Galina García
email: galina.garcia@usach.cl

Takéo Takahashi
email: takeo.takahashi@inria.fr

Numerical observers with vanishing viscosity for the 1d wave equation

HAL is

Introduction and main results

The aim of this article is to consider and analyze an iterative method based on observers to reconstruct initial conditions of some evolution equations. More precisely, we focus on a method developed in [START_REF] Ramdani | Recovering the initial state of an infinitedimensional system using observers[END_REF] to recover the initial conditions for reversible infinite-dimensional systems. Our aim is to analyze this method when we discretize the evolution equations. In order to do this, we recall the method considered in [START_REF] Ramdani | Recovering the initial state of an infinitedimensional system using observers[END_REF] for a 1d wave equation and we then discretize this equation by using the finite difference method on a uniform mesh. Nevertheless, our approach can be extended to other systems and to other numerical discretization methods (see Section 7).

Let us consider the 1d wave equation:

       ∂ tt v -∂ xx v = 0
in (0, τ ) × (0, 1), v(t, 0) = 0 for t ∈ (0, τ ), ∂ x v(t, 1) = 0 for t ∈ (0, τ ), v(0, x) = v 0 (x), ∂ t v(0, x) = v 1 (x) for x ∈ (0, 1). [START_REF] Auroux | Back and forth nudging algorithm for data assimilation problems[END_REF] It is well-known that system (1) is well-posed for

(v 0 , v 1 ) ∈ H 1 L (0, 1) × L 2 (0, 1), (2)
where H 1 L (0, 1) := v ∈ H 1 (0, 1) ; v(0) = 0 (see, for instance, [START_REF] Brezis | Analyse fonctionnelle[END_REF] for the definitions of the Lebesgue and the Sobolev spaces). In particular, for initial conditions satisfying [START_REF] Banks | Exponentially stable approximations of weakly damped wave equations[END_REF], there exists a unique solution v of system (1) verifying v ∈ C([0, τ ]; H 1 L (0, 1)) ∩ C 1 ([0, τ ]; L 2 (0, 1)).

Moreover, if we define the observation y by y(t) := ∂ t v(t, 1) for t ∈ (0, τ ), [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] then, from the admissibility of the observation operator (see, for instance, [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]), we have y ∈ L 2 (0, τ ).

The corresponding inverse problem consists in recovering the initial data (v 0 , v 1 ) of (1) by using the observation y defined by [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF], where τ is a positive time to be determined.

In order to solve this inverse problem, Ramdani, Tucsnak and Weiss propose in [START_REF] Ramdani | Recovering the initial state of an infinitedimensional system using observers[END_REF] an iterative algorithm. They define the two following forward and backward operators F y , B y :

F y : H 1
L (0, 1) × L 2 (0, 1) → H 1 L (0, 1) × L 2 (0, 1), (q 0 , q 1 ) → (q(τ ), ∂ t q(τ )),

where

      
∂ tt q -∂ xx q = 0 in (0, τ ) × (0, 1), q(t, 0) = 0 for t ∈ (0, τ ), (∂ x q + ∂ t q)(t, 1) = y(t)

for t ∈ (0, τ ), q(0, x) = q 0 (x), ∂ t q(0, x) = q 1 (x) for x ∈ (0, 1), [START_REF] Brezis | Analyse fonctionnelle[END_REF] and B y : H 1 L (0, 1) × L 2 (0, 1) → H 1 L (0, 1) × L 2 (0, 1), (q 0 b , q 1 b ) → (q b (0), ∂ t q b (0)),

where

       ∂ tt q b -∂ xx q b =
0 in (0, τ ) × (0, 1), q b (t, 0) = 0 for t ∈ (0, τ ), (∂ x q b -∂ t q b )(t, 1) = -y(t)

for t ∈ (0, τ ), q b (τ, x) = q 0 b (x), ∂ t q b (τ, x) = q 1 b (x) for x ∈ (0, 1). [START_REF] Chapelle | Exponential convergence of an observer based on partial field measurements for the wave equation[END_REF] Then, they consider the sequences

a (n) , b (n) = (B y • F y ) a (n-1) , b (n-1) , n ∈ N * , (8)
where a (0) , b (0) is an initial guess of (v 0 , v 1 ). The convergence analysis developed in [START_REF] Ramdani | Recovering the initial state of an infinitedimensional system using observers[END_REF] shows that for τ large enough, the sequence a (n) , b (n) converges towards (v 0 , v 1 ). More precisely, we have the following result: Proposition 1.1. For τ > 2, there exists α ∈ (0, 1) such that for all n ∈ N,

a (n) -v 0 2 H 1 L (0,1) + b (n) -v 1 2
L 2 (0,1)

α 2n a (0) -v 0 2 H 1 L (0,1) + b (0) -v 1 2 L 2 (0,1) . ( 9
)
Although the proof of the above proposition is a consequence of the general results of [START_REF] Ramdani | Recovering the initial state of an infinitedimensional system using observers[END_REF], we give it in Section 2 since it is quite short in our case and contains hints for the proof of the main result.

Let us note that the condition τ > 2 is related to the exact observability of (1) with the observation (3). Indeed, (1), ( 3) is exactly observable if and only if τ > 2. For the wave equation in dimension N > 1, the time τ is related to "Geometric Control Condition" (see [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF]). We recall that exact observability is related to exact controllability and to stabilization property (see [START_REF] Komornik | Exact controllability and stabilization[END_REF], [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]).

In this paper, we focus on the discretized counterpart of the above result. Let N ∈ N and

h = 1 N + 1 . ( 10
)
We consider the uniform subdivision of (0, 1): x j = jh, j = 0, . . . , (N + 1). Finally, we consider a positive constant ν that is used in the numerical scheme to add a numerical viscosity.

The numerical adaptation of the method of [START_REF] Ramdani | Recovering the initial state of an infinitedimensional system using observers[END_REF] is given below. First, we consider spatial approximations of the forward and backward operators:

F y,h : R N +2 × R N +2 → R N +2 × R N +2 , (q 0 h , q 1 h ) → (q h (τ ), q ′ h (τ )), (11)
where q h = (q j ) 0 j N +1 verifies

             q ′′ j - q j+1 -2q j + q j-1 h 2 -ν q ′ j+1 -2q ′ j + q ′ j-1 = 0 in (0, τ ), 1 j N, q 0 = 0 in (0, τ ), q N +1 -q N h + q ′ N +1 = y in (0, τ ), q j (0) = q 0 j , q ′ j (0) = q 1 j for j = 0, • • • , N + 1, (12)
and

B y,h : R N +2 × R N +2 → R N +2 × R N +2 , (q 0 b,h , q 1 b,h ) → (q b,h (0), q ′ b,h (0)), (13)
where q b,h = (q b,j ) 0 j N +1 verifies

             q ′′ b,j - q b,j+1 -2q b,j + q b,j-1 h 2 +ν q ′ b,j+1 -2q ′ b,j + q ′ b,j-1 = 0 in (0, τ ), 1 j N, q b,0 = 0 in (0, τ ), q b,N +1 -q b,N h -q ′ b,N +1 = -y in (0, τ ), q b,j (τ ) = q 0 b,j , q ′ b,j (τ ) = q 1 b,j for j = 0, • • • , N + 1. ( 14
)
Recall that y(t) = ∂ t v(t, 1) is the observation of the system (1) defined by ( 3). The terms -ν q ′ j+1 -2q ′ j + q ′ j-1 in ( 12) and +ν q ′ b,j+1 -2q ′ b,j + q ′ b,j-1 in ( 14) are numerical viscosities that play crucial role in our analysis. The idea of adding numerical viscosities in the numerical schemes for the wave equation was already considered by [START_REF] Roder | Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity[END_REF], [START_REF] Ramdani | Uniformly exponentially stable approximations for a class of second order evolution equations-application to LQR problems[END_REF], [START_REF] Ramdani | Internal stabilization of the plate equation in a square: the continuous and the semi-discretized problems[END_REF], [START_REF] Nicaise | Quasi exponential decay of a finite difference space discretization of the 1-d wave equation by pointwise interior stabilization[END_REF], etc. They use these viscosities to obtain uniform observability or stabilization properties for the discretization of the wave equation. Without these viscosities, these properties may fail because of some spurious modes which are not well approximated ( [START_REF] Banks | Exponentially stable approximations of weakly damped wave equations[END_REF], [START_REF] Infante | Boundary observability for the space-discretizations of the 1-d wave equation[END_REF], [START_REF] Antonio | Boundary observability for the space semidiscretizations of the 1-D wave equation[END_REF], etc.). Here, to reconstruct the initial conditions, a key point in the method is to have a uniform decay of the energy in time τ and this property is strongly related to the stabilization, which explain that we also add some numerical viscosity in the above systems. The change of sign is simply coming from the method proposed in [START_REF] Ramdani | Recovering the initial state of an infinitedimensional system using observers[END_REF], where both the forward and the backward steps need these uniform decay of the energy.

Using these operators F y,h and B y,h , we can recover the approximations of v 0 and v 1 by using the following sequences:

a (n) h , b (n) h = (B y,h • F y,h ) a (n-1) h , b (n-1) h , n ∈ N * , where a (0) h , b (0) h is an initial guess of (v 0 h , v 1 h ). Our main result is Theorem 1.2. Assume that v 0 ∈ H 2 (0, 1)∩ H 1
L (0, 1) and that v 1 ∈ H 1 L (0, 1). For τ > 2 and for ν > 0, there exists γ = γ(τ, ν) ∈ (0, 1) such that for all N ∈ N (and h given by [START_REF] Deguenon | Infinite dimensional observers for vibrating systems[END_REF]) and for all n ∈ N,

h 2 N j=0   b (n) j -v 1 j 2 + a (n) j+1 -a (n) j h - v 0 j+1 -v 0 j h 2   + νh 2 2 b (n) N +1 -v 1 N +1 2 1 1 -γ 2 ε h + γ 2n   h 2 N j=0   b (0) j -v 1 j 2 + a (0) j+1 -a (0) j h - v 0 j+1 -v 0 j h 2   + νh 2 2 b (0) N +1 -v 1 N +1 2   , (15)
where v 0 j = v 0 (jh), v 1 j = v 1 (jh) for 0 j N + 1 and where

ε h = ε h v 0 H 2 (0,1)∩H 1 L (0,1) , v 1 H 1 L (0,1)
is independent of n and goes to 0 as h → 0.

Remark 1.3. In the above result, let us note that in (15)

h 2 N j=0   b (n) j -v 1 j 2 + a (n) j+1 -a (n) j h - v 0 j+1 -v 0 j h 2   is a discrete counterpart of b (n) -v 1 2 L 2 (0,1) + a (n) -v 0 2 H 1 L (0,1)
in [START_REF] Curtain | An introduction to infinite-dimensional linear systems theory[END_REF]. The term 1 1-γ 2 ε h is independent of the number of iterations and can only be reduced by decreasing h, whereas the other term in the right-hand side of (15) goes to 0 if the number of iterations goes to infinity.

Remark 1.4. It is important to notice that the numerical viscosity we use in our numerical scheme allows to obtain in the above result a time τ and a decay γ independent of h. Remark 1.5. In Theorem 1.2, we need the following regularity on the initial conditions:

v 0 ∈ H 2 (0, 1) ∩ H 1 L (0, 1) and v 1 ∈ H 1 L (0, 1).
In particular, we need more regularity than in Proposition 1.1. This assumption allows us to deal with the numerical error for the observation (at the boundary), and it is not clear that it is possible to obtain [START_REF] Haine | Observateurs itératifs en horizon fini. application à la reconstruction de données initiales pour des edp d'évolution[END_REF] for v 0 ∈ H 1 L (0, 1) and v 1 ∈ L 2 (0, 1). For locally distributed observation, we would obtain the result for v 0 ∈ H 1 0 (0, 1) and v 1 ∈ L 2 (0, 1).

Let us make some comments on the main result. We consider here a finite difference approximations of the iterative algorithm proposed by [START_REF] Ramdani | Recovering the initial state of an infinitedimensional system using observers[END_REF]. This algorithm use back and forth in time observers to reconstruct the initial conditions. The use of observers is classical in finite dimensional systems (see, for instance, [START_REF] Kailath | Linear Estimation[END_REF] ) and has been tackled in infinite dimensional systems ( [START_REF] Auroux | Back and forth nudging algorithm for data assimilation problems[END_REF], [START_REF] Bensoussan | Filtrage optimal des systèmes linéaires[END_REF], [START_REF] Chapelle | Improving convergence in numerical analysis using observers-the wave-like equation case[END_REF], [START_REF] Chapelle | Exponential convergence of an observer based on partial field measurements for the wave equation[END_REF], [START_REF] Curtain | An introduction to infinite-dimensional linear systems theory[END_REF], [START_REF] Dang | Time reversal focusing of the initial state for Kirchhoff plate[END_REF], [START_REF] Deguenon | Infinite dimensional observers for vibrating systems[END_REF], [START_REF] Ito | A time reversal based algorithm for solving initial data inverse problems[END_REF], [START_REF] Li | On the numerical investigation of a Luenberger type observer for infinite-dimensional vibrating systems[END_REF], [START_REF] Moireau | Joint state and parameter estimation for distributed mechanical systems[END_REF]). The algorithm using back and forth in time observers for finite dimensional systems was considered in [START_REF] Haine | Observateurs itératifs en horizon fini. application à la reconstruction de données initiales pour des edp d'évolution[END_REF].

The method we use here could be adapted to other systems (Schrödinger equation, plate equation, Maxwell system, etc.) Instead of adding a vanishing viscosity, we could use one of the several methods which have been developed in the literature to handle the high frequency spurious modes: Tychonoff regularization [START_REF] Glowinski | A numerical approach to the exact boundary controllability of the wave equation. I. Dirichlet controls: description of the numerical methods[END_REF], mixed finite elements [START_REF] Banks | Exponentially stable approximations of weakly damped wave equations[END_REF], filtering of high frequencies [START_REF] Antonio | Boundary observability for the space semidiscretizations of the 1-D wave equation[END_REF], [START_REF] Micu | Uniform boundary controllability of a semidiscrete 1-D wave equation with vanishing viscosity[END_REF], etc. (see the review paper [START_REF] Zuazua | Propagation, observation, and control of waves approximated by finite difference methods[END_REF] for more details and extensive references). Let us note that in [START_REF] Cîndea | An approximation method for exact controls of vibrating systems[END_REF], the authors, prove that for regular solutions, the numerical viscosity is not needed. We present in Section 7 some possible generalizations of the studied done for the 1d wave equation.

