Sebastien Prothin
email: sebastien.prothin@ecole-navale.fr

Henda Djeridi
email: henda.djeridi@univ-brest.fr

Jean-Yves Billard
email: jean-yves.billard@ecole-navale.fr

EFFECT OF LONGITUDINAL VORTEX ON BOUNDARY LAYER STATE AND SEPARATION ON NACA SYMMETRIC FOIL

 vortex generators are embedded in the boundary layer and the vortex height, with regards to the wall, is of the boundary layer thickness. The objective of this configuration is obviously far from being the effects of a single longitudinal vortex (generated upstream by an elliptical plan form profile) on the turbulent boundary layer shape over a Naca0015 symmetric foil at different incidences at high Reynolds number 5 10 5 . The vortex is situated outside the boundary layer (ten times the BL thickness over the wall) taking into account the small value of the thickness in our hydrodynamic application. Obviously, this situation is optimum as the vortex delays separation and increases the maximum lift but introduces drag penalty at small incidence. This is nevertheless frequently encountered in hydrodynamic applications (hub vortex upstream of a rudder) and of interest. To point out the mechanism of the boundary layer manipulation, both global efforts using gauge balance and velocity measurements using LDV and PIV have been performed and compared with and without vortex. The base flow is an APG boundary layer characterized by a predominant wake area. Effect of the vortex is analyzed via the shape factor both in inflow and outflow regions. The longitudinal vortex suppress the hysteretic loop classically described in this Reynolds number range (Djeridi et al., 2009) but an increase of the drag is observed in the range of incidence just before stall. Velocity measurements indicated that, for incidences near the stall appearance, the shape factor is decreased both in the inflow and in the outflow regions. Even for large incidences, in the inflow region the value of the shape factor is equivalent to the one found in the turbulent BL over a flat plate. In this region

the vortex modifies the equilibrium state of the BL as attested by the Clauser parameter. Even for large distances between the vortex and the wall, the ability of the vortex to suppress the detachment of the BL is observed on the evolution of the backflow coefficient. This effect is greater pronounced in inflow area near the trailing edge region where the flow is locally reattached due to the high momentum fluid displacement.

NOMENCLATURE (x, y, z)

cartesian coordinate [

INTRODUCTION

Interaction between vortex and boundary layer can be used in control applications. In spite of this industrial interest very few studies deal with the physical mechanism which takes place in this interaction and most of the studies focus on aerodynamic applications where vortex generators are in the boundary layer. In hydrodynamic applications the vortex is generated out of the boundary layer (BL), for instance, hub vortex issued from a propeller in interaction with the downstream rudder.

Many studies have been experimentally conducted to test the effectiveness of different configuration of interactions between single vortex or vortices issue from vortex generators (VGs) and BL. Shabaka et al. [START_REF] Shabaka | Longitudinal vortices imbedded in turbulent boundary layers. Part 1. Single vortex[END_REF] investigated the mean and turbulent flow of the low speed BL in zero pressure gradient (ZPG) disturbed by a longitudinal isolated vortex imbedded in it. They shown that the circulation around the vortex was almost maintained, the region of flow affected by the vortex continued to grow downstream and remained proportional to the BL thickness (δ). Mehta and Bradshaw [START_REF] Mehta | Longitudinal vortices imbedded in turbulent boundary layers. Part 2. Vortex pair with common flow upwards[END_REF] in the same configuration but with a pair of counter-rotating vortices have shown that BL was lift-up by the vortices and has shown that BL was lifted and carried away by the vortices.

Recently, Logdberg [START_REF] Löbdberg | Turbulent boundary layer separation and control[END_REF], Angele and Muhammad-Klingmann [START_REF] Angele | The effect of streamwise vortices on the turbulence structure of a separating boundary layer[END_REF], Angele and Grewe [START_REF] Angele | Instantaneous behaviour of streamwise vortices for turbulent boundary layer separation control[END_REF], Velte et al. [START_REF] Velte | Flow analysis of vortex generators on wing sections by stereoscopic particle image velocimetry measurements[END_REF] and Velte [START_REF] Velte | Characterization of Vortex Generator Induced Flow[END_REF] investigated the effect of VGs on separating BL for control applications. For all these studies, VGs have been imbedded in the BL with height ratio (d/δ) about unity.

The present study investigates experimentally the effects of a single longitudinal vortex generated by an elliptical plane form NACA0020 and upstream the BL with adverse pressure gradient (APG) which developed on a 2D NACA0015 foil. In the actual case, the vortex is located outside the BL and corresponding to the hydrodynamic configuration (hub vortex/ rudder) and leads each time to an efficient separation control.

This study focuses on the mean velocity distribution of turbulent boundary layers before and after static stall. Several incidences (5, 10, 12.5, 15, 20 and 25°) have been investigated and results at 15° will be presented below since this incidence value corresponds to the hysteretic behaviour associated with lift decrease (Sarraf et al. [START_REF] Sarraf | Thickness effect of NACA foils on hydrodynamic global parameters, boundary layer states and stall establishment[END_REF]).

EXPERIMENTAL SETUP Hydrodynamic Tunnel

The experiments have been conducted in the hydrodynamic tunnel of the French Naval Academy. This facility is fitted with a 1m long and 0.192 x 0.192 m 2 square cross test section, in which a maximum velocity of 15ms -1 can be achieved. The turbulence intensity upstream at the entrance of the test section is 2%.

The flow characteristics correspond to the upstream Reynolds number based on NACA0015 chord length of 5.10 5 . The vortex is generated by a NACA0020 foil with elliptical plan form located 2 chords upstream of the 2D foil. The characteristics of the vortex are described by Pichon [START_REF] Pichon | Contribution à l'étude de la Cavitation de Tourbillon Marginal[END_REF]. The experimental configuration is shown on Figure 1. On the plan form of the 2D foil the vortex determines 2 zones namely an outflow area where the vortex induces a velocity field directed from the wall to undisturbed flow and an inflow area where the vortex induces a velocity field directed toward the foil.

