/2 +|g a 1,t (X 0 , Z 0 ), ∂ t g a 2,1 (X 0 , Z 0 )| 2 -1/2 ) ≤ Cε 4 . We have then proved the following proposition. At this step, there are two strategies: Either we use a Banach fixed point argument and in this case we need estimates for higher Sobolev norms that are obtained in a way similar to our previous calculation. In this case, we can prove that the sequence (u k , p k ) is a Cauchy sequence respectively in the space C(0, T ; H 2+η (Ω)) and L 2 ((0, T ), H 2+η (Ω)) with 0 < η < 1/2, see [START_REF] Beale | The Initial Value Problem for the Navier-Stokes Equations with a Free Surface[END_REF] for further references. In this case, the result is not sharp. To obtain the well posedness in critical spaces, one has to use Tychonoff fixed point procedure as in [START_REF] Coutand | Unique solvability of the free-boundary Navier-Stokes equations with surface tension[END_REF]. There is no difficulty here to follow both strategies to obtain the well posedness result in our situation since the estimate in high norms is proved.

Conclusion

In this paper, we have proved the well-posedness of free surface Navier-Stokes equations with a slip condition at the bottom and convergence to a viscous shallow water model in the shallow water scaling ε → 0, ε being the aspect ratio. In contrast to our mathematical justification of a shallow water model for a fluid flowing down an inclined plane, we dropped the tension surface effect (which is important in order to deal with realistic situations). This is due to the fact that we worked in the more suitable Lagrangian coordinates.

There is no difficulty to extend this result for fluids flowing down an inclined plane when the slope θ is asymptotically small, θ ∼ ε and when the uniform flow is stable (see [START_REF] Bresch | Mathematical justification of a shallow water model[END_REF] for more details).

When the uniform flow is unstable, it is known that instabilities so called roll-waves appear: these are periodic travelling waves. Small amplitude rollwaves are proved to exist both in viscous shallow water equations [START_REF] Merkin | An infinite period bifurcation arising in roll waves down an open inclined channel[END_REF] and free surface Navier-Stokes equations [START_REF] Nishida | Hopf bifurcation in viscous incompressible flow down an inclined plane[END_REF]. However, the result of Nishida et al. is proved in the presence of surface tension whereas roll-waves are proved to exist in viscous shallow water equations without surface tension and are of large amplitude when the viscosity is small. In order to obtain a larger range of validity of shallow water equations, this would be of interest to prove the existence of roll-waves in Navier-Stokes equations without surface tension that are close to "shallow water roll-waves" in the shallow water scaling.

Proposition 6 ε 3 2 2 2

 632 Assume that u 0 2 ≤ C-1 C 2 , then for ε small enough, one has, for any k ≥ 1 (u k , p k ) 2 X ≤ C( u 0 + ε 3 ).

v 2 .The proof of this lemma is found in[START_REF] Coutand | Unique solvability of the free-boundary Navier-Stokes equations with surface tension[END_REF] in a fixed domain: we combine this proof and proposition 1 to obtain the estimate of lemma


R Kollengode Easwaran

L Longchambon

P.-E Pottie

V Lorent

H Perrin

B M Garraway

RF spectroscopy in a resonant RF-dressed trap

Keywords: numbers: 39, 25, +k, 32, 80, Pj

We study the spectroscopy of atoms dressed by a resonant radiofrequency (RF) field inside an inhomogeneous magnetic field and confined in the resulting adiabatic potential. The spectroscopic probe is a second, weak, RF field. The observed line shape is related to the temperature of the trapped cloud. We demonstrate evaporative cooling of the RF-dressed atoms by sweeping the frequency of the second RF field around the Rabi frequency of the dressing field.

