Edoardo Bocchi
email: edoardo.bocchi@math.u-bordeaux.fr

Floating structures in shallow water: local well-posedness in the axisymmetric case

Floating structures in shallow water: local

well-posedness in the axisymmetric case

Introduction

The floating structure problem was first formulated by John in his two famous papers [START_REF] John | On the motion of floating bodies. I[END_REF][START_REF]On the motion of floating bodies. II. Simple harmonic motions[END_REF], in which he considered a linear flow. It is a particular example of fluid-structure interaction, when a partially immersed body is floating on the fluid free surface. The fluid is supposed to be incompressible and the flow irrotational, and we can consider both the case of a prescribed solid motion or of a free motion governed by Newton's law for the solid object. In this problem we have to treat two free boundary problems.

The first free boundary problem is the typical water waves problem, which consists in describing the evolution of the fluid surface in contact with the air (or another fluid whose density can be neglected). In the absence of floating bodies, this is the standard water waves problem, which has been studied by many authors in the last years and whose local well-posedness theory is well-known. For instance we refer to Wu [START_REF] Wu | Well-posedness in Sobolev spaces of the full water wave problem in 2-D[END_REF][START_REF]Well-posedness in Sobolev spaces of the full water wave problem in 3-D[END_REF], Lannes [START_REF] Lannes | Well-posedness of the water-waves equations[END_REF][START_REF]Mathematical Surveys and Monographs[END_REF] , Alazard, Burq and Zuily [START_REF] Alazard | On the water-wave equations with surface tension[END_REF][START_REF]On the Cauchy problem for gravity water waves[END_REF] and Iguchi [START_REF] Iguchi | A shallow water approximation for water waves[END_REF]. A notable formulation is the one introduced by Zakharov, Craig and Sulem [START_REF] Zakharov | Stability of periodic waves of finite amplitude on the surface of a deep fluid[END_REF][START_REF] Craig | Numerical simulation of gravity waves[END_REF]: they consider the potential velocity and they remove the dependence on the vertical variable z working with new unknowns, the free surface elevation and the trace of the potential on the free surface. The second free boundary problem is given by the fact that the portion of the body in contact with the fluid depends on time, so that the contact line between is a free boundary problem. For this difficulty John studied a more simplified problem. He considered a linear model in order to describe the evolution of the free surface waves and he used the potential velocity formulation. Then, he assumed that the motion of the solid is of small amplitude and he neglected the variations of the contact line in time. These assumptions permitted him to avoid the free boundary problem associated with the contact line. Moreover he studied a one-dimensional problem (where d is the horizontal dimension). Another way to avoid this free boundary problem is to consider a structure with vertical side-walls and to assume its motion to be only vertical. Even if John's approach is simplified because it does not take into account nonlinear effects, the linear approach has been used extensively in hydrodynamic engineering. In particular we refer to Cummins who, dealing with ship motion, proposed in [START_REF] Cummins | The Impulse Response Function and Ship Motions[END_REF] his celebrated delay differential equation on the six modes of response: surge, sway, heave, roll, pitch and yaw. In order to take into account nonlinear effects to better describe the real motion of floating bodies, Lannes proposed a different approach in his recent paper [START_REF]On the dynamics of floating structures[END_REF]. He modelled the problem not with the velocity potential theory but using a new formulation. In order to remove the dependence on the vertical variable z, he considered the horizontal discharge, i.e. the horizontal component of the velocity field integrated vertically between the free surface and the fluid domain bottom. He showed that the equations of the problem have a "compressibleincompressible"structure, in which the interior pressure exerted by the fluid on the body is a Lagrange multiplier that one can determine via the resolution of a d-dimensional elliptic equation. He also implemented the same approach on asymptotic models, such as the nonlinear shallow water equations and Boussinesq equations.

In this paper we address the two-dimensional floating body problem, where a cylindrically symmetric structure with vertical side-walls is floating only vertically on an incompressible fluid with irrotational motion. These assumptions on the shape of the solid and its motion permit us to avoid the free boundary problem associated with the contact line and to simplify the problem. Indeed in this case the projection of the portion of the body in contact with the fluid does not depend on time. We suppose that the flow is axisymmetric and without swirl, i.e. we consider a rotation-invariant velocity field with no azimuthal component. Moreover we consider the shallow water regime, which means that the wavelength of the waves is larger than the depth. Consequently, we work with the nonlinear shallow water equations for the flow model, instead of the much more involved free surface Euler equations. The one-dimensional case with any assumption on the solid motion and shape has been studied by Iguchi and Lannes in [START_REF] Iguchi | Hyperbolic free boundary problems and applications to wave-structure interactions[END_REF], where they proposed a general approach to study the wellposedness of initial boundary value problems with a free boundary. The aim of this paper is to prove the local in time well-posedness of this coupled fluid-structure problem in Sobolev spaces. We need enough regular initial data, satisfying some compatibility conditions, in order to get the solution. We consider here a two-dimensional problem, but the axisymmetry keeps the boundary condition maximally dissipative. We use this condition to get better trace estimates which in general we do not have in a two-dimensional case. With all these assumptions we can reduce the problem to a one-dimensional radial problem, and we adapt the classical theory with a reformulation for weighted spaces. The important point is that, instead of John's model, in this paper we take into account nonlinear terms. On the other hand our model still has some limitations. For instance, one should allow the solid to move also in the horizontal direction and to rotate. In these cases, one cannot bypass the study of the evolution of the contact line; moreover the flow would cease to be axisymmetric. The axisymmetric situation considered here is however relevant and can be used to validate the shallow water approach to the floating body problem: indeed, several experimental data with an axisymmetric geometry are available.

Outline of the paper

In Section 2 we write the free surface Euler equations with the constraint that the solid must be in contact with the fluid during all the motion, avoiding air holes between them. Then, we write the nonlinear shallow water approximation for this floating structure problem using the same formulation as Lannes [START_REF]On the dynamics of floating structures[END_REF], with the introduction of the horizontal discharge. Once we have assumed the axisymmetry and the absence of swirl, introducing cylindrical coordinates we reformulate the problem to get a one-dimensional set of equations.

In Section 3 we focus on the "fluid part"of the problem. We write the floating structure problem in the exterior domain (R, +∞) as a quasilinear hyperbolic initial boundary value problem, namely

       ∂ t h e + ∂ r q e + 1 r q e = 0 ∂ t q e + ∂ r q 2 e h e + q 2
e rh e + gh e ∂ r h e = 0 in (R, +∞)

coupled with the boundary condition

q e | r=R = q i| r=R . ( 2
)
where h e = ζ e + h 0 and ζ e are the fluid height and the free surface elevation in the exterior domain respectively (the flat bottom is parametrized by -h 0 ). The exterior horizontal discharge q e is defined as

q e (t, r) = ζ e -h 0 u r (t, r, z)dz in (R, +∞),
where u r is the radial component of the fluid velocity field U. In the boundary condition (2) q i is the interior horizontal discharge, defined as before but for in the interior domain (0, R).

As usual we first show the well-posedness of the associated linear problem in L 2 (rdr). In order to have more regular solutions, the data of the problem must satisfy some compatibility conditions. Then, we apply a standard iterative scheme argument to get the existence and uniqueness of the linear solution.

We treat the solid motion in Section 4. We write the vertical component of Newton's law for the conservation of linear momentum, which describes the motion of the structure, as a nonlinear second order ODE on the displacement of the vertical position of the solid center of mass from its equilibrium position, denoted by δ G . We show that this ODE can be written under the form

(m + m a (δ G )) δG (t) = -cδ G (t) + cζ e (t, R) + b h 2 e (t, R) + β(δ G ) δ2 G (t). (3)
In this equation the terms ζ e (t, R) and h e (t, R) are responsible for the coupling with the fluid equations ( 1). An important point is the presence of the added mass term m a (δ G ) which can be explained by the fact that, in order to move, the solid has to accelerate itself and also the portion of fluid around it. We can find the added mass effect also in other fluid-structure interaction problems. For instance, in the case of a totally submerged solid we refer to Glass, Sueur and Takahashi [START_REF] Glass | Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid[END_REF] and Glass, Munnier and Sueur [START_REF] Glass | Dynamics of a point vortex as limits of a shrinking solid in an irrotational fluid[END_REF]. Moreover, this effect has an important role for the stability of numerical simulations [START_REF] Causin | Added-mass effect in the design of partitioned algorithms for fluid-structure problems[END_REF].

The value of the elevation of the fluid surface at the boundary ζ e (t, R) in ( 3) is the coupling term with the fluid motion. On the other hand we show that the boundary condition in (1) can be written as

q e | r=R = q i| r=R = - R 2 δG ,
showing the retro-action of the solid on the fluid motion.

Finally in Section 5 we write the coupled system modelling the problem and we show the local in time existence and uniqueness introducing an iterative scheme on the coupled fluid-structure system, looking for the solution via a fixed point argument, and we get the following result (see Theorem 5.3 for the rigorous statement):

Theorem 1.1. The coupled system (1) -( 3) is local in time well-posed, provided the initial data are regular enough and verify some compatibility conditions.

In Appendix A we show the details in the case of a non-flat solid bottom, considering that the contact between the solid and the fluid is still on the vertical side-walls, and we derive the corresponding solid motion ODE. In Appendix B we show the proof of a product estimate for functions in H k ((0, T)).

tion. We call C(t) the region occupied by the solid at time t, ∂C(t) the boundary and ∂ w C(t) the portion of the boundary in contact with the fluid, called the wetted surface. The presence of the solid naturally allows to divide the horizontal plane R 2 into two regions, the projection I, of the wetted surface on it, and E := R 2 \ I. We call them interior and exterior domain respectively. The boundary Γ := ∂I = ∂E is called the projection of the contact line, where the solid, the fluid and the exterior air interact. For simplicity we call Γ itself the contact line. These domains do not depend on time since the solid is moving only vertically and is assumed to have vertical side-walls. We consider a wetted surface that can be parametrized as graph of some function ζ w (t, X) for X ∈ I and, like in the water waves theory, we assume that the surface of the fluid is the graph of a function ζ(t, X) for X ∈ R 2 , as shown in Figure 1.

We assume that the fluid is incompressible, irrotational, with constant density ρ and inviscid. For simplicity we consider a flat bottom which can be parametrized by -h 0 with h 0 > 0 and the fluid domain is

Ω(t) = {(X, z) ∈ R 2+1 | -h 0 < z < ζ(t, X)}.
Then, the motion of the fluid is given by the incompressible Euler equation

∂ t U + U • ∇ X,z U = - 1 ρ ∇ X,z P -ge z in Ω(t) (4)
div

U = 0 (5) curl U = 0. ( 6
)
The boundary conditions for the Euler equation on the velocity field U in the fluid domain are the traditional kinematic equation at the surface and the impermeability condition at the bottom, respectively

z = ζ, ∂ t ζ -U • N = 0 with N = -∇ζ 1 (7) z = -h 0 , U • e z = 0 (8)
We consider a configuration when the fluid is completely attached to the solid. Hence we have the following contact constraint:

ζ(t, X) = ζ w (t, X) in I.
Let us denote the restrictions to the interior domain and the exterior domain of a function f defined on R 2 as

f i := f | I f e := f | E . y ζ w (t, X) h(t, X) x z ζ(t, X) I Γ E ζ e ζ i Ω(t)
Figure 1: A cylindrically symmetric floating structure with vertical side-walls

According to this notation the contact constraint becomes the following

ζ i = ζ w in I (9)
Remark 2.1. Since the solid has vertical side-walls, the free surface is not continuous across Γ, i.e.

