Magali Mercier

Global smooth solutions of Euler equations for Van der Waals gases

Keywords: 2000 Mathematics Subject Classification: 35L60, 35Q31, 76N10 Euler compressible equations, smooth solutions, special symmetrisation

We prove global in time existence of solutions of the Euler compressible equations for a Van der Waals gas when the density is small enough in H m , for m large enough. To do so, we introduce a specific symmetrisation allowing areas of null density. Next, we make estimates in H m , using for some terms the estimates done by M. Grassin, who proved the same theorem in the easier case of a perfect polytropic gas. We treat the remaining terms separately, due to their non-linearity.

Introduction

We are interested in the Cauchy problem for Euler compressible equations, describing the evolution of a gas whose thermodynamical and kinetic properties are known at time t = 0.

More specifically, we are concerned with the life span of smooth solutions. Various authors, in particular T. C. Sideris [20,21], T. Makino, S. Ukai & S. Kawashima [13], J.-Y. Chemin [START_REF] Chemin | Dynamique des gaz à masse totale finie[END_REF]3] have given criteria for mathematical explosion. We know also that there exist global in time solutions for well chosen initial data. Li Ta Tsien [12], D. Serre [START_REF] Serre | Solutions classiques globales des équations d'Euler pour un fluide parfait compressible[END_REF], M. Grassin [7] prove for example the global in time existence of regular solutions under some hypotheses of "expansivity".

Most of these results were obtained within the framework of Perfect Polytropic Gases. A natural question is to determine whether these results extend to more realistic gases, following for example the Van der Waals law. This law takes into account the volume of molecules, which is important in physical situations like explosions or implosions. In such limits, the gas is highly compressed and the Van der Waals law fits better with the behaviour of real gases than the Perfect polytropic gases in such limits. The Van der Waals law is also used to modelise dusty gases, seen as perfect gases with dust pollution [START_REF] Gretler | Variable-energy blast waves generated by a piston moving in a dusty gas[END_REF][START_REF] Saffman | On the stability of laminar flow of a dusty gas[END_REF]22,[START_REF] Vishwakarma | A self-similar solution of a shock propagation in a mixture of a non-ideal gas and small solid particles[END_REF]. This law is given by the relationship

p(v -b) = RT ,
where p is the pressure, v the massic volume, T the temperature and b, R are given constants.

The addition of the covolume b, representing the compressibility limit of the molecules in the gas, modifies non-trivially the analysis of the Euler equations. However, we are going to show the global in time existence of regular solutions thus generalising a theorem of M. Grassin [7].

Theorem 1.1. Let m > 1 + d/2. Let (ρ 0 , u 0 , s 0 ) be the initial conditions for the Cauchy problem associated to the Euler compressible equations (2.1) for a Van der Waals gas with constant c v and c v > 0. Let us assume 0 ρ 0 1/b. Then we can define γ 0 = 1 + R cv ; furthermore there exists ε 0 > 0 such that if

(H1) (π 0 , s 0 ) H m (R d ) ε 0 , where π 0 = ρ 0 1-bρ 0 γ 0 -1 2
exp( s 0 cv ), (H2) the initial speed u 0 belongs to the space X = {z : R

+ × R d → R d ; Dz ∈ L ∞ , D 2 z ∈ H m-1 }, (
H3) there exists δ > 0 such that for all x ∈ R d , dist(Spec(Du 0 )(x), R -) δ, (H4) the initial density ρ 0 and the initial entropy s 0 have compact support, then the problem

∂ t ū + (ū • ∇)ū = 0 on R + × R d , ū(0, x) = u 0 (x) on R d . (1.1)
admits a global classical solution. If furthermore γ 0 = ν+1 ν-1 with ν ∈ N and ν 2, or if γ 0 and m satisfy ν = γ 0 +1 γ 0 -1 m > 1 + d 2 , then there exists a global classical solution (ρ, u, s) to the Euler compressible equations (2.1) satisfying

( ρ 1 -bρ γ 0 -1 2 , u -u, s) ∈ C 0 (R + ; H m (R d ; R d+2 )) ∩ C 1 (R + ; H m-1 (R d ; R d+2 )) .
To do so, we have first to extend to Van der Waals gases a symmetrisation obtained by Makino, Ukai & Kawashima [13], which allows null density areas. Next we will derive energy estimates in H m .

Since the Van der Waals gases have a behaviour close to perfect polytropic gases for weak densities, this result is not much surprising. However, the non linear terms introduced by the Van der Waals law have to be treated carefully.

In Section 2, we describe the thermodynamic properties of a compressible gas and we state some important properties such as the Friedrichs' symmetrisation. In Section 3, we give the detailed proof of this result and in Section 4 we give the proofs of some technical lemmas used in Section 3.

Thermodynamic and first properties 2.1 Conservation law

Compressible fluid dynamics, without viscosity or heat transfer, is described by the Euler equations, which are made of the conservation of mass, of momentum and of energy (see [5, chap. 2]) :

     ∂ t ρ + div(ρu) = 0 , ∂ t q + div(ρu ⊗ u) + ∇p = 0 , ∂ t E + div (E + p) u = 0 , (2.1)
where ρ is the mass of the fluid per unit of volume, q = ρu is the momentum per unit of volume and E = 1 2 ρu 2 + ρe is the total energy per unit of volume, sum of the kinetic energy and internal energy. This is a system of (d + 2) equations and (d + 3) unknowns: the density ρ ∈ R + , the speed u ∈ R d , the internal energy e ∈ R and the pressure p ∈ R. In order to complete this system, we have to add a state law, for example an incomplete state law, also called pressure law (ρ, e) → p(ρ, e). Definition 2.1. We call Euler compressible equations the system made of the conservation laws (2.1) and of an incomplete state law p = p(ρ, e).

A simplified model is often considered, conserving only the conservation of mass and momentum, assuming that the fluid is isentropic. This simplified system is

∂ t ρ + div(ρu) = 0 , ∂ t q + div(ρu ⊗ u) + ∇p = 0 , (2.2)
and the state law is a given function ρ → p(ρ).

Definition 2.2. We call isentropic Euler equations the system (2.2) with a given state law p = p(ρ) such that p = -∂e ∂v s , where v = 1/ρ is the specific volume, T is the temperature, s is the specific entropy and f = e -T s is the specific free energy, assuming we are given a complete equation of state (v, s) → e(v, s).

The thermodynamical variables v, s, e, T, p must satisfy some relations described in Section 2.2. The Euler equations are a system of first order conservation law, whose study is developed in particular in the books [4,[START_REF] Serre | Hyperbolicité, entropies, ondes de choc[END_REF][START_REF] Serre | Structures géométriques, oscillation et problèmes mixtes[END_REF].

State law

The state law has a strong influence on the mathematical analysis of the compressible Euler equations. The state law of "real" gases can reveal particular behaviour and introduce existence and/or uniqueness troubles which do not appear for perfect gases, see [14]. We describe below the physical principles a state law has to satisfy.

2.2.A Definitions

We consider a fluid, whose internal energy is a regular function of its specific volume 1 v = 1/ρ and of its specific entropy s. This means that the gas is entitled with a complete state law, or energy law e = e(v, s). The fundamental thermodynamic principle is then de = -pdv + T ds (2.3)

1 specific is a synonym of massic
where p is the pressure and T the temperature of the gas. Consequently, the pressure p and the temperature T can be defined as

p = - ∂e ∂v s , T = ∂e ∂s v , (2.4)
where the notation | precises the variable maintained constant in the partial derivation.

The greater order derivatives of e have also an important role; we introduce the following adimensional quantities:

γ = - v p ∂p ∂v s , Γ = - v T ∂T ∂v s , δ = pv T 2 ∂T ∂s v , G = - v 2 ∂ 3 e ∂v 3 s ∂ 2 e ∂v 2 s
.

(2.5)

The coefficient γ is called the adiabatic exponent, and Γ is the Grüneisen coefficient. The quantities γ, δ, Γ and G characterise the geometrical properties of the isentropic curves in the (v, p) plan (see [14]). They can be expressed in function of e through the relationships:

γ = v p ∂ 2 e ∂v 2 , Γ = - v T ∂ 2 e ∂s∂v , δ = pv T 2 ∂ 2 e ∂s 2 .
We also introduce the calorific capacity at constant volume c v and the calorific capacity at constant pressure c p by

c v = ∂e ∂T v = T ∂ 2 e ∂s 2 v , c p = T ∂s ∂T p . (2.6)
These two quantities are linked with pv T and with γ, δ, Γ through

δc v = pv T , c p = pv T γ γδ -Γ 2 .
(2.7)

The quantity γ * = cp cv can besides be expressed as γ * = γδ γδ-Γ 2 . It is not equal to γ in the general case, but for an ideal gas we have δ = Γ = γ -1, so that γ * = γ.

2.2.B Thermodynamical constraints

It is very natural to assume v 0. We assume furthermore that the pressure p and the temperature T are positive, which imposes that e is a function increasing in T and decreasing in v.

A classical thermodynamical hypothesis requires furthermore e to be a convex function of s and v, which means:

γδ -Γ 2 0 , δ 0 , γ 0 .
In particular, γ 0 means that p increases with the density ρ = 1/v, which allows us to define the adiabatic sound speed by

c = ∂p ∂ρ s = γ p ρ . (2.8)
Then, we show that G can be expressed in function of ρ and c through the expression

G = 1 c ∂(ρc) ∂ρ s .
Furthermore, we require usually Γ > 0 and G > 0. The condition Γ > 0 is not thermodynamically required but is satisfied for many gases and ensures that the isentropes do not cross each other in the (v, p) plan. The condition G > 0 means that the isentropes are strictly convex in the (v, p) plan.

2.2.C Van der Waals Gas

Definition 2.3. A gas is said to follow the Van der Waals law, if it satisfies the following pressure law:

p (v -b) = RT , (2.9)
where v is the massic2 volume and b is the covolume, representing the compressibility limit of the fluid, due to the volume of the molecules.

The Van der Waals law is a modification of the perfect gas law, in which b = 0. In opposition to the perfect gas law, it takes into account the proper size of the molecules, which is important in some situations when the gas is strongly compressed. In this model, the density must be bounded and the maximal density is ρ max = 1 b . The fundamental relationship (2.3) gives us the PDE:

∂ v e + R v-b ∂ s e = 0. Thus, we introduce new variables w = (v -b) -R , σ = (v -b) -R
exp(s) and ê(w, σ) = e(v, s). We obtain ∂ w ê = 0, so that e = E((v -b) -R exp(s)) for any regular function E.

If we assume furthermore that c v is constant, thanks to the definition of c v and (2.3), we get that

∂ 2 e ∂s 2 v = 1 cv ∂e ∂s v , hence σE ′′ = ( 1 cv -1)E ′ and E(σ) = Cσ 1/cv which leads to: e = (v -b) -R cv exp( s c v ) , p = R c v e v -b .
Some computations allow us finally to obtain

γ = γ 0 v v -b , Γ = δ = (γ 0 -1) v v -b , G = γ 0 + 1 2 v v -b
,

where γ 0 = R c v + 1 . (2.10)
The conditions of Section 2.2.B are then satisfied for γ 0 > 1.

Remark 2.4. 1. A perfect gas can be seen as a Van der Waals gas with b = 0. A perfect gas for which c v is constant is called polytropic.

2. The Van der Waals law coincides with the dusty gas law [START_REF] Gretler | Variable-energy blast waves generated by a piston moving in a dusty gas[END_REF][START_REF] Jena | Self-similar shocks in a dusty gas[END_REF][START_REF] Saffman | On the stability of laminar flow of a dusty gas[END_REF]22,[START_REF] Vishwakarma | A self-similar solution of a shock propagation in a mixture of a non-ideal gas and small solid particles[END_REF]. In this model, we consider that the gas is perfect but polluted by dust particles that are equidistributed and have a non-negligible volume.

Very often in the literature, the perfect polytropic gases are considered as a canonical example. However, their adequation with physical observations is not as good as for Van der Waals gases, for example in explosion phenomena, or in the sonoluminescence phenomenon [START_REF] Blake | Acoustic cavitation and sonoluminescence[END_REF][START_REF] Evans | Instability of converging shock waves and sonoluminescence[END_REF][START_REF] Lauterborn | Acoustic cavitation, bubble dynamics and sonoluminescence[END_REF].

In the following, we only consider Van der Waals fluids with constant and strictly positive calorific capacity c v :

c v > 0 , (2.11)
which implies γ 0 := R cv + 1 > 1.

Symmetrisation

An important property of the Euler equation is their symmetrisability.

2.3.A General case, without vacuum

If ρ > 0 and ∂p ∂ρ s > 0 then the system (2.1) can be written in the variables (p, ρ, s). Then, the system is almost symmetric, since it can be written matricially

∂ t Ṽ + k Ãk ( Ṽ )∂ k Ṽ = 0, with Ṽ = (p, u T , s) T and Ã(ξ, V ) = k ξ k Ãk ( Ṽ ) =    u • ξ ρc 2 ξ T 0 1 ρ ξ u • ξI d 0 0 0 u • ξ    .
This matrix is almost symmetric since we obtain a symmetric matrix by multiplying it on the left by D := Diag 1 ρc 2 , ρ, . . . , ρ, 1 . Consequently, we have the following Proposition 2.5. The system (2.1) is Friedrichs symmetrisable when (ρ, u, s) takes values in a compact subset of R * + × R d × R. Indeed, for such values of (ρ, u, s), D Ã(ξ, V ) is symmetric with D symmetric definite positive.