Laser Doppler Anemometry

To characterize the structure of the boundary layer and the near wake that develop on the 2D-hydrofoils, detailed velocity measurements was performed using a refined spatial grid by LDV (Figure 2). On each normal line measurements were performed from the unperturbed flow down to the wall. The spatial dimension of the probe volume (LDV) in vertical and longitudinal directions is 0.05 mm allows measurements from the outer region y+ ~ 1000, to the wall y+ = 3 ~ 5. This value corresponds to a position of the last validate point on each the normal which is positioned at 30 µm from the wall. Velocity measurements are performed from the leading to the trailing edge (0<x/c<1). The time histories were registered with 10 000 to 5 000 samples acquired in the range of 20 to 60 seconds corresponding to the mean data rates of 2 000 to 130 Hz. Mean and rms values of the velocity are obtained with an uncertainty of 1% and 1.5 % respectively.

Particule Image Velocimetry 2D3C

The PIV 2D3C measurements have been carried out for the NACA0015 profile at incidence 15°, 20° and 25°. A double pulse ND:YAG laser DualPower TR Lasers has been used. This delivers an energy of 2x30 mJ that illuminates silver glass particles used as flow seeding with a 1 mm thick light sheet.

The particle size is about 10 µm. Two SpeedSense 9072 cameras have been used with a resolution of 1280x800 pixels, recording double full-frame particle image. The cameras are equipped with a 60 mm objective lens at a diaphragm aperture of 1.2. The system, both camera and laser were operated at a frequency of 1000Hz. The setup is shown in Figure 3. The size of the measurement area was -0.115 < x/c < 1.185 and -0.395 < y/c < 0.413. The flow was analyzed by crosscorrelating 50% overlapping windows of 32x32 pixels.

This yield of 77x49 vectors and 1000 pairs of images was analyzed to generate converged flow fields statistics. The convergence is reached beyond an order of 500 images. Therefore, the results corresponding to 1000 images per each PIV plane provide fully converged turbulence statistics.

Gauge balance measurements

The lift and drag measurements were performed using a resistive gauge hydrodynamic balance calibrated by ourselves, with the maximum range within 0-1800 N for the lift force and 0-170 N for the drag force. The mean and root mean square (rms) values are calculated from 30 seconds test measurement carried out at 2048Hz. The determination of the hydrodynamic parameters was performed for incidence angles in the range [-20°, 30°].

FLOW CHARACTERISATION Boundary layer with APG

For each point of measurement, 10 000 samples have been validated during a maximum of 60 s (corresponding to the near wall locations). With this number of samples, corrected convergence of mean and rms values of velocity components has been achieved. According to the previous refined grid, a survey of the distribution of mean tangential and normal velocity components is presented. Firstly, our primary objective was to characterize the boundary shape parameters on a Naca0015 hydrofoil. Velocity profiles were numerically integrated to compute the displacement and momentum thicknesses δ 1 and δ 2 , from which shape factor H 12 is deduced. The following formulas are applied:

dy U u e ∫ ∞         - = 0 1 1 δ dy U u U u e e ∫ ∞         - = 0 2 1 δ 2 1 12 δ δ = H
In our confined case this velocity is the maximum velocity measured at the location.

This factor is classically used to identify the transition from laminar to turbulent. However, as H 12 value is very high at separation, h parameter defined by Kline et al. [START_REF] Kline | Correlation of the Detachment of Two-dimensional Turbulent Boundary Layers[END_REF] is preferred. This parameter is defined by: Evolution of this parameter is indicated in table 1 for the case without vortex. It can be noticed that incipient separation occurs for x/c slightly lower than 0.4 and full separation for x/c = 0.45. The thickness parameter δ, characteristic of the boundary layer thickness is defined by the speed at which corresponds to 99% of the infinite speed. in our confined case this thickness is taken to the thickness at which the speed is equal to the maximum speed. The backflow coefficient χ is defined by the percentage of time where speed is in opposite direction of that of the reference velocity. The separated region is defined by Lögdberg [START_REF] Löbdberg | Turbulent boundary layer separation and control[END_REF] as the area where the backflow coefficient is larger than 0.5. This parameter is estimated using PIV data, since the interrogation areas must be large enough to contain approximately 5 particles. The first available χ values are determined 1mm from the wall and are significant when the BL thickness is large enough (i.e. in incipient or fully developed separated area).

Longitudinal Vortex

The vortex is generated by a 3D wing positioned 2 chord (2 c ell ) upstream of the zero incidence leading edge of the main hydrofoil. This generation method and foil characteristics have been chosen, first, because previous studies in the laboratory gave numerous data concerning the characteristics of such vortex (Pichon, [START_REF] Pichon | Contribution à l'étude de la Cavitation de Tourbillon Marginal[END_REF]), and secondly, for convenience of implementation in the vein of the water tunnel. Indeed, the implementation of a mechanical system allowed us to control the characteristics of the vortex (intensity and position).

An experimental validation was performed, to verify that the wake of the 3D wing did not disturb the flow on the suction side of the profile. The incidence of the foil for the generation of the tip vortex was far enough from the separation to develop a thin wake. The validation showed that the signature of the 3D wing on the flow on the suction side of the 2D foil is hardly observable, and that therefore the assumption of isolated vortex is justified.

Mean axial and tangential velocity obtained by LDV was fitted using a theoretical model of vortex. The chosen model due to Batchelor [START_REF] Batchelor | Axial flow in trailing line vortices[END_REF], and developed to model the aircraft tip vortices, is a 3D extension of Lamb-Oseen model. This model is defined by the following equations: Considering that the aim of this study was to focus on the BL modifications, only one configuration of vortex generation has been achieved. The incidence angle of the 3D NACA0020 was 8° for the upstream velocity of U inf = 5.56 m/s.

0 = r U                 - - Γ = 2 2 exp 1 2 a r r U π θ                 - - Γ - =
The vortex trajectory was determined in the (x, y, z) coordinates where the origin of these axes was located at the extremity of the 3D foil. The vortex trajectory is plotted in Figure 6. For selected points of the trajectory, circulation evolution (Γ) of the vortex has been done with normalized x-axis as it is shown in Figure 7. An increase of the vortex circulation with downstream direction associated with a growth of the vortex size and the decrease of the maximum tangential velocity and an increase of the helicity pitch was noticed. This behavior characterizes the vortex diffusion, ultimately leading to its disintegration. To estimate the axial flow compared to a characteristic velocity of the vortex rotation, the non dimensional number W 0 was used. This number characterized the vortex stability and is defined by:

Γ - = a U U W π 2 ) ( inf 0 0
and plotted in Figure 8. Note that |W 0 | <0.6 along the xaxis, this value is considered as stable by Fabre and Jacquin [START_REF] Fabre | Viscous instabilities in trailing vortices at large swirl numbers[END_REF].