I. INTRODUCTION

Since the initial proposal of Zobay and Garraway [1], adiabatic potentials for ultracold atoms, resulting from the radiofrequency (RF) dressing of atoms in a inhomogeneous magnetic field, have been demonstrated [2] and are used by many groups to propose and realize unconventional traps [3][4][5][6][START_REF] Extavour | Thy-wissen[END_REF][START_REF] Jo | [END_REF][9], in particular in the framework of atom chips [6][START_REF] Extavour | Thy-wissen[END_REF][START_REF] Jo | [END_REF]. They offer the great advantage of a simple tailoring of the trap potential, allowing the realisation of a double well [6] or a ring trap [4,9] with dynamically adjustable parameters. The potential seen by the dressed atoms is due to the spatial variation of the RF coupling strength or the RF detuning, or both. These RF-dressed traps are usually loaded directly from a conventional magnetic trap, by ramping up the RF amplitude or sweeping the RF frequency.

Depending on the conditions, the adiabatic potential may result essentially from a light shift induced by a variation of the RF coupling, as was first demonstrated in a pioneering experiment at NIST with microwave radiation [10]. In this case, denoted here as the 'off-resonant configuration', the detuning of the RF field to the magnetic Zeeman splitting is negative everywhere and the atoms are expelled from the most resonant central region, leading for example to a double well structure well suited for matter wave interferometry [11]. On the other hand, the trapping potential may be due essentially to the variation of the detuning, the atoms being then attracted towards the regions of zero detuning [1]. In this 'resonant configuration', the atoms sit at points where the RF field is resonant with the magnetic level spacing, which makes them more sensitive to fluctuations of the RF frequency, and heating due to frequency fluctuations was indeed observed [2,12,13]. This heating may be avoided if a low noise RF source is used [13], leading to traps compatible with Bose-Einstein condensates. However, the transfer of the atoms from the initial magnetic trap to the RF-dressed trap is not usually adiabatic, * Electronic address: helene [dot] perrin [at] univ -paris13 [dot] fr which results in an initial heating in the loading phase and the destruction of the condensate [2,12].

This limitation motivated a theoretical study of evaporative cooling inside the dressed trap, by means of a second RF field [14]. In the case of a non-resonant RF trap, this scheme was used to prepare independent condensates in a double well [11]. In this paper, we present results of spectroscopy in the dressed trap, obtained by recording the atom losses induced by a very weak second RF field [15,16]. The shape of the resonance is directly linked to the energy distribution of the atoms in the dressed trap. We discuss the position of the expected lines and the strength of the second RF coupling. Finally, we discuss the choice of the second field frequency for evaporative cooling of the atoms dressed by the main RF field.

II. SPECTROSCOPY OF THE RF-DRESSED TRAP

The spectroscopy of an RF-dressed trap is performed in the following way: A weak probing field can induce transitions between dressed states, which results in a loss of atoms from the adiabatic potential. The losses are monitored as a function of the frequency of the probing field, as has been investigated in detail in the off-resonant configuration [15]. In this section, the position and strengths of the expected resonances are discussed.

A. Effect of the dressing field

In the following, we denote as (x, y, z) the axes of the laboratory frame, z being the vertical axis. In the experiment, the adiabatic potential is created by a strong RF field of frequency ω 1 and Rabi frequency Ω 1 , linearly polarised along y, which dresses the atoms placed in an inhomogeneous magnetic field B(r). The direction of this static magnetic field at position r is denoted as Z. The two important parameters which determine the resulting adiabatic potentials are the RF detuning δ(r) = ω 1ω L (r) from the local Larmor frequency and the local effective RF coupling Ω eff 1 (r). The latter depends on the angle θ(r) between the RF polarisation y and the static magnetic field direction Z, being zero for collinear fields: Ω eff 1 (r) = Ω 1 sin θ(r). Hence, the static magnetic field geometry is the skeleton of the adiabatic potentials.

