Gaelle Campagne
email: gaelle.campagne@ensica.fr

Jean-Bernard Cazalbou

Laurent Joly

Patrick Chassaing

DIRECT NUMERICAL SIMULATION OF THE INTERACTION BETWEEN UNSHEARED TURBULENCE AND A FREE-SLIP SURFACE

Keywords: Turbulence, Direct numerical simulation, Free surface, Random forcing

In this paper, direct numerical simulation is used to study the interaction between turbulence and a free surface. The configuration is original due to the fact that a random force generates turbulence in the vicinity of a plane parallel to the free surface. Turbulence is therefore statistically steady and nearly isotropic at some distance of the surface. A detailed description of the flow is provided, including second-order statistics and full Reynolds-stress budgets. It is shown that the results obtained in this configuration can help the understanding of intercomponent energy-transfer mechanisms.

INTRODUCTION

Turbulence interacting with a free-slip surface is encountered in many flows of engineering interest. The most obvious are those of a liquid near a gas-liquid interface like, for instance, the surface layer of the ocean and lakes, stirred fluids within vessels, and many flows which occur in chemical engineering processes. On another hand, it can be viewed as a first approach to the physics of near-wall turbulence since it helps to evaluate the effects of kinematic blocking independently of the viscous effects, and of those effects induced by the almost systematic presence of mean shear in the vicinity of a solid wall. Kinematic blocking is known to affect the mechanism of intercomponent energy transfer (embodied in the Reynolds-stress-transport equations through the pressure-strain correlations): In an unbounded turbulent flow, at moderate levels of anisotropy, the mechanism of the intercomponent energy transfer can be viewed as a simple return-to-isotropy phenomenon. The picture changes drastically in the immediate vicinity of a blocking surface where, whatever the anisotropy, energy transfer is always at the expense of the normal component of the fluctuations. Such a modification has long been attributed to the so-called splat effect: blobs of fluid moving toward the surface transfer energy from the normal to the tangential directions at the impact. This simple phenomenology has been questioned by Perot and Moin 1 who noticed that splat events should necessarily be balanced by "anti-splat" events with an opposite effect on the energy transfer. As a result, they suggested that the imbalance is small -hence the pressure-strain correlation-and can only be determined by viscous effects along the surface. This was supported by their direct numerical simulation of decaying turbulence near a free surface at short times. In another simulation of the same flow at larger times, Walker et al. [START_REF] Walker | Shear-free turbulence near a flat free surface[END_REF] came to a slightly different conclusion: According to them, the imbalance between splat and anti-splat events results from conflicting contributions from the kinematic blocking and the return-to-isotropy mechanism which is present at large times (when anisotropy due to the presence of the surface becomes significant.) The results based on rapid distortion theory obtained by Magnaudet 3 confirm this last statement, and highlight the fact that the intercomponent energy transfer mechanism in the presence of a blocking surface strongly depends on the nature of turbulence at a distance: whether it is steady or unsteady, isotropic or not, etc.

The present study aims at complementing our view of these transfer phenomena when turbulence at a distance remains steady and quasi-isotropic. We believe that this case is more representative of engineering applications for which turbulence-production regions are present in the flow field, and continuously feed the surface regions via turbulent transport. To this end, we have defined a numerical analog of the oscillating-grid experiments (see, for instance De Silva and Fernando 4 ). This configuration is characterized by statistical steadiness and continuous feeding of the surface by turbulent diffusion from a turbulence-production layer. It bears some similarities with the free-surface of an openchannel flow (see, for instance, the numerical simulations of Handler et al. [START_REF] Handler | Length scales and the energy balance for turbulence near a free surface[END_REF] ), but ensures strictly-zero mean shear in the surface region and, from a computational point of view, maximizes the number of grid points placed there. The final goal of our study will be to identify the structures responsible for the modifications induced by the presence of a free surface to the intercomponent energy transfer, and to give some informations about a physically-consistent formulation of a statistical model for the pressure-strain correlations.

The paper is organized as follows: Details about the computational method and the configuration are given in section 2. A description of the flow based on the examination of the second-order statistics is then provided in section 3, while section 4 is devoted to the presentation of the Reynolds-stress budgets. General comments and direction for future work are provided as the conclusion.

CALCULATION METHOD AND DEFINITION OF THE FLOW CASE

Our results are obtained from a numerical solution of the three-dimensional timedependent Navier-Stokes equations. We discuss the computational method with a particular attention to the forcing technique -how turbulence is generated-and then, describe the way statistics are obtained. All the parameters of the solution are then given, together with a description of its specific features.

Top surface

Bottom surface Energy-input profile

E ' © c T c T L x L y L f L z L x L y L z L f 2π 2π 7π/4 7π/12
Figure 1: Definition of the computational domain. Turbulence production is strictly zero outside of the forcing region.

Computational method

As one can see in figure 1, the computational domain is a rectangular box in which the grid spacing is equal and uniform in each direction. The two horizontal directions, x and y, are periodical, with a period of 2π. The wave numbers are made nondimensional with L x = 2π, and all the resolved wave numbers κ are therefore integer numbers. Symmetry conditions are imposed in the z direction to force free-surface boundary conditions. A random volume force, confined in the vicinity of a central plane parallel to the free surfaces, represents the steady source of turbulence.

The incompressible Navier-Stokes equations are solved under the form

∇. u = 0 ∂ t u = u × (∇ × u) -∇ (P/ρ + u 2 /2) + ν ∆ u (1)
using a pseudo-spectral method, similar to that of Orszag and Patterson. [START_REF] Orszag | Numerical simulation of three-dimensional homogeneous isotropic turbulence[END_REF] The momentum equation is advanced in time using a third-order Runge-Kutta scheme with implicit treatment of the diffusive term. Unlike the periodic x and y directions, odd and even symmetries along the z direction deserve a specific treatment. The (even) tangential velocities and (odd) normal velocity are decomposed using cosine and sine transforms, respectively. Such a procedure enforces the impermeability and free-slip boundary conditions at the top and bottom surfaces:

w| z=0 = w| z=2π = 0, ∂u ∂z z=0 = ∂u ∂z z=2π = 0 and ∂v ∂z z=0 = ∂v ∂z z=2π = 0.
Forcing method Among the available methods for the numerical generation of statistically steady turbulent flows, we have followed the method of Alvelius. 7 A fully random three-dimensional force field is generated in the whole computational domain. It is implemented in spectral space and concentrated at small wave numbers according to a given spectrum

E(κ) ∝ exp[-(κ -κ f ) 2 /c].
We choose the forcing wave number κ f = 4 and a concentration parameter c = 0.05, the result of which is to confine the input of energy in the range κ = 2 to κ = 6. By construction, the random force is divergence free so that it does not influence the pressure field. Confinement of the forcing in the vicinity of a central plane parallel to the free surfaces is performed using a Heaviside function modulated by a sine function (see figure 1). The energy density injected by the forcing function at each time step is then equal to P = 0.025. After a transient stage needed for the dissipation to adapt to the amount of energy-power input, turbulence reaches a statistically steady state.

Calculation of the statistics Calculation of the statistics benefits from several properties of the solution: (i) statistical homogeneity in planes parallel to the surfaces, (ii) invariance under reflection about the midplane, (iii) invariance under rotation about the z axis, and (iv) statistical steadiness. For a given time, statistical quantities are calculated by first averaging over planes parallel to the free surface. Symmetry about the midplane is then used to average the data between planes located at an equal distance from each of the free surfaces. Finally, rotational invariance about z is used to average correlations involving tangential components of the fluctuation. The results of this spatial averaging phase are further averaged in time with samples spaced by half the large eddy turnover time τ . Hence, for a calculation performed on a n x ×n y ×n z domain during a time interval equal to n τ ×τ , the z-dependent statistics rely on 2n x n y n τ samples for correlations involving w, and twice that number for those involving at least one of the tangential velocity components. In the case presented below, we have:

n x = n y = 192, n z = 168 and n τ = 498.

Parameters and general properties of the solution

Definition of the computed flow Given the general dimensions and discretization of the computational domain, the flow will be fully defined by the value of the viscosity (ν). This value must be selected according to a resolution criterion for the smallest scales (κ max ). We use κ max η = 1.5 (η is the Kolmogorov lengthscale), and κ max is taken as n x /3 in order to avoid antialiasing in agreement with the two-third rule. Analysis of the result in the next sections will need meaningful references, we can use several sets of characteristic scales,

• surface references: the values k s and s of the turbulent kinetic energy and dissipation rate at the surface, the corresponding turbulent Reynolds number Re s is defined as k 2 s /(ν s ); • forcing references: the values k f and f at the edge of the forcing region and Quantities made nondimensional with the surface characteristic scales will be denoted by a star as a superscript, and those using the forcing scales by a tilde. Table 1 below gives the characteristics of the computed flow according to these two sets of references. Table 1: Definition of the flow case. The distance between the bottom surface and the edge of the forcing region is denoted as z f = (L z -L f )/2 and is given in both surface and forcing references.

Re f = k 2 f /(ν f ).
Establishment of the fluctuating field Figure 2 shows the time evolutions of the volume-averaged turbulent kinetic energy k and dissipation rate normalized by their final steady-state values (angle brackets denote volume average at a given time). The initial field is at rest when the forcing is switched on. Turbulence initiated in the forcing region is then diffused in the whole domain. After a transient stage equal to approximately 5 characteristic timescales, the flow reaches a statistical steady state: the figure shows that the volume averages of the turbulent kinetic energy and dissipation rate rise up to significant levels and fluctuate around these (constant) levels. In our study, turbulence production does not rely on the existence of mean-velocity gradients. The absence of a mean flow can be checked in figure 3 where the time evolutions of the volume-averaged components of the velocity field normalized by their r.m.s. final values have been plotted. One can see that these averages are less than 0.15 % at any time. The energy-spectrum function obtained at steady state in the domain is plotted in figure 4, it presents several characteristics worth to mention. First, the input of energy at κ = 4 is clearly visible on the plot. With increasing wave numbers, a region of limited extension exhibiting the Kolmogorov -5/3 slope follows, before the dissipation cut-off. At the high end of the spectrum, a slight pile-up of kinetic energy results from a good compromise between the value reached by the Reynolds number and the resolution of the smallest scales. This pile-up can be reduced by increasing the resolution criterion, but we have checked in a grid-independence study that the main part of the spectrum was unaffected as soon as κ max η = 1.5.

SECOND-ORDER STATISTICS

Figure 5 shows the profiles of the turbulent kinetic energy and Reynolds stresses, made nondimensional by k s . The top of the figure corresponds to the lower edge of the forcing region. The maximum of each of these three quantities is located there. Then, they decrease toward the bottom surface. As the surface is approached, the normal Reynolds stress (w 2 ) is forced to zero by the impermeability condition, while the tangential Reynolds stress (u 2 ) increases significantly to reach a local maximum at the surface. Intercomponent energy transfer probably contributes to this local maximum, but the turbulent kinetic energy profile also shows such a maximum at the surface, this latter effect cannot be explained by intercomponent energy transfer. The simulations of Perot and Moin 1 and Walker et al. [START_REF] Walker | Shear-free turbulence near a flat free surface[END_REF] for an ideal free surface also showed such a peak, in both studies it was suggested that the peak was due to a decrease in the dissipation rate. Our Reynolds-stress budgets presented in section 4 will support this interpretation.

The difference in the behaviour of the two Reynolds stresses is evidenced by the evolution of the isotropy factor defined as I = w /u , where u and w are the r.m.s. values of the tangential and horizontal components of the fluctuation. Figure 6 presents the vertical profile of this quantity. In the middle of the production region z c , turbulence is slightly anisotropic (I ≈ 0.9) at the expense of the vertical component of the fluctuation. As the random force field is isotropic, this is probably due to the z-confinement of the forcing. At the lower edge of the production region and moving toward the surface, the isotropy factor starts to increase to reach a maximum value about 1.16 at z * s = 0.79. At this location, w is therefore higher than u which is consistent with the experiments of turbulent diffusion from a plane source (oscillating-grid experiments, see De Silva and Fernando 4 for instance) where the isotropy factor is in the range 1.1-1.3. After reaching this maximum, the isotropy factor goes to zero at the free surface. We shall consider that the region located between the surface and z s is the surface-influenced region and call it the surface layer.

We present in figure 7(a,b) the evolutions of different lengthscales: the turbulence lengthscale = k 3/2 / , the longitudinal and transverse microscales λ f and λ g . The longitudinal microscale λ f = λ u

x is associated to gradients of tangential velocities and the transverse microscale λ g = λ w

x to the gradients of normal velocities. Between the lower edge of the forcing region and the top of the surface region, the turbulent lengthscale and the microscales exhibit different behaviours, the former slightly decreases while the latters increase. In this region, the two microscales remain approximately proportional, their ratio being about 1.30 is slightly lower than √ 2 (the value obtained in isotropic turbulence). In purely-diffusive turbulence, all lengthscales should increase with the distance from the Gaelle Campagne, Jean-Bernard Cazalbou, Laurent Joly and Patrick Chassaing source. However, such behaviour relies on a self-similarity hypothesis which is obviously not satisfied here due to the limited size of the pure-diffusion region. That is probably the reason why decreases there. In the surface region, the evolution of the different lengthscales exhibit some similarities. All of them steadily increase as the surface is approached except very near the surface where they experience a rapid change: exhibits a sharp peak and the microscales decrease. This region of abrupt change should correspond to what is usually called the viscous layer (see Magnaudet 3 ). We can estimate the height z v of this viscous layer with the location of the maximum of λ f , and retain z * v ≈ 0.09. The evolutions of the Reynolds numbers based on the turbulent lengthscale and microscales across the flow are plotted in figure 8(a,b). The Reynolds numbers based on the microscales are defined by Re λ = λ √ 1.5k/ν. All three Reynolds numbers decrease consistently across the pure-diffusion region. In the surface layer, their evolutions share the characteristics of those of the lengthscales: a mild variation across most of the layer followed by rapid changes in the viscous-layer. At this point, we note that the Reynolds numbers are not small in the so-called viscous layer: the turbulence Reynolds number and the longitudinal microscale Reynolds number rise up to values which are comparable to those obtained at the top of the surface layer. Examination of the Reynolds-stress budgets in the next section will also show that turbulent diffusion is not small as compared to viscous diffusion in this region, the term "viscous layer" will be kept in the following but should therefore be used with caution.

w 2 /k s u 2 /k s k/k s z *

REYNOLDS-STRESS BUDGETS

Transport equations for the components of the Reynolds-stress tensor u i u j are easily derived from the incompressible Navier-Stokes equations. When the flow is statistically steady with zero mean, they take the form

0 = - ∂u i u j u k ∂x k D u ij - 1 ρ ∂pu i ∂x j + ∂pu j ∂x i D p ij + ν ∂ 2 u i u j ∂x k ∂x k D ν ij + p ρ ∂u i ∂x j + ∂u j ∂x i Π ij -2ν ∂u i ∂x k ∂u j ∂x k ij , (2)
where D u ij , D p ij and D ν ij represent, respectively, diffusion by velocity fluctuations, pressure diffusion and viscosity; Π ij represents the pressure-strain correlation and ij the dissipation-rate tensor. These equations are valid outside of the forcing region, otherwise a production term corresponding to the random force would have to be added at the right hand side. Considering the symmetries of our problem, only u 2 (≡ v 2 ) and w 2 are nonzero among the components of the Reynolds-stress tensor, their transport equations take the form As k = u i u i /2, its transport equation can be deduced from the above:

0 = - ∂ u 2 w ∂z + ν ∂ 2 u 2 ∂z 2 + 2 p ρ ∂u ∂x -2 ν ∂u ∂x k ∂u ∂x k , ( 3
) 0 = - ∂ w 3 ∂z -2 ∂ ∂z p ρ w + ν ∂ 2 w 2 ∂z 2 + 2 p ρ ∂w ∂z -2 ν ∂w ∂x k ∂w ∂x k . (4)
0 = - 1 2 ∂ u i u i w ∂z D u - ∂ ∂z p ρ w D p D t + ν ∂ 2 k ∂z 2 D ν -ν ∂u i ∂x k ∂u i ∂x k . (5)
All the terms in equations (3-5) can be computed from the simulation results. The detailed structure of the budgets will be presented in the next paragraphs.

Tangential-stress budget

The budget of u 2 across the surface layer is presented in figure 9. In the pure-diffusion region (between z s and z f ) it closely resembles to that observed at the top of the surface layer and is therefore not reproduced here. At this location, the two dominant terms are the loss due to dissipation and the gain due to turbulent transport from the forcing region (by the velocity fluctuations only, since pressure diffusion is zero in this equation). A slight but noticeable positive contribution to the budget is also provided by the pressure-strain correlation. Recalling that the flow is slightly anisotropic in this region (w ≈ 1.15u ), it appears that the pressure-strain term acts conventionally as a return-to-isotropy term. As the surface is approached, dissipation and turbulent transport remain the dominant terms while the pressure-strain term goes to zero at z * ≈ 0.3, note that this is precisely the location where the isotropy factor reaches unity. The return-to-isotropy character of the intercomponent-energy-transfer process is thus remarkably verified down to this location.

Things really change in the viscous layer: all the terms in the budget become of the same order of magnitude. The viscous processes (diffusion and dissipation) both contribute negatively to the budget, they are balanced by the turbulent transport and pressurestrain terms. This exact balance between viscous and nonviscous terms confirms that the "viscous layer" is not fully dominated by viscosity and that such denomination may be misleading under certain circumstances. The value of the pressure-strain term clearly shows that intercomponent energy transfer is significant there, and acts in such a way that anisotropy is increased. Referring to the splat/anti-splat phenomenology introduced by Perot and Moin, 1 this would mean that the balance between splats and anti-splats is clearly in favour of the formers. A noticeable feature in the evolution of the dissipation rate in this region is the fact that, after reaching a moderate local maximum at the edge of the viscous layer ( 11 ≈ 1.2 s ), it decreases down to half this maximum at the surface (≈ 0.6 s ). This result is in agreement with the observations of Perot and Moin 1 and Walker et al., [START_REF] Walker | Shear-free turbulence near a flat free surface[END_REF] and supports the suggestion of these authors that such effect contributes to the increase of u 2 at the surface. Note, however, that the value at the surface of the other contributor, the pressure-strain term, is nearly twice the "dissipation deficit" (Π 11 = 1.16 s ). One last point worth to mention is the fact that, within the viscous layer, turbulent transport is antidiffusive. At present we are unable to provide an explanation of this result except that the counter-gradient character of diffusion by velocity fluctuations follows from Prandtl phenomenology which cannot be invoked when the characteristic scales of the fluctuating motion become of the same order as the distance to the blocking surface.

Normal-stress budget

Figure 10 shows the budget of w 2 across the surface layer. As compared to the u 2 budget, it involves one more term: pressure diffusion. The addition of this term and diffusion by velocity fluctuations will be denoted as the turbulent-transport term. At the top of the surface layer and down to z * ≈ 0.3, the budget shares the same characteristics as that of the tangential Reynolds stress: (i) it is dominated by turbulent transport from the forcing region and dissipation, (ii) the imbalance is accounted for by the pressure-strain correlation that acts as a return-to-isotropy term. We can notice, however, that turbulent diffusion by velocity fluctuations reaches much higher values than the corresponding term in the u 2 budget. This is also true, although to a lesser extend, when turbulent transport as a whole is considered, and should explain why the isotropy factor is above unity in this region. Note also that pressure diffusion is antidiffusive, as usually recognized for this term (see the model of Lumley 8 ).

In the viscous layer, all the terms contribute significantly to the balance except very near the surface where diffusion by velocity fluctuations goes to zero. In this region, the dissipation and the pressure-strain correlation both contribute negatively to the budget, the latter acting -as expected -against a return to isotropy. The positive contributions to the budget come from pressure diffusion and viscous diffusion. Nonviscous contributions (D p 33 and Π 33 ) are equal in magnitude and significantly higher (×3 at the surface) than the viscous contributions ( 33 and D u 33 ) which are also equal in magnitude. Another characteristic of the budget in this region is that pressure diffusion, which is antidiffusive in the top of the surface layer, now becomes diffusive. As mentioned above, this is at odds with the usual behaviour of this term.

Turbulent-kinetic-energy budget

In the turbulent-kinetic-energy budget, plotted in figure 11, we have regrouped turbulent transport in a single term. Consistently with the previously presented budgets, the level of turbulent kinetic energy level in the top of the surface region mainly results from turbulent transport from the forcing region and dissipation, viscous diffusion being negligible. In the viscous region, viscous diffusion and dissipation are comparable and contribute negatively to the budget, they are balanced by turbulent transport which is about twice as large as each of the latters. As in the u 2 budget, we can observe a dissipation deficit at the surface which is most probably one of the reasons for the increase of k there. Regrouping diffusion by velocity and pressure fluctuations in a single term helps to show that turbulent transport as a whole behaves as an antidiffusive process in the viscous layer. This unusual behaviour also contributes to the increase of the turbulent-kinetic-energy level at the surface.

SUMMARY AND CONCLUSION

The results presented in this paper shed some light on the interaction between turbulence and a free surface. Direct numerical simulation is performed in an original configura- tion where turbulence is continuously generated without mean shear in a finite-width layer parallel to the free surface. As a result the flow is statistically steady, turbulence remains nearly isotropic at some distance of the surface, and the surface layer is continuously fed by turbulent diffusion.

Examination of the statistics allows to identify a surface layer where the isotropy factor responds to the two-component limit imposed at the surface. Rapid changes in the profiles of the Reynolds stresses and characteristic lengthscales suggest the presence of a subregion located close to the surface within the surface layer. This region is usually called the viscous layer. Given the parameters of the simulation, both regions are easily distinguished, the surface layer being nearly ten times thicker than the viscous layer.

Inspection of the Reynolds-stress budget supports this partition of the flow: The upper part of the surface layer exhibits roughly the same kind of balance as that of a purelydiffusive turbulence, while the structure of the budgets radically changes in the viscous layer.

On the overall, three major conclusions can be drawn from our study:

• The pressure-strain correlation is an important term close to the wall, this makes our results closer to those obtained by Perot and Moin 1 in the simulation of decaying turbulence near a free surface at short times than to those of Walker et al. [START_REF] Walker | Shear-free turbulence near a flat free surface[END_REF] in the same situation at large times. It supports the interpretation of Walker et al. according to which anisotropy, present at large time in their simulation, is responsible for a return-to-isotropy mechanism superposed onto the splat/anti-splat phenomenology. This mechanism alters the balance between the two kinds of events and results in a lower level of pressure-strain correlation at the surface. In our con-figuration, anisotropy remains close to unity far from the surface, the pressure-strain correlation is therefore small outside of the viscous region, but constitutes a major contributor to the budget near the surface. We may thus suspect a clear imbalance between splats and anti-splats in favour of the formers.

• The so-called viscous layer is not dominated by viscous effects: the viscous terms are not dominant in the budget, and the turbulent Reynolds numbers based on the tangential velocity fluctuations or √ k are equal or even higher than those recorded at the top of the surface layer.

• The turbulent-diffusion terms present a rather unconventional behaviour in the viscous layer. Turbulent diffusion by velocity fluctuations is "antidiffusive" in the tangential Reynolds-stress budget, and pressure diffusion is "diffusive" in the normal Reynolds-stress budget. As a result, turbulent transport of turbulent kinetic energy taken as a whole is antidiffusive in the viscous layer. From a modelling point of view, it may be a delicate task to reproduce such a behaviour with a gradient diffusion scheme: it was shown by Cazalbou and Chassaing 9 that such a model would lead to severe realizability problems.

Figure 2 :

 2 Figure 2: Time evolutions of the volume-averaged turbulent kinetic energy (--) and dissipation rate (-), normalized by their final volume averages.

Figure 3 :

 3 Figure 3: Time evolution of the volume average of each of the three components of the velocity field, normalized by their root-mean-square final values: U (-), V (--), W ( ... ).

7 slope

 7 Gaelle Campagne, Jean-Bernard Cazalbou, Laurent Joly and Patrick Chassaing slope -

Figure 4 :

 4 Figure 4: Energy-spectrum function at t * = 50.

Figure 5 :Figure 6 :Figure 7 :

 567 Figure 5: Profiles of the turbulent kinetic energy and Reynolds stresses across the zero-production region.

Figure 8 :

 8 Figure 8: (a) Profile of the turbulent Reynolds number based on the turbulent lengthscale . (b) Profiles of the Reynolds numbers based on the longitudinal and transverse Taylor microscales λ f (-) and λ g (--). At the top of the surface layer (z = z s ), we have Re T = 48 and Re λf = 13 and, at the surface (z=0), Re T = 76 and Re λf = 13.

Figure 9 :

 9 Figure 9: Tangential-stress (u 2 ) budget plotted across the surface layer. (All data are normalized with surface references.)

0

  Figure 10: Normal-stress (w 2 ) budget plotted across the surface layer. (All data are normalized with surface references.)

0

  Figure 11: Turbulent-kinetic-energy budget plotted across the surface layer. (All data are normalized with surface references.)

Future work on the data obtained here will aim at investigating in more details the elementary structures responsible for the intercomponent energy transfer. Questions remain on these structures, whether they are all of the generic types "splat" and "anti-splat", if there is some possibility to quantify them separately. Answers to these questions would help to understand the influence of the nature of the blocking surface (rigid or free-slip) on the level of pressure-strain correlation.


Development of a Multiobjective Scheduler for Semiconductor Manufacturing

Keywords: Discrete Event Simulation, Artificial Neural Networks, Multicriteria Genetic Algorithm, Semiconductor manufacturing

Scheduling of semiconductor wafer fabrication system is identified as a complex problem, involving multiple and conflicting objectives (meeting due dates and minimizing waiting time for instance) to satisfy. In this study, we propose an effective approach based an artificial neural network technique embedded in a multiobjective optimization loop for multi-decision scheduling problems in a semiconductor wafer fabrication environment.

Introduction

Scheduling of semiconductor wafer fabrication system is identified as a difficult task, mainly because of the typical features of the process scheme, such as complex product flows (the so-called wafer fab is indeed a multipurpose plant), high uncertainties in operations, rapidly changing products and technologies (Ellis et al., 2004). It is thus a significant challenge to develop effective scheduling methods in wafer fabrication. Discrete-event simulation (DES) is one of the most widely used methods to study, analyze, design, and improve manufacturing systems. The combined used of a DES and an optimization procedure based on a genetic algorithm was an efficient solution to short-term job-shop scheduling problems and was adopted in our previous works (Charles et al., 2003). In spite of its acknowledged benefits, this kind of approach often reaches its limits in the industrial practice because of the highly combinatorial nature of the problem. In addition, the main emphasis of much of the work on scheduling has been on the development of predictive methodologies with a single objective. Actually, production managers have to cope with various objectives, which contributes to scheduling complexity: meeting due dates is an important goal in low-volume and high variety production circumstances within competitive market environments; another major objective in scheduling of semiconductor wafer fabrication is reducing waiting time for work-in-process (WIP) inventory to improve responsiveness to customers: in addition, the shorter the period that wafers are exposed to contaminants while waiting for process, the smaller the yield loss. Increasing the throughput is also an important stake, since the investment in fabrication equipment is capital intensive. In this study, we propose an approach based on an artificial neural network (ANN) technique coupled with a multiobjective genetic algorithm (MUGA) for multi-decision scheduling problems in semiconductor wafer fabrication. The paper is organized as follows. Section 2 is devoted to the general modelling framework. Then, the principles of the multiobjective optimization procedure are presented with some typical results. Then, we draw the conclusions on this work. O. Baez Senties, C. Azzaro-Pantel, L. Pibouleau, S. Domenech Laboratoire de Génie Chimique, UMR 5503, ENSIACET INPT. 5, rue Paulin Talabot -BP 1301 -31106 Toulouse, Cedex 01 -France

General framework and plant modelling

The main emphasis of much of the work on scheduling has been on static systems with a single objective. As noted above, wafer fabrication is complex, dynamic and highly stochastic. Satisfying the multiple objectives might be more important than only optimally meeting a single objective. In this study, we propose an effective approach based on an artificial neural network technique for multiobjective and multi-decision scheduling problems in semiconductor wafer fabrication. Figure 1 shows the role of the proposed methodology, which will be presented in more detail what follows.

Discrete-event simulation

To model the wafer fab in a high degree of detail, Discrete Event Simulation (DES) techniques were previously implemented, leading to the development of MELISSA software (see (Charles et al., 2003) for more detail). Typical events taken into account and managed in the simulation core have been widely presented in (Bérard et al., 1999) and will not be recalled here. It was largely used and validated in previous works and reflects the behaviour of the plant. The different runs performed with MELISSA allow identifying the more sensitive variables on some performance criteria, such as makespan, cycle time, respect of due dates, limitation of WIP etc... The example which serves here to illustrate the methodology is presented in Figure 2 and imitates the fabrication plant of a typical semiconductor manufacturing process. sequence, to reproduce the so-called re-entrant flow. Some units are in parallel, which confers more flexibility to the process. We consider the production of 5 different families of products with their associated recipe identified by a number (1,2,3,4,5). A batch is constituted by 50 wafers and 3 levels of productions are to be considered according to market demand, i.e. 16, 24 and 32 batches. A horizon time of 10000 min was chosen as a reference.

Decision variables

The definition of each decision variable is proposed in Table 1, where an example is developed. Each wafer batch is identified by a number corresponding to its family. A campaign is constituted by the release of several products belonging to various family sequences into the wafer fab. 3 discrete values (2,3,4) ex : 2 This variable sets the number of batches within a campaign [1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4] campaign 1 [1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4] campaign 2 Time Between Batches (TBB) (identical from a batch to the following one, whenever the campaign) 3 discrete values (120, 750, 1250) min ex : ex : [1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4] campaign 1 TBB = 120 min Time between Campaigns (TBC) 5 discrete values (500,1500,1900,2400,2800) [1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4] campaign 1

Separated by TBC = 500 min from [1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4] campaign 2

Performance criteria

There are multiple criteria that can be used in evaluating the system performance and system status of the semiconductor fabricator. The criteria are mainly based on completion times, due-dates, inventory level, or machine utilization. Two criteria were selected here, involving the computations of delay/advance of the due date (C 1 ) called ADV/DEL and of the average waiting time (AWT) of all products (n) at each processing unit (C 2 ) (WT i refers to the waiting time of product i).

(

)

2 1 / ) ( Delays Advance DEL ADV C + = = n WT AWT C n i i ∑ = = = 1 1 2 ) ( (2)
(1)

Multiobjective optimization

The neural networks have then been embedded in a multiobjective genetic algorithm (MOGA) to optimize the decision variables and to deal with the set of compromise solutions for the studied criteria, thus giving the optimal Pareto zone solutions.

Lately, there has been a large development of different types of multiobjective genetic algorithms, which are reflected in the literature. The big advantage of genetic algorithms

It must be emphasized that the expression selected for criterion C 1 will penalize more the delays than the advances of products. These expressions will then been used at the optimization step with the objectives of Min (C 1 ) and Min (C 2 ).

ANN modelling

We then resort to an artificial neural network technique (Dreyfus et al., 2004) to model the semiconductor plant, since its efficiency has been successfully demonstrated in semiconductor processing by many researchers (Min and Yih, 2003) (Sun and Choung, 1999). The training phase was carried out by a classical backpropagation algorithm. For this purpose, the MATLAB software package dedicated to neural network modelling was used. The neurons of the input layer correspond to the wafer manufacture process parameters (decision variables). The output neurons correspond to the wafer manufacture performance parameters to be evaluated (criteria). The same network structure was adopted for each criterion. The network also includes one hidden layer, which helps the network in learning the non-linear mapping between the input and output layers. The training was carried out systematically with varying the number of hidden neurons (a number of 30 was finally selected). All these models were based on the multilayer perceptron architecture (Cheng and Billings, 1992). Tan-sigmoid transfer function was used as an activation function for hidden and output layers. The values of the test data were normalized within the range from -1 to 1. The root mean square error (RMSE) was chosen as a criterion for supervised training. The error is computed as the difference between the target and network output values. The total combination set of decision variables was generated to build the neural network, i.e. 5400 (training and test data, 2:1) at the preliminary stage of the study. Typical results obtained with MELISSA and ANN computations can be shown in Figure 3 and exhibit a good agreement between the two sets of values, thus showing the efficiency of ANN to model the batch plant.

over other methods, particularly over other stochastic procedures such as Simulated Annealing, is that a GA manipulates a population of individuals.

It is therefore tempting to develop a strategy in which the population captures the whole Pareto front in one single optimization run. Following the guidelines proposed by (Dietz et al., 2005a, b), this approach was adopted in this work. The same encoding procedure was used since only integer variables are involved in the problem formulation (more detail concerning crossover and mutation can be found in (Dietz et al., 2005a, b). He can thus validate his decision by using the discrete-event simulator to determine precisely all the scheduling parameters. A major interest is that the approach may find wide applications in practice since the data base may be updated with the feedback from the real process, thus improving continuously the model accuracy in time, as seen in Figure 1.

Conclusion

Semiconductor wafer fabrication involves an important number of decision problems. The objective of this paper was to propose an optimization strategy in order to assign appropriate decision variables. More precisely, a scheduler for selection of decision

Fig

  Fig. 1. General methodology

Fig. 2 .

 2 Fig. 2. Typical semiconductor plant It consists in 14 different equipment units used in a process involving 24 operating stages. Two shared equipment units are used up to 6 times in the manufacturing

Fig. 3 -

 3 Fig. 3-Comparison between ANN and MELISSA computed values for ADV/DEL criterion and for AWT criterion [min]

Fig 4 .

 4 Fig 4. Combination between neural networks models and multiobjective genetic algorithm

  4b 5b 6b 7b 1a 2a 3a 4a 5a 6a 7a 1b 2b 3b 4b 5b 6a 7a 1a 2a 3a 4a 5a 6b 7bParent «A» Parent «B» Enfant «AB» Enfant «BA»

Table 1

 1 Decision variables

	Decision variable	Number and type of parameter
	Maximum number of recipes	5
	Maximum number of recipes treated simultaneously (NC)	ex . 4 among the 5 available recipes
		24 combinations, ex : 1,2,3,4
	Identification of family sequence (FS)	Batches are always released with this sequence
		order into production
	Total of batches (NP)	3 discrete values (16, 24, 32) ex : 16
	Number of campaigns (NC)

variables in order to obtain desired performance measures at the end of a certain production interval was developed. In the proposed methodology, a three-level strategy based on the combined use of a simulation technique, a neural network and a multiobjective genetic algorithm is suggested. The results indicate that this methodology is an effective method considering the complexity of semiconductor wafer fabrication systems, as a time-saving way to achieve a prompt response in a dynamically changing environment. The computational time for reaching the Pareto's solutions is about 2000 seconds, that is much faster than the process real-time (about 6 days). If the DES model is used instead of the ANN the time is multiplied by 100, and the computational time is in the same order of magnitude than the real time.


Joachim Escher
email: escher@ifam.uni-hannover.de

Boris Kolev
email: kolev@cmi.univ-mrs.fr

THE DEGASPERIS-PROCESI EQUATION AS A NON-METRIC EULER EQUATION

Keywords: 2000 Mathematics Subject Classification. 35Q53, 58D05 Euler equation, diffeomorphisms group of the circle, Degasperis-Procesi equation

In this paper we present a geometric interpretation of the periodic Degasperis-Procesi equation as the geodesic flow of a right invariant symmetric linear connection on the diffeomorphism group of the circle. We also show that for any evolution in the family of b-equations there is neither gain nor loss of the spatial regularity of solutions. This in turn allows us to view the Degasperis-Procesi and the Camassa-Holm equation as an ODE on the Fréchet space of all smooth functions on the circle.

Introduction

Due to the highly involved structure of the full governing equations for the classical water wave problem, it is intriguing to approximate them by mathematically easier models. In the shallow-water medium-amplitude regime, introduced to capture stronger nonlinear effects than in the smallamplitude regime (which leads to the famous Korteweg-de Vries equation), the Camassa-Holm (CH) equation [START_REF] Camassa | An integrable shallow water equation with peaked solitons[END_REF] and the Degasperis-Procesi (DP) equation [START_REF] Degasperis | Asymptotic integrability, Symmetry and perturbation theory[END_REF] attracted a lot of attention due to their integrable structure as infinite bi-Hamiltonian systems [START_REF] Constantin | Inverse scattering transform for the Camassa-Holm equation[END_REF][START_REF] Degasperis | A new integrable equation with peakon solutions[END_REF] and to the fact that their solitary wave solutions are solitons. Moreover it is known that this medium-amplitude regime allows for breaking waves, cf. [START_REF] Constantin | The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations[END_REF][START_REF] Johnson | Camassa-Holm, Korteweg-de Vries and related models for water waves[END_REF][START_REF] Constantin | Wave breaking for nonlinear nonlocal shallow water equations[END_REF] Both of these equations are members of the so-called family of 'b-equations' [START_REF] Degasperis | A new integrable equation with peakon solutions[END_REF][START_REF] Hone | Prolongation algebras and Hamiltonian operators for peakon equations[END_REF] (1)

m t = -(m x u + bmu x )
where m = uu xx . For b = 2 it is the Camassa-Holm equation (CH) [START_REF] Arnold | Topological methods in hydrodynamics[END_REF] u tu txx + 3uu x -2u x u xxuu xxx = 0 and for b = 3 it is the Degasperis-Procesi equation (DP)

(3)

u t -u txx + 4uu x -3u x u xx -uu xxx = 0.
The hydrodynamic relevance of the b-equations is described e.g. in [START_REF]The classical problem of water waves: a reservoir of integrable and nearlyintegrable equations[END_REF][START_REF]Water waves and integrability[END_REF]. It is integrable only for b = 2 and b = 3 (see [START_REF] Hone | Prolongation algebras and Hamiltonian operators for peakon equations[END_REF][START_REF] Mikhailov | Perturbative symmetry approach[END_REF][START_REF] Ivanov | On the integrability of a class of nonlinear dispersive waves equations[END_REF]). The periodic Camassa-Holm equation is known to correspond to a geodesic flow on the diffeomorphism group of the circle, cf. [START_REF] Kouranbaeva | The Camassa-Holm equation as a geodesic flow on the diffeomorphism group[END_REF]. Local existence of the geodesics and properties of the Riemannian exponential map were studied in [START_REF] Constantin | On the geometric approach to the motion of inertial mechanical systems[END_REF][START_REF]Geodesic flow on the diffeomorphism group of the circle[END_REF].

The main objective of the present paper is to extend this work to the whole family of b-equations and in particular for (DP).

Analogous to the Camassa-Holm equation, see [START_REF]Well-posedness, global existence, and blowup phenomena for a periodic quasilinear hyperbolic equation[END_REF][START_REF] Rodríguez-Blanco | On the Cauchy problem for the Camassa-Holm equation[END_REF], the Cauchy problem for the b-equation is locally well-posed in the Sobolev space H s for any s > 3/2 and the solution depends continuously on the initial data, cf. [START_REF] Escher | Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation[END_REF][START_REF] Escher | Well-posedness, blow-up phenomena, and global solutions for the b-equation[END_REF]. The geometric approach of the present paper enables us to complete this picture by showing that the b-equation is well-posed in the smooth category C ∞ (J, C ∞ (S 1 )), where J is an open interval containing 0. The situation of analytic initial data and the question of analyticity of the corresponding solution will be addressed in a forthcoming paper.

We already mentioned several common properties of the (CH) and the (DP) equation. Besides the fact that the connection on Diff ∞ (S 1 ) in the geometric re-expression of the (DP) equation is non-metric, it is worth pointing out that the (DP) equation possesses bounded but discontinuous solitons, so-called shock peakons, cf. [START_REF] Escher | Wave breaking and shock waves for a periodic shallow water equation[END_REF]. Indeed, it is possible to verify that

u c (t, x) = sinh (x -[x] -1/2) t cosh (1/2) + c sinh (1/2) , x ∈ R/Z,
is for any c > 0 a weak solution to the (DP) equation. Note that the (CH) equation has no discontinuous weak solutions, due to the fact that the H 1 -norm of any weak solution is preserved. Another significant difference becomes clear by comparing the isospectral problem in the Lax pair for both equations. Letting again m = uu xx , we have the third order equation

ψ x -ψ xxx -λmψ = 0
as the isospectral problem for the (DP) equation (cf. [START_REF] Degasperis | A new integrable equation with peakon solutions[END_REF]), while in the case of the (CH) equation, the second order equation

ψ xx - 1 4 ψ -λmψ = 0
is the relevant spectral problem in the Lax pair, cf. [START_REF] Camassa | An integrable shallow water equation with peaked solitons[END_REF].

The paper is organized as follows. In Section 2, we introduce the notion of non-metric Euler equation. Some basic material about the diffeomorphism group Diff ∞ (S 1 ) is recalled in Section 3. In Section 4, we show that the bequation may be recast as the geodesic flow of a symmetric linear connection on Diff ∞ (S 1 ) and that the corresponding second order vector field is smooth. In Section 5, we show that there is neither gain nor loss of the spatial regularity of solutions and we establish short time existence and smooth dependence on the initial data for this geodesic flow in the smooth category. In Section 6, we show that the exponential map of this symmetric linear connection is a smooth local diffeomorphism. In the last Section 7, we discuss the non-periodic case.

Non-metric Euler equations

The idea of studying geodesic flows in order to analyze the motion of hydrodynamical systems is attributed to Arnold [START_REF] Arnold | Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits[END_REF] (despite the fact that a previous short paper from Moreau in 1959 already discussed the problem [START_REF] Moreau | Une méthode de "cinématique fonctionnelle" en hydrodynamique[END_REF]). He pointed out that the Euler equations of the motion of a rigid body and the Euler equations of hydrodynamics (with fixed boundary) could both be recast as the geodesic equations of a one-sided invariant Riemannian metric on a Lie group.

This structure is the prototype for the mathematical treatment of physical systems which underlying configuration space can be identified with a Lie group G. The general Euler equation was derived initially for the Levi-Civita connection of a one-sided invariant Riemannian metric on a Lie group G (see [START_REF] Arnold | Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits[END_REF] or [START_REF] Arnold | Topological methods in hydrodynamics[END_REF]) but the theory is valid in the more general setting of a one-sided invariant linear connection on G, as we shall explain now.

A linear connection

(covariant derivative) on a manifold M is a bilinear map Vect(M ) × Vect(M ) → Vect(M ), denoted by (X, Y ) → ∇ X Y , such that: ∇ f X Y = f ∇ X Y, and
∇ X (f Y ) = (X • f )Y + f (∇ X Y ), for all X, Y ∈ Vect(M ) and f ∈ C ∞ (M ). The linear connection ∇ is symmetric if moreover ∇ X Y -∇ Y X = [X, Y ]
for all X, Y ∈ Vect(M ), where [X, Y ] is the usual bracket of vector fields. Given a local chart (x j ) (or more generally, any local frame), a linear connection ∇ is completely defined by the Christoffel symbols

Γ k ij = (∇ ∂ i (∂ j )) k .
A covariant derivative ∇ can be uniquely extended for smooth vector fields X(t) defined only along a smooth path x(t) on M (see [START_REF] Lang | Fundamentals of differential geometry[END_REF] for instance). This operator, denoted by D/Dt, is an extension of ∇ in the sense that if the vector field X(t) is the restriction to the curve x(t) of a globally defined vector field X, that is X(t) = X(x(t)) then we have

DX Dt (t) = (∇ ẋ X) (x(t)).
In a local chart (x j ), D/Dt is defined by

DX Dt k = Ẋk + Γ k ij ẋi X j . Definition 1. A geodesic for the linear connection ∇ is a smooth curve x(t) on M such that D ẋ Dt = 0.
Given a local chart (x j ) of M , the smooth curve x(t) is a geodesic iff it satisfies the following second order differential equation ẍk + Γ k ij ẋi ẋj = 0.

Remark 1. If two linear connections differ by a skew-symmetric tensor, they have the same geodesics.

Let ϕ be a diffeomorphism of M . We say that the linear connection ∇ is invariant under ϕ if ϕ * (∇ X Y ) = ∇ ϕ * X ϕ * Y where ϕ * X is the vector field defined by

(ϕ * X)(x) = T ϕ(x) ϕ -1 X(ϕ(x)).
Suppose now that M is a Lie group G. Then, we say that a linear connection on G is right invariant (resp. left invariant) if it is invariant under each right translation R g (resp. left translation L g ) of G. On any Lie group G, there is a canonical connection which is both right and left invariant (we say it is bi-invariant). It is defined by

∇ 0 ξu ξ v = 1 2 [ξ u , ξ v ],
where ξ u is the right invariant vector field on G generated by the vector u in the Lie algebra g of G.

Remark 2. This definition does not change if we use left invariant vector fields rather than right invariant vector fields in the preceding formula.

Given a right-invariant linear connection ∇ on G, then

B(X, Y ) = ∇ X Y -∇ 0 X Y is a right-invariant
tensor field on G and is therefore completely defined by its value at the unit element e of G, that is by a bilinear map g × g → g. Conversely, each bilinear operator B : g × g → g defines uniquely a right invariant linear connection on the Lie group G, which is given by ( 4)

∇ ξu ξ v = 1 2 [ξ u , ξ v ] + B(ξ u , ξ v ),
where we use the same notation for B and the right invariant tensor field it generates on G. Given a basis (e k ) 1≤k≤n of the Lie algebra g, let (ξ k ) be the global right invariant frame on G generated by the basis (e k ) of g and let (ω k ) be the dual co-frame of (ξ k ) (notice that each ω k is itself a right invariant 1-form on G).

Let g(t) be a smooth path in G. We define its Eulerian velocity by

u(t) = R g -1 (t) ġ(t)
which is a smooth path in the Lie algebra g of G. Let u k be the components of u in the basis (e k ). Then

u k = ω k e (u) = ω k g (R g u) = ω k g ( ġ)
are the components of the vector ġ in the frame (ξ k ). The linear operator D/Dt defined along g(t) is then given by

DX Dt k = Ẋk + 1 2 c k ij + b k ij u i X j
where c k ij are the structure constants of the Lie algebra g and b k ij are the components of the tensor B. In particular, a smooth curve g(t) in G is a geodesic if and only if uk + b k ij u i u j = 0 because c k ji = -c k ij for all i, j, k. Proposition 2. A smooth curve g(t) on a Lie group G is a geodesic for a right invariant linear connection ∇ defined by (4) iff its Eulerian velocity u = R g -1 ġ satisfies the first order equation ( 5)

u t = -B(u, u).
known as the Euler equation.

In particular, every quadratic operator in g corresponds to the geodesic flow of a right-invariant symmetric linear connection on G. Notice however that this connection is not necessarily Riemannian -there may not exist a Riemannian metric which is preserved by this connection. We will therefore call such an equation as a non-metric Euler equation. Remark 3. In the metric case, that is when ∇ is the Levi-Civita connection of a right invariant Riemannian metric on G, the bilinear operator B is related to the metric by the following formula

B(u, v) = 1 2 (ad u ) * (v) + (ad v ) * (u)
where u, v ∈ g and (ad u ) * is the adjoint (relatively to the given Riemannian metric) of the natural action of the Lie algebra on itself given by

ad u : v → [u, v].
3. The Fréchet Lie group Diff ∞ (S 1 )

In this paper, we are interested in a (non metric) Euler equation on the infinite dimensional Lie group Diff ∞ (S 1 ) of smooth, orientation preserving diffeomorphisms of the unit circle S 1 .

Since the tangent bundle

T S 1 ≃ S 1 × R is trivial, Vect ∞ (S 1
), the space of smooth vector fields on S 1 , can be identified with C ∞ (S 1 ), the space of real smooth functions on S 1 . Given a diffeomorphism ϕ ∈ Diff ∞ (S 1 ), we define ϕ x ∈ C ∞ (S 1 ) by the following construction. Let p : R → S 1 , x → exp(2iπx) be the universal cover of the circle. A lift of ϕ is a smooth map f : R → R such that

ϕ(exp(2iπx)) = exp(2iπf (x)).
Since ϕ preserves the orientation we have f (x + 1) = f (x) + 1. A lift is not unique but two lifts of ϕ differ by an integer. For any lift f of ϕ, its derivative f ′ is a smooth periodic map on S 1 and we define ϕ x as

ϕ x := f ′ ,
which is independent of the choice of a particular lift of ϕ.

The space C ∞ (S 1 ) is a Fréchet space 1 . A family of semi-norms which defines the topology of C ∞ (S 1 ) is just given by the C n -norms. A sequence u k converges to u in C ∞ (S 1 ) if and only if

u k -u C n (S 1 ) → 0 as k → ∞ for all n ≥ 0.
Remark 4. In a Fréchet space, only directional or Gâteaux derivative is defined

Df (x)u = lim ε→0 1 ε (f (x + εu) -f (x)). A map f : X → Y between Fréchet spaces X, Y is continuously differentiable (C 1 ) on U ⊂ X if the directional derivative Df (x)
u exists for all x in U and all u in X, and the map (x, u) → Df (x)u is continuous. Note that even in the case where X and Y are Banach spaces this definition of continuous differentiability is weaker than the usual one [START_REF] Hamilton | The inverse function theorem of Nash and Moser[END_REF]. Higher derivatives and C n classes in Fréchet spaces are defined inductively.

The group Diff ∞ (S 1 ) is naturally equipped with a Fréchet manifold 2 structure modeled over the Fréchet vector space C ∞ (S 1 ). A smooth atlas with only two charts may be constructed as follows (see [START_REF] Guieu | Algèbre de Virasoro: aspects géométriques et algébriques[END_REF]). Given ϕ ∈ Diff ∞ (S 1 ), it is always possible to find a lift f : R → R of ϕ such that either

(6) -1/2 < f (0) < 1/2, or (7) 0 < f (0) < 1,
these conditions being not exclusive. Let V 1 and V 2 be the subsets of ϕ ∈ Diff ∞ (S 1 ) for which there is a lift satisfying respectively (6) or [START_REF] Constantin | On the geometric approach to the motion of inertial mechanical systems[END_REF]. Notice that

V 1 and V 2 are open and that V 1 ∪ V 2 = Diff ∞ (S 1 ). For any lift f of ϕ ∈ Diff ∞ (S 1 ), the function u = f -id is 1-periodic and hence lies in C ∞ (S 1 ) and moreover u ′ (x) > -1. Let U 1 = u ∈ C ∞ (S 1 ) : -1/2 < u(0) < 1/2 and u ′ > -1 and U 2 = u ∈ C ∞ (S 1 ) : 0 < u(0) < 1 and u ′ > -1 .
For j = 1, 2, the maps

Φ j : U j → V j , u → f = id + u, j = 1, 2 define charts of Diff ∞ (S 1 ) with values in C ∞ (S 1
). The change of charts corresponds to a change of lift and is just a translation in C ∞ (S 1 ) by ±1.

1 A topological vector space E has a canonical uniform structure. When this structure is complete and when the topology of E may be given by a countable family of semi-norms, we say that E is a Fréchet vector space. A Fréchet space is a Banach space if and only if it is locally bounded, which is not the case of C ∞ (S 1 ). 2 Fréchet manifolds are defined as sets which can be covered by charts taking values in a given Fréchet space and such that the change of charts are smooth.

Since the composition and the inverse are smooth maps we say that Diff ∞ (S 1 ) is a Fréchet-Lie group [START_REF] Hamilton | The inverse function theorem of Nash and Moser[END_REF]. In particular, Diff ∞ (S 1 ) is itself parallelizable

T Diff ∞ (S 1 ) ≃ Diff ∞ (S 1 ) × C ∞ (S 1
). The tangent space at the unit element T id Diff ∞ (S 1 ) -the Lie algebra of Diff ∞ (S 1 ) -is defined as follows. Let t → ϕ(t) be a smooth path in Diff ∞ (S 1 ) with ϕ(0) = id. Then

ϕ t (0, x) ∈ T x S 1 , ∀x ∈ T x S 1
and therefore ϕ t (0, •) is a vector field on S 1 . In other words, the Lie algebra of Diff ∞ (S 1 ) corresponds to Vect ∞ (S 1 ). The Lie bracket on Vect

∞ (S 1 ) ≃ C ∞ (S 1 ) is given by 3 [u, v] = u x v -uv x .
Remark 5. For n ≥ 1, we define the group Diff n (S 1 ) of orientation-preserving diffeomorphisms of class C n . This group is equipped with a smooth Banach manifold structure modeled on the Banach vector space C n (S 1 ). However, Diff n (S 1 ) is only a topological group and not a Banach Lie group since the composition and the inverse are continuous but not differentiable [START_REF] Ebin | Groups of diffeomorphisms and the notion of an incompressible fluid[END_REF].

4. The b-equation as a non-metric Euler equation on Diff ∞ (S 1 )

We can recast equation ( 1) as ( 8)

u t = -A -1 [u(Au) x + b(Au)u x ] ,
where

A = I - d 2
dx 2 is an invertible, linear, differential operator on C ∞ (S 1 ). As a quadratic evolution equation on the Lie algebra Vect ∞ (S 1 ), each member of the bfamily corresponds to a the Euler equation of a right invariant symmetric linear connection on Diff ∞ (S 1 ).

On a Fréchet manifold, only covariant derivatives along curves are meaningful. In the present case, the covariant derivative of the vector field

ξ(t) = ϕ(t), w(t) along the curve ϕ(t) ∈ Diff ∞ (S 1 ) is defined as Dξ Dt (t) = ϕ(t), w t + 1 2 [u(t), w(t)] + B u(t), w(t)
where

u(t) = ϕ t • ϕ -1 and B(u, w) = 1 2 A -1 [u(Aw) x + w(Au) x + b(Au)w x + b(Aw)u x ] .
This connection is Riemannian only for b = 2 (see [START_REF] Kolev | Some geometric investigations on the degasperis-procesi shallow water equation[END_REF]), that is for (CH). In that case it corresponds to the Levi-Civita connection of the right invariant metric on Diff ∞ (S 1 ) generated by the H 1 inner product on C ∞ (S 1 )

< u, v > H 1 = S 1 (uv + u x v x ) dx = S 1
uA(v) dx 3 Notice that this bracket differs from the usual bracket of vector fields by a sign.

where u, v ∈ C ∞ (S 1 ) and the operator

B(u, u) = A -1 [(Au x )u + 2(Au)u x ]
is defined as the H 1 -adjoint of the natural action ad

u v = u x v -uv x . That is S 1 (u x v -uv x )A(w) dx = S 1 B(w, u)A(v),
where u, v, w ∈ C ∞ (S 1 ).

Euler equation ( 8) is not an ordinary differential equation (ODE) on C n (S 1 ) because of the term (Au) x which is not regularized by the operator A -1 of order -2 (the right hand side of [START_REF]Geodesic flow on the diffeomorphism group of the circle[END_REF] does not belong to C n (S 1 )). As we shall show now, the introduction of Lagrangian coordinates allows to rewrite equation ( 8) as a second order vector field on the Banach manifold Diff n (S 1 ) for n ≥ 2.

Let J be an open interval in R and u be a time dependent vector field on J × S 1 of class C n (n ≥ 1). Then, the flow ϕ of u is defined on J × S 1 and is of class C n as well as ϕ t = u • ϕ. Conversely, let (ϕ, v) be defined, of class C n on J × S 1 and such that ϕ(t) is a diffeomorphism for all t ∈ J. Then ϕ -1 is of class C n on J × S 1 and similarly for u = v • ϕ -1 . Proposition 3. The C 3 time dependent vector field u is a solution of (8) if and only if (ϕ, v) is solution of

(9) ϕ t = v, v t = -P ϕ (v),
where

P ϕ = R ϕ • P • R ϕ -1 and P (u) = A -1 [3u x u xx + b(Au)u x ]
Proof. Let u be a time dependent vector field on J × S 1 and ϕ its flow. Let v := ϕ t = u • ϕ, then v t = (u t + uu x ) • ϕ. Therefore, u is a solution of equation ( 8) if and only if

u t + uu x = -A -1 [u(Au) x -A(uu x ) + b(Au)u x ] = -A -1 [3u x u xx + b(Au)u x ] .
That is u is a solution of equation ( 8) if and only if (ϕ, v) is a solution of equation ( 9).

Since P is a pseudo-differential operator of order 0, the mapping

F (ϕ, v) = v, P ϕ (v) sends Diff n (S 1 ) × C n (S 1 ) into C n (S 1 ) × C n (S 1
). Notice however, that since the trivialization T Diff n (S 1 ) ≃ Diff n (S 1 )×C n (S 1 ) is only topological and not smooth, equation ( 9) is meaningful only locally (strictly speaking, ϕ should be replaced by its lift f in this expression). In any case, F is a well-defined second order vector field on the Banach manifold Diff n (S 1 ).

Remark 6. By its very definition, F is equivariant by the right action of Diff n (S 1 ) on Diff n (S 1 ) × C n (S 1 ). That is

(10) F (ϕ • ψ, v • ψ) = F (ϕ, v) • ψ for all (ϕ, v) ∈ Diff n (S 1 ) × C n (S 1
) and ψ ∈ Diff n (S 1 ).

Theorem 4. The vector field

F : Diff n (S 1 ) × C n (S 1 ) → C n (S 1 ) × C n (S 1 )
is smooth for all n ≥ 2.

We cannot conclude directly from the smoothness of P that F is smooth because neither the composition nor the inversion are smooth maps on Diff n (S 1 ). The Banach manifold Diff n (S 1 ) is only a topological group. The main argument in the proof of Theorem 4 is the following lemma.

Lemma 5. Let P be a polynomial differential operator of order r on C ∞ (S 1 ), with constant coefficients. Then

P ϕ = R ϕ • P • R ϕ -1
is a polynomial differential operator of order r whose coefficients are rational functions of ϕ x , . . . ϕ (r)

x . Proof. It suffices to prove the lemma for a monomial

P (u) = u p 0 (u ′ ) p 1 • • • (u (r) ) pr .
We have

P ϕ (u) = u p 0 (u • ϕ -1 ) ′ • ϕ p 1 • • • (u • ϕ -1 ) (r) • ϕ pr . Let a k = (u • ϕ -1 ) (k) • ϕ, k = 1, 2, . . . Since a 1 = u x ϕ x , and
a k+1 = 1 ϕ x (a k ) ′
the proof is achieved by a recursive argument.

Proof of Theorem 4. To prove the smoothness of F , we will show that for each n ≥ 2, the operator

P (ϕ, v) = ϕ, R ϕ • P • R ϕ -1 (v)
is a smooth map from Diff n (S 1 ) × C n (S 1 ) to itself. To do so, we write P as the composition P = Ã-1 • Q, where

Ã(ϕ, v) = ϕ, R ϕ • A • R ϕ -1 (v) and Q(ϕ, v) = ϕ, R ϕ • Q • R ϕ -1 (v)
where Q is the quadratic differential operator defined by

Q(u) := 3u x u xx + b(Au)u x .
By virtue of Lemma 5, both à and Q are smooth maps from Diff n (S 1 ) × C n (S 1 ) to Diff n (S 1 )×C n-2 (S 1 ), since the spaces C k (S 1 ) are Banach algebras.

