Olivier Glass

Franck Sueur

On the motion of a rigid body in a two-dimensional irregular ideal flow

 to a case where the flow is irregular.

Introduction

In this paper, we consider the movement of a rigid body immersed in a perfect incompressible fluid in the plane. The solutions that we will consider are weak solutions in the spirit of the solutions of Yudovich [START_REF] Yudovich | Non-stationary flows of an ideal incompressible fluid[END_REF] concerning the fluid alone. The goal of this paper is to study time regularity issues for the solid and fluid flows associated to such solutions of the system.

Let us be more specific on the problem under view. We consider the motion of a rigid body which occupies at time t the domain S(t) ⊂ R 2 . The motion of this solid is rigid (and as we will see, driven by the pressure force on its boundary), so that S(t) is obtained by a rigid movement (that is a translation and a rotation) from its initial position S 0 , which is supposed to be a closed, simply connected domain in the plane with smooth boundary. In the rest of the plane, that is in the open set F(t) := R 2 \ S(t), evolves a planar ideal fluid driven by the Euler equations. We denote correspondingly F 0 := R 2 \ S 0 the initial fluid domain.

The complete system driving the dynamics reads

∂u ∂t + (u • ∇)u + ∇p = 0 for x ∈ F(t), (1.1)
div u = 0 for x ∈ F(t),

u • n = u S • n for x ∈ ∂S(t), (1.2)
)

mh (t) = ∂S(t)
pn ds, (1.4)

J θ (t) = ∂S(t) p(x -h(t)) ⊥ • n ds, (1.5
)

u| t=0 = u 0 , (1.6)
h(0) = h 0 , h (0) = 0 , θ(0) = 0, r(0) = r 0 .

(1.7)

Here u = (u 1 , u 2 ) and p denote the velocity and pressure fields defined on F(t) for each t, m > 0 and J > 0 denote respectively the mass and the inertia of the body while the fluid is supposed to be homogeneous of density 1, in order to simplify the equations (and without loss of generality).

1 When x = (x 1 , x 2 ) the notation x ⊥ stands for x ⊥ = (-x 2 , x 1 ), n denotes the unit outward normal on ∂F(t), ds denotes the integration element on the boundary ∂S(t) of the body. In the equations (1.4) and (1.5), h(t) is the position of the center of mass of the body, (t) := h (t), is the velocity of the center of mass and r(t) := θ (t), is the angular velocity of the body. Accordingly, the solid velocity is given by u S (t, x) := (t) + r(t)(x -h(t)) ⊥ .

(1.8)

Since S(t) is a rigid body its position is obtained thanks to the rotation matrix Q(t) := cos θ(t) -sin θ(t) sin θ(t) cos θ(t) .

(1.9)

More precisely the position Φ S (t, x) ∈ S(t) at the time t of the point fixed to the body with an initial position x is Φ S (t, x) := h(t) + Q(t)(x -h 0 ), so that S(t) = Φ S (t, S 0 ).

In Eulerian description the velocity u S (t, x) of the body S at time t at the position x is u S (t, x) = ∂ t Φ S (t, (Φ S ) -1 (t, x)).

The flow Φ S corresponding to the solid is a rigid movement, that can be considered as a function of t ∈ R with values in the special Euclidean group SE(2) of rigid movement in the plane. The equations (1.1) and (1.2) are the incompressible Euler equations, the condition (1.3) means that the boundary is impermeable, the equations (1.4) and (1.5) are the Newton's balance laws for linear and angular momenta: the fluid acts on the body through pressure forces.

A key quantity in the analysis is the vorticity

ω := curl u = ∂ 1 u 2 -∂ 2 u 1 ,
which satisfies the transport equation:

∂ t ω + (u • ∇)ω = 0.
(1.10)

One has the following result concerning the Cauchy problem for the above system, the initial position of the solid being given. This result describes both Yudovich and classical solutions.

Theorem 1. For any u 0 ∈ C 0 (F 0 ; R 2 ), ( 0 , r 0 ) ∈ R 2 × R, such that:

div u 0 = 0 in F 0 and u 0 • n = ( 0 + r 0 (x -h 0 ) ⊥ ) • n on ∂S 0 , (1.11)
w 0 := curl u 0 ∈ L ∞ c (F 0 ), (1.12) lim |x|→+∞ u 0 (x) = 0,
there exists a unique solution ( , r, u) of

(1.1)-(1.7) in C 1 (R; R 2 ×R)×L ∞ loc (R; LL(F(t))) with ∂ t u, ∇p ∈ L ∞ loc (R; L q (F(t))) for any q ∈ (1, +∞).
Moreover such a solution satisfies that for all t > 0, w(t) := curl(u(t)) ∈ L ∞ c (F(t)), and w(t, •) L q (F (t)) (for any q ∈ [1, ∞]), F (t) w(t, x) dx and the circulation

γ := ∂S(t) u • τ ds, (1.13) are preserved over time. If moreover u 0 ∈ C λ+1,ν (F 0 ; R 2 ) for λ in N and ν ∈ (0, 1), then u is L ∞ loc (R; C λ+1,ν (F(t))).
The notation LL(Ω) refers to the space of log-Lipschitz functions on Ω, that is the set of functions f ∈ L ∞ (Ω) such that

f LL(Ω) := f L ∞ (Ω) + sup x =y |f (x) -f (y)| |x -y|(1 + ln -|x -y|) < +∞. (1.14)
On the other hand the notation C λ,ν (F 0 ) denotes the Hölder space, endowed with the norm:

u C λ,ν (F0) := sup |α| λ ∂ α u L ∞ (F0) + sup x =y∈F0 |∂ α u(x) -∂ α u(y)| |x -y| ν < +∞.
Above, we used the abuse of notation L ∞ (R; X(F(t))) (resp. C 0 (R; X(F(t)))) where X is a functional space; by this we refer to functions defined for almost each t as a function in the space X(F(t)), and which can be extended as a function in L ∞ (R;

X(R 2 )) (resp. C 0 (R; X(R 2 ))).
Theorem 1 is proven in [START_REF] Glass | On the motion of a small body immersed in a two dimensional incompressible perfect fluid[END_REF]. Let us also mention that the existence and uniqueness of finite energy and classical solutions to the problem (1.1)-(1.7) has been tackled by Ortega, Rosier and Takahashi in [START_REF] Ortega | On the motion of a rigid body immersed in a bidimensional incompressible perfect fluid[END_REF].

Consider a solution ( , r, u) given by Theorem 1. The corresponding fluid velocity field u is log-Lipschitz in the x-variable; consequently there exists a unique flow map

Φ F continuous from R × F 0 to F(t) such that Φ F (t, x) = x + t 0 u(s, Φ F (s, x))ds.
(1.15)

Moreover there exists c > 0 such that for any t, the vector field Φ(t, •) lies in the Hölder space

C 0,exp(-c|t| ω0 L ∞ (F 0 ) ) (F 0 ),
and an example due to Bahouri and Chemin [START_REF] Bahouri | Équations de transport relatives à des champs de vecteurs non-lipschitziens et mécanique des fluides[END_REF] shows that this estimate is optimal.

For M 1, we will denote by G M ((-T, T ); E) the Gevrey space of order M of smooth functions f : (-T, T ) → E with values in a Banach space E such that for any compact set K ⊂ (-T, T ) there exist L, C > 0 such that for all s ∈ N and for all t ∈ K,

∂ s t f (t) CL s (s!) M . (1.16)
Let us recall that for M = 1 the Gevrey space G M ((-T, T ); E) is the space C ω ((-T, T ); E) of real-analytic functions.

The main result of this paper is the following.

Theorem 2. Assume that the boundary ∂S 0 is Gevrey of order M and that the assumptions of Theorem 1 are satisfied. Associate the solid and fluid flows Φ S and Φ F to the solution ( , r, u). Then for each T > 0,

(Φ S , Φ F ) ∈ G M +2 ((-T, T ); SE(2) × C 0,exp(-cT ω0 L ∞ (F 0 ) ) (F 0 )).
(1.17)

Remark 3. Since we will proceed by induction the proof of Theorem 2 also yields that assuming that the boundary ∂S 0 is only C k+1,ν , with k ∈ N and ν ∈ (0, 1), then the flow

(Φ S , Φ F ) are C k from (-T, T ) to SE(2) × C 0,exp(-cT ω0 L ∞ (F 0 ) ) (F 0 ).
Remark 4. Considering the particular case M = 1, we see that when the boundary is real-analytic, the flows (Φ S , Φ F ) belong to the Gevrey space G 3 . This extends the results obtained in [START_REF] Gamblin | Système d'Euler incompressible et régularité microlocale analytique[END_REF] for a fluid filling the whole plane and in [START_REF] Sueur | Smoothness of the trajectories of ideal fluid particles with Yudovich vorticities in a planar bounded domain[END_REF] for a fluid bounded by fixed boundaries. Moreover, mixing the techniques of the present paper and the ones of [START_REF] Glass | Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid[END_REF], one could prove that for strong solutions of the system, that is when u 0 ∈ C λ+1,ν (F 0 ; R 2 ) in Theorem 1, and when the boundary ∂S 0 is real-analytic, then the flows

(Φ S , Φ F ) belong to C ω ((-T, T ); SE(2) × C λ+1,ν (F 0 )).
Actually it is expected that this also holds true in three dimensions locally in time. This is the equivalent in the case of an exterior domain of what is proven in [START_REF] Glass | Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid[END_REF] in a bounded one.