The static magnetic field B(r) is that of a compact Ioffe-Pritchard trap [17] characterised by the bias field B 0 = 1.8 G at the centre, the radial gradient b ′ = 223 G•cm -1 and the longitudinal curvature b ′′ = 250 G•cm -2 along the trap axis x. We define as ω 0 = |g F |µ B B 0 / the Larmor frequency at the trap centre and as α = |g F |µ B b ′ / the magnetic gradient in units of frequency, where g F is the Landé factor, σ(g F ) = g F /|g F | its sign and µ B is the Bohr magneton. The typical lengthscales in this magnetic trap are L = B 0 /b ′′ = 850 µm and R = B 0 /b ′ = ω 0 /α = 81 µm. The Larmor frequency at position r is then approximately given by ω L (r) = ω 0 1 + x 2 /L 2 + (y 2 + z 2 )/R 2 in the vicinity of the magnetic trap centre. On the other hand, the local effective coupling to the RF field is approximately

Ω eff 1 (r) = Ω 1 1 -α 2 y 2 ωL(r) 2 .
As a result of the near resonant dressing field, the energy levels of the atom plus RF field system are grouped into manifolds separated by an energy ω 1 . Inside a manifold, the energy spacing of the 2F + 1 levels is denoted Ω(r). The adiabatic potential V mF 1 (r) = σ(g F ) m F Ω(r) then varies in space through the spatial variation of both the detuning δ(r) and the effective RF coupling Ω eff 1 (r) [3]. Given the static magnetic field geometry, the adiabatic potential for a given F, m F adiabatic state reads:

V mF 1 (r) = σ(g F ) m F Ω(r), where (1)
Ω(r) = δ(r) 2 + Ω 2 1 1 - α 2 y 2 ω L (r) 2 .
This expression corresponds to the eigenenergies at point r of the Hamiltonian H 1 = -δ(r)F Z + Ω eff 1 (r)F X , obtained after applying the rotating wave approximation (RWA) to the RF-atom coupling, which is valid when Ω 1 ≪ ω 1 and |δ(r)| ≪ ω 1 . Here, X is direction orthogonal to the static field direction Z, in the y -Z plane. The total potential energy is V mF (r) = V mF 1 (r)+M gz, where g is the gravitational acceleration and M the atomic mass.

The potential minimum for the extreme adiabatic state m F = σ(g F ) F is located on the isomagnetic surface S 0 defined by δ(r) = 0 or equivalently ω L (r) = ω 1 [START_REF]Strictly speaking[END_REF]. Schematically, this surface is an elongated ellipsoid, cylindrically symmetric around x with a radius r 0 = ω 2 1ω 2 0 /α and a long axis x 0 = r 0 L/R. On the surface S 0 , ω L (r) = ω 1 and the RF coupling only depends on the coordinate y:

Ω eff 1 (y) = Ω 1 1 -α 2 y 2 ω 2 1
. The total potential energy on S 0 then reads F Ω eff 1 (y) + M gz. It varies through the RF shift between F Ω 1 1 -

α 2
m F = +2 m F = -2
FIG. 1: Adiabatic potential energy (V m F 1 (z) + M gz)/h for the F = 2 ground state of 87 Rb in our magnetic field configuration, in units of frequency of the dressed states mF = 2, ...-2, plotted along the vertical direction. The dressing frequency is set to ω1 = 2π ×8 MHz with a Rabi frequency of Ω1 = 2π ×50 kHz, resulting in a dressed trap centred at position z = -506 µm. The two extreme dressed levels are emphasised. The origin z = 0 corresponds to the static magnetic field minimum, and gravity is taken into account. The probing field ω2 induces transitions between dressed states at a position z such that Ω(z) = |ω2 -nω1|, n ∈ N. These positions correspond to two points z± belonging to two surfaces S±,

defined respectively by δ(r) = ± (nω1 -ω2) 2 -Ω eff 1 (r) 2
, on both sides of the trapping surface S0 defined by δ(r) = 0.

F ω0

ω1 Ω 1 at y = ±r 0 on the y axis and F Ω 1 in the y = 0 plane. On the other hand, the gravitational shift also varies as a function of z between -M gr 0 at the bottom (z = -r 0 ) and 0 at the equator (z = 0). Depending on the relative strength of the gravitational shift M gr 0 with respect to the RF shift F Ω 1 (1 -ω0 ω1 ), there are either two minima at the equator at z = 0 and y = ±r 0 [6] or a single minimum at the bottom of this surface z = -r 0 [START_REF] Morizot | [END_REF]. With the parameters of our experiment, we are always in the second situation [2]. At the minimum, where the atomic density is largest, the RF coupling reaches its maximum value Ω 1 . Away from the minimum, it decreases down to a value depending on the energy and on the RF frequency through the ellipsoid radius r 0 . At a given energy above the potential minimum, the coupling strength reduction is more pronounced for smaller RF frequencies ω 1 . For example, at the operational RF frequency of 8 MHz the reduction is less than 20% at energies below 10 µK. By contrast, at 3 MHz, the reduction remains below 7% at an energy corresponding to 1 µK, and reaches 50% for 10 µK. Nevertheless, the effective coupling Ω eff 1 never vanishes over S 0 . If necessary, the coupling inhomogeneity could be reduced by increasing the bias field B 0 .