To show that Ã-1 :

Diff n (S 1 ) × C n-2 (S 1 ) → Diff n (S 1 ) × C n (S 1
) is smooth, we compute the derivative of à at an arbitrary point (ϕ, v), obtaining

D Ã(ϕ, v) = Id 0 * R ϕ • A • R ϕ -1
which is clearly a bounded linear operator from C n (S 1 )×C n (S 1 ) to C n (S 1 )× C n-2 (S 1 ) by virtue of Lemma 5. It is moreover a topological linear isomorphism since it is bijective (open mapping theorem). The application of the inverse mapping theorem [START_REF] Lang | Fundamentals of differential geometry[END_REF] in Banach spaces achieves the proof.

Short time existence of the geodesics on Diff

∞ (S 1 )
From the smoothness of F and the Cauchy-Lipschitz theorem in Banach spaces (see [START_REF] Lang | Fundamentals of differential geometry[END_REF] for instance) we deduce existence of integral curves for the vector field F on Diff n (S 1 ) × C n (S 1 ). Proposition 6. Let n ≥ 2. Then there exists a open interval J n centered at 0 and an open ball B n (0, δ n ) in C n (S 1 ) such that for each u 0 ∈ B(0, δ n ) there exists a unique solution (ϕ, v) ∈ C ∞ J n , Diff n (S 1 )×C n (S 1 ) of ( 9) such that ϕ(0) = id and v(0) = u 0 . Moreover, the flow (ϕ, v) depends smoothly on (t, u 0 ).

Classical results like the Cauchy-Lipschitz theorem or the local inverse theorem for Banach spaces are no longer valid in general for Fréchet spaces. To establish a short time existence result in the C ∞ category, it is useful to introduce the notion of Banach approximation of Fréchet spaces. Definition 7. A Banach approximation of a Fréchet space X is a sequence of Banach spaces (X n , • n ) n≥0 such that

X 0 ⊇ X 1 ⊇ X 2 ⊇ • • • ⊇ X and X = ∞ n=0 X n
where { • n } n≥0 is a sequence of norms inducing the topology on X and

x 0 ≤ x 1 ≤ x 2 ≤ • • • for any x in X.
For Fréchet spaces admitting Banach approximations one has the following lemma. Lemma 8. Let X and Y be Fréchet spaces with Banach approximations (X n ) n≥0 , and (Y n ) n≥0 respectively. Let Φ 0 : U 0 → V 0 be a smooth map between the open subsets U 0 ⊂ X 0 and V 0 ⊂ Y 0 . Let U = U 0 ∩X, V = V 0 ∩Y and for each n ≥ 0

U n = U 0 ∩ X n , V n = V 0 ∩ Y n .
Assume that, for each n ≥ 0, the following properties are satisfied:

(1

) Φ 0 (U n ) ⊂ Y n , (2) the restriction Φ n := Φ 0   Un : U n → Y n is a smooth map. Then one has Φ 0 (U ) ⊂ V and the map Φ := Φ 0   U : U → V is smooth.
Proof. By property (1) we have Φ 0 (U ) ⊂ V and Φ = Φ 0   U : U → V is clearly continuous. Property (2) ensures that the Gâteaux derivative of Φ is well-defined for every u ∈ U and for any direction w in X and that (u, w) → DΦ(u)w is continuous. Inductively, property [START_REF] Arnold | Topological methods in hydrodynamics[END_REF] shows that Φ is of class C n for all n, and therefore is a smooth map.

A remarkable property of equation ( 1) whose interpretation remains mysterious (unless for b = 2 where it corresponds to the conservation of the momentum for metric Euler equations) is the following conservation law. Proposition 9. Let u be a solution of (1) of class C 3 on [0, T ] × S 1 and ϕ the flow of the time-dependent vector field u. Then we have

(m(t) • ϕ(t))ϕ b x (t) = m(0) for all t ∈ [0, T ].
Proof. We have

d dt (m • ϕ)ϕ b x = ϕ b x m t ϕ + (m x • ϕ)(u • ϕ) + b(m • ϕ)(u x • ϕ) since ϕ t = u • ϕ and ϕ xt = (u x • ϕ)ϕ x . Hence d dt (m • ϕ)ϕ b x = 0 if m t = -(m x u + bmu x ).
Lemma 10. Let (ϕ(t), v(t)) be a solution of (9) in Diff 3 (S 1 )×C 3 (S 1 ) defined on [0, T ] with initial data (id, u 0 ). Then we have for all t ∈ [0, T ] (11)

ϕ xx (t) = ϕ x (t) t 0 v(s)ϕ x (s) ds -m 0 t 0 ϕ x (s) 1-b ds and (12) v xx (t) = v x (t) t 0 v(s)ϕ x (s) ds -m 0 t 0 ϕ x (s) 1-b ds + ϕ x (t) v(t)ϕ x (t) -m 0 ϕ x (t) 1-b .
Proof. We have d dt

ϕ xx ϕ x = (u xx • ϕ)ϕ x since ϕ xt = (u x • ϕ)ϕ x and ϕ xxt = (u xx • ϕ)ϕ 2 x + (u x • ϕ)ϕ xx . Using Proposition 9, we get u xx • ϕ = u • ϕ -m 0 ϕ -b x and hence d dt ϕ xx ϕ x = (u • ϕ)ϕ x -m 0 ϕ 1-b x .
Integrating this last relation on the interval [0, t] leads to equation [START_REF] Degasperis | Asymptotic integrability, Symmetry and perturbation theory[END_REF] and taking the time derivative of (11) leads to equation [START_REF] Djebali | Existence de géodésiques d'un groupe de difféomorphismes muni d'une métrique de Sobolev[END_REF].

The meaning of Lemma 10 is that there can be "neither loss, nor gain" in spatial regularity: the solution at time t > 0 has exactly the same regularity as it has at time t = 0. Corollary 11. Let n ≥ 3 and let (ϕ, v) be a solution of [START_REF] Constantin | The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations[END_REF] in Diff 3 (S 1 ) × C 3 (S 1 ) with initial data (id, u 0 ).

(

) If u 0 ∈ C n 1 ) then (ϕ(t), v(t)) ∈ Diff n (S 1 )×C n (S 1 ) for all t ∈ [0, T ]. (2) If there exists t ∈ (0, T ] such that ϕ(t) ∈ Diff n (S 1 ) or v(t) ∈ C n (S 1 ) then u 0 ∈ C n (S 1 ). Proof. 1) Let n ≥ 3. From Lemma 10 we get that if (ϕ(t), v(t)) ∈ Diff n (S 1 )× C n (S 1 ) for all t ∈ [0, T ] and u 0 ∈ C n+1 (S 1 ) then (ϕ(t), v(t)) ∈ Diff n+1 (S 1 ) × C n+1 (S 1 1
) for all t ∈ [0, T ]. A recursive argument completes the proof of the first assertion of the corollary.

2) Let n ≥ 3 and suppose that u 0 ∈ C n (S 1 ). Then, as we just verified, (ϕ(t), v(t)) ∈ Diff n (S 1 ) × C n (S 1 ) for all t ∈ [0, T ]. If moreover there exists t ∈ (0, T ] such that ϕ(t) ∈ Diff n+1 (S 1 ) or v(t) ∈ C n+1 (S 1 ), then since ϕ x is strictly positive, we get from Lemma 10 that u 0 ∈ C n+1 (S 1 ). Again, a recursive argument completes the proof of the second assertion of the corollary. We are know able to state our main theorem.

Theorem 12.

There exists an open interval J centered at 0 and δ > 0 such that for each u 0 ∈ C ∞ (S 1 ) with u 0 C 3 (S 1 ) < δ, there exists a unique solution (ϕ, v) ∈ C ∞ J, Diff ∞ (S 1 ) × C ∞ (S 1 ) of (9) such that ϕ(0) = id and v(0) = u 0 . Moreover, the flow (ϕ, v) depends smoothly (in the smooth category) on (t, u 0 ) in J × C ∞ (S 1 ).

Proof. By application of Proposition 6 for n = 3, we obtain the existence of an open interval J centered at 0 and an open ball U 3 = B 3 (0, δ) in C 3 (S 1 ) such that for each u 0 ∈ U 3 there exists a unique solution (ϕ, v) ∈ C ∞ J, Diff 3 (S 1 ) × C 3 (S 1 ) of ( 9) with initial data (id, u 0 ) and such that the flow Φ 3 :

J × U 3 → Diff 3 (S 1 ) × C 3 (S 1 ) is smooth. Let U n = U 3 ∩ C n (S 1 ) and U ∞ = U 3 ∩ C ∞ (S 1 ).
By Corollary 11, we have

Φ 3 (J × U n ) ⊂ Diff n (S 1 ) × C n (S 1 ),
for each n ≥ 3 and by Cauchy-Lipschitz theorem, the map

Φ n := Φ 3   J×Un : J × U n → Diff n (S 1 ) × C n (S 1
) is smooth. Therefore, applying Lemma 8, we get first that

Φ 3 (J × U ∞ ) ⊂ Diff ∞ (S 1 ) × C ∞ (S 1 )
which shows the short time existence in the smooth category and then that the map

Φ ∞ := Φ 3   J×U∞ : J × U ∞ → Diff ∞ (S 1 ) × C ∞ (S 1 )
is smooth, which shows the smooth dependence upon initial condition in this category. This achieves the proof.

In the smooth category, the map

Diff ∞ (S 1 ) × C ∞ (S 1 ) → C ∞ (S 1 ), (ϕ, v) → v • ϕ -1
is smooth and we immediately get the following corollary.

Corollary 13.