ζ e = ζ i on Γ.
In the presence of a floating structure we have to change the standard condition on the value of the pressure on the free surface. In the exterior domain it is given by the constant atmospheric pressure P atm , i.e. P e = P atm . [START_REF] Glass | Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid[END_REF] with P = P | z=ζ . In the interior domain the pressure on the free surface is an unknown of the problem, depending on the dynamics of the solid but we know its value on Γ. Indeed, by integrating the vertical component of Euler's equation (4) between z = ζ i and z = ζ e , we have

P i (t, •) = P atm + ρg(ζ e -ζ i ) + ρ ζ e ζ i (∂ t w + U • ∇ X,z w) on Γ, ( 11
)
where w is the vertical component of the velocity field U. The second and the third term do not vanish due to the discontinuity of the free surface on Γ (see Remark 2.1). Moreover one has the continuity of the normal velocity at the vertical side-walls, i.e.

V • ν = V C • ν ( 12
)
where ν is the unit normal vector to Γ pointing towards E , V and V C are the horizontal velocities of the fluid and the solid respectively. As in the standard water waves theory we suppose that the height of the fluid h e (t) does not vanish during all the motion. Hence we have the following assumption:

∃ h m > 0 : h e (t, X) ≥ h m ∀t ∈ [0, T), ∀X ∈ E . ( 13
)
From the physics of the floating structure problem we suppose also that the solid does not touch the bottom of the domain during its motion. This is equivalent to assuming that the height of the fluid h i (t, X) under the solid does not vanish, i.e.

∃ h min > 0 : h w (t, X) ≥ h min ∀t ∈ [0, T), ∀X ∈ I. ( 14
)
with h w (t, X) = h i (t, X) in I due to [START_REF] Glass | Dynamics of a point vortex as limits of a shrinking solid in an irrotational fluid[END_REF]. This assumption is completely relevant for the situation investigated here; we refer to [START_REF] De Poyferr É | A priori estimates for water waves with emerging bottom[END_REF][START_REF] Ming | Elliptic estimates for the Dirichlet-Neumann operator on a corner domain[END_REF] (Euler equation) and [START_REF] Metivier | The shoreline problem for the one-dimensional shallow water and green-naghdi equations[END_REF] (nonlinear shallow water and Green-Naghdi equations) for the analysis of the vanishing depth problem.

Averaged free surface Euler equations with a floating structure

Because of the difficulty to deal with a moving domain, we want to obtain a set of evolution equations on R 2 , without the dependence on the vertical variable z. This idea has been used also for the water waves problem without a floating structure by Zakharov-Craig-Sulem (see [START_REF] Zakharov | Stability of periodic waves of finite amplitude on the surface of a deep fluid[END_REF], [START_REF] Craig | Numerical simulation of gravity waves[END_REF] and [START_REF]Mathematical Surveys and Monographs[END_REF]), who introduce the trace of the velocity potential on the free surface. In the presence of a floating structure we use another formulation, the one that Lannes implemented in his paper [START_REF]On the dynamics of floating structures[END_REF]. We will see that this formulation permits us to write directly the boundary condition [START_REF] Iguchi | Hyperbolic free boundary problems and applications to wave-structure interactions[END_REF]. We define the horizontal discharge

Q(t, X) := ζ(t,X) -h 0 V(t, X, z)dz = h(t, X)V(t, X)
where h(t, X) = h 0 + ζ(t, X) is the fluid height and V is the vertical average of V. Then, we can reformulate the problem: Proposition 2.2. Using the (ζ, Q)-formulation, the water waves problem with a floating structure is modelled by the following system

       ∂ t ζ + ∇ • Q = 0, ∂ t Q + ∇ • 1 h Q ⊗ Q + gh∇ζ + ∇ • R + ha NH (h, Q) = - h ρ ∇P, (15)
with the "Reynolds"tensor R and the non-hydrostatic acceleration a NH as

R(h, Q) = ζ -h 0 V -V ⊗ V -V , a NH (h, Q) = 1 h ζ -h 0 ∇ ζ z (∂ t U + U • ∇ X,z U) • e z .
The surface pressure P is given by P e = P atm and

     -∇ • h ρ ∇P i = -∂ 2 t ζ w + a FS (h, Q) in I P i| Γ(t) = P atm + ρg(ζ e -ζ i ) + P NH on Γ, (16)
where

a FS (h, Q) = ∇ • ∇ • 1 h Q ⊗ Q + gh∇ζ + ∇ • R + ha NH (h, Q) , P NH = ρ ζ e ζ i (∂ t U + U • ∇ X,z U) • e z
and the transition condition at the contact line is

Q e • ν = Q i • ν on Γ. ( 17
)
Remark 2.3. Note that the expressions for R and a NH in the statement above involve the velocity field U = ( V w ). It is shown in [18] that the velocity field is fully determined by the knowledge of ζ and Q, hence the notations R(h, Q) and a NH (h, Q). Sketch of the proof. In order to get [START_REF]On the motion of floating bodies. II. Simple harmonic motions[END_REF] we just need to integrate over the vertical coordinate the horizontal component of the Euler equation ( 4). The condition on the surface in the exterior domain is [START_REF] Glass | Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid[END_REF]. One can note that, in the interior domain, P i is the Lagrange multiplier associated with the constraint [START_REF] Glass | Dynamics of a point vortex as limits of a shrinking solid in an irrotational fluid[END_REF].Then, using the standard argument for incompressible Euler, the elliptic system in ( 16) is obtained by applying the horizontal divergence to [START_REF]On the motion of floating bodies. II. Simple harmonic motions[END_REF] in the interior domain. The boundary condition for the elliptic system comes from [START_REF] Iguchi | A shallow water approximation for water waves[END_REF]. Finally for [START_REF]Mathematical Surveys and Monographs[END_REF] we use the definition of Q e and Q i and ( 12) (see Proposition 10 in [START_REF]On the dynamics of floating structures[END_REF]).

Remark 2.4. The fluid energy is

E fluid = ρ 2 g R 2 ζ 2 + ρ 2 Ω(t) |U| 2 .

The shallow water regime

We are interested here in a simplified model of these equations in the shallow water regime, when the horizontal scale of the problem is much larger than the depth. In the water waves problem the horizontal scale is given by the typical wavelength of the waves.

Remark 2.5. In the floating structure problem a third relevant length in the floating structures problem is the solid width. In particular the ratio of the solid width and the wavelength naturally appears in the adimensionalized equations. Here we consider this quantity as a parameter independent of the other parameters, such as the shallowness and the nonlinear parameters, and it takes no role in the derivation of the asymptotic model.

Following [START_REF]On the dynamics of floating structures[END_REF], the same approximations as in the case without floating object are made, namely R ≈ 0 and a NH ≈ 0.

Then, we get the following equations:

Proposition 2.6. The nonlinear shallow water equations with a floating structure in the (ζ, Q)-formulation are

       ∂ t ζ + ∇ • Q = 0, ∂ t Q + ∇ • 1 h Q ⊗ Q + gh∇ζ = - h ρ ∇P. (18)
with the surface pressure P given by P e = P atm and

     -∇ • h ρ ∇P i = -∂ 2 t ζ w + a FS (h, Q) in I P i| Γ = P atm + ρg(ζ e -ζ i ) | Γ + P cor , (19)
where

a FS (h, Q) = ∇ • ∇ • 1 h Q ⊗ Q + gh∇ζ ,
coupled with the transition condition at the contact line

Q e • ν = Q i • ν on Γ. (20)
Differently from the case considered by Lannes in [START_REF]On the dynamics of floating structures[END_REF], where the jump of pressure P i | Γ(t)

-P atm at the boundary of the object is assumed to be only due to the hydrostatic pressure, i.e.

P i | Γ(t) -P atm = ρg (ζ e -ζ i ) | Γ(t) ,
we add here a non-hydrostatic correction term P cor . This corrector is determined later in Proposition 2.9 below to ensure exact energy conservation (the mathematical analysis if we remove this term can be performed in the same way).

As for the kinematic condition [START_REF] Cummins | The Impulse Response Function and Ship Motions[END_REF], we have that

∂ t ζ w -U w • N w in I with N w = -∇ζ w 1 ( 21
)
where U w is the velocity of the solid on the wetted surface. Let us denote the center of mass of the solid

G(t) = (X G (t), z G (t)) and U G (t) = (V G (t), w G (t)
) its velocity and ω the angular velocity of the solid. From the solid mechanics we have

U w = U G + ω × r G with r G (t, X) = X -X G (t) ζ w (t, X) -z G (t) .
Then, [START_REF] Étivier | Stability of multidimensional shocks[END_REF] gives

∂ t ζ w = (U G + ω × r G ) • N w in I. ( 22
)
Because of the linearity of the elliptic problem we can decompose the interior pressure as P i = P I i + P II i + P III i where: • P I i is the pressure we would have in the case of a fixed solid, solution to

     -∇ • h ρ ∇P I i = a FS (h, Q) in I, P I i | Γ = P atm , (23)
where a FS (h, Q) is the free surface acceleration in the absence of a floating structure;

• P II i is the part of the pressure due to the acceleration of the solid

     -∇ • h ρ ∇P II i = -∂ 2 t ζ w , in I, P II i | Γ = 0, (24)
where w G is the vertical component of the velocity of the center of mass G(t) of the solid;

• P III i is the part of the pressure due to the pressure discontinuity at the contact line

     -∇ • h ρ ∇P III i = 0 in I, P III i | Γ = ρg(ζ e -ζ i ) | Γ + P cor . ( 25
)
ζ w (t, r) h(t, r) z ζ(t, r) R r > R ζ e ζ i Ω(t) R r > R z = -h 0 r < R

Axisymmetric without swirl setting

Without loss of generality we suppose the center of mass to have coordinates

G(t) = (0, 0, z G (t)
) and let R be the radius of the interior domain I. Introducing a cylindrical coordinates system with the z-axis coincident with the axis of symmetry of the solid (see Figure 2) we write the velocity field U as

U(t, r, θ, z) = (u r (t, r, θ, z), u θ (t, r, θ, z), u z (t, r, θ, z)) .
From now on and throughout the paper we consider an axisymmetric flow without swirl, which means that the flow has no dependence on the angular variable θ, i.e. U = U(t, r, z), and u θ = 0 respectively. Hence the horizontal discharge can be written as

Q(t, r) = (q r (t, r), 0) with q r (t, r) = ζ -h 0 u r (t, r, z)dz
and the tangential component vanishes since

q θ (t, r) = ζ -h 0 u θ (t, r, z)dz = 0.
For simplicity we write q instead of q r for the radial component of the horizontal discharge. Moreover, since the solid moves only vertically and the swirl is neglected in the flow, V G = 0 and ω = 0. Hence from [START_REF] Étivier | Small viscosity and boundary layer methods, Modeling and Simulation in Science[END_REF] we have

∂ 2 t ζ w = ẇG .
In the new system of reference the shallow water model ( 15) -( 17) becomes

       ∂ t h + ∂ r q + q r = 0 ∂ t q + ∂ r q 2 h + q 2 rh + gh∂ r h = - h ρ ∂ r P in (0, +∞) (26)
coupled with the transition condition

q e | r=R = q i | r=R . ( 27
)
We have P e = P atm and ( 23) -( 25) become

               -∂ r + 1 r h w ρ ∂ r P I i = ∂ r + 1 r ∂ r q 2 i h w + q 2 i rh w + gh w ∂ r h w in (0, R) P I i | r=R = P atm , (28)
     -∂ r + 1 r h w ρ ∂ r P II i = -ẇG in (0, R) P II i | r=R = 0, (29)
     -∂ r + 1 r h w ρ ∂ r P III i = 0 in (0, R) P III i | r=R = ρg(ζ e -ζ i ) | r=R + P cor , (30)
where we replace h i = ζ i + h 0 with h w = ζ w + h 0 due to the contact constraint [START_REF] Glass | Dynamics of a point vortex as limits of a shrinking solid in an irrotational fluid[END_REF]. Using axisymmetry and absence of swirl together with this change of coordinates we pass from a two-dimensional to a one-dimensional problem, where explicit calculations can be done (see Section 4). With these assumptions, the horizontal discharge is no more a vectorial quantity but a scalar quantity, making the problem easier to handle.