Another way to express the result obtained in Theorem 2 is that we prove that for any k ∈ N, for any T > 0, any τ ∈ (0, T ), there exists L > 0 such that for any t ∈ [-τ, τ ],

∂ k+1 t Φ F (t, •) C 0,exp(-cT ω 0 L ∞ (F 0 ) ) (F0) + (k) (t) + |r (k) (t)| L k+1 (k!) M +2 . (1.18)

Preliminaries

In this section, we introduce the basic material that we use in order to prove Theorem 2.

Basic material

We introduce the distance to ∂S 0 : ρ(x) := dist(x, ∂S 0 ).

Using the assumption on S 0 , we deduce that there exists c ρ > 0 such that on some bounded neighborhood W 0 of the boundary ∂S 0 in F 0 , one has for all s ∈ N,

∇ s ρ c s ρ (s!) M , (2.1)
as a function (on W) with values in the set of symmetric s-linear forms. Without loss of generality, we may assume that c ρ > 1.

Since the motion of the body is rigid we have that, for all t, the distance to S(t) in

W(t) := Φ S (t, W 0 ), is given by ρ(t, x) = ρ 0 ((Φ S ) -1 (t, x)). (2.
2)

It will useful to have in mind the following form of the Hölder inequality: for any integer k, for any θ := (s, α) in

A k := {θ ∈ N * × N s / 2 s k + 1 and α := (α 1 , . . . , α s ) ∈ N s / |α| = k + 1 -s},
where the notation |α| stands for |α| := α 1 + . . . + α s , and for any p 1,

s i=1 f i L p k+1 (F (t)) s i=1 f i L p α i +1 (F (t))
.

(2.3)

Conventions. We will use Einstein's repeated index convention. Given A ∈ M 2 (R), we denote by tr{A} its trace and by as{A} := A -A * its antisymmetric part. Given ϕ a smooth vector field, curl(ϕ) can be considered either as a scalar ∂ 1 ϕ 2 -∂ 2 ϕ 1 or as a matrix, whose entry at the k-th row and l-th

column is [curl(ϕ)] kl = ∂ k ϕ l -∂ l ϕ k .
The translation between the two is immediate. Also, ∇ϕ is the matrix (∇ϕ) kl = ∂ k ϕ l . Hence curl ϕ = as{∇ϕ}. Throughout this paper, we denote by N the set of nonnegative intergers and by N * the set of positive integers.

Added mass

We will use the following decomposition of the pressure, which is the two-dimensional counterpart of [5, Lemma 3].

Lemma 5. Equation (1.1) can be written as

Du = -∇µ + ∇ Φ • r , (2.4)
with Φ := (Φ a ) a=1,2,3 , where the functions Φ a = Φ a (t, x) and µ = µ(t, x) are the solutions of the following problems:

-∆Φ a = 0 for x ∈ F(t), (2.5)

Φ a (x) → 0 for x → ∞, (2.6
)

∂Φ a ∂n = K a for x ∈ ∂S(t), (2.7)
where

K a := n a if a = 1, 2, (x -h(t)) ⊥ • n if a = 3, (2.8)
and -∆µ = tr{∇u • ∇u} for x ∈ F(t), (2.9) µ(x) → 0 as x → ∞, (2.10)

∂µ ∂n = σ, for x ∈ ∂S(t), (2.11)
where u S = u S (t, x) is given by (1.8) and where

σ := ∇ 2 ρ {u -u S , u -u S } -n • r (2u -u S -) ⊥ .
(2.12)

Let us observe that Φ a (t, x) = Φ a (0, (Φ S ) -1 (t, x)), so that the matrix

M 2 = F (t) ∇Φ a • ∇Φ b dx a,b∈{1,2,3}
is time-independent. It is also easy to see that the matrix M 2 is symmetric and positive, as a Gram matrix (actually, when S 0 is not a ball, it is even positive definite). Consequently the matrix

M := M 1 + M 2 , M 1 := m Id 2 0 0 J ,
is symmetric and positive definite. Actually M is referred as the "virtual inertia tensor", it incorporates the "added inertia tensor" M 2 which, loosely speaking, measures how much the surrounding fluid resists the acceleration as the body moves through it. Its relevance in that context is highlighted by the following property, which is the two-dimensional counterpart of [5, Lemma 4].

Lemma 6. The equations (1.4)-(1.5) can be written as

M r = F (t) ∇µ • ∇Φ a dx a∈{1,2,3}
.

(2.13)

Regularity lemma

We will use the following elliptic regularity estimate. It is a classical regularity estimate for the div-curl elliptic system in L p , except for what concerns the treatment of low frequencies, for which we rely on an approach due to T. Kato [START_REF] Kato | On the smoothness of trajectories in incompressible perfect fluids[END_REF].

Lemma 7. There exists c > 0 such that for any p ∈ (2, +∞), for any smooth vector field

f ∈ L p (F 0 ; R 2 ) satisfying (i) div f and curl f are in L p (F 0 ), (ii) div f = ∂ i ∂ h a ih and curl f := ∂ i ∂ h b ih with the a ih and the b ih in L p (F 0 ), (iii) there exists φ in W 1,p (W) such that (n • f )| ∂F0 = φ| ∂F0 , then f ∈ W 1,p (F 0 ; R 2 ) and c f W 1,p (F ) p div f L p (F ) + curl f L p (F0) + φ W 1,p (W) + a ih L p (F0) + b ih L p (F0) + ∂S0 f • τ ds . (2.14)
To obtain Lemma 7, we are first going to prove the following lemma.

Lemma 8. Let η ∈ C ∞ (R 2 ) such that 1 -η ∈ C ∞ c (R 2 ).
There exists c > 0 such that for any p ∈ (2, +∞), for any smooth vector field

f ∈ L p (R 2 ; R 2 ) satisfying (i) div f and curl f are in L p (R 2 ), (ii) div f = η∂ i ∂ h a ih and curl f := η∂ i ∂ h b ih with the a ih and the b ih in L p (R 2 ), then f ∈ W 1,p (R 2 ) and c f W 1,p (R 2 ) p( div f L p (R 2 ) + curl f L p (R 2 ) ) + a ih L p (R 2 ) + b ih L p (R 2 ) .
Proof of Lemma 8. According to the Biot-Savart formula, we have, for any

x ∈ R 2 , f (x) = R 2 H(x -y) div f (y) dy + R 2 H ⊥ (x -y) curl f (y) dy, (2.15)
where

H(x) := x ⊥ 2π|x| 2 .
Here the difficulty lies in the estimate of the L p norm of f , since for the gradient we have the classical Calderón-Zygmund estimates:

c ∇f L p (R 2 ) p( div f L p (R 2 ) + curl f L p (R 2 ) ).
To overcome this difficulty we will follow the strategy of the proof of [START_REF] Kato | On the smoothness of trajectories in incompressible perfect fluids[END_REF]Lemma 9.1]. Let us only deal with the first term in Eq. (2.15); the second one can be tackled in a similar way. We introduce a smooth function ζ ∈ C ∞ c (R 2 ) such that ζ = 1 in the unit ball B(0, 1) and ζ = 0 in the complementary set of B(0, 2). Now we decompose the integral in two parts:

R 2 H(x -y) div f (y)dy = R 2 ζ(x -y)H(x -y) div f (y)dy + R 2 (1 -ζ(x -y))H(x -y)η(y)∂ i ∂ h a ih (y)dy. (2.16)
Using the classical Calderón-Zygmund theory we obtain that the norm in W 1,p (R 2 ) of the first term in Eq. (2.16) is bounded by cp div f L p (R 2 ) , where the constant c does not depend on p > 2 (the dependence of the constants on p is crucial here as well as in Yudovich's argument [START_REF] Yudovich | Non-stationary flows of an ideal incompressible fluid[END_REF]). Observe in particular that the kernel ζH is integrable at infinity which yields the L p part of the previous estimate. Now, for the second part we integrate by parts twice, so that we get

R 2 (1 -ζ(x -y))H(x -y)η∂ i ∂ h a ih (y)dy = R 2 ∂ i ∂ h η(y)(1 -ζ(x -y))H(x -y) a ih (y)dy.
Since the kernel above is smooth we easily get by Young's inequality that the norm of this term in

W 1,p (R 2 ) is bounded by C a ih L p (R 2 )
. This completes the proof of Lemma 8.