2.3.B Van der Waals gas

We are not completely satisfied with the previous formulation as it does not authorise ρ to vanish, or even to tend to 0 at infinity. For example, ρ cannot be taken into H m for m 0. Makino et al. have introduced in [13] a symmetrisation for perfect polytropic gases allowing the null density areas. We generalise here their method to the case of Van der Waals gases.

First, let us remind that for Van der Waals gases, we have γ = γ 0 1-bρ and p = (γ 0 -1)

ρ 1-bρ γ 0 exp s cv . We now introduce the new variable π = 2 γ 0 γ 0 -1 p γ 0 -1 γ 0 -1 2γ 0 ,
and we re-write the system (2.1) in the variables (π, u, s). In order to do that, we first write the system (2.1) in (p, u, s) variables:

     ∂ t p + u • ∇p + ρc 2 div u = 0 , ∂ t u + (u • ∇)u + 1 ρ ∇p = 0 , ∂ t s + u • ∇s = 0 . (2.12) Since π = f (p) = Cp α , it is sufficient to multiply the first line by f ′ (p) = Cαp α-1 to obtain an equation in π:      ∂ t π + u • ∇π + Cαp α-1 ρc 2 ∇ • u = 0 , ∂ t u + (u • ∇)u + 1 ρCαp α-1 exp( s γ 0 cv )∇π = 0 , ∂ t s + u • ∇s = 0 . Besides, we know that c 2 = γ 0 1-bρ p ρ , C = 2 γ 0 γ 0 -1 ( 1 γ 0 -1 ) γ 0 -1 2γ 0
and α = γ 0 -1 2γ 0 . It remains to evaluate the coefficients:

Cαp α-1 ρc 2 = γ 0 α 1 -bρ π = γ 0 -1 2 π 1 -bρ , 1 ρCαp α-1 = exp(s/(γ 0 c v )) 1 -bρ (γ 0 -1) 1/γ 0 αC (γ 0 -1)/(γ 0 α) π 1-α-1/γ 0 α = exp s γ 0 c v γ 0 -1 2 π 1 -bρ . Thus,      ∂ t π + u • ∇π + γ 0 -1 2 π 1-bρ ∇ • u = 0 , ∂ t u + (u • ∇)u + γ 0 -1 2 π 1-bρ exp( s γ 0 cv )∇π = 0 , ∂ t s + u • ∇s = 0 . (2.13) Moreover, 1/(1 -bρ) = 1 + b γ 0 -1 4γ 0 1 γ 0 -1 exp -s γ 0 cv π 2 γ 0 -1 . Therefore, denoting ν = γ 0 + 1 γ 0 -1 > 1 , and b = b γ 0 -1 4γ 0 1 γ 0 -1 , (2.14)
we obtain π 1-bρ = π(1+ be -s/(γ 0 cv) π ν-1 ), which is well defined for all π 0 and in particular for p = 0, since we have assumed in (2.11) γ 0 > 1. The matrix associated to the system (2.13) writes now

A(ξ, π, u, s) =    u • ξ γ 0 -1 2 π 1-bρ ξ T 0 γ 0 -1 2 π 1-bρ exp( s γ 0 cv )ξ u • ξI d 0 0 0 u • ξ    .
It is once again "almost symmetric" in the sense that SA is symmetric, S being the diagonal definite positive matrix

S = Diag 1, exp(-s/(γ 0 c v )), . . . , exp(-s/(γ 0 c v )), 1 .
Furthermore, this symmetriser is independent from p and in particular is well defined and definite positive even when p or ρ vanishes. Finally, we have the proposition:

Proposition 2.6. For a Van der Waals gas with constant c v and c v > 0, the system of Euler equations can be written for regular solutions as (2.13)

which is Friedrichs sym- metrisable for (p, u, s) ∈ K, where K is a compact subset of R + × R d × R.
Note that in the variables (π, u, s), the system (2.13) can be written

     ∂ t π + u • ∇π + γ 0 -1 2 (1 + b exp( -s γ 0 cv )π ν-1 )π∇ • u = 0 , ∂ t u + (u • ∇)u + γ 0 -1 2 exp( s γ 0 cv )(1 + b exp( -s γ 0 cv )π ν-1 )π∇π = 0 , ∂ t s + u • ∇s = 0 , (2.15)
where ν and b are defined as in (2.14).

Local existence

The symmetrisation of Proposition 2.6 is very useful to show local existence of regular solutions with vanishing density.

Theorem 2.7. We consider a Van der Waals gas with constant c v such that

γ 0 ∈]1, 3]. Let π 0 = 2 γ 0 γ 0 -1 ρ 0 1 -bρ 0 (γ 0 -1)/2 exp( γ 0 -1 2γ 0 c v s 0 )
,

where ρ 0 ∈ C 1 (R 3 ; [0, 1/b[). We introduce also ν = γ 0 +1 γ 0 -1 , b = b γ 0 -1 4γ 0 1 γ 0 -1 . We assume that (π 0 , u 0 , s 0 ) ∈ H m (R d ) for m > 1 + d 2 .
Then there exists T > 0 and a unique solution

(π, u, s) ∈ C 1 ([0, T ] × R 3 ) to the Cauchy problem for      ∂ t π + u • ∇π + γ 0 -1 2 (1 + bπ ν-1 )π∇ • u = 0 , ∂ t u + (u • ∇)u + γ 0 -1 2 exp( s γ 0 cv )(1 + b exp( -s γ 0 cv )π ν-1 )π∇π = 0 , ∂ t s + u • ∇s = 0 , (2.16)
with initial condition (π 0 , u 0 , s 0 ). Furthermore,

(π, u, s) ∈ C ([0, T ]; H m (R d ; R d+2 )) ∩ C 1 ([0, T ]; H m-1 (R d ; R d+2 )) .

Positivity of the density

For regular solutions, the positivity of the density is given by the following

Proposition 2.8. Let (ρ, ū, s) ∈ C 1 ([0, T ex [×R d )

be a regular solution of the Cauchy problem (2.1) associated to the regular initial conditions

(ρ 0 , u 0 , s 0 ) ∈ C 1 (R d ). If ∇ū ∈ L ∞ ([0, T ] × R d ) for all T < T ex , ρ 0 ∈ L ∞ (R d , R) and if ρ 0 (x) 0 for all x ∈ R d , then for all t ∈ [0, T ex [, ρ(t, x) 0 .
Proof. We use the characteristics method to obtain an expression of the solution of the Cauchy problem (1.1). Assuming ∇ū is bounded, we obtain:

ρ(t, x) = ρ 0 (X(0; t, x)) exp - t 0 div ū(τ, X(τ ; t, x)) dτ > 0 ,
where X is a solution of the Cauchy problem

dX dt = ū(t, X) , X(t 0 ; t 0 , x 0 ) = x 0 ,
which is global in time since ∇ū is bounded.

We also prove that for a Van der Waals gas with constant c v > 0, the variable π = 2

γ 0 γ 0 -1 p γ 0 -1 γ 0 -1
2γ 0 introduced in section 2.3.B remains non-negative if it is non-negative at initial time. This property implies in particular that if ρ 0 < 1/b, then, as long as the regular solution exists, this property will be satisfied. Proposition 2.9. We consider a Van der Waals gas with constant c v . We denote π = 2

γ 0 γ 0 -1 p γ 0 -1 γ 0 -1 2γ 0 , ν = γ 0 +1 γ 0 -1 and b = b γ 0 -1 4γ 0 1 γ 0 -1 . Let (π, ū, s) ∈ C 1 ([0, T ex [×R d ) be a
regular solution of (2.15) satisfying the initial conditions

π(0, x) = π 0 (x) , u(0, x) = u 0 (x) , s(0, x) = s 0 (x) , with (π 0 , u 0 , s 0 ) ∈ C 1 (R d ) and s 0 ∈ L ∞ . If div ū ∈ L ∞ ([0, T ] × R d ) for all T < T ex , π 0 ∈ L ∞ (R d , R) and 0 ρ 0 (x) < 1/b for all x ∈ R d , then for all t ∈ [0, T ex [, π 0.
Then, we can define ρ, and we have:

0 ρ(t, x) < 1/b .
Proof. Let T 0 < T ex . We introduce the Cauchy problem

∂ t w + div (wū) = g(t, x, w) , (2.17) w(0, x) = w 0 (x) , where g(t, x, w) = 1 - γ 0 -1 2 (1 + b exp( -s(t, x) γ 0 c v )w ν-1 ) w div ū(t, x)
We can apply the Kružkov theorem [10,[START_REF] Serre | Hyperbolicité, entropies, ondes de choc[END_REF]. Indeed, the hypotheses ensure that g(t, x, w)w div (ū(t, x)) = -γ 0 -1 2 (1 + be -s/(γ 0 cv) w ν-1 )w div ū is uniformly bounded with respect to x ∈ R d when w is considered as a variable taking values in a compact set. Furthermore, π 0 ∈ L ∞ , so the regular solution π coincides with the entropy solution w 1 of (2.17) associated to w 0,1 = π 0 .

Besides, the entropy solution w 2 of (2.17) associated to the initial condition w 0,2 ≡ 0 is the function constantly equal to 0.

After Kružkov Theorem w 0,1 w 0,2 implies w 1 w 2 for all (t, x)

∈ [0, T 0 ] × R d , that is to say π(t, x) 0 for all (t, x) ∈ [0, T 0 ] × R d . The formula: ρ = 1 b   1 - 1 1 + b exp( -s γ 0 cv )π 2 γ 0 -1   
allows us to conclude.

Proof of Theorem 1.1

In order to prove this theorem, we adapt M. Grassin's idea [7]. First, we look to the isentropic case, which allows us to simplify the estimates. For a Van der Waals gas, nonlinear terms now appear in the estimate which we need to treat separately.

Isentropic case

Let us consider first the isentropic case

∂ t π + u • ∇π + γ 0 -1 2 (1 + bπ ν-1 )π∇ • u = 0 , ∂ t u + (u • ∇)u + γ 0 -1 2 (1 + bπ ν-1 )π∇π = 0 , (3.1)
with initial conditions

π(0, x) = π 0 (x) , u(0, x) = u 0 (x) . (3.2)
which is technically simpler than the general case, but provides estimates very useful in order to treat the general case.

We consider besides the problem

∂ t ū + (ū • ∇)ū = 0 on R + × R d , ū(0, x) = u 0 (x) on R d , (3.3
= Id 1+t + K(t,x) (1+t) 2 , for all x ∈ R d and t ∈ R + , 2. D l u(t, •) L 2 K l (1 + t) d/2-(l+1) for l ∈ 2, M + 1 , 3. D 2 u(t, •) L ∞ C(1 + t) -3 , where K : R + × R d → M d (R), K L ∞ (R + ×R d ) K 1 ,
C and K l for l ∈ 1, m + 1 being non-negative constants not depending on m, d, δ, u 0 X .

3.1.A Local uniqueness

Proposition 3.2. Let U 0 = (π 0 , u 0 ) T ∈ H m (R d ) and Ũ0 be two initial data for (3.1). Let U = (π, u) T , Ũ be two corresponding solutions, defined for 0 t T 0 . We assume that

D Ũ L ∞ ([0,T 0 ]×R d ) < ∞. Let x 0 ∈ R d and R 0. We denote M = sup{( γ 0 -1 2 |π|(1 + bπ ν-1 ) + |u|)(t, x) , (t, x) ∈ [0, T 0 ] × B(x 0 , R)} , (3.4)
C T = {(t, x) ∈ [0, T ] × B(x 0 , R -M t)} for T ∈ [0, T 1 ] , (3.5)
where

T 1 = min(T 0 , η/M ). If U 0 = Ũ0 on B(x 0 , R) then U = Ũ on C T 1 .
The proof of this proposition is classical for hyperbolic systems (see for example [START_REF] Racke | Lectures on nonlinear evolution equations[END_REF]), the constant M being the maximal propagation speed.