Remark 2.7. Under the shallow water approximation and in the axisymmetric without swirl setting the fluid energy in Remark 2.4 becomes

E SW = 2π ρ 2 g +∞ 0 ζ 2 rdr + 2π ρ 2 +∞ 0 q 2 h rdr. (31)
In the presence of a floating structure the fluid energy E SW is no more conserved by the equations in [START_REF]On the dynamics of floating structures[END_REF]. Let us define the energy for a solid moving only vertically as

E sol = 1 2 mw 2 G + mgz G . ( 32
)
and the total fluid-structure energy

E tot := E SW + E sol . ( 33
)
Notation 2.8.

f := f e | r=R -f i | r=R
is the jump between the exterior and the interior domain at the contact line r = R.

We can now state the following proposition:

Proposition 2.9.

Choosing P cor = ρ 2 q 2 i | r=R 1 h 2 , the total fluid-structure energy is conserved, i.e. d dt E tot = 0. ( 34
)
Proof. Multiplying the first equation in [START_REF] Shibata | On the mixed problem for some quasilinear hyperbolic system with fully nonlinear boundary condition[END_REF] by ρgζr and the second by qr h and summing up, we use the fact that ∂ t h = ∂ t ζ to write the system under the conservative form

∂ t e + ∂ r F = -rq∂ r P, ( 35
)
where e is the local fluid energy

e = ρ 2 gζ 2 r + ρ 2 q 2 h r
and F is the flux

F = ρ q 3 2h 2 r + gζqr .
The conservative form (35) reads in the interior domain (0, R)

∂ t e i + ∂ r F i = -rq i ∂ r P i (36)
and in the exterior domain (R, +∞)

∂ t e e + ∂ r F e = 0 ( 37
)
We integrate (36) on (0, R) and (37) on (R, +∞) and multiplying by 2π we obtain

d dt E SW -2πρR q 3 2h 2 + gζq = -2π R 0 rq i ∂ r (P i -P atm ) dr, (38)
By integration by parts we get

d dt E SW = 2πρR q 3 2h 2 + gζq -2πR (P i -P atm ) | r=R q i | r=R + 2π R 0 (P i -P atm ) ∂ r (rq i )dr.
On the other hand, from the definition of E sol , we have

d dt E sol = mw G ẇG + mgw G = w G (m ẇG + mg) = w G 2π R 0 (P i -P atm ) rdr = 2π R 0 (P i -P atm ) ∂ t ζ w rdr
where we have used Newton's law for the conservation of the linear momentum in the axisymmetric configuration

m ẇG = -mg + 2π R 0 (P i -P atm ) rdr
and [START_REF] Étivier | Small viscosity and boundary layer methods, Modeling and Simulation in Science[END_REF]. From the contact constraint (9) and the mass conservation equation in [START_REF] Shibata | On the mixed problem for some quasilinear hyperbolic system with fully nonlinear boundary condition[END_REF] the following yields:

d dt E sol = -2π R 0 (P i -P atm ) ∂ r (rq i )dr. ( 39
) Therefore d dt E sol = - d dt E SW + 2πρR q 3 2h 2 + gζq -2πR (P i -P atm ) | r=R q i | r=R .
Using the expression of the interior pressure P i on the boundary r = R in ( 28) - [START_REF] Zakharov | Stability of periodic waves of finite amplitude on the surface of a deep fluid[END_REF] and the transition condition [START_REF] Taylor | Partial differential equations III[END_REF] we obtain

d dt (E SW + E sol ) = 2πR ρ 2 q 3 i | r=R 1 h 2 -q i | r=R P cor .
If q i | r=R = 0 the result follows directly. Otherwise, we choose the pressure corrector term P cor in [START_REF] Zakharov | Stability of periodic waves of finite amplitude on the surface of a deep fluid[END_REF] as

P cor = ρ 2 q 2 i | r=R 1 h 2 , ( 40
)
and we get (34).

The fluid equations

In this section we focus on the "fluid part "of the coupled problem. We show that the exterior part of ( 26) can be written as a one-dimensional quasilinear hyperbolic initial boundary value problem in an exterior domain and we shall prove the local in time well-posedness. Like frequently in the literature, throughout this paper we also use the term mixed problem: this comes from the fact that we have, as data of the problem, both the initial (in time) and the boundary (in space) values. Hyperbolic problems in exterior domains have been treated in many works. Métivier [START_REF] Étivier | Small viscosity and boundary layer methods, Modeling and Simulation in Science[END_REF], Benzoni and Serre [START_REF] Serre | Multi-dimensional hyperbolic partial differential equations: First-order Systems and Applications[END_REF] have studied hyperbolic initial boundary value problems in exterior domains with constant coefficients and maximally dissipative boundary condition. Isozaki [START_REF] Isozaki | Singular limits for the compressible Euler equation in an exterior domain. II. Bodies in a uniform flow[END_REF] and Alazard [START_REF] Alazard | Incompressible limit of the nonisentropic Euler equations with the solid wall boundary conditions[END_REF] have studied the singular incompressible limit for the compressible Euler equation in an exterior domain. Concerning the quasilinear hyperbolic mixed problems, Schochet [START_REF] Schochet | The compressible euler equations in a bounded domain: existence of solutions and the incompressible limit[END_REF] has proved the local in time existence in the case of bounded domains and Shibata and Kikuchi [START_REF] Shibata | On the mixed problem for some quasilinear hyperbolic system with fully nonlinear boundary condition[END_REF] have showed the local in time existence for some second order problem in bounded and unbounded domains. Differentiability of solutions to hyperbolic mixed problems has also been studied by Rauch and Massey [START_REF] Rauch | Differentiability of solutions to hyperbolic initial-boundary value problems[END_REF]. The case we are considering here has not been treated in the literature yet. We consider a two-dimensional problem, but the axisymmetry keeps the boundary condition maximally dissipative which in general for a two-dimensional problem is not true. This property is essential for the coupling with the solid motion as it provides us better trace estimates than the ones for the general two-dimensional shallow water equations, but in other cases it is not necessary (see [START_REF] Majda | Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary[END_REF] for elementary examples). Hence we reduce the problem to a onedimensional radial problem, then we must adapt the classical theory.

Let us recall that in the exterior domain we have

       ∂ t h e + ∂ r q e + q e r = 0
∂ t q e + ∂ r q 2 e h e + q 2
e rh e + gh e ∂ r h e = 0.

in (R, +∞)

coupled with the boundary condition

q e | r=R = q i| r=R (42)
Defining u = (ζ e , q e ) T and adding the Cauchy data we can write (41) -(42) as the following quasilinear hyperbolic mixed problem

     ∂ t u + A(u)∂ r u + B(u, r)u = 0 in (R, +∞) e 2 • u | r=R = q i| r=R u(0) = u 0 ( 43
)
with

A(u) =   0 1 gh e - q 2 e h 2 e 2q e h e   , B(u, r) =    0 1 r 0 q e rh e   
and u 0 = (ζ e,0 , q e,0 ) T .

The linear hyperbolic mixed problem

In order to construct the solution to the floating structure problem, which is a quasilinear mixed problem of the form (43) coupled with Newton's equation for the solid motion, we shall use an iterative scheme based on the following linearization of (43),

     L(u)u = ∂ t u + A(u)∂ r u + B(u, r)u = f , e 2 • u | r=R = g, u(0) = u 0 , (44)
with some u and g = q i | r=R . Since the coefficients of A and B are rational fractions, we have

A(•), B(•, r) ∈ C ∞ (U )
for some open set U ∈ R 2 where h e does not vanish, which represents a phase space of u.

Let us assume that we are in the subsonic regime, which means that for u = (ζ e , q e ) the following holds:

q e 2 h e 2 < gh e (45)
with h e = h 0 + ζ e . In the case of water waves in oceans, where the water depth is much bigger than the water velocity, this assumption is satisfied. Then, for the linear initial boundary problem (44) we have the following result: Proposition 3.1. Assume that u satisfies (45). Then, the linear exterior hyperbolic mixed problem (44), which is the linearization of the floating structure equations (43), satisfies the following properties :

(P3) A(u) has one strictly positive eigenvalue λ + (u) and one strictly negative eigenvalue λ -(u), (P4) P -(u)e ⊥ 2 = (0, 0) except in (0, 0), where P -(u) is the projector on the eigenspace associated with the negative eigenvalue of A(u) and e ⊥ 2 is the orthogonal complement of e 2 .

Proof. The eigenvalues of A(u) are λ ± (u) = ± gh e + q e h e and the associated unit eigenvectors are

e ± (u) = 1 1 + λ 2 ± (u) (1, λ ± (u)).
The assumption (45) gives property (P3). We prove now property (P4). Let us denote P + (u) and P -(u) the projectors on the eigenspaces associated with λ + (u) and λ -(u) respectively. They are given explicitly by

P + (u) = A(u) -λ -(u)Id λ + (u) -λ -(u) P -(u) = - A(u) -λ + (u)Id λ + (u) -λ -(u) (46)
Since e ⊥ 2 is of the form ae 1 with a ∈ R, from the definition of P -(u) in ( 46) we have

P -(u)e ⊥ 2 = - a λ + (u) -λ -(u) (λ + (u), -λ + (u)λ -(u)) T , a ∈ R (47)
which is different from zero, except for a = 0, since λ + (u) = 0.

The following lemma is a direct consequence of Proposition 3.1: Lemma 3.2. Assume that u satisfies (45). The linear hyperbolic exterior mixed problem (44) satisfies the following properties:

(P1) The system is Friedrichs symmetrizable, i.e. there exists a symmetric matrix S(u), called the symmetrizer, such that there exist α > 0 such that S(u) ≥ αId and S(u)A(u) is symmetric.

(P2) The boundary condition is maximally dissipative: S(u)A(u) is negative definite on the kernel of the boundary condition e ⊥ 2 , where e ⊥ 2 is the orthogonal complement of e 2 .. Proof. From (P3) we have that λ + (u) > 0 and λ -(u) < 0. We define the symmetrizer S(u) := MP T -(u)P -(u) + P T + (u)P + (u) for some constant M > 0. We compute that

(S(u)v, v) = ((MP T -(u)P -(u) + P T + (u)P + (u))v, v) = M(P -(u)v, P -(u)v) + (P + (u)v, P + (u)v).
Hence , from the decomposition v = P + (u)v + P -(u) we get

(S(u)v, v) ≥ α(v, v) (48)
with α = min(M, 1)/2. The symmetry of S(u) is trivial. We have the following spectral decomposition

A(u) = λ + (u)P + (u) + λ -(u)P -(u). ( 49
)
By the definition of the projectors (46), S(u)A(u) reads

S(u)A(u) = λ -MP T -(u)P -(u) + λ + P T + (u)P + (u) (50)
which is clearly symmetric and we get property (P1). We refer to Taylor (see Prop. 2.2 of [START_REF] Taylor | Partial differential equations III[END_REF]) for a different proof with a general notion of symmetrizer involving pseudo-differential operators.

Let us consider e ⊥ 2 , the one dimensional orthogonal complement of e 2 ∈ R 2 , which is the kernel of the boundary condition. Then, we compute that

(S(u)A(u)e ⊥ 2 , e ⊥ 2 ) = λ -(u)M(P -(u)e ⊥ 2 , P -(u)e ⊥ 2 ) + λ + (u)(P + (u)e ⊥ 2 , P + (u)e ⊥ 2 )
.