INTERACTION Interaction parameters

Before analyse the interaction between vortex and BL, it was necessary to define three local parameters: d/δ, Γ/δU e , a/δ. These parameters represent respectively the relative normal position of the vortex to BL, the non-dimensional circulation and the aspect ratio between the core vortex size and BL size. The evolutions of these three parameters with the position x/c are shown Figure 10 & 11. It can be noticed that d/δ is greater than 1, corresponding to the vortex outside the BL thickness.

Several studies (Angele et al. [START_REF] Velte | Flow analysis of vortex generators on wing sections by stereoscopic particle image velocimetry measurements[END_REF]) concerning the effect of vortices on BL pattern considered this parameter rather constant and always lower than 1 for all configurations. In our case, d/δ decreases along the boundary layer as the vortex is guided in the NACA0015 flow and perturbed by the pressure gradient induced by the 2D foil. It is smallest value (∼1) is reached near the trailing edge of the 2D foil.

The presence of the NACA0015 foil induces a vortex deviation in the vertical (x-y) plane associated with the lifting flow generated by the 2D foil. This trajectory is shown in Figure 9. In the transversal (y-z) plane the vortex is slightly deviated due to the image effect. As the vortex is located quite far from the wall this effect remains of little importance. The relative normal position of the vortex and its relative circulation to BL, are shown Figure 10. The same trends as compared to the non-dimensional distance can be observed. These values can be compared with those obtained by Shabaka et al. [START_REF] Shabaka | Longitudinal vortices imbedded in turbulent boundary layers. Part 1. Single vortex[END_REF] in the case of longitudinal vortices imbedded in turbulent boundary layers. They determined the ratio of Γ to Ue.δ downstream the VGs of about 0.054. In our case, this value is obtained through an asymptotic behaviour near the trailing edge. As it is expected, these values will provide a strong influence on the separation area.

Global efforts

Lift and drag measurements have been performed with and without vortex and are reported on Figure 12. Obviously the values of both lift and drag coefficients are not affected by the upstream vortex for incidences lower than 7°. Between 7° and 15° the lift coefficient remains unchanged but the drag coefficient increases rapidly.

For larger values of the angle of attack the hysteretic loop, characteristic of the static stall, disappears (Sarraf [START_REF] Sarraf | Thickness effect of NACA foils on hydrodynamic global parameters, boundary layer states and stall establishment[END_REF]) and the lift coefficient have a plateau like behaviour. The presence of the vortex induces a displacement of the detachment point, and their oscillations are inhibited.

Interaction of the BL on the vortex

It can be seen in Figure 10 that the vortex gets nearer the BL at trailing edge. This trend is accentuated when incidence angle is increased.

The vortex circulation has been determined in the presence of the BL. Its evolution is shown Figure 13 versus normalized downstream direction x/cell and compared to the one obtained without BL. The grey area represents the NACA zone in the xaxis. Note that the vortex is disturbed upstream the 2D foil and the circulation is globally increased. This behavior is accentuated in the wake of the 2D foil probably due to APG influence.

Concerning the vortex stability, the W 0 value is increased in the NACA0015 area (corresponding to the Cp min location on the 2D foil) and the vortex is strongly destabilized. Furthermore, W 0 is decreased upstream the 2D foil near the stagnation point. It can be noticed that the circulation (Figure 13) and the stability (Figure 14) parameter have been determined until x/Cell= 4 (corresponding to the location near the NACA0015 trailing edge).

Even the existence of the vortex is confirmed by the axial and tangential velocity, it is destructured and the Batchelor model is not justified. These results can be linked with the vortex breakdown associated to the APG due to the NACA0015 design and incidence angle. As described by Gursul and Yang [START_REF] Gursul | Vortex breakdown over a pitching delta[END_REF] it is well known that there are two parameters which determine the breakdown location: the swirl angle and the external pressure gradient outside the vortex core. Hall [START_REF] Hall | Vortex breakdown[END_REF] suggested that the vortex breakdown occurs as a result of APG along the vortex axis which leads to a stagnation point.

Nevertheless, in our case the vortex coherence is sufficient to compensate the APG effects and continue to exist downstream the 2D foil. In fact, the breakdown is observed for higher incidences.

Interaction of the vortex on the BL

Shape factor is now analyzed to point out the effect of the vortex on the BL. The h values corresponding to classical incipient detachment and complete separation are also reported. Figure 15 shows the h evolution versus x/c with and without vortex at different transverse position (inflow and outflow zones).

It can be noted that, in the inflow region the boundary layer is attached until x/c=0.7 and the shape value is close to the one obtained in turbulent boundary layer case.

In the outflow region the complete separation is reached for x/c= 0.65 instead of x/c=0.45 without vortex. As it is expected in the inflow region the vortex considerably delays the separation near the trailing edge by the transfer of momentum in the boundary layer with a delay to reach separation (Velte et al. [START_REF] Velte | Flow analysis of vortex generators on wing sections by stereoscopic particle image velocimetry measurements[END_REF]).

SUMMARY AND CONCLUSION

The results of the actual work provide us a survey of a boundary layer that develops on a 2D NACA0015 foil with and without longitudinal vortex passing in the vicinity of the boundary layer.

As a conclusion the presence of the vortex has a major effect both on the boundary layer and on the global parameters. We have shown that the hysteretic behavior of the global parameters (lift and drag) is inhibited by the presence of the vortex. The lift stress is suppressed and replaced by a plateau where the lift value is equivalent to the maximum lift value in the case without vortex.

This effect is related to the modification of the flow on the suction side of the foil where, both in the inflow or outflow region where the extension of the separated area, both is decreased. This effect is greater in the inflow region even if it remains observable in the outflow region. These effects are induced by the momentum exchanges induced by the vortex between the external and internal regions of the boundary layer with an input of external fluid in the boundary layer in the inflow region and an extraction of slow fluid from the boundary layer in the outflow region.