Proof. Let U 0 = (π 0 , u 0 ), Ũ0 = (π 0 , ũ0 ) be two initial data for (3.1) such that U 0 ∈ H m . Let U, Ũ be the solutions of the associated Cauchy problems. We assume that these solutions are defined on [0, T 0 ] with T 0 > 0. Let also x 0 ∈ R d , η ∈ R * + and M , C T be as in (3.4) and (3.5). Then we have:

∂ t U + j a j (U )∂ j U = 0 , where a j (U ) =          u j 0 γ 0 -1 2 (π + bπ ν ) 0 . . . γ 0 -1 2 (π + bπ ν ) 0 . . . . . . 0 0 0 u j          . Consequently, ∂ t (U -Ũ ) + j a j (U )∂ j (U -Ũ ) + (a j (U ) -a j ( Ũ ))∂ j Ũ = 0.
Then, we make the scalar product with (U -Ũ ) and we integrate on

C T for T ∈ [0, T 0 ]. We get 1 2 C T ∂ t U -Ũ 2 + j ∂ j (U -Ũ ) • a j (U )(U -Ũ ) - j (U -Ũ ) • ∂ j (a j (U ))(U -Ũ ) dx dt = - C T (U -Ũ ) • (a j (U ) -a j ( Ũ ))∂ j Ũ dx dt .
Then using the Stokes formula and noting that

∂C T = ({0} × B(x 0 , η)) ∪ ({T } × B(x 0 , η - M T )) ∪ C, we obtain 1 2 B(x 0 ,R-M T ) U -Ũ 2 (T, x) dx - 1 2 B(x 0 ,η) U -Ũ 2 (0, x) dx + 1 2 1 + 1/M 2 C U -Ũ 2 + j (U -Ũ ) • a j (U )(U -Ũ ) x j M |x| dσ = C T j (U -Ũ ) • ∂ j (a j (U ))(U -Ũ ) -(U -Ũ ) • (a j (U ) -a j ( Ũ ))∂ j Ũ dx dt . Besides ∂ j a j (u) L ∞ C DU L ∞ (C T ) (1 + π ν-1 L ∞ (C T ) ). Hence, C T (U -Ũ ) • ∂ j (a j (U ))(U -Ũ ) -(U -Ũ ) • (a j (U ) -a j ( Ũ ))∂ j Ũ dx dt C DU L ∞ (C T ) (1 + π ν-1 L ∞ (C T ) ) T 0 B(x 0 ,R-M t) U -Ũ 2 dx dt .
Furthermore, the choice of M implies

C U -Ũ 2 + j (U -Ũ ) • a j (U )U -Ũ ) x j M |x| dσ 0 ,
so finally we get the estimate

1 2 B(x 0 ,R-M T ) U -Ũ 2 (T, x) dx - 1 2 B(x 0 ,η) U -Ũ 2 (0, x) dx C DU L ∞ (C T ) (1 + π ν-1 L ∞ (C T ) ) T 0 B(x 0 ,R-M t) U -Ũ 2 dx dt .
We conclude thanks to Gronwall lemma that

1 2 B(x 0 ,R-M T ) U -Ũ 2 (T, x) dx 1 2 e C ′ T B(x 0 ,R) U 0 -Ũ0 2 (x) dx
where

C ′ = C DU L ∞ (C T ) (1 + π ν-1 L ∞ (C T ) ).

3.1.B Local Existence

We construct a local solution of (3.1)-(3.2) such that the difference between this solution and (0, u) be in

C 0 (R + ; H m (R d ; R d+1 )) ∩ C 1 (R + ; H m-1 (R d ; R d+1 ))
. The first step is the symmetrisation of the system, given by Proposition 2.6. This result allows us to use a general theorem (see Theorem 2.7) giving the local existence of solution. Let us define as above,

π = 2 γ 0 γ 0 -1 p γ 0 -1 γ 0 -1 2γ 0 = 2 γ 0 γ 0 -1 ρ 1 -bρ γ 0 -1 2 , (3.6)
where we assume 0 ρ < 1/b. Then we use the same method as M. Grassin [7] to prove that the system (3.1) admits a local in time solution, with initial condition u 0 in the space X and not in a Sobolev space (in particular, u 0 does not tend to 0 at infinity). We use here the compactness of the support of ρ 0 (hypothesis (H4)) and the finite propagation speed of the solutions for an hyperbolic system. More precisely, we assume that Supp(ρ 0 ) ⊂ B(0, R) for R > 0. Let η > 0 and ϕ ∈ C ∞ c (R d ; R + ) be such that ϕ ≡ 1 on B(0, R+2η). We obtain a local in time solution (π ϕ , u ϕ ) of the problem (3.1) with initial conditions (π 0 , ϕu 0 ) ∈ H m for t ∈ [0, T [. The Propositions 2.8 and 2.9 ensure that the condition 0 ρ < 1/b is satisfied.

Let ε ∈]0, T [ , we introduce the maximal propagation speed M = sup{ γ 0 -1

2 (|π ϕ | + |π ϕ | ν ) + |u ϕ | ; t ∈ [0, T -ε] , x ∈ R d }. We also introduce ε ′ ∈]0, η 2M [ and T 1 = min(T - ε, η
2M -ε ′ ) the time for which this construction is available. We finally obtain a solution (π, u) of (3.1)-(3.2) by denoting

(π, u) = (π ϕ , u ϕ ) in K , (0, u) out of K ,
where K is the cone K = {(t, x) ; 0 t T 1 , x ∈ B(0, R+η +M t)}. Then it is sufficient to show that the solutions can be glued smoothly along ∂K. We use here the property of local uniqueness given below by Proposition 3.2. Let indeed x 0 ∈ S(0, R + η) be the sphere of radius R + η of centre 0 and

E x 0 = {(t, x) ; t ∈ [0, T 1 ], x ∈ B(x 0 , η -M t)}.
The choice of T 1 implies in particular ∂K ⊂ ∪ x 0 ∈S(0,R+η) E x 0 (see Fig. 1). The initial conditions of (π ϕ , u ϕ ) and (0, u) coincide on E x 0 since the support of π 0 is included in B(0, R). Proposition 3.2 allows us then to claim that (π ϕ , u ϕ ) = (0, u) on E x 0 and consequently on a neighbourhood of ∂K.

3.1.C Energy Estimates

When we compare to the Perfect Polytropic Gas case [7], we observe that the system (3.1) has an additional term, which will modify the estimates. Let us denote U = (π, u -u), w = u -u and U = (0, u). We also introduce

A j (U ) =          u j -ūj 0 . . . γ 0 -1 2 π 0 0 u j -ūj 0 0 γ 0 -1 2 π 0 . . . . . . . . . 0 0 0 . . . 0 u j -ūj          , B(DU , U ) = γ 0 -1 2 π div u (w • ∇)u ,
(3.7) so that the system (3.1) can be written

∂ t U + d j=1 A j (U )∂ j U = -B(DU , U ) - d j=1 u j ∂ j U -F (DU , DU , U ) (3.8)
where

F (DU , DU , U ) = γ 0 -1 2 bπ ν div (w + u) ∇π , with ν = γ 0 + 1 γ 0 -1 . (3.9)
Observing the properties of u described in Proposition 3.1, we expect the terms

D k U L 2
for k ∈ 0, m to decrease with respect to time with a rate depending on k. Consequently, we introduce

Y k (t) = D k U (t, •) L 2 , Z(t) = m k=0 (1 + t) g k Y k (t) , (3.10)
with g k = k + c, in which c has to be chosen so that all the terms of Z have the same decreasing in time. In order to estimate Z, we apply the operator D k to (3.8), we make the scalar product with D k U and we integrate on R d . The system (3.8) is different from the one considered by M. Grassin through the term F (DU , DU , U ) defined in (3.9). We use now [7, Prop. 3.2 and 3.3] to estimate the terms in common. We remind these results here:

Proposition 3.3. Let α ∈ N d be a d-uplet of size k 0 (that is to say |α| 1 = α 1 +. . .+α d = k). Let us denote, for U = (π, u -ū), R k = - R d D k U • D k   d j=1 A j (U )∂ j U   dx , S k = - R d D k U • D k   B(DU , U ) + d j=1 u j ∂ j U   dx ,
then there exists a constant C > 0 depending only on m, d such that

|R k | C(1 + t) -g 1 -d/2 Y 2 k Z , S k C(1 + t) -g k -2 Y k Z - k + r 1 + t Y 2 k ,
where

r = min(1 - d 2 , ( γ 0 2 -1)d) .
(3.11)

We have now to estimate -

R d D k U • D k (F (DU , DU , U )). Let us denote b = γ 0 -1 2 b and I = -b R d D k U • D k π ν div w ∇π , J = -b R d D k U • D k π ν div u 0 so that -R d D k U • D k (F (DU , DU , U )) = I + J.
A priori, J is easier to estimate than I.

However the estimate of I is possible since the matrix π ν 0 ξ T ξ 0 is symmetric.

a) Estimate of I. We show in Section 4.1 the following Lemma 3.4. With the notations introduced in Section 3.1

|I| C π ν-1 L ∞ DU L ∞ D k U 2 L 2 . (3.12) b) Estimate of J.
Here, we divide J in two parts: a first part J 1 which contains only first order derivatives of u, and a second part J 2 in which all the derivatives of u are at least of order 2. More precisely,

J 1 = -b R d α∈N d ||α| 1 =k (∂ α π)(∂ α π ν ) div u , J 2 = J -J 1 .
For J 1 , we use the first point of Proposition 3.1 giving the decreasing in time of Du, and Lemma 4.7 giving the estimate:

∂ α (π ν ) L 2 π ν-1 L ∞ D k π L 2 ,
for all d-uplet α ∈ N d of size k, that is to say satisfying |α| 1 = α i = k.

We obtain, using the Cauchy-Schwarz inequality:

|J 1 | b ∂ α π L 2 ∂ α (π ν ) L 2 div u L ∞ C 1 + t D k U 2 L 2 π ν-1 L ∞ . (3.13)
For J 2 , we prove in Section 4.1 the following Lemma 3.5. With the notations introduced in Section 3.1, there exists a constant C > 0 such that

|J 2 | C(1 + t) d k D k U L 2 Z ν , (3.14)
where

d k = (-g 1 -d 2 + 1)(ν -1) -g k -2.
c) Re-assembling of the estimates. Assembling the results of Proposition 3.3, the estimates (3.12)-(3.13)-(3.14) of I and J, and finally using Lemma 4.5, we obtain

1 2 dY 2 k dt + k + r 1 + t Y 2 k C(1 + t) β Y 2 k Z + C ′ (1 + t) -g k -2 Y k Z + C(1 + t) β+(β+1)(ν-1) Y 2 k Z ν + C(1 + t) (β+1)(ν-1)-1 Y 2 k Z ν-1 + C(1 + t) (β+1)(ν-1)-g k -2 Y k Z ν , (3.15)
where we have denoted

β = -g 1 - d 2 .
Here, we choose the constant c introduced in g k = k + c in order to have β = 0 and consequently a good decreasing in time. This means to require g 1 + d/2 = 0 and c = -1 -d/2. Consequently, we have

g k = k - d 2 -1 .
We introduce now a = 1 + d/2 + r > 1 , so that k + r = g k + a. We can now divide by Y k in (3.15), multiply by (1 + t) g k and summate on k to obtain a differential inequality in Z (defined in (3.10)):

dZ dt + a 1 + t Z C Z 2 + Z (1 + t) 2 + (1 + t) ν-1 Z ν+1 . Then, we introduce ζ(t) = (1 + t) a exp C
1+t Z(t) and we deduce from the inequality just above dζ dt

C (1 + t) a (ζ 2 + ζ ν+1 ). Besides, ζ 2 + ζ ν+1 2ζ(1 + ζ ν ) for ν 2. Therefore 1 ζ(1 + ζ ν ) dζ dt C (1 + t) a , that is to say d dt (f (ζ(t))) C (1 + t) a , with f (x) = 1 ν ln x ν 1+x ν . By integration, we obtain f (ζ(t)) + C a-1 (1 + t) -(a-1) f (ζ(0)) + C a-1 .
As f is strictly increasing and one-by-one from R * + to R * -, if f (ζ(0)) + C/(a -1) belongs to the set on which f -1 is well-defined, we obtain 0) is small enough, since f (x) tends to -∞ when x tends to 0, and

ζ(t) f -1 f (ζ(0)) + C/(a -1) . But f (ζ(0))+ C/(a-1) 0 is only possible if ζ(
ζ(0) = exp (C) Z(0) = exp(C) π 0 H m . The smallness condition is satisfied thanks to the hypothesis (H1) with 0 < ε 0 < f -1 -C a-1 . d) Conclusion.
We have obtained the following inequalities

Z(t) 1 (1 + t) a exp -C 1 + t f -1 f (e C Z(0)) + C a -1 , Y k (t) (1 + t) -g k Z(t) 1 (1 + t) k+r exp -C 1 + t f -1 f (e C Z(0)) + C a -1 .
The L 2 norms of the derivatives of the local solution U consequently do not explode in finite time since t → 1 (1+t) a exp -C 1+t do not explode in finite time. Let us assume that the regular solution exists to time T . Our estimates give us, for all t ∈ [0, T [, for C T not depending on T , (π, uū)(t) H m C T .

Since Supp π ⊂ B(0, R), our construction is possible when the norm of (π, u) is bounded in H m (R d ) and

(π, u)(t) H m (B(0,R)) C T + ū(t) H m (B(0,R)) K T .
We can associate to the constant K T a time of existence T * (K T ) for the local in time solution. Let t 1 ∈]0, T [ be such that t T -T * (K T ). Introducing the solution with initial condition (π(t 1 ), u(t 1 )), we succeed in prolongating the solution up to time T , which finishes the proof.