Due to property (P4) we obtain property (P2) choosing

M > - λ + (u)(P + (u)e ⊥ 2 , P + (u)e ⊥ 2 ) λ -(u)(P -(u)e ⊥ 2 , P -(u)e ⊥ 2 )
. Remark 3.3. Property (P4) is a reformulation of the uniform Kreiss-Lopatinski ȋ condition. Then, we have just proved that the system (44) admits a Kreiss symmetrizer, which transforms the system into a symmetric one with the additional property that the boundary condition for this symmetric system is maximally dissipative. This property will permit to control the trace of the solution at t he boundary by the standard energy estimate.

Let us now introduce the following space:

X k (T) := k j=0 C j ([0, T], H k-j r ((R, +∞)))
endowed with the norm

u X k (T) := sup t∈[0,T] u(t) X k , u(t) X k = k ∑ j=0 ∂ j t u(t) H k-j r ((R,+∞)) (51)
with • H k r ((R,+∞)) the norm of the weighted Sobolev space H k ((R, +∞), rdr). We first show the following a priori estimate useful to find strong solutions of the problem (44). Proposition 3.4. Let T > 0 and u ∈ X 2 (T) be such that (45) is satisfied. With α as in Lemma 3.2 there are constants c α,R and a non-decreasing function C R (•) on [0, +∞) such that all the solutions u ∈ H 1 r ((0, T) × (R, +∞)) solving (44) satisfy

u(t) 2 L 2 r ((R,+∞)) + u | r=R 2 L 2 ((0,t)) ≤ c α,R e tC α,R (u) × × u 0 2 L 2 r ((R,+∞)) + g 2 L 2 ((0,t)) + t 0 f (τ) 2 L 2 r ((R,+∞)) dτ (52) for all t ∈ [0, T], with C α,R (u) = 1 + α -1 C R ( u X 2 (T) ).
Proof. Following Proposition 2.2 of [START_REF] Serre | Multi-dimensional hyperbolic partial differential equations: First-order Systems and Applications[END_REF], we have from property (P1) and by integrations by parts

d dt (S(u)u, u) L 2 r ((R,+∞)) = -2(S(u)A(u)∂ r u, u) L 2 r ((R,+∞)) -2(S(u)B(u, r)u, u) L 2 r ((R,+∞)) + ((∂ t S(u))u, u) L 2 r ((R,+∞)) + 2(S(u) f , u) L 2 r ((R,+∞)) = S(u)A(u)u | r=R • u | r=R R + (W(u)u, u) L 2 r ((R,+∞)) + (S(u)u, f ) L 2 r ((R,+∞)) . with W(u) = ∂ t S(u) + ∂ r (S(u)A(u)) + 1 r S(u)A(u) -2S(u)B(u, r) (53)
Property (P2) permits us to control the first term on the right-hand side of the inequality, using the following Lemma from Métivier [START_REF] Étivier | Small viscosity and boundary layer methods, Modeling and Simulation in Science[END_REF]:

Lemma 3.5. The symmetric matrix S(u)A(u) is negative definite on e ⊥ 2 , the set of all vectors orthogonal to e 2 , if and only if there are constants c 1 , c 2 > 0 such that for each vector h ∈ C 2 :

-

(S(u)A(u)h, h) ≥ c 1 |h| 2 -c 2 |e 2 • h| 2 .
Choosing h = u | r=R , integrating in time and using property (H1) we have

(S(u)u(t), u(t)) L 2 ((R,+∞)) ≤ (S(u(0))u 0 , u 0 ) L 2 ((R,+∞)) + t 0 (S(u) f (τ), f (τ)) L 2 ((R,+∞)) + c 2 R |g(τ)| 2 -c 1 R u | r=R (τ) 2 dτ + (1 + α -1 W(u) L ∞ ((0,T)×(R,+∞)) ) t 0 (S(u)u(τ), u(τ)) L 2 ((R,+∞)) dτ. ( 54
)
where we have used the fact that S(u) ≥ αId. Moreover, we get the following estimate:

W(u) L ∞ ((0,T)×(R,+∞)) = S (u)∂ t u + S (u)∂ r uA(u) + S(u)A (u)∂ r u + 1 r S(u)A(u) -2S(u)B(u, r) L ∞ ((0,T)×(R,+∞)) ≤ C R ( u W 1,∞ ((0,T)×(R,+∞)) ) ≤ C R u X 2 (T) (55)
for some non-decreasing function C R (•) on [0, +∞), where we have used the fact that r ∈ (R, +∞), the embedding H 2 r ((R, +∞)) → W 1,∞ ((R, +∞)) and the definition of X 2 (T). By Gronwall's Lemma we have

α u(t) 2 X 0 +c 1 R u | r=R 2 L 2 ((0,t)) ≤ e t 1+α -1 C R ( u X 2 (T) ) × × c( u(0) X 2 ) u 0 2 X 0 + c 2 R g 2 L 2 ((0,t)) + α -1 t 0 f (τ) 2 X 0 dτ .
We get (52) for

C α,R (u) = 1 + α -1 C R ( u X 2 (T) ), c α,R = max(c( u(0) X 2 ), c 2 R) min(α, c 1 R) .
Then, following Theorem 2.4.5 of [START_REF] Étivier | Small viscosity and boundary layer methods, Modeling and Simulation in Science[END_REF], one can show that there is a unique solution u ∈ C 0 [0, T], L 2 r ((R, +∞)) for the initial datum u 0 in L 2 r ((R, +∞)) and boundary value g in L 2 ((0, T)) . This solution satisfies the energy estimate (52).

Regular solutions. To solve the mixed problem in Sobolev spaces we need some compatibility conditions. For instance, the initial and the boundary conditions imply that necessarily

e 2 • u 0 | r=R = g | t=0 = e 2 • u | t=0,r=R , (56)
if the traces are defined. Let us consider the generic equation

∂ t u = f -A(u)∂ r u -B(u, r)u.
Then, we can formally (this is the meaning of the brackets "") define

"∂ t u | t=0 " = -A(u 0 ) | t=0 ∂ r u 0 -B(u 0 , r) | t=0 u 0 + f | t=0 .
Hence, provided traces are defined,

e 2 • "∂ t u | t=0 " | r=R = e 2 • ( f 0 -A(u 0 )∂ r u 0 -B(u 0 , r)u 0 ) | r=R = g 1
where g 1 := ∂ t g | t=0 . These conditions are necessary for the existence of a smooth solution. We can continue the expansion to higher orders looking for more compatibility conditions. Let us introduce the following notation (as in [START_REF] Étivier | Stability of multidimensional shocks[END_REF] and [START_REF] Schochet | The compressible euler equations in a bounded domain: existence of solutions and the incompressible limit[END_REF]):

Notation 3.6. Let us write the linear equation in (44)

∂ t u = F(u, ∂)u + f with F(u, ∂) = -A(u)∂ r -B(u, r).
We define formally the traces u j := "∂ j t u | t=0 " as functions of u 0 determined inductively by u 0 = u | t=0 u j+1 = F j (u 0 , ...u j ) + f j (57)

with

F j (u 0 , ...u j ) = ∑ p+|k|≤j A j,p,k (u 0 )u (k) ∂ r u p + ∑ p+|k|≤j B j,p,k (u 0 , r)u (k) u p ( 58
)
where we use the notation

for k = (k 1 , . . . , k r ), u (k) = u k 1 . . . u k r .
Note that u j is not the derivative of a known function but rather the value that the derivative of u will have if u exists. Therefore necessarily smooth enough solutions to (44) must satisfy

e 2 • u j | r=R = g j (59) with g j := ∂ j t g | t=0 .
Definition 3.7. The data u 0 ∈ H k r (R + ), g ∈ H k ((0, T)) and f ∈ H k r ((0, T) × R + ) satisfy the compatibility conditions up to order s ≤ k -1 if (59) holds for each j = 0, ..., s.

For the linear floating structure mixed problem (44) the compatibility conditions (59) on the initial data u 0 = (ζ e,0 , q e,0 ) and the boundary value g = q i | r=R can be written as:

q e | r=R,t=0 = q i | r=R,t=0
, F j 2 (ζ e,0 , ..., ζ e,j , q e,0 , ..., q e,j )

| r=R = ∂ j t q i | r=R,t=0 , j ≥ 1.
(60) where F j 2 is the second component of F j . As in the L 2 case, our goal is to find an a priori estimate for the linear problem (44) in order to get existence and uniqueness of the solution in some more regular space. Proposition 3.8. Let T > 0 and k ≥ 1 be an integer. Assume that u ∈ X s (T) with s = max(k, 2) satisfies (45). With α as in Lemma 3.2 there are a constant c α,R and non-decreasing functions

C R (•), C 1,k,R (•) and C 2,k,R (•) on [0, +∞) such that all the solutions u ∈ H k+1 r ((0, T) × (R, +∞)) solving (44) satisfy: u(t) 2 X k + u | r=R 2 H k ((0,t)) ≤ c α,R e tC α,R,k ( u X s (T) ) × × u 0 2 X k + g 2 H k ((0,t)) + K k,R ( u X s (T) ) t 0 f (τ) 2 X k dτ (61)
for all t ∈ [0, T] with

C α,R,k ( u X s (t) ) = 1 + α -1 C R ( u X 2 (t) ) + α -1 (k + 1)C 1,k,R ( u X s (t) ) and K k,R ( u X s (T) ) = C 2,k,R ( u X s (T) ) max(c( u(0) X 2 ), c 2 R) .
Proof. We adapt here the argument presented in [START_REF] Étivier | Small viscosity and boundary layer methods, Modeling and Simulation in Science[END_REF]. We denote by u i the tangential derivative ∂ i t u for i ≤ k, which in the one dimensional case is simply the time derivative, and we introduce the tangential norm

u(t) X k := k ∑ i=0 ∂ i t u(t) L 2 r ((R,+∞))
We apply ∂ i t to the equation of (44) and we get

∂ t u i + A(u)∂ r u i + B(u, r)u i = [A(u), ∂ i t ]∂ r u + [B(u, r), ∂ i t ]u + f i , ( 62
)
e 2 • u i | r=R = g i .
As we have done in the previous L 2 case we consider d dt (S(u)u i , u i ) L 2 r ((R,+∞)) . The only difference from the previous case is the presence of the two commutator terms in (62). We need to control their L 2 r norms in a different way. The first term can be written under the form ∑ α=1,...,i

∂ α t (A(u))∂ i-α t ∂ r u L 2 r .
For α ≤ k -1 every term of the sum is controlled by

∂ α t (A(u)) L ∞ ∂ i-α t ∂ r u L 2 r ≤ c R ∂ α t (A(u)) H k-α r u X k-α+1 ≤ c R A(u) X k u X k ( 63
)
using the fact that 1 ≤ α ≤ k -1 for the Sobolev embedding and that

• X λ ≤ • X δ if λ < δ.
Recall that Sobolev embeddings H k r → W s,∞ still hold for the weighted spaces H k r since we are considering the exterior domain (R, +∞). For α = k we directly have

∂ k t (A(u)) L 2 r ∂ r u L ∞ ≤ c R A(u) X k u X k since u ∈ H k+1 r with k ≥ 1.
We can find the same estimate for the commutator term with B(u, r). We recall the following Moser-type estimate for the • X k norm: Lemma 3.9 (Schochet). For A(•) smooth enough, the following holds

A(u) X k ≤ C k (1 + u k X k ) ( 64
)
with k ≥ 1.