There exists an open interval J centered at 0 and δ > 0 such that for each u 0 C ∞ (S 1 ) with u 0 C 3 (S 1 ) < δ, there exists a unique solution u ∈ C ∞ J, C ∞ (S 1 ) of (1) such that u(0) = u 0 . Moreover, the solution u depends smoothly (in the smooth category) on (t, u 0 ) in J × C ∞ (S 1 ).

The exponential map

The geodesic flow of a symmetric linear connection on a Banach manifold M (also called a spray in [START_REF] Lang | Fundamentals of differential geometry[END_REF]) satisfies the following remarkable property ϕ(t, x 0 , sv 0 ) = ϕ(st, x 0 , v 0 ), like in the Riemannian case. This is in fact a consequence of the quadratic nature of the geodesic equation. Therefore, the exponential map exp x 0 , defined as the time one of the flow is well defined in a neighbourhood of 0 in T x 0 M for each point x 0 . On a Banach manifold, it can be shown moreover (see [START_REF] Lang | Fundamentals of differential geometry[END_REF] for instance) that the differential of exp x 0 at 0 is equal to id and therefore that exp x 0 is a local diffeomorphism from a neighbourhood of 0 in T x 0 M to a neighbourhood of x 0 in M . This privileged chart, called the normal chart plays a very special role in classical differential geometry, especially to establish convexity results.

On a Fréchet manifold and in particular on Diff ∞ (S 1 ), the existence of this privileged chart is far from being granted automatically. One may find useful to recall on this occasion that the group exponential of Diff ∞ (S 1 ) is not a local diffeomorphism [START_REF] Milnor | Remarks on infinite-dimensional Lie groups, Relativity, groups and topology[END_REF]. Moreover, the Riemannian exponential map for the L 2 metric (Burgers equation) on Diff ∞ (S 1 ) is not a local C 1diffeomorphism near the origin [START_REF] Constantin | On the geometric approach to the motion of inertial mechanical systems[END_REF].

However, it has been established in [START_REF] Constantin | On the geometric approach to the motion of inertial mechanical systems[END_REF], that for the Camassa-Holm equation -which corresponds to the Euler equation of the H 1 metric on Diff ∞ (S 1 ) -and more generally for H k metrics (k ≥ 1) (see [START_REF]Geodesic flow on the diffeomorphism group of the circle[END_REF]), the Riemannian exponential map was in fact a smooth local diffeomorphism. We extend this result here for the general b-equation (a non metric Euler equation).

Theorem 14. The exponential map exp at the unit element id for the bequation on Diff ∞ (S 1 ) is a smooth local diffeomorphism from a neighborhood of zero in Vect ∞ (S 1 ) to a neighborhood of id on Diff ∞ (S 1 ).

The proof of this theorem relies mainly on a linearized version of Lemma 10. Let n ≥ 3 and (ϕ ε (t), v ε (t)) be the local expression of an integral curve in T Diff n (S 1 ) of ( 9), defined on [0, T ], with initial data (id, u + εw), where u, w ∈ C n (S 1 ). We define

ψ(t) = ∂ ∂ε ε=0 ϕ ε (t).
Then ψ(t) = L n (t, u)w where L n (t, u) is a bounded linear operator of C n (S 1 ).

Lemma 15. Suppose u ∈ C n+1 (S 1 ). Then, we have

L n (t, u)(C n (S 1 )\C n+1 (S 1 )) ⊂ C n (S 1 )\C n+1 (S 1 ))
for all t ∈ (0, T ],

Proof. Writing [START_REF] Degasperis | Asymptotic integrability, Symmetry and perturbation theory[END_REF] for (ϕ ε (t), v ε (t)) and taking the derivative with respect to ε at ε = 0, we get

ψ xx (t) = a(t)ψ x (t) + b(t)ψ(t) -c (
∈ C n (S 1 ) \ C n+1 (S 1 ) then ψ(t) = L n (t, u)w ∈ C n (S 1 ) \ C n+1 (S 1
), ∀t ∈ (0, T ], which achieves the proof of the lemma.

Proof of Theorem 14. First, we can find neighborhoods U 3 of 0 in C 3 (S 1 ) and V 3 of id in Diff 3 (S 1 ) such that

exp 3 : U 3 → V 3 is a smooth diffeomorphism (for the C 3 norm). For n ≥ 3, let U n = U 3 ∩ C n (S 1 ) and V n = V 3 ∩ Diff n (S 1 ).
By Corollary 11, exp 3 (U n ) = V n and exp n := exp 3

 

Un : U n → V n is a bijection. By virtue of the Cauchy-Lipschitz theorem in Banach spaces, exp n is smooth (for the C n norm). We are going to show that it is a diffeomorphism. For each u ∈ C n (S 1 ), Dexp n (u) is a bounded linear operator of C n (S 1 ). Notice that

Dexp n (u) = Dexp 3 (u)   C n (S 1 )
. It is therefore one-to-one. We now prove inductively that Dexp n (u) is surjective. For n = 3 this is so by our hypothesis. If it is true for 3 ≤ j ≤ n, then it true also for n + 1 (with u ∈ C n+1 (S 1 )) because of Lemma 15 and the fact that Dexp n (u) = L n (1, u). Therefore, according to the open mapping theorem, we get that for every n ≥ 3 and each u ∈ C n (S 1 ) Dexp n (u) : C n (S 1 ) → C n (S 1 ) is a topological linear isomorphism. Applying the inverse function theorem for exp n , we deduce that exp n is a diffeomorphism from U n to V n . Since this is true for all n ≥ 3, we conclude, using Lemma 8 that exp ∞ := exp 3   U∞ : U ∞ → V ∞ as well as exp -1 ∞ : V ∞ → U ∞ are smooth maps. That is exp ∞ is a diffeomorphism (in the smooth category) between U ∞ and V ∞ .

The non-periodic case

It would be interesting to extend the whole work done in this paper for the non-periodic case -that is for the group Diff ∞ (R) of smooth orientationpreserving diffeomorphisms of the real line. Unfortunately and contrary to Diff ∞ (S 1 ), this Fréchet Lie group is not regular as defined by Milnor in [START_REF] Milnor | Remarks on infinite-dimensional Lie groups, Relativity, groups and topology[END_REF]. In particular, not every of its Lie algebra, Vect ∞ (R) can be integrated into a one parameter subgroup.

In order to extend straightforwardly the present work, one needs therefore to restrict to a subgroup of Diff ∞ (R) which is a regular Fréchet Lie group. Moreover, the Fréchet space on which the differentiable structure is modeled must admit a Banach approximation, as defined in Section 5. And last but not least, these Banach spaces must be Banach algebras (for pointwise multiplication of functions) in order to prove Theorem 4.

One candidate was proposed in [START_REF] Michor | Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach, Phase space analysis of partial differential equations[END_REF]. It consists of the subgroup of rapidly decreasing diffeomorphisms Diff S (R) = id + f ; f ∈ S(R) and f ′ > -1

where S(R) is the Schwartz space of rapidly decreasing functions. This group is a regular Fréchet Lie group (see [START_REF] Michor | Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach, Phase space analysis of partial differential equations[END_REF]).

Another and simpler solution has been proposed in [START_REF] Djebali | Existence de géodésiques d'un groupe de difféomorphismes muni d'une métrique de Sobolev[END_REF]. It consists of the subgroup of Diff ∞ (R), defined as

Diff H ∞ (R) = id + f ; f ∈ H ∞ (R) and f ′ > -1
where

H ∞ (R) = +∞ n=1 H n (R)
and H n (R) are the Sobolev spaces on the line. It has been shown in [START_REF] Djebali | Existence de géodésiques d'un groupe de difféomorphismes muni d'une métrique de Sobolev[END_REF] that this group is a regular Fréchet Lie group and that the theory extends well, at least in the metric case. Notice that Diff S (R) ⊂ Diff H ∞ (R) and that this inclusion is strict.

Remark 7 .

 7 Proposition 9, Lemma 10 and Corollary 11 are of course true if we replace the time interval [0, T ] by [-T, 0].


Leonardo De

come

Electrochemical properties of various enzyme / electrode material interfaces for biofuel cell application

Electrochemical properties of various enzyme / electrode material interfaces for biofuel cell application

De Silva Muñoz Leonardo*, Basseguy Régine † Laboratoire de Génie Chimique, UMR5503, 5 rue Paulin Talabot, 31106 Toulouse France Biological fuel cells are a type of fuel cells that contain biological components, like enzymes, proteins and microorganisms that help achieving the transformation of chemical energy into electricity (1). In this kind of fuel cells, a large range of fuels can be used, either organic like sugars, organic acids and alcohols (2,3,4) or inorganic like hydrogen (5). Biofuel cells may be a good alternative to platinum based fuel cells where the oxygen reduction and the hydrogen oxidation are both catalyzed thanks to platinum properties. But this metal is rare on Earth and expensive. Reducing its load in fuel cells remains one of the major issues to solve in order to achieve a large scale fuel cell commercialization. One way to do this can be the use of enzymes as catalysts in fuel cell internal processes.

Enzymes in biofuel cells can either directly catalyze the electrochemical reactions involved (6,7) or intervene in intermediate reactions like the regeneration of mediators (2) or in metabolic processes inside microorganisms (2,5,8). The direct, or also called mediatorless, electron transfer between enzyme and electrode is the subject of a significant number of works in the field of bioelectrocatalysis and more particularly in the biosensor domain. This is because this kind of mechanisms may simplify the enzymatic electrochemical system and the preparation of the electrode/enzyme interface.

In the present work, several enzymeelectrode couples are electrochemically studied with the aim to find new alternatives for both cathodic and anodic reactions in biofuel cells by exploiting the direct electron transfer capability of the enzymes. The results were compared to the platinum behavior and allowed to classify the different systems for an energetic use. The study was made combining a biocatalyst (laccase, hydrogenase, glucose oxidase and hemin) with different electrode materials (titanium, stainless steel, glassy carbon, graphite and DSA). The biocatalysts were confined near the electrode by a dialysis membrane or were adsorbed using a Dimethyl Sulfoxide solution (DMSO). The electrochemical tests were performed using galvanostatic mode and voltammetry.