Proof of Lemma 7. Let R > 0 be large enough for S 0 ⊂ B(0, R) and let us consider a smooth function η such that η = 0 in a neighborhood of ∂S 0 and η = 1 in the complementary set of B(0, R). According to the previous lemma, the solution

F in W 1,p (R 2 ) of div F = η div f and curl F = η curl f in R 2 ,
satisfies the estimate (2.14) (Here we extend the functions η, f , a ih and b ih by 0 inside S 0 to be in position to apply Lemma 8). Now, the function f := f -F, has its divergence and rotational supported in the ball B(0, R), and its circulation around S 0 is given by

∂S0 f • τ ds = ∂S0 f • τ ds - S0 div F dx.
Consequently there holds (extending n in the neighborhood W 0 of ∂S 0 ):

f L p (F0) Cp div f L p (F0) + curl f L p (F0) + f • n W 1,p (W0) + C ∂S0 f • τ ds Cp div f L p (F0) + curl f L p (F0) + φ W 1,p (W) + a ih L p (F0) + b ih L p (F0) + C ∂S0 f • τ ds .
This concludes the proof of Lemma 7.

Remark 9. Thanks to the invariance properties of the divergence and of the curl with respect to rotation and translation the previous lemma holds for the domain F(t) with the same constant for any time t.

Formal identities

We will use some formal identities, which have already been obtained in [START_REF] Glass | Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid[END_REF], as a combinatorial refinement of the ones obtained by Kato in [START_REF] Kato | On the smoothness of trajectories in incompressible perfect fluids[END_REF]. They concern the iterated material derivatives (D k u) k∈N * , where

D := ∂ t + (u • ∇),
under the assumption that ( , r, u) is a smooth solution of the above system. We use the following notations: for α := (α 1 , . . . , α s ) ∈ N s we will denote α! := α 1 ! . . . α s !. We denote for any integer k,

A k := {θ := (s, α) ∈ N * × N s / 2 s k + 1 and α := (α 1 , . . . , α s ) ∈ N s / |α| = k + 1 -s},
where the notation |α| stands for |α| := α 1 + . . . + α s .

Let us be given a smooth vector field ψ. We first recall some formal identities for div D k ψ, for curl D k ψ of the iterated material derivatives (D k ψ) k∈N * as combinations of the functionals

f (θ)[u, ψ] := ∇D α1 u • . . . • ∇D αs-1 u • ∇D αs ψ,
(2.17)

with θ := (s, α) ∈ A k . The precise statement is the following (see [START_REF] Glass | Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid[END_REF]Prop. 6]).

Proposition 10. For k ∈ N * , we have in F(t)

div D k ψ = D k (div ψ) + tr F k [u, ψ] where F k [u, ψ] := θ∈A k c 1 k (θ) f (θ)[u, ψ], (2.18) curl D k ψ = D k (curl ψ) + as G k [u, ψ] where G k [u, ψ] := θ∈A k c 2 k (θ) f (θ)[u, ψ], (2.19)
and where for i = 1, 2 and θ = (s, α), the c i k (θ) are integers satisfying

|c i k (θ)| k! α! . ( 2

.20)

Remark 11. In particular for ψ = u and k = 1 we obtain

div(Du) = -div(∇µ) = tr(F 1 [u, u]).
Now we recall some formal identities for normal traces on ∂S(t) of the rigid body of iterated material derivatives D k ψ, and for iterated material derivatives of the functions K i defined in (2.8). To a scalar r ∈ R we associate the matrix R(r) := r 0 -1 1 0 .

To any β ∈ N s and r ∈ C |β| ((-T, T ); R) we define the functional R β [r] which associates to the time-dependent function r the time-dependent rotation matrix

R β [r] := R r (β1) • . . . • R r (βs) . (2.21)
For any s ∈ N * , we will use some multi-indices s := (s 1 , ..., s s ) in N s . Then we will denote s := |s | = s 1 + ... + s s , (α 1 , ..., α s ) will be in N s 1 × ... × N s s and α := (α 1 , ..., α s , α s +1 , ..., α s +s ) will be an element of N s +s . The bricks of the following formal identities will be the functionals, defined for smooth vector fields ϕ and ψ and a multi-index

ζ := (s, s , α) ∈ N * × N s × N s+s : h(ζ)[r, ϕ, ψ] := ∇ s ρ(t, x){R α 1 [r]D α s +1 ϕ, ..., R α s-1 [r]D α s +s-1 ϕ, R α s [r]D α s +s ψ}. (2.22)
In (2.22) the term R α i [r] should be omitted when s i := 0. We introduce the following set

B k := {ζ = (s, s , α)/ 2 s + s k + 1 and |α| + s + s = k + 1}. (2.23)
We have the following formal identity.

Proposition 12. Given a smooth vector field ψ, for k ∈ N * , there holds on the boundary ∂S(t)

n • D k ψ = D k (n • ψ) + H k [r, u -u S , ψ] where H k [r, u -u S , ψ] := ζ∈B k d 1 k (ζ) h(ζ)[r, u -u S , ψ],
(2.24)

D k K i = H k [r, u -u S , σ i ] where H k [r, u -u S , σ i ] := ζ∈B k d 2 k (ζ) h(ζ)[r, u -u S , σ i ], (2.25)
where the K i are defined in (2.8),

σ i := e i if i = 1, 2,
and
σ i := (x -h(t)) ⊥ if i = 3, (2.26)
and where the d j k (ζ), j = 1, 2, are integers satisfying, for any

ζ := (s, s , α) ∈ B k , |d j k (ζ)| 3 s+s k! α!(s -1)! , (2.27)
The proof of this proposition is completely identical to the proof of [START_REF] Glass | Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid[END_REF]Prop. 8] and is therefore omitted.

We can also establish identities for the gradient ∇D k ψ for a smooth scalar-valued function ψ:

Proposition 13. For k 1, we have in the domain F(t)

D k ∇ψ = ∇D k ψ + K k [u, ψ], (2.28)
where for k 1,

K k [u, ψ] := - k r=1 k r ∇D r-1 u • D k-r ∇ψ. (2.29)
The proof of this proposition is completely identical to the proof of [6, Prop. 3.5], and is therefore omitted.

Further formal identities

In this subsection, we give some other formal identities aimed at dealing with the far-field/low frequencies. This is inspired by [START_REF] Kato | On the smoothness of trajectories in incompressible perfect fluids[END_REF]Section 6]. Again, we assume here that ( , r, u) is a smooth solution of the system.

We first recall the following commutation rules which allow to exchange D and other differentiations. They are valid for ψ a smooth scalar/vector field defined in the fluid domain:

D(ψ 1 ψ 2 ) = (Dψ 1 )ψ 2 + ψ 1 (Dψ 2 ),
(2.30)

∂ k (Dψ) -D(∂ k ψ) = (∂ k u j )(∂ j ψ), (2.31) div Dψ -D div ψ = tr {(∇u) • (∇ψ)} , (2.32)
curl Dψ -D curl ψ = as {(∇u) • (∇ψ)} .

(2.33)

An identity concerning the divergence

The first formal identity of this section is given in the following statement.

Proposition 14. Let us consider ψ a smooth vector field, and suppose that for some family ( φij ) i,j=1...d of smooth functions, one has in F(t) div(ψ) = ∂ i φ i where φ i = ∂ j φij .

(2.34)

Then for n ∈ N * , we have in

F(t) div D n ψ = ∂ i ∂ j φn ij [u, ψ], where φn ij [u, ψ] := ξ∈ Â1 n ĉn,1 i,j (ξ) ĝ(ξ)[u, ψ] + ξ∈ Â2 n ĉn,2 i,j (ξ) ǧ(ξ)[u, φ], (2.35)
with, for δ = 1 or 2,

A δ n := {θ := (s, α) ∈ N * × N s / 3 -δ s n + 1 and α := (α 1 , . . . , α s ) ∈ N s / |α| = n + 1 -s}, (2.36) Âδ n := {ξ := (s, α, k, λ)/ (s, α) ∈ A δ n and (k, λ) ∈ {1, ..., d} [s+δ-3]+s }, (2.37) ĝ(ξ)[u, ψ] := D α1 u λ1 • ∂ k2 D α2 u λ2 • ... • ∂ ks-1 D αs-1 u λs-1 • D αs ψ λs , (2.38) ǧ(ξ)[u, φ] := D α1 u λ1 • ∂ k2 D α2 u λ2 • ... • ∂ ks-1 D αs-1 u λs-1 • ∂ ks D αs φλsks , (2.39)
and where, for δ = 1, 2, the ĉn,δ i,j (ξ) are integers satisfying

|ĉ n,δ i,j (ξ)| 4 s n! α! . (2.40)
We first state and prove two lemmas before establishing Proposition 14.