We refer to the Appendix B of [START_REF] Schochet | The compressible euler equations in a bounded domain: existence of solutions and the incompressible limit[END_REF] for the details of the proof based on Gagliardo-Nirenberg inequalities. Better estimates can be used, in particular if one wants to derive blow up conditions as Métivier in [START_REF] Étivier | Stability of multidimensional shocks[END_REF], but here we are not interested in this problem. Hence we get the following inequality:

S(u)u i (t), u i (t) L 2 ((R,+∞)) ≤ S(u(0))u i 0 , u i 0 L 2 ((R,+∞)) + t 0 S(u) f i (τ), f i (τ) L 2 ((R,+∞)) + c 2 R g i (τ) 2 -c 1 R u i | r=R (τ) 2 dτ + 1 + α -1 C R ( u X 2 (T) ) t 0 (S(u)u i (τ), u i (τ)) L 2 ((R,+∞)) dτ + C k,R ( u X s (T) ) t 0 u(τ) 2 X k dτ (65)
with s = max(k, 2) and some non-decreasing function C k,R (•) on [0, +∞). Here u i 0 = u i where the u i are defined in (57). We note that for solutions to (44), we have

u(0) X k = k ∑ j=0 u j L 2 r , u(0) X k = k ∑ j=0 u j H k-j r .
For u satisfying the equation ( 44), we have

∂ r u = A -1 (u)( f -∂ t u -B(u, r)u)
and we can show that the X k norm is controlled by the tangential one. The following holds for 0 ≤ t ≤ T:

u(t) 2 X k ≤ C 1 ( u X s (T) ) u(t) X k 2 + C 2 ( u X s (T) ) f (t) 2 X k ≤ C 1 ( u X s (T) )α -1 k ∑ i=0 S(u)u i , u i L 2 ((R,+∞)) + C 2 ( u X s (T) ) f (t) 2 X k
(66) with some non-decreasing functions C 1 (•), C 2 (•) on [0, +∞). In the second inequality we have used the fact that S(u) ≥ αId. By taking the sum for i from 0 to k and by applying Gronwall's Lemma we obtain

α u(t) X k 2 + c 1 R u | r=R 2 H k ((0,t)) ≤ e t 1+α -1 C R ( u X 2 (T) )+α -1 (k+1)C 1,k,R ( u X s (T) ) × × c( u(0) X 2 ) u(0) X k 2 + c 2 R g 2 H k ((0,t)) + C 2,k,R ( u X s (T) ) t 0 f (τ) X k 2 dτ
(67) with some non-decreasing functions C 1,k,R (•), C 1,k,R (•) on [0, +∞). By definition of the tangential norm we have u 0 X k 2 ≤ u 0 X k 2 . We use (66) and the estimate (61) follows with

c α,R = max(c( u(0) X 2 ), c 2 R) min(α, c 1 R) , C α,R,k ( u X s (T) ) = 1 + α -1 C R ( u X 2 (T) ) + α -1 (k + 1)C 1,k,R ( u X s (T) )
and

K k,R ( u X s (T) ) = C 2,k,R ( u X s (T) ) max(c( u(0) X 2 ), c 2 R) .
Equivalently to the L 2 -case, we can state the following theorem:

Theorem 3.10. Let k ≥ 1 be an integer and T > 0. Suppose u 0 ∈ H k r ((R, +∞)), g ∈ H k ((0, T)) and f ∈ H k r ((0, T) × (R, +∞)) satisfy the compatibility conditions (59) up to the order k -1. Assume that u ∈ X s (T) with s = max(k, 2) satisfies (45). Then, there is a unique solution u ∈ X k (T) to (44). Its trace on r = R belongs to H k ((0, T)) and u satisfies the energy estimate (61).

Proof. We show only the idea of the proof of the existence. For more details and the proof of uniqueness see [START_REF] Étivier | Stability of multidimensional shocks[END_REF]. First we solve the equation with a loss of smoothness. We consider the data u 0 , f and g in H k+2 r satisfying the compatibility conditions up to order k. One can prove that there is a solution in H k+1 r ((0, T) × (R, +∞)) ⊂ X k (T), by extending the data by 0 for t < 0 and then by applying the existence result for the mixed problem in (-∞, T] × (R, +∞) of [START_REF] Étivier | Small viscosity and boundary layer methods, Modeling and Simulation in Science[END_REF]. The second step is to consider H k -data: we use the compatibility conditions up to order k -1 to approximate u 0 , f and g in H k r and H k with sequences

u n 0 ∈ H k+2 r ((R, +∞)), f n ∈ H k+2 r ((0, T) × (R, +∞
)) and g n ∈ H k+2 ((0, T)) satisfying the compatibility conditions up to order k + 1. From the previous argument and the energy estimate (61) we have that u n is a Cauchy sequence in X k (T) and therefore converges to the limit u ∈ X k (T), which is a solution to (44) since k ≥ 1.

The quasilinear problem and application to the case of a solid with prescribed motion

In the particular case of the floating structure problem, the boundary condition in (43) is g = q i | r=R , the value of the horizontal discharge in the interior domain at the boundary r = R. We will see in the next section that this quantity is strictly linked to the solid motion, in particular to the vertical component of the velocity of the center of mass w G (t).

In the case of a solid with prescribed motion, the boundary condition g is a datum of the problem. Hence, after having studied the linear problem (44), one can use a standard iterative scheme argument in order to get existence and uniqueness of the solution to (43). Theorem 3.11. Consider a solid with a prescribed motion. For k ≥ 2, let u 0 ∈ H k r ((R, +∞)) and w G ∈ H k ((0, T)) satisfy the compatibility conditions in Definition 5.2 up to order k -1. Assume that u 0 satisfies (45). Then, the fluid problem (76) with boundary condition -R 2 w G (t) admits a unique solution u ∈ X k (T) with X k (T) as in (51).

Sketch of the proof. We introduce the iterative scheme by defining the sequence (u n ) n with u n solution to the linear problem L(u n-1 )u n = 0. The existence of such a sequence is given by Theorem 3.10. Once we have showed the control of the sequence in some "big norm"and the convergence in some "small norm", the limit u of (u n ) n is the solution to (43). For more details we refer to [START_REF] Étivier | Stability of multidimensional shocks[END_REF]. We will show a detailed proof in the case of a free motion in Theorem 5.3 below.

From this point on we consider a solid with free motion. Therefore the boundary condition is still an unknown of the problem and we must adapt the classical argument used in Theorem 3.11 to our problem introducing an iterative scheme for the fluid-structure coupled system. The details of this coupled iterative scheme argument are given in Section 5. Before, we deal with the solid problem and we deduce an ordinary differential equation describing the motion of its center of mass.

The solid equation

In this section we address the motion of the solid. We recall that we are considering a floating structure moving only vertically. Denoting m the mass of the body, g the gravity acceleration and z G the vertical position of the center of mass, we consider only the vertical component of Newton's law for the conservation of linear momentum:

m zG = -mg + F fluid (68)
where F fluid = 2π R 0 (P i -P atm )rdr is the resulting vertical force exerted by the fluid on the solid. Let us introduce the displacement δ G (t) := z G (t)z G,eq between the vertical position of the center of mass at time t and at its equilibrium. In the case of vertical motion h w (t, r) = h w,eq (r) + δ G (t), where h w,eq is the fluid height at the equilibrium. For simplicity we consider a cylindrically symmetric solid with flat bottom, which means that the wetted surface ζ w (hence h w ) does not depend on the spatial coordinate in the interior domain (0, R). See Appendix A for the general case with a cylindrically symmetric solid with a non-flat bottom. Proposition 4.1. Newton's law (68) can be written under the following form:

(m + m a (δ G )) δG (t) = -cδ G (t) + cζ e (t, R) + b h 2 e (t, R) + β(δ G ) δ2 G (t) (69) with c = ρgπR 2 , b = πρR 4 8 , m a (δ G ) = b h w (δ G ) = b h w,eq + δ G (t) , β(δ G ) = b 2h 2 w (δ G ) = b 2(h w,eq + δ G (t)) 2 .
Remark 4.2. In (69) m a (δ G ) is called the added mass term and it represents the fact that, in order to move in the fluid, the solid has to accelerate itself but also the portion of fluid next to it. This effect appears in other hydrodynamical configurations, in particular for totally submerged solids studied by Glass, Sueur and Takahashi [START_REF] Glass | Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid[END_REF] and Glass, Munnier and Sueur [START_REF] Glass | Dynamics of a point vortex as limits of a shrinking solid in an irrotational fluid[END_REF]. It has an important role for the stability of numerical simulations of fluid-structure interactions [START_REF] Causin | Added-mass effect in the design of partitioned algorithms for fluid-structure problems[END_REF].

The coupling with the fluid motion is given by the term ζ e (t, R) and b h 2 e (t, R)

, which means that the solid motion depends on the value of the elevation of the exterior free surface at the boundary r = R.

Proof. We decompose F fluid according to the decomposition ( 28) -( 30 Using the elementary potential Φ r I defined in [START_REF]On the dynamics of floating structures[END_REF] we can write

F II fluid = -2π R 0 P II i ∂ r + 1 r (h w ∂ r Φ r I ) rdr,
and, after integration by parts,

F II fluid = -2π R 0 ∂ r + 1 r h w ∂ r P II i Φ r I rdr = -2πρ R 0 ẇG Φ r I rdr,
where the second equality comes from the definition (29) of P II i . Using again the definition of elementary potential we obtain

F II fluid = 2πρ R 0 Φ r I ∂ r + 1 r (h w Φ r I ) ẇG rdr = -2πρ R 0 1 h w (h w ∂ r Φ r I ) 2 ẇG rdr.
From the definition of the elementary potential we explicitly have that

h w ∂ r Φ r I = - r 2 .
It follows that F II fluid = -m a (h w ) ẇG , with m a (h w ) as in (69). Proceeding similarly we can write also

F I fluid = -2πρ R 0 r 2h w h w ρ ∂ r P I i rdr.
Then, (68) becomes

(m + m a (h w )) ẇG = -mg + F I fluid + 2π R 0 P III i rdr. (70)
Moreover, ( 28) can be written as

∂ r y(r) + y(r) r = - b(r) r in (0, R) with y(r) = h w ρ ∂ r P I i + ∂ r q 2 i h w and b(r) = ∂ r q 2 i h w .
Hence we have

y(r) = - 1 r q i (r) 2 h w - q i (0) 2 h w (0) . ( 71
)
Because of the constraint (9), the mass conservation equation of [START_REF] Shibata | On the mixed problem for some quasilinear hyperbolic system with fully nonlinear boundary condition[END_REF] in the interior domain becomes

∂ r q i + 1 r q i = -δG (t) in (0, R) then we have q i (t, r) = - r 2 δG (t). ( 72
)
Hence q i (t, 0) = 0 and (71) becomes

h w ρ ∂ r P I i = -∂ r q 2 i h w - 1 r q 2 i h w = - 3 4h w r δ2 G .
Replacing the expression of

These two equalities, together with the constraint

ζ i = ζ w , give -mg = cζ i (t, R) -cδ G (t).
Solving the elliptic problem [START_REF] Zakharov | Stability of periodic waves of finite amplitude on the surface of a deep fluid[END_REF] whose solution is the constant (in space) boundary value, we obtain the nonlinear second order ordinary differential equation (69). with particular p and q = Φ r I , where div r = ∂ r + 1 r is the divergence in the axisymmetric configuration.

Recall that in (72) we have

q i (t, R) = - R 2 δG (t).
This term is the boundary value in the fluid mixed problem (41). It follows that this is the coupling term between the fluid and the solid motion in the fluid system, as ζ e (t, R) has the same property in the solid equation (see Remark 4.2).