General Case

3.2.A Local in time Existence

As in the isentropic case, we first seek to symmetrise the system. Let us denote

π = γ 0 -1 γ 0 p γ 0 -1 γ 0 -1 2γ 0 = γ 0 -1 γ 0 ρ 1 -bρ γ 0 -1 2 exp γ 0 -1 2γ 0 s c v .
The system (2.1) can be written in variables (π, u, s):

       e s/(γ 0 cv) ∂ t π + e s/(γ 0 cv) u • ∇π + γ 0 -1 2 e s/(γ 0 cv) π div u = -γ 0 -1 2 bπ γ 0 +1 γ 0 -1 div u , ∂ t u + (u • ∇)u + γ 0 -1 2 e s/(γ 0 cv) π∇π = -γ 0 -1 2 bπ γ 0 +1 γ 0 -1 ∇π , (1 + t) -θ (∂ t s + u • ∇s) = 0 , (3.16)
where b = b γ 0 -1 4γ 0 1 γ 0 -1 . We introduce furthermore a parameter θ to be determined so that (1 + t) -θ s has a decreasing in time similar to the estimates obtained in the isentropic case.

In order to obtain local existence of a solution, we construct a solution by following the same strategy as in the isentropical case and using once again a property of local uniqueness, given by Proposition 3.6.

3.2.B Local in time Uniqueness

We show here a similar result to the one obtained in Section 3.1.A in the isentropical case.

Proposition 3.6. Let U 0 = (π 0 , u 0 , s 0 ) T ∈ H m (R d ) and Ũ0 be two initial data for (3.1). Let U = (π, u, s) T , Ũ be the two corresponding solutions defined for 0 t T 0 . We

assume that D Ũ L ∞ ([0,T 0 ]×R d ) < ∞. Let x 0 ∈ R d and R 0. We denote M = sup{e s 2γ 0 cv ( γ 0 -1 2 |π|(1 + b|π| ν-1 ) + |u|)(t, x) , (t, x) ∈ [0, T 0 ] × B(x 0 , R)} , (3.17)
C T = {(t, x) ∈ [0, T ] × B(x 0 , R -M t)} for T ∈ [0, T 1 ] , (3.18)
where T 1 = min(T 0 , η/M ).

If U 0 = Ũ0 on B(x 0 , R) then U = Ũ on C T 1 .
Proof. Let U 0 = (π 0 , u 0 , s 0 ), Ũ0 = (π 0 , ũ0 , s0 ) be two initial data for (3.16) such that U 0 ∈ H m . Let U, Ũ be the two solutions of the associated Cauchy problem. We assume that these solutions are defined on [0, T 0 ] with T 0 > 0. Let furthermore

x 0 ∈ R d , η ∈ R * +
and M , C T be as in (3.17) and (3.18). Then we have

α 0 (U )∂ t U + j α j (U )∂ j U = 0 ,
where α 0 = Diag(e s γ 0 cv , 1, . . . , 1, 1) and, for all j ∈ {1, . . . , d},

α j (U ) =         e s γ 0 cv u j 0 e s γ 0 cv γ 0 -1 2 (π + bπ ν ) 0 u j e s γ 0 cv γ 0 -1 2 (π + bπ ν ) 0 . . . . . . 0 0 0 u j         .
We introduce, for T ∈ [0, T 1 ],

I = C T   ∂ t [(U -Ũ ) • α 0 (U )(U -Ũ )] + j ∂ j [(U -Ũ ) • α j (U )(U -Ũ )]   dx dt . (3.19) Denoting U -Ũ 2 0 = (U -Ũ )α 0 (U )(U -Ũ ) and U -Ũ t = B(x 0 ,η-M t) (U -Ũ )(t, x) 2 0
dx, we obtain from the Stokes formula

I = ∂C T   (U -Ũ ) • α 0 (U )(U -Ũ )n t + j (U -Ũ ) • α j (U )(U -Ũ )n j   dσ = U -Ũ T -U -Ũ 0 + 1 1 + 1/M 2 C   U -Ũ 0 + j (U -Ũ ) • α j (U )(U -Ũ ) x j M |x|   dσ .
But we have also

j (U -Ũ )α j (U )(U -Ũ ) x j M |x| = U -Ũ 2 0 u • x M |x| + 2e s γ 0 cv γ 0 -1 2 (π + bπ ν )(π -π) (u -ũ) • x M |x| U -Ũ 2 0 |u| M + e s/(2γ 0 cv) γ 0 -1 2 (|π| + b|π| ν )(e s/(γ 0 cv) |π -π| 2 + |u -ũ| 2 ) 1 M 1 M U -Ũ 2 0 (|u| + e s/(2γ 0 cv) γ 0 -1 2 (|π| + b|π| ν )) U -Ũ 2 0 , therefore I U -Ũ T -U -Ũ 0 .
Besides, we have

α 0 (U )∂ t (U -Ũ ) + j α j (U )(U -Ũ ) = j α 0 (U )(α 0 ( Ũ ) -1 α j ( Ũ ) -α 0 (U ) -1 α j (U ))∂ j Ũ , thus I = C T (U -Ũ )∂ t α 0 (U )(U -Ũ ) + j (U -Ũ )∂ j α j (U )(U -Ũ ) + 2 C T j α 0 (U )(α 0 ( Ũ ) -1 α j ( Ũ ) -α 0 (U ) -1 α j (U ))∂ j Ũ .
Let us denote

I 1 = C T (U -Ũ )∂ t α 0 (U )(U -Ũ ) , I 2 = j (U -Ũ )∂ j α j (U )(U -Ũ ) , I 3 = 2 C T j α 0 (U )(α 0 ( Ũ ) -1 α j ( Ũ ) -α 0 (U ) -1 α j (U ))∂ j Ũ .
We obtain by computing explicitly ∂ t α 0 and ∂ j α j

I 1 = - C T 1 γ 0 c v e s/(γ 0 cv) (π -π) 2 (u • ∇s)
and

I 2 = C T 1 γ 0 c v e s/(γ 0 cv) (π -π) 2 (u • ∇s) + C T div (u) U -Ũ 2 0 + C T (γ 0 -1)e s/(γ 0 cv) (π -π)(u -ũ) • 1 γ 0 c v (π + bπ ν )∇ s + (1 + bνπ ν-1 )∇π -I 1 + DU L ∞ (1 + C(1 + π L ∞ )(1 + bν π ν-1 L ∞ )) T 0 U -Ũ t dt .
Finally, we bound I 3 by computing

α 0 ( Ũ ) -1 α j ( Ũ ) -α 0 (U ) -1 α j (U ) = -    u j -ũj γ 0 -1 2 (π -π + b(π ν -πν ))e j T 0 Ψe j (u j -ũj )I d 0 0 0 u j -ũj    , where Ψ = γ 0 -1 2 (e s γ 0 cv (π + bπ ν ) -e s γ 0 cv (π + bπ ν )). We denote R = max( s L ∞ , s L ∞ ).
The exponential function being convex, we have:

e s/(γ 0 cv) -e s/(γ 0 cv) 1/(γ 0 c v )e R/(γ 0 cv) |s -s| .
So, e s/(γ 0 cv) (π + bπ ν ) -e s/(γ 0 cv) (π + bπ ν ) C(1 + R ν-1 )(|π||s -s| + |π -π|) , which gives us

I 3 D Ũ L ∞ (1 + C(1 + π L ∞ )(1 + bν π ν-1 L ∞ )) + DU L ∞ (1 + C(1 + π L ∞ )(1 + bν π ν-1 L ∞ )) T 0 U -Ũ t dt .
Finally, we obtain

U -Ũ T -U -Ũ 0 C(1 + C(1 + π L ∞ )(1 + bν π ν-1 L ∞ )) T 0 U -Ũ t dt .
We conclude thanks to Gronwall Lemma

1 2 B(x 0 ,R-M T ) U -Ũ 2 (T, x) dx 1 2 e C ′ T B(x 0 ,R) U 0 -Ũ0 2 (x) dx
where

C ′ = C DU L ∞ (C T ) (1 + π ν-1 L ∞ (C T ) ).

3.2.C Estimates

The system (3.16) can be written

A 0 ∂ t V + d j=1 A j ∂ j = -B(DV , V ) - C j (V )∂ j V - γ 0 -1 2 bπ ν    div (w + u) ∇π 0    , (3.20)
where V = (π, uū, s) ∈ R d+2 , V = (0, ū, 0) and, denoting (e j ) 1 j d the standard orthonormal basis of R d ,

A 0 = Diag(e s/(γ 0 cv) , 1, . . . , 1, (1 + t) -θ ) ∈ M d+2 (R) , C j = ūj A 0 , A j =        e s/(γ 0 cv) w j γ 0 -1 2 e s/(γ 0 cv) πe j T 0 γ 0 -1 2 e s/(γ 0 cv) πe j w j I d 0 0 0 (1 + t) -θ w j        , B =        γ 0 -1 2 e s/(γ 0 cv) π div ū (w • ∇)ū 0        . We also introduce N k (t) = R d D k V • A 0 (V ) D k V dx 1/2
and Z(t) = m k=0 (1 + t) g k N k (t) with g k = k + r -a and r = θ/2 -d/2, θ ∈]0, min(1, γ 0 -1

2 )]. In order to obtain energy estimates, we apply D k to (3.20) and we multiply it by D k V . Then, we integrate on R d . The additional term with respect to the Perfect Polytropic gases considered by M. Grassin [7] is now

F * (DV , DV , V ) = γ 0 -1 2 bπ ν div (w + u), ∇π, 0 ∈ R d+2 .
With the notations U = (π, w) ∈ R d+1 and F as in (3.9), the last component of F * being 0, we have

R d ∂ k V ∂ k (F * (DV , DV , V )) dx = R d ∂ k U ∂ k (F (DU , DU , U )) dx .
Using the estimates (3.12)-(3.13)-(3.14) we finally get an estimate on Y k . The definition of the norm is slightly changed with respect to isentropical case, however, for all

v = (v 1 . . . , v d+1 ) ∈ R d+1 , if v * = (v 1 , . . . , v d+1 , z) ∈ R d+2 , we have v 2 e s 0 L ∞ /(γ 0 cv)t v * A 0 v * .
Consequently Y k e s 0 L ∞ /(γ 0 cv) N k and the estimates on Y k obtained in the isentropical case give an estimate on N k in the general case.

Finally, we obtain, adding the estimate on F obtained in the isentropical case to the estimates from M. Grassin in the general case:

1 2 dN 2 k dt + k + r 1 + t N 2 k C(1 + t) β N 2 k Z + C ′ (1 + t) -g k -2 N k Z + C(1 + t) β+(β+1)(ν-1) N 2 k Z ν + C(1 + t) (β+1)(ν-1)-1 N 2 k Z ν-1 + C(1 + t) (β+1)(ν-1)-g k -2 N k Z ν + C ξ∈E k N k Z 2+ξ (1 + t) -g k +β+ξ(β+1+θ/2)
where β = -g 1 -d

2 and

E k = 0, k ∪ l k -1 ; l ∈ 1, k -1 .
Then, we choose a so that β = 0, i.e.

a = 1 + θ/2 > 1 with θ ∈]0, min(1, γ 0 -1 2 )] .
Next, we simplify by N k , we multiply by (1 + t) g k and we summate on k to obtain

dZ dt + a 1 + t Z C Z 2 + Z (1 + t) 2 + (1 + t) ν-1 Z ν+1 + Z 2+m (1 + t) am .
We denote now ζ(t) = (1+ t) a exp C 1+t Z(t) and we deduce from the inequality just above dζ dt

C (1 + t) a (ζ 2 + ζ ν+1 + ζ m+2 ).
We conclude in the same way we did in the isentropical case, replacing ν by ν * = max(ν, m+ 1) 2, since

ζ 2 + ζ ν+1 + ζ m+2 2(ζ 2 + ζ ν * +1 ).
4 Technical tools 4.1 Lemmas 3.4 and 3.5

We show here Lemma 3.4, which states, with the notations introduced in Section 3.1:

|I| C π ν-1 L ∞ DU L ∞ D k U 2 L 2 .
Proof of Lemma 3.4.

If k = 0, I = -b R d π ν (π div w + w • ∇w) dx.
By integration by parts, we obtain

I = - b ν ν + 1 R d π ν+1 div w dx C π ν-1 L ∞ π 2 L 2 div w L ∞ . If k 1, I is such that I = -b α∈N d ||α| 1 =k R d ∂ α π∂ α (π ν div w) + j ∂ α w j ∂ α (π ν ∂ j π) .
Expanding, we find

I = -b α∈N d ||α| 1 =k R d ∂ α (π)π ν ∂ α (div w) + j ∂ α (w j )π ν ∂ α (∂ j π) + Σ ,
where Σ is a sum of terms as

R d ∂ α 0 U ∂ α 1 (π ν )∂ α 2 U
, where U is any of its component and

|α 0 | 1 = k, |α 1 | 1 = l, |α 2 | 1 = k + 1 -l, with l ∈ 1
, k so that the derivatives are of order less than k. We treat first one of the terms of the preceding sum for a d-uplet α ∈ N d , of size k, that is to say |α| 1 = α 1 + . . . + α d = k. By integration by parts, we find

R d ∂ α (π)π ν ∂ α (div w) + j ∂ α (w j )π ν ∂ α (∂ j π) = R d π ν j ∂ j ∂ α (π)∂ α (w j ) = - j R d ∂ j (π ν )∂ α (π)∂ α (w j ) ,
Hence, I is a sum of terms as

∂ k U ∂ l (π ν )∂ k-l+1 U where 1 l k -1.
Note that we used here the notation after which ∂ k U means ∂ α U for a given α ∈ N d such that |α| 1 = α j = k. Furthermore, we write U for any of its component. Consequently, ∂ k U can mean ∂ α w j . We will use by now this notation.