In the following, we concentrate on the case of rubidium 87 in the state F = 2. A plot of the adiabatic poten-tials around z = -r 0 as a function of the vertical coordinate z in the laboratory frame is given in figure 1 for a typical RF frequency ω 1 = 2π×8 MHz and for the parameters of our Ioffe-Pritchard magnetic trap. The resulting trap in the dressed state m F = F = 2 is very anisotropic, with oscillation frequencies near the trap centre of the order of 1 kHz along z and of 5 to 20 Hz in the horizontal directions. In this way, the atom cloud has the shape of a crêpe [2].

B. Coupling to the probing field: resonance frequencies

A second RF field of frequency ω 2 and Rabi frequency Ω 2 is applied to the dressed atoms. The polarisation of this second field is also linear along y, and again its orientation relative to the static magnetic field slightly varies across the dressed trap. As a result, both polarisations X and Z are present and can couple to the atomic spin. In the following, we neglect the position dependence of Ω 2 and concentrate in each case on the polarisation component having a significant coupling to the atomic spin. Ω 2 is typically a factor 100 times smaller than the dressing Rabi frequency Ω 1 . This second field allows a coupling between dressed states m F and m ′ F with |m Fm ′ F | = 1: see the arrows in figure 1. This coupling occurs either within the same manifold or between different manifolds. The expected resonant frequencies for this coupling at position r are thus ω 2 = nω 1 ± Ω(r), n being an integer [15].

In the 'resonant configuration', the points r in space where the probing field is resonant, i.e. where the condition Ω(r) = |ω 2nω 1 | is satisfied, consist of two surfaces S + and S -on both sides of the trapping surface S 0 (see figure 2a). The detuning of the local Larmor frequency with respect to ω 1 has an opposite sign for these two probing surfaces, δ(r) = ± (nω 1ω 2 ) 2 -Ω eff 1 (r) 2 for the surface S ± respectively.

The shape of the resonant surfaces S ± strongly depends on the value of the probing frequency ω 2 around nω 1 . First, let's remark that the resonance can be reached at point r for probing frequencies ω 2 close to nω 1 only if |ω 2nω 1 | > Ω eff 1 (r). Three cases can then be distinguished. First, for frequencies ω 2 satisfying |ω 2nω 1 | > Ω 1 , the resonant surfaces are very close to isomagnetic ellipsoids, see figure 2a. The atoms need a minimum non zero energy to reach these surfaces, and the point where the necessary energy is the lowest lies at the bottom of the external surface S -. Second, for |ω 2nω 1 | = Ω 1 , the two resonant surfaces merge at the trapping surface S 0 all over the vertical plane y = 0, in particular at the trap bottom, and all atoms may reach this point and be lost, see figure 2b In summary, the energy in the dressed trap necessary for an atom to escape through a point resonant with the probing field depends on the probe detuning to nω 1 . Figure 3 gives, as a function of |ω 2nω 1 |, the minimum energy E min (ω 2 ) with respect to the trap bottom for an atom to escape, for a dressing frequency of ω 1 /2π = 3 MHz and a Rabi frequency of Ω 1 /2π = 75 kHz. The three regimes discussed above are clearly visible.