Fluid-structure coupling

From the previous two sections, it follows that the fluid-structure interaction problem considered in this paper is described by the following mathematical model:

Proposition 5.1. The nonlinear shallow water equations with a floating structure for an axisymmetric flow without swirl take the form

       ∂ t u + A(u)∂ r u + B(u, r)u = 0 in (R, +∞) e 2 • u | r=R = - R 2 δG (t) u(0) = u 0 , (76)
with A(u), B(u, r) as in (43). Moreover the solid motion is given by the Cauchy problem

     (m + m a (δ G )) δG (t) = -cδ G (t) + ce 1 • u | r=R + (b(u) + β(δ G )) δ2 G (t), δ G (0) = δ 0 , δG (0) = δ 1 , (77) with m a (δ G ), β(δ G ) as in (69), b(u) = b e 1 • u | r=R + h 0 2 = b h e (t, R) 2 ,
using the fact that

ζ e (t, R) = e 1 • u | r=R and h e (t, R) = h 0 + ζ e (t, R).
Let us give the notion of compatibility conditions in the case of this particular fluid-structure coupled problem. We recall the equation in (76)

∂ t u = F(u, ∂)u with F(u, ∂) = A(u)∂ r + B(u, r) (78)
We define formally the traces u j := "∂ j t u | t=0 " as functions of u 0 determined inductively by

u 0 = u | t=0 u j+1 = F j (u 0 , ...u j ) (79)
with

F j (u 0 , ...u j ) = ∑ p+|k|≤j A j,p,k (u 0 )u (k) ∂ r u p + ∑ p+|k|≤j B j,p,k (u 0 , r)u (k) u p .
where we use the notation

for k = (k 1 , . . . , k r ), u (k) = u k 1 . . . u k r .
Definition 5.2. The data u 0 ∈ H k r ((R, +∞)), δ 0 ∈ R and δ 1 ∈ R of the floating structure coupled system (76) -(77) satisfy the compatibility conditions up to order k -1 if, for 0 ≤ j ≤ k -1, the following holds:

e 2 • u j | r=R = - R 2 δ j+1 ,
where δ j+1 are the formal traces " d j+1 dt j+1 δ G (0) " defined from the ODE in (77) as

d j-1 dt j-1 1 (m + m a (δ G )) -cδ G + ce 1 • u | r=R + (b(u) + β(δ G )) δ2 G (0).
In the following theorem we prove that the coupled model ( 76) -( 77) is locally in time well-posed: Theorem 5.3. For k ≥ 2, let u 0 = (ζ e,0 , q e,0 ) ∈ H k r ((R, +∞)), δ 0 and δ 1 satisfy the compatibility conditions in Definition 5.2 up to order k -1. Assume that there exist some constants h min , c sub > 0 such that ∀r ∈ (R, +∞) : h e,0 (r) ≥ h min , gh e,0 -q 2 e,0 h 2 e,0 (r) ≥ c sub ,

with h e,0 = h 0 + ζ e,0 , and that δ 0 > -h w,eq with the constant h w,eq as in Section 4. Then, the coupled problem (76) -(77) admits a unique solution (u, δ G ) ∈ X k (T) × H k+1 ((0, T)) with X k (T) as in (51).

Remark 5.4. Considering an initial datum u 0 ∈ H 2 r ((R, +∞)), we need the following compatibility conditions satisfied:

q e (0, R) = - R 2 δ 1 and -∂ r q 2 e h e (0, R) - 1 R q 2 e h e (0, R) -gh e (0, R)∂ r ζ e (0, R) = - R 2 (m + m a (δ 0 )) -cδ 0 + cζ e (0, R) + b h 2 e (0, R) + β(δ 0 ) δ 2 1 .
For instance let us take the initial configuration of the fluid-structure interaction as the following: the solid displaced from its equilibrium position with no initial velocity, which means δ 0 = 0 and δ 1 = 0, and the fluid such that

h e (0, R) = h 0 , q e (0, R) = 0, ∂ r ζ e (0, R) = - cδ 0 R 2 (m + m a (δ 0 )) gh 0 .
Then, the initial conditions are compatible and we can apply Theorem 5.3.

Proof. We adapt here the argument that Métivier used in [START_REF] Étivier | Stability of multidimensional shocks[END_REF] for the existence and uniqueness of the solution to the fluid mixed problem and we couple it with the solid equation. Similar techniques are used in [START_REF] Iguchi | Hyperbolic free boundary problems and applications to wave-structure interactions[END_REF] by Iguchi and Lannes.

We introduce the following iterative scheme for the coupled system (76) -(77). For u 0 ∈ H k r ((R, +∞)), let us consider the linear mixed problem

       ∂ t u n + A(u n-1 )∂ r u n + B(u n-1 , r)u n = 0, in (R, +∞) e 2 • u n | r=R = - R 2 δn-1 G (t) u n (0) = u 0 .
(80) and the linear ODE

       (m + m a (δ n-1 G )) δn G = -cδ n G + ce 1 • u n | r=R + b(u n-1 ) + β(δ n-1 G ) δn-1 G δn G , δ n G (0) = δ 0 , δn G (0) = δ 1 ,
(81) Our goal is to find the solution of the coupled system as the limit of the previous iterative scheme. Hence we need to show the existence and the convergence of the sequence V n = (u n , δ n G ). We consider the product space X k (T) × H k+1 ((0, T)) endowed with the norm

V n coup,k := u n X k (T) + δ n G H k+1 ((0,T))
with the X k (T) norm defined as in (51). We denote by E the subspace

E = {V = (u, δ G ) ∈ X k (T) × H k+1 ((0, T)) | V coup,k ≤ R},
for some R > 0 to determine later, such that ∀t ∈ [0, T), ∀r ∈ (R, +∞) :

h e (t, r) ≥ C 0 , gh e - q 2 e h 2 e (t, r) ≥ c 0 and δ G -δ 0 L ∞ ((0,T)) ≤ M 0 for some constants 0 < C 0 ≤ h min , 0 < c 0 ≤ c sub and M 0 = δ 0 +h w,eq 2
> 0. We choose the first element of the sequence

V 0 = (u 0 , δ 0 G ) with u 0 ∈ H k+ 1 2 r (R × (R, +∞)) such that ∂ j t u 0 | t=0 = u j , 0 ≤ j ≤ k
with u j as in (79). We can assume that u 0 vanishes for |t| ≥ 1, hence u 0 ∈ X k (T) for all T. There exists a constant K 0 = K 0 (u 0 , δ 0 , δ 1 ) depending only on the data such that u

0 X k (T) + δ 0 G H k ((0,T)) ≤ K 0 . (82)
We have that

V 0 ∈ E choosing R ≥ K 0 . We suppose that V n-1 = (u n-1 , δ n-1 G ) is constructed in E ⊆ X k (T) × H k+1 ((0, T)) for some T > 0 with ∂ j t u n-1 | t=0 = u j , j ≤ k. (83)
For n = 1 this is true. By the definition (58) of F j and by (83),

∂ j t (F(u n-1 , ∂)u n ) | t=0 = F j (u 0 , ..., u j )
with u = u n-1 . Now we consider the linear problem (80). We compute u n j using (57). We can see that u n j = u j with the u j defined before. Then, the compatibility conditions e 2 • u j | r=R = -R 2 δ j+1 imply that the data δn-1 G and u 0 are compatible for the linear problem. From Theorem 3.10 the system (80) has a unique solution

u n ∈ X k (T) and ∂ j t u n | t=0 = u n j = u j . Moreover, u n (0) X k = ∑ j≤k u j H k r ≤ K 0 .
in the following way:

δn

G H k-1 √ T c m + m a (δ n-1 G ) H k-1 δ n G H k-1 + e 1 • u n | r=R H k-1 + √ T b(u n-1 ) + β(δ n-1 G ) m + m a (δ n-1 G ) δn-1 G H k-1 δn G H k-1 + C 0 c m + m a (δ n-1 G ) H k-1 + D 0 δ n G H k-1 + e 1 • u n | r=R H k-1 + C 1 b(u n-1 ) + β(δ n-1 G ) m + m a (δ n-1 G ) δn-1 G H k-1 + D 1 δn G H k-1 with C 0 = C 0 |δ n 0 | , ..., δ n k-2 , e 1 • u n | r=R (0) , ..., d k-2 dt k-2 e 1 • u n | r=R (0) , D 0 = D 0 c m + m a (δ n-1 G ) (0) , ..., d k-2 dt k-2 c m + m a (δ n-1 G ) (0) , C 1 = C 1 |δ n 1 | , ..., δ n k-1 , D 1 = D 1 b(u n-1 ) + β(δ n-1 G ) m + m a (δ n-1 G ) δn-1 G (0) , ..., d k-2 dt k-2 b(u n-1 ) + β(δ n-1 G ) m + m a (δ n-1 G ) δn-1 G (0) .
By applying Lemma 5.5 several times to the products in the previous estimate and using the fact that for V n-1 = (u n-1 , δ n-1 G ) ∈ E the denominators are bounded from below, we get δn

G H k-1 C(T, δ n-1 G H k+1 , u n-1 H k ) δ n G H k+1 + C(T, δ n-1 G H k+1 ) u n
| r=R H k (88) Here the constants may not tend to zero as T goes to zero but they are bounded. Then, using the control for V n-1 coup,k , for T small enough we can move the first term in the right hand side of (88) to the left of the inequality (87) and (86) follows.

From (84) and (86), we get the following estimate for the coupled norm:

V n coup,k ≤ u n X k (T) + δ n G H k+1 ((0,T)) ≤ u n X k (T) + α(T, R) + β(T, R) u n
| r=R H k ((0,T))

≤ C(T, R, K 0 , u n-1 X k (T) )(K 0 + δ n-1 G H k+1 ((0,T)) ) + α(T, R).

Using again the control for V n-1 coup,k , we can find some R ≥ K 0 such that for T small enough V n coup,k ≤ R. For u 0 = (ζ e,0 , q e,0 ) we have

≤ T A(u n ) X k (T) ≤ TC(1 + R k )
where in the last inequality we have used (64). Finally,

δ n G -δ 0 L ∞ ((0,T)) ≤ √ T δ n G H k+1 ((0,T)) ≤ √ T R.
Hence, using the assumption on the initial data, the time existence T can be shorten to get ∀t ∈ [0, T), ∀r ∈ (R, +∞)

h n e (t, r) ≥, gh n e -q n e h n e 2 (t, r) ≥ c 0 and δ n Gδ 0 L ∞ ((0,T)) ≤ M 0 . Now we look for the convergence of the sequence V n in a "smaller" norm. We consider the space X 0 (T) × H 1 ((0, T)) = C 0 ([0, T], L 2 r ((R, +∞))) × H 1 ((0, T)). We have that u nu n-1 satisfies

             ∂ t u n -u n-1 + A(u n-1 )∂ r u n -u n-1 + B(u n-1 , r) u n -u n-1
= -A(u n-1 ) -A(u n-2 ) ∂ r u n-1 -B(u n-1 , r) -B(u n-2 , r) u n-1 ,

e 2 • u n -u n-1 | r=R = - R 2 δn-1 G (t) -δn-2 G (t) , u n -u n-1 (0) = 0.

A Solid with non-flat bottom

We derive here the equation for the solid motion, as in Section 4, in the more general case of a solid with non-flat bottom. Due to the fact that the interior and exterior domains do not change during the motion, we suppose that the contact between the free surface and the floating structure takes place on the vertical side-walls during all the motion. Then, we can state the following proposition: Proof. We derive only the expression of F I fluid and F III fluid in the case of a solid with a non-flat bottom. The added mass term comes from the fact that F II fluid can be written as [START_REF] Glass | Dynamics of a point vortex as limits of a shrinking solid in an irrotational fluid[END_REF]. Putting all these expressions in Newton's law (68) and integrating by parts, we get (93).

F II fluid = -

B Proof of Lemma 5.5

We prove here the product estimate (94): Proposition B.1. Let k ≥ 1 be an integer. For f , g ∈ H k ((0, T)) the following estimate holds: (94)

Figure 2 :

 2 Figure 2: Vertical cross section of the axisymmetric configuration

  ) of the pressureF fluid = F I fluid + F II fluid + j = II, III.

  definition of the equilibrium state, we have -mg -2πρg R 0 ζ w,eq rdr = 0. (73) Since the solid has vertical side-walls the following equality holds 2πρg R 0 ζ w (t)rdr -2πρg R 0 ζ w,eq rdr = cδ G (t).