Lemma 15. Under the hypothesis of Proposition 14, we have that for n ∈ N, for x in F(t)

div D n ψ = ∂ i φ n i
(2.41)

φ n i = ∂ j φn i,j , (2.42)
where the sequences (φ n i ) n∈N and ( φn i,j ) n∈N , are respectively defined by

φ 0 i = ∂ j φij and φ n+1 i := Dφ n i -(∂ k u i )(φ n k -D n ψ k ), ( 2

.43) φ0

i,j = φij and φn+1 i,j

:= D φn i,j -φn i,k • ∂ k u j + u i (∂ k φn j,k -D n ψ j ). ( 2

.44)

Proof. This is an induction argument.

• Let us first prove by iteration that (2.41) holds true when the sequence (φ n i ) n∈N is defined by (2.43). The case n = 0 is precisely the hypothesis (2.34). Let us now assume that (2.41) and (2.43) hold true for some n ∈ N and let us show the same for the rank n + 1.

We first use the commutation rule (2.32) to exchange D and div to get

div D n+1 ψ = D div D n ψ + tr(∇u • ∇D n ψ).
Then we use (2.41) and the commutation rule (2.31) so that div

D n+1 ψ = D∂ i φ n i + (∂ i u k )(∂ k D n ψ i ), = ∂ i Dφ n i -(∂ i u k )(∂ k φ n i -∂ k D n ψ i ), Now, since div u = 0 we get div D n+1 ψ = ∂ i Dφ n i -∂ k ((∂ i u k )(φ n i -D n ψ i )), = ∂ i Dφ n i -∂ i ((∂ k u i )(φ n k -D n ψ k )),
after exchanging the dummy indices i and k. This yields (2.43).

• We go on with (2.42) and (2.44). We also proceed by iteration. The case n = 0 is satisfied by assumption, see again the hypothesis (2.34). Now let us assume that (2.42) holds true for some n ∈ N when the sequence ( φn i,j ) n∈N is defined by (2.44), and show the same at rank n + 1.

Plugging (2.42) (at rank n) in (2.43) we have

φ n+1 i = D∂ j φn i,j -(∂ k u i )(∂ j φn k,j -D n ψ k ),
Then we use the commutation rule (2.31) and Leibniz' rule to get

φ n+1 i = ∂ j D φn i,j -(∂ j u k )(∂ k φn i,j ) -∂ k (u i (∂ j φn k,j -D n ψ k )) + u i (∂ j ∂ k φn k,j -∂ k D n ψ k ).
This last term vanishes according to (2.41) so that, using div u = 0 and exchanging some dummy indices, we get

φ n+1 i = ∂ j D φn i,j -∂ k ((∂ j u k )( φn i,j + u i (∂ j φn k,j -D n ψ k )), = ∂ j D φn i,j -∂ j ((∂ k u j )( φn i,k + u i (∂ k φn j,k -D n ψ j )), = ∂ j φn+1 i,j ,
where φn+1 i,j is given by (2.44).

The second step of the proof of Proposition 14 is the following lemma.

Lemma 16. Under the hypothesis of Proposition 14, we have that for n ∈ N, we have in F(t)

φ n i = ξ∈A 1 n c n,1 i,j (ξ) g(ξ)[u, ψ] + ξ∈A 2 n c n,2 i,j (ξ) g(ξ)[u, φ], (2.45) with A
g(ξ)[u, ψ] := ∂ k1 D α1 u λ1 • ∂ k2 D α2 u λ2 • ... • ∂ ks-1 D αs-1 u λs-1 • D αs ψ λs , (2.47) g(ξ)[u, φ] := ∂ k1 D α1 u λ1 • ∂ k2 D α2 u λ2 • ... • ∂ ks-1 D αs-1 u λs-1 • ∂ ks D αs φλsks , (2.48)
and where, for δ = 1, 2, the c n,δ i,j (ξ) are integers satisfying

|c n,δ i,j (ξ)| 2 s n! α! . ( 2

.49)

Proof. Let us first explain why formula (2.45) holds without yet estimating the coefficients c n,δ i,j (ξ). When n = 0, it is clear that one can put φ 0 i = ∂ j φij in the form (2.45). Next, assuming that φ n i has this form, we apply (2.43). It is obvious that the terms (∂ k u i )(φ n k -D n ψ k ) can enter the right hand side of (2.45); only the term Dφ n i is not trivial. But actually, that Dφ n i can be put of the form (2.45) is a consequence of the induction assumption, Leibniz's formula and the commutation rule (2.31).

It remains to make the claim quantitative, that is to say, to prove that the coefficients c n,δ i,j (ξ) that we recover satisfy (2.49). For n = 0 the claim is trivial. Let us now discuss the passage from n to n + 1. To obtain φ n+1 i , we apply relation (2.43) and consider the resulting terms coming from those of (2.45). We first consider a term coming from the first sum of (2.45), that is, corresponding to δ = 1. The modifications needed in the case δ = 2 will be explained below. of the second kind (one has added a (∂ kj u i ) factor to some g(ξ )[u, ψ]).

Let ξ ∈ A

either from (∂

k u i )φ n k .
Again there are at most (s -1)2 s-1 n! α! contributions. Since (

s j=1 α j ) + s -1 = n + 1, in total, we have |c n+1,δ i,j (ξ)| 2 s (n + 1)! α! .
Concerning the particular case s = 2, we have to take into account the additional term (∂ k u i )D n ψ k in relation (2.43). But since δ = 1, there are no terms corresponding to s = 1, that is, in this case, there are only contributions "of the first kind" as referred to above. It follows that here the total contributions can be estimated from above by

n2 2 n! + 1 2 2 (n + 1)!.
The case δ = 2 is similar, but two modifications are in order: 1. here s starts from s = 1, 2. there are no particular additional contributions for s = 2.

We can now get back to the proof of Proposition 14.

Proof of Proposition 14. Again the case n = 0 is clear, and we discuss the passage from n to n + 1. The induction relation on which we will rely is the following, directly deduced from Lemma 15:

φn+1 i,j = D φn i,j -φn i,k • ∂ k u j + u i (φ n j -D n ψ j ).
We begin with δ = 1. A term ĝ(ξ)[u, ψ] with ξ ∈ Â1 n+1 (with s 3) has been obtain through several ways:

1. either from D φn i,j : either from a term whose one factor has one less material derivative or from a term having a factor of the type ∂ kj u i less. There are at most For what concerns the case s = 2, we have again an additional term u i D n ψ j , but here there is no contribution coming from φn i,k • ∂ k u j because there are no terms corresponding to s = 1. The conclusion follows as previously.

Again, the case δ = 2 is mutatis mutandis the same, with s starting from s = 1, but no particular additional term for s = 2.

An identity concerning the curl

The second formal identity of this section is given in the following statement.

Proposition 17. Let us consider ψ a smooth gradient vector field. Then for n ∈ N * , we have in F(t) ) is due to the fact that we suppose ψ to be a gradient field, so that (2.50) is trivial for n = 0, which simplifies the analysis. This is sufficient to our purpose.

(curl D n ψ) kl = ∂ i ∂ j Γn ij,kl [u, ψ], (
Again we will need two preliminary lemmas before proving Proposition 17. The following lemma is the counterpart of Lemma 17.

Lemma 19. Under the hypothesis of Proposition 17, we have that for n ∈ N, for x in F(t)

(curl D n ψ) kl = ∂ i Γ n i,kl (2.53) Γ n i,kl = ∂ j Γn ij,kl , (2.54)
where the sequences (Γ n i,kl ) n∈N , ( Γn ij,kl ) n∈N are respectively defined by We first use the commutation rule (2.33) to exchange D and curl to get

Γ 0 i,kl = 0 and Γ n+1 i,kl := DΓ n i,kl -(∂ k u i )Γ n i,kl + (∂ k u i )D n ψ l -(∂ l u i )D n ψ k , (2.55) Γ0 ij,kl = 0 and Γn+1 ij,kl := D Γn ij,kl -(∂ k u j ) Γn ih,kl -u i Γ n j,kl -δ j,k u i D n ψ l -δ j,l u i D n ψ k . ( 2
(curl D n+1 ψ) kl = (D curl(D n ψ) + as {∇u • ∇D n ψ}) kl . (2.57)
Then we use (2.53), the commutation rule (2.31) and that div u = 0 to get

(curl D n+1 ψ) kl = D∂ i Γ n j,kl + (∂ k u h )(∂ h D n ψ l ) -(∂ l u h )(∂ h D n ψ k ) = ∂ i DΓ n j,kl -(∂ i u h )(∂ h Γ n i,kl ) -∂ h ((∂ k u h )D n ψ l -(∂ l u h )D n ψ k ) = ∂ i DΓ n j,kl -∂ h ((∂ i u h )Γ n i,kl -(∂ k u h )D n ψ l -(∂ l u h )D n ψ k ).
After exchanging the dummy indices i and h this yields