The final shape of the recorded spectrum can be inferred from this minimum energy. If, over the duration of the experiment, the RF weak probe couples atoms efficiently out of the dressed trap when a resonant surface is reached, all the atoms with an external energy E larger than E min (ω 2 ) should be lost. Hence, the expected fraction of remaining atoms in the trap f rem (ω 2 ) is a function of E min (ω 2 ), namely

f rem = E min (ω2) 0 p(E) dE (2)
where p(E) is the energy distribution in the dressed trap, and the zero energy level for E is taken at the trap bottom. In particular, we expect f rem (nω 1 ± Ω 1 ) = 0 and an asymmetric shape of the spectrum around this value. The exact shape of p(E) is related to the exact energy potential. Its expression is not analytical, as the trap is far from being harmonic. However, we propose as a simple ansatz an exponential expression for p of the form

p(E) = (k B T ) -1 exp(-E/k B T ) and compare the simple prediction f rem (ω 2 ) = 1 -exp[-E min (ω 2 )/k B T ] to the experimental data, see section III B.

C. Coupling to the probing field: coupling strengths

In the previous section, the two resonant surfaces S ± at which the transitions ω 2 = nω 1 ± Ω(r) can occur have been presented. Let us point out that the strength of these different transitions is not the same, and depends on position in the dressed trap. This is true even if we assume a homogeneous Rabi frequency Ω 2 , which we do in the following for simplicity. For example for n = 1, the coupling strength at position r is Ω2 2

1 ∓ δ(r) Ω(r)
for the transitions at frequency ω 2 = ω 1 ±Ω(r), respectively [14]. The coupling strength can be interpreted in terms of the number of photons n from the dressing field necessary for the transition to occur. If we consider detunings δ(r) which are large compared to the dressing amplitude Ω 1 (such that Ω(r) ≃ |δ(r)| = |ω 1ω L (r)|), then the atoms are far from the trapping surface S 0 . In this case, the two RF couplings become independent and the probing field essentially couples undressed states together. As a result, the coupling is of order Ω 2 for the single photon process at frequency ω 2 ≃ ω L (r), and drops to a vanishing value Ω 2 Ω eff 1 (r) 2 4δ 2 for the three photon process at frequency ω 2 ≃ 2ω 1ω L (r). Hence, transitions to an untrapped state occur essentially through one of the two surfaces resonant with the probing field: the inner surface S + , closer to the magnetic field minimum, for the ω 2 = ω 1 -Ω(r) transition, and the outer surface S -, below the trapped atoms, for the ω 2 = ω 1 + Ω(r) transition.

Let us now focus on the low frequency transition n = 0, at ω 2 = Ω(r). In contrast to the previous case, this transition presents (as we shall see below) a symmetric coupling Ω eff 1 (r)Ω 2 /ω 2 at the two resonant surfaces S ± , between which the trapped atoms lie. This has some importance in the implementation of evaporative cooling in the dressed trap.

As this transition was not discussed in Ref. [14], we will briefly present its origin. First, we remark that the RF probe field should be π-polarised, meaning that the RF field component parallel to the static magnetic field direction Z is responsible for the transition. This can be understood by considering the transition induced by the probe away from the trapping surface S 0 , where the two RF fields decouple and the resonant condition is simply ω 1 ± ω 2 = ω L (r): the probe and the circularly polarized dressing field ω 1 induce a two-photon transition, that will change m F by 1 unit if the probe field is π-polarized. The Hamiltonian describing the spin evolution at position r in the presence of the two RF fields then reads

H(t) = ω L (r)F Z + 2Ω eff 1 (r)F X cos ω 1 t + 2Ω 2 F Z cos ω 2 t
(3) where Z is the direction of quantization imposed by the static magnetic field and X is the effective polarisation of the dressing RF field ω 1 of Rabi frequency Ω eff 1 (r). To deal with this double frequency RF coupling, we apply a rotation transformation to the Hamiltonian in the spirit of Ref. [14], at an angle θ(t) = (ω 1 t + 2Ω2 ω2 sin ω 2 t) around the Z axis. The idea of this transformation is that the term in θF Z is chosen to cancel the new term 2Ω 2 F Z cos ω 2 t. After the rotation, product terms of the form cos ω 1 t cos ω 1 t + 2Ω2 ω2 sin ω 2 t appear. Since Ω 2 ≪ ω 2 , these terms can be developed as products of sine and cosine functions. Finally, application of the ro-tating wave approximation allows the elimination of fast rotating terms (rotating at about 2ω 1 ). The Hamiltonian in the rotated frame finally simplifies to