If k 1 and l = 1, l = k, using the Hölder inequality and the Gagliardo-Nirenberg Lemma 4.1, we have:

R d ∂ k U ∂ l (π ν )∂ k-l+1 U dx ∂ k U L 2 ∂ l (π ν )∂ k-l+1 U L 2 ∂ k U L 2 ∂ l (π ν ) L 2 k-1 l-1 ∂ k-l+1 U L 2 k-1 k-l ∂ k U L 2 ∂(π ν ) 1-l-1 k-1 L ∞ ∂ k (π ν ) l-1 k-1 L 2 × ∂U 1-k-l k-1 L ∞ ∂ k U k-l k-1
L 2 . We use next Lemma 4.7 and the inequality

∂(π ν ) L ∞ = νπ ν-1 ∂π L ∞ C π ν-1 L ∞ ∂π L ∞ , to obtain R d ∂ k U ∂ l (π ν )∂ k-l+1 U dx C ∂ k U 2 L 2 ∂U L ∞ π ν-1 L ∞ .
If k 1 and l = k, we have to estimate

R d ∂ k U ∂ k (π ν )∂U ∂ k U L 2 ∂ k (π ν ) L 2 ∂U L ∞ ∂ k U L 2 π ν-1 L ∞ ∂ k π L 2 ∂U L ∞ .
If k 1 and l = 1, we have to estimate

R d ∂ k U ∂(π ν )∂ k U ∂ k U L 2 ∂ k (π ν ) L 2 ∂U L ∞ ∂ k U 2 L 2 ∂ 1 (π ν ) L ∞ . The inequality ∂ 1 (π ν ) L ∞ C π ν-1 L ∞ ∂ 1 π L ∞ allows us to conclude.
We prove now Lemma 3.5, which states, with the notations introduced in Section 3.1

J 2 C(1 + t) d k D k U L 2 Z ν ,
where

d k = (β + 1)(ν -1) -g k -2 and β = -g 1 -d 2 .
Proof of Lemma 3.5. For k = 0, J 2 = 0. We are considering here k

1 ; J 2 is then a sum of terms ∂ k U ∂ l (π ν )∂ k-l+1 u dx for 0 l k -1. The choice of initial conditions gives us U ∈ H m , but we do not know if D 2 U ∈ L ∞ . So we can distinguish two cases: m > 2 + d/2 and D 2 U ∈ L ∞ , or m 2 + d/2.
Case m > 2 + d/2. We study now different cases, after the values of k and l.

1. If k 1 and l = 0, we use Proposition 3.1 and Lemma 4.7 to obtain the estimate

R d π ν ∂ k π∂ k+1 u dx π ν L ∞ ∂ k U L 2 ∂ k+1 u L 2 C(1 + t) (β+1)ν Z ν D k U L 2 (1 + t) d/2-k-2 C(1 + t) d k D k U L 2 Z ν , since k -d/2 + 2 = β + 1 + g k + 2
we are done.

2. If k 2 and l = 1, we have to estimate

R d π ν-1 ∂π∂ k π∂ k u dx ∂ k π L 2 ∂ k u L 2 π ν-1 L ∞ ∂π L ∞ C(1 + t) d/2-k-1 (1 + t) (β+1)(ν-1)+β Z ν ∂ k U L 2 , which concludes that case, since d/2-k-1+β +(β +1)(ν -1) = (β +1)(ν -1)-g k -2. 3. If k = 3 and l = 2, R d ∂ 3 π∂ 2 (π ν )∂ 2 u dx ∂ k π L 2 ∂ 2 u L ∞ ∂ 2 (π ν ) L 2 (1 + t) -3+(β+1)(ν-1)-g 2 ∂ k U L 2 Z ν ,
and we are done as g 2 + 3 = g 3 + 2.

4. If k > 3 and l ∈ 2, k -1 , we use Lemma 4.2. Denoting q = 2 k-3 l-2 and q ′ = 2 k-3 k-l-1 so that 1/q + 1/q ′ = 1/2, we obtain:

R d ∂ k π∂ l (π ν )∂ k-l+1 u dx ∂ k U L 2 ∂ l (π ν ) L q ∂ k-l+1 u L q ′ ∂ k U L 2 D 2 (π ν ) 1-2/q L ∞ D k-1 (π ν ) 2/q L 2 D 2 u 1-2/q ′ L ∞ D k-1 u 2/q ′ L 2 . Since D 2 u L ∞ C(1 + t) -3 , D k-1 u L 2 C(1 + t) d/2-k , and
D 2 (π ν ) L ∞ C π ν-2 L ∞ Dπ 2 L ∞ + π ν-1 L ∞ D 2 π L ∞ C(1 + t) (β+1)ν-2 Z ν , D k-1 (π ν ) L 2 C π ν-1 L ∞ D k-1 π L 2 C(1 + t) (β+1)(ν-1)-g k +1 Z ν .
we obtain

J 2 C(1 + t) m k D k π L 2 Z ν , where m k = -3 1 - 2 q - 2 q (1 + β + g k ) + 2 q ((β + 1)ν -2) + 1 - 2 q ((β + 1)(ν -1) -g k + 1) = (β + 1)(ν -1) -g k -2 = d k .
Case m 2 + d/2. First, we note that the computations of the case k 1 and l = 0, k 2 and l = 1, k = 3 and l = 2 are similar. There remains to treat the case k > 3 and 2 l k -1. Since k m 2 + d/2 we have necessarily k 3, if d = 1, 2 or 3, and we are done. We assume now d 4. Let us denote h = 1 2 (k + 1 + d/2) > 2. Then we have h m and

0 < h -l d -1 2 , 1 2 h -(k + 1 -l) d -1 2 . We introduce h 1 = h -l, h 2 = h -(k -l + 1) and 1/q 1 = 1/2 -h 1 /d, 1/q 2 = 1/2 -h 2 /d.
Therefore 1/q 1 + 1/q 2 = 2, which allows to use Hölder inequality

J 2 (k, l) = R d ∂ k π∂ l (π ν )∂ k-l+1 u dx ∂ k π L 2 ∂ l (π ν ) L q 1 ∂ k-l+1 u L q 2 .
Next, we apply Lemma 4.3 to find

J 2 (k, l) C ∂ k π L 2 D n (π ν ) L 2 D h u L 2 .
Finally, we use Lemma 4.7 and Proposition 3.1

J 2 (k, l) C ∂ k U L 2 (1 + t) d/2
-h-1-g h +(β+1)(ν-1) Z ν . As d/2 -h -1 -g h + (β + 1)(ν -1) = (β + 1)(ν -1) -g k -2, we are done.

4.2

The Gagliardo-Nirenberg inequality and its consequences 4.2.A The Gagliardo-Nirenberg inequality Lemma 4.1 (Gagliardo-Nirenberg). (See [START_REF] Taylor | Partial differential equations[END_REF]Prop. 3.5,p. 4]) Let r > 0, i ∈ [0, r] and z ∈ (L ∞ ∩ H r )(R d ). Then ∂ i z ∈ L 2r/i (R d ) and

∂ i z L 2r/i C i,r z 1-i/r L ∞ D r z i/r L 2 .
We deduce easily from Lemma 4.1 the following result.

Lemma 4.2. Let z ∈ H m be such that D 2 z ∈ L ∞ , then for all k ∈ [4, m], for all i ∈ [2, k], we have D i z ∈ L q for q = 2 k-3 i-2 and

∂ i z L q C D 2 z 1-2/q L ∞ D k-1 z 2/q L 2 .
Thanks to the Sobolev imbedding (see [23, p.4]) and Lemma 4.1 we also prove Lemma 4.3. Let ℓ ∈]0, d/2[ and 1/q = 1/2 -ℓ/d. There exists C > 0 depending on ℓ, q, d such that for all z ∈ H ℓ (R d ) we have

z L q C D ℓ z L 2 .
Proof. The space H ℓ (R d ) is endowed with the norm • L 2 + D ℓ • L 2 . The Sobolev imbedding between H ℓ and L q can the be written, for a given C > 0,

z L q C( z L 2 + D ℓ z L 2 )
, for all z ∈ H ℓ . (4.1)

Let us define now, for z ∈ H ℓ , λ ∈ R + , the function z λ ∈ H ℓ such that z λ (x) = z(λx). Applying (4.1) to z λ , and noting that

z λ L q = λ -d q z L q , D ℓ z λ L 2 = λ ℓ-d 2 D ℓ z L 2 , (4.2)
we obtain

z L q Cλ d q -d 2 ( z L 2 + λ ℓ D ℓ z L 2 ) ,
where, by definition d q -d 2 = -ℓ. Consequently, introducing λ =

z L 2 D ℓ z L 2 1/ℓ
, we have

z L q 2C D ℓ z L 2 .
Similarly

U L ∞ C U 1-d 2m L 2 D m U d 2m L 2 .
But we also have

U L 2 (1 + t) -g 0 Z , D m U L 2 (1 + t) -gm Z ,
∂ α (f ϕ) L 2 C( f L ∞ D k ϕ L 2 + ϕ L ∞ D k f L 2 ) .
∂ α f ν L 2 C f ν-1 L ∞ D k f L 2 , (4.3)
where C > 0 is a constant independent from f , α, ν.

Proof. For ν ∈ N, ν 2, we proceed by iteration on ν, using Lemma 4.

∂ α (f ν ) L 2 C k j=1 f ν-j L ∞ j i=1 b i =k ∂ b i β i f L 2 k b i C k j=1 f ν-j L ∞ j i=1 b i =k f 1-b i /k L ∞ D k f b i /k L 2 C f ν-1 L ∞ D k f L 2 ,
using besides that for all j ∈ 1, m , f ν-j L ∞ f ν-j L ∞ since ν -j 0.

Figure 1 :

 1 Figure 1: Gluing of the solutions along ∂K.

so U L ∞ C( 1 2 .

 12 + t) s Z , where s = -g 0 (1 -d 2m ) -(g 0 + m) d 2m = -g 0 -d 2 = β + 1. 2. ApplyingLemma 4.4 to DU , with p = m -1, we have DU L ∞ C DU In the same way as before, we obtain DU L ∞ C(1 + t) s Z where s = -g 1 (1 -d 2(m -1) ) -(g 1 + m -Lemma 4.4 to D 2 U with p = m -2, which is possible since m -2 > d/2, we finally prove the third inequality. Lemma 4.6. Let f, ϕ ∈ H m ∩ L ∞ (R d ), let α ∈ N d such that |α| = k m. Then

Lemma 4. 7 .

 7 Let f ∈ (L ∞ ∩ H m )(R d ). If ν ∈ N and ν 2, or ν ∈ R and ν ∈ [m, +∞[, we have f ν ∈ H m (R d) and, for all α ∈ N d such that |α| = k m, we have

6 .

 6 For ν ∈ [m, +∞[, we have∂ α (f ν (x)) = 1 j k β 1 +...+β j =α |β i |=b i 1 c α,β f ν-j ∂ β 1 f . . . ∂ β j f .Then we take∂ β i f ∈ L 2 kb i , we apply Hölder inequality and Lemma 4.1 to obtain:

,

  Lemma 4.4. Let p > d/2. There exists C > 0 such that for all z ∈ H p (R d )z L ∞ C z Proof. We use now the continuous imbedding H p (R d ) ⊂ L ∞ (R d ). Thus, there exists C > 0 such that z L ∞ C( z L 2 + D p z L 2 ) ,for all z ∈ H p .Applying the inequality to z λ : x → z(λx), we obtain, sincez λ satisfies z λ L ∞ = z L ∞ and (4.2), z L ∞ Cλ -d 2 ( z L 2 + λ p D p z L 2 ) . Let m > 1 + d/2, U ∈ H m (R d ),r, a ∈ R and Z be the norm defined by (3.10) :

				1-d 2p L 2	D p z	d 2p L 2 .
	Taking λ =	z L 2 D p z L 2	1/p	, we have finished the proof.
	4.2.B Estimates
	Lemma 4.5. Z(t) =

m k=0 (1 + t) g k D k U (t) L 2 ,

with g k = k + r -a. Then we have:

1. U (t) L ∞ C(1 + t) β+1 Z(t), 2. DU (t) L ∞ C(1 + t) β Z(t), 3. If m > 2 + d/2, then D 2 U (t) L ∞ C(1 + t) β-1 Z(t), with β = -g 1 -d/2.

Proof. 1. Applying Lemma 4.4 to U , we obtain:

also called specific

Acknowledgement :

The author thanks Sylvie Benzoni-Gavage for initiating this work and for providing useful advises.