Remark 4 . 3 .

 43 All the computations in the proof of Proposition 4.1 reduce to particular cases of the Green's identity: R 0 pdiv r (h w ∂ r q)rdr = r=R h w p∂ r qr -r=R h w q∂ r pr + R 0 qdiv r (h w ∂ r p)rdr (75)

0 ∂ 2 L

 02 t h n e ≤ T ∂ t h n e L ∞ ((0,T))L ∞ r ((R,+∞)) ≤ T u n X k (T) ≤ T R. ∞ ((0,T))L ∞ r ((R,+∞))

Proposition A. 1 . 3 w

 13 In the case of a solid with non-flat bottom, Newton's law (68) can be written under the following form:(m + m NF a (δ G )) δG (t) = -cδ G (t) + cζ e (t, R) + b h 2 e (t, R) + β NF (δ G ) δ2 G (t) (93) with c = ρgπR 2 , (δ G , r) ∂ r h w (δ G , r) dr,and the dependence on δ G given by h w (δ G , r) = δ G (t) + h w,eq (r).

Remark A. 2 .

 2 One can note that in the case of a solid with flat bottom (∂ r h w (δ G , r) = 0)m NF a (δ G ) = m a (δ G ), β NF (δ G ) = β(δ G ),and (93) coincides with (69).

0 gr 2 ∂

 02 t, r)drcδ G (t). Now we observe that the term πρ R 0 gr 2 ∂ r h w dr can be written by integration by parts as πρ R r h w dr = cζ i (t, R) -2πρg R 0 rζ i (t, r)dr where ∂ r h w = ∂ r ζ w = ∂ r ζ i using the constraint

  f g H k ((0,T)) √ T f H k ((0,T)) g H k ((0,T)) + (| f (0)| + | d dt f (0)| + ... + | d k-1 dt k-1 f (0)|) g H k ((0,T)) + (|g(0)| + | d dt g(0)| + ... + | d k-1 dt k-1 g(0)|) f H k ((0,T))

  m NF a (h w ) ẇG with Also in the case of a solid with non-flat bottom, (30) admits the unique constant solutionP III i (t, r) = ρg(ζ e (t, R)ζ i (t, R)) + (t, R) = -R 2 δG (t). By definition of F III fluid we haveF III fluid = c(ζ e (t, R)ζ i (t, R)) + bThe relations (73) and (74) still hold but in this case we obtain -mg = 2πρg

	we get
	-	h w ρ	∂ r P I i = ∂ r		q 2 i h w	+	q 2 i rh w	+ gh w ∂ r h w .
	Using the formula for the horizontal discharge in the interior domain q i (t, r) = -r 2 δG (t), we obtain that
	F I fluid = πρ	0	R	3r 3 4h 2 w	δ2 G + (gr 2 -	r 4 4h 3 w	δ2 G )∂ r h w dr.
								ρ 2	q 2 i (t, R)	1 e (t, R) h 2	-	1 w (t, R) h 2	,
	with q i 1 h 2 e (t, R)	-	1 w (t, R) h 2	δ2 G (t).
				m NF a (h w ) =	ρπ 2	0	R	r 3 h w	dr.
	By definition,
	F I fluid = 2πρ	0	R	r 2h w	-	h w ρ	∂ r P I i rdr
	with P I i defined as the solution to (28). Since we want
				P I i -P atm ∈ H 1 0,r ((0, R))

Acknowledgements

The author addresses his sincere gratitude to D. Lannes for the supervision of this paper and his precious advices. The author wants to acknowledge C. Prange for his useful remarks and T. Iguchi for some interesting conversations. He also wants to thank M. Tucsnak, who noticed that, in the configuration we consider in this paper, a correction pressure term must be introduced to have exact energy conservation.

The author was supported by the French National Research Agency project NABUCO, grant ANR-17-CE40-0025, the Fondation Simone et Cino Del Duca and the Conseil Régional Nouvelle Aquitaine.

Floating structure equations 2.1 Constrained free surface Euler equations

Let us consider a floating body, typically a wave energy converter, with vertical side-walls and a cylindrical symmetry, forced to move only in the vertical direc-Therefore we can continue the construction and this permits to define a sequence u n ∈ X k (T) solving the linear problem (80) and, from (61), satisfying

(84) The existence and uniqueness of δ n G ∈ W 2,∞ ((0, T)) is given by the Cauchy-Lipschitz-Picard theorem since the coefficients in (81) are bounded when

To do that, we now provide a control of product estimates in Sobolev spaces in time; of course one has the standard estimate

but the constant C(T) blows up as T → 0 which raises some issues since we are led to choose T small enough in the proof. We therefore use the following more precise lemma where the time dependence of the constants is made explicit (see Proposition B.1 in Appendix B for the proof): Lemma 5.5. Let k ≥ 1 be an integer. For f , g ∈ H k ((0, T)) the following holds:

The previous lemma yields the following estimate for the solution δ n G to (81): Proposition 5.6. The solution δ n G to (81) satisfies

with C 1 (T), C 2 (T) → 0 as T → 0. Using the equation on δn G we can estimate δn

Then, by (52) it follows

) On the other hand, we have

with C 2 (T) → 0 as T → 0. Since in the ODE (77) the terms

are Lipschitz continuous on (u, δ G ) ∈ E from L 2 to L 2 and considering the equation for δ n G and δ n-1 G , we obtain the following estimate for T small enough:

for some constants α(T, R), β(T, R). Therefore, using (90) and (92), we get

where we have used u n-1 X 2 (T) ≤ u n-1 X k (T) and the inductive hypothesis (??). Then, we can choose T small enough such that K(T, K 0 , R) < 1 and we obtain that V n is a convergent sequence in X 0 (T) × H 1 ((0, T)) with limit V = (u, δ G ). By standard arguments (see [START_REF] Schochet | The compressible euler equations in a bounded domain: existence of solutions and the incompressible limit[END_REF]) we have that V ∈ E ⊆ X k (T) × H k+1 ((0, T)) is the unique solution of the coupled problem (76) -(77).

Proof. We write f (t) as

We prove (94) by induction. For k = 1 we have

and using (95) we get

Let us suppose that (94) is true for k -1. Then, we have

d k-i dt k-i f L 2 ((0,T)) .

From the estimate (95) for f , g, d k-1 dt k-1 g and d i dt i f we get and (94) follows using the inductive hypothesis.


Zhi Yan

Nicolas Jouandeau

Arab Ali Cherif

Sampling-based Multi-robot Motion Planning

Keywords: Multi-robot systems, Coordination, Motion planning, Sampling-based method

This paper describes a sampling-based approach to multi-robot motion planning. The proposed approach is centralized, which aims to reduce interference between mobile robots such as collision, congestion and deadlock, by increasing the number of waypoints. The implementation based on occupancy grid map is decomposed into three steps: the first step is to identify primary waypoints by using the Voronoi diagram, the second step is to generate additional waypoints by sampling the Voronoi diagram, and the last step is to assign the waypoints to robots by using the Hungarian method. The approach has been implemented and tested in simulation and the experimental results show a good system performance for multi-robot motion planning.

INTRODUCTION

In robotics, the motion planning problem is to produce a continuous motion from one configuration to another in configuration space, while avoiding collision with obstacles. The motion is represented as a path in configuration space. Motion planning is eminently necessary for mobile robot since, by definition, a robot accomplishes tasks by moving in the real world [START_REF] Latombe | Robot Motion Planning[END_REF]. Environments for autonomous mobile robot are usually represented by an occupancy grid map. For most of motion planning algorithms, the robot is reduced to a point in a twodimensional plane.

Multi-robot motion planning is one of the most challenging tasks in multi-robot systems. It should consider not only the obstacles (whether static or dynamic) in the environment, but also the possible interference between robots. When robots in a team are used to perform independent tasks in a shared workspace, each one will become a mobile obstacle for the other. Thus each robot should take into account the movement of other robots. One well-studied example is the multi-robot space sharing problem [START_REF] Jäger | Decentralized collision avoidance, deadlock detection, and deadlock resolution for multiple mobile robots[END_REF] [START_REF] Luo | Realtime path planning with deadlock avoidance of multiple cleaning robots[END_REF] (Marcolino and Chaimowicz, 2009a) [START_REF] Marcolino | Traffic control for a swarm of robots: Avoiding target congestion[END_REF] [START_REF] Luna | Efficient and complete centralized multi-robot path planning[END_REF].

Multi-robot environment must definitely be dynamic, in which motion planning is inherently difficult. Even for a simple case in two dimensions, the planning is NP-hard and is not solvable in polynomial time [START_REF] Guo | A distributed and optimal motion planning approach for multiple mobile robots[END_REF]. A very common problem in the motion planning for a group of mobile robots is when multiple robots move to the same waypoint, causing collision, congestion and deadlock, and thus affects system performance. We define this kind of dynamic standstill of system caused by waypoint conflict as the multi-robot waiting situation problem. Because, for better or worse, a robot should wait until the other pass first, or all robots should wait for planner to replan their local trajectories. Therefore, some kind of mechanism is needed to coordinate the motion of robots. In this paper, we present a sampling-based approach based on centralized decision making mechanism to multi-robot motion planning. The basic idea of the proposed approach is to reduce the problem of waiting situation between robots by increasing the number of waypoints. The implementation includes three critical steps:

• The first step is toward identifying primary waypoints on top of an occupancy grid map by using the Voronoi diagram.

• The second step is generating additional waypoints on top of the Voronoi diagram by using a sampling-based method.

• The third step is toward assigning the waypoints to robots by using the Hungarian method.

The remainder of the paper is organized as follows: Section 2 describes an overview of some related works; Section 3 describes our sampling-based approach; Section 4 describes the experimental results obtained with our approach; and the paper is concluded in Section 5 at last.

RELATED WORK

The major approaches to multi-robot motion planning are expanded from the results of single-robot system. [START_REF] Bennewitz | Optimizing schedules for prioritized path planning of multi-robot systems[END_REF] presented an approach based on A* path planning algorithm to optimize the priorities for decoupled and prioritized path planning methods for groups of mobile robots. Their approach is a randomized method which repeatedly reorders the robots to find a sequence for which a plan can be computed and to minimize the overall path lengths. [START_REF] Guo | A distributed and optimal motion planning approach for multiple mobile robots[END_REF] proposed an approach based on D* path planning algorithm which contain two modules: path planning and velocity planning. The detailed process can be described as follows: first, each robot plans its own path independently, then a coordination diagram is constructed based on collision checks among all robot paths. One approach named artificial potential field (APF) is used extensively for multirobot formation control which is usually linked with multi-robot motion planning problem. [START_REF] Tanner | Towards decentralization of multi-robot navigation functions[END_REF] presented a strategy which can ensure almost global asymptotic convergence of the robots to a particular oriented formation shape, while guaranteeing collision avoidance in the process.