(curl D n+1 ψ) kl = ∂ i (DΓ n i,kl -(∂ h u i )Γ n h,kl + (∂ k u i )D n ψ l -(∂ l u i )D n ψ k ) = ∂ i Γ n+1 i,kl ,
where Γ n+1 i,kl is given by (2.55). Now,

Γ n+1 i,kl = DΓ n i,kl -(∂ h u i )Γ n h,kl + (∂ k u i )D n ψ l -(∂ l u i )D n ψ k = D∂ j Γn+1 ij,kl -∂ h (u i Γ n h,kl ) + u i ∂ h Γ n h,kl + ∂ k (u i D n ψ l ) -u i ∂ k (D n ψ l ) -∂ l (u i D n ψ k ) + u i ∂ l D n ψ k .
Summing the third term with the fifth term and the last one yields 0 so that

Γ n+1 i,kl = ∂ j D Γn+1 ij,kl -∂ h (∂ j u h Γn+1 ij,kl ) -∂ h (u i Γ n h,kl ) -∂ k (u i D n ψ l ) -∂ l (u i D n ψ k ) = ∂ j Γn+1 ij,kl
where Γn+1 ij,kl is given by (2.56). Lemma 20. Under the hypothesis of Proposition 17, we have that for n ∈ N, we have in F(t)

Γ n i,kl = ξ∈A 1 n d n i,kl (ξ) g(ξ)[u, ψ], (2.58)
where

A 1 n is defined in (2.46), g(ξ)
is defined in (2.47) and where the d n i,kl are integers satisfying

|d n i,kl | 2 s n! α! . (2.59)
Proof of Lemma 20. This is exactly the same proof as for Lemma 16 (even, it is simpler since there is only one type of terms here), except for what concerns s = 2. For s = 2, we see from (2.55) that a term g(ξ)[u, ψ] with ξ ∈ A 1 n+1 comes either by adding a material derivative to a factor in a term of Γ n i,kl , or from the additional terms (∂ k u i )D n ψ l and -(∂ l u i )D n ψ k . Hence the total contributions can be estimated from above by

n2 2 n! + 2 = 2 2 (n + 1)! -4n! + 2 2 2 (n + 1)!.
Proof of Proposition 17. One can observe that, except for what concerns s = 2, the induction relations (2.55)-(2.56) are exactly the same as (2.43)-(2.44). The only difference consists in the number of additional terms which can be 2 in (2.56) when j = k = l. But the particular argument for s = 2 works again: we have (at most) two additional terms, but the other contributions come only from adding a material derivative to an existing term. The conclusion follows as previously.

Proof of Theorem 2

We are now in position to prove Theorem 2.

We proceed by regularization. Consequently, we will work from now on a smooth solution of the equation, without changing the notation. Since the estimates that we are going to establish are uniform with respect to the regularization parameter, the general result follows. We refer to [START_REF] Glass | Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid[END_REF] for more details on this step. Note in particular that we can use the formal identities of Section 2 that were derived under the assumption that ( , r, u) is smooth.

The main argument is to prove by induction an estimate on the k-th material derivative of the fluid and the body velocities.

Iteration

Let k 0 ∈ N * , ν ∈ (0, 1), p 1 2 1 -ν and p 2 p 1 • (k 0 + 1). (3.1)
We are going to prove recursively that for L > 0 large enough, for any integer k k 0 ,

D k u W 1, p 2
k+1 (F (t))

+ (k) + |r (k) | V k , (3.2)
with

V k := p k 2 (k!) M L k (k + 1) 2 V k+1 , where V := u W 1,p 2 (F (t)) + + |r|.
The norm on vectors of R 2 (here and its derivatives) is the usual Euclidean one. We will also the notation • for the associated matrix norm.

The inequality (3.2) is true for k = 0. Now let us assume that Eq. (3.2) is proved up to k -1 k 0 -1.

We will first prove the following proposition, which, under the induction hypothesis, allows to estimate the next iterated material derivative of the pressure field, as it decomposed in Paragraph 2.2.

Proposition 21. The functions Φ a (a = 1, 2, 3) and µ satisfy the following assertions.

• There exists a positive constant C 0 = C 0 (S 0 ) such that

1 a 3 ∇Φ a W 1,p 2 (F (t)) C 0 . (3.3)
• There exists γ 1 a positive decreasing function with lim L→+∞ γ 1 (L) = 0 such that if for all j k -1,

D j u W 1, p 2
j+1 (F (t))

+ (j) + r (j) V j , (3.4)
then for all 1 j k,

1 a 3 D j ∇Φ a W 1, p 2 j+1 (F (t)) γ 1 (L) V j V . (3.5)
• There exists a positive constant

C 0 = C 0 (S 0 ) such that ∇µ W 1, p 2 2 (F (t)) C 0 V 2 . (3.6)
• There exists γ 2 a positive decreasing function with lim L→+∞ γ 2 (L) = 0 such that if for all j k -1, (3.4) holds true then for all 1 j k -1,

D j ∇µ W 1, p 2
j+2 (F (t)) γ 2 (L)V V j . (3.7)
Proposition 21 is proven in Subsections 3.2 and 3.3.

The second proposition allows to propagate the induction hypothesis on the solution ( , r, u) itself.

Proposition 22. There exist a positive decreasing functions γ 3 with lim L→+∞ γ 3 (L) = 0 such that if for all j k -1, (3.4) holds, then

(k) + |r (k) | + D k u W 1, p 2
k+1 (F (t))

V k γ 3 (L).

(3.8)

The proof of Proposition 22 is given in Subsection 3.4.

Once Proposition 22 established, the claim that (3.2) holds true up to k k 0 is a direct induction argument.

Proof of Proposition 21, functions ∇Φ a

Recall that the functions ∇Φ a (a = 1, 2, 3) defined by (2.5)-(2.7) satisfy div ∇Φ a = 0 in F(t), curl ∇Φ a = 0 in F(t), n • ∇Φ a = K a on ∂S(t), Φ a (x) → 0 for x → ∞. (3.9)

Then by applying the regularity lemma (Lemma 7), we obtain immediately (3.3).

Let us now prove the second point of Proposition 21 by induction on j. Let us therefore assume that the estimate (3.5) holds true up to rank j -1 and prove that then it is also true at rank j. By applying D j to (3.9) and by using Propositions 10, 12, 13, 14 (with ψ = ∇Φ a , which satisfy the assumption (2.34) with identically vanishing functions φij ) and 17 (also with ψ = ∇Φ a ) we obtain that D j ∇Φ a satisfies the following relations div

D j ∇Φ a = tr F j [u, ∇Φ a ] in F(t), (3.10) curl D j ∇Φ a = as G j [u, ∇Φ a ] in F(t), (3.11
)

n • D j ∇Φ a = D j K a + H j [r, u -u S , ∇Φ a ] on ∂S(t) (3.12) div D j ∇Φ a = ∂ i ∂ h a j ih [u, ∇Φ a ] in F(t), (3.13) curl D j ∇Φ a = ∂ i ∂ h b j ih [u, ∇Φ a ] in F(t), (3.14)
∂S(t) D j ∇Φ a • τ ds = ∂S(t) K j [u, Φ a ] • τ ds, (3.15)
where

a j ih [u, ∇Φ a ] := ξ∈ Â1 j a j ih (ξ) ĝ(ξ)[u, ∇Φ a ], b j ih [u, ∇Φ a ] := ξ∈ Â1 j b j ih (ξ) ĝ(ξ)[u, ∇Φ a ].
Above the set Â1 j and the functionals ĝ(ξ) are the one defined in (2.37); and the integer coefficients a j ih (ξ) and b j ih (ξ) satisfy respectively the estimates

|a j ih (ξ)| 4 s j! α! , |b j ih (ξ)| 4 s j! α! . 3.2.1 Estimate of F j [u, ∇Φ a ] and G j [u, ∇Φ a ]
Applying the Hölder inequality (2.3) to f (θ)[u, ∇Φ a ] (whose definition is given in (2.17)) for θ ∈ A j , we obtain that

f (θ)[u, ∇Φ a ] L p 2 j+1 (F (t)) s-1 i=1 D αi u W 1, p 2
α i +1 (F (t)) D αs ∇Φ a W 1, p 2 αs+1 (F (t))
.