Amandine Michelas
email: michelas@lpl-aix.fr

Mariapaola D'imperio
email: mariapaola.dimperio@lpl-aix.fr

Mariapaola D'imperio

Accentual Phrase Boundaries and Lexical Access in French

Keywords: speech segmentation, lexical access, prosodic boundary, Phonological Phrase, Accentual Phrase, French

, in that a PP boundary might remove a temporary lexical ambiguity. In a set of two experiments, we show here that a temporary lexically ambiguous sequence can also be removed by the presence of an AP boundary. Specifically, reaction times for word monitoring were faster for ambiguous sequences when an AP boundary was present. These results suggest that tonal cues and other phonetic/phonological properties of the auditory stimuli have an impact on word recognition and must be considered for lexical access in French.

Introduction

For many recognition models, the word recognition process consists of resolving a competition between several potential lexical candidates. Words that are phonologically consistent with the acoustic signal of the input are activated (see [START_REF] Frauenfelder | Simulating the Time Course of Word Recognition: an Analysis of Lexical Competition in TRACE[END_REF] for a review), and partial input is often consistent with several word interpretations. For instance the sequence [vi] in French is temporary ambiguous as being word initial in either vipère /vipϯల/ ('viper') and virée /viలe/ ('drive').

However, the acoustic signal also contains cues deriving from the presence of potential prosodic boundaries. These boundaries can significantly alter the phonetic representation for the same lexical item. For instance, in French, the last syllable of a word, immediately preceding an accentual phrase boundary (AP), can be lengthened and appears to possess a great degree of prominence [START_REF] Jun | A Phonological Model of French Intonation[END_REF]. We know that listeners are capable of taking into account such cues in word segmentation. For example, Bagou et al. [START_REF] Bagou | Contribution of Prosody to the Segmentation and Storage of "Words" in the Acquisition of a New Mini-language[END_REF] showed that Swiss French listeners use lengthening of AP-final syllables and F0 rises to detect words in an artificial language. Spinelli et al. [START_REF] Spinelli | Fine-grained Access to Targets and Competitors in Phonemically Ambiguous Spoken Sequences : the Case of French Elision[END_REF], [START_REF] Spinelli | An Intonational Cue to Word Segmentation in Phonemically Identical Sequences[END_REF] showed that tonal cues can help listeners find word beginnings in the speech stream. Thus the activation of a lexical candidate also depends on how well the phonetic realization matches input representation. Although the influence of prosodic boundaries on lexical competition between candidates has been shown (see for example [START_REF] Salverada | The Role of Prosodic Boundaries in the Resolution of Lexical Embedding in Speech Comprehension[END_REF]), word recognition models refer only to the phonemic overlap between the input and the potential lexical candidates.

Moreover, the domain within which segmentation strategies operate across languages is still unclear. Christophe et al. [START_REF] Christophe | Phonological Phrase Boundaries Constrain Lexical Access I. Adult Data[END_REF] proposed that segmentation strategies operate within a domain smaller than the utterance, yet larger than the lexical word in French, i.e. the phonological phrase (PP). As Christophe et al. showed, a phonological phrase boundary (as in le chat grimpait /lђЀa/ PP /gలϯ pe/ PP 'the cranky cat climbed up') can remove a temporary lexical ambiguity with the word chagrin ('sadness') in French. This ambiguity would instead remain within a PP, even across prosodic word boundaries, as supported by reaction time data in a word monitoring task. According to Prosodic Phonology theories, such as that proposed by Selkirk [START_REF] Selkirk | The Interaction of Constraints on Prosodic Phrasing[END_REF], the right boundary of a phonological phrase can be aligned with the right edge of a major syntactic phrase boundary. A PP can have internal phonological properties, such as being the domain of sandhi phenomena. On the other hand, a tone unit such as the AP can either be isomorphic or non-isomorphic with a syntactically, edge-based PP grouping. In French [START_REF] Jun | The Accentual Phrase and the Prosodic Structure of French[END_REF], an AP is typically characterized by a final rise (LH*) and an optional initial rise (LHi). Generally, an AP has approximately the size of a PP, but AP (defined according to tonal cues) and PP (defined in syntactic terms) boundaries do not necessarily overlap. AP boundaries strictly depend on the number of final rises (LH*) actually produced by the speaker. For instance in the sentence Je crois que Marie t'a parlé des pins somptueux de cette forêt ('I think that Mary told you about this forest's sumptuous pine trees'), the noun phrase (NP) pins somptueux ('sumptuous pine trees') can be pronounced as either one or two APs according to several factors such as speech rate [START_REF] Fougeron | Rate Effects on French Intonation : Phonetic Realization and Prosodic Organisation[END_REF] or speaking style (see Figure 1). As mentioned above, tonal cues and other phonetic/phonological properties of the auditory stimuli do appear to have an impact on lexical access in French. Hence, we specifically predicted that the final F0 rise characterizing a LH* as well as the rhyme lengthening associated with an APfinal boundary in French would influence lexical access. We questioned whether PP and AP boundaries induce the same effect in removing temporary lexical ambiguity. Specifically we predicted that: 1. lexical access would be slower for items containing a local ambiguity at a prosodic word (PW) boundary than for items without lexical ambiguity; 2. as in Christophe et al. [START_REF] Christophe | Phonological Phrase Boundaries Constrain Lexical Access I. Adult Data[END_REF], PP boundaries would speed up the lexical decision task, and crucially that 3. AP boundaries would induce the same facilitatory effect.

Method

Corpus

In order to separately study the influence of PP and AP boundaries we employed potentially ambiguous sentences. We specifically predicted that, in a word monitoring task, participants will be slower in detecting the word pins ('pine trees') in items such as [START_REF] Jun | A Phonological Model of French Intonation[END_REF] than in items such as [START_REF] Christophe | Phonological Phrase Boundaries Constrain Lexical Access I. Adult Data[END_REF], because of the presence of a temporary lexical ambiguity (pin /pϯ /] can be temporarily ambiguous with the competitor pinsons ('finches') because of the following word somptueux /sѐ ptyø/.

(1) Je crois que Marie t'a parlé des pins somptueux de cette forêt. 'I think that Mary told you about this forest's sumptuous pine trees.' Ambiguous: competitor pinsons /pϯ sѐ/ (2) Je crois que Marie t'a parlé des pins luxuriants de cette forêt 'I think that Mary told you about the somptuous pine trees of this forest.' Non-ambiguous: no word starts in French with /pϯ ly/ We ran a set of two experiments using a similar experimental protocol to the one employed by Christophe et al. [START_REF] Christophe | Phonological Phrase Boundaries Constrain Lexical Access I. Adult Data[END_REF]. In Experiment 1, we focused on the temporary lexical ambiguity which occurs at two types of prosodic boundaries (i) at a PP boundary and (ii) at prosodic word boundary (PW). In Experiment 2, we studied the effect of lexical ambiguity associated with (i) a PW boundary vs. (ii) an AP boundary, neither of which overlapped with a phonological phrase (PP) boundary. The different prosodic conditions in each experiment are shown in Figure 2. For each experiment, 24 pairs of experimental sentences were constructed. In each pair one member contained a temporary lexical ambiguity. Target position within the sentence was kept constant. The sentences had a Subject Verb Object structure and were produced as all-focus utterances, all matched in number of syllables. The first part of the sentence was constructed in such a way that it did not semantically favor the activation of either the target word or its competitor (neutral preceding context). Ten native speakers of French judged the plausibility of the sentences on a scale from 0 (completely plausible) to 10 (highly plausible). Sentences containing competitors were all found to be plausible (mean rating: 6.2).

EXPERIMENT
All noun phrases (NP) were composed of preposition+(determiner)+ noun + adjective and were matched in number of syllables. In both experiments, the noun phrase of the Prosodic Word (PW) boundary condition was always produced as a single AP with a clear final rise (LH*) plus preboundary lengthening. In Experiment 1, PP boundary utterances always contained a LH* on the last syllable of the target word and another LH* associated to the last syllable of the following verb, so that PPs boundaries always overlapped with APs boundaries. In Experiment 2, within AP boundary utterances, the targets NPs were always produced as two APs within one PP so that PP boundaries never overlapped with AP boundaries.

The target word was always a monosyllabic word. The following word was necessarily different in the ambiguous and non-ambiguous conditions, although it was matched in number of syllables and frequency (ambiguous vs. non-ambiguous, for experiment 1, mean frequency: 8.3 vs. 7.6, t(23)<1; for experiment 7 vs. 7.7, t(23)<1). Frequencies were obtained from the database Lexique 2, [START_REF] New | Lexique 2: A New French Lexical Database[END_REF]. We also obtained the frequencies of words following the target in both experiments to ensure that the following context would not influence participants' decisions in the two conditions (ambiguous vs. non-ambiguous; for Experiment 1, following word mean frequencies: 8.3 vs. 7.6, t(23)<1; for Experiment 2, 7 vs. 7.7 t(23)<1). Finally we also computed diphone statistics in order to verify whether the diphone spanning the word boundary (e.g. /al/ in chat légendaire) was more likely to occur within a word or at a word boundary. For both experiments, we observed that, independently of the ambiguity, the diphones were more likely to occur at a word boundary than within a word (within a word vs. at a word boundary mean frequencies, for Experiment 1, 0.002 vs. 0.004 t(47)=-5.7, p<0.0001; for Experiment 2, t(47)=-13, p<0.0001). In both experiments 96 fillers were added to the 48 experimental sentences.

A 25-years old female, native speaker of French read all sentences 6 times at normal speech rate. We examined f 0 curves through Praat [START_REF] Boersma | Praat: doing phonetics by computer[END_REF] in order to select utterances which best corresponded to our prosodic conditions. In this way we obtained 24 natural speech sentences for the PW condition of both experiments and 24 natural speech sentences with a clear LH* associated with the last syllable of the target word for the PP and AP conditions of Experiments 1 and 2.

The selected experimental sentences were interspersed with fillers in 4 blocks. These 4 blocks were built with the following constraints (i) each target word appeared only once in each block (ii) each block contained sentences of each level for each condition (ambiguity with two levels: ambiguous and non-ambiguous and prosodic boundary with two levels: PW and PP for Experiment 1 and PW and AP for Experiment 2). Within each block, order of presentation of the sentences was random and different for each subject. The blocks were presented to participants following a Latin square design. This process was intended in order to neutralize a potential list effect. In this way, participants heard only 12 experimental sentences (3 for each level of each condition) and each target word was heard only once.

Procedure

40 native speakers of French took part in each experiment. In both experiments, the task was a word-monitoring task. Participants were tested individually in the sound proof room of the Laboratoire Parole et Langage (U. of Provence). Participants were seated in front of a computer. Items were presented over Sennheiser HD 212 Pro headphones at a comfortable listening level. Participants were instructed to click a button as soon as they heard the target word in the utterance. The target words were first visually presented and the sentences were played over the headphones at a comfortable sound level after one second while the computer screen was left blank. Before the experiments began, participants listened to a few sentences to test the material and the procedure. Reaction times where recorded relative to target word onset.

Results

Out of 48 utterances used for each experiment, 46 were retained. Two items were excluded because participants systematically answered before the beginning of the target word onset. Incorrect responses (no response or response before the beginning of the target word) were removed (for Experiment 1:1.52%; for Experiment 2: 2.61%). Because of the intrinsic variability in word duration, RTs were corrected by subtracting the duration of each target word from the RT for the word. For each subject and experiment, both RTs longer than 1200 ms and those greater than 2.5 standard deviations above the participants' overall response time relative to target word offset were removed from the latency analyses (for Experiment 1:5.15%; for Experiment 2:7.59%). Results in each condition for the two experiments are presented in Figures 3 and4. Mixed model analyses were conducted for both experiments with ambiguity (ambiguous/non-ambiguous) and prosodic condition (PW and PP for Experiment 1; PW and AP for Experiment 2) as fixed effects and participant and target word as random effects.

Mixed models showed that there is no significant effect of temporary lexical ambiguity (independent of prosodic condition) in neither experiment 1 nor 2 (for Experiment 1, t=-0.900, p=0.3687; for Experiment 2, t=-1.305, p=0.1926). However the reaction time analyses revealed a significant main effect of Prosodic boundary for both experiments (for experiment 1, t=-8.114, p<0.0001, effect size: 115ms; for experiment 2, t=-9.584, p<0.0001, effect size: 61ms). In experiment 1, participants responded earlier when a PP boundary was associated with the end of the target word, and the same effect was obtained for the AP boundary condition in experiment 2. In both experiments, the interaction between the two fixed effects was not significant (for experiment 1, t=1.374, p=0.172; for experiment 2, t=1.694, p=0.091). The data analyses showed that the percentage of responses given before the target word offset ("early responses") was preferentially matched with long target words and fit well with reaction times distribution. Thus early responses did not necessarily correspond to errors. The percentage of early responses for each experiment is shown in Figures 5 and6.