H ′ (t) ≃ -δ(r)F Z + Ω eff 1 (r)F X - 2Ω eff 1 (r)Ω 2 ω 2 sin ω 2 tF Y .
(4) This consists of the Hamiltonian H 1 = -δ(r)F Z + Ω eff 1 (r)F X for the state dressed by the main RF field, to which a transverse RF coupling of frequency ω 2 and effective amplitude

Ω eff 1 (r)Ω2 ω2
is added. This coupling allows transitions at a frequency ω 2 = Ω(r). Again, the condition is fulfilled at the two resonant surfaces S ± , but now with the same coupling amplitude at two points symmetric with respect to the surface S 0 , in contrast to the n = 1 case.

III. EXPERIMENTAL RESULTS

In this section, experimental results of RF spectroscopy of a RF-dressed trap are presented. The results are obtained in a dressed Ioffe-Pritchard trap in the resonant configuration.

A. Experimental procedure

The loading procedure into the adiabatic potential was described in detail in a previous paper [13]. Rubidium 87 atoms are initially confined in their internal state F = 2, m F = 2 in a static magnetic trap and evaporatively cooled to a temperature T 0 . The dressed trap is loaded into the dressed state m F = 2 with a 500 ms RF ramp from 800 kHz to the final value of ω 1 , ranging between 3 and 8 MHz. At the end of this loading stage, a second RF field with a fixed frequency ω 2 and a Rabi frequency Ω 2 is switched on for 2 seconds, to perform the spectroscopy of the adiabatic potential. The Rabi frequency Ω 1 of the dressing field is kept constant during the loading stage and the probing stage [20]. The RF fields are produced by two identical antennas (9 windings, 10 mm in diameter) placed face to face across the vacuum chamber, at a distance of 17 mm from the atomic cloud. At the input of each antenna, the RF amplitude lies between 160 and 640 mA pp for the dressing field, and between 0.8 and 5 mA pp for the probing field. Finally, the atom number and the optical density are after a time of flight of 7 or 10 ms, and a spectrum is recorded as a function of the probing frequency ω 2 . This procedure is repeated for different values of Ω 1 , Ω 2 and ω 1 . As discussed in the previous section, resonances are expected at frequencies around Ω 1 , ω 1 -Ω 1 , ω 1 + Ω 1 , 2ω 1 -Ω 1 , etc. [15]. FIG. 4: Spectroscopy of the dressed trap, around the dressing frequency ω1 = 2π × 8 MHz. The number of atoms remaining after the application of the probe field is plotted (squares) as a function of the detuning ∆2 = ω2 -ω1 to the frequency ω1. The dressing amplitude is 400 mApp at the input of the RF antenna, corresponding to a Rabi frequency Ω1 = 2π ×50 kHz as deduced from the spectrum. The probing amplitude is 1.6 mApp, smaller by a factor 250. The calculated number of remaining atoms for T = 11 µK is plotted as a full grey line, see text.

B. Experimental spectra

A typical spectrum recorded around ω 1 = 2π × 8 MHz is plotted in figure 4. For this data set, the dressing amplitude is set to 400 mA pp while the probing amplitude is 1.6 mA pp . The two resonances at ω 1 ± Ω 1 are visible, broadened by the energy distribution in the trap at temperature T . For probing frequencies between ω 1 -Ω 1 and ω 1 + Ω 1 , the atomic loss rate rapidly decreases to zero, and no transition is observed at a frequency ω 2 = ω 1 , as expected from the previous discussion. At ω 2 = ω 1 ± Ω 1 , the probing frequency is resonant with the bottom of the adiabatic potential and almost all the atoms are lost (80% for this particular choice of the probe coupling Ω 2 ).