Using the induction hypothesis and since for θ ∈ A j , |α| = j + 1 -s, we have

f (θ)[u, ∇Φ a ] L p 2 j+1 (F (t)) L j V j (α!) M L 1-s p |α| 2 s i=1 1 (1 + α i ) 2 .
Now thanks to Proposition 10, we obtain

F j [u, ∇Φ a ] L p 2 j+1 (F (t)) j!L j V j j+1 s=2 L 1-s α s.t. |α|=j+1-s (α!) M -1 p |α| 2 s i=1 1 (1 + α i ) 2 .
When θ ∈ A j , 2 s j + 1 and |α| = j + 1 -s, then |α| j -1 so that

F j [u, ∇Φ a ] L p 2 j+1 (F (t)) p j-1 2 (j!) M L j V j j+1 s=2 L 1-s j M -1 α/ |α|=j+1-s s i=1 1 (1 + α i ) 2 .
(3.16)

We now use the following lemma (cf. [START_REF] Chemin | Fluides parfaits incompressibles[END_REF]Lemma 7.3.3]).

Lemma 23. For any couple of positive integers (s, m) we have

α∈(N * ) s |α|=m Υ(s, α) 20 s (m + 1) 2 , where Υ(s, α) := s i=1 1 (1 + α i ) 2 .
(3.17)

We deduce from (3.16) and from the above lemma that

F j [u, ∇Φ a ] L p 2 j+1 (F (t)) p j-1 2 (j!) M L j (j + 1) 2 V j j+1 s=2 L 1-s j M -1 20 s (j + 1) 2 (j -s + 2) 2 .
We

h(ζ)[r, u -u S , ∇Φ a ] W 1, p 2
j+1 (W(t)) c s ρ (s!) M s i=1 s i l=1 |r (j) | s-1 i=1 D α s +i (u -u S ) W 1, p 2 α s +i +1 (W(t)) D α s +s ∇Φ a W 1, p 2 α s +s +1 (F (t))
. By using the induction hypothesis and Lemma 24, we have

h(ζ)[r, u -u S , ∇Φ a ] W 1, p 2
j+1 (W(t)) L j V j (α!) M (s!) M L 1-s c s ρ p |α| 2 s i=1 1 (1 + α i ) 2 .
Thanks to Proposition 10 and Lemma 23 we obtain

H j [r, u -u S , ∇Φ a ] W 1, p 2
j+1 (W(t)) p j-1 2 (j!) M L j (j + 1) 2 V j j+1 s=2 s M L 1-s c s ρ 20 s (j + 1) 2 (j -s + 2) 2 . (3.19) 3.2.3 Estimate of K j [u, Φ a ]
Applying Hölder's inequality to the terms appearing in (2.29) yields for any j 2,

K j [u, Φ a ] L p 2 j+1 (F (t)) j-1 s=1 j -1 s D s-1 u W 1, p 2 s (F (t)) D j-s ∇Φ a W 1, p 2
j+1-s (F (t))

.

By using the induction hypothesis we get

K j [u, Φ a ] L p 2 j+1 (F (t)) p j-1 2 (j!) M L j (j + 1) 2 V j+1 L -1 j-1 s=1 j -s js M j + 1 s(j -s + 1) 2 ,
and the same estimate holds true for ∂S(t) (D j ∇Φ a ) • τ ds up to a multiplicative geometric constant, thanks to (3.15).

Estimate of a j

ih [u, ∇Φ a ] and of b j ih [u, ∇Φ a ] Applying the Hölder inequality (2.3) to the definition of ĝ(ξ)[u, ∇Φ a ] in (2.37), for ξ ∈ Â1 j , yields that

ĝ(ξ)[u, ∇Φ a ] L p 2 j+1 (F (t)) s-1 i=1 D αi u W 1, p 2
α i +1 (F (t)) D αs ∇Φ a W 1, p 2 αs+1 (F (t))
.

Then we proceed as for F j [u, ∇Φ a ], and obtain

a j ih [u, ∇Φ a ] L p 2 j+1 (F (t)) p j-1 2 (j!) M L j (j + 1) 2 V j j+1 s=2 L 1-s j M -1 80 s (j + 1) 2 (j -s + 2) 2 .
The analysis is the same for b j ih [u, ∇Φ a ].

Conclusion

It remains to gather the above estimates. We apply Lemma 7 (observing that, thanks to (3.1), we have p2 k+1 > 2) and we use the previous estimates to get (3.5) at rank j, with

γ 1 (L) := C(S 0 ) sup j 1 j+1 s=2 L 1-s j M -1 80 s c s ρ (j + 1) 2 (j -s + 2) 2 .

Proof of Proposition 21, function ∇µ

We now turn to the claims concerning µ. The function ∇µ defined by (2.9)-(2.11) satisfies

div ∇µ = -tr F 1 [u, u] = -tr {∇u • ∇u} , curl ∇µ = 0 in ∂F(t), n • ∇µ = σ on ∂S(t),
where σ is defined by (2.12). Hence (3.6) follows again from Lemma 9.

By applying D j to (3.9) and by using Propositions 10, 12, 13, 14 with ψ = ∇µ, which satisfies the assumption (2.34) with φih := -u i u h , and Proposition 17 (also with ψ = ∇µ) we obtain that D j ∇µ satisfies the following relations

div D j ∇µ = -D j tr F 1 [u, u] + tr F j [u, ∇µ] in F(t), curl D j ∇µ = as G j [u, ∇µ] in F(t), n • D j ∇µ = D j σ + H j [r, u -u S , ∇µ] on ∂S(t) div D j ∇µ = ∂ i ∂ h a j ih [u, ∇µ] in F(t), curl D j ∇µ = ∂ i ∂ h b j ih [u, ∇µ] in F(t), ∂S(t) D j ∇µ • τ ds = ∂S(t) K j [u, µ] • τ ds, where a j ih [u, ∇µ] := ξ∈ Â1 j a j,1 ih (ξ) ĝ(ξ)[u, ∇µ] + ξ∈ Â2 j a j,2 ih (ξ) ĝ(ξ)[u, φ], b j ih [u, ∇µ] := ξ∈ Â1 j b j ih (ξ) ĝ(ξ)[u, ∇µ],
and the integer coefficients a j,1 ih (ξ), a j,2 ih (ξ) and b j ih (ξ) satisfy the estimate (3.16). The proof that the validity of (3.4) for j k -1 implies the one of (3.7) for 1 j k -1 is completely similar to the equivalent proof for Φ a .

Proof of Proposition 22

We now turn to the proof of Proposition 22. Under the same assumption that (3.4) is valid for all j k -1, we first prove In order to obtain (3.21), we write

D k u = -D k-1 ∇p = -D k-1 ∇µ + D k-1 ∇Φ • r . ( 3

.23)

We notice that

D k-1 ∇Φ • r = k-1 i=0 k -1 i D i ∇Φ • (k-i) r (k-i) .
Thus, by using (3.4) (valid up to rank k) and (3.5) (valid up to rank k due to Proposition 21) to estimate the terms of the above sum corresponding to i 1 and by using (3.20) for the term corresponding to i = 0, we deduce, together with

k-1 i=1 1 (i + 1) 2 (k -i + 1) 2 k-1 i=1 1 (i + 1) 2 π 2 6 , k-1 i=0 k -1 i (i!) M ((k -i)!) M (k!) M , that D k-1 ∇Φ • r π 2 6 γ 1 (L) + γ 4 (L)C 0 V k .
Combining the above inequality, (3.23) and (3.7), we obtain (3.21), and the proof is complete.

End of the proof

Let ν ∈ (0, 1). We now apply Eq. (3.2) with p 2 (k + 1) instead of p 2 , and we use Stirling's formula to obtain that for any k ∈ N * , for any

p 2 2 1 -ν , (3.24)
one has for L sufficiently large (depending on the geometry only)

D k u W
for some constant L L independent of k and ( , r, u).

So far time has intervened only as a parameter, and the inequality (3.25) holds for any time. We will now estimate its right hand side with respect to the initial data. First thanks to Lemma 7 there exists c > 0 such that for any k, u W We consider T > 0 and we obtain by composition that ∂ k+1 t Φ F is in C 0,ν exp(-cT ω0 L ∞ (F 0 ) ) (F 0 ), with an estimate

∂ k+1 t Φ F C 0,ν exp(-cT ω 0 L ∞ (F 0 ) ) (F0) p k 2 (k!) M +1 L k+1 k!p k+1 2 + |γ| k+1 + k+1 + |r| k+1 , (3.28)
for some L depending on the geometry only.

δn

  := {ξ := (s, α, k, λ)/ (s, α) ∈ A δ n and (k, λ) ∈ {1, ..., d}[s+δ-2]+s },(2.46)

1 n+1 with s 3 . 1 n

 131 When computing φ n+1 i with the relation (2.43) the term involving g(ξ)[u, ψ] comes: 1. either from Dφ n i : the term appears after applying D to a function g(ξ )[u, ψ], ξ ∈ A , and using the commutation rule (2.31). The function function g(ξ )[u, ψ] has one less material derivative or a term having a factor of the type ∂ kj u i less. There are at most s j=1 α j 2 s n! α! , contributions of the first kind (one has added a materiel derivative to some g(ξ )[u, ψ]), and at most (s -1)2 s-1 n! α! ,

  first kind, and at most (s -1)4 s-1 n! α! , of the second kind, 2. either from φn i,k • ∂ k u j , which gives again at most (s -1)4 s-1 n! α! contributions, 3. either from u i φ n j , giving a contribution at most 2 s-1 n! α! , according to Lemma 16. Summing these contributions, this gives the conclusion.