Discussion

In the two experiments presented here, the effect of a temporary ambiguity on lexical access was not observed, which runs counter to the results of [START_REF] Christophe | Phonological Phrase Boundaries Constrain Lexical Access I. Adult Data[END_REF]. Specifically participants did not respond more slowly to the target word in context where the local ambiguity occurred at a PW boundary. Different from Christophe et al.'s study, these results suggest that the acoustic/prosodic cues at PW boundary may be sufficient in removing lexical ambiguity. However the absence of ambiguity could be partly explained by the fact that our fillers did not contain possible competitors to the target words (i.e. words in which the target would be embedded, such as CHApeau /Ѐapo/ 'hat' if the target were CHAT /Ѐa/ 'cat') therefore ambiguity was only potential and not explicitly reinforced by the experimental design. The absence of such items could partly explain our fast reaction times (compared to [START_REF] Christophe | Phonological Phrase Boundaries Constrain Lexical Access I. Adult Data[END_REF]) and the absence of an ambiguity effect. Nevertheless, irrelevant of ambiguity, results show that participants respond earlier than in PW condition when the target word is followed by either a PP (Experiment 1) or an AP boundary (Experiment 2). These results suggest that participants exploited prosodic boundary cues in order to segment speech as already proposed in the literature for several languages (see [START_REF] Spinelli | Processing resyllabified Words in French[END_REF] for French). This is in line with Christophe et al's main findings suggesting that prosodic structure might influence lexical activation online. Crucially, AP boundaries appear to behave similarly to PP boundaries in speeding up lexical activation, independent of the presence of a syntactic constituent boundary. These results showed that the AP-final rise (LH*) plus preboundary lengthening associated with an AP-right boundary [START_REF] Jun | A Phonological Model of French Intonation[END_REF] can influence lexical decision.

These results have implications both for the linguistic definition of prosodic units and for models of lexical access. First, our results show that tonal and duration cues have to be taken into account in the definition of prosodic units, which are not entirely dependent on syntactic phrasing. Although the results of Christophe et al. [START_REF] Christophe | Phonological Phrase Boundaries Constrain Lexical Access I. Adult Data[END_REF] suggested that the PP (defined in syntactic terms according to prosodic phonology) constrains lexical access, our results indicate that the AP boundary (defined taking into account both prosodic cues and syntactic constrains) speeded participant responses in a cross-modal monitoring task. Analyses of percentage of responses given before the offset of the target word confirmed this AP boundary effect.

The comparison between the early responses obtained in Experiments 1 and 2 seemed to indicate that the effect of AP boundary is reinforced when the syntactic structure is aligned with the prosodic structure. Specifically we predicted that when an AP boundary is aligned with a major syntactic break, such as the boundary between an NP subject and a VP (as it was the case in PP condition of Experiment 1) the effect of the boundary would be reinforced. This hypothesis is in line with recent studies on the prosodic hierarchy in French, which support the existence of an intermediate level of phrasing between the AP and the Intonation Phrase (IP) [START_REF] Michelas | Durational Cues and Prosodic Phrasing in French: Evidence for the Intermediate Phrase[END_REF], [START_REF] D'imperio | Embedded Register Levels and Prosodic Phrasing in French[END_REF], i.e; an intermediate phrase (ip). In fact, an ip can occur within all focus utterances when the right edge of a prosodic boundary is aligned with the right edge of a major syntactic break [START_REF] D'imperio | Embedded Register Levels and Prosodic Phrasing in French[END_REF].

Second, our results also have an impact on models of lexical access. The domain within which segmentation strategies operate is still unclear. Christophe et al. have proposed that segmentation strategies might operate within the Phonological Phrase domain in French. Since we did not find an ambiguity effect, we cannot distinguish between domain hypotheses at this point. However, our results offer new insight about how and when prosodic information intervenes within the lexical access process. Two possible roles of prosodic information in the lexical process have been proposed: (i) prosodic cues are called upon to help in removing ambiguities when segmental cues are not sufficient to recognize a word (ii) prosodic cues are activated in parallel with the other cues (segmental/acoustic/semantic/syntactic etc…) during lexical activation process.

Our results, in line with Christophe et al. findings support the idea that prosodic boundaries are among the cues that contribute, in parallel with other cues, to the activation of lexical candidate since prosodic boundaries speed up participant responses even if there is no ambiguity in the utterance. While our results suggest that prosodic boundaries cues are computed at the same time as lexical activation and can influence it, additional studies are necessary to examine whether the prosodic information is computed simultaneously with a segmental analysis or if they are encoded in the lexical representations themselves.

Conclusion

In this study we have shown that AP boundaries can induce the same facilitation on lexical retrieval as PP boundaries. AP boundaries appear to speed participants' responses in a crossmodal word monitoring task. This suggests that both duration and tonal cues to prosody need to be controlled separately from syntactic structure in order to assess the role of phrasing in lexical access strategies. The results also support the hypothesis of an active role of fine phonetic detail in candidate activation mediated by rich lexical representations. Finally, since the PP boundary was accompanied by specific tonal and duration cues that might differ from mere AP boundary cues, the study indirectly supports the existence of the ip in French.

Figure 1 :

 1 Figure 1 : F0 curves of the noun phrase (NP) pins somptueux ('sumptuous pine trees') produced as either 1 APs (left) or 2 APs (right) excised from the utterance Je crois que Marie t'a parlé des pins somptueux de cette forêt.

Figure 3 :

 3 Figure 3: Reaction times for Experiment 1. Interaction between the two fixed factors (Ambiguity and Prosodic boundary). PW = prosodic word; PP = phonological phrase.

Figure 4 :

 4 Figure 4: Reaction times for Experiment 2. Interaction between the two fixed factors (Ambiguity and Prosodic boundary). PW = prosodic word; AP = accentual phrase.

Figure 5 :

 5 Figure 5: Percentage of early responses for experiment 1 in PW and PP condition.

Figure 6 :

 6 Figure 6: Percentage of early responses for experiment 2 in PW an AP condition.

1 PW boundary PP boundary Ambiguous

…[sur ce CHAT légendaire]PP… …[son beau CHAT]PP [léchait]PP… H* H* H* Non ambiguous …[sur ce chat fa*buleux]PP… …[son beau CHAT]PP [mor*dait]PP… H*

H* H* EXPERIMENT 2 PW boundary AP boundary within a PP Ambiguous

…[des PINS somptueux]PP… …[(des PINS)AP (somptueux)AP]PP… H* H* H* Non ambiguous …[des PINS lu*xuriants]PP… …[(des PINS)AP (lu*xuriants)]AP]PP… H* H* H* Figure 2: Sample of corpus utterances, with prosodic condition crossed with ambiguity.


Amandine Michelas
email: michelas@lpl-aix.fr

Mariapaola D'imperio
email: mariapaola.dimperio@lpl-aix.fr

Durational Cues and Prosodic Phrasing in French: Evidence for the Intermediate Phrase

Keywords: intermediate phrase, prosodic phrasing, prosodic boundary, degree of lengthening, speech rate, French

Studies addressing prosodic constituency in French generally agree on two levels of phrasing (accentual phrase, AP, and intonation phrase, IP), while the existence of an intermediate level of phrasing (intermediate phrase, ip) is still controversial. In this study we examine durational cues in a read speech corpus at normal and fast rates in which the target syllable was either adjacent to a prosodic boundary or word-internal. Additional evidence for the existence of an intermediate level of phrasing between the AP and the IP was found: the vicinity to an ip-boundary is signaled by durational cues that are stronger than the ones associated to an AP-boundary, yet this lengthening is weaker than the one found in the vicinity of an IP boundary.

Introduction

Authors generally agree on the existence of two prosodic units in French (though they are labeled differently according to the approach and author). A smaller unit called rhythmic unit [START_REF] Di Cristo | Intonation in French[END_REF], accentual phrase (AP, [START_REF] Jun | A Phonological Model of French Intonation[END_REF]), phonological phrase [START_REF] Post | Tonal and Phrasal Structures in French Intonation[END_REF], or prosodic word [START_REF] Martin | Une théorie syntaxique de l'accentuation en français[END_REF][START_REF] Vaissière | Rhythm, Accentuation and Final Lengthening in French[END_REF], which is characterized by an obligatory F0 rise on the last syllable of the phrase (LH* in the the autosegmental-metric model of Jun & Fougeron [START_REF] Jun | A Phonological Model of French Intonation[END_REF]) plus a small degree of preboundary lengthening [START_REF] Hirst | French Intonation: a Parametric Approach[END_REF][START_REF] Pasdeloup | Modèle de règles rythmiques du français appliqué à la synthèse de parole[END_REF]. This constituent is also characterized by the presence of an optional initial rise (Hi) on its first syllable. Authors also agree on the existence of a larger unit, i.e. the Intonation Phrase (IP, [START_REF] Jun | A Phonological Model of French Intonation[END_REF][START_REF] Post | Tonal and Phrasal Structures in French Intonation[END_REF][START_REF] Jun | The Realizations of the Accentual Phrase in French Intonation[END_REF], see also intonation unit [START_REF] Di Cristo | Vers une typologie des unités intonatives du français[END_REF][START_REF] Di Cristo | Vers une modélisation de l'accentuation du français : première partie[END_REF][START_REF] Di Cristo | Vers une modélisation de l'accentuation du français seconde partie[END_REF]). The intonation phrase (IP) is demarcated by a major intonation contour (marked by a phrase final boundary tone realized on the last syllable of the phrase) and a marked preboundary lengthening [START_REF] Pasdeloup | Modèle de règles rythmiques du français appliqué à la synthèse de parole[END_REF][START_REF] Di Cristo | Des indices prosodiques aux traits perceptuels: Application d'un modèle d'analyse prosodiques à l'étude du vocative en français[END_REF]. Thus IP-final syllables show a greater degree of lengthening than AP-final syllables [START_REF] Jun | A Phonological Model of French Intonation[END_REF][START_REF] Di Cristo | Des indices prosodiques aux traits perceptuels: Application d'un modèle d'analyse prosodiques à l'étude du vocative en français[END_REF][START_REF] Di Cristo | De la microprosodie à l'intonosyntaxe[END_REF]. Additionally, IP can be optionally followed by a pause. From a syntactic point of view, the IP is the domain of the root sentence corresponding to the grouping of a noun phrase (NP) and a verb phrase (VP) [START_REF] Selkirk | On Prosodic Structure and its Relation to Syntactic Structure[END_REF]. According to this definition specific syntactic constructions such as dislocations or parentheticals, as in [START_REF] Jun | A Phonological Model of French Intonation[END_REF] may form an autonomous IP [START_REF] Delais-Roussarie | Unités prosodiques et grammaires de l'intonation : vers une nouvelle approche[END_REF].

(1) Le mari d'Amanda réclamait sa bicyclette.

[ ]IP "Amanda's husband asked for his bicycle." (2) Le mari d'Amanda, d'après ce qu'on m'a dit, était marin.

[ ]IP [ ]IP[

]IP "As far as I was told, your husband was a sailor" Speech rate has also been shown to affect phrasing in a significant way [START_REF] Bartkova | Speaking Rate Modelisation in French : Application to Speech Synthesis[END_REF], [START_REF] Keller | A Statistical Timing Model for French[END_REF], [START_REF] Fougeron | Rate Effects on French Intonation: Phonetic Realization and Prosodic Organisation[END_REF]. Specifically, Fougeron and Jun [START_REF] Fougeron | Rate Effects on French Intonation: Phonetic Realization and Prosodic Organisation[END_REF] show that speakers reduce the number of phrases within an utterance at fast rate, by deleting or reducing the strength of prosodic boundaries. The study uncovered three specific examples of reduction at fast rate: the IP boundary can be reduced to an AP boundary, (2) two APs can be grouped at fast rate and (3) the AP initial Hi tone can be suppressed. .

An intermediate level of phrasing, the intermediate phrase (ip) or major phonological phrase (MAP, [START_REF] Selkirk | The Interaction of Constraints on Prosodic Phrasing[END_REF]) has been shown for several languages, including English [START_REF] Beckman | Intonational Structure in Japanese and English[END_REF], Italian [START_REF] D'imperio | Italian Intonation: an Overview and some Questions[END_REF][START_REF] D'imperio | Intonational Phrasing and Constituent Length in Romance[END_REF], Catalan [START_REF] Prieto | The Intonational Phonology of Catalan[END_REF], and Cairene Arabic [START_REF] Hellmuth | The Relationship between Prosodic Structure and Pitch Accent Distribution: evidence from Egyptian Arabic[END_REF]. Arguments for the existence of an intermediate level smaller than the IP but bigger than the AP in French have been previously proposed even if its status remains controversial. Di Cristo and Hirst [START_REF] Di Cristo | Vers une typologie des unités intonatives du français[END_REF] remarked the existence of an intermediate constituent (called an intonation phrase segment, or S.UI) which would be stronger than the rhythmic unit boundary yet smaller than an intonation unit boundary in specific prosodic structures (such as tag-questions, dislocations or postpositions). A similar domain (which can be found in tag or wh-questions, dislocations as well as vocatives, lists, or implicative contours) has been uncovered by Jun and Fougeron [START_REF] Jun | The Realizations of the Accentual Phrase in French Intonation[END_REF].

In this study we focused on the durational cues of prosodic constituents of different sizes at both normal and fast speech rates (for tonal phonetic and phonological cues, see [START_REF] D'imperio | Embedded Register Levels and Prosodic Phrasing in French[END_REF]). Specifically, we predicted that the prosodic cues associated with an ip right boundary would be stronger than the ones which are associated with an AP-right boundary and yet weaker than those associated with an IP-right boundary.