The approaches often used for high-dimensional motion planning problems can also be efficiently applied to deal with low-dimensional problems. There are two major families of approaches known as cell decomposition and skeletonization. The cell decomposition method often applied for multi-robot area coverage problem. [START_REF] Hazon | Towards robust on-line multi-robot coverage[END_REF] presented an online robust multi-robot spanning tree coverage (ORMSTC) algorithm based on an approximate cellular decomposition for covering an unknown environment. The algorithm is called online because in which the robots do not have apriori knowledge of the work area, and it is robust and complete because in which as long as a single robot is able to move, the coverage will be completed. A typical skeletonization method is Voronoi diagram which is often applied to the problem of robotic exploration. [START_REF] Wurm | Coordinated multi-robot exploration using a segmentation of the environment[END_REF] presented a strategy for coordinating a team of exploring robots by using Voronoi diagram for the segmentation of the environment map. The strategy extracts the critical points [START_REF] Thrun | Learning metric-topological maps for indoor mobile robot navigation[END_REF] as the exploration targets to assign to robots. An alternative to the Voronoi diagrams is the probabilis-tic roadmap (PRM) [START_REF] Kavraki | Analysis of probabilistic roadmaps for path planning[END_REF] which is a sampling-based method. [START_REF] Švestka | Coordinated motion planning for multiple car-like robots using probabilistic roadmaps[END_REF] presented a PRM-based approach for multiple nonholonomic car-like robots motion planning in the same static workspace, in which the roadmaps for the composite robot are derived from roadmaps for the underlying simple robots, and the latter is computed by a probabilistic single-robot learning method.

The sampling-based methods are currently considered state-of-the-art for robot motion planning in complex environments. Because unlike the incremental heuristic search (such as A* and D*) and the topological map (such as Voronoi diagram) methods, their running time does not grow exponentially with the dimension of the configuration space, but also easier to implement. Yet, the sampling-based methods are probabilistically complete, and can not determine if no solution exists.

OUR APPROACH

Voronoi Diagram

The first step of our approach is to identify primary waypoints on top of the occupancy grid map by applying the Voronoi diagram. The Voronoi diagram is a very popular graph partitioning method for the problem of map segmentation [START_REF] Thrun | Learning metric-topological maps for indoor mobile robot navigation[END_REF] [START_REF] Zivkovic | Hierarchical map building and planning based on graph partitioning[END_REF] [START_REF] Wurm | Coordinated multi-robot exploration using a segmentation of the environment[END_REF]. For a given map m, we define for each point p in the free space S of m, a set O p (m) which is made up of closest obstacle points to p. While the Voronoi diagram can be represented by a graph G(m) = (V, E), where V is the set of points in O p (m) that are equidistant to at least two nearest obstacles, and E is the set of edges connecting two adjacent points in V of m:

V = p ∈ S| |O p (m)| ≥ 2 (1) E = (p, q)|p, q ∈ V, p adjacent to q (2)
Figure 1 shows the Voronoi diagrams of two experimental maps.

Then the topological map is built based on the Voronoi diagram. The grid map is decomposed into a small set of regions separated by narrow passages. These narrow passages, which are called critical lines, are found by analyzing a skeleton of the environment. The partitioned map is mapped into an isomorphic graph, where nodes correspond to regions and arcs connect neighboring regions. This graph is the topological map [START_REF] Thrun | Learning metric-topological maps for indoor mobile robot navigation[END_REF].

On the basis of the topological map, a series of points called critical points is generated, at which the distance to the nearest obstacle in the map is a local minimum. In other words, the critical points are the midpoints of the critical lines. These points are considered as the primary waypoints for mobile robot motion planning. Figure 2 illustrates the results of the first step.

Sampling-based Method

The second step of the approach is to generate additional waypoints on top of the Voronoi diagram by applying a PRM-like method. The original PRM is presented by [START_REF] Kavraki | Analysis of probabilistic roadmaps for path planning[END_REF], of which the basic algorithmic framework can be summarized as follows:

• Firstly, randomly sample a sufficient number of points in the configuration space (denoted by C) of the robot, keeping any that are not in collision with obstacles (denoted by C obs ). This creates a point set in the the free configuration space (denoted by C f ree ). • Secondly, using a local planner (such as straight line planner), attempt to connect pairs of samples that are relatively close to each other by thoroughly sampling and collision checking configurations between them. This creates a graph data structure called a roadmap.

• Finally, to query the roadmap, first attempt to connect the start and goal configurations to the existing graph. If that is successful, search the graph for a path from start to goal using any standard graph search method (such as A* algorithm).

Like most of motion planning algorithms, the PRM-like method often computes a collision free roadmap in a configuration space where the object is reduced to a point. This point represents the robot's model in the environment and the dimension of configuration space is equal to the number of degrees of freedom of the mechanical system. Then the algorithm searches a free path for a point in the roadmap. Figure 3 illustrates the process of generating a probabilistic roadmap for an example occupancy grid map with 80 random samples.

An extension of the original PRM, which is called medial axis based PRM (i.e. MAPRM) [START_REF] Wilmarth | MAPRM: A probabilistic roadmap planner with sampling on the medial axis of the free space[END_REF], aims to efficiently retract any sampled configuration, free or not, onto the medial axis of the free space without having to compute the medial axis explicitly. This strategy is applied to our approach with some small modifications. The implementation details can be seen in Algorithm 1. When the random configuration p is sampled, the nearest obstacle configuration q to p should be found, and another configuration r is set to fix p at the center of q and r. If r is a free configuration, then p will be moved to their center and r will be pushed to a farther location. This step is repeated until r becomes an obstacle configuration. The algorithm produces a self adaptive medial axis synthesis, which is fitted to the clearance of each local free space. The results of applying this algorithm can be seen in Figure 4. Find the nearest point q ∈ C obs to p. p is added to the set of medial axis.

12:

end if 13:

end if 14: until N points have been generated.

Hungarian Method

The last step is to assign the waypoints to robots by using the Hungarian method. The Hungarian method is a combinatorial optimization algorithm which can solve the assignment problem in polynomial time (O(n 3 )). It was developed and published by [START_REF] Kuhn | The hungarian method for the assignment problem[END_REF]. In the context of our research, given n way-points and n robots, we would like to find the best way (cost minimizing assignment) to assign the waypoints to the robots. The implementation details on this issue by using the Hungarian method can be summarized in the following four steps:

1. Construct a n × n matrix containing the cost of assigning each waypoint to a robot. The cost could be the distance or energy required for the robot to reach the waypoint.

2. The minimum element in each row is subtracted from all the elements in that row, and the minimum element in each column is subtracted from all the elements in that column.

3. Find the minimum number of lines required to cover all the zeros across rows and columns. If the number of the lines is n, then the optimal waypoint allocation plan can be given by selecting a combination from the modified cost matrix in such a way that the sum is zero. Otherwise, continue with Step 4.

4. Find the smallest element which is not covered by any of the lines. Then subtract it from each element which is not covered by the lines and add it to each element which is covered by a vertical and a horizontal line. Continue with Step 3.

The implementation of the Hungarian method described above requires a square cost matrix, which means the number of waypoints and the number of robots should be equal, thus each robot can only move to a waypoint. On the contrary, if the cost matrix is not square, then we have to augment it into a square matrix by adding zero rows or columns. Suppose there are three robots, r 0 , r 1 and r 2 , and only two waypoints, p 0 and p 1 , then the 3 × 3 cost matrix can be represented as follows:

C =   cost(r 0 , p 1 ) cost(r 0 , p 2 ) 0 cost(r 1 , p 1 ) cost(r 1 , p 2 ) 0 cost(r 2 , p 1 ) cost(r 2 , p 2 ) 0   (3)
Finally we combine the three methods described above (i.e. the Voronoi diagram, the probabilistic roadmap and the Hungarian method), and take into account both the primary waypoints and the additional waypoints to coordinate multi-robot motion planning.

EXPERIMENTS 4.1 Experimental Setup

Our approach has been implemented and tested in simulation with the 2D multiple robots simulator Player/Stage [START_REF] Gerkey | The Player/Stage project: Tools for multi-robot and distributed sensor systems[END_REF]. The experiment is to explore an unknown indoor environment by using a group of virtual Pioneer 2-DX robot equipped with a laser range finder which can provide 361 samples with 180 degrees field of view and a maximum range of 8 meters. Every robot is equipped with an abstract localization device which models the implementation of SLAM (Simultaneous Localization And Mapping), and a navigation device which performs the wavefront propagation algorithm [START_REF] Latombe | Robot Motion Planning[END_REF] for global path planning and the vector field histogram algorithm [START_REF] Ulrich | VFH+: Reliable obstacle avoidance for fast mobile robots[END_REF] for goal seeking and local obstacle avoidance. A central planning server is assumed, which is able to communicate with all robots and assign the exploration tasks to each one. In addition, all the robots share an occupancy grid map with respect to the structure of the environment during their exploration mission.

A different number of robots (from two to six) combining with two environments is used to evaluate our approach. In addition, a comparison with experimental results of our approach and a Voronoi-based approach is given. Figure 1 ((a) and (b)) depicts two experimental maps used in our simulation which are enclosed spaces with 14 meters long and 16 meters wide. The ratio between real-world time and simulation time is about 1:3. All experiments reported in this paper were carried out on a system with an Intel Core 2 Duo E8400 3.00GHz processor, an Intel Q43 Express chipset and two DDR2 800MHz 1024MB dual channel memory. A screenshot of our implementation can be seen in Figure 5.

Experimental Results

The results of the experiments are given in Figure 6. We measured the exploration time gained by our sampling-based approach and compared to the Voronoi-based approach. In each plot, the abscissa denotes the team size of the mobile robots, the ordinate denotes the time gain in percent of the total exploration time, and the error bar indicates the confidence interval of each corresponding gain of robot team size with the 0.95 confidence level. 10 runs are performed for each team size. Figure 6 shows that an exploration time saving of 5.8% to 16.1% in map A and 6.4% to 15.3% in map B is obtainable under our sampling-based approach compared to the Voronoi- based approach. These results show that our technique could improve performance of the multi-robot motion planning. We also counted the average number of waypoint conflicts in each experiment map, as shown in Table 1. This table shows that the waypoint conflict is obviously reduced by using our approach compared to the Voronoi-based approach. For this reason, the problem of waiting situation such as collision, congestion and deadlock has been alleviated and the system performance is improved.

Conclusion

In this paper, we presented a multi-robot motion planning approach based on sampling method. This approach is designed to relieve multi-robot waiting situation problem such as collision, congestion and deadlock, by increasing the number of waypoints for mobile robots. The proposed approach includes three main steps: the first step is to identify primary waypoints on top of an occupancy grid map by using the Voronoi diagram, the second step is to generate additional waypoints on top of the Voronoi diagram by using the sampling-based method, and the third step is to assign the waypoints to robots by using the Hungarian method. The efficiency of our approach was verified by simulation and experimental results.

  map B Figure 1: Voronoi diagrams of two experimental maps. (a) and (b) depict the original maps. (c) and (d) depict the generated Voronoi diagrams. (a) map A (b) map B Figure 2: The primary waypoints extracted by using the Voronoi diagram.

  Generation of the probabilistic roadmap. (a) shows a configuration space in which we will build a probabilistic roadmap. (b) shows the random points generated in C f ree . (c) shows the roadmap constructed by connecting the samples relatively close using the straight line planner. (d) shows a path found by querying the constructed roadmap using the A* algorithm.

  The additional waypoints extracted by using the sampling-based method with 400 random samples.

Algorithm 1

 1 Medial axis based PRMRequire: N, the sufficient number of points to generate. Ensure: N points in C f ree by sampling on the medial axis. random point p in C.

  r ∈ C and r ∈ C f ree do

Figure 5 :

 5 Figure 5: Four robots explore an unknown environment cooperatively. The green robot and the orange robot are exploring different rooms. The red robot and the yellow robot are exploring corridors along different paths.

  Exploration time gained by using our samplingbased approach compared with the Voronoi-based approach.

Table 1 :

 1 The Number of Waypoint Conflicts by Using Our Sampling-based Approach Compared with The Voronoi-

	based Approach
		(a) map A
	# robots	2	3	4	5	6
	Sampling-based 2.5 2.9 4.3 3.1 3.4
	Voronoi-based	6.2 6.7 11.6 8.1 10.2
		(b) map B
	# robots	2	3	4	5	6
	Sampling-based 0.0 1.4 2.7 2.9 2.4
	Voronoi-based	4.1 6.2 6.4 7.7 10.7