. 56 )

 56 Proof of Lemma 19. Let us first prove by iteration that (2.53) holds true when the sequences (Γ n i,kl ) n∈N are defined by (2.55). The case n = 0 holds true since ψ is a gradient vector field by hypothesis. Let us now assume that (2.53)-(2.55) holds true for some n ∈ N and let us show the same for the rank n + 1.

F.

  (k) + |r(k) | V k γ 4 (L),(3.20) and then proveD k u V k γ 5 (L),(3.21)for positive decreasing functions γ 4 , γ 5 with limL→+∞ γ 4 (L) + γ 5 (L) = 0.In order to prove (3.20) it suffices to differentiate the equations (2.13) k times with respect to the time (recall that the matrix M is constant): (t)(D i ∇µ • D k-1-i ∇Φ a ) dx a∈{1,2,3}Then we use that M is invertible and we apply the estimates (3.3), (3.5), (3.6), (3.7) to obtain that there exists a positive decreasing function γ 4 with limL→+∞ γ 4 (L) = 0 such that (k) + |r (k) | γ 4 (L) i V i V k-i-1 .(3.22)Next we use Lemma 23 in the case s = 2 to get (3.20), where the function γ 4 has been modified to incorporate the constant coming from (3.17

  ).

  obtain the same bound onG j [u, ∇Φ a ] Estimate of H j [r, u -u S , ∇Φ a ]To estimate the body velocity u S in h(ζ)[r, u -u S , ∇Φ a ], we will use the following result, which is the twodimensional counterpart of [5,Lemma 8] (with a different norm which has no importance here, since we estimate the solid velocity which belongs to a finite-dimensional space).

	L	p 2 j+1 (F (t))	by using (2.19) instead of (2.18).
	3.2.2 Lemma m+1 (W(t))	CV m .	(3.18)
	Applying (2.3) to h(ζ) (whose formula is given in (2.22)) and using (2.1), we obtain:

24. Under the same assumptions as Proposition 21, there exists a geometric constant C > 1 such that for any m k D m u S W 1, p 2

  1,p 2 (F (t))+ (k) + |r (k) | (p 2 (k + 1)) k (k!) M L k (k + 1) 2 u W 1,p 2 (k+1) (F (t)) + + |r|

k+1

(3.25)

p k 2 (k!) M +1 Lk u W 1,p 2 (k+1) (F (t)) + + |r| k+1 ,

  Now conservation of the L p norms of the vorticity and Kelvin's circulation theorem yields that at any positive timeu W 1,p 2 (k+1) (F (t)) cp 2 (k + 1) ω 0 L p 2 (k+1) (F (t)) + c(|γ| + + |r|), cp 2 (k + 1) + c(|γ| + + |r|),since ω 0 is bounded with compact support (enlarging c if necessary). Plugging this into (3.25) and using again Stirling's formula, we obtain that there exists L > 0 depending only on F(t) such that for any k,D k u W 1,p 2 (F (t)) p k 2 (k!) M +1 L k+1 k!p k+1 2 + |γ| k+1 + k+1 + |r| k+1 .-ν), thanks to Morrey's inequality, there exists C > 0 such that for any smooth function f onF(t), f C 0,ν (F (t)) C f W 1,p 2 (F (t)) .(3.27)This allows to bound D k u C 0,ν (F (t)) thanks to D k u W 1,p 2 (F (t)) . Then we differentiate Eq. (1.15) to get

	Using p 2	2/(1 ∂ k+1 t	Φ

1,p 2 (k+1) (F (t)) cp 2 (k + 1) curl u L p 2 (k+1) (F (t)) + c(|γ| + + |r|). F (t, x) = D k u(t, Φ F (t, x)).

Acknowledgements. The authors were partially supported by the Agence Nationale de la Recherche, Project CISIFS, grant ANR-09-BLAN-0213-02.

Now, in order to prove (1.18), we have to absorb the ν factor in the Hölder exponent in (3.28). To do this we use the fact that these estimates are valid whatever the choice of the time interval and of ν ∈ (0, 1) (note that p 2 satisfies (3.24) and that the constant in (3.27) is uniform for ν ∈ (1/2, 1)). In particular, we consider τ ∈ (0, T ), and we apply the above result on the interval (-τ, τ ) and ν ∈ (0, 1) such that ν exp(-cτ ω 0 L ∞ (F0) ) > exp(-cT ω 0 L ∞ (F0) ). Then one can choose the constant L > 0 containing the (•) k factors and the proof of Theorem 2 is over.


Named entity extraction for ontology

enrichment Yayoi Nakamura-Delloye †1 We propose in this paper two pattern-based named entity extraction methods for ontology enrichment. The proposed methods are characterized by the use of entity relation patterns obtained by our unsupervised extraction method.

These patterns correspond to syntactic paths that connect two named entities in dependency trees. This work aims to take advantage of parsing benefits and also offers solutions for parsing disadvantage.

Introduction

Knowledge organization using ontological resources is a valuable issue that arouses the interest of researchers in natural language processing (NLP). World knowledge is important in many NLP tasks, such as automatic text understanding. NLP technologies are also used to acquire knowledge from text in order to enrich such resources. This paper presents knowledge extracting methods using NLP technologies for ontological resource enrichment. The proposed methods aim to acquire from large corpora new named entities (NE) such as people or organizations using automatically extracted patterns.

Pattern-based extraction methods are used for relation discovery 7),10) and also for entity extraction 1),2),4), [START_REF] Pennacchiotti | Entity extraction via ensemble semantics[END_REF] . Most methods use patterns provided by a semisupervised approach based on induction using some seeds. The key feature of our approach is the use of syntactic patterns obtained by an unsupervised technique of relation extraction (RE). We also try to show how our approach takes advantage of parsing benefits and offers solutions for parsing disadvantage.

We first describe the framework for ontology enrichment (Section 2). We then present our named entity extraction methods using entity relation patterns tion 3). Finally, we examine the results of experiments we conducted and discuss the performance of each approach (Section 4).

Framework

Within the semantic enrichment experiments of Agence France-Presse (AFP) dispatches, we are currently developing an ontological resource [START_REF] Stern | Resources for named entity recognition and resolution in news wires[END_REF] . This resource is designed to collect, to organize and to exploit the information extracted from dispatches for purposes of categorization, reference retrieval or thematic filtering, etc. Named entities are the main data of this resource. The population is progressively carried out by the module called "Resolution" [START_REF] Stern | Détection et résolution d'entités nommées dans des dépêches d'agence, Actes de TALN 2010[END_REF] , using NEs previously extracted from AFP corpora.

Fig. 1 presents the framework of the ontology enrichment procedure.

To populate the ontological resource, we first worked on relation extraction from texts. For the automatic acquisition of relations and relation patterns, we implemented a clustering-based unsupervised method [START_REF] Nakamura-Delloye | Extraction non-supervisée de relations basée sur la dualité de la représentation[END_REF] .

We use AFP parsed corpora for the extraction of relations and relation patterns. The parsing (see 1 in Fig. 1) is carried out by the FRMG parser [START_REF] Villemonte De La Clergerie | FRMG : evolutions dun analyseur syntaxique tag du francais Journée ATALA "Quels analyseurs syntaxiques pour le francais ?[END_REF] , which uses a named entity recognition module, SxPipe 9) , to identify and classify NEs. From the parsing result, our RE system (see 2 in Fig. 1) builds syntactic dependency trees and identifies all NE pairs and the paths we call "relation syntactic paths" that connect these NE pairs. We assume that the paths connecting NE pairs represent their relation. In the sentence "Xavier Bertrand succède officiellement Patrick Devedjian" (Xavier Bertrand formally succeeds Patrick Devedjian) we identify an NE pair (Xavier Bertrand, Patrick Devedjian) and the path, (X) → SU J-V succder v ← CP L-V (à) (Y ), which connects this pair and represents its relation.

The system provides two types of results: on the one hand a list of classes containing relation paths (see 4 in Fig. 1) and on the other a list of classes containing NE pairs connected by a path which belongs to one of the path classes (see 3 in Fig. 1). The classes correspond to the relations and the paths correspond to patterns of the relation the class represents. The NE pairs correspond to instances of the relation represented by the class to which the pair belongs.