Method

Corpus

In this study we investigated the properties of SVO utterances in which the target syllable /na/ occurred in four different contexts: 1) within an AP, 2-at an AP boundary (ip-internal), 3) at a potential ip boundary (at the boundary between the subject NP and VP) and 4) at an IP boundary (i.e., before a parenthetical). The target vowel was always the low front unrounded vowel /a/, in the context of a CV syllable, in which the onset consonant was always voiced and could be either a nasal or a liquid, while the following onset consonant was always /d/. The 4 contexts are shown in table 1.

Target vowel /a/ 1-within a Prosodic Word Les grenadiers]AP de marrakesh]ip ne poussent pas bien vers chez nous.]IP

2-at an AP boundary

Le sauna]AP de Paolo]ip deviendra incontournable.]IP

3-at an ip boundary

Le sauna]ip deviendra incontournable.]IP

4-at an IP boundary

Le sauna]IP d'après ce qu'on m'a dit]IP n'est pas très loin.]IP 20 fillers items were added to the 20 experimental sentences. The order of presentation of the sentences was randomized separately for each subject.

Procedure

Two native speakers of French (both male: ages 28 and 27 years old) read the sentences aloud four times at both normal and fast speech rate for a total of 1600 experimental sentences (20 sentences x 4 repetitions x 2 speech rates x 2 speakers = 320). Speakers were instructed to read each sentence as naturally as possible, first at a normal rate and then at a fast speech rate. The sentences were visually presented on a computer screen. Participants were instructed to press the space bar on the keyboard in order to advance between items. Speakers were recorded onto digital audio tape (DAT), using a Shure SM10A head-mounted microphone in a soundattentuated roomat the Laboratoire Parole et Langage. Before the experiment, participants read six sentences at both normal and fast speech rate in order to test the material and the procedure. The resulting-sound files were segmented and each utterance was saved as a separate file. Spectrograms were created using Praat [START_REF] Boersma | Praat: Doing Phonetics by Computer[END_REF]. Target vowels, target syllables and target AP boundaries were labeled by inspecting both waveforms and spectrograms.

Measures

Both syllable and target vowel durations were measured. The total duration of the utterance was calculated in order to verify that the rate manipulation was significant [F(3.252)=345.2, p<0.001]. Rate was similarly calculated for each target AP in order to test whether the adjustments to speech rate affected the target AP and the overall utterance in the same way. Results of two factor ANOVA (rate and speaker) show that there is a strong effect of speech rate (see Figure 1 [START_REF] Welby | Anchored down in Anchorage: Syllable Structure and Segmental Anchoring in French[END_REF]. Each utterance was inspected auditorily, and the base count was adjusted on the actual pronunciation (the syllable base count of each sentence depended on the context condition: 15 syllables for the APinternal vowel condition, 10 syllables for the AP-final vowel condition, 13 syllables for the ip and IP-final vowel conditions. This syllable count was divided by the utterance duration (including pauses) to obtain a rate measurement for each utterance. These results confirmed that the two speakers successfully augmented their rate of speech in going from a normal to a fast speaking rate condition, for both the target AP and the overall sentence. The rate increase was approximately the same for the two speakers. Out of 320 utterances, 256 (16 sentences x 4 repetitions x 2 speech rates x 2 speakers = 256) were used for the analyses. The items containing the syllable /la/ were excluded because these items presented some segmentation difficulties (despite the liquid nature of the consonant there were not an abrupt change in intensity).

Hypothesis

We examined the effect of the four contexts on both vowel and syllable duration for the two speech rates. At normal speech rate, we predict that units higher in the prosodic hierarchy will undergo a greater degree of preboundary lengthening. If our hypothesis is correct, we will obtain a four-step vowel lengthening as shown in Figure 3. We also examined the effect of speech rate on vowel and syllable duration. Our hypothesis was that a fast speech rate would result in a modification of the prosodic organization of an utterance. Specifically we predicted that IP boundaries would be demoted to ip or IP boundaries.

Results

Two mixed models were separately performed for syllable duration and vowel duration. Both models included rate (normal/fast), prosodic boundary type (Unaccented, AP-final, ip-final, IP-final) and speaker (VA and FP) as fixed effects and the type of preceding onset consonant as random effect. These statistical models were intended to normalize the speech rate variability found both within and across speakers and allowed us to evaluate the effect of speech rate on prosodic phrasing (see [START_REF] Baayen | Analyzing Linguistic Data: a Pratical Introduction to Statictics[END_REF] for more information). Duration measurements were log-transformed in order to obtain a normal distribution. Vowel and syllable duration (in ms) by boundary type at both the normal and fast speech rate are shown for both speakers in Figures 4 and5.

Consistent with our predictions, the statistical analysis showed that at the normal speech rate, both vowel and syllable length increases with prosodic boundary strength. At the normal speech rate, AP-final vowels and syllables were significantly longer than unaccented vowels (for vowel lengthening: t=4.40, p<0.05, effect size: 19 ms; for syllable lengthening: t=2.15, p=0.05, effect size: 10ms) AP-final vowels and syllables were significantly longer than ip-final syllables (for vowel lengthening: t=2.64, p<0.05, effect size: 14ms; for syllable lengthening: t=2.4, p<0.05, effect size: 15ms) and ip-finals vowels and syllables were significantly longer than IP-final vowels (for vowel lengthening, t=3.83, p<0.05, effect size:24ms; for syllable lengthening, t=4.81, p<0.05, effect size:30ms). The large distribution of the duration values observed for the IP-final syllables boundary may be explained by the optional presence a silent pause after the IP boundary At the fast speech rate, vowel durations were much more similar, so that no significant effect of boundary type is found between the levels of IP, ip and AP. The mixed model analyses showed that only the IP-final vowels (t=2.47, p<0.05, effect size: 9ms) and ip-final vowels (t=2.87, p<0.05, effect size: 10.5ms) were significantly longer than unaccented vowels. For syllable lengthening we similarly observed a significant difference between ip-final syllables and unaccented vowels (t=4.75, p<0.05, effect size: 18ms) and between IP-final syllables and unaccented vowels (t=2.60, p<0.05, effect size: 10ms). Durational differences were not significant for the comparison between unaccented vowels and AP-final vowels (t=1.06, p=0.2895) nor for the comparison between AP-final syllables and unaccented syllables (t=1.57, p=0.118, size effect: 10ms).

Discussion

Consistent with the predictions of Jun and Fougeron [START_REF] Jun | A Phonological Model of French Intonation[END_REF] and Pasdeloup [START_REF] Pasdeloup | Modèle de règles rythmiques du français appliqué à la synthèse de parole[END_REF], our results showed that, at the normal speech rate, AP-final vowels were significantly longer than APinternal syllables, while they were significantly shorter than IP-final syllables. As Pasdeloup showed, the lengthening of AP-final vowels does not exceed 50% of mean AP-internal vowel duration (the mean for the two speakers was 71 ms for the normal speech rate and 84 ms for the fast rate). Vowel lengthening associated with an IP-boundary exceeded 50% of mean AP-internal vowel duration (the mean for IP-final vowels for the two speakers at the fast speech rate was 128 ms). Nevertheless, the results also seem to support the existence of an intermediate level of phrasing (ip) which would be higher than AP but lower than IP. Indeed at the normal speech rate, ip-final vowels were significantly longer than AP-final vowels while also significantly shorter than IPfinal syllables. This ip would not be restricted to marked constructions as it was originally proposed by Jun & Fougeron [START_REF] Jun | The Realizations of the Accentual Phrase in French Intonation[END_REF], since the syntactic construction employed here is unmarked. We propose that an alignment constraint between syntactic and prosodic structure conspires to place an ip boundary to the right edge of a major syntactic break (see [START_REF] D'imperio | Embedded Register Levels and Prosodic Phrasing in French[END_REF]). Specifically, we propose that an ip boundary can appear also in all-focus utterances and that its right boundary will be signaled through significant preboundary lengthening relative to the AP-final domain. Moreover, our results show that the degree of vowel lengthening associated with an ip-boundary is about 50% of the mean for AP-internal vowels.

Although we showed that preboundary lengthening associated with an ip-right boundary is significantly different from that associated with an AP-right or IP-right boundary, the question of the nature (discret/gradual) of the phonetic cues associated with the ip-boundary arises. Most studies addressing prosodic constituency have been conducted in the Prosodic Phonology framework, in which prosodic constituency is viewed as a hierarchy of domains. We know that length affects prosodic phrasing. If the preboundary lengthening associated with the ip-right boundary reflects phonological structure, it should not vary relative to the length of the ip. In a study to appear [START_REF] Michelas | Syntactic and Length Effects on French Prosodic Phrasing[END_REF], we manipulated the length of the ip in order to test the hypothesis that this manipulation would affect the "relative strength" [START_REF] Gee | Performance Structures: a Psycholinguistic and Linguistic Appraisal[END_REF] of the duration cues associated with the boundaries. The results showed that the phonetic and phonological cues associated with an ip break were independent of the length manipulation, and they generally support the existence of an intermediate level of phrasing in French.

At the fast speech rate, preboundary lengthening seems to differentiate only AP-internal vs. ip and the IP-final vowels. Despite the absence of significant lengthening of AP-final vowels at fast rate, this level nevertheless appears to be tonally marked by the presence of a LH* accent. Indeed in 75 out of 80 utterances in our AP-final condition, we observed a final LH* rise associated with the primary stressed syllable (/na/ in Figure 6). Since no significant vowel lengthening is observed here, it could be objected that the rise observed on /na/ is an instance of an initial LHi rise (and not a final LH* rise), which is an optional left edge marker of the AP [START_REF] Jun | A Phonological Model of French Intonation[END_REF] subject to conditions of rhythm, style or speaker (see also [START_REF] Welby | The Slaying of Lady Mondegreen, Being a Study of French Tonal Association and Alignment and their Role in Speech Segmentation[END_REF]). However two facts allow us to claim that the H tone we observed in our AP-final condition is a LH* pitch accent and not an instance of Hi. First, the H tone is often realized on a primary stressed syllable, which is a phrase-final full vowel (accent primaire) of a phrase and not on the initial stressed syllable of the first lexical word. Secondly, the L tone in the APs tends to be realized on the syllable preceding the H* marked syllable (as shown in Figure 6), and not at the onset of the same syllable, as it is usually the case for LHi.

Thus, in contrast with previous results [START_REF] Fougeron | Rate Effects on French Intonation: Phonetic Realization and Prosodic Organisation[END_REF], we conclude that in our corpus, fast speech rate did not induce complete AP-boundary. Moreover, we did not observe ip-boundary erasure, since a slight lengthening was always found for this level as well as a tonal marker of this level (i.e., a significant return to the phrase register line, cf. [START_REF] D'imperio | Embedded Register Levels and Prosodic Phrasing in French[END_REF]). Finally, the duration cues are less marked at the fast speech rate than at normal speech rate. These results seem to support the idea of mixed marking for prosodic boundaries, in that there appears to be a trade-off between lengthening cues and tonal (and possibly even spectral) cues in order to induce a perceptible phrase break. We plan to test our proposal through future perception experiments testing the reality of the phrasing levels addressed in the present analysis.

Conclusion

In this paper, we have shown that, in French, an ipboundary may occur in all-focus utterances showing a nonmarked syntactic structure. The right edge of an ip boundary appears to be marked by significant vowel and syllable lengthening which are both stronger than those observed for AP-final syllables and yet weaker than those observed for IPfinal syllables. Our results suggest that prosodic cues are reinforced when there is an alignment between prosodic and syntactic boundaries, and they support the existence of an intermediate prosodic level in French. Finally, our results show a tendency for prosodic organization to be modified at a fast speech rate, since duration cues can be weaker than at a normal speech rate, while being compensated for other cues, such as phrase accents and/or edge tones.

  and 2 [by utterances: F(3,252)=345.2, p<0.001; by target APs: F(3,252)=85.8, p<0.001)]. The effect of speaker was not significant [for sentences F(3,252)=3.8, p>0.001; for target APs F(3,252)=3.6528, p > 0.001], nor was the rate by speaker interaction [for sentences: F(3,252)=7,9, p>0,001; for target APs: F(3,252)=0,1 p>0.001]. Average speaking rate in syllables per second was calculated for each speaker according to the procedure used by Welby & Loevenbruck

Figure 1 :

 1 Figure 1: Speaking rates in syllable/s for entire utterances for both speakers..

Figure 2 :

 2 Figure 2: Speaking rates in syllable/s for target APs for both speakers.

Figure 3 :

 3 Figure 3: Expected target vowel duration (ms) as a function of boundary type at normal speech rate.

Figure 4 :

 4 Figure 4: Vowel duration (in ms) by boundary type for both speakers at fast and normal speech rates.

Figure 5 :

 5 Figure 5: Syllable duration (in ms) by boundary type for both speakers at fast and normal speech rates.

Figure 6 :

 6 Figure 6 : F0 curve for the utterance La villa de Simon ressemblait à un château. 'Simon's house looked like a castle' produced by one of our speakers at fast speech rate.

Table 1 :

 1 Example of corpus items.