It is important to notice that our work is related to another approach of this problem developed in [START_REF] Haine | Reconstructing initial data using observers: error analysis of the semi-discrete and fully discrete approximations[END_REF]. In that case, the authors don't add any vanishing viscosity and analyze the corresponding convergence of the numerical observers towards the initial conditions. The main differences with our work are that in their cases, the initial data are more regular and the observability operator is bounded with an extra hypothesis of regularity. Indeed their proofs, based on finite element method, consists in using these regularity assumptions to derive estimates errors and to follow the result obtain in the continuous system. In particular, it allows them to treat with the same approach a large class of systems and also to obtain similar results for the full-discretized case.

Here, we work directly on the discrete systems, and thus we don't need such assumptions on the regularity of the initial conditions. In fact, we could adapt our work to the case of internal observation and in that case, we could work with initial condition only in H 1 0 (0, 1) × L 2 (0, 1) instead of H 2 (0, 1) ∩ H 1 0 (0, 1) × H 1 0 (0, 1). Another difference with the result of [START_REF] Haine | Reconstructing initial data using observers: error analysis of the semi-discrete and fully discrete approximations[END_REF] is that in their case, they need to choose a particular number of steps in the iterative method with respect to the mesh size.

Here, since we add this numerical viscosity, such a relation is not needed.

The paper is organized as follows: in Section 2, we present a short proof of Proposition 1.1. An important ingredient to obtain this result is the study of a damped wave equation. The discretization of this damped wave equation is analyzed in Section 3. Then Section 4 is devoted to some properties of the numerical schemes associated to (1) if we add a numerical viscosity. We study in particular the convergence results and an admissibility property. This allows to prove Theorem 1.2 in Section 5. Section 6 gives some numerical examples of this method. Finally, Section 7 presents some possible extensions of the result obtained for the 1d wave equation to more general systems.

Analysis in the theoretical framework

This section is devoted to the proof of Proposition 1.1. This analysis is already done in [START_REF] Ramdani | Recovering the initial state of an infinitedimensional system using observers[END_REF], but here we precise some aspects which help to understand the numerical analysis of the next section. Let us first consider the following damped wave equation.

       ∂ tt w -∂ xx w = 0 in (0, ∞) × (0, 1), w(t, 0) = 0 for t ∈ (0, ∞), (∂ x w + ∂ t w)(t, 1) = 0 for t ∈ (0, ∞), w(0, x) = w 0 (x), ∂ t w(0, x) = w 1 (x) for x ∈ (0, 1). ( 16
)
Then we have the following classical result: Lemma 2.1. Assume τ > 2. Then there exists α ∈ (0, 1) such that for all w 0 , w 1 ∈ H 1 L (0, 1) × L 2 (0, 1), w(τ ) 2

H 1 L (0,1) + ∂ t w(τ ) 2 L 2 (0,1) α w 0 2 H 1 L (0,1) + w 1 2 L 2 (0,1) . ( 17
)
Remark 2.2. The proof we give for this lemma is classical and is based on the multiplier method (see, for instance, [START_REF] Komornik | Exact controllability and stabilization[END_REF]). It is used to obtain stabilizability properties. We recall the proof here since we use a similar method in the numerical case. Let us also notice that this method can be used to prove the same result for the wave equation in several dimensions (see, for instance, [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]). For interior domain measurements, the same result can be obtained again by using a multiplier method but a compactness argument is needed (see [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF] for the continuous case, [START_REF] Roder | Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity[END_REF] for the discretized case).

Proof. We first multiply the first equation of ( 16) by ∂ t w and integrate over (0,1):

d dt 1 2 1 0 (∂ t w) 2 + (∂ x w) 2 dx -∂ x w(1)∂ t w(1) = 0,
using the third equation in [START_REF] Infante | Boundary observability for the space-discretizations of the 1-d wave equation[END_REF] we obtain

d dt 1 2 1 0 (∂ t w) 2 + (∂ x w) 2 dx + (∂ t w) 2 (1) = 0, (18)
which implies in particular that the energy

E = 1 2 1 0 (∂ t w) 2 + (∂ x w) 2 dx ( 19
)
is a nonincreasing function of time.

Then we multiply the first equation of ( 16) by x∂ x w, integrate over (0,1) and after some computation, we deduce

d dt 1 0 ∂ t wx∂ x w dx - 1 0 x∂ x (∂ t w) 2 2 dx - 1 0 x∂ x (∂ x w) 2 2 dx = 0.
The above equation implies

d dt 1 0 (∂ t w)x(∂ x w) dx -(∂ t w) 2 (1) + E(t) = 0. ( 20
)
Let us set

V ε (t) := E(t) + ε 1 0 ∂ t w(t)x∂ x w(t) dx. (21)
We deduce from ( 18) and ( 20) that

V ′ ε (t) = (ε -1)(∂ t w) 2 (1) -εE(t). (22)
On the other hand, we have that

1 0 ∂ t wx∂ x w dx E, which implies that (1 -ε)E V ε (1 + ε)E. (23)
Combining the above equation and ( 22) yields that for ε ∈ (0, 1),

V ′ ε (t) - ε 1 + ε V ε (t) for all t 0.
We deduce from the above equation and ( 23) that

E(t) 1 + ε 1 -ε exp - ε 1 + ε t E(0). ( 24
)
Let us consider the function

f (ε) := 1 + ε ε ln 1 + ε 1 -ε .
For τ > f (ε), we have

α := 1 + ε 1 -ε exp - ε 1 + ε τ < 1,
which implies [START_REF] Infante | Boundary observability for the space-discretizations of the 1-d wave equation[END_REF]. Since lim ε→0 + f (ε) = 2, we conclude the proof of the lemma.

Using the above result, we deduce the proof of Proposition 1.1:

Proof of Proposition 1.1. Let us set w = v -q where v and q are respectively the solutions of ( 1) and [START_REF] Brezis | Analyse fonctionnelle[END_REF]. Then it can be easily checked that w is the solution of ( 16) with the initial conditions w 0 = v 0 -q 0 and w 1 = v 1 -q 1 . As a consequence, using Lemma 2.1, we deduce that for τ > 2,

v(τ ) -q(τ ) 2 H 1 L (0,1) + ∂ t v(τ ) -∂ t q(τ ) 2 L 2 (0,1) α v 0 -q 0 2 H 1 L (0,1) + v 1 -q 1 2 L 2 (0,1) . (25)
Now let us set w = v -q b where v and q b are respectively the solutions of ( 1) and [START_REF] Chapelle | Exponential convergence of an observer based on partial field measurements for the wave equation[END_REF]. Then it can be easily checked that w is the solution of

       ∂ tt w -∂ xx w = 0 in (0, ∞) × (0, 1), w(t, 0) = 0 for t ∈ (0, ∞), (∂ x w -∂ t w)(t, 1) = 0 for t ∈ (0, ∞), w(τ, x) = v(τ, x) -q 0 b (x), ∂ t w(τ, x) = ∂ t v(τ, x) -q 1 b (x) for x ∈ (0, 1). ( 26
)
Thus, w(t, x) := w(τ -t, x) satisfies ( 16) with w 0 (x) = v(τ, x) -q 0 b (x) and with w 1 (x) = -∂ t v(τ, x) + q 1 b (x). As a consequence, using Lemma 2.1, we deduce that for τ > 2,

v(0) -q b (0) 2 H 1 L (0,1) + ∂ t v(0) -∂ t q b (0) 2 L 2 (0,1) α v(τ ) -q 0 b 2 H 1 L (0,1) + ∂ t v(τ ) -q 1 b 2 L 2 (0,1) . (27)
In particular for (q 0 b , q 1 b ) = (q(τ ), ∂ t q(τ )) = F y (q 0 , q 1 ), we deduce from ( 25) and ( 27) that

v 0 -q b (0) 2 H 1 L (0,1) + v 1 -∂ t q b (0) 2 L 2 (0,1) α 2 v 0 -q 0 2 H 1 L (0,1) + v 1 -q 1 2 L 2 (0,1) . (28)
Consequently, we deduce that

(v 0 , v 1 ) -B y • F y (q 0 , q 1 ) 2 H 1 L (0,1)×L 2 (0,1) α 2 (v 0 , v 1 ) -(q 0 , q 1 ) 2 H 1 L (0,1)×L 2 (0,1)
and by induction, we deduce (9).

Numerical analysis of a damped wave equation

In order to prove Theorem 1.2, we first need, as in the theoretical part, to analyze a damped wave equation. More precisely, let us first consider the following finite-difference discretization of system ( 16) with the presence of a "pertubation" η = η(t) at x = 1:

               w ′′ j - w j+1 -2w j + w j-1 h 2 -ν w ′ j+1 -2w ′ j + w ′ j-1 = 0 in (0, τ ), 1 j N, w 0 = 0 in (0, τ ), w N +1 -w N h + w ′ N +1 = η in (0, τ ), w j (0) = w 0 j , w ′ j (0) = w 1 j 0 j N + 1. ( 29
)
The above system, with the add of the numerical viscosity term -ν w ′ j+1 -2w ′ j + w ′ j-1 and in the case η ≡ 0, was studied in [START_REF] Tebou | Uniform boundary stabilization of the finite difference space discretization of the 1 -d wave equation[END_REF]. They show the uniform exponential stability of the above system. Here we prove a uniform decay of the energy with presence of the perturbation η.

Let us define the following energy associated to the system (29):

E h (t) = h 2 N j=0 w ′ j (t) 2 + w j+1 (t) -w j (t) h 2 + νh 2 2 w ′ N +1 (t) 2 . ( 30
)
Lemma 3.1. The energy E h defined by (30) satisfies

E ′ h = -(w ′ N +1 ) 2 -hν N j=0 w ′ j+1 -w ′ j 2 + ηw ′ N +1 + h 2 νη ′ w ′ N +1 (31)
in (0, ∞).

Proof.

In what follows, we use the classical formula (Abel transformation)

N j=1 (b j -b j-1 )a j = b N a N +1 -b 0 a 0 - N j=0 b j (a j+1 -a j ). (32)
Applying [START_REF] Ramdani | Semi-discrétisation en espace du problème de la stabilisation interne de l'équation des poutres[END_REF], we deduce the following relations:

N j=1 (w j+1 -2w j + w j-1 )w ′ j = N j=1 [(w j+1 -w j ) -(w j -w j-1 )]w ′ j = (w N +1 -w N )w ′ N +1 -(w 1 -w 0 )w ′ 0 - N j=0 (w j+1 -w j )(w ′ j+1 -w ′ j ),
and

N j=1 (w ′ j+1 -2w ′ j + w ′ j-1 )w ′ j = N j=1 [(w ′ j+1 -w ′ j ) -(w ′ j -w ′ j-1 )]w ′ j = (w ′ N +1 -w ′ N )w ′ N +1 -(w ′ 1 -w ′ 0 )w ′ 0 - N j=0 (w ′ j+1 -w ′ j ) 2 .
Then, we multiply the first equation of ( 29) by hw ′ j , and by using the two above formulas, we obtain

h 2 N j=1 d dt w ′ j 2 - w ′ N +1 (w N +1 -w N ) h + h N j=0 (w j+1 -w j )(w ′ j+1 -w ′ j ) h 2 -hν(w ′ N +1 -w ′ N )w ′ N +1 + hν N j=0 (w ′ j+1 -w ′ j ) 2 = 0
The above equation, combined with the boundary condition in [START_REF] Nicaise | Quasi exponential decay of a finite difference space discretization of the 1-d wave equation by pointwise interior stabilization[END_REF], yields

h 2 N j=0 d dt w ′ j 2 + w j+1 -w j h 2 + hν N j=0 w ′ j+1 -w ′ j 2 + (w ′ N +1 ) 2 -ηw ′ N +1 + 1 2 h 2 ν d dt w ′ N +1 2 -h 2 νη ′ w ′ N +1 = 0,
which implies [START_REF] Ramdani | Internal stabilization of the plate equation in a square: the continuous and the semi-discretized problems[END_REF].

The previous lemma shows that, without the "perturbation" η, the function E h is a nonincreasing function of time. We improve this result in the following lemma. Lemma 3.2. Assume τ > 2. Then there exist β ∈ (0, 1) and C = C(ν) such that

E h (τ ) βE h (0) + C τ 0 η 2 + h 2 (η ′ ) 2 ds. ( 33
)
Proof. Let us define the following function of time

W ε := E h + ε N j=1 w ′ j jh w j+1 -w j-1 2 - εh 4 w ′ N +1 2 . ( 34
)
From [START_REF] Ramdani | Internal stabilization of the plate equation in a square: the continuous and the semi-discretized problems[END_REF] and the first equation of ( 29), we have

W ′ ε = -(w ′ N +1 ) 2 -hν N j=0 w ′ j+1 -w ′ j 2 + ηw ′ N +1 + h 2 νη ′ w ′ N +1 + ε N j=1 w j+1 -2w j + w j-1 h 2 jh w j+1 -w j-1 2 + ε N j=1 ν w ′ j+1 -2w ′ j + w ′ j-1 jh w j+1 -w j-1 2 + ε N j=1 w ′ j jh w ′ j+1 -w ′ j-1 2 - εh 2 w ′ N +1 w ′′ N +1 . ( 35
)
First, using [START_REF] Ramdani | Semi-discrétisation en espace du problème de la stabilisation interne de l'équation des poutres[END_REF] and the third equation of ( 29), one obtains

ε N j=1 w j+1 -2w j + w j-1 h 2 jh w j+1 -w j-1 2 = ε 2h N j=1 j [w j+1 -w j ] 2 -[w j -w j-1 ] 2 = ε 2 w N +1 -w N h 2 -ε h 2 N j=0 w j+1 -w j h 2 = ε 2 (η -w ′ N +1 ) 2 -ε h 2 N j=0 w j+1 -w j h 2 . (36)
Second, we have

ενh N j=1 w ′ j+1 -2w ′ j + w ′ j-1 j w j+1 -w j-1 2 = ενh 1 2 N j=1 w ′ j+1 -2w ′ j + w ′ j-1 j (w j+1 -w j ) + ενh 1 2 N j=1 w ′ j+1 -2w ′ j + w ′ j-1 j (w j -w j-1 ) = ενh 1 2 N j=1 w ′ j+1 -w ′ j j (w j+1 -w j ) -ενh 1 2 N j=1 w ′ j -w ′ j-1 j (w j+1 -w j ) +ενh 1 2 N j=1 w ′ j+1 -w ′ j j (w j -w j-1 ) -ενh 1 2 N j=1 w ′ j -w ′ j-1 j (w j -w j-1 ) . ( 37
)
We have

ενh 1 2 N j=1 w ′ j+1 -w ′ j j (w j+1 -w j ) νh 8 N j=1 w ′ j+1 -w ′ j 2 + νh 2 N j=1 ε 2 w j+1 -w j h 2
and we compute similarly the three other terms of (37) to obtain

ενh N j=1 w ′ j+1 -2w ′ j + w ′ j-1 j w j+1 -w j-1 2 νh 2 N j=0 w ′ j+1 -w ′ j 2 + 4νε 2 h 2 N j=0 w j+1 -w j h 2 . (38)
Finally, standard computations (and again the third equation of ( 29)) yield

ε 1 2 N j=1 w ′ j jh w ′ j+1 -w ′ j-1 = ε 1 2 N j=1 w ′ j jhw ′ j+1 -ε 1 2 N j=1 w ′ j jhw ′ j-1 = -ε h 2 N j=0 w ′ j w ′ j+1 + ε 2 w ′ N w ′ N +1 = -ε h 2 N j=0 (w ′ j ) 2 + ε h 2 N j=0 w ′ j w ′ j -w ′ j+1 + ε 2 w ′ N +1 2 + εh 2 w ′ N +1 w ′′ N +1 - εh 2 η ′ w ′ N +1 . ( 39
)
On the other hand,

ε h 2 N j=0 w ′ j w ′ j -w ′ j+1 1 2 νh N j=0 w ′ j -w ′ j+1 2 + ε 2 h 8ν N j=0 w ′ j 2 .
Gathering ( 35), ( 36), ( 38), (39) and the above equation yields

W ′ ε -(w ′ N +1 ) 2 -hν N j=0 w ′ j+1 -w ′ j 2 + ηw ′ N +1 + h 2 νη ′ w ′ N +1 -ε h 2 N j=0 w j+1 -w j h 2 + ε 2 (w ′ N +1 ) 2 + εη 2 2 -εηw ′ N +1 + νh 2 N j=0 w ′ j+1 -w ′ j 2 + 4νε 2 h 2 N j=0 w j+1 -w j h 2 -ε h 2 N j=0 (w ′ j ) 2 + ε 2 w ′ N +1 2 - εh 2 η ′ w ′ N +1 + 1 2 νh N j=0 w ′ j -w ′ j+1 2 + ε 2 h 8ν N j=0 w ′ j 2 .
The above relation and the definition (30) imply

W ′ ε ε(1 + νh 2 2 ) -1 (w ′ N +1 ) 2 -εE h (t) + ε 2 4ν + 1 4ν E h (t) -ε 2 4ν + 1 4ν νh 2 2 (w ′ N +1 ) 2 +ηw ′ N +1 + h 2 νη ′ w ′ N +1 + ε η 2 2 -ηw ′ N +1 - h 2 η ′ w ′ N +1 ,
and thus

W ′ ε -ε 2 4ν + 1 4ν νh 2 2 + ε 2 + νh 2 2 - 1 4 (w ′ N +1 ) 2 -εE h (t) + ε 2 4ν + 1 4ν E h (t) +(1 + 2ε)η 2 + (h 4 ν 2 + ε 2 h 2 )(η ′ ) 2 . ( 40
)
In particular, there exists ε 0 = ε 0 (ν) uniform in h such that for ε ∈ (0, ε 0 ), we have

W ′ ε -εE h + ε 2 4ν + 1 4ν E h + (1 + 2ε)η 2 + (h 4 ν 2 + ε 2 h 2 )(η ′ ) 2 . ( 41
)
On the other hand, we have

N j=1 w ′ j jh w j+1 -w j-1 2 = N j=1 w ′ j jh h 2 w j+1 -w j h + w j -w j-1 h N j=1 w ′ j jh w j+1 -w j-1 2 h 2 N j=0 w ′ j 2 + h 2 N -1 j=0 w j+1 -w j h 2 + h 4
w N +1 -w N h 2 (42) E h - νh 2 2 w ′ N +1 2
and thus

W ε E h + εE h -ε νh 2 2 w ′ N +1 2 - εh 4 w ′ N +1 2 (1 + ε)E h . (43)
Gathering ( 41) and ( 43) and taking ε ∈ (0, ε 0 ), we get

W ′ ε (-ε + ε 2 4ν + 1 4ν 1 + ε W ε + (1 + 2ε)η 2 + (h 4 ν 2 + ε 2 h 2 )(η ′ ) 2 .
The above equation implies

W ε (t) exp -ε + ε 2 4ν + 1 4ν 1 + ε t W ε (0) + t 0 (1 + 2ε)η 2 + (h 4 ν 2 + ε 2 h 2 )(η ′ ) 2 ds. (44)
In order to estimate from below W ε (in function of E h ), we use the third equation of ( 29) to obtain

εh 4 w ′ N +1 2 εh 4 (1 + ε) w N +1 -w N h 2 + (1 + 1 ε )η 2 .
We deduce from (42) and from the above equation that

W ε E h -ε   h 2 N j=0 w ′ j 2 + h 2 N -1 j=0 w j+1 -w j h 2 + h 4
w N +1 -w N h 2   - εh 4 (1 + ε) w N +1 -w N h 2 + (1 + 1 ε )η 2
and thus

W ε E h -εE h - ε 2 2 E h - h 4 (ε + 1)η 2 .
Combining the above equation with (43) and (44) yields

1 -ε - ε 2 2 E h (t) (1 + ε) exp -ε + ε 2 4ν + 1 4ν 1 + ε t E h (0) + (ε + 1)h 4 η 2 (t) + t 0 (1 + 2ε)η 2 + (h 4 ν 2 + ε 2 h 2 )(η ′ ) 2 ds.
and thus

E h (t) (1 + ε) 1 -ε -ε 2 2 exp -ε + ε 2 4ν + 1 4ν 1 + ε t E h (0) + (ε + 1)h 4 1 -ε -ε 2 2 η 2 (t) + t 0 (1 + 2ε)η 2 + (h 4 ν 2 + ε 2 h 2 )(η ′ ) 2 1 -ε -ε 2 2 ds.
Using that

η 2 (t) 1 t + 1 h t 0 η 2 ds + h t 0 (η ′ ) 2 ds,
we deduce that there exists a constant C = C(t, ν, ε 0 ) such that for ε ∈ (0, ε 0 ),

E h (t) (1 + ε) 1 -ε -ε 2 2 exp -ε + ε 2 4ν + 1 4ν 1 + ε t E h (0) + C t 0 η 2 + h 2 (η ′ ) 2 ds.
Consequently, if we set

f ν (ε) := 1 + ε ε + ε 2 4ν + 1 4ν ln (1 + ε) 1 -ε -ε 2 2
then, for τ > f ν (ε), we have

E h (τ ) βE h (0) + C τ 0 η 2 + h 2 (η ′ ) 2 ds, with
β := (1 + ε) 1 -ε -ε 2 2 exp -ε + ε 2 4ν + 1 4ν 1 + ε τ ∈ (0, 1). ( 45
)
Since lim ε→0 + f ν (ε) = 2, we conclude the proof of the lemma.

Numerical schemes for (1) with vanishing viscosity

In this section, we consider two numerical approximations of (1):

             v ′′ j - v j+1 -2v j + v j-1 h 2 -ν(v ′ j+1 -2v ′ j + v ′ j-1 ) = 0 in (0, τ ), 1 j N, v 0 = 0 in (0, τ ), v N +1 = v N in (0, τ ), v j (0) = v 0 j , v ′ j (0) = v 1 j 0 j N + 1, (46)
and

             v ′′ b,j - v b,j+1 -2v b,j + v b,j-1 h 2 + ν(v ′ b,j+1 -2v ′ b,j + v ′ b,j-1 ) = 0 in (0, τ ), 1 j N, v b,0 = 0 in (0, τ ), v b,N +1 = v b,N in (0, τ ), v b,j (τ ) = v 0 b,j , v ′ b,j (τ ) = v 1 b,j 0 j N + 1. ( 47
)
Remark 4.1. We need two different approximations of (1) since we add a numerical viscosity in these numerical schemes and since there is a change of sign between the forward and the backward systems. The idea to prove Theorem 1.2 is to make the differences between theses systems and systems ( 12) and ( 14) (respectively) and to reduce to the system (29) studied in the previous subsection.

We recall first some results of convergence (proved in [START_REF] Tebou | Uniform boundary stabilization of the finite difference space discretization of the 1 -d wave equation[END_REF]) of these numerical schemes. Then we consider a numerical property associated to the admissibility of the observation operator considered here.

Convergence of numerical schemes with vanishing viscosity

Here we recall results proved in [START_REF] Tebou | Uniform boundary stabilization of the finite difference space discretization of the 1 -d wave equation[END_REF] of systems ( 12) and ( 14).

Let us define

E h (a h , b h ) := h 2 N j=0 b 2 j + a j+1 -a j h 2 (a h = (a j ) 0 j N +1 , b h = (b j ) 0 j N +1 ∈ R N +2 ). ( 48
)
Let us also consider the operator

P h (a) := (a(jh)) 0 j N +1 ,
for any function a (regular enough). It is proved in [START_REF] Tebou | Uniform boundary stabilization of the finite difference space discretization of the 1 -d wave equation[END_REF] that if v h := (v j ) 0 j N +1 is the solution of (46), if v is the solution of (1) and if E h (0) (defined in [START_REF] Dang | Time reversal focusing of the initial state for Kirchhoff plate[END_REF]) converges towards E(0) (defined in ( 19)) as h → 0 then

E h (P h (v(τ )) -v h (τ ), P h (∂ t v(τ )) -v ′ h (τ )) → 0, ( 49
)
νh 2 2 ∂ t v(τ, 1) -v ′ N +1 (τ ) 2 → 0, ( 50
) τ 0 v ′ N +1 -∂ t v(1) 2 ds → 0, (51)
as h → 0. In particular,

E h (τ ) → E(τ ) as h → 0. ( 52
)
We can note that ṽ defined by ṽ(t, x)

:= v(τ -t, x) is solution of        ∂ tt ṽ -∂ xx ṽ = 0 in (0, τ ) × (0, 1), ṽ(t, 0) = 0 for t ∈ (0, τ ), ∂ x ṽ(t, 1) = 0 for t ∈ (0, τ ), ṽ(0, x) = v(τ, x), ∂ t ṽ(0, x) = -∂ t v(τ, x) for x ∈ (0, 1). (53) One can check that ṽ(τ ) = v 0 , ∂ t ṽ(τ ) = -v 1 . Moreover, if we set ṽb,j (t) := v b,j (τ -t) (0 j N + 1, t ∈ (0, τ )),
where v b,h := (v b,j ) 0 j N +1 is solution of (47) then

             ṽ′′ b,j - ṽb,j+1 -2ṽ b,j + ṽb,j-1 h 2 -ν(ṽ ′ b,j+1 -2ṽ ′ b,j + ṽ′ b,j-1 ) = 0 in (0, τ ), 1 j N, ṽb,0 = 0 in (0, τ ), ṽb,N+1 = ṽb,N in (0, τ ), ṽb,j (0) = v 0 b,j , ṽ′ b,j (0) = -v 1 b,j 0 j N + 1. (54)
One can check ṽb,h (τ ) = v b,h (0) and ṽ′ b,h (τ ) = -v ′ b,h (0). Consequently, from (52) and from the result of convergence of [START_REF] Tebou | Uniform boundary stabilization of the finite difference space discretization of the 1 -d wave equation[END_REF], we deduce that

E h (P h (v 0 ) -v b,h (0), P h (v 1 ) -v ′ b,h (0)) → 0, ( 55
)
νh 2 2 v 1 (1) -v ′ b,N +1 (0) 2 → 0, ( 56
) τ 0 v ′ b,N +1 -∂ t v(1) 2 ds → 0. ( 57
)

Numerical admissibility

In this section, we show the numerical admissibility of the observation operator. This is the numerical counterpart of the admissibility result (or hidden regularity) for the system (1) for systems ( 12) and ( 14):

τ 0 |∂ t v(t, 1)| 2 dt CE(0). Lemma 4.2. Assume that v h = (v j ) 0 j N +1 is solution of (46). Then there exists a constant C = C(τ ) independent of h such that τ 0 (v ′ N +1 ) 2 dt C h 2 N j=0   v 1 j 2 + v 0 j+1 -v 0 j h 2   . (58)
In the above result, it is important to notice that the constant C is independent of h. Indeed, for a fixed h, since we deal with a finite dimensional system, the observation operator is admissible and thus (58) holds for a constant C = C(h). In the proof of the main result (Section 5), we need the admissibility estimate (58) for a constant C independent of h.

Before proving Lemma 4.2, let us show how to deduce from it similar inequalities for v ′ h and v ′ b,h . Since v ′ h is also solution of (46) if v h is solution of (46) (but with different initial conditions), then we can apply Lemma 4.2 to v ′ h and we obtain

τ 0 (v ′′ N +1 ) 2 dt C h 2   N j=1 v 0 j+1 -2v 0 j + v 0 j-1 h 2 + ν(v 1 j+1 -2v 1 j + v 1 j-1 ) 2 + N j=0 v 1 j+1 -v 1 j h 2   and thus τ 0 (v ′′ N +1 ) 2 dt C h 2   N j=1 v 0 j+1 -2v 0 j + v 0 j-1 h 2 2 + N j=0 v 1 j+1 -v 1 j h 2   . (59)
In particular, if v 0 ∈ H 2 (0, 1), v 1 ∈ H 1 (0, 1), and if

v 0 h = P h (v 0 ), v 1 h = P h (v 1 ), then τ 0 (v ′′ N +1 ) 2 dt C v 0 2 H 2 (0,1) + v 1 2 H 1 (0,1) , ( 60
)
where C is a positive constant independent of h. Moreover, if v b,h is the solution of (47) with v 0 b,h = v h (τ ) and v 1 b,h = v ′ h (τ ), we deduce from (59) that

τ 0 (v ′′ b,N +1 ) 2 dt C h 2   N j=1 v j+1 (τ ) -2v j (τ ) + v j-1 (τ ) h 2 2 + N j=0 v ′ j+1 (τ ) -v ′ j (τ ) h 2   .
Combining the above estimate with Lemma 4.3 below we deduce that

τ 0 (v ′′ b,N +1 ) 2 dt C v 0 2 H 2 (0,1) + v 1 2 H 1 (0,1) . ( 61
)
Proof of Lemma 4.2. Let us multiply the first equation of ( 46) by hv ′ j . Following the proof of Lemma 3.1, we obtain

d dt   h 2 N j=0 v ′ j (t) 2 + v j+1 (t) -v j (t) h 2   0. ( 62
)
Now let us multiply the first equation of (46) by jh(v j+1 -v j-1 )/2. Using (32), we deduce

N j=1 v j+1 -2v j + v j-1 h 2 jh v j+1 -v j-1 2 = - h 2 N j=0 v j+1 -v j h 2 . ( 63
)
We also obtain, in similar way as [START_REF] Zuazua | Propagation, observation, and control of waves approximated by finite difference methods[END_REF], that

νh N j=1 v ′ j+1 -2v ′ j + v ′ j-1 j v j+1 -v j-1 2 3νh 2   N j=0 v ′ j 2 + N j=0 v j+1 -v j h 2   . ( 64
)
Finally, standard calculations yield

τ 0 N j=1 v ′′ j (jh) v j+1 -v j-1 2 dt = N j=1 v ′ j (τ )(jh) v j+1 (τ ) -v j-1 (τ ) 2 -v ′ j (0)(jh) v j+1 (0) -v j-1 (0) 2 - τ 0 N j=1 v ′ j (jh) v ′ j+1 -v ′ j-1 2 dt. (65)
Moreover,

N j=1 v ′ j (jh)(v ′ j+1 -v ′ j-1 ) = (v ′ N +1 ) 2 -h N j=1 v ′ j v ′ j+1 .
Gathering the above equation, ( 63), ( 64) and (65) and using (62), we deduce (58).

Lemma 4.3. Assume that v h = (v j ) 0 j N +1 is solution of (46). Then there exists a constant C independent of h such that

h 2 N j=1 v j+1 (t) -2v j (t) + v j-1 (t) h 2 2 + h 2 N j=0 v ′ j+1 (t) -v ′ j (t) h 2 C h 2   N j=1 v 0 j+1 -2v 0 j + v 0 j-1 h 2 2 + N j=0 v 1 j+1 -v 1 j h 2   .
Proof. Since v ′ h is solution of (46), we can use (62) to obtain

h 2 N j=0 v ′′ j (t) 2 + v ′ j+1 (t) -v ′ j (t) h 2 h 2 N j=0 v ′′ j (0) 2 + v ′ j+1 (0) -v ′ j (0) h 2 .
(66)

Moreover, v ′′ 0 (t) = 0 and for j = 1,

• • • , N , v ′′ j (t) = v j+1 -2v j + v j-1 h 2 + ν(v ′ j+1 -2v ′ j + v ′ j-1 )
and thus

N j=1 v j+1 -2v j + v j-1 h 2 2 2 N j=0 v ′′ j (t) 2 + 2 N j=1 ν 2 (v ′ j+1 -2v ′ j + v ′ j-1 ) 2 C N j=0 v ′′ j (t) 2 + v ′ j+1 (t) -v ′ j (t) h 2 . ( 67
)
In a similar way,

N j=0 v ′′ j (0) 2 C   N j=1 v j+1 (0) -2v j (0) + v j-1 (0) h 2 2 + N j=0 v ′ j+1 (0) -v ′ j (0) h 2   .
Combining the above equation with ( 66) and (67), we conclude the proof of the lemma.

Proof of the main result

This section is devoted to the proof of Theorem 1.2.

Proof of Theorem 1.2. Using the same arguments as in Proposition 1.1, let us define w h := v h -q h , where v h and q h are respectively the solutions of ( 46) and [START_REF] Glowinski | A numerical approach to the exact boundary controllability of the wave equation. I. Dirichlet controls: description of the numerical methods[END_REF]. It is easy to see that w h = (w j ) 0 j N +1 satisfies [START_REF] Nicaise | Quasi exponential decay of a finite difference space discretization of the 1-d wave equation by pointwise interior stabilization[END_REF] with the initial conditions w 0 j = v 0 j -q 0 j and w 1 j = v 1 j -q 1 j , for 0 j N + 1 and with

η := v ′ N +1 -y .
From Lemma 3.2, we obtain the following estimate

h 2 N j=0 v ′ j (τ ) -q ′ j (τ ) 2 + v j+1 (τ ) -v j (τ ) h - q j+1 (τ ) -q j (τ ) h 2 + νh 2 2 v ′ N +1 (τ ) -q ′ N +1 (τ ) 2 β   h 2 N j=0   v 1 j -q 1 j 2 + v 0 j+1 -v 0 j h - q 0 j+1 -q 0 j h 2   + νh 2 2 v 1 N +1 -q 1 N +1 2   + C τ 0 v ′ N +1 -y 2 + h 2 v ′′ N +1 -y ′ 2 ds. ( 68
)
Let us define w b,h := v b,h -q b,h , where v b,h and q b,h are the solutions of ( 47) and ( 14), respectively. We can check that w b,h = (w b,j ) 0 j N +1 satisfies

               w ′′ b,j - w b,j+1 -2w b,j + w b,j-1 h 2 + ν w ′ b,j+1 -2w ′ b,j + w ′ b,j-1 = 0 in (0, τ ), 1 j N, w b,0 = 0 in (0, τ ), w b,N +1 -w b,N h -w ′ b,N +1 = -(v ′ b,N +1 -y) in (0, τ ), w b,j (τ ) = v b,j (τ ) -q 0 b,j , w ′ b,j (τ ) = v ′ b,j (τ ) -q 1 b,j 0 j N + 1. ( 69
)
Thus, w h (t) := w b,h (τ -t, x) satisfies ( 29) with w 0 j := v b,j (τ ) -q 0 b,j and w 1 j := -v ′ b,j (τ ) + q 1 b,j , for 0 ≤ j N + 1, and with

η(t) := -(v ′ b,N +1 -y)(τ -t).
Using Lemma 3.2, we obtain

h 2 N j=0 v ′ b,j (0) -q ′ b,j (0) 2 + v b,j+1 (0) -v b,j (0) h - q b,j+1 (0) -q b,j (0) h 2 + νh 2 2 v ′ b,N +1 (0) -q ′ b,N +1 (0) 2 β h 2 N j=0   v ′ b,j (τ ) -q 1 b,j 2 + v b,j+1 (τ ) -v b,j (τ ) h - q 0 b,j+1 -q 0 b,j h 2   + νh 2 2 v ′ b,N +1 (τ ) -q 1 b,N +1 2 + C τ 0 v ′ b,N +1 -y 2 + h 2 v ′′ b,N +1 -y ′ 2 ds. (70)
Notice that for (q 0 b,h , q 1 b,h ) = (q h (τ ), q ′ h (τ )) = F y,h (q 0 h , q 1 h ) and taking

v b,h (τ ) := v h (τ ) v ′ b,h (τ ) := v ′ h (τ )
in (47), we get from ( 69) and (70) that

h 2 N j=0 v ′ b,j (0) -q ′ b,j (0) 2 + v b,j+1 (0) -v b,j (0) h - q b,j+1 (0) -q b,j (0) h 2 + νh 2 2 v ′ b,N +1 (0) -q ′ b,N +1 (0) 2 β 2   h 2 N j=0   v 1 j -q 1 j 2 + v 0 j+1 -v 0 j h - q 0 j+1 -q 0 j h 2   + νh 2 2 v 1 N +1 -q 1 N +1 2   + C τ 0 v ′ N +1 -y 2 + h 2 v ′′ N +1 -y ′ 2 ds + C τ 0 v ′ b,N +1 -y 2 + h 2 v ′′ b,N +1 -y ′ 2 ds. (71)
From the above estimate and using that for θ 0

(a + b) 2 (1 + θ)a 2 + 1 + 1 θ b 2 , we deduce h 2 N j=0   v 1 j -q ′ b,j (0) 2 + v 0 j+1 -v 0 j h - q b,j+1 (0) -q b,j (0) h 2   + νh 2 2 v 1 N +1 -q ′ b,N +1 (0) 2 1 + 1 θ h 2 N j=0   v 1 j -v ′ b,j (0) 2 + v 0 j+1 -v 0 j h - v b,j+1 (0) -v b,j (0) h 2   + νh 2 2 v 1 N +1 -v ′ b,N +1 (0) 2 + (1 + θ)β 2   h 2 N j=0   v 1 j -q 1 j 2 + v 0 j+1 -v 0 j h - q 0 j+1 -q 0 j h 2   + νh 2 2 v 1 N +1 -q 1 N +1 2   + C τ 0 v ′ N +1 -y 2 + h 2 v ′′ N +1 -y ′ 2 ds + C τ 0 v ′ b,N +1 -y 2 + h 2 v ′′ b,N +1 -y ′ 2 ds, (72)
with θ small enough so that γ := (1 + θ) 1/2 β ∈ (0, 1).

Let us set

ε h = 1 + 1 θ h 2 N j=0   v 1 j -v ′ b,j (0) 2 + v 0 j+1 -v 0 j h - v b,j+1 (0) -v b,j (0) h 2   + νh 2 2 v 1 N +1 -v ′ b,N +1 (0) 2 + C τ 0 v ′ N +1 -y 2 + h 2 v ′′ N +1 -y ′ 2 ds + C τ 0 v ′ b,N +1 -y 2 + h 2 v ′′ b,N +1 -y ′ 2 ds. (73)
Note that ε h does not depend on (q 0 h , q 1 h ). Using (51), ( 55), ( 56), (57), ( 60) and (61) we obtain

ε h → 0 as h → 0. (74)
From ( 72) and (73), we deduce that

E h ((v 0 h -q b,h (0)), (v 1 h -q ′ b,h (0))) + νh 2 2 v 1 N +1 -q ′ b,N +1 (0) 2 ε h + γ 2 E h ((v 0 h -q 0 h ), (v 1 h -q 1 h )) + νh 2 2 v 1 N +1 -q 1 N +1 2 . (75)
Consequently, we can proceed by induction, considering that (q b,h (0), q ′ b,h (0)) = B y,h • F y,h (q 0 h , q 1 h ), we deduce the result.

Numerical illustrations

This section is devoted to some numerical illustrations of the method proposed in this paper.

The numerical scheme that we use is slightly different from ( 12), [START_REF] Haine | Recovering the observable part of the initial data of an infinite-dimensional linear system[END_REF]. We replace them by

             q ′′ j - q j+1 -2q j + q j-1 h 2 -ν q ′ j+1 -2q ′ j + q ′ j-1 = 0 in (0, τ ), 1 j N, q 0 = 0 in (0, τ ), q N +1 -q N h + µq ′ N +1 = µy in (0, τ ), q j (0) = q 0 j , q ′ j (0) = q 1 j for j = 0, • • • , N + 1, (76)
and

             q ′′ b,j - q b,j+1 -2q b,j + q b,j-1 h 2 +ν q ′ b,j+1 -2q ′ b,j + q ′ b,j-1 = 0 in (0, τ ), 1 j N, q b,0 = 0 in (0, τ ), q b,N +1 -q b,N h -µq ′ b,N +1 = -µy in (0, τ ), q b,j (τ ) = q 0 b,j , q ′ b,j (τ ) = q 1 b,j for j = 0, • • • , N + 1, (77)
where µ is a positive constant that is chosen to be 1 in ( 12), [START_REF] Haine | Recovering the observable part of the initial data of an infinite-dimensional linear system[END_REF]. The reason for this change is that all the results we proved are still valid in that case, but for µ = 1, we may have a better result. Indeed, for system [START_REF] Haine | Reconstructing initial data using observers: error analysis of the semi-discrete and fully discrete approximations[END_REF], one can prove that for τ > 2, w(τ ) = ∂ t w(τ ) = 0. This means that in particular in Proposition 1.1, we only need one iteration (n = 1) to recover exactly v 0 and v 1 . For the numerical observers, based on this result, we can hope that in this particular example, we may not need a numerical viscosity in this particular example and with µ = 1. Nevertheless, this property of equation ( 1) is quite particular and our aim is to precise this example in order to understand how we can use numerical viscosity in such a method. We developed some extensions in Section 7.

Another important aspect for the numerical simulation is the discretization in time. We have not dealt with this problem here, but this is an important issue that was considered in the context of numerical viscosity in [START_REF] Negreanu | Uniform boundary controllability of a discrete 1-D wave equation[END_REF], [START_REF] Negreanu | Discrete Ingham inequalities and applications[END_REF], [START_REF] Münch | A uniformly controllable and implicit scheme for the 1-D wave equation[END_REF], etc. Here we have discretized the second derivatives by using the classical scheme

q ′′ j (t n ) ≈ q n+1 j -2q n j + q n-1 j (δt) 2 (78)
and for the first derivative in (76), we take

q ′ j (t n ) ≈ q n j -q n-1 j δt . ( 79
)
This leads to an explicit scheme in time and thus we take in the examples below (δt)/h small enough (CFL condition).

In our first example, we take

v 0 (x) = x(1 -x) 2 sin(15x), v 1 (x) = -x (x ∈ [0, 1]).
We show in Figure 1 several iterations of the iterative method (n ∈ {2, 4, 6, 8, 10}).

In this test, we take µ = 5, ν = 3 and a mesh size h = 1/(N + 1) with N = 240. We also discretized the time interval, with a time step δt = τ /M , with M = 1200, and τ = 2.5. We notice that the reconstruction of the velocity is less accurate that the reconstruction of the position. This may come from the discretization of velocity by using(79). Let us recall the error term considered in our analysis:

E h = h 2 N j=0   b (n) j -v 1 j 2 + a (n) j+1 -a (n) j h - v 0 j+1 -v 0 j h 2   (80)
where v 0 j = v 0 (jh), v 1 j = v 1 (jh) for 0 j N + 1 and

a (k) h , b (k)
h = (B y,h • F y,h ) a (k-1) h , b (k-1) h , k ∈ {1, . . . , n},
where here, we take for initial guess a

(0) h , b (0) h
= (0, 0). We represent in Figure 2 the relative error made after each iteration. We can see the influence of the viscosity ν: let us take

v 0 (x) = sin 31πx 2 , v 1 (x) = -x (x ∈ [0, 1]).
For instance in Figure 3, we show the relative error made for µ = 5, N = 240, M = 1200, τ = 2.5 and n = 10. This illustrates the fact that the factor of the vanishing viscosity need to be chosen carefully. Here we see that there is an optimal choice of ν * ≈ 3. Such an optimal factor may depend on the other parameters of the problem.

We also show how the the time τ plays a role in the method. We take

v 0 (x) = x sin 31πx 2 , v 1 (x) = -x (x ∈ [0, 1])
and µ = 5, N = 240, M = 1200, ν = 3 and n = 10. We represent in Figure 4 the final reconstruction of the initial condition for the position and the relative error for τ ∈ {1. In particular, we notice that if τ < 2, then the method does not give a good approximation of the initial conditions. In [START_REF] Haine | Recovering the initial data of an evolution equation. Application to thermoacoustic tomography[END_REF] and [START_REF] Haine | Recovering the observable part of the initial data of an infinite-dimensional linear system[END_REF], the author tackles the problem of reconstructing the initial conditions if we lose the exact observability (as for τ < 2) and shows that a set of initial conditions can still be reconstructed.

Even if our result (Theorem 1.2) needs regular initial conditions, we can use our method to try to reconstruct initial conditions with less regularity than in the main result. For instance, we take

v 0 (x) = x, v 1 (x) = -1 (0,1/2) (x).
We still have v 0 ∈ H 1 L (0, 1) and v 1 ∈ L 2 (0, 1) in that case. We obtain in that case, with µ = 5, τ = 2.5, N = 240, M = 1200, ν = 3 and n = 10 the figure 5 where we represent the reconstruction of the initial conditions and the relative error. We notice that in that case, the algorithm is still converging Finally, we present a last experiment where we add in the observation y a random perturbation. More precisely, instead of giving y(t k ) to the numerical scheme, we give y(t k ) + X k , where (X k ) k is a random vector, normally distributed with mean 0 and deviation σ.

0 0.
We take µ = 5, τ = 2.5, N = 240, M = 1200, ν = 3 and n = 10 and

v 0 (x) = x(1 -x) 2 sin(15x), v 1 (x) = -x (x ∈ [0, 1]).
Figure 6 represents the mean of ten experiments with independent perturbations.

Remarks on the vanishing viscosity for numerical observers

In this paper, we focus on the numerical counterpart of the method developed in [START_REF] Ramdani | Recovering the initial state of an infinitedimensional system using observers[END_REF] for recovering initial conditions by using observers. More precisely, we detailed a way to discretize their method for the 1d wave equation. However, this approach may be applied to many different problems. In this section, we present some possible generalizations.

v + A 0 v = 0 in (0, τ ), (81)
v(0) = v 0 , v(0) = v 1 , (82)
y = C 0 v in (0, τ ). ( 83
)
Let us make the hypothesis that

ẅ + A 0 w + C * 0 C 0 ẇ = 0 in (0, τ ), (84)
w(0) = w 0 , ẇ(0) = w 1 , (85)
is exponentially stable:

ẇ(t) H + A 1/2 0 w(t) H M 0 e -α 0 t w 1 H + A 1/2 0 w 0 H ∀(w 0 , w 1 ) ∈ D(A 1/2 0 ) × H. ( 86
)
Such an hypothesis is sufficient to obtain the convergence of the algorithm of [START_REF] Ramdani | Recovering the initial state of an infinitedimensional system using observers[END_REF]. Indeed they assume

C 0 ∈ L(D(A 1/2 0 ), U
) is an observation admissible operator (not necessary bounded) but we choose to write this abstract generalization only in the case of a bounded observation operator.

Let us present how to recover approximations v 0 h and v 1 h of v 0 and v 1 by using an approximation of y with a method of numerical observers.

Assume that A 0h ∈ L(V h ) (resp. C 0h ∈ L(V h , U h )) are finite dimensional approximations of A 0 (resp. C 0 ) obtained by using a classical method of discretization (finite element method, finite difference method, etc.). Let us consider ν 0 and θ > 0 some real numbers and y h ∈ U h an approximation of y. Then the numerical observers of (81)-( 83) can be written as

F h : V h × V h → V h × V h , (q 0 h , q 1 h ) → (q h (τ ), qh (τ )) (87)
and

B h : V h × V h → V h × V h , (q 0 b,h , q 1 b,h ) → (q b,h (0), qb,h (0)) (88)
where

qh + A 0h q h + C * 0h C 0h qh -C * 0h y h + νh θ A 0h qh = 0 in (0, τ ), (89)
q h (0) = q 0 h , qh (0) = q 1 h , (90) and qb
,h + A 0h q b,h -C * 0h C 0h qb,h + C * 0h y h -νh θ A 0h qb,h = 0 in (0, τ ), (91)
q b,h (τ ) = q 0 b,h , qb,h (τ ) = q 1 b,h , (92)
We assume that the numerical viscosity terms νh θ A 0h qh and -νh θ A 0h qb,h are used to obtain the numerical counterpart of (86). More precisely, let us consider the finite-dimensional approximations of the system (84)-(85):

ẅh + A 0,h w h + C * 0h C 0h ẇh + νh θ A 0h ẇh = 0 in (0, τ ), (93)
w h (0) = w 0 h , ẇh (0) = w 1 h . (94)
We assume that ν 0 and θ > 0 are chosen so that there exist M > 0 and α > 0 so that for all h,

ẇh (t) V h + A 1/2 0h w h (t) V h M e -αt w 1 h V h + A 1/2 0h w 0 h V h ∀(w 0 h , w 1 h ) ∈ V h × V h . (95)
Remark 7.1. Let us emphasize that without the numerical viscosity νh θ A 0h ẇh , the property (95) may be false (see, for instance, [START_REF] Glowinski | A numerical approach to the exact boundary controllability of the wave equation. I. Dirichlet controls: description of the numerical methods[END_REF], [START_REF] Infante | Boundary observability for the space-discretizations of the 1-d wave equation[END_REF], [START_REF] Antonio | Boundary observability for the space semidiscretizations of the 1-D wave equation[END_REF]). The use of artificial numerical viscosity to ensure the uniform exponential decay (95) was considered in several partial differential equations or classes of partial differential equations. Let us quote some of them that can be used as examples of application of this section:

• 2D wave equation with finite-difference space semi-discretization ( [START_REF] Roder | Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity[END_REF]);

• 2D plate equation with finite-difference space semi-discretization ( [START_REF] Ramdani | Internal stabilization of the plate equation in a square: the continuous and the semi-discretized problems[END_REF]);

• a class of second order evolution equations (with C 0 bounded and the spectrum of A 0 consisting of isolated eigenvalues with a gap between them) ( [START_REF] Ramdani | Uniformly exponentially stable approximations for a class of second order evolution equations-application to LQR problems[END_REF]).

Let us also remark that for the 1D plate equation with finite-difference space semi-discretization, it is proved in [START_REF] Ramdani | Semi-discrétisation en espace du problème de la stabilisation interne de l'équation des poutres[END_REF] that (95) holds without the add of a numerical viscosity (ν = 0).

Let us consider the sequence

a (n) h , b (n) h = (B h • F h ) a (n-1) h , b (n-1) h , n ∈ N * ,
where a

(0) h , b (0)
h is an initial guess of (v 0 h , v 1 h ). We also assume the following technical hypothesis: there exists M > 0 such that for all h small enough C * 0h L(U h ,V h ) M.

(96)

Finally, we assume the following hypothesis on the numerical schemes with numerical viscosity: let us consider v h and v b,h the solutions of vh + A 0h v h + νh θ A 0h vh = 0 in (0, τ ), (97)

v h (0) = v 0 h , vh (0) = v 1 h , (98)
and vb,h + A 0h v b,h -νh θ A 0h vb,h = 0 in (0, τ ), (99)

v b,h (τ ) = v 0 b,h , vb,h (τ ) = v 1 b,h . (100)
) -v 1 h V h + A 1/2 0h (v b,h (0) -v 0 h ) V h → 0. (102)
Note that the above condition always holds true if ν = 0. Recall that v 0 h and v 1 h are approximations of v 0 and v 1 . Theorem 7.2. Assume the above hypotheses. Then there exist τ > 0, β ∈ (0, 1) and ε h > 0 such that for all n,

b n h -v 1 h V h + A 1/2 0h (a n h -v 0 h ) V h β 2n b 0 h -v 1 h V h + A 1/2 0h (a 0 h -v 0 h ) V h + 1 1 -β 2 ε h ,
where ε h is independent of n and goes to 0 as h → 0. Remark 7.3. Let us emphasize that in this abstract result, the time τ used to construct the numerical observers may be quite large and is obtained in the proof by using (95). The minimal time τ should be related, as in the case of the 1d wave equation, to the observability inequality.

Proof of Theorem 7.2. We notice that w h := v h -q h satisfies ẅh + A 0h w h + C * 0h C 0h ẇh + νh θ A 0h ẇh = C * 0h η h in (0, τ ), (103) and using (95), we deduce that there exist τ > 0 and β < 1 such that for all t τ , ẇh (t)

w h (0) = v 0 h -q 0 h , ẇh (0) = v 1 h -q 1 h , (104)
V h + A 1/2 0h w h (t) V h β v 1 h -q 1 h V h + A 1/2 0h (v 0 h -q 0 h ) V h + t 0 C * 0h η h V h ds. ( 107
)
In particular, vh (τ ) -qh (τ )

V h + A 1/2 0h (v h (τ ) -q h (τ )) V h β v 1 h -q 1 h V h + A 1/2 0h (v 0 h -q 0 h ) V h + M τ 0 η h V h ds. ( 108
)
In a similar way, vb,h (0) -qb,h (0

) V h + A 1/2 0h (v b,h (0) -q b,h (0)) V h β vb,h (τ ) -qb,h (τ ) V h + A 1/2 0h (v b,h (τ ) -q b,h (τ )) V h + M τ 0 ηh V h ds, (109)
where ηh := C 0h vb,h -y h .

Taking v b,h (τ ) = v h (τ ), vb,h (τ ) = vh (τ ) q b,h (τ ) = q h (τ ), qb,h (τ ) = qh (τ )

we deduce from (108) and from (109) that vb,h (0) -qb,h (0

) V h + A 1/2 0h (v b,h (0) -q b,h (0)) V h β 2 v 1 h -q 1 h V h + A 1/2 0h (v 0 h -q 0 h ) V h + M τ 0 η h V h ds + M τ 0 ηh V h ds. ( 111
)
The above estimate yields

v 1 h -qb,h (0) V h + A 1/2 0h (v 0 h -q b,h (0)) V h β 2 v 1 h -q 1 h V h + A 1/2 0h (v 0 h -q 0 h ) V h + M τ 0 η h V h ds + M τ 0 ηh V h ds + vb,h (0) -v 1 h V h + A 1/2 0h (v b,h (0) -v 0 h ) V h . (112)
Using ( 101) and (102) and proceeding by induction, we deduce the result.

Figure 1 :

 1 Figure 1: Reconstruction of v 0 (at the left) and of v 1 (at the right)

Figure 2 :

 2 Figure 2: Relative error made after n iterations

Figure 3 :

 3 Figure 3: Relative error with respect to ν

Figure 4 :

 4 Figure 4: Reconstruction of v 0 and relative error with respect to τ

Figure 5 :

 5 Figure 5: Reconstruction of v 0 , v 1 and relative error made after n iterations and allows to reconstruct such initial conditions.Finally, we present a last experiment where we add in the observation y a random perturbation. More precisely, instead of giving y(t k ) to the numerical scheme, we give y(t k ) + X k , where (X k ) k is a random vector, normally distributed with mean 0 and deviation σ.We take µ = 5, τ = 2.5, N = 240, M = 1200, ν = 3 and n = 10 and

Figure 6 :

 6 Figure 6: Reconstruction of v 0 , v 1 and relative error with random perturbation

τ 0 C 0 C

 00 0h vh -y h U h ds + τ 0h vb,h -y h U h ds → 0 (101) and if v h (τ ) = v b,h (τ ), vh (τ ) = vb,h (τ ), thenvb,h (0

  with η h := C 0h vh -y h . 0h -C * 0h C 0h -νh θ A 0h w h ẇh + 0 C * 0h η h(106)

Acknowledgments

The first author's work was partially supported by Fondecyt Conicyt 1120560 project and CONICYT Anillo ACPA ACT1106 grants. The second author's work was partially supported by Fondecyt Conicyt 1120560 project and by ANR Grant CISIFS (09-BLAN-0213-02).


Ana Leonor Silvestre

Takéo Takahashi

The motion of a fluid-rigid ball system at the zero limit of the rigid ball radius

We study the limiting motion of a system of a rigid ball moving in a Navier-Stokes fluid flow in R 3 as the radius of the ball goes to zero. Recently, Dashti and Robinson solved this problem in the 2-D case, in the absence of rotation of the ball [Arch. Rational Mech. Anal. 200 (2011) 285-312]. This restriction was caused by the difficulty in obtaining appropriate uniform bounds on the second order derivatives of the fluid velocity when the rigid body can rotate. In this paper, we show how to obtain the required uniform bounds on the velocity fields in the 3-D case. These estimates then allow to pass to the zero limit of the ball radius and show that the solution of the coupled system converges to the solution of the Navier-Stokes equations describing the motion of only fluid in the whole space. The trajectory of the centre of the ball converges to a fluid particle trajectory, which justifies the use of rigid tracers for finding Lagrangian paths of fluid flow.

Introduction

Suppose that, relatively to an inertial reference frame, a homogeneous three-dimensional rigid body, represented by a closed ball Br(t) := {y ∈ R 3 : |yh(t)| r}, is moving in a Navier-Stokes liquid which occupies the exterior domain Ωr(t) := R 3 \ Br(t). The function h = h(t) describes the trajectory of the center of mass of the rigid body, which is also allowed to rotate, ω = ω(t) being its angular velocity. Denoting by v = v(t, y) the Eulerian velocity of the fluid and q = q(t, y) the corresponding pressure in the same inertial frame, the evolution of the motion of the system body-liquid is described by the initial boundary value problem ∂tv(t, y) + (v(t, y) • ∇y) v(t, y) = ∇y • T (v(t, y), q(t, y)), y ∈ Ωr(t), t ∈ (0, T ), [START_REF] Conca | Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid[END_REF] ∇y • v(t, y) = 0, y ∈ Ωr(t), t ∈ (0, T ),

v(t, y) = h ′ (t) + ω(t) × (yh(t)) y ∈ ∂Br(t), t ∈ (0, T ),

mrh ′′ (t) = -

∂Br (t)
T (v(t, y), q(t, y))n(t, y) dS, t ∈ (0, T ), (4)

Jrω ′ (t) = - ∂Br (t)
(yh(t)) × T (v(t, y), q(t, y))n(t, y) dS, t ∈ (0, T ), [START_REF] Desjardins | Existence of weak solutions for the motion of rigid bodies in a viscous fluid[END_REF] v(0, y) = v 0 (y), y ∈ Ωr(0), ( 6)

h(0) = h 0 , h ′ (0) = h ′0 , ω(0) = ω 0 . (7)
In the above system, we have denoted by n(t) = n(t, y) the external unit normal to Ωr(t), and by T (v, q) the stress tensor of the liquid, defined by

T (v, q) = 2νD(v) -q I3, (8)
D(v)ij = 1 2 ∂vi ∂yj + ∂vj ∂yi i, j = 1, 2, 3, (9)
where ν is the viscosity coefficient of the liquid and where I3 is the identity matrix of order 3. Finally, mr and Jr are respectively the mass of the body, and the moment or the matrix of inertia of the body. We have assumed that the density of the ball is constant and we denote it by ρ. Then mr and Jr have explicit formulas:

mr = 4 3 πr 3 ρ (10)
and

Jr = 8 15 πr 5 ρ I3 . (11)
Note that Jr is independent of time since the structure is a homogeneous ball. By means of the changes of variables

x := yh(t), u(t, x) := v(t, x + h(t)), p(t, x) := q(t, x + h(t)), ℓ(t) := h ′ (t), [START_REF] Grandmont | Existence for an unsteady fluid-structure interaction problem[END_REF] system ( 1)-( 7) can be written in a frame attached to the rigid ball:

∂tu + ((u -ℓ) • ∇) u = ∇ • T (u, p) in (0, T ) × Ωr, (13)
∇ • u = 0 in (0, T ) × Ωr, (14)
u(t, x) = ℓ(t) + ω(t) × x for t ∈ (0, T ), x ∈ ∂Br, (15)
mrℓ ′ = -∂Br T (u, p)n dS in (0, T ), ( 16)

Jrω ′ = - ∂Br
x × T (u, p)n dS in (0, T ), [START_REF] Iftimie | Remarks on the vanishing obstacle limit for a 3D viscous incompressible fluid[END_REF] u(0, x) = u 0 (x), x ∈ Ωr ( 18)

ℓ(0) = ℓ 0 , ω(0) = ω 0 . ( 19
)
The advantage of this formulation is that now the fluid domain is known and time-independent Ωr = {x ∈ R 3 : |x| > r}. As concerns the well-posedness of the system ( 13)- [START_REF] Lamb | Hydrodynamics[END_REF], we refer to the works of Galdi and Silvestre [START_REF] Galdi | Strong solutions to the problem of motion of a rigid body in a Navier-Stokes liquid under the action of prescribed forces and torques[END_REF], Takahashi and Tucsnak [START_REF] Takahashi | Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid[END_REF]. In particular, for all r, for any

(u 0 , ℓ 0 , ω 0 ) ∈ H 1 (Ωr) × R 3 × R 3 such that u 0 (x) = ℓ 0 + ω 0 × x on ∂Br,
the system ( 13)-( 19) admits a unique local strong solution

ur ∈ H 1 (0, T ; L 2 (Ωr) 3 ) ∩ L ∞ (0, T ; H 1 (Ωr) 3 ) ∩ L 2 (0, T ; H 2 (Ωr) 3 ), ℓr ∈ H 1 (0, T ) 3 , ωr ∈ H 1 (0, T ) 3 .
Note that the maximal time of existence may be finite and may depend on r. In the last years, fluid-structure systems such as system ( 13)-( 19) have been extensively studied. After the pioneer results of Judakov [START_REF] Judakov | The solvability of the problem of the motion of a rigid body in a viscous incompressible fluid[END_REF], Serre [START_REF] Serre | Chute libre d'un solide dans un fluide visqueux incompressible[END_REF], and Weinberger [START_REF] Weinberger | On the steady fall of a body in a Navier-Stokes fluid[END_REF], many results have been published in the case of exterior domain (see, for instance, [START_REF] Leonor | On the slow motion of a self-propelled rigid body in a viscous incompressible fluid[END_REF][START_REF] Cumsille | Wellposedness for the system modelling the motion of a rigid body of arbitrary form in an incompressible viscous fluid[END_REF][START_REF] Galdi | On the steady self-propelled motion of a body in a viscous incompressible fluid[END_REF][START_REF] Galdi | Slow motion of a body in a viscous incompressible fluid with application to particle sedimentation[END_REF][START_REF] Galdi | On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications[END_REF]) or in the case of bounded domain (see, for instance, [START_REF] Conca | Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid[END_REF][START_REF] Desjardins | Existence of weak solutions for the motion of rigid bodies in a viscous fluid[END_REF][START_REF] Desjardins | On weak solutions for fluid-rigid structure interaction: compressible and incompressible models[END_REF][START_REF] Feireisl | On the motion of rigid bodies in a viscous fluid[END_REF][START_REF] Grandmont | Existence for an unsteady fluid-structure interaction problem[END_REF][START_REF] Gunzburger | Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions[END_REF][START_REF] Hoffmann | On a motion of a solid body in a viscous fluid. Two-dimensional case[END_REF]).

As in the work of Dashti and Robinson [START_REF] Dashti | The motion of a fluid-rigid disc system at the zero limit of the rigid disc radius[END_REF], our aim is to investigate the limit of system ( 13)- [START_REF] Lamb | Hydrodynamics[END_REF] when the radius r of the ball goes to 0. We show that the limiting system is the following Navier-Stokes system in R 3 :

∂tu + ((u -ℓ) • ∇) u = ∇ • T (u, p) in (0, T ) × R 3 , (20)
∇ • u = 0 in (0, T ) × R 3 , (21)
u(0, x) = u 0 (x), x ∈ R 3 . ( 22
)
The function ℓ is such that u(t, 0) = ℓ(t).

(23) In particular, using the change of variables [START_REF] Grandmont | Existence for an unsteady fluid-structure interaction problem[END_REF] we conclude that

∂tv + (v • ∇) v = ∇ • T (v, q) in (0, T ) × R 3 , (24)
∇ • v = 0 in (0, T ) × R 3 , (25)
v(0, x) = v 0 (x), x ∈ R 3 , (26)
and h ′ (t) = v(t, h(t)) in (0, T ). ( 27) Dashti and Robinson [START_REF] Dashti | The motion of a fluid-rigid disc system at the zero limit of the rigid disc radius[END_REF] have solved this problem in R 2 , in the absence of rotation, considering ω = 0 by imposing an external torque and zero initial angular velocity of the disc. This last restriction has been made because of the difficulty in obtaining L 2 -estimates independent of r for D 2 ur when the particle can rotate, see Lemma 4 of [START_REF] Dashti | The motion of a fluid-rigid disc system at the zero limit of the rigid disc radius[END_REF]. Such estimates are crucial to pass to the limit of the ball radius in the equations of motion of the fluid and show that the solution of the coupled system converges to the solution of the Navier-Stokes equations describing the motion of only fluid in the whole space.

The study of this type of asymptotic analysis problems is also considered in [START_REF] Iftimie | Two-dimensional incompressible viscous flow around a small obstacle[END_REF], [START_REF] Iftimie | Remarks on the vanishing obstacle limit for a 3D viscous incompressible fluid[END_REF] for the case of a fixed obstacle. In [START_REF] Robinson | A coupled particle-continuum model: well-posedness and the limit of zero radius[END_REF], a simplified model of combustion is analyzed with this approach.

In this paper we extend the results of Dashti and Robinson, considering the three-dimensional fluid-ball interaction problem (13)- [START_REF] Lamb | Hydrodynamics[END_REF]. We show that the required uniform bounds on the velocity fields can be obtained by a careful analysis of a steady Stokes-type problem associated with the problem ( 13)- [START_REF] Lamb | Hydrodynamics[END_REF]. Dashti and Robinson derived the estimates for D 2 ur based on the classical exterior steady Stokes problem with a rigid velocity on the boundary. Here, we decompose the solution of the Stokes problem associated with the fluid-structure interaction problem in such a way that the explicit solutions for Stokes flow past a translating ball and around a rotating ball can be used to obtain a precise information about the dependence in r when combined with the Newton's laws-type equations of the modified Stokes problem (see Theorem 3.1).

To state our main result more precisely, we extend the fluid velocities ur in Br by the rigid velocities:

ur(x) = ℓr + ωr × x in Br. This convention is used in all that follows. We also use some classical notation for functional spaces of divergence-free velocities H and V . We define them in Section 2 (see (28), (29)).

Theorem 1.1. Assume (u 0 r )0<r<r * is a bounded sequence of V with u 0 r (x) = ℓ 0 r + ω 0 r × x in Br, for all r and assume that there exists u 0 ∈ V such that u 0 r ⇀ u 0 in H as r → 0. Then there exists T independent of r such that for all r, the strong solution (ur, ℓr, ωr) of (13)- [START_REF] Lamb | Hydrodynamics[END_REF] associated to (u 0 r , ℓ 0 r , ω 0 r ) exists on (0, T ). Moreover, for all r0, R > 0,

ur ⇀ u in H 1 (0, T ; L 2 (R 3 ) 3 ) ∩ L 2 (0, T ; H 2 (Ωr 0 ) 3 ) -weak, ur ⇀ u in L ∞ (0, T ; H 1 (R 3 ) 3 ) -weak*, ur → u strongly in L 2 (0, T ; L 2 (BR) 3 ), ℓr → ℓ strongly in L 2 (0, T ) 3 ,
where (u, ℓ) is the solution of (20)-( 22), [START_REF] Leonor | On the slow motion of a self-propelled rigid body in a viscous incompressible fluid[END_REF] associated to u 0 . The plan of the paper is the following. In Section 2, we introduce some notation and prove some auxiliary interpolation inequalities. Section 3 is devoted to the analysis of the above mentioned steady Stokes-type problem, where we obtain an uniform estimate in r for the second derivatives of the solution of such a problem in terms of a norm of the associated Stokestype operator. In Section 4, we obtain a number of uniform estimates for the fluid-structure interaction problem (13)- [START_REF] Lamb | Hydrodynamics[END_REF] and, finally, in Section 5, we pass to the zero limit of the ball radius to obtain the Navier-Stokes flow ( 20)- [START_REF] Serre | Chute libre d'un solide dans un fluide visqueux incompressible[END_REF]. For the reader's convenience, we include in an Appendix (Section A) some calculations for the explicit Stokes solutions that have been used in Section 3.

Notation and auxiliary results

For a sufficiently regular vector field v : R 3 → R 3 , we denote by ∇v the second order tensor field whose components (∇v) jk are given by ∂v k ∂x j , and by D(v) we denote the symmetric part of ∇v (see [START_REF] Galdi | Strong solutions to the problem of motion of a rigid body in a Navier-Stokes liquid under the action of prescribed forces and torques[END_REF]). The second order derivatives of v will be indicated by D 2 v, more precisely, for each j ∈ {1, 2, 3}, D 2 vj is the Hessian matrix of vj .

We shall use standard notation for function spaces. So, for instance, L s (D) and H m (D) will denote the usual Lebesgue and Sobolev spaces on the domain D, with norms • s,D and

• m,2,D , respectively. Notations like L s (D)

3 (resp. L s (D) 3×3 ), H m (D) 3 (resp. H m (D) 3×3
), etc. will be used to denote analogous spaces of vector-valued functions (resp. tensor-valued functions).

We also use the following classical notation

H := {φ ∈ L 2 (R 3 ) 3 : ∇ • φ = 0 in R 3 }, (28)
V := H ∩ H 1 (R 3 ) 3 . ( 29
)
Finally, we consider in this paper the following subsets of R 3 :

Ωr = {x ∈ R 3 : |x| > r}, Br = {x ∈ R 3 : |x| r}, Br,R = {x ∈ R 3 : R |x| > r},
where 0 < r < R.

In the next lemma, we establish a useful interpolation inequality in the exterior domain Ωr, where the embedding constant is independent of r. with a constant C independent of u and r.

Proof. Let D ⊆ R 3 be an exterior domain. In [START_REF] Crispo | An interpolation inequality in exterior domains[END_REF], Crispo and Maremonti proved, in particular, that any function

u ∈ L 2 (D) with D 2 u ∈ L 2 (D) 3×3 satisfies u ∞,D C D 2 u 3 4 2,D u 1 4 2,D , (30)
where C is a constant independent of u. Let u ∈ L 2 (Ωr). Then, by a simple change of variables, we find the relation

u 2,Ωr = r 3 2 u r 2,Ω 1 where u r (x) := u(rx) (x ∈ Ω1). Analogously, if the second derivatives of u belong to L 2 (Ωr), we have D 2 u 2,Ωr = r -1 2 D 2 u r 2,Ω 1 . Hence, applying (30) to u r and Ω1 yields u ∞,Ωr = u r ∞,Ω 1 C D 2 u r 3 4 2,Ω 1 u r 1 4 2,Ω 1 with C independent of r. But D 2 u r 3 4 2,Ω 1 = r 3 4 (-1 2 ) D 2 u 3 4 2,Ωr = r -3 8 D 2 u 3 4 2,Ωr and u r 1 4 2,Ω 1 = r 1 4 3 2 u 1 4 2,Ωr = r 3 8 u 1 4 2,Ωr so that u ∞,Ω C D 2 u 3 4 2,Ωr u 1 4

2,Ωr

with a constant C independent of r.

We have also the following estimate.

Lemma 2.2. Let u ∈ H 2 (Ωr). Then the following inequality holds

u C 0,1/2 (Ωr ) C D 2 u 2,Ωr
with a constant C independent of u and r.

Proof. We proceed as in the proof of Lemma 2.1. More precisely, using [START_REF] Crispo | An interpolation inequality in exterior domains[END_REF] again, we know that for an exterior domain

D ⊆ R 3 , a function u ∈ L 2 (D) with D 2 u ∈ L 2 (D) 3×3 satisfies u C 0,1/2 (D) C D 2 u 2,D (31)
where C is a constant independent of u. Then for u ∈ L 2 (Ωr) with D 2 u ∈ L 2 (Ωr) 3×3 , we set u r (x) := u(rx) (x ∈ Ω1), and we recall

D 2 u 2,Ωr = r -1 2 D 2 u r 2,Ω 1 .
We can also prove

u C 0,1/2 (Ωr ) = r -1/2 u r C 0,1/2 (Ω 1 )
. Hence, applying (31) to u r and Ω1 yields the result.

As in Lemma 2 in [START_REF] Dashti | The motion of a fluid-rigid disc system at the zero limit of the rigid disc radius[END_REF], we can prove a useful result which shows how the trace inequality for the domain Ωr depends on r in the three-dimensional case.

Lemma 2.3. Let w ∈ L 1 loc (Ωr) be such that ∇w ∈ L 2 (Ωr) 3 . Then there exist constants w0 and

C such that w -w0 L 2 (∂Br ) Cr 1/2 ∇w L 2 (Ωr ) 3
where the constant C is independent of w and r. If w ∈ L 6 (Ωr) then w0 = 0.

Proof. Let w ∈ L 1 loc (Ωr) be such that ∇w ∈ L 2 (Ωr) 3 and set w r (y) := w(ry) (y ∈ Ω1). It is well known (see [START_REF] Crispo | An interpolation inequality in exterior domains[END_REF]) that there exist constants w0 and C such that

w r -w0 L 6 (Ω 1 ) C ∇w r L 2 (Ω 1 ) 3 (32)
where the constant C is independent of w and r, and w0 = 0 if w ∈ L 6 (Ωr). Now

∂Br |w -w0| 2 dS = r 2 ∂B 1 |w r -w0| 2 dS Cr 2 B 1,2 |w r -w0| 2 dx + Ω 1 |∇w r | 2 dx
and since

B 1,2 |w r -w0| 2 dx C B 1,2 |w r -w0| 6 dx 1/3 C w r -w0 2 6,Ω 1 , from (32), we get ∂Br |w -w0| 2 dS Cr 2 ∇w r 2 2,Ω 1 .
Finally, by changing variables again,

∂Br |w -w0| 2 dS Cr 2 ∇w r 2 2,Ω 1 Cr ∇w 2 2,Ωr .
Now we recall some function spaces appropriate for studying the system ( 13)- [START_REF] Lamb | Hydrodynamics[END_REF]. The spaces

Hr := {φ ∈ L 2 (R 3 ) 3 : ∇ • φ = 0 in R 3 , D(φ) = 0 in Br} (33) Vr := {φ ∈ H 1 (R 3 ) 3 : ∇ • φ = 0 in R 3 , D(φ) = 0 in Br} (34)
are Hilbert spaces for the inner products of L 2 (R 3 ) 3 and H 1 (R 3 ) 3 , respectively. It is well known that, for each φ ∈ Hr, there exist ξ φ ∈ R 3 and ω φ ∈ R 3 such that

φ |Br (x) = ξ φ + ω φ × x.
We also define a global density by

ρr = 1 in Ωr, ρ in Br,
and the following inner product of

L 2 (R 3 ) 3 (u, v)H r = R 3 ρru • vdx.
The corresponding norm is equivalent to the usual L 2 norm. The notation we choose for this inner product comes from the fact that if u, v ∈ Hr, then

(u, v)H r = Ωr u • vdx + mrξu • ξv + (Jrωu) • ωv.
It is also common (see [START_REF] Galdi | On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications[END_REF]) to consider the space

{v ∈ C ∞ c (R 3 ) 3 : ∇ • v = 0, D(v) =
∇v 2 2,R 3 = 2 D(v) 2 2,R 3 = 2 D(v) 2 2,Ωr , ∀v ∈ Wr. ( 35
)
Let us recall (see [START_REF] Galdi | On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications[END_REF]Lemma 4.11])

Wr = {v ∈ L 6 (R 3 ) 3 : ∇v ∈ L 2 (R 3 ) 3×3 , ∇ • v = 0, D(v) = 0 in Br}.
We also recall the useful relations (see [START_REF] Galdi | On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications[END_REF]Lemma 4.9])

r 1/2 |ξv| C ∇v 2,R 3 , ∀v ∈ Wr, (36)
r 3/2 |ωv| C ∇v 2,R 3 , ∀v ∈ Wr (37)
for positive constants C independent of v and r. Finally let us give the following result. Proof. We follow an idea of [START_REF] Dashti | The motion of a fluid-rigid disc system at the zero limit of the rigid disc radius[END_REF]. Since the normal to Ωr is given by n(x) = -x r , we have

u • n = ℓ • n on ∂Br.
Now, by taking a point x ∈ ∂Br such that ℓ • n(x) = |ℓ|, we obtain

|ℓ| = |u(x) • n(x)| |u(x)| u ∞,Ωr .
3 The Stokes-type problem and the operator A r

The semigroup approach of [START_REF] Takahashi | Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid[END_REF] for problem ( 13)-( 19) is based on the operator defined by

D(Ar) := {φ ∈ H 1 (R 3 ) 3 : φ |Ωr ∈ H 2 (Ωr) 3 , ∇ • φ = 0 in R 3 , D(φ) = 0 in Br} Ar := PrAr (38)
with Pr the orthogonal projector in L 2 (R 3 ) 3 onto Hr (with respect to the inner product (•, •)H r ) and

Aru :=      -ν∆u in Ωr 2ν mr ∂Br D(u)n dS + 2νJ -1 r ∂Br y × D(u)n dS × x in Br. (39)
In this section, we analyze the operator Ar. Given u ∈ D(Ar), there exist ℓ ∈ R 3 and ω ∈ R 3 such that

u |Br (x) = ℓ + ω × x.
Let f := Aru. Since f |Ωr ∈ L 2 (Ωr) 3 , there exists (see, for instance, [START_REF] Takahashi | Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid[END_REF]) p ∈ L 2 loc (Ωr) with ∇p ∈ L 2 (Ωr) 3 such that

-∇ • T (u, p) = f |Ωr in Ωr, ( 40
)
∇ • u = 0 in Ωr, (41)
u(x) = ℓ + ω × x for x ∈ ∂Br, (42)
lim |x|→∞ u(x) = 0, (43)
mrℓ f = - ∂Br T (u, p)n dS, (44)
Jrω f = - ∂Br x × T (u, p)n dS. ( 45
)
Our aim is to show that

D 2 u 2,Ωr C ∇u 2,R 3 + ( f |Ωr 2 2,Ωr + mr|ℓ f | 2 + Jrω f • ω f ) 1 2
with C independent of u, f and r. Recall that mr = O(r 3 ) and Jr = O(r 5 ).

Theorem 3.1. There exists a positive constant C independent of r such that

D 2 u 2,Ωr C( Aru Hr + ∇u 2,R 3 ) (u ∈ D(Ar)). (46)
Proof. We use the following decomposition (u, p) = (u (1) , p (1) ) + (u (2) , p (2) ) + (u (3) , p (3) ) (47)

where (u (1) , p (1) ) satisfies

-∇ • T (u (1) , p (1) ) = f |Ωr in Ωr, (48)
∇ • u (1) = 0 in Ωr, (49)
u (1) (x) = 0 for x ∈ ∂Br, (50)
lim |x|→∞ u (1) (x) = 0 (51) and (u (2) (x), p (2) (x)) = 3r 4 ℓ |x| + (ℓ • x)x |x| 3 + r 3 4 ℓ |x| 3 -3 (ℓ • x)x |x| 5 , 3νr 2
ℓ • x |x| 3 (52) (u (3) (x), p (3) (x)) = r 3 ω × x |x| 3 , 0 . (53)
Using the results of Section A, we see that (u (2) , p (2) ) and (u (3) , p (3) ) satisfy ∇ • T (u (2) , p (2) ) = 0, in Ωr, 2) , p (2) )ndS = 6πνrℓ, ∂Br

∇ • u (2) = 0, in Ωr, u (2) (x) = ℓ, x ∈ ∂Br, lim |x|→∞ u (2) (x) = 0, ∂Br T (u (
x × T (u (2) , p (2) )ndS = 0 (54) and ∇ • T (u (3) , p (3) ) = 0, in Ωr, 3) , p (3) )ndS = 0, ∂Br x × T (u (3) , p (3) )ndS = 8πνr 3 ω .

∇ • u (3) = 0, in Ωr, u (3) (x) = ω × x, x ∈ ∂Br, lim |x|→∞ u (3) (x) = 0, ∂Br T (u (
(55)

From ( 52), (53) and the results of Section A, we deduce the existence of a positive constant C independent of r such that

D 2 u (2)

2,Ωr

C|ℓ|r -1/2 (56)

D 2 u (3) 2,Ωr C|ω|r 1/2 . ( 57
)
Writing ( 44) and (45) in terms of the decomposition (47) of (u, p) yields 6πνrℓ = -mrℓ f -∂Br T (u (1) , p (1) )n dS 1) , p (1) )n dS.

8πνr 3 ω = -Jrω f - ∂Br x × T (u (
This implies

ℓ = - 2r 2 9ν ℓ f - 1 6πνr ∂Br
T (u (1) , p (1) )n dS (58) (1) , p (1) )n dS .

ω = - r 2 15ν ω f - 1 8πνr 3 ∂Br x × T (u
(59)

By Lemma 1 of [START_REF] Heywood | The Navier-Stokes equations: on the existence, regularity and decay of solutions[END_REF],

D 2 u (1) 2,Ωr C( f |Ωr 2,Ωr + ∇u (1) 2,Ωr ), (60)
with a constant C independent of r. Moreover, we have ∇u (1) 2,Ωr ∇u 2,Ωr + ∇u (2) 2,Ωr + ∇u (3) 2,Ωr ∇u 2,Ωr + C|ℓ|r 1/2 + C|ω|r 3/2 and from (36) and (37) it follows

∇u (1) 2,Ωr C ∇u 2,R 3 so that D 2 u (1) 2,Ωr C( f |Ωr 2,Ωr + ∇u 2,R 3 ). ( 61
)
In order to appropriatly bound ℓ and ω from (58) and (59), we begin by showing that the inequality ∂Br x × T (u (1) , p (1) )n dS Cr 5/2 f |Ωr 2,Ωr ,

holds with C independent of r. Indeed, if we consider the functions Hi(x) := r 3 e i ×x |x| 3 , which belong to L 2 (Ωr) 3 , and note that for each i = 1, 2, 3, ei • ∂Br x × T (u (1) , p (1) )n dS = ∂Br ei × x • T (u (1) , p (1) )n dS = ∂Br Hi • T (u (1) , p (1) )n dS, by Green's formula for the Stokes equations, we have ei • ∂Br x × T (u (1) , p (1) )n dS = ∂Br Hi • T (u (1) , p (1) )n dS = -

Ωr f • Hi dx .
Then (62) follows from the fact that Hi 2 Cr 5/2 (see Section A). It remains to show that ∂Br T (u (1) , p (1) )n dS

Cr 3/2 ( f |Ωr 2,Ωr + ∇u 2,Ωr ), ( 63
)
where, again, C is independent of r. Since ∂Br T (u (1) , p (1) )n dS = ∂Br T (u (1) , p (1) -p0)n dS Cr( ∇u (1) 2,∂Br + p (1) -p0 2,∂Br )

for any p0 ∈ R, we can modify the pressure p (1) in such a way that, according to Lemma 2.3,

p (1) -p0 2,∂Br Cr 1/2 ∇p (1) 2,Ωr .
Also by Lemma 2.3, ∇u

2,∂Br

Cr 1/2 D 2 u (1) 2,Ωr . Now, we only have to note that ∇p (1) 2,Ωr C( f 2,Ωr + D 2 u (1) 2,Ωr )

and use (61) to bound D 2 u (1) 2,Ωr . Finally, (58) and (59) combined with (63) and (62) yield

|ℓ| Cr 2 |ℓ f | + Cr 1/2 ( f |Ωr 2,Ωr + ∇u 2,Ωr ), |ω| Cr 2 |ω f | + C 1 r 1/2 f |Ωr 2,Ωr ,
for constants C independent of r. Coming back to (56) and (57) with the above estimates for |ℓ| and |ω|, we get

D 2 u (2) 2,Ωr Cr 3/2 |ℓ f | + C( f |Ωr 2,Ωr + ∇u 2,R 3 ) D 2 u (3) 2,Ωr Cr 5/2 |ω f | + C f |Ωr 2,Ωr
which together with (61) and (47) yields

D 2 u 2,Ωr Cr 3/2 |ℓ f | + Cr 5/2 |ω f | + C f |Ωr 2,Ωr + C ∇u 2,R 3 C( Aru Hr + ∇u 2,R 3 ).
4 Estimates independent of r for the solution of ( 13)- [START_REF] Lamb | Hydrodynamics[END_REF] Assume that u 0 r ∈ Vr for some r > 0. Then, writing u 0 r = ℓ 0 r + ω 0 r × x in Br, and taking (u 0 r , ℓ 0 r , ω 0 r ) for the initial conditions of ( 13)-( 19), we obtain (see, for instance, [START_REF] Takahashi | Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid[END_REF]) the existence and uniqueness of a strong solution

ur ∈ H 1 (0, T ; L 2 (R 3 ) 3 ) ∩ L ∞ (0, T ; H 1 (R 3 ) 3 ) ∩ L 2 (0, T ; H 2 (Ωr) 3 ).
Here and in what follows, we extend ur to Br by ℓr + ωr × x. Let us note that the maximal time of existence T may be finite and may depend on r.

We can write ( 13)-( 17) in an abstract way by using the operator Ar defined by ( 38)-( 39) and the following nonlinear operator:

Nr(u) = -((u -ℓ) • ∇) u in Ωr 0 in Br (u ∈ Vr),
and its projection in Hr Nr(u) := PrNr(u).

With this notation, we can write ( 13)-( 17) as

∂tur + Arur = Nr(ur). ( 64
) Theorem 4.1. Assume (u 0 r )0<r<r * is bounded in H 1 (R 3 ) 3 , u 0 r ∈ Vr for all r.
Then there exists T independent of r such that

(i) (ur)0<r<r * is bounded in H 1 (0, T ; L 2 (R 3 ) 3 ) ∩ L ∞ (0, T ; H 1 (R 3 ) 3 ) ∩ L 2 (0, T ; H 2 (Ωr) 3 ), (ii) (ℓr)0<r<r * is bounded in L 2 (0, T ) 3 .
Proof. Let us multiply (64) by ur and integrate over Ωr. We obtain

ur(t) 2 2,R 3 + ν T 0 ∇ur 2 2,R 3 dt C u 0 r 2 2,R 3 . (66)
+ u 0 r 8/9 2,R 3 + u 0 r 2 2,R 3 )( ∇ur 2 2,R 3 + 1) 4 . (73)
Let y(t) := 1 + ∇ur(t) 2 2,R 3 . Then, from (73) we have

y ′ Cy 4
with C independent of r. We conclude that y(t) y(0) 3 1 -3y(0) 3 Ct as long as t < 1/(3y(0) 3 C). Let us consider T small enough so that

T < 1 3C(1 + ∇u 0 r 2 2,R 3 ) 3 . Since (u 0 r )0<r<r * is bounded in H 1 (R 3 ) 3 ,
T can be chosen so that it is independent of r. We deduce that for all t ∈ [0, T ],

∇ur(t) 2 2,R 3 C(1 + ∇ur(0) 2 2,R 3 ) K1
with K1 independent of r. Now from (73), T 0 Arur(s) 2 Hr ds K2 with K2 independent of r, which implies with Theorem 3.1

D 2 ur L 2 (0,T ;L 2 (Ωr ) 3×3×3 ) K3
with K3 independent of r. We can find constants K4 and K5 independent of r such that ℓr L 2 (0,T ;R 3 ) K4 and ∂tur L 2 (0,T ;L 2 (Ωr ) 3 ) K5.

5 Passage to the zero limit of the ball radius in (13)-

The aim of this section is to prove the main result of the paper. Applying Theorem 4.1 and using a diagonal process, we deduce that for all r0 > 0,

ur ⇀ u in H 1 (0, T ; L 2 (R 3 ) 3 ) ∩ L 2 (0, T ; H 2 (Ωr 0 ) 3 ) -weak, ( 74
) ur ⇀ u in L ∞ (0, T ; H 1 (R 3 ) 3 ) -weak*, ( 75
) ℓr ⇀ ℓ in L 2 (0, T ) -weak. ( 76
)
In particular for any r0, u ∈ L 2 (0, T ; H 2 (Ωr 0 ) 3 ) and its norm in these spaces is bounded independently of r0. Following the proof of [START_REF] Dashti | The motion of a fluid-rigid disc system at the zero limit of the rigid disc radius[END_REF], we deduce that u ∈ L 2 (0, T ; H 2 (R 3 ) 3 ). Applying Aubin-Lions-Simon Lemma, and using the fact that (ur)0<r<r * is bounded in L 2 (0, T ; H 1 (BR) 3 ) ∩ H 1 (0, T ; L 2 (BR) 3 ), we conclude that there exists a subsequence such that ur → u strongly in L 2 (0, T ; L 2 (BR) 3 ).

(77)

In order to pass to the limit as r → 0, we write an integral formulation of ( 13)- [START_REF] Iftimie | Remarks on the vanishing obstacle limit for a 3D viscous incompressible fluid[END_REF]. Assume ϕ ∈ C 1 (R 3 ) 3 is divergence free, with compact support in Bs,R, R > s > 0. Then for r < s, we multiply (64) by ϕ to obtain By integrating by parts, we deduce from the above equation that

R 3 (∂tur -ν∆ur) • ϕ dx - R 3 [(ur -ℓr) • ∇) ϕ] • ur dx = 0 in (0, T ). ( 78
)
Using ( 74)-( 76) and (77), we deduce that for all ϕ ∈ C 1 (R 3 ) 3 divergence free, with compact support in Bs,R, R > s > 0,

R 3 (∂tu -ν∆u + ((u -ℓ) • ∇) u) • ϕ dx = 0 in (0, T ). ( 79
)
We observe that the functions in C 1 (R 3 ) 3 which are divergence free, with compact support in Bs,R, R > s > 0 are dense in H. To be complete we give below a proof of this result based on [20, pp. 165-166]. Consequently, we deduce that (79) holds for all ϕ ∈ L 2 (0, T ; H). In particular, (see, for instance, [26, pp. 14-15]), there exists p with ∇p ∈ L 2 (0, T ; L 2 (R 3 )) such that

∂tu -ν∆u + ((u -ℓ) • ∇) u = -∇p in (0, T ) × R 3 .
To recover the initial condition, we apply a classical method: using (74) and (77), we deduce that for any function ϕ ∈ C 1 ([0, T ) × R 3 ) 3 with compact support, we have

R 3 ur(0) • ϕ(0) dx = (0,T )×R 3 ur • ∂tϕ dt dx - (0,T )×R 3 ∂tur • ϕ dt dx → (0,T )×R 3 u • ∂tϕ dt dx - (0,T )×R 3 ∂tu • ϕ dt dx = R 3 u(0) • ϕ(0) dx.
Since u 0 r ⇀ u 0 weakly in L 2 (R 3 ) 3 , then we deduce that u(0) = u 0 . Let us now prove the strong convergence of (ℓr)0<r<r * . Let us consider a decreasing sequence r k → 0. Then, for k > m and for any n ∈ ∂B1,

|(ℓr k -ℓr m ) • n| = |(ur k (r k (-n)) -ur m (rm(-n))) • n| |(ur k (rm(-n)) -ur m (rm(-n))) • n| + |(ur k (r k (-n)) -ur k (rm(-n))) • n| ur k -ur m ∞,B rm,2rm + |r k -rm| 1/2 ur k C 0,1/2 (Ωr k ) . (80)
Let us note that, with an argument similar to the proof of Lemma 2.1, we can prove that for w ∈ H 2 (Ωr) the following inequality holds

w ∞,B r,2r C w 1/4 2,B r,2r D 2 w 3/4 2,B r,2r
with a constant C > 0 independent of w and r.

Combining (80) with the above result, Lemma 2.2 and with Theorem 4.1, we conclude that ℓr → ℓ in L 2 (0, T ) 3 . To obtain the relation between ℓ and u, we consider a similar calculation: for a fixed e ∈ ∂B1,

|u(t, 0) -ℓ(t)| |u(t, 0) -u(t, re)| + |u(t, re) -ℓr(t)| + |ℓr(t) -ℓ(t)| . (81)
The first term of the right-hand side of (81) can be estimated by u C 0,1/2 (R 3 ) , the last term has just been treated, and the second term can be handle through the following manipulation: for any n, e ∈ ∂B1,

|(u(t, re) -ℓr(t)) • n| = |(u(t, re) -ur(t, -rn)) • n| |(u(t, re) -u(t, -rn)) • n| + |(u(t, -rn) -ur(t, -rn)) • n| r 1/2 u C 0,1/2 (R 3 ) + u -ur ∞,B r,2r ,
and we conclude as above. This concludes the proof of Theorem 1.1.

We give here the proof of the density in H of the subspace of the functions C 1 (R 3 ) 3 which are divergence free, with compact support in Bs,R, R > s > 0. The proof is based on [20, pp. 165-166].

Lemma 5.1. Given u ∈ H and δ > 0, there exist 0 < s < 1 and u s ∈ V with supp u s ⊂ B s,1/s such that uu s 2,R 3 < δ.

Proof. Given u ∈ H and δ > 0, using the density of V in H, there exists v ∈ V such that

u -v 2,R 3 < δ 2 . For 0 < ε < 1 and v ∈ V , consider the problem of finding vε ∈ V such that ∆vε -vε -∇pε = ∆v -v in B ε,1/ε , ∇ • vε = 0 in B ε,1/ε . (82)
This problem has a unique solution (vε, pε)

∈ V × L 2 (B ε,1/ε )/R such that vε 2,B ε,1/ε + ∇vε 2,B ε,1/ε v V (83)
Moreover, pε can be normalized in such a way that

pε 2,B ε,1/ε C v V (84)
with C independent of ε and v. From (83), extending each vε by zero outside B ε,1/ε , we deduce the existence of v ∈ V such that vε ⇀ v weakly in V, as ε → 0, while (84) implies the existence of p ∈ L 2 (R 3 ) such that

pε ⇀ p weakly in L 2 (R 3 ), as ε → 0. The function v -v ∈ V satisfies ∆(v -v) -(v -v) -∇p = 0 in R 3 , ∇ • (v -v) = 0 in R 3 and therefore v = v. Actually, since vε ⇀ v weakly in V and vε V v V , it follows that vε → v strongly in V, as ε → 0.
This results allows us to take vs ∈ V with support in B s,1/s such that vvs 2,R 3 < δ 2 .

A Computations involving the auxiliary Stokes solutions

Stokes derived in 1851 [START_REF] Gabriel | On the effect of the internal friction of fluids on the motion of pendulums[END_REF] (see also [START_REF] Lamb | Hydrodynamics[END_REF]) an explicit solution for tridimensional Stokes flow past a translating sphere. Specifically, the solution to the Stokes problem As concerns the summability properties of u and its derivatives, we have u ∈ L q (Ωr) 3 , ∀q > 3, with u q,Ωr C(q)|ξ|r 3/q , ∇u ∈ L q (Ωr) 3×3 , ∀q > 3/2, with ∇u q,Ωr C(q)|ξ|r 3/q-1 D 2 u ∈ L q (Ωr) 3×3×3 , ∀q > 1, with D 2 u q,Ωr C(q)|ξ|r 3/q-2 .

In particular, ∇u where ω ∈ R 3 , is (see [START_REF] Gabriel | On the effect of the internal friction of fluids on the motion of pendulums[END_REF][START_REF] Lamb | Hydrodynamics[END_REF])

u(x) = r 3 ω × x |x| 3 p(x) = c
where c is a constant. The gradient of u and its symmetric part are given by ∇u(x) = r 3 |x| 3 W - We have u ∈ L q (Ωr) 3 for all q > 3/2 and u q,Ωr C(q)|ω|r 1+3/q .

In particular, u has finite kinetic energy with u 2,Ωr C|ω|r 5/2 . For the derivatives of u, we have ∇u ∈ L q (Ωr) 3×3 , ∀q > 1, with ∇u q,Ωr C(q)|ω|r 3/q , D 2 u ∈ L q (Ωr)

3×3×3

, ∀q ∈ [1, ∞[, with D 2 u q,Ωr C(q)|ω|r 3/q-1 .

Lemma 2 . 1 .

 21 Let u ∈ H 2 (Ωr). Then the following inequality holds u

  0 in Br} and its completion Wr with respect to the seminorm D(v) 2,R 3 . In what follows, we use several times the following classical relation

Lemma 2 . 4 .

 24 Let u ∈ Vr be such that u |Ωr ∈ H 2 (Ωr) 3 and u |Br (x) = ℓ + ω × x. Then |ℓ| u ∞,Ωr .

R 3 ρr∂tur

 3 • ϕ dx + (Arur, ϕ) Hr = (Nr(ur), ϕ) Hr in (0, T ).

∇ 4 ξ ⊗ x -x ⊗ ξ |x| 3 + 7 and 7 .

 4377 • T (u, p) = 0, in Ωr,∇ • u = 0, in Ωr, u(x) = ξ for x ∈ ∂Br, |x| 3 + cwhere c is a constant. By direct calculations, we find that the gradient of u is∇u(x) = 3r x)I3 + ξ ⊗ x + x ⊗ ξ |x| 5 + 15(ξ • x)x ⊗ x |x| x)I3 + ξ ⊗ x + x ⊗ ξ |x| 5 + 15(ξ • x)x ⊗ x |x|Therefore, since the external unit normal to Ωr is given by n(x) = -x r and(ξ ⊗ x)n(x) = -rξ, (x ⊗ ξ)n(x) = (ξ • x)n(x), (x ⊗ x)n(x) = r 2 n(x) (x ∈ ∂Br)

  2,Ωr C|ξ|r 1/2 and D 2 u 2,Ωr C|ξ|r -1/2 . Now we consider the Stokes flow around a rotating ball. The solution to the Stokes problem ∇ • T (u, p) = 0, in Ωr, ∇ • u = 0, in Ωr, u(x) = ω × x for x ∈ ∂Br, lim |x|→∞ u(x) = 0

3r 3 (ω × x) ⊗ x |x| 5 5

 355 [x ⊗ (ω × x) + (ω × x) ⊗ x] We have Du(x)n(x) = 3ω × x 2r , x × Du(x)n(x) = 3x × (ω × x) 2r (x ∈ ∂Br)and then, due to the symmetry of the domain Ωr, ∂Br T (u, p)n dS = 0, ∂Br x × T (u, p)n dS = 2ν ∂Br x × (Du(x)n(x)) dS = 3ν r ∂Br x × (ω × x) dS = 8πνr 3 ω.

  Hr Arur Hr ((urℓr) • ∇) ur 2,Ωr Arur Hr .

	Now, using Theorem 3.1, and (35) implies
	ν 2	d dt	∇ur 2 2,R 3 + Arur 2 Hr	C ur	1/4 2,Ωr ∇ur 2,Ωr Arur	7/4 Hr + C ur	1/4 2,Ωr ∇ur	7/4 2,Ωr Arur Hr .
	Applying Young's inequality to the above estimate yields
				ν	d dt	∇ur 2 2,R 3 + Arur 2 Hr	C( ur 2 2,Ωr + 1) ∇ur 8 2,Ωr + C ur	8/9 2,Ωr
	with C independent of r. Using the estimate (66) we get
			ν	d dt	∇ur 2 2,R 3 + Arur 2 Hr	C(1
		Now we multiply (64) by Arur and integrate by parts over Ωr to get
						ν	d dt	D(ur) 2 2,Ωr + Arur 2 Hr = (Nr(ur), Arur)H r ,	(67)
	where
			(Nr(ur), Arur)H r		Nr(ur) (68)
	By Hölder's inequality,
								(ur • ∇)ur 2,Ωr	ur ∞,Ωr ∇ur 2,Ωr	(69)
	and by Lemma 2.4
								(ℓr • ∇)ur 2,Ωr	ur ∞,Ωr ∇ur 2,Ωr .	(70)
		Again we use (69), (70) and Lemma 2.1 to get
						((ur -ℓr) • ∇) ur 2,Ωr	2 ur ∞,Ωr ∇ur 2,Ωr
									C ur	1/4 2,Ωr D 2 ur	3/4 2,Ωr ∇ur 2,Ωr	(71)
			ν	d dt	D(ur) 2 2,Ωr + Arur 2 Hr	C ur	1/4 2,Ωr D 2 ur	3/4 2,Ωr ∇ur 2,Ωr Arur Hr .	(72)

with a constant C independent of r. By combining (71) with (

67

) and (68), we deduce that