Away from this value, the atoms satisfying Ω(r) > |ω 2ω 1 | are lost. As expected, the observed line has an asymmetric shape, with a dip at ω 2 = ω 1 ± Ω 1 and approximately exponential outer wings with a width related to the temperature. Indeed, in a harmonic trap, the lineshape should be exponential, with a 1/e-width k B T /(2 ) and a threshold at an energy corresponding to the trap bottom. Using an exponential trial function p(E) = (k B T ) -1 exp(-E/k B T ) for the energy distribution, the resulting expression for f rem derived from Eq.( 2), and the temperature of T = 11 µK deduced from an independent time-of-flight measurement, we obtain the predicted spectrum plotted with a grey line in figure 4. f rem is normalised to total the atom number, and the limited maximum contrast of 80% of atom loss for 2 s at resonance is also taken into account. The experi- mental spectrum is in good agreement with the predicted lineshape, indicating that the exponential model used to estimate the energy distribution is indeed sufficient.

The sharp edge of the spectrum allows a good determination of the dressing Rabi frequency Ω 1 at the bottom of the adiabatic potential, equal to 50 kHz in the spectrum of figure 4. The determination is more accurate when a smaller probe Rabi frequency is used. A larger amplitude Ω 2 leads to broadening of the resonance line. Experimentally, an amplitude of 1.6 to 3.2 mA pp was found to be a good compromise between a good probing efficiency (80% of atom loss after 2 s on resonance) and a negligible line broadening. The RF dressing amplitude was varied with a RF attenuator and the position of the peak atom loss was recorded to check the validity of the determination of Ω 1 by this technique. The result is plotted on figure 5. The detuning at the maximum atom loss is expected to be directly proportional to the RF amplitude. At a dressing frequency ω 1 = 2π × 8 MHz, the deduced Rabi frequency is Ω 1 = ηI 1 , with a scaling factor η = 2π × 135 kHz/A pp , I 1 being the RF amplitude in amperes peak-to-peak. This is in agreement with the value of 130 kHz/A pp calculated from the geometric characteristics of the antenna.

The determination of Ω 1 from the spectra around ω 2 = ω 1 ± Ω 1 is also confirmed by the data recorded at low probing frequency. Figure 6 presents a typical spectrum at frequency ω 2 ≃ Ω 1 . We recover a Rabi frequency of 75 kHz at 3 MHz of dressing frequency. Again, the asymmetric shape is due to energy distribution of the atoms confined in the dressed trap. The overall shape of the spectrum is in reasonable agreement with the calculated spectrum for a temperature of 4 µK.

C. Evaporative cooling

Garrido Alzar et al. [14] proposed to use this second RF in this trap to force evaporation and cool the atoms. Indeed, the energy sensitivity of the atom loss induced by the second RF can be used to selectively remove the most energetic atoms from the trap. We implemented this forced evaporative cooling inside the RF-dressed trap. For this purpose, the second RF was scanned from an initial frequency away from the resonance down to a frequency close to the resonance at the trap bottom. The experiments where performed with a dressing RF amplitude of Ω 1 = 2π × 50 kHz, at ω 1 = 2π × 8 MHz. To optimize the symmetry of the coupling through the two resonant surfaces S ± , the low frequency resonance was preferred. The frequency of the weak field ω 2 /2π was lowered from 600 kHz to a final value ν end . We observed a decrease of the cloud temperature and the atom number as ν end is lowered, see figure 7. The selective out-coupling of the atoms with a large vertical energy E z works as expected around the low frequency resonance, whereas almost no effect was obtained when scanning the frequency around ω 1 , where the out-coupling is asymmetric. In our dressed trap, the phase space density increases only slightly, due to a poor thermalisation efficiency with the low horizontal frequencies (5 Hz × 20 Hz). With larger horizontal frequencies in the dressed trap, as is the case in a dressed quadrupole trap for example [START_REF] Morizot | [END_REF], and thus with a higher collision rate, thermalisation in the trap would lead to an increase of the phase space density [21], as shown by molecular dynamics simulations. From the point of view of the out-coupling efficiency, our results show that evaporative cooling is feasible in a resonant dressed trap in the same conditions as in a conventional magnetic trap.