These effects can be used to control the flow on the 2D hydrofoil.

Figure 1 :

 1 Figure 1: Experimental setup.

Figure 2 :

 2 Figure 2: Refined spatial grid.

Figure 3 :

 3 Figure 3: PIV 2D3C setup.

  These authors use this parameter to characterize incipient and fully develop separations. The turbulent boundary layers which develop in ZPG conditions are characterized by a constant H 12 value of 1.4, in those conditions h = 0.29. Kline determined two values of h that characterize incipient and fully developed separation: h ≥ 0.63: incipient separation (H 12 ≥ 2.7) h ≥ 0.75: complete separation (H 12 ≥ 4).

π

  This Batchelor model is illustrated in Figure4. The experimental mean axial and tangential profiles are plotted and superimposed to the model in figure5. As it is shown, Batchelor vortex is one of the most realistic models to represent a single vortex with axial flow.

Figure 4 :

 4 Figure 4: Scope of vorticity and the velocity induced by a model type Batchelor

Figure 5 :

 5 Figure 5: Example of tangential and axial velocity fit

Figure 6 :

 6 Figure 6: Trajectory of the vortex. Incidence 8° and Uinf = 5.56 m/s.

Figure 7 :

 7 Figure 7: Evolution of Vortex Circulation depending on the distance to the free end

Figure 8 :

 8 Figure 8: Stability of the vortex.

Figure 9 :Figure 10 :Figure 11 :

 91011 Figure 9: Trajectory of the tip vortex with downstream foil (solid line) and without (dotted line)

Figure 12 :

 12 Figure 12: Global coefficient of lift Cl and Drag Cd

Figure 13 :

 13 Figure 13: Evolution of Vortex Circulation with and without vortex.

Figure 14 :

 14 Figure 14: Evolution of stability parameter with and without vortex.

Figure15: h coefficient

  Figure15: h coefficient for the case without and with vortex.The following figures (Figure16(a)(b)(c)) show the evolution of the backflow coefficient χ superimposed with normalized velocity U/U inf , the boundary layer thickness (δ) and the vortex position (d) with and without vortex. It can be seen from this figure that the trends described by Logdberg are not reproduced in the case of the present results. Essentially, the criteria of separation (χ=0.5) is not in agreement with our results even if the principal trends concerning correlation between the mean velocity defect and the maximum of backflow coefficient remain the same. Obviously the vortex induces a decrease of the maximum backflow value observed even in the outflow region and this maximum is observed further downstream in this case showing the effectiveness of the vortex to decrease the extent of the separated area.

  (a) Without Vortex (b) With vortex Inflow region (b) With vortex Outflow region Figure16: Boundary layer and separation zone for three cases, without vortex (a), with vortex Inflow (b) and Outflow (c). The figure is not to scale in the X direction for more visibility. The grey full lines show U/Uinf, the grey dashed-dotted lines show the backflow coefficient χ. The grey full thick lines show the δ parameter of boundary layer. The black full lines represent the trajectory of the vortex, and the black dashed-dotted lines show the virtual position of the vortex in the case without vortex.

Table 1 :

 1 Parameters of BL without vortex

	x/c	δ (mm) δ 1 (mm) δ 2 (mm)	H 12	h	χ max (%)
	0,3	2,71	0,56	0,32	1,73	0,42	0
	0,4	6,32	1,57	0,55	2,86	0,65	0,19
	0,45	6,28	2,47	0,61	4,08	0,76	1,12

12

ACKNOWLEDGMENTS

We would like to thank the company Dantec Dynamics for their great help and more particularly J-J. Lasserre, O. Pust, A. Poidatz, and P. Galletier, for their support through the sharing of their know-how and the provision of hardware necessarily for the realization of measurements of PIV 2D3C.


François Dubois
email: francois.dubois@math.u-psud.fr

On hyperbolic systems with entropy velocity covariant under the action of a group

Keywords: Galileo group, Lorentz group, circular group, mathematical entropy. AMS classification: 35L40, 58J45

For hyperbolic systems of conservation laws in one space dimension with a mathematical entropy, we define the notion of entropy velocity. Then we give sufficient conditions for such a system to be covariant under the action of a group of space-time transformations. These conditions naturally introduce a representation of the group in the space of states. We construct such hyperbolic system from the knowledge of data on the manifold of null velocity. We apply these ideas for Galileo, Lorentz and circular groups. We focus on particular non trivial examples for two by two systems of conservation laws.

1) Introduction

• The first link between covariance under the action of a group and conservation laws is the classic result of Emmy Noether [START_REF] Noether | Invariante Variationsprobleme[END_REF]: when the action is invariant under the action of a group, an associated moment is conserved along the time evolution. This result is well understood for systems that possess some action. This result is also fundamental for all the modern physics. It has been intensively studied and generalized by Jean-Marie Souriau in his fundamental book [START_REF] Souriau | Structure des systèmes dynamiques[END_REF] (see also the work of Kostant [START_REF] Kostant | Quantization and Unitary Representations[END_REF]). The fact to consider Galilean relativity is natural in this kind of work. In order to explicit the mass as an invariant, the Bargmann group [START_REF] Bargmann | On unitary ray representations of continuous groups[END_REF] is necessary in this framework and we refer to the work of de Saxcé and Vallée [START_REF] De Saxcé | Bargmann group, momentum tensor and Galilean invariance of Clausius-Duhem inequality[END_REF] for the introduction of the second principle of thermodynamics. Observe also that very early Souriau has considered not only Galilean relativity but also Einstein's relativity [START_REF] Souriau | Géométrie et relativité[END_REF][START_REF] Souriau | Thermodynamique relativiste des fluides[END_REF][START_REF] Souriau | Thermodynamique et géométrie[END_REF]. An other contribution is due to Vallée [START_REF] Vallée | Lois de comportement des milieux continus dissipatifs compatibles avec la physique relativiste[END_REF] relative to continuum media in the framework of special relativity.

• Our scientific background is first concerned with the approximation of gas dynamics problems with numerical methods [START_REF] Després | Systèmes hyperboliques de lois de conservation ; application à la dynamique des gaz[END_REF]. The notion of first order hyperbolic system of conservation laws is fundamental for this study and the contribution of Peter lax (see e.g. [START_REF] Lax | Hyperbolic Partial Differential Equations[END_REF]) is essential for all that follows. The introduction of the second principle of thermodynamics in this framework is due to Godunov [START_REF] Godunov | An interesting class of quasilinear systems[END_REF] and to Friedrichs and Lax [START_REF] Friedrichs | Systems of Conservation Equations with a Convex Extension[END_REF] with the notion of "mathematical entropy". A fundamental result of existence and uniqueness of an entropy solution in the scalar case is due to Kruzkov [START_REF] Kruzkov | First Order Quasi-Linear Systems in Several Independant Variables[END_REF]. Galilean conservation for hyperbolic systems has been proposed by Ruggeri [START_REF] Ruggeri | Galilean invariance and entropy principle for systems of balance laws; the structure of extended thermodynamics[END_REF]. We have considered the same question with enforcing the structure of the mathematical entropy in [START_REF] Dubois | Conservation laws invariants for Galileo group; CEMRACS preliminary results[END_REF]. We consider an extension of this approach in this contribution.

• The summary of our study is the following. We introduce the entropy velocity at Section 2 for hyperbolic systems with a mathematical entropy. Then we define in Section 3 the notion of covariance for an hyperbolic system with entropy velocity under the action of a group and propose sufficient conditions that make in evidence an algebraic structure. The manifold of null velocity defined at Section 4 is a natural notion in ths context. Then precise constraints for the entropy variables are derived at Section 5. In Section 6, we study the particular case of systems of order two. Some words of conclusion are proposed at Section 7.

2) Entropy velocity

• Let N be a positive integer and Ω a non void convex open part of IR N . The physical flux f (W ) ∈ IR N of a state W ∈ Ω is a regular function f : Ω -→ IR N . We study in the following the system of conservation laws in one space dimension with unknown IR × [0, +∞[ (x, t) -→ W (x, t) ∈ Ω associated with this nonlinear flux function:

(1)

∂W ∂t + ∂ ∂x f (W ) = 0 .
• As proposed by Godunov [START_REF] Godunov | An interesting class of quasilinear systems[END_REF], Friedrichs and Lax [START_REF] Friedrichs | Systems of Conservation Equations with a Convex Extension[END_REF] and Boillat [START_REF] Boillat | Sur l'existence et la recherche d'équations de conservation supplémentaires pour les systèmes hyperboliques[END_REF], a mathematical entropy for the system (1) is a regular strictly convex function η : Ω -→ IR such that for  any regular solution W (x, t) of the conservation law [START_REF] Bargmann | On unitary ray representations of continuous groups[END_REF], there exists a complementary conservation equation

(2) ∂η ∂t + ∂ζ ∂x = 0 .
The space derivative of the second term in (2) is the entropy flux, a regular function ζ : Ω -→ IR that satisfies the identity

(3) dζ ≡ dη • df .
It is well known (see e.g. Godlewski -Raviart [START_REF] Godlewski | Numerical Approximation of Hyperbolic Systems of Conservation Laws[END_REF] or [START_REF] Després | Systèmes hyperboliques de lois de conservation ; application à la dynamique des gaz[END_REF]) that a system of conservation law which admits a mathematical entropy is hyperbolic: the jacobian df (W ) is diagonalisable in IR N . Observe also that the restriction to only one space dimension is not restrictive in principle if we adopt a point of view founded on mathematical entropy. The generalization to several space dimensions does not add in priciple new fundamental difficulties. As well known, the equality (2) has to be replaced by an inequality (taken in an appropriate weak sense) when discontinuous solutions of the conservation law (1) are considered. The co-vector ϕ of entropy variables is the jacobian of the entropy relatively to the variation of the state W :

(4) dη ≡ ϕ • dW , ∀ W ∈ Ω .
The dual entropy η * is the Legendre [START_REF] Legendre | Mémoire sur l'intégration de quelques équations aux différences partielles[END_REF] -Fenchel [START_REF] Fenchel | On conjugate convex functions[END_REF] -Moreau [START_REF] Moreau | Fonctions convexes duales et points proximaux dans un espace hilbertien[END_REF] transform of the convex function η. It is a function of the entropy variables:

(5)

η * (ϕ) ≡ sup W ∈Ω ϕ • W -η(W )
and we have

(6) dη * (ϕ) = dϕ • W .
• In the following, we define the "entropy velocity" u as the quotient of the entropy flux divided by the entropy. In other terms, we suppose that the system (1) (2) admits an entropy velocity u if there exists a regular function u : Ω -→ IR such that the entropy flux ζ can be written under the form η u:

(7) ζ(W ) ≡ η(W ) u(W ) , ∀ W ∈ Ω .
This definition has been previously proposed in [START_REF] Dubois | Conservation laws invariants for Galileo group; CEMRACS preliminary results[END_REF]. For gas dynamics this entropy velocity is the usual velocity because the specific entropy is advected by the flow (see e.g. [START_REF] Després | Systèmes hyperboliques de lois de conservation ; application à la dynamique des gaz[END_REF] or [START_REF] Godlewski | Numerical Approximation of Hyperbolic Systems of Conservation Laws[END_REF]). Once the velocity is defined, it is natural, as suggested by Ruggeri [START_REF] Ruggeri | Galilean invariance and entropy principle for systems of balance laws; the structure of extended thermodynamics[END_REF], to decompose the physical flux f into a convective part u W and a complementary contribution g. We define the "thermodynamic flux" g(W ) by the relation

(8) f (W ) ≡ u(W ) W + g(W ) , ∀ W ∈ Ω .  • Proposition 1.
Compatibility relation With the above framework, the compatibility relation ( 3) is satisfied if and only if [START_REF] Godlewski | Numerical Approximation of Hyperbolic Systems of Conservation Laws[END_REF] ϕ • dg + η * du ≡ 0 .

Proof of Proposition 1.

The proof has been explicited in [START_REF] Dubois | Conservation laws invariants for Galileo group; CEMRACS preliminary results[END_REF]; we detail it for completeness. We have, due to the relations (3) to ( 8), the following calculus:

dζ -dη • df = d(η u) -dη • d u W + g = η du + u dη -ϕ • (du) W + u dW + dg = η -ϕ • W du + u dη -ϕ • dW -ϕ • dg = -η * du -ϕ • dg
and the conclusion is clear.

3) Covariance under the action of a group

• We introduce a group G with one real parameter θ composed by transformations G θ of space-time defined by a two by two matrix:

(10) G θ = α θ δ θ β θ γ θ , G θ ∈ G , θ ∈ IR .
At a fixed θ, the transformation (x, t) -→ (x , t ) is defined by the relations:

(11) x t = G θ x t ; x = α θ x + δ θ t t = β θ x + γ θ t .
In the following, we consider three particular groups, defined by the conditions

α θ = γ θ = C θ , β θ = ε c S θ , δ θ = c S θ
id est by the following matrices

(12) G θ = C θ c S θ ε c S θ C θ .
The Galileo group corresponds to

(13) ε = 0 , C θ ≡ 1 , S θ ≡ θ , the Lorentz group to (14) ε = 1 , C θ ≡ cosh θ , S θ ≡ sinh θ ,
and a third one here called the "circular group" to

(15) ε = -1 , C θ ≡ cos θ , S θ ≡ sin θ .
We remark that we have the elementary relations, ( 16)

C 2 θ -ε S 2 θ ≡ 1 , dS θ dθ ≡ C θ , dC θ dθ ≡ ε S θ , ε ∈ {-1, 0, 1} .
• We say here that the system (1) associated with the conservation of entropy

     ∂W ∂t + ∂ ∂x f (W ) = 0 ∂ ∂t η(W ) + ∂ ∂x η(W ) u(W ) = 0
with the same flux function f (•), the same mathematical entropy η(•) and the same entropy velocity u(•).

• We present now an elementary calculus. We have the chain rule

       ∂ ∂x = ∂x ∂x ∂ ∂x + ∂t ∂x ∂ ∂t = α θ ∂ ∂x + β θ ∂ ∂t ∂ ∂t = ∂x ∂t ∂ ∂x + ∂t ∂t ∂ ∂t = δ θ ∂ ∂x + γ θ ∂ ∂t .
We inject these operators inside the relation (1):

δ θ ∂ ∂x + γ θ ∂ ∂t W + α θ ∂ ∂x + β θ ∂ ∂t f (W ) = 0
and we put in evidence the new partial derivatives:

∂ ∂t γ θ W + β θ f (W ) + ∂ ∂x δ θ W + α θ f (W ) = 0 .
We introduce a bijective linear operator Y θ depending on θ ∈ IR, acting from IR N into IR N and independent on x and t :

(19) Y θ ∈ GL(IR N ) , θ ∈ IR .
After applying Y θ to the previous equation, we obtain a new conservation law equivalent to (1):

(20) ∂ ∂t γ θ Y θ • W + β θ Y θ • f (W ) + ∂ ∂x δ θ Y θ • W + α θ Y θ • f (W ) = 0 .
A sufficient condition to establish the first equation of ( 18) is to identify the arguments of the two partial derivatives ∂ t and ∂ x in the previous equation:

(21) W = γ θ Y θ • W + β θ Y θ • f (W ) f (W ) = δ θ Y θ • W + α θ Y θ • f (W ) .
The relation [START_REF] Souriau | Thermodynamique et géométrie[END_REF] can be written in a more compact form:

(22) f (W ) W = G θ Y θ • f (W ) W ≡ G θ Y θ • f (W ) Y θ • W .
Under the "linearity hypothesis" [START_REF] Souriau | Structure des systèmes dynamiques[END_REF], the relation [START_REF] Vallée | Lois de comportement des milieux continus dissipatifs compatibles avec la physique relativiste[END_REF] gives geometrical constraints for the transformation Ω W -→ W ∈ Ω associated with the action of the group G.

• We make an analogous calculus for the second equation of ( 18) relative to entropy. After the transformation of partial derivatives, the relation [START_REF] Ruggeri | Galilean invariance and entropy principle for systems of balance laws; the structure of extended thermodynamics[END_REF] takes the form

δ θ ∂ ∂x + γ θ ∂ ∂t η(W ) + α θ ∂ ∂x + β θ ∂ ∂t η(W ) u(W ) = 0  François Dubois
and can be written as

(23) ∂ ∂t γ θ η(W ) + β θ η(W ) u(W ) + ∂ ∂x δ θ η(W ) + α θ η(W ) u(W ) = 0 .
This is a scalar conservation law. This equation is identical to the second equation of [START_REF] Souriau | Géométrie et relativité[END_REF] if the following sufficient conditions are satisfied:

(24) η(W ) = γ θ η(W ) + β θ η(W ) u(W ) η(W ) u(W ) = δ θ η(W ) + α θ η(W ) u(W ) .
With a notation that makes the action of the group G explicit:

(25) η(W ) u(W ) η(W ) = G θ η(W ) u(W ) η(W ) .
• Proposition 2. Linear representation If the relation ( 22) is satisfied for every θ ∈ IR, then the application G G θ -→ Y θ ∈ GL(IR N ) is a linear representation of the group G:

(26) Y θ+θ = Y θ • Y θ , θ, θ ∈ IR .
Proof of Proposition 2.

We first consider the relation [START_REF] Vallée | Lois de comportement des milieux continus dissipatifs compatibles avec la physique relativiste[END_REF] and an analogous relation obtained by replacing the variable θ by the variable θ :

f (W ) W = G θ Y θ • f (W ) W ≡ G θ Y θ • f (W ) Y θ • W .
We compose this relation with [START_REF] Vallée | Lois de comportement des milieux continus dissipatifs compatibles avec la physique relativiste[END_REF] and we obtain

f (W ) W = G θ G θ Y θ • Y θ • f (W ) Y θ • Y θ • W .
When we apply the relation [START_REF] Vallée | Lois de comportement des milieux continus dissipatifs compatibles avec la physique relativiste[END_REF] with the argument θ + θ , we have directly

f (W ) W = G θ+θ Y θ+θ • f (W ) Y θ+θ • W .
We look precisely to the two previous relations. The scalars α θ , β θ , γ θ and δ θ commute with the linear operator Y θ and we obtain the relation (26) because the two previous equalities are satisfied for an arbitrary state W .

4) Manifold of null velocity

• In the following we denominate by Ω 0 the set of states with a velocity equal to zero.

Because the mapping Ω W -→ u(W ) ∈ IR is scalar, it is natural to suppose that Ω 0 is a manifold in IR N of codimension one. We observe also that the flux function f (•) is reduced to its thermodynamic contribution g(•) on the manifold of null velocity. Our program is to construct the states W , the flux function f (•) and the mathematical entropy η(•) when we suppose that the flux function Ω 0 W 0 -→ g 0 (W 0 ) ∈ IR N and the mathematical entropy Ω 0 W 0 -→ σ(W 0 ) ∈ IR are given on the manifold of null velocity.

The idea is to use the group invariance to link a given state W with an appropriate state W 0 on the manifold Ω 0 as presented in Figure 1. The conditions ( 22) and ( 25) are geometric constraints that will allow essentially to solve the problem.

Figure 1.

Fiber above the manifold of null velocity Ω 0 . Thanks to the action of the group, a state W 0 of null velocity Ω 0 generates a current state W .

•

Proposition 3. Velocity field With the hypothesis ( 21) and if the mathematical entropy σ on the manifold of null velocity is not the null function, the velocity field u(W ) is necessarily given by the relation

(27) u(W ) = δ θ γ θ .
Proof of Proposition 3.

We consider a given state W 0 on the manifold Ω 0 and a running state W ∈ Ω obtained by the relations [START_REF] Souriau | Thermodynamique et géométrie[END_REF]. With the actual notations, we have

(28) W = γ θ Y θ • W 0 + β θ Y θ • g 0 (W 0 ) , f (W ) = δ θ Y θ • W 0 + α θ Y θ • g 0 (W 0 ) .
The analogous relations (24) relative to the entropy take a simple form

(29) η(W ) = γ θ σ(W 0 ) , η(W ) u(W ) = δ θ σ(W 0 ) .
because the velocity is also null on Ω 0 . We consider a state

W 0 such that σ(W 0 ) = 0. If γ θ is null, then η(W ) = 0 is null and δ θ = 0 because σ(W 0 ) = 0.
This contradicts the fact that the matrix defining G θ by the relation ( 10) is invertible. Then γ θ = 0 and the relation ( 27) is obtained by taking the ratio of the two relations of (29).

• With the example proposed in [START_REF] Kruzkov | First Order Quasi-Linear Systems in Several Independant Variables[END_REF], we have

(30) u = c S θ C θ .
We observe that we have also established the following property: 

François Dubois

• Proposition 4. Entropy field With the above hypotheses, if the state W is given from W 0 ∈ Ω 0 with the help of the relations (28), we have necessarily

(31) η(W ) = γ θ σ(W 0 ) .
• Hypothesis 1. The null-velocity manifold Ω 0 is included in an hyperplane We remark that if a convex restriction of the mathematical entropy Ω 0 W 0 -→ σ(W 0 ) ∈ IR is given on the manifold Ω 0 , a natural hypothesis is to suppose Ω 0 convex. We will do this hypothesis in the following. Moreover, we suppose that Ω 0 is flat; this is expressed by our hypothesis 1.

•

Proposition 5. Thermodynamic flux With the above framework (28), the thermodynamic flux g(W ) is given from the datum g 0 (W 0 ) according to (32)

g(W ) = α θ -u(W ) β θ Y θ • g 0 (W 0 ) .
Proof of Proposition 5.

The second relation of ( 28) and the definition of the thermodynamic flux (8) allows us to evaluate the function g(•). We have

g(W ) = f (W ) -u(W ) W = δ θ Y θ • W 0 + α θ Y θ • g 0 (W 0 ) -u(W ) γ θ Y θ • W 0 + β θ Y θ • g 0 (W 0 ) = α θ -u(W ) β θ Y θ • g 0 (W 0 ) due to (27)
and the relation (32) is established.

• For the groups proposed in [START_REF] Kruzkov | First Order Quasi-Linear Systems in Several Independant Variables[END_REF], we have α θ -u(W ) [START_REF] Noether | Invariante Variationsprobleme[END_REF]. Then the relation (32) takes the form

β θ = C θ -c S θ C θ ε c S θ = 1 C θ due to
(33) g(W ) = 1 C θ Y θ • g 0 (W 0 ) .

5) Entropy variables

• Even if the state and the entropy are only partially known with the help of relations (28) and (31), its jacobian ϕ introduced in (4) relative to the conserved variables W can be essentially determined. We first consider the dynamical equations present in this study. First, the relation

(34) G θ+θ = G θ • G θ
and the relation (26) for the matrices Y θ can be differentiated relatively to θ. We have

(35) dG θ dθ = G 0 G θ = G θ G 0 , dY θ dθ = Y 0 Y θ = Y θ Y 0 .
With the choice (12) of one classical group, we have

(36) G 0 = 0 c ε c 0 
On hyperbolic systems with entropy velocity... and we can precise how to extend the differential relatations (34) and ( 35).

• Proposition 6. Differential relations for the entropy variables With the above context, we have

(37) G 0 u η * + ϕ g η * dθ + ϕ Y 0 f (W ) ϕ Y 0 W dθ + G θ ϕ Y θ dg 0 ϕ Y θ dW 0 -dσ = 0
Proof of Proposition 6.

We write the relation (28) under a matricial form

(38) f (W ) W = G θ Y θ g 0 (W 0 ) Y θ W 0 .
We differentiate the relation (38), taking into account the two contributions relative to the group G on one hand and along the manifold Ω 0 on the other hand, as illustrated on the figure 1. Then

df dW = G 0 G θ Y θ g 0 Y θ W 0 dθ + G θ Y 0 Y θ g 0 Y 0 Y θ W 0 dθ + G θ Y θ dg 0 Y θ dW 0 and because G θ Y 0 Y θ g 0 Y 0 Y θ W 0 = α θ Y 0 Y θ g 0 + u γ θ Y 0 Y θ W 0 β θ , Y 0 Y θ g 0 + γ θ Y 0 Y θ W 0 = Y 0 α θ Y θ g 0 + u γ θ Y θ W 0 Y 0 β θ Y θ g 0 + γ θ Y θ W 0 = Y 0 f (W ) Y 0 W , we have (39) df dW = G 0 f W dθ + Y 0 f Y 0 W dθ + G θ Y θ dg 0 Y θ dW 0 .
We multiply each line of the relation (39) by the line vector ϕ. We remark that

ϕ G 0 df dW = ϕ α 0 (u γ) 0 β 0 γ 0 df dW = ϕ α 0 f + (u γ) 0 W β 0 f + γ 0 W = α 0 ϕ f + (u γ) 0 ϕ W β 0 ϕ f + γ 0 ϕ W = G 0 ϕ f ϕ W and in a similar way, ϕ G θ Y θ dg 0 dW 0 = ϕ α θ Y θ dg 0 + (u γ) θ Y θ dW 0 β θ Y θ dg 0 + γ θ Y θ dW 0 = α θ ϕ Y θ dg 0 + (u γ) θ ϕ Y θ dW 0 β θ ϕ Y θ dg 0 + γ θ ϕ Y θ dW 0 = G θ ϕ Y θ dg 0 ϕ Y θ dW 0 . Then (40) ϕ df ϕ dW = G 0 ϕ f ϕ W dθ + ϕ Y 0 f ϕ Y 0 W dθ + G θ ϕ Y θ dg 0 ϕ Y θ dW 0 .
• We develop the same calculus for the entropy. We start with a matricial form of the relation (29):

(41) η u η = G θ 0 σ  François Dubois Then (42) d(η u) dη = G 0 η u η dθ + G θ 0 dσ .
We make the difference between the relations (42) and (40). Due to (3) and ( 4) the left hand side is equal to zero! Then the right hand side can be written as

G 0 ϕ f -η u ϕ W -η dθ + ϕ Y 0 f ϕ Y 0 W dθ + G θ ϕ Y θ dg 0 ϕ Y θ dW 0 -dσ .
The relation ( 37) is now easily obtained by the explicitation of the dual entropy introduced in ( 5) and the thermodynamic flux defined in [START_REF] Friedrichs | Systems of Conservation Equations with a Convex Extension[END_REF].

• Proposition 7. Constraints for the entropy variables With the above framework, if we suppose moreover that the derivative of the group G for θ = 0 is given by the relation (36), we have the following relations for the entropy variables

(43) c η * + ϕ Y 0 f (W ) = 0 (44) ε c u η * + ϕ g(W ) + ϕ Y 0 W = 0 (45) ϕ Y θ dg 0 = 0 (46) ϕ Y θ dW 0 -dσ = 0 .
Proof of Proposition 7.

With the matrix G 0 chosen according to (36), the relation (37) becomes

c η * ε c u η * + ϕ g(W ) dθ + ϕ Y 0 f (W ) ϕ Y 0 W dθ + G θ ϕ Y θ dg 0 ϕ Y θ dW 0 -dσ = 0
and the relations (43) to (46) are clear.

• The relation (46) can be expressed with the help of the gradient ϕ 0 of the entropy restricted on the manifold Ω 0 :

(47) dσ ≡ ϕ 0 • dW 0 .

Then the relation (46) express that the difference ϕ Y θ -ϕ 0 belongs to the polar set Ω 0 0

of Ω 0 : we can write ϕ Y θ -ϕ 0 = µ r 0 for some scalar µ and some non-null linear form r 0 identically equal to zero on Ω 0 . We make also a new hypothesis.

• Hypothesis 2. The entropy σ and the thermodynamic flux g 0 are weakly decoupled on the hyperplane Ω 0 In other terms, (48)

(ϕ 0 , dg 0 • ρ 0 ) = 0 , ∀ρ 0 ∈ Ω 0 .
• Proposition 8. Explicitation of the entropy variables We make the hypotheses 1 and 2 and suppose moreover

(49) ∃ ρ 0 ∈ Ω 0 , dg 0 • ρ 0 = 0 .
Then the relations (43) to (46) imply a simple form for the entropy variables:

(50) ϕ Y θ = ϕ 0 .

Proof of Proposition 8.

Consider an arbitrary vector ρ 0 ∈ Ω 0 . When we apply the relation ϕ Y θ -ϕ 0 = µ r 0 established previously to the vector dg 0 • ρ 0 , we find zero due to the constraint (45) and the hypothesis 2 with (48). Then µ < r 0 , dg 0 • ρ 0 > = 0. With the choice of ρ 0 suggested in (49), the duality product < r 0 , dg 0 • ρ 0 > is not null. So µ = 0 and the relation ( 50) is established.

• Hypothesis 3. The entropy σ and the thermodynamic flux g 0 are strongly decoupled on the hyperplane Ω 0 In other terms, (51)

ϕ 0 (W 0 ) • g 0 ( W 0 ) = 0 , ∀ W 0 , W 0 ∈ Ω 0 .
We observe that this strong hypothesis (51) implies (48) by differentiation relatively to the variable W 0 in Ω 0 . In consequence, we have a new property for the entropy variables.

• Proposition 9. The entropy variables are in the polar set of the thermo flux We make the hypotheses 1 and 3 and suppose also the technical condition (49) proposed previously. Then we have a complete decoupling:

(52) ϕ(W ) • g( W ) ≡ 0 ∀ W, W ∈ Ω .
Proof of Proposition 9.

We just have to precise the expression (33) of the thermodynamic flux. Then we have

ϕ(W ) • g( W ) = ϕ(W ) Y θ Y -θ g( W ) = ϕ 0 (W 0 ) 1 C θ g 0 ( W 0 ) = 0
due to (52).

• Proposition 10. Constraints for the representation Y Under the same hypotheses that the ones done for Proposition 9, we have

(53) c η * + C θ ϕ 0 Y 0 g 0 = 0 (54) ϕ 0 Y 0 W 0 = 0 on Ω 0 .
Proof of Proposition 10.

We first have the following calculus:

ϕ Y 0 g = ϕ Y θ Y -θ Y 0 g = ϕ Y θ Y 0 Y -θ g = ϕ 0 Y 0 1 C θ g 0 = 1 C θ ