Boris Haspot
email: haspot@ceremade.dauphine.fr

Global strong solution for the compressible Navier Stokes system with large rotationnally invariant initial data in dimension N ≥ 2

We show the existence of global strong solution for compressible Navier-Stokes equations with shallow water viscosity coefficients when the initial data are large and rotationally invariant. In dimension N = 2 we consider general initial data of the form (ρ 0 (|x|), u 0 (x)) with u 0 (x) = ∇θ(|x|) + ∇ ⊥ ψ(|x|) with θ and ψ sufficiently regular. In dimension N = 3 we deal with radially symmetric initial data (ρ 0 (|x|), ∇θ(|x|)).

Introduction

The conservation of mass and of momentum write:

     ∂ t ρ + div(ρu) = 0,
∂ t (ρu) + div(ρu ⊗ u) -div(2µ(ρ) Du) -∇(λ(ρ)divu) + ∇P(ρ) = 0, (ρ(0, •), u(0, •)) = (ρ 0 , u 0 ). (1.1) Here u = u(t, x) ∈ R N stands for the velocity field, ρ = ρ(t, x) ∈ R + is the density and D(u) = 1 2 (∇u + t ∇u) is the strain tensor. We denote by λ and µ the two viscosity coefficients of the fluid, which are assumed to satisfy µ > 0 and λ + 2µ > 0. Such a condition ensures ellipticity for the momentum equation and is satisfied in the physical cases where λ + 2µ N > 0. We supplement the problem with initial condition (ρ 0 , u 0 ). Throughout the paper, we assume that the space variable x ∈ R N . We restrict ourselves to the case N ≥ 2. In the sequel we shall only consider the viscous shallow-water system which corresponds to: µ(ρ) = µρ with µ > 0 and λ(ρ) = 0.

(1.

2)

It is motivated by the physical consideration that in the derivation of the NavierStokes equations from the Boltzmann equation through the ChapmanEnskog expansion to the second order (see [START_REF] Grad | Asymptotic theory of the Boltzmann equation II[END_REF]), the viscosity coefficient is a function of the temperature. If we consider the case of isentropic fluids, this dependence is reduced to the dependence on the density function.

Following the new formulation proposed in [START_REF] Haspot | New formulation of the compressible Navier-Stokes equations and parabolicity of the density[END_REF] (see also [START_REF] Bresch | Two-velocity hydrodynamics in fluid mechanics: Part II existence of global κ-entropy solutions to compressible Navier-Stokes systems with degenerate viscosities[END_REF][START_REF] Haspot | From the highly compressible Navier-Stokes equations to fast diffusion and porous media equations, existence of global weak solution for the quasi-solutions[END_REF][START_REF] Haspot | From the highly compressible Navier-Stokes equations to the Porous Media equation, rate of convergence[END_REF]), we can rewrite the previous system as follows setting v = u + 2µ∇ ln ρ:

∂ t ρ -2µ∆ρ + div(ρv) = 0, ρ∂ t v + ρu • ∇v -µdiv(ρcurlv) + ∇P (ρ) = 0.

(1.3)

Let us express now the energy of the system (when P (ρ) = aρ γ with a > 0 and γ > 1) and when the density is close from a constant state ρ > 0, multiplying the momentum equation by u we have:

E(ρ, u)(t) = R N 1 2 ρ(t, x)|u(t, x)| 2 + (Π(ρ)(t, x) -Π(ρ)) dx + t 0 R N 2µ ρ(t, x)|Du| 2 (t, x)dtdx ≤ R N ρ 0 (x)|u 0 (x)| 2 + Π(ρ 0 )(x) dx.
(1.4) with Π(ρ) defined as follows:

Π(s) = s s ρ P (z) z 2 dz -P (ρ) ρ , so that P (s) = sΠ (s) -Π(s), Π (ρ) = 0 and:

∂ t Π(ρ) + div(uΠ(ρ)) + P (ρ)divu = 0 in D ((0, T ) × R N ).
Notice that Π is convex in the case of the γ law. Multiplying now momentum equation of (1.3) by v when P (ρ) = aρ γ we obtain the so called BD entropy:

E 1 (ρ, v)(t) = R N 1 2 ρ(t, x)|v(t, x)| 2 + (Π(ρ)(t, x) -Π(ρ)) dx + t 0 R N µ ρ(t, x)|curlv| 2 (t, x)dtdx + 8µ γ t 0 R N |∇ρ γ 2 | 2 dxdt ≤ R N ρ 0 (x)|v 0 (x)| 2 + (Π(ρ 0 )(x) -Π(ρ) dx.
(1.5) In the sequel we will assume that ρ = 1 in order to simplify the notations. Combining (1.5) and (1.4) allows to prove at least heuristically (if in particular the solution of (1.1 is sufficiently regular) that for any T > 0 we have (see [START_REF] Lions | Mathematical Topics in Fluid Mechanics[END_REF] for the definition of the Orlicz space):

• ρ -1 ∈ L ∞ ((0, T ), L γ 2 (R N )), ∇ √ ρ ∈ L ∞ ((0, T ), L 2 (R N )), √ ρ -1 ∈ L ∞ ((0, T ), H 1 (R N )).
• ∇ρ γ 2 ∈ L 2 ((0, T ), L 2 (R N )).

• √ ρu ∈ L ∞ ((0, T ), L 2 (R N )), √ ρ∇u ∈ L 2 ((0, T ), L 2 (R N )).
In a first time we will consider the case of axisymmetric solution in dimension N = 2. In particular we assume that we can write (ρ, v) as follows:

v(t, x) = v 1 (t, |x|) x |x| + v 2 (t, |x|)
x ⊥ |x| and ρ(t, x) = ρ(t, |x|).

(1.6)

It implies that (ρ, v 1 , u 2 ) verify the following system (see the appendix for the details of the calculus):

           ∂ ∂t ρ -2µ + div(ρv 1 ) = 0, ρ∂ t v 1 + ρu 1 • ∇v + ∇P (ρ) = ρ u 2 2 |x| x |x| , ρ∂ t u 2 + ρu 1 • ∇u 2 + ρu 2 • ∇u 1 -2µdiv(ρDu 2 ) = 0.
(1.7)

We will also consider the case of radially symmetric initial data when N ≥ 3, it means:

u(t, x) = x |x| u 1 (t, |x|) and ρ(t, x) = ρ 1 (t, |x|), (1.8)
The system (1.3) can be written as follows since v is irrotational in this case:

∂ t ρ -2µ∆ρ + div(ρv = 0, ρ∂ t v + ρu • ∇v + ∇P (ρ) = 0.

(1.9)

The main goals of the present paper are to prove global existence and uniqueness of axisymmetric solutions to the systems (1.7) and (1.9) with non-negative initial densities and large initial data. There is huge literature on the studies about global existence and behaviors of solutions to (1.1) in the case that the viscosity coefficients µ and λ are both constants. The global existence of weak solution with large initial data of the system (1.1) has been proved for the first time by Lions in [START_REF] Lions | Mathematical Topics in Fluid Mechanics[END_REF] using the so called notion of renormalized solution combined with the introduction of the effective pressure which allows to test the compactness of the solutions. Feireisl et al in [START_REF] Feireisl | On the existence of globally defined weak solutions to the Navier-Stokes equations of isentropic compressible fluids[END_REF] have been generalized the previous results in the case of general pressure of the form P (ρ) = ρ γ with γ > N 2 (see also Plotnikov and Weigant in [START_REF] Plotnikov | Isothermal Navier-Stokes Equations and Radon Transform[END_REF] for the case N = 2, γ = 1). This last result have been generalized by Bresch and Jabin for anisotropic viscosity coefficient (see [START_REF] Bresch | Global Existence of Weak Solutions for Compresssible Navier-Stokes Equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor[END_REF]). We refer also to [START_REF] Danchin | Global existence in critical spaces for compressible Navier-Stokes equations[END_REF][START_REF] Danchin | A Lagrangian approach for the compressible Navier-Stokes equations. to appear[END_REF][START_REF] Mucha | The Cauchy problem for the compressible Navier-Stokes equation in the L p framework[END_REF][START_REF] Chen | Global well-posedness for the compressible Navier-Stokes equations with the highly oscillating initial velocity[END_REF][START_REF] Charve | A global existence result for the compressible Navier-Stokes equations in the critical L p framework[END_REF][START_REF] Haspot | Existence of global strong solutions in critical spaces for barotropic viscous fuids[END_REF][START_REF] Charve | Existence of strong solutions in a larger space for the shallowwater system[END_REF][START_REF] Mucha | Compressible NavierStokes system in 1-D[END_REF][START_REF] Danchin | The incompressible limit in L p type critical spaces[END_REF] for the existence of global strong solution with small initial data in critical spaces for the scaling of the equations. Let us focus now on the case of the system (1.1) with degenerate viscosity coefficients, we can first mention that the study is completely different from the constant viscosity case. Indeed the dependence of viscosity coefficients on the flow density makes difficult the analysis, in particular it is not clear how to obtain uniformly a-priori estimates for the velocity and trace the motion of particle paths near vacuum regions, which is the essential difference of (1.1) from compressible Navier-Stokes equations with constant viscosity coefficients where the particle path can be defined a-priori even near vacuum. Recently different authors have obtained significant progress on the existence of global weak solutions with degenerate viscosity coefficients. Bresch and Desjardins in [START_REF] Bresch | Existence of global weak solutions for a 2D Viscous shallow water equations and convergence to the quasi-geostrophic model[END_REF][START_REF] Bresch | On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids[END_REF][START_REF] Bresch | Existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids[END_REF] discovered a new entropy structure which ensures estimate on the gradient of the density. More precisely if µ and λ verifies the following algebraic relation:

λ(ρ) = 2µ (ρ) -2µ(ρ), (1.10)
then at least heuristically Bresch and Desjardins have observed that the quantity µ (ρ)

√ ρ ∇ρ is conserved in L ∞ T (L 2 (R N
)) norm for any T > 0. It enables us (see [START_REF] Bresch | Existence of global weak solutions for a 2D Viscous shallow water equations and convergence to the quasi-geostrophic model[END_REF][START_REF] Bresch | On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids[END_REF][START_REF] Bresch | Existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids[END_REF] and also [START_REF] Bresch | Two-velocity hydrodynamics in fluid mechanics: Part II existence of global κ-entropy solutions to compressible Navier-Stokes systems with degenerate viscosities[END_REF][START_REF] Mucha | Approximate solutions to model of twocomponent reactive flow[END_REF][START_REF] Zatorska | On the flow of chemically reacting gaseous mixture[END_REF]) to prove the existence of global weak solution with either a drag friction or a cold pressure term (a pressure that is singular at the vacuum). The fact to work with a friction term is crucial in the sense that it provides simply a gain of integrability on the velocity, combined with the BD entropy it allows to obtain the stability of the global weak solution and to construct global approximated solution which verify uniformly the energy and the BD entropy. The stability of the global weak solutions for a general γ law (P (ρ) = aρ γ with 1γ < +∞ when N = 2 and 1 < γ < 3 when N = 3) was proved by Mellet and Vasseur in [START_REF] Mellet | On the barotropic compressible Navier-Stokes equations[END_REF] using new energy estimate improving the integrability of the velocity. A particular choice of viscosity coefficients µ(ρ) = µρ, λ(ρ) = 0 satisfying (1.10) leads to the so-called shallow water system, for which the proof of existence of global weak solutions has been recently delivered independently by Vasseur and Yu [START_REF] Vasseur | Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations[END_REF][START_REF] Vasseur | Global weak solutions to compressible quantum Navier-Stokes equations with damping[END_REF] and Li and Xin in [START_REF] Li | Global Existence of Weak Solutions to the Barotropic Compressible Navier-Stokes Flows with Degenerate Viscosities[END_REF]. Concerning the existence of global strong solution with large initial data, there is a huge literature for the one dimension case. We refer to [START_REF] Ya | On a model system of equations of one-dimensional gas motion[END_REF][START_REF] Hoff | Global existence for 1D compressible, isentropic Navier-Stokes equations with large initial data[END_REF] for the case of constant viscosity coefficients. . In the case of degenerate viscosity coefficient of the form µ(ρ) = µρ α with α > 0, when 0 < α < 1 2 the BD entropy allows to bound the density from below. Indeed, in [START_REF] Mellet | Existence and Uniqueness of global strong solutions for onedimensional compressible Navier-Stokes equations[END_REF] Mellet and Vasseur proved the existence of global strong solution to (1.1) for initial density far away from the vacuum. The main ingredient of the proof was to use entropy from [START_REF] Bresch | Existence of global weak solutions for a 2D Viscous shallow water equations and convergence to the quasi-geostrophic model[END_REF] to estimate ∂ x (ρ α-1 2 ) in L ∞ (0, T ; L 2 (R)) for all T > 0. This implies boundedness of ρ -1 in L ∞ (0, T ; L ∞ (R)) which allows to show sufficient regularity of solution in order to prove the uniqueness. We have recently extended this result in [START_REF] Haspot | Existence of global strong solution for the compressible Navier-Stokes equations with degenerate viscosity coefficients in 1D[END_REF] to the general case. The main idea was to prove boundedness of v appearing in (1.3) by using the structure of the transport equation for v. It allows to bound ρ -1 by application of the maximum principle to the continuity equation. Concerning the existence of global strong solution in the multi-d case, there is very few works and the problem remains essentially open. However Kazhikhov and Vaigant proved the existence of global strong solution with large initial data in two dimension for the torus with particular viscosity coefficient of the form:

µ(ρ) = µ and λ(ρ) = λρ α , (1.11)
with λ, µ > 0 and α > 3. This result is remarkable and use strongly the structure of effective velocity (see [START_REF] Lions | Mathematical Topics in Fluid Mechanics[END_REF]) which consists to estimate (µ + λ(ρ))divu -P (ρ) + P (ρ) with ρ. Indeed roughly speaking it is possible for such viscosity coefficients by applying the operator (∆) -1 div to the momentum equation. An additional key point is the form of the viscosity coefficient λ(ρ) which enables for α > 3 to show a gain of integrability on ρ and more precisely ρ is bounded in L ∞ T (L p (T)) for any T > 0 and p ≥ 1. This result has slightly been improved by weakening the condition on α (see [START_REF] Haspot | Existence of global strong solution for the compressible Navier-Stokes system and the Korteweg system in two-dimension[END_REF][START_REF] Jiu | Global well-posedness of 2D compressible NavierStokes equations with large data and vacuum[END_REF][START_REF] Huang | Global well-posedness of classical solutions to the Cauchy problem of two-dimensional barotropic compressible Navier-Stokes system with vacuum and large initial data[END_REF][START_REF] Huang | Yun Global strong solution with vacuum to the two dimensional density-dependent Navier-Stokes system[END_REF]). Let us also mention [START_REF] Haspot | Global existence of strong solution for shallow water system with large initial data on the irrotational part[END_REF] where we prove the existence of global strong solution for irrotational data with a smallness condition which is supercritical. It allows in particular to exhibit initial data which are large for the energy estimate in dimension two such that they furnish global strong solution. In all its generality, the problem of the existence of global strong solution with large initial data for the shallow. It is then a natural and interesting problem to investigate the existence of global strong solution by adding symmetry condition on the initial data. It is well known that in dimension N = 3 (see [START_REF] Ladyzhenskaya | Unique solvability in large of a three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry[END_REF][START_REF] Ukhovskii | Axially symmetric flows of ideal and viscous fluids filling the whole space[END_REF]) there exists global strong solution for the incompressible Navier-Stokes equation and the incompressible Euler equation for axisymmetric initial data without swirl. There exists a large literature on the existence of global weak or strong solution with smallness assumption involving symmetry condition on the initial data, let us point out that systematically the authors consider initial data which are radially symmetric in dimension N ≥ 2. It yields that they study initial data such that u 2 = 0 in (1.7) in dimension N = 2. The first existence result was proved by Hoff [START_REF] Hoff | Spherically symmetric solutions of the NavierStokes equations for compressible, isothermal flow with large, discontinuous initial data[END_REF]; he proved the global existence of radially symmetric weak solutions to problem (1.1) with strictly positive initial densities in annular domains and with constant viscosity coefficients. Then it was extended by Jiang and Zhang in [START_REF] Jiang | On spherically symmetric solutions of the compressible isentropic NavierStokes equations[END_REF] to the Cauchy problem with general non-negative initial densities. Roughly speaking, they proved the global existence of radially symmetric weak solutions for general pressure P (ρ) = ρ γ with γ > 1 generalizing the results of Feireisl et al (see [START_REF] Feireisl | On the existence of globally defined weak solutions to the Navier-Stokes equations of isentropic compressible fluids[END_REF]). Choe and Kim in [START_REF] Choe | Global existence of the radially symmetric solutions of the NavierStokes equations for the isentropic compressible fluids[END_REF] proved the existence of global strong solution with large initial data for radially symmetric initial data when N ≥ 2 in an annular domain. Let us mention that it is important here to work in an annular domain since it avoids the difficulties related to the control of the vacuum around 0 which could blow-up. Indeed as emphasized in many related papers, the possible appearance of vacuum is one of the main difficulties, which indeed leads to the singular behaviors of solutions in the presence of vacuum, such as the failure of continuous dependence of weak solutions on initial data [START_REF] Hoff | The failure of continuous dependence on initial data for the NavierStokes equations of compressible flow[END_REF] and the finite time blow-up of smooth solutions [START_REF] Xin | Blow-up of smooth solution to the compressible NavierStokes equations with compact density[END_REF]. That is why, it is a natural and interesting problem to investigate the influence of the vacuum state on the existence and dynamics of global solutions to (1.1). One of the prototype problems which has been largely studied is the time-evolution of the compressible viscous flow of finite mass expanding into infinite vacuum. This corresponds to free boundary value problems for the compressible Navier-Stokes equations (1.1) for general initial data and variant boundary conditions imposed on the free surface. The study is a fundamental issue of fluid mechanics and has attracted lots of research interests, we refer to [START_REF] Chen | Global solutions to the NavierStokes equations for compressible heat-conducting flow with symmetry and free boundary[END_REF][START_REF] Ducomet | Viscous compressible barotropic symmetric flows with free boundary under general mass force. I. Uniform-in-time bounds and stabilization[END_REF][START_REF] Ducomet | Lyapunov functional method for 1D radiative and reactive viscous gas dynamics[END_REF][START_REF] Matusu-Necasov | Free boundary problem for the equation of spherically symmetric motion of viscous gas (II)(III)[END_REF][START_REF] Okada | Free boundary value problems for the equation of spherically symmetrical motion of viscous gas[END_REF][START_REF] Okada | Free boundary problem for the equation of one-dimensional motion of compressible gas with density-dependent viscosity[END_REF][START_REF] Straskraba | Global behavior of 1d-viscous compressible barotropic fluid with a free boundary and large data[END_REF][START_REF] Vong | Compressible NavierStokes equations with degenerate viscosity coefficient and vacuum (II)[END_REF][START_REF] Yang | Compressible NavierStokes equations with degenerate viscosity coefficient and vacuum[END_REF][START_REF] Zhang | Global behavior of compressible NavierStokes equations with a degenerate viscosity coefficient[END_REF][START_REF] Li | Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations[END_REF] and the reference therein. In this paper we are interested in proving the existence of global strong solution for the system (1.1) with large axisymmetric initial data verifying in dimension N = 2 the condition (1.6) and (??) in dimension N = 3. The purpose is triple, indeed in comparison with the previous works cited below we would like to avoid any condition of smallness on the initial data, to work in the complete euclidean space R N (and not only in an annular which allows to avoid all the difficulties related to some phenomena of blow-up of the density around the origin) and to take into account the rotational component (u 2 when N = 2, we refer to (1.6), ). Let us explain briefly now the strategy of the proof in the case N = 2 (the case N = 3 follows the same ideas up to small modifications). As we explained previously, one of the main difficulty in order to prove the existence of global strong solution consist in controlling the vacuum or in other word to estimate the L ∞ norm of1 ρ . If we look at the mass equation in (1.7), it seems natural to apply a maximum principle theorem since the density ρ verifies a parabolic equation as observed in [START_REF] Haspot | New formulation of the compressible Navier-Stokes equations and parabolicity of the density[END_REF]. We have then to estimate the effective velocity v 1 in L p ((0, T ), L q (R N )) space with (p, q) suitably chosen such that (see [START_REF] Ladyzhenskaya | Linear and quasilinear equations of parabolic type[END_REF]):

2 p + N q < 1. (1.12)
In order to obtain this type of information, we are going to estimate the L ∞ ((0, T ), L p (R N )) norm of ρ 1 p v 1 for any p ≥ 2 by multiplying the second equation of (1.7) by v 1 |v 1 | p-2 . The main difficulty consists then in evaluating the remainder term in the second equation of (1.7)

u 2 2 |x|
x |x| . Indeed this terms tends to blow-up in a neighborhood of the origin, in order to overcome this difficulty we are going to establish new estimates on u 2 by involving new weight estimate combining with some maximum principle arguments on u 1 in order to show that u 1 is essentially the sum of a positive function and of an L ∞ t,x function in space and time. It will enables us to show that

u 2 2 |x| x |x| is bounded in L ∞ ((0, T ), L p (R N )) for any p ≥ 2.
The last difficulty is to deal with the pressure term which can be rewritten as the sum of v 1 and -u 1 . It implies that the pressure produces a damping effect on the effective velocity v 1 provided that we estimate u 1 in L ∞ ((0, T ), L p (R N )) for any p ≥ 2 (it will be done by using a gain of integrability as in [START_REF] Mellet | On the barotropic compressible Navier-Stokes equations[END_REF]). Below we are now giving our main results on the global existence of strong solution with large initial data for axisymmetric initial data. In a first time let us recall a general 1 theorem of strong solution in finite time in critical Besov space for the scaling of the equations (see [START_REF] Danchin | A Lagrangian approach for the compressible Navier-Stokes equations. to appear[END_REF][START_REF] Haspot | Well-posedness in critical spaces for the system of compressible NavierStokes in larger spaces[END_REF][START_REF] Haspot | Cauchy problem for viscous shallow water equations with a term of capillarity[END_REF][START_REF] Charve | Existence of strong solutions in a larger space for the shallowwater system[END_REF]). We set in the sequel ρ = q + 1.

Theorem 1.1. Let N ≥ 2, the viscosity coefficients verify (1.2). Assume now that (q 0 , u 0 ) ∈ B N p p,1 × B N p -1 p,1
with 1 ≤ p < 2N and that there exists c such that 0 < c ≤ ρ 0 . Then there exists a time T such that system (1.1) has a unique solution on [0, T ] with:

q ∈ C T (B N p p,1 ), 1 ρ , ρ ∈ L ∞ T (L ∞ (R N )) and u ∈ C T (B N p -1 p,1 ) ∩ L 1 T (B N p +1 p,1 ). (1.13)
If in addition (q 0 , u 0 ) belongs to B s p,1 × B s-1 p,1 for any s > N p then we have:

q ∈ C T (B s p,1 ) and u ∈ C T (B s-1 p,1 ) ∩ L 1 T (B s+1 p,1 ). (1.14) Assume now that (1.1) has a solution (q, u) ∈ C([0, T * ), B N p p,1 × B N p -1 p,1
) on the time interval [0, T * ) which satisfies the following three conditions:

1. the function q belongs to L ∞ ([0, T * ], B N p p,1 ), 2. 1 ρ ∈ L ∞ ([0, T * ], L ∞ (R N )),

we have

T * 0 ∇u(s) L ∞ ds < +∞,
Then (q, u) may be continued beyond T * .

Remark 1. This theorem is a direct consequence of [START_REF] Haspot | Global existence of strong solution for shallow water system with large initial data on the irrotational part[END_REF][START_REF] Haspot | Cauchy problem for viscous shallow water equations with a term of capillarity[END_REF][START_REF] Danchin | Global existence in critical spaces for compressible Navier-Stokes equations[END_REF][START_REF] Danchin | A Lagrangian approach for the compressible Navier-Stokes equations. to appear[END_REF]. In [START_REF] Haspot | Global existence of strong solution for shallow water system with large initial data on the irrotational part[END_REF] the existence

is proved for (q 0 , u 0 ) ∈ B N p p,1 × B N p -1 p,1
for 1 ≤ p < 2N with uniqueness for 1 ≤ p ≤ N . The uniqueness is extended in [START_REF] Danchin | A Lagrangian approach for the compressible Navier-Stokes equations. to appear[END_REF] to the case N ≤ p < 2N . The blow up criteria is given in [START_REF] Haspot | Global existence of strong solution for shallow water system with large initial data on the irrotational part[END_REF] following an argument developed in [START_REF] Danchin | Global existence in critical spaces for compressible Navier-Stokes equations[END_REF].

Let us define now the maximal time T * of existence:

T * = sup{T ∈ R ; there exists a strong solution (q, u) of the system (1.1) on [0, T ] verifying (1.13)} Theorem 1.2. Let N = 2, P (ρ) = aρ γ with γ > 1, the viscosity coefficients veryfy (1.2) and assume that the initial data can be written under the form (1.6):

ρ 0 (x) = (ρ 1 ) 0 (|x|), u 0 = u 1 0 + u 2 0 = x |x| (u 1 ) 0 (|x|) + x ⊥ |x| (u 2 ) 0 (|x|).
In addition we assume that (q 0 , v 0 ) ∈ (B

N p p,1 ∩ B N p + p,1 ) × (B N p -1 p,1 ∩ B N p -1+ p,1
) with > 0 such that

N 1-< p < 2N and 0 < c ≤ ρ 0 . Furthermore we suppose that v 1 0 ∈ L ∞ , u 1 0 , u 2 0 ∈ L ∞ , u 2 0 |•| ∈ L ∞ and: E(ρ 0 , u 0 ) < +∞, E 1 (ρ 0 , v 0 ) < +∞,
then we have T * = +∞. Furthermore the global solution so defined (ρ, u) is unique and verifies locally in time (1.13).

Remark 2. As we explained previously there is essentially one result of global strong solution in dimension N = 2 for the compressible Navier-Stokes system which is due to Vaigant and Kazhikhov [START_REF] Vaigant | On existence of global solutions to the two-dimensional NavierStokes equations for a compressible viscosity fluid[END_REF] when the viscosity coefficients verify µ(ρ) = µ and λ(ρ) = ρ α with α > 3. The proof is based on the structure of effective velocity which allows such a choice of coefficients of viscosity (see [START_REF] Haspot | Existence of global strong solution for the compressible Navier-Stokes system and the Korteweg system in two-dimension[END_REF][START_REF] Jiu | Global well-posedness of 2D compressible NavierStokes equations with large data and vacuum[END_REF][START_REF] Huang | Global well-posedness of classical solutions to the Cauchy problem of two-dimensional barotropic compressible Navier-Stokes system with vacuum and large initial data[END_REF][START_REF] Huang | Yun Global strong solution with vacuum to the two dimensional density-dependent Navier-Stokes system[END_REF] for some refinements). However the problem remains actually completely open for the shallow water system (it means with the viscosity coefficients verifying (1.2) or more generally for a system verifying the BD condition (1.10).

To the best of our knowledge, it is the first generic result of global strong solution in dimension N = 2 for the shallow water system with large axisymmetric initial data in R 2 . Indeed the most of the results which deal with geometric initial data are restrict to the case u 2 0 = 0 (the rotational part) which corresponds to radially symmetric initial data and are focused on annular domain in order to avoid all the possible blow-up around the origin. We can obviously mention that the existence of global strong solution with general initial data (without any geometric assumption) remains open for the shallow water system.

To finish we conclude with our theorem on the system (1.9). Theorem 1.3. Let N = 3, P (ρ) = aρ γ with 1 ≤ γ < 7 6 and assume that the initial data can be written under the form (1.8):

u 0 (x) =
x |x| u 1 0 (|x|) and ρ 0 (x) = (ρ 1 ) 0 (r, z).

In addition we assume that (q 0 , v 0 ) ∈ (B

N p p,1 ∩ B N p + p,1 ) × (B N p -1 p,1 ∩ B N p -1+ p,1
) with > 0 such that N 1-< p < 2N and 0 < c ≤ ρ 0 . Furthermore we suppose that v 0 ∈ L ∞ , u 0 ∈ L ∞ and:

E(ρ 0 , u 0 ) < +∞, E 1 (ρ 0 , v 0 ) < +∞,
then we have T * = +∞. Furthermore the global solution so defined (ρ, u) is unique and verifies locally in time (1.13).

Remark 3. Let us mention that it would be easy to extend this result to the case of general dimension N ≥ 4, the only modification would concern the exponent γ which should be small enough. The case γ = 1 works (see [START_REF] Haspot | Global strong solution for the Korteweg system with quantum pressure in dimension N ≥ 2, to appear in Mathematische Annalen[END_REF]).

The paper unfolds as follows. In the next section, we introduce the homogeneous Besov spaces and the tools (the Littlewood-Paley decomposition and paradifferential calculus). In the section 3, we use the theorem 1.1 that we describe briefly to obtain the existence of strong solution in finite time. Furthermore we show that the solution remains axysymmetric. Section 4 is devoted to the proof of a gain of integrability on v 1 which allows us to apply a maximum principle in order to bound in L ∞ norm ρ and 1 ρ . The proof of the theorem 1.2 is carried over to the section 5. The proof os the theorem 1.3 is given in section 6. We postpone in appendix some tedious computations on the writing of the system (1.7) and (??).

Littlewood-Paley theory and Besov spaces

Throughout the paper, C stands for a constant whose exact meaning depends on the context. The notation A B means that A ≤ CB. This section is intended to briefly recall some classical results on homogeneous Besov spaces which are defined via the so-called Littlewood-Paley decomposition (for a recent account of the theory we refer to the excellent book [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF]). To build the Littlewood-Paley decomposition, one may fix a smooth radial function χ supported in (say) the ball B(0, 4 3 ), equal to 1 in a neighborhood of B(0, 3 4 ) and such that r → χ(r.e r ) is nondecreasing over R + . So that if we define ϕ(ξ) = χ(ξ/2) -χ(ξ), then ϕ is compactly supported in the annulus {ξ ∈ R d , 3 4 ≤ |ξ| ≤ 8 3 } and we have

∀ξ ∈ R d \ {0}, l∈Z ϕ(2 -l ξ) = 1. (2.15)
Then we can define the dyadic blocks

(∆ l ) l∈Z by ∆ l := ϕ(2 -l D) (that is ∆ l u = ϕ(2 -l ξ) u(ξ)) so that, formally, we have u = l ∆ l u. (2.16)
As (2.15) is satisfied for ξ = 0, the previous formal equality holds true for tempered distributions modulo polynomials. A way to avoid working modulo polynomials is to consider the set S h of tempered distributions u such that lim l→-∞ S l u L ∞ = 0, where S l stands for the low frequency cut-off defined by S l := χ(2 -l D). If u ∈ S h , (2.16) becomes true and one may write S l u = q≤l-1 ∆q u. One can now define the homogeneous Besov spaces used in this article:

Definition 2.1. For s ∈ R and 1 ≤ p, r ≤ ∞, we set

u B s p,r := l 2 rls ∆ l u r L p 1 r if r < ∞ and u B s p,∞ := sup l 2 ls ∆ l u L p .
We then define the space B s p,r as the subset of distributions u ∈ S h such that u B s p,r is finite.

Bernstein lemma (describing how derivatives act on spectrally localized functions, see [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF]) implies the following embedding result:

Proposition 2.1. For all s ∈ R, 1 ≤ p 1 ≤ p 2 ≤ ∞ and 1 ≤ r 1 ≤ r 2 ≤ ∞, the space B s p 1 ,r 1 is continuously embedded in the space B s-N ( 1 p 1 -1 p 2 ) p 2 ,r 2
. In addition we have:

B N p p,1 → B 0 ∞,1 → L ∞ and B 0 p,1 → L p → B 0 p,∞ .
In this paper, we shall mainly work with functions or distributions depending on both the time variable t and the space variable x. We shall denote by C(I; X) the set of continuous functions on I with values in X. For p ∈ [1, ∞], the notation L p (I; X) stands for the set of measurable functions on I with values in X such that t → f (t) X belongs to L p (I). In the case where I = [0, T ], the space L p ([0, T ]; X) (resp. C([0, T ]; X)) will also be denoted by L p T X (resp. C T X). Finally, if I = R + we shall alternately use the notation L p X. The Littlewood-Paley decomposition enables us to work with spectrally localized (hence smooth) functions rather than with rough objects. We naturally obtain bounds for each dyadic block in spaces of type L ρ T L p . Going from those type of bounds to estimates in L ρ T B s p,r requires to perform a summation in r (Z). When doing so however, we do not bound the L ρ T B s p,r norm for the time integration has been performed before the r summation. This leads to the following notation:

Definition 2.2. For T > 0, s ∈ R and 1 ≤ r, σ ≤ ∞, we set u L σ T (B s p,r ) := 2 js ∆ l u L σ T (L p ) r (Z) .
One can then define the space L σ T (B s p,r ) as the set of tempered distributions u over (0,

T )×R d such that lim l→-∞ S l u = 0 in L σ ([0, T ]; L ∞ (R d )) and u L σ T (B s p,r ) < ∞ 2 . The spaces L σ T (B s p,r
) may be compared with the spaces L σ T (B s p,r ) using the Minkowski inequality:

u L σ T (B s p,r ) ≤ u L σ T (B s p,r ) if r ≥ σ and u L σ T (B s p,r ) ≥ u L σ T (B s p,r ) if r ≤ σ.
All the properties of continuity for the product and composition which are true in Besov spaces remain true in the above spaces. The time exponent just behaves according to Hölder's inequality.

Formally, any product of two distributions u and v may be decomposed into

uv = T u v + T v u + R(u, v), (2.17)
T u v := l S l-1 u∆ l v, T v u := l S l-1 v∆ l u and R(u, v) := l |l -l|≤1 ∆ l u ∆ l v.
The above operator T is called "paraproduct" whereas R is called "remainder". The decomposition (2.17) has been introduced by Bony in [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF] (see also [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF]). In this article we will deal with the following classical paraproduct estimates (we refer to [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF]).

Proposition 2.2. Under the same assumptions there exists a constant C > 0 such that if 1/p 1 + 1/p 2 = 1/p, and 1/r 1 + 1/r 2 = 1/r:

T u v B s p,1 ≤ C u L ∞ v B s p,1 , T u v B s+t p,r ≤ C u B t p 1 ,r 1 v Ḃs p 2 ,r 2
(t < 0), R(u, v) B s 1 +s 2 -d 2 p,r ≤ C u B s 1 p 1 ,r 1 v B s 2 p 2 ,r 2 (s 1 + s 2 > 0).
(2.18)

Let us now turn to the composition estimates (see [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF]).

Proposition 2.3. Let s > 0, u ∈ B s p,1 ∩ L ∞ and F ∈ W [s]+2,∞ loc (R d ) such that F (0) = 0. Then F (u) ∈ B s
p,1 and there exists a function of one variable C 0 only depending on s, p, d and F such that

F (u) B s p,1 ≤ C 0 ( u L ∞ ) u B s p,1 .
A maximal regularity estimate for the heat equation

According to [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF], the following result holds:

Proposition 2.4. Let s ∈ R, (p, r) ∈ [1, +∞] 2 and 1 ≤ ρ 2 ≤ ρ 1 ≤ +∞. Assume that u 0 ∈ B s p,r and f ∈ L ρ 2 T (B s-2+2/ρ 2 p,r
). Let u be a solution of:

∂ t u -µ∆u = f, u /t=0 = u 0 ,
where µ > 0. Then there exists C > 0 depending only on N, µ, ρ 1 and ρ 2 such that:

u L ρ 1 T (B s+2/ρ 1 p,r ) ≤ C u 0 B s p,r + f L ρ 2 T (B s-2+2/ρ 2 p,r
) .

If in addition r is finite then u belongs to C([0, T ], B s p,r ).

Strong solution in finite time

In this section we are interested in proving the existence of strong solution on a maximal time interval (0, T * ) for initial data verifying the assumptions of the theorem 1.2. Furthermore we will verify that the solution remains axisymmetric on (0, T * ).

3.1 Existence of strong solution in finite time on a maximal time interval (0, T * )

The theorem 1.1 is a direct consequence of [START_REF] Haspot | Global existence of strong solution for shallow water system with large initial data on the irrotational part[END_REF][START_REF] Haspot | Cauchy problem for viscous shallow water equations with a term of capillarity[END_REF][START_REF] Danchin | Global existence in critical spaces for compressible Navier-Stokes equations[END_REF][START_REF] Danchin | A Lagrangian approach for the compressible Navier-Stokes equations. to appear[END_REF]. In [START_REF] Haspot | Global existence of strong solution for shallow water system with large initial data on the irrotational part[END_REF] the existence is proved for

(q 0 , u 0 ) ∈ B N p p,1 × B N p -1 p,1
for 1 ≤ p < 2N with uniqueness for 1 ≤ p ≤ N . The uniqueness is extended in [START_REF] Danchin | A Lagrangian approach for the compressible Navier-Stokes equations. to appear[END_REF] to the case N ≤ p < 2N . The blow up criteria is given in [START_REF] Haspot | Global existence of strong solution for shallow water system with large initial data on the irrotational part[END_REF] following an argument developed in [START_REF] Danchin | Global existence in critical spaces for compressible Navier-Stokes equations[END_REF]. We can simply recall that the condition p < 2N is a restriction due to the paraproduct law when we want to deal with the term of the form ∇ ln ρ • Du.

Existence of strong solution in finite time which remains axisymmetric

We have seen that there exists a strong solution (ρ, u) on a maximal time interval (0, T * ) of the system (1.1) for the initial data (ρ 0 , u 0 ) verifying the assumptions of the theorem 1.2. We are now interested in proving that this solutions remains radially symmetric in the case N = 3 or rotationally invariant in the case N = 2 on the time interval (0, T * ) provided that the initial data are radially symmetric or rotationally invariant. Let us start with the first case. In other way we wish to verify that (ρ, u) can be written under the following form:

ρ(t, x) = ρ 1 (t, |x|) and v(t, x) = x |x| v 1 (t, |x|). (3.19)
To see this, it suffices to check that any isometry

A ∈ M N (R N ) (whith A t A = AA t = Id N )
we have for all t ∈ (0, T * ):

ρ(t, x) = ρ(t, Ax) and v(t, x) = Av(t, A t x). (3.20)
Indeed (3.20) implies that for every t ∈ (0, T * ):

v(t, x) = xf (t, x), (3.21)
with f a scalar function. By contradiction assume that it is not true, then there exists t ∈ (0, T * ), y 1 and y 0 which are not collinear such that:

v(t, y 0 ) = λy 1 , with λ ∈ R * . (3.20
) is true for any A an isometry, let us now choose A 0 a rotation such that:

A 0 y 0 = y 1 |y 1 | |y 0 |.
Now we have for all the rotations A (we recall that the rotations are commutative), we have:

v(t, Ay 0 ) = λAy 1 = λ |y 1 |y 0 | AA 0 y 0 = λ |y 1 | |y 0 | A 0 Ay 0 .
We deduce then that:

v(t, x) = λ |y 1 | |y 0 | A 0 x.
We take now any isometry B and we deduce that:

Bv(t, x) = λ |y 1 | |y 0 | BA 0 x = λ |y 1 | |y 0 | A 0 Bx = v(t, Bx).
It implies that for any isometry B we have BA 0 = A 0 B which is false. Then by contradiction (3.21) is true. We now prove that f in (3.21) is necessary a radial function. From (3.20) we observe that for any isometry A:

H(t, x) = (v(t, x), x) = (Av(t, t Ax), x) = (v(t, t Ax), t Ax) = H(t, t Ax).
It implies that H is radial furthermore we have:

H(t, x) = |x| 2 f (x).
We deduce that f is radial and it concludes the fact that v remains radially symmetric.

Let us prove now (3.20), since ρ and v are regular on (0, T * ) and solution of (1.3) we can easily verify that:

     ∂ t (ρ(t, A t x)) -2µ∆(ρ(t, A t x)) + div ρ(t, A t x)Av(ρ(t, A t x) = 0 ρ(t, A t x)∂ t (Av(t, A t x)) + ρ(t, A t x)Au(t, A t x) • ∇(Av(t, A t x)) -µdiv(ρ(t, A t x)curl[Av(t, A t x)]) + ∇[P (ρ)(t, A t x)] = 0 (3.22)
It implies that (ρ(t, A t x), Av(t, A t x)) is a solution of (1.3) with initial data (ρ 0 (x), v 0 (x)) since ρ 0 is radial and v 0 radially symmetric. By uniqueness of the strong solution we deduce the result (3.20).

We prove similar result for the case of rotationally invariant initial data in dimension N = 2 (it means initial data verifying the condition 1.6). To do this it suffices to rewrite the system in function of divu and curlu and to observe that these functions are radial. More precisely for initial data verifying (1.6) we consider solution of the form: 4 Maximum principle on the density ρ and control of the vacuum

v(t, x) = t Av 1 (t, Ax) + A ⊥ v 2 (t,
In the sequel we are interested in dealing with radially symmetric solutions such that the initial data (ρ(0, •), u(0, •)) verify:

ρ 0 (x) = ρ 1 0 (|x|) and u 0 (x) = x |x| u 1 0 (|x|) + x ⊥ |x| u 2 0 (|x|).
It is then natural to search solution to the system (1.3) under the form:

ρ(t, x) = ρ 1 (t, |x|) et v(t, x) = x |x| v 1 (t, |x|) + x ⊥ |x| u 2 (t, |x|).
It implies that (ρ, v 1 , u 2 ) has to solve the following system:

             ∂ t ρ -2µ∆ρ + div(ρv 1 ) = 0 ρ∂ t v 1 + ρu 1 • ∇v 1 + ∇P (ρ) - u 2 2 |x| x |x| = 0 ρ∂ t u 2 + ρu 1 • ∇u 2 + ρu 2 • ∇u 1 -2µdiv(ρDu 2 ) = 0 (ρ, v 1 , u 2 ) t=0 = (ρ 1 0 , v 1 0 , u 2 0 ). (4.23)
We have seen in the previous section that for initial data verifying the assumption of the theorem 1.2 there exists strong solution (ρ, v 1 , u 2 ) on a finite time interval (0, T * ). We are interested now in proving that T * = +∞. By absurd we assume now that T * < +∞. In order to prove a contradiction, we are going to show that the norm (q, u)(T, •) X does not blow up when T goes to T * . In a first time we have to show that 1 ρ and ρ are bounded in L ∞ ([0, T * ], L ∞ (R N )). In order to show this result, we are going to prove a gain of integrability on v 1 which allows us to apply a maximum principle on the first equation of (4.23).

Estimate on u 1

We recall that P (ρ) = aρ γ in addition since u 1 is irrotational we have Du 1 = ∇u 1 , it implies that on (0, T * ) we have:

ρ∂ t u 1 + ρu 1 • ∇u 1 -2µdiv(ρ∇u 1 ) + ∇P (ρ) = ρu 2 2 |x| x |x| . (4.24)
It gives in particular since 1 ρ is bounded in L ∞ ((0, T * ) × R N ):

∂ t u 1 + (u 1 -2µ∇ ln ρ) • ∇u 1 -2µ∆u 1 + ∇F (ρ) = u 2 2 |x| x |x| . (4.25)
We now define by S(t) the semi-group associated to parabolic the equation:

∂ t u 1 + w • ∇u 1 -2µ∆u 1 = 0,
with w = u 1 -µ∇ ln ρ. From Duhamel formula, it yields:

u 1 (t, •) = S(t)u 1 0 - t 0 S(t -s)∇F (ρ)(s, •)ds + t 0 S(t -s)( u 2 2 (s, •) | • | • | • | )ds.
Let us estimate now w 1 the solution of the problem:

∂ t w 1 + w • ∇w 1 -2µ∆w 1 + ∇F (ρ) = 0 w 1 (0, •) = u 1 0 , (4.26)
then we show that for any p ≥ 2, ρ

1 p w 1 is uniformly bounded in L ∞ T (L p (R N )) which implies that w 1 belongs to L ∞ T (L ∞ (R N )
). Indeed we observe that:

ρ∂ t w 1 + ρu 1 • ∇w 1 -2µdiv(ρ∇w 1 ) + ∇P (ρ) = 0 w 1 (0, •) = u 1 0 , (4.27)
Multiplying the previous equation by w 1 |w 1 | p-2 with p ≥ 2 and integrating over (0, t) × R N with 0 < t < T * we obtain:

1 p R N (ρ|w 1 | p )(t, x)dx + t 0 R N ρ|w 1 | p-2 |∇w 1 | 2 (s, x)dsdx + (p -2) 4 t 0 R N ρ|∇(|w 1 | 2 )| 2 |w 1 | p-4 (s, x)dsdx ≤ 1 p R N (ρ 0 |u 1 0 | p )(x)dx + | t 0 R N |w 1 | p-2 w 1 • ∇aρ γ (s, x)dsdx|.
(4.28)

Following the same procedure than in [START_REF] Haspot | Global strong solution for the Korteweg system with quantum pressure in dimension N ≥ 2, to appear in Mathematische Annalen[END_REF] we prove that exists a continuous function C such that for any 0 < T < T * and 2 ≤ p < +∞ we have:

ρ 1 p w 1 L ∞ T (L p (R N )) ≤ C(T ). (4.29)
In addition we can observe that w 1 is bounded via the maximum principle in

C([0, T ], L ∞ (R N )) when 0 ≤ T < T * (with eventually w 1 L ∞ ([0,T * ],L ∞ (R N )) = +∞).
We have now for any t ∈ [0, T ] and > 0:

{x, |w(t,w)|≥ w(t,•) L ∞ -} ρ(t, x)|w(t, x)| p dx 1 p ≥ ( w(t, •) L ∞ -) min x∈R N |ρ(t, x)| λ({x, |w(t, w)| ≥ w(t, •) L ∞ -}) 1 p ,
with λ the Lebesgue measure. We deduce that since min

x∈R N |ρ(t, x)| λ({x, |w(t, w)| ≥ w(t, •) L ∞ -}) = 0, we have then for any > 0: lim inf p→+∞ ρ 1 p w 1 L ∞ T (L p (R N )) ≥ ( w(t, •) L ∞ -).
Using (4.29), we deduce that for any T ∈ (0, T * ) we have:

w 1 L ∞ ([0,T ],L ∞ (R N )) ≤ C(T ). ( 4

.30)

We are now interested in proving that:

x |x| • t 0 S(t -s)( u 2 2 (s, •) | • | • | • | )ds ≥ 0.
To do this it suffices to study the solution w 2 (t, x) = x |x| w 2 (t, |x|) of the linear equation:

   ∂ t w 2 + w • ∇w 2 -2µ∆w 2 = u 2 2 |x| x |x| w 2 (0, •) = 0. (4.31) Simple calculus gives:                      (w • ∇w 2 ) • x |x| = w • ∇(w 2 • x |x| ) ∆w 2 = x |x| (3∂ r ( w 2 r ) + |x|∂ rr ( w 2 r )) ∆w 2 • x |x| = ∂ rr w 2 + ∂ r w 2 r - w 2 r 2 ∆w 2 = ∂ rr w 2 + ∂ r w 2 r .
It means in particular that:

   ∂ t w 2 + w • ∇w 2 -2µ∆w 2 + 2µ w 2 |x| 2 = u 2 2 |x| . w 2 (0, •) = 0. (4.32)
Let us consider now the unknown w k 2 (t, x) = max 0, max x∈R N (-w 2 (t, x)) . Multiplying (4.32) by w k 2 and integrating over (0, t) × R N for t ∈ (0, T * ) we obtain:

1 2 w k 2 (t, •) 2 L 2 + 2µ t 0 R N |∇w k 2 |(s, x)dsdx ≤ - t 0 R N w i ∂ i w k 2 (s, x)ds dx. (4.33)
The theorem 1.1 ensures that on [0, t] we have w

∈ L 2r ([0, t], L 2q (R N )) with 1 r + N 2q = 1 -κ with 0 < κ1 and q ∈ [ N 2(1-κ) , +∞], r ∈ [ 1 1-κ , +∞]
. By Höder inequality, Young inequality and Gagliardo Niremberg inequalities, we show that:

| t 0 R N w i ∂ i w k 2 (s, x)ds dx| ≤ t 0 R N 2 |∇w k 2 |(s, x) + 1 2 N i=1 w i (s, x)(w k 2 ) 2 (s, x) ds dx ≤ 2 t 0 R N |∇w k 2 |(s, x)dsdx + C 2 N i=1 w 2 i L r ((0,t),L q (R N )) ( w k 2 (t, •) 2 L 2 + t 0 R N |∇w k 2 |(s, x)dsdx).
(4.34) Plugging (4.34) in (4.33), we obtain for small enough, there exists C > 0 such that:

w k 2 (t, •) 2 L 2 + t 0 R N |∇w k 2 |(s, x)dsdx ≤ C N i=1 w 2 i L r ((0,t),L q (R N )) w k 2 (t, •) 2 L 2 + t 0 R N |∇w k 2 |(s, x)dsdx .
(4.35) Taking t sufficiently small we obtain that:

w k 2 (t, •) 2 L 2 + t 0 R N |∇w k 2 |(s, x)dsdx ≤ 0. (4.36)
It implies that w 2 (s, x) ≥ 0 for (s, x) ∈ (0, t) × R N . Repeating the procedure we show that:

w 2 (t, x) ≥ 0 on (0, T * ) × R N . (4.37)
Combining (4.30) and (4.37), we deduce that for all t ∈ (0, T * ) we have:

u 1 (t, •) = f 1 (t, •) + f 2 (t, •), (4.38) with f 1 ∈ L ∞ ([0, T * ], L ∞ (R N )) and f 2 ≥ 0.

Estimate on v 1

We can rewrite the equation on v 1 as follows:

ρ∂ t v 1 + ρu 1 • ∇v 1 + a 2µ ρ γ v 1 - ρu 2 2 |x| x |x| = a 2µ ρ γ u 1 . (4.39)
It gives that:

∂ t v 1 + u 1 • ∇v 1 + a 2µ ρ γ-1 v 1 = ( a 2µ ρ γ-1 u 1 + u 2 2 |x| ) x |x| . (4.40)
Applying the characteristic method, we now observe that:

∂ t [v 1 (t, χ(t, x))] + a 2µ ρ γ-1 (t, χ(t, x))v 1 (t, χ(t, x)) = ( a 2µ ρ γ-1 (t, χ(t, x))u 1 (t, χ(t, x)) + ρu 2 2 (t, χ(t, x)) |(t, χ(t, x))| ) χ(t, x)) |χ(t, x)| , (4.41)
with:

∂ t χ(t, x) = u 1 (t, χ(t, x)).
We have now:

∂ t [v 1 (t, χ(t, x))] = ∂ t [v 1 (t, χ(t, x))] χ(t, x) |χ(t, x)| + v 1 (t, χ(t, x))[ ∂ t χ(t, x) |χ(t, x)| + χ(t, x)∂ t ( 1 |χ(t, x)| )]
Next we have:

∂ t ( 1 |χ(t, x)| ) = - 1 2 1 |χ(t, x)| 3 (2χ 1 (t, x)∂ t χ 1 (t, x) + 2χ 2 (t, x)∂ t χ 2 (t, x)) = - 1 2 1 |χ(t, x)| 3 2u 1 (t, χ(t, x))|χ(t, x)| = - 1 |χ(t, x)| 2 u 1 (t, χ(t, x)).
It implies that:

∂ t [v 1 (t, χ(t, x))] = ∂ t [v 1 (t, χ(t, x))] χ(t, x) |χ(t, x)|
Finally we have obtained that:

∂ t [v 1 (t, χ(t, x))] + a 2µ ρ γ-1 (t, χ(t, x))v 1 (t, χ(t, x)) = ( a 2µ ρ γ (t, χ(t, x))u 1 (t, χ(t, x)) + ρu 2 2 (t, χ(t, x)) |(t, χ(t, x))|
).

(4.42)

It implies that according to (4.38) and since (v 1 ) 0 belongs to L ∞ (R N ) that for any (t, x) we have:

v 1 = g 1 + g 2 , (4.43) with g 1 ∈ L ∞ ([0, T * ], L ∞ (R N
)) and g 2 (t, x) ≥ 0 for any (t, x) ∈ (0, T * ) × R N .

Energy estimate

We recall now that we have:

       ρ∂ t v 1 + ρu 1 • ∇v 1 + ∇P (ρ) + ρu 2 2 |x| x |x| = 0 ρ∂ t u 2 + ρu 1 • ∇u 2 -µdiv(ρ∇u 2 ) + ρu 1 2|x| u 2 + ρv 1 2|x| u 2 = 0
We multiply the first equation by v 1 , and the second by 2u 2 and we obtain:

1 2 R N ∂ t (ρ|v 1 | 2 + ρ|u 2 | 2 )dx + R N ∇P (ρ) • v 1 dx + 2µ R N ρ|∇u 2 | 2 dx + ρ u 1 |x| |u 2 | 2 dx = 0.
We can check now that (w = curlu

2 = ∂ 1 u 2 2 -∂ 2 u 2 1 )
:

|∇u 2 | 2 = (∂ r u 2 ) 2 + (u 2 ) 2 |x| 2 , w = u 2 |x| + ∂ r u 2 and divu = u 1 |x| + ∂ r u 1 .
We have finally:

1 2 R N ∂ t (ρ|v 1 | 2 + ρ|u 2 | 2 + j γ (ρ))dx + 2µ R N ∇P (ρ) • ∇ ln ρdx + 2µ R N ρ(∂ r u 2 ) 2 dx + 2µ R N ρ (u 2 ) 2 |x| 2 dx + R N ρ u 1 |x| |u 2 | 2 = 0.

Weight energy estimate on u 2

We recall that:

ρ∂ t u 2 + ρu 1 • ∇u 2 -µdiv(ρ∇u 2 ) + ρ(u 1 + µ∂ r ln ρ) |x| u 2 = 0. (4.44)
Let f a radial function that we will determine later, we verify that:

f (|x|)div(ρ∇u 2 ) = div(ρ∇(f u 2 )) -ρu 2 (∆f + f ∂ r ln ρ) -2ρ∇ ln f • ∇(f u 2 ) + 2ρf |∇ ln f | 2 u 2 .
And:

f ρu 1 • ∇u 2 = ρu 1 • ∇(f u 2 ) -ρu 2 u 1 • ∇f.
We can then rewrite (4.44) as follows:

ρ∂ t (f u 2 ) + ρ(u 1 + 2µ∇ ln f ) • ∇(f u 2 ) -µdiv(ρ∇(f u 2 )) + ρ(f u 1 + µf ∂ r ln ρ) |x| u 2 + µρu 2 (µ∆f + f ∂ r ln ρ -2f |∇ ln f | 2 ) -ρu 2 u 1 • ∇f = 0.
It gives in particular:

ρ∂ t (f u 2 ) + ρ(u 1 + 2µ∇ ln f ) • ∇(f u 2 ) -µdiv(ρ∇(f u 2 )) + µρu 2 (∆f -2f |∇ ln f | 2 ) + ρu 2 (u 1 ( f |x| -f ) + µ∂ r ln ρ( f |x| + f )) = 0 (4.45)
Multiplying the previous equation by f u 2 we get:

1 2 R N ∂ t (ρ|f u 2 | 2 )dx -µ R N ρ(∆ ln f + ∂ r ln ρ f f )|f u 2 | 2 dx + µ R N ρ|∇(f u 2 )| 2 dx + µ R N ρf |u 2 | 2 (∆f -2 (f ) 2 f )dx + R N ρ|f u 2 | 2 (u 1 ( 1 |x| - f f ) + µ∂ r ln ρ( 1 |x| + f f ))dx = 0.
It gives:

1 2 R N ∂ t (ρ|f u 2 | 2 )dx + µ R N ρ|∇(f u 2 )| 2 dx + µ R N ρf |u 2 | 2 (∆f -2 (f ) 2 f -f ∆ ln f )dx + R N ρ|f u 2 | 2 (u 1 ( 1 |x| - f f ) + µ∂ r ln ρ( 1 |x| ))dx = 0.
We have in particular:

1 2 R N ∂ t (ρ|f u 2 | 2 )dx + µ R N ρ|∇(f u 2 )| 2 dx + µ R N ρf |u 2 | 2 (∆f -2 (f ) 2 f -f ∆ ln f )dx + R N ρ|f u 2 | 2 (u 1 ( 1 2|x| - f f ) + 1 2|x| v 1 )dx = 0.
Furthermore a basic calculus yields:

∆f -2 (f ) 2 f -f ∆ ln f = ∆f -2 (f ) 2 f -(∆f - |∇f | 2 f ) = - (f ) 2 f . (4.46)
Using the fact that

|∇u 2 | 2 = (∂ r u 2 ) 2 + (u 2 ) 2
|x| 2 , we deduce from (4.46):

1 2 R N ∂ t (ρ|f u 2 | 2 )dx + µ R N ρ|∂ r (f u 2 )| 2 dx + µ R N ρ |f u 2 | 2 |x| 2 dx -µ R N ρ|f u 2 | 2 (f ) 2 f 2 dx + R N ρ|f u 2 | 2 (u 1 ( 1 2|x| - f f ) + 1 2|x| v 1 )dx = 0. (4.47)
Now taking f (|x|) = |x| -1 with > 0 small enough we have:

1 2 R N ∂ t (ρ |u 2 | 2 |x| 2-2 )dx + µ R N ρ|∂ r (f u 2 )| 2 dx + µ(2 -2 ) R N ρ |u 2 | 2 |x| 4-2 dx + R N ρ |u 2 | 2 |x| 2-2 (u 1 ( 3 2 -) 1 |x| + 1 2|x| v 1 )dx = 0. (4.48)
Applying Young inequality we have now for α 1 > 0 small enough and C > 0:

1 2 R N ∂ t (ρ |u 2 | 2 |x| 2-2 )dx + µ R N ρ|∂ r (f u 2 )| 2 dx + µ(2 -2 ) R N ρ |u 2 | 2 |x| 4-2 dx + R N ρ |u 2 | 2 |x| 2-2 (f 2 ( 3 2 -) 1 |x| + 1 2|x| g 2 )dx ≤ α 1 2 R N ρ |u 2 | 2 |x| 4-2 dx + 1 2α 1 R N ρ |u 2 | 2 |x| 2-2 C(|f 1 | 2 + |g 2 | 2 )dx. (4.49)
Since f 2 and g 2 are bounded in L ∞ ([0, T * ], L ∞ (R N )) we deduce by Gronwall lemma that:

√ ρ u 2 |x| 1-∈ L ∞ ([0, T * ], L 2 (R N )). √ ρ u 2 |x| 2-∈ L 2 ([0, T * ], L 2 (R N )).
1 p + 2 R N ∂ t (ρ|f u 2 | p+2 )dx + µ R N ρ|f u 2 | p |∇(f u 2 )| 2 + µp 4 ρ|∇|f u 2 | 2 | 2 |f u 2 | p-4 dx + µ R N ρ|f u 2 | p+2 ( ∆f f -2|∇ ln f | 2 ) + ρ|f u 2 | p+2 (u 1 ( 1 2|x| - 3f 2f ) + v 1 2 ( 1 |x| + f f ))dx + 2µ R N ρ∇ ln f • ∇(f u 2 )(f u 2 )|f u 2 | p dx = 0.
Next we have:

R N ρ∇ ln f • ∇(f u 2 )(f u 2 )|f u 2 | p dx = - 1 p + 2 R N ρ(∂ r ln ρ f f + ∆ ln f )|f u 2 | p+2 dx.
We deduce that:

1 p + 2 R N ∂ t (ρ|f u 2 | p+2 )dx + µ R N ρ|f u 2 | p |∇(f u 2 )| 2 + µp 4 ρ|∇|f u 2 | 2 | 2 |f u 2 | p-4 dx + µ R N ρ|f u 2 | p+2 ( ∆f f -2|∇ ln f | 2 - 2 p + 2 ∆ ln f ) + ρ|f u 2 | p+2 (u 1 ( 1 |x| - f f ) + µ∂ r ln ρ( 1 |x| + f f (1 - 2 p + 2
))dx = 0.

We have in particular:

1 p + 2 R N ∂ t (ρ|f u 2 | p+2 )dx + µ R N ρ|f u 2 | p |∂ r (f u 2 )| 2 dx + µ R N ρ |f u 2 | p+2 |x| 2 dx + µp 4 R N ρ|∇|f u 2 | 2 | 2 |f u 2 | p-4 dx + µ R N ρ|f u 2 | p+2 ( ∆f f (1 - 2 p + 2 ) -2(1 - 1 p + 2 )|∇ ln f | 2 ) + ρ|f u 2 | p+2 (u 1 ( 1 2|x| - f f ( 3 2 - 1 p + 2 )) + v 1 2 ( 1 |x| + f f (1 - 2 p + 2
))dx = 0.

(4.51) Taking f (|x|) = |x| α we have:

∆f (x) = α 2 |x| α-2 . ∆f f (1 - 2 p + 2 ) -2(1 - 1 p + 2 )|∇ ln f | 2 = - α 2 |x| 2 . ( 1 2|x| - f f ( 3 2 - 1 p + 2 )) = 1 |x| ( 1 2 - 3α 2 + α p + 2 ) 1 |x| + f f (1 - 2 p + 2 ) = 1 |x| (1 + α - 2α p + 2
).

Let us deal now with the case α = -1 + with > 0 small enough, it implies that:

1 p + 2 R N ∂ t (ρ |u 2 | p+2 |x| (p+2)(1-) )dx + µ R N ρ|f u 2 | p |∂ r (f u 2 )| 2 + µ(2 -2 ) R N ρ |u 2 | p+2 |x| 2+(p+2)(1-) dx + µp 4 R N ρ|∇|f u 2 | 2 | 2 |f u 2 | p-4 dx + µ R N ρ |u 2 | p+2 |x| (p+2)(1-) u 1 |x| (2 - 3 2 - 1 - p + 2 ) + v 1 2|x| ( + 2 - p + 2 ) dx = 0.
(4.52) Using (4.38), (4.43) and Young inequality there exists C > 0 and α 1 > 0 small enough such that:

1 p + 2 R N ∂ t (ρ |u 2 | p+2 |x| (p+2)(1-) )dx + µ R N ρ|f u 2 | p |∂ r (f u 2 )| 2 + µ(2 -2 ) ρ |u 2 | p+2 |x| 2+(p+2)(1-) dx + µp 4 R N ρ|∇|f u 2 | 2 | 2 |f u 2 | p-4 dx + µ R N ρ |u 2 | p+2 |x| (p+2)(1-) f 2 |x| (2 - 3 2 - 1 - p + 2 ) + g 2 2|x| ( + 2 - p + 2 ) dx ≤ α 1 2 R N ρ |u 2 | p+2 |x| 2+(p+2)(1-) dx + C 2α 1 R N ρ |u 2 | p+2 |x| (p+2)(1-) (|f 1 | 2 + |g 1 | 2 )dx (4.53)
By Gronwall lemma we deduce that for any > 0 small enough and any p ≥ 2, there exists a continuous function C independent on p such that for any T ∈ (0, T * ):

ρ 1 p u 2 |x| 1-L ∞ (0,T ),L p (R N )) ≤ C(T ). (4.54)
4.4 Gain of integrability on u 2

We recall now that u 2 verifies the following equation:

ρ∂ t u 2 + ρu 1 • ∇u 2 -µdiv(ρ∇u 2 ) + ρu 1 2|x| u 2 + ρv 1 2|x| u 2 = 0
We multiply this equation by u 2 |u 2 | p and we integrate over R N :

1 p + 2 R N ∂ t (ρ|u 2 | p+2 )dx + µ R N ρ|u 2 | p |∂ r u 2 | 2 dx + µ R N ρ|u 2 | p |u 2 | 2 |x| 2 dx + µp 4 ρ|∇|u 2 | 2 | 2 |u 2 | p-4 dx + R N ρu 1 2|x| |u 2 | p+2 dx + R N ρv 1 2|x| |u 2 | p+2 dx = 0.
Using (4.38), (4.43) and Young inequality there exists C > 0 and α 1 > 0 small enough such that:

1 p + 2 R N ∂ t (ρ|u 2 | p+2 )dx + µ R N ρ|u 2 | p |∂ r u 2 | 2 dx + µ R N ρ|u 2 | p |u 2 | 2 |x| 2 dx + µp 4 ρ|∇|u 2 | 2 | 2 |f u 2 | p-4 dx + R N ρf 2 2|x| |u 2 | p+2 dx + R N ρg 2 2|x| |u 2 | p+2 dx ≤ α 1 2 R N ρ|u 2 | p |∂ r u 2 | 2 dx + C 2α 1 R N ρ|u 2 | p+2 (|f 1 | 2 + |g 1 | 2 )dx.
By Gronwall lemma we deduce that for any > 0 small enough and any p ≥ 2, there exists a continuous function C independent on p such that for any T ∈ (0, T * ):

ρ 1 p u 2 L ∞ ((0,T ),L p (R N )) ≤ C(T ). (4.55)

Gain of integrability on u 1

Combining (4.54) and (4.55), we deduce that for any T ∈ (0, T * ) and any p ≥ 2:

ρ 2 p u 2 2 |x| x |x| (T, •) p 2 L p 2 ≤ ρ 1 p u 2 |x| 1- x |x| (T, •) p L p + ρ 1 p u 2 (T, •) p L p .
It implies that there exists a continuous function C such that for any p ≥ 2 we have:

ρ 2 p u 2 2 |x| x |x| (T, •) L p 2 ≤ C(T ). ( 4

.56)

We recall now that u 1 verifies the following equation:

ρ∂ t u 1 + ρu 1 • ∇u 1 -2µdiv(ρ∇u 1 ) + ∇P (ρ) = ρu 2 2 |x| x |x| . (4.57)
Multiplying the previous equation by w 1 |w 1 | p-2 with p ≥ 2 and integrating over (0, t) × R N with 0 < t < T * we obtain:

1 p R N (ρ|u 1 | p )(t, x)dx + t 0 R N ρ|u 1 | p-2 |∇u 1 | 2 (s, x)dsdx + (p -2) 4 t 0 R N ρ|∇(|u 1 | 2 )| 2 |u 1 | p-4 (s, x)dsdx ≤ 1 p R N (ρ 0 |u 1 0 | p )(x)dx + | t 0 R N |u 1 | p-2 w 1 • ∇aρ γ (s, x)dsdx| + | t 0 R N ρ |u 2 | 2 |x| u 1 |u 1 | p-2 dxds|.
(4.58)

The pressure term can be treated as in [START_REF] Haspot | Global strong solution for the Korteweg system with quantum pressure in dimension N ≥ 2, to appear in Mathematische Annalen[END_REF], let us focus on the second term in the right hand side of (4.62). By Hölder inequality and Young inequality, we have when q = 2p for > 0 small enough and C > 0:

| t 0 R N ρ |u 2 | 2 |x| u 1 |u 1 | p-2 dxds| ≤ t 0 ρ p-1 p u 1 (s) L p p-1 ρ 1 p (s, •) |u 2 | 2 (s, •) | • | L p ds ≤ t 0 ρ p-1 p u 1 (s) L p p-1 ρ 2 q (s, •) |u 2 | 2 (s, •) | • | L q 2 ds ≤ t 0 ρ 1 p u 1 (s) p p-1 L p ρ 2 q (s, •) |u 2 | 2 (s, •) | • | L q 2 ds ≤ t 0 ρ 1 p u 1 (s) p L p ds + C t 0 ρ 2 q (s, •) |u 2 | 2 (s, •) | • | p p-1 L q 2 ds (4.59)
From (4.58) and (4.63) we deduce that the exists a continuous function C such that for any p ≥ 2 and T ∈ (0, T * ) we have:

ρ 1 p u 1 L ∞ ((0,T ),L p (R N )) ≤ C(T ). (4.60)

Gain of integrability on v 1

Let us recall now that v 1 verifies the following equation:

ρ∂ t v 1 + ρu 1 • ∇v + a 2µ ρ γ v 1 = a 2µ ρ γ u 1 + ρ u 2 2 |x| x |x| . (4.61)
Multiplying the previous equation (4.61) by v|v| p-2 for p ≥ 2, we have:

1 p R N ∂ t (ρ|v 1 | p )dx + a 2µ R N ρ γ |v 1 | p dx = a 2µ R N ρ γ p-1 p |v 1 | p-2 v 1 • ρ γ p u 1 dx + R N ρ u 2 2 |x| v 1 |v 1 | p-2 dx. (4.62)
We have now by Hölder inequalities and Young inequality:

| t 0 R N ρ |u 2 | 2 |x| v 1 |v 1 | p-2 dxds| ≤ t 0 ρ p-1 p v 1 (s) L p p-1 ρ 1 p (s, •) |u 2 | 2 (s, •) | • | L p ds ≤ t 0 ρ 1 p v 1 (s) p L p ds + C t 0 ρ 2 q (s, •) |u 2 | 2 (s, •) | • | p p-1 L q 2 ds (4.63)
Next by Höder, Young inequality and energy inequality we have for > 0 small enough:

| R N ρ γ p-1 p |v 1 | p-2 v 1 • ρ γ p u 1 dx| ≤ p -1 p R N ρ γ |v 1 | p dx + 1 p R N ρ γ |u 1 | p dx ≤ p -1 p R N ρ γ |v 1 | p dx + 1 p [ ρ 1 p u 1 p L p (R N ) + √ ρ|u 1 | p L 2 (R N ) ρ γ-1 2 - √ ρ L 2 (R N )) ] ≤ p -1 p R N ρ γ |v 1 | p dx + 1 p [ ρ 1 p u 1 p L p (R N ) + C ρ 1 2p u 1 2 L 2p (R N ) ]. (4.64)
Combining (4.63), (4.63) and (4.64) It yields that for any p ≥ 2 and T ∈ (0, T * ) there exists a continuous increasing function C such that we have:

ρ 1 p v L ∞ t (L p ) C(T ). ( 4

Maximum principle

Let us recall the different estimates that we have obtained, from (4.65), (4.60), (4.55) there exists a continuous increasing function C such that for any p ≥ 2 and T ∈ (0, T * ) we have:

ρ 1 p u 1 L ∞ ((0,T ),L p (R N )) ≤ C(T ), ρ 1 p u 2 L ∞ ((0,T ),L p (R N )) ≤ C(T ), ρ 1 p v 1 L ∞ ((0,T ),L p (R N )) ≤ C(T ). (4.66)
Our goal is now to show that we can extend the solution (ρ, u) beyond T * using the estimates (4.66). From the previous estimates (4.66) we deduce using the maximum principle the following proposition (see [START_REF] Haspot | Global strong solution for the Korteweg system with quantum pressure in dimension N ≥ 2, to appear in Mathematische Annalen[END_REF] for a detailed proof).

Proposition 4.5. Under the assumption of theorem 1.2, the density ρ verifies for any T ∈ (0, T * ):

1 ρ L ∞ T (L ∞ (R N )) ≤ C(T ), (4.67)
with C an increasing continuous function in T .

5 Proof of the theorem 1.2

We are going to show that we can extend the solution (ρ, u) beyond T * using the estimates (4.66).

Estimate on u(T, •)

B N p -1+ p,1
for any T < T * and > 0

Let us recall that we have proved according to (4.60), (4.55) and proposition 4.5 that for any p ≥ 2 there exists increasing continuous functions C and C 1 depending only on the initial data (ρ 0 , u 0 ) such that:

ρ 1 p u L ∞ T (L p ) ≤ C(T ) and 1 ρ 1 p L ∞ T (L ∞ ) ≤ C 1 p 1 (T ).
(5.68)

By Besov embedding we observe that for p ≥ 2 there exists C > 0 such that:

u L ∞ T (B 0 p,∞ ) ≤ CC(T )C 1 p 1 (T ) and u L ∞ T (B -N ( 1 2 -1 p ) p,∞ ) ≤ CC(T )C 1 2
1 (T ).

(5.69)

For p > N 1-we have -N 2 + N p < N p -1 + < 0; we deduce then by interpolation in the Besov space that there exists M an increasing continuous function depending only on the initial data (ρ 0 , u 0 ) verifying for any 0 < T < T * 3 :

u(T, •) B N p -1+ p,1 ≤ M (T ).
(5.70)

5.2 Estimate on q 1 (T, •)

B N p + p,1
for any T < T * and > 0

Using the first equation in (1.3) and the proposition 2.4, there exists C > 0 such that for any T ∈ (0, T * ) we have (where we set q 1 = ρ -ρ):

q 1 L ∞ T (B N p + p,1 ) ≤ C( q 1 0 B N p + p,1 + ρv 1 L ∞ T (B N p -1+ p, 1
)

).

(5.71)

In the same way as in the previous section we have for any p ≥ 2:

ρv 1 L ∞ T (B 0 p,∞ ) = ρv 1 L ∞ T (B 0 p,∞ ) ≤ ρ 1-1 p L ∞ T (L ∞ ) ρ 1 p v 1 L ∞ T (L p ) , ρv 1
L ∞ T (B -N ( 1 2 -1 p ) p,∞ ) = ρv 1 L ∞ T (B -N ( 1 2 -1 p ) p,∞ ) ≤ ρ 1 2 L ∞ T (L ∞ ) ρ 1 2 v 1 L ∞ T (L 2 ) .
(5.72) Since the inequality -N 2 + N p < N p -1 + < 0 holds, we get by interpolation with N p -1 + = θ(-N 2 + N p ):

ρv 1 L ∞ T (B N p -1+ p,1 ) ≤ ρ θ 2 L ∞ T (L ∞ ) ρ 1 2 v 1 θ L ∞ T (L 2 ) ρ (1-θ)(1-1 p ) L ∞ T (L ∞ ) ρ 1 p v 1 1-θ L ∞ T (L p ) ≤ ρ 1-1- N L ∞ T (L ∞ ) ρ 1 2 v 1 θ L ∞ T (L 2 ) ρ 1 p v 1 1-θ L ∞ T (L p ) .
(5.73)

According to (4.65) and the energy inequality, there exists a continuous increasing function M 1 such that:

ρv L ∞ T (B N p -1+ p,1 ) ≤ ρ 1-1- N L ∞ T (L ∞ ) M 1 (T ). ( 5

.74)

Plugging the previous estimate in (5.72) it gives:

q 1 L ∞ T (B N p + p,1 ) ≤ C( q 1 0 B N p + p,1 + ( q 1 L ∞ T (L ∞ ) + ρ) 1-1- N M 1 (T )).
(5.75)

From Besov embedding and interpolation, we know that there exists C, C > 0 such that: ).

q 1 L ∞ T (L ∞ ) ≤ C q 1 L ∞ T (B N p p,1 ) ≤ C q 1 θ L ∞ T (B -N ( 1 2 -1 p ) p,∞ ) q 1 1-θ L ∞ T (B N p + p,∞ ) , ( 5
with F (ρ) = P (ρ) ρ . It remains now to estimate F

L 1 T (B N p -1 p,1
)

, by proposition 2.2,2.3, interpolation and Young inequality we have for C a continuous function and with θ ∈ (0, 1):

∇P (ρ) ρ L 1 T (B N p -1 p,1 ) ≤ T F (ρ) -1 L ∞ T (B N p p,1 )
≤ T ( q )

L ∞ T (B N p + p,1 ) + q L ∞ T (L 2 ) ) u • ∇u
≤ C T 0 C( 1 ρ L ∞ T (L ∞ ) , ρ L ∞ T (L ∞ ) ) q(s)
+ u L 1 T (B N p +1 p,1
)

≤ C(T ).

(5.82)

We have proved in particular that for any T ∈ (0, T * ) we have:

∇u L 1 ((0,T ),L ∞ (R N )) ≤ C(T ).

(5.83)

In addition by classical estimate on transport equation we prove also that there exists a function C continuous and increasing such that for any T ∈ (0, T * ):

q L ∞ T (B N p p,1 )
≤ C(T ).

(5.84) Proposition (4.5), (5.83) and (5.84) allows to apply the blow-up criterion of the theorem 1.1.

In particular it contradicts the assumption T * < +∞ and we have then seen that necessary T * = +∞. It implies that the solution (q, u) is unique and verify locally in time (1.13). In particular since the proof of the uniqueness in [START_REF] Danchin | A Lagrangian approach for the compressible Navier-Stokes equations. to appear[END_REF] requires only local estimates in time, we have proved that (q, u) is a global strong solution of the system (1.1).

6 Proof of the theorem 1.3

The proof follows exactly the same lines than the previous section and is even simpler. We are going to explain how to adopt the previous proof. First since radially the solution is radially symmetric symmetric we observe that the strong solution (ρ, v) of the system (1.3) verifies on the maximal time interval (0, T * ):

ρ∂ t v + ρu • ∇v + ∇P (ρ) = 0. (6.85)

Using an argument developed in [START_REF] Haspot | Existence of global strong solution for the compressible Navier-Stokes equations with degenerate viscosity coefficients in 1D[END_REF], we can rewrite the previous equation as follows:

ρ∂ t v + ρu • ∇v + aγ 2µ v = aγ 2µ u. (6.86)
The rest of the proof consists in getting L ∞ T (L p (R N )) estimates on ρ 1 p v for any p ≥ 2 and T ∈ (0, T * ). It is then sufficient as in the previous section and [START_REF] Haspot | Global strong solution for the Korteweg system with quantum pressure in dimension N ≥ 2, to appear in Mathematische Annalen[END_REF] to apply a maximum principle and to prove that ρ and 1 ρ is bounded in L ∞ ([0, T * ], L ∞ (R N )). To do this, it suffices to multiply the equation (6.86) by v|v| p and to integrate over (0, T ) × R N . It gives: In order to conclude, it remains to bound ρ 1 q u in any L ∞ ([0, T * ], L q (R N )) with q ≥ 2. We recall now that since u is radially symmetric, we have Du = ∇u and we can rewrite the momentum equation on u as follows: ρ∂ t u + ρu • ∇u -2µdiv(ρ∇u) + a∇ρ γ = 0. (6.88)

1 p + 2 R N ∂ t (
Multiplying the previous equation by u|u| p and integrating over r (0, T ) × R N we get: (6.89) Using same arguments than in [START_REF] Haspot | Global strong solution for the Korteweg system with quantum pressure in dimension N ≥ 2, to appear in Mathematische Annalen[END_REF], we show that we obtain L ∞ ([0, T * ], L q (R N )) on ρ 1 q u for any q ≥ 2 when 1 ≤ γ < 7 6 . The rest of the proof follows the same arguments than the previous section.

1 p + 2 R N ∂ t (

Appendix

Assume that v has the following form:

v = x ⊥ |x| v 2 (t, |x|) + x |x| v 1 (t, |x|) = v 2 + v1.
We verify then that:

u 1 • ∇v 1 = x ⊥ |x| (u 1 ∂ r v 1 ), u 2 • ∇u 2 = - x ⊥ |x| ( u 2 2 r
),

u 1 • ∇v 2 = x ⊥ |x| (u 1 ∂ r v 2 ), u 2 • ∇v 1 = x ⊥ |x| ( v 1 u 2 r
).

(4. 50 ) 4 . 3 L p weight estimate on u 2

 50432 Multiplying the previous equation (4.45) by f u 2 |f u 2 | p , we have:

  .[START_REF] Zhang | Global behavior of compressible NavierStokes equations with a degenerate viscosity coefficient[END_REF]

1

 1 is well defined since u belongs to CT * (B N p -1+ p,1

  From proposition 4.5 and according to (5.70) and (5.79), plugging the previous estimate (5.81) and using Gronwall lemma we prove that there exists a function C continuous and increasing such that: u

  ρ|v| p+2 (t, x))dx + aγ 2µ R N |v| p+2 (t, x)dx = aγ 2µ R N u • v|v| p (t, x)dx. (6.87)

  ρ|u| p+2 (t, x))dx + 2µ |∇u| 2 |u| p (t, x)dx + µp 2 R N ρ|∇(|u| 2 )| 2 |v| p-2 (t, x)dx

	≤ |

R N a∇ρ γ • u|u|p(t, x)dx|.

It means that there is no geometric assumption on the initial data.

The letter T is omitted for functions defined over R + .

Acknowledgement:

The author has been partially funded by the ANR project INFAMIE ANR-15-CE40-0011.

with N p = -θN ( 1 2 -1 p ) + (1 -θ)( N p + ). Since q 1 = ρ -ρ is bounded in L ∞ ((0, T ), L 2 (R N )) for any T ∈ (0, T * ) from the energy estimate and by Sobolev embedding, we can now show that by Young inequality and (5.76) there exists M an increasing function such that:

) .

(5.77) From (5.75), (5.77) and by Young inequality we have for C > 0 and any > 0:

)

)

.

(5.78)

Choosing sufficiently small we deduce that there exists C > 0 such that:

(5.79) From ( 5.77) it implies in particular that the L ∞ T (L ∞ ) norm of ρ depends only on T and on the initial data.

Lipschitz control on the velocity u

We recall that we have on (0, T * ):

Applying the proposition 2.4 we have for any T ∈ (0, T * ):

)

), (5.80) with:


Raharijaona Nivoniaina Fahendrena

Romaine Ramananarivo

Professeur Titulaire

Keywords: publication scientifiques, références bibliographiques, ressource web scientific publication, references, web resource, search

Par l'avènement de l'internet, les ressources disponibles qui s'accroissent de jour au lendemain, sont utilisés par les universitaires. L'objectif de cette recherche est l'utilisation des ressources Internet par les étudiants des Universités publiques de Madagascar. La bibliométrie a été utilisé afin d'analyser les citations, provenant de 168 mémoires et 63 thèses malgaches publiés entre 2003 et 2013. Comme résultat, l'utilisation des ressources sur internet s'accroit au fil du temps. Cette pratique a un impact positif dans l'enseignement supérieur par l'amélioration de la représentation des références bibliographiques. Il en découle aussi que le site le plus consulté étant le Wikipédia. Malgré les ressources sur web, la consultation des documents papier reste encore prioritaire.

Introduction

Depuis les années 1990, un nouveau media, le Word Wide Web, a vu le jour, et a provoqué la popularisation de l'Internet [START_REF] Megnigbeto | Les ressources internet dans les citations des mémoires en Science de l'information au Bénin[END_REF]. Ceci a bouleversé fortement les pratiques documentaires, avec la mise à disposition, en ligne, d'un nombre croissant de ressources (ELISA, 2008). Le web, plus qu'un sérieux concurrent, devance aujourd'hui les systèmes d'information documentaires, et devient le point d'entrée principal pour la recherche d'information.

Sur Internet, la diversité des ressources disponibles s'en trouve accrue ; parallèlement à cela, la publication scientifique évolue avec le développement de systèmes d'archives ouvertes, permettant aux chercheurs de mettre en accès libre leurs publications sur des serveurs dédiés.

Par conséquent, de plus en plus d'articles sont ainsi disponibles gratuitement. A cela s'ajoute qu'Internet est plus souple que tout autre support. Internet propose une panoplie de possibilités de loin supérieure à ce que peut offrir tout autre support ; il permet de travailler sur des documents authentiques (journaux, radios, documents vidéo, TV…).

Ces effets de l'avènement des TIC ont un impact de plus en plus marqué sur l'enseignement supérieur et la recherche. Ils vont également favoriser l'accès à de multiples et diverses ressources documentaires tout en entraînant des modifications importantes dans le rapport que les individus et les groupes ont avec le savoir.

Les TIC sont également d'excellents instruments pour faciliter aux scientifiques des pays en développement, l'accès aux revues spécialisées, aux bibliothèques, aux bases de données et aux services scientifiques de pointe. Elles présentent aussi l'avantage de pouvoir améliorer la collecte et l'analyse de données scientifiques complexes.

Le développement des TIC apparaît selon de nombreuses organisations internationales (Banque mondiale, UNESCO, AUF…) intervenant sur des projets de développement de l'enseignement supérieur et de la recherche scientifique en Afrique, comme une solution pouvant permettre aux universités africaines, d'atteindre les objectifs que la plupart d'entre elles se sont fixées. Le Web est devenu le canal privilégié de l'information, des transactions, de la gestion des documents et autres [START_REF] Lakos | Portals in libraries[END_REF].  Les ressources web développent l'apprentissage permanent des usagers.

Résultats attendus

 L'exploitation des ressources numériques sera déterminée.

 L'utilisation des ressources numériques par les étudiants sera évaluée.

Matériels et méthodes

Résultats

Utilisation des ressources documentaires

Les ressources documentaires sont de plusieurs types soit sous forme papier dit document physique, soit accessible en ligne ou sur internet dénommé ressource web.

Utilisation documents physiques

Pour leurs travaux de recherche, les étudiants des Universités citent dans leurs bibliographie différents type de documents physiques tels que : monographies, les chapitres de monographies, les revues, les articles de périodique, les mémoires, les thèses, les notes de

Norme de présentation des ressources Web

Internet provoque des changements profonds dans la recherche et l'exploitation de l'information, ainsi qu'une grande évolution dans les pratiques intellectuelles [START_REF] Bernard | Ressources Electroniques pour les Etudiants, la Recherche et l'Enseignement (REPERE)[END_REF]. Dans l'ensemble, les étudiants chercheurs estiment que les ressources numériques constituent une plus-value par rapport aux supports traditionnels.

Évolution de la description bibliographique web

Discussions

Conclusion

Cette étude apporte des données quantitatives justifiant l'utilisation de l'Internet par les étudiants de l'Enseignement Supérieur, surtout dans la réalisation de leurs travaux de recherche.

En effet, grâce au développement des TIC et plus particulièrement l'Internet, l'utilisation de ressources électroniques dans les travaux universitaires est de plus en plus fréquente. Cet outil a engendré des changements dans la présentation de la bibliographie, dans les comportements de citations et également dans le mode de diffusion de la recherche. D'une manière générale, le changement est palpable à tous les niveaux de l'élaboration des travaux de recherche.

Par ailleurs une des grandes conquêtes de ce nouvel outil consiste à sa contribution d'une manière certaine au développement de l'auto-formation, qui pour l'enseignant en plus de son rôle s'ajoute l'encadrement des étudiants au processus d'apprentissage.

Il reste à savoir si Internet constitue une alternative à la pauvreté des fonds universitaires et permet d'accéder à des documents non disponibles dans les bibliothèques.

Dans cette perspective, pour que l'utilisation de cet outil soit effective dans tous les domaines de la recherche, la connexion à l'Internet doit être à très haut débit, fluide et abordable.

  Pour l'investigation, les Universités publiques de Madagascar ont été pris comme centre d'étude. Ces Universités forment chaque année des milliers d'étudiants qui soumettent un mémoire ou une thèse sanctionné par des diplômes académiques (ingénieur, DEA 1 ou Master 2, Doctorat et HDR 2 ) : travail de recherche nécessitant des affirmations pour appuyer leurs idées, et ce, avec différents type de ressources d'information, résultats des travaux de recherche antérieurs à partir des documents papiers ou consulté sur Internet en faisant référence à d'autres travaux de recherche. La question suivante se pose : Les étudiants de ces universités utilisent-ils effectivement les ressources numériques de l'Internet misent à leur disposition ? L'objectif de ce travail est d'évaluer l'utilisation des ressources Internet par les étudiants des Universités publiques de Madagascar. Cet objectif occasionne deux objectifs spécifiques à savoir si les étudiants utilisent vraiment les ressources web, puis ces ressources électroniques sont-elles bien citées ? Comme hypothèses :  l'internet constitue un atout pour la recherche d'information.

  Pour la réalisation de cette étude, dans un premier temps un inventaire des mémoires soutenus publiquement aux Universités publiques de Madagascar, ont été effectué, et ceci depuis 2002, date de dépôt du Portail malgache des Thèses et Mémoires, par de la bibliothèque universitaire d'Antananarivo. Ensuite, une constitution du corpus de l'étude a été élaborée par la définition de l'échantillonnage en tenant compte des références bibliographies des mémoires. Ce corpus est constitué de 231 mémoires et thèses. Les ressources Internet recensées, issues des références bibliographiques des mémoires et des thèses en question, constituent le matériel de notre recherche. Pour ce faire, nous considérons la bibliographie et les références Web, cités dans leurs travaux de fin d'étude entre 2003 et 2013, comme indicateur de l'usage d'une ressource Internet, et au delà, comme mesure de l'usage de l'Internet. Enfin une base de données a été créée pour collecter : le nom, prénoms de l'auteur, l'option de l'étudiant, l'année de fin d'étude, l'année de soutenance, le diplôme, le nombre de références bibliographiques, le nombre de références numériques (Web) tirées de l'Internet et cités dans chaque document (mémoire ou thèse), et la normalisation de la présentation des références Internet. La base de données est créée avec le logiciel documentaire WinIsis ; les fichiers ainsi produits sont importés dans une base de données sous Excel pour analyser les variables. Dans le cadre de cette étude, seuls les mémoires soutenus entre 2003 et 2013 sont considérés, et ceux qui ont utilisé des ressources issues du net. Après collecte, la présence de webiographie a été ressentie sur les 114 publications dont 25 thèses et 89 mémoires sur les 231.

  cours, les documents d'archives, les publications officielles. Entre 2003 et 2013, les mémoires et thèses en question (231) ont utilisé dans leur bibliographie 7 498 documents physiques.

Graphe 1 : 2 :

 12 Évolution de l'utilisation des documents papiers au fil du tempsCe Graphe montre que l'utilisation des documents papiers reste encore prioritaire pour justifier les recherches faites par les étudiants malagasy. Ils étaient utilisés à 84,4% et avec une moyenne de 612 au cours des dix dernières années. En effet, pour la réalisation des travaux de recherche, les documents papier sont utilisés en premier lieu par les étudiantschercheurs, puisqu'en général, ils ont l'habitude de travailler essentiellement avec les articles des revues et les ouvrages. Aussi, les étudiants ne sont pas prêts à abandonner les livres en format papier car ces derniers sont perçus comme étant faciles à employer et faciliteraient la concentration.2.1.2. Utilisation des ressources WebUne progression régulière de l'utilisation des ressources Web dans la bibliographie est remarquée entre 2003 et 2013. Comme le montre le graphe 2, il se trouve qu'elles sont utilisées en complément, comme une nouvelle source d'informations, avec l'avantage de donner la possibilité d'accéder aux textes intégraux. Évolution de l'utilisation des ressources web au fil du temps En effet, une forte utilisation des ressources Web, dit aussi « sitation » est notée entre 2010 et 2013. Cette remarque explique que les étudiants font, de plus en plus, recours aux ressources numériques. Internet constitue pour les étudiants, un outil de repérage et de balayage le plus large possible dans le domaine de la recherche d'information. De plus, les étudiantschercheurs sont intéressés par les références bibliographiques. Ces dernières constituent un outil indispensable pour la réalisation de tout travail de recherche.

3 :

 3 La bibliographie, présentation normalisée de la description bibliographique, est l'ensemble des informations relevées sur un document et servant à son identification. Elles doivent être présentées de la même façon.Le graphe 3 ci-dessous montre qu'il y a évolution dans la pratique de l'utilisation de l'internet, pour se procurer des ressources Web afin d'enrichir la bibliographie. En effet l'augmentation est palpable de 2003 à 2013. normalisée des références web dans la bibliographie Toutes les ressources qui sont issues d'Internet(livre, revue, vidéo, etc.), devraient être cité suivant les modèles en fonction du type de la ressource. Ajoutez après le titre la mention : [en ligne] et à la fin de la référence bibliographique, indiquez l'url et la date de consultation.Suivant la norme de la description bibliographique ISBD3 , les ressources se présentent comme suit : Site internet : AUTEUR ou ORGANISME. Titre de la page. la page d'accueil en italique[en ligne]. Lieu d'édition : Editeur 4 , date de publication. Date de mise à jour si possible. (Consulté le jj/mm/aaaa). Disponible à l'adresse : URL Page d'un site : AUTEUR de la page. Titre de la page [en ligne]. In : AUTEUR ou ORGANISME. Titre de la page. la page d'accueil en italique. Lieu d'édition : Editeur, date de publication. Date de mise à jour si possible. (Consulté le jj/mm/aaaa). Disponible à l'adresse : URL (BIBMIOTHECAIRES, 2000).2.2.2 Présentation des références par les étudiants 3 ISBD Description bibliographique internationale normalisée élaboré par IFLA Graphe Nombre de webiographie comparé avec sa présentation normalisée Nous remarquons que l'écriture des citations par les étudiants, suivant la norme de la description bibliographique se concrétise au fil des années (FONDIN, 2002) ce qui explique l'auto formation des étudiants. Ces derniers considèrent ainsi le Web comme une ressource de connaissances importante et faisant de l'internet un outil à part entière dans la recherche d'information. En effet, le Graphe 4 montre que de nouveaux modes de présentation de bibliographie comme la webographie se développent et se concrétisent chez eux à partir de l'utilisation des ressources Web. Entre 2003 et 2013, cette variation se situe entre 8 et 259 références. des ressources Web Les ressources numériques sont utilisées par les étudiants chercheurs pour avoir plus de données afin de justifier leurs recherches. Ainsi, la répartition des ressources Web (Graphe 4), dévoile les différentes ressources numériques utilisées par les étudiants. La nature des ressources, leurs usages et leur fréquence varient d'une discipline à l'autre. Ces ressources sont issues des bases de données de rang A, des ressources francophones telles que lamicrofinance.org, path.org, revues.org…, des différents sites nationales et internationale : sahonagasy.org, dotissimo.fr etc. et des encyclopédies. Sur ce Graphe, nous notons néanmoins une utilisation prépondérante des références Web issue de Wikipedia. En effet, cette domination en nombre se justifie par la crédibilité et la fiabilité (LOPES & CARRICO, 2008 ; REMMERSAAL, 2010), de Wikipedia « Avec Wikipédia, on échange la garantie de fiabilité des encyclopédies classiques contre la gratuité et la possibilité d'un lien aux sources immédiates que Wikipédia impose à ses auteurs de citer », (O'NEIL, ). Aussi, différents commentaires sont émis « Sur la confiance Wikipedia » (MAGNUS, 2009). Ainsi, l'utilisation publique de Wikipedia est remarquable. Depuis sa création en 2001, le site web "Wikipedia.org" est devenu un phénomène Internet et est rapidement classé parmi les top-10 sites les plus visités dans le monde.

Graphe 5 :

 5 Cumul de l'utilisation des ressources web au fil du temps Lorsque l'on considère les proportions de ressources Internet citées dans les mémoires publiés entre 2003 et 2013, seuls 15,6% des références bibliographiques utilisées sont des ressources numériques. Il en découle que les étudiants citent moins les ressources électroniques que les ressources traditionnelles.

L

  'utilisation du réseau Internet à des fins de recherches d'information nécessite l'usage permanent des matériels informatiques et des connaissances adéquates ; c'est dans ce cadre que des formations à utilisation pédagogique et méthodologique du réseau telles que : l'initiation à l'informatique, l'utilisation de l'Internet, la recherche d'information sur Internet, la gestion des ressources documentaires, l'exploitation des ressources numériques… s'imposent. De plus « la connaissance de l'Internet est utile pour mieux comprendre les résultats des recherches d'informations, pour l'évaluation des ressources d'informations et ». (DEFOSSE & NOEL, 2008). Ces formations sont introduites actuellement parmi les disciplines enseignées aux Universités et aux grandes Écoles pour que les étudiants ne soient pas tenus à l'écart de la révolution en cours, ils devront oeuvrer dans un monde où ces technologies domineront. La prolifération des sites Web, l'abondance des services documentaires, le développement exponentiel des sources d'information et la diversité de formats de présentation de celles-ci, impliquent de nouvelles pratiques estudiantines afin de satisfaire leur besoin informationnel en général et ceci pour l'élaboration de leurs travaux de recherche académiques. Suite à ces formations, la présentation de références numériques et plus précisément la Webiographie devient de plus en plus clair et utilisable pour d'autres fins (étude bibliométrique, scientométrique). Il découle de l'évolution de l'utilisation des documents papiers, que les étudiants citent plus de ressources traditionnelles, ce qui veut dire qu'ils ont une nette préférence pour les documents en format papier et l'utilisation des livres reste pour eux, une source incontournable d'information graphe 5. Pour consulter ces documents, ils ont recours aux différents Centres de documentation et bibliothèques universitaires. Cela confirme les conclusions de ZHANG (2001) qui, dans une étude similaire sur les publications universitaires, dégage que les ressources électroniques sont moins citées que les ressources imprimées ; de même, KUSHKOWSKI (2005) après avoir comparé les citations d'une part dans les thèses soumises électroniquement et d'autre part dans les thèses imprimées, aboutit à la conclusion que la proportion des citations aux ressources du web en pourcentage du total des citations tous types de documents confondus s'est accrue avec le temps, mais est restée faible. (KUSHKOWSKI, 2005) Pour les étudiants des Universités publiques de Madagascar, cette faiblesse est due aussi par l'arrivée récente de la connexion Internet dans la Grande Île car l'histoire d'internet à Madagascar est encore jeune. En effet, c'était en mars 1996 que la société DTS, filiale de l'opérateur national des télécommunications TELMA et de France Câble et Radio a lancé officiellement ses services Internet. (ANDRIAMPARANY, et al., 2001) En ce qui concerne les ressources Web, les étudiants des Universités les utilisent pour acquérir des informations et compléter leurs bibliographies. D'après ces résultats, les étudiants se préoccupent de plus en plus de ces ressources, ils sont intéressés par les références bibliographiques, car ces dernières constituent un outil indispensable pour la réalisation de leur travail de recherche. Les bibliographies permettront de vérifier si l'auteur d'un article diffusé sur Internet fait l'objet de citations, ou bien si les références qu'ils citent dans son article sont citées dans d'autres publications. L'intérêt que les chercheurs accordent thèses, articles et revues, peut s'expliquer par le fait que les chercheurs sont intéressés par toute information qui peut concerner leur domaine de recherche. Ils consultent, tout au long de la réalisation de leur travail, Internet dans le but de trouver plus d'informations pour compléter ou comparer leurs travaux avec ce qui se fait ailleurs, et pour appuyer ou soutenir un point de vue.

  Le cumul de l'utilisation des ressources web au fil du temps, indique que l'utilisation des ressources Web par rapport à celle des documents physiques s'améliore et s'accroît très rapidement entre 2003 et 2013 mais reste faible (graphe 5).

	35
	30
	25
	20
	15
	10 Cette variation s'explique par une appropriation plus intense des pratiques de recherches
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DEA : Diplôme d'études approfondies