IV. CONCLUSION AND PROSPECTS

In this paper, we have demonstrated RF spectroscopy in a RF dressed trap operated in a regime where the trapped atoms are resonant with the dressing RF source. Thus we use two RF fields: a strong dressing RF source, and a second weak probing field which couples the atoms out of the dressed trap. Spectra are recorded either around the dressing RF frequency (a few MHz), or at low frequencies near the Rabi frequency (a few tens of kHz). The experimental data fit well with a simple model based on the minimum energy for a trapped atom to be outcoupled by the weak probe. The low frequency transition is used to selectively evaporate the hotter atoms from the dressed trap. A temperature decrease is demonstrated in the transverse direction of the dressed trap together with an atomic population decrease. The resulting phase space density increase is modest at the present stage of the experiment since the very low motional frequencies of this anisotropic trap imply a low collision rate and thus a poor thermalisation.

The results of this paper emphasise the interest in utilizing resonant RF fields to realise unusual atomic potentials and show that we can apply evaporative cooling inside these potentials. Unlike light based traps, a resonant RF-based trap is not affected by spontaneous emission. On the other hand it is sensitive to the noise of the RF source. However, heating, and the ensuing losses, can be kept at very low values provided there is a good control of the parameters of the source [13]. Finally, we conclude that with reasonable trap frequencies (in all directions) it should be possible to have an efficient RF evaporative cooling of an atomic population in a RF trap.

FIG. 2 :

 2 FIG. 2: Cut of the resonant surfaces S± in the plane x = 0 in the three cases discussed in the paper. (a): |ω2 -nω1| = 2Ω1 > Ω1, (b): |ω2 -nω1| = Ω1, (c): |ω2 -nω1| = 0.7Ω1, i.e. Ω1ω0/ω1 < |ω2 -nω1| < Ω1. The isomagnetic surface S0 to which the atoms are trapped is also represented. ω1 is set to 2π × 3 MHz to show the potentials clearly and Ω1 is set to 2π × 75 kHz. (d): Corresponding outer resonant surface Sin a 3D representation in the case (c).

FIG. 3 :

 3 FIG. 3: Minimum energy required for an atom to leave the dressed trap by reaching a resonant surface, as a function of the probe detuning |ω2 -nω1|. The parameters are: ω1 = 2π × 3 MHz and Ω1 = 2π × 75 kHz.

FIG. 5 :

 5 FIG. 5: Rabi frequency Ω1, as deduced from the position of the edge of the line, plotted as a function of the dressing RF amplitude. RF dressing frequency: ω1 = 2π × 8 MHz.

FIG. 6 :

 6 FIG.6: Spectroscopy of the dressed trap, around the dressing amplitude Ω1 = 2π × 75 kHz, for a dressing frequency ω1 = 2π×3 MHz. The horizontal axis indicates the probe frequency ω2. The dressing amplitude is 580 mApp at the input of the RF antenna, corresponding to a Rabi frequency Ω1 = 2π × 75 kHz (see text). The probing amplitude is 9 mApp at the input of the second antenna, but only the component parallel to Z couples to the atoms (see text). The calculated number of remaining atoms for T = 4 µK is plotted as a full grey line.

FIG. 7 :

 7 FIG.7: Final atom number N and kinetic temperature Tz, deduced from the vertical size of the cloud after a time-offlight expansion, after an evaporation ramp of the second RF field between 600 kHz and ν end .

  . Third, for |ω 2nω 1 | < Ω 1 , resonant surfaces still exist due to the spatial dependence of Ω eff 1 , but each of them is split into two curved surfaces centred around the y axis, see figure 2c and 2d. Again a minimum energy is necessary to

Acknowledgments

We thank V. Dini for assistance in data processing. This work was supported by the Région Ile-de-France (contract number E1213 and IFRAF), by the PPF 'Manipulation d'atomes froids par des lasers de puissance', by the European Community through the Marie Curie Training Network 'Atom Chips' under contract number MRTN-CT-2003-505032, and by Leverhulme Trust. B. M. G. thanks CNRS for support. Laboratoire de physique des lasers is UMR 7538 of CNRS and Paris 13 University. LPL is member of the Institut Francilien de Recherche sur les Atomes Froids (IFRAF).