The integration of acquired relation instances into the ontology is already in progress [START_REF] Nakamura-Delloye | Extraction de relations et de patrons de relations entre entités nommées en vue de lenrichissement dune ontologie[END_REF] . In the next section, we show how the extracted patterns are exploited for the new named entity acquisition.

Named entity acquisition using relation patterns

We propose two NE acquisition methods using relation patterns. The first uses relation syntactic paths to identify NEs in a parsed corpus (see 5 in Fig. 1). The second method (see 6 in Fig. 1) consists in transforming the relation syntactic paths into string patterns in order to identify NEs in unparsed texts.

Using syntactic paths as extraction patterns

The first approach exploits, as in the work of Snow et al. 10) , relation syntactic paths to locate new NEs by pattern matching. The operation consists of ( 1 ) constructing syntactic trees from a parsed corpus, ( 2 ) identifying the paths corresponding to an input pattern, and ( 3 ) acquiring the items located at both ends of the paths. When we identify the elements connected by a path corresponding to one of the input patterns, we assume that on the one hand they are NEs and on the other the type of these entities corresponds to one defined in the pattern. Consider, for example, a path representing the "CEO" relation that links two NEs, person type one X (NE IND) and company type one Y (NE COM):

X(N E IN D) → AP P OS PDG ← M OD-N (de) Y (N E COM )
In the parse tree representing the following sentence, the syntactic structure of the underlined strings corresponds to this CEO pattern.

M. Rousset, qui reconnaît avoir "une divergence d'appréciation" sur le dossier Toyal avec M.Lassalle, doit rencontrer le PDG de Toyo, Masao Imasu, vendredi à Osaka (ouest du Japon) pour le "rassurer". The pattern matching result is X = Masao Imasu (pn) and Y = Toyo (pn). As the types of the elements connected by the path do not correspond to the types defined in the pattern, the system proposes a reclassification and provides two new NEs as a result: X = Masao Imasu (NE IND) and Y = Toyo (NE COM).

The elements linked by the paths can be not only simple units but also compound units. In addition to pure simple proper names, there are also NEs composed of pure proper names and common nouns, or only common nouns, possibly with adjectives. They are sometimes called mixed proper names or descriptivebased proper names [START_REF] Jonasson | Le nom propre. Constructions et interprtations[END_REF] . In order to identify these compound units, all dependent items are included in the identified NEs connected by a path. Consider the following sentence:

France Gamerre, 64 ans, présidente de Génération Ecologie. By the matching it with a "President" pattern (person-organization relation), France (location) and Génération (n) are indentified as the two connected items. But after including dependent items, the result finally becomes X = France Gamerre (NE IND) and Y = Génération Ecologie (NE ORG).

Transformation of syntactic paths into string patterns

The second approach consists of the transformation of syntactic paths into string patterns for acquiring new NEs during the dynamic procedure. String patterns allow the identification of NEs in unparsed corpora.

The transformation of syntactic paths into string patterns is carried out using the contexts in which these patterns occur. The operation consists in locating in context examples two NEs connected by the path and extracting the strings which contain all words included on the syntactic path. These strings can be located either

• between the two identified NEs or 2 c ⃝ 2011 Information Processing Society of Japan

• between a word that consistutes a node of the syntactic path used for matching and one of two identified NEs. Take, for example, a path of the "Director" relation that links two NEs, person type one X (NE IND) The result of the experiment is presented in Table 1. Each column corresponds to the result obtained with each input pattern. The frequency indicates the number of extracted paths that provide new NEs and the precision is defined as the proportion of the correct results against all the extracted new ENs.

As can be seen in the result (Table 1) and in Fig. 2, high-frequency patterns provide interesting results, and not all low-frequency patterns are necessarily inefficient. Some low-frequency patterns also provide very interesting results. To identify effective patterns is a delicate task and selection criteria remain to be defined.

This approach using syntactic paths has an advantage of extracting distant NE pairs in the linear representation of texts which are difficult to identify. Another advantage of this approach is to make it possible to identify compound NEs (see Fig. 3). Compound NEs are not often contained in the lexical resources, and it is difficult to identify these units by an annotation method based on lexical resources.

Most errors come from incorrect parsing results. Another source of error is the ambiguity of the element types for certain relationships. Relations such as "Patron (boss)" or "Entraîneur (coach)" involve not only personorganization pairs (person X is head of organization Y ) but also person-person pairs (person X is boss of person Y ). In most cases, the relations "Directeur (director)" and "Président" refer to person-organization pairs, but sometimes they involve person-event pairs (ex. Aldo Tassone, directeur artistique du festival du film français de Florence).

In addition, some extracted elements are anaphoric and require additional processing of anaphora resolution before these extracted NEs can be integrated into the resource.

Transformation-based approach

In order to examine what we could obtain by transformation, we transformed syntactic paths that belong to nine classes obtained by our RE method. Table 2 shows the result of this experiment. The "Total obtained" row indicates the total number of obtained string patterns. The "NE tag error" and "Parse error" rows contain the number of incorrect patterns, due respectively to NE annotation error and parse error. The last row, "Patterns too specific", indicates the number of patterns which contain phrases too specific, especially with parenthetical clauses Except for the class "Président" which contains many paths created with parsing errors, we obtained a large number of correct and interesting patterns (see Fig. 4, Fig. 5).

These string patterns could make it possible to identify unknown NEs in unparsed texts and avoid errors due to incorrect parsing results. However, our work has not yet been completed. We have to define a grammar to integrate the unknown NEs identification into the dynamic procedure for AFP dispatch processing and then evaluate the performance of our string pattern-based identification method.

Conclusion

We have yet traced in outline our automatic named entity acquisition methods. This whole process is twofold. The exploitation of syntactic paths as extraction patterns allows the discovery of distant NE pairs which are difficult to identify in the linear representation of texts. But the use of parsing results is always accompanied by risks due to parsing errors. However, the transformation strategy produces string patterns, which allow us to avoid the influence of parse quality, giving robustness against the inconveniences of parsing to our model.

Fig. 1

 1 Fig. 1 Framework for ontology enrichment

Fig. 2

 2 Fig. 2 Relationship between frequency and precision

Fig. 3

 3 Fig. 3 Examples of extracted compound NEs

Fig. 4

 4 Fig. 4 Examples of obtained string patterns: Class PDG

  Christian Larièpe and Nantes, are connected by the path in text 1 and Christian Charpy and ANPE in text 2. Extracting the strings between the two linked NEs, we can derive the following string patterns (1b from text 1, 2b from text 2) for this syntactic path: From these examples, the system extracts strings between the word "directeur" and the NEs which occur in the right context of the other, so that all words located on the syntactic path are included (3b from text 3, 4b from text 4):

	and organization type one Y (NE ORG):
	X(N E IN D) → AP P OS directeur ← M OD-N (de) Y (N E ORG).
	The path links instances of this relation in many different contexts. Consider
	the following texts:
	1. Christian Larièpe, directeur technique du FC Nantes, écope d'un
	match d'interdiction de banc de touche et de vestiaire d'arbitres pour
	son comportement envers l'arbitre.
	2. Il s'agit de Christian Charpy, actuel directeur général de l'ANPE,
	ex-directeur de cabinet de Philippe Douste-Blazy.
	Two NEs, also occurs in many different contexts and links instances in the following con-
	texts:
	3. Le directeur des recherches, Carles Lalueza Fox, de l'Université de
	Barcelone, émet l'hypothèse qu'ils avaient la possibilité de détecter
	d'autres composants pas encore identifiés, ce qui permettrait d'expliquer
	la raison de la survivance du gène récessif.
	4. Le directeur départemental de La Poste, Jean-Claude Sénat, a toute-
	fois évoqué auprès de l'AFP une "première étape d'organisation dès
	2009".

1b. [EN IND] , directeur technique de [EN ORG] 2b. [EN IND] , actuel directeur général de [EN ORG]

In the same way, another path of the "Director" relation

X(N E IN D) ← AP P OS directeur ← M OD-N (de) Y (N E ORG) 3b. directeur des recherches , [EN IND] ,

de [EN ORG]

Table 2

 2 Result of pattern transformation Examples of obtained string patterns: Class Professeur

	Class	Président	Directeur	Directrice	Entraîneur	Économiste	Professeur	PDG	Patron
		(president)	(director)	(director)	(coach)	(economiste)	(profesor)	(CEO)	(boss)
	Total obtained	111	170	59	102	24	56	81	90
	NE tag error	0.05	0.09	0.05	0.04	0.04	0.02	0.05	0.09
	Parse error	0.35	0.04	0	0.04	0.13	0.05	0.07	0
	Patterns too specific	0.27	0.11	0.32	0.13	0.08	0.04	0.15	0.12

c ⃝ 2011 Information Processing Society of Japan

Acknowledgment

This work is funded by and conducted under the SCRIBO project and the EDyLex project.

Appendix

A.1 Syntactic patterns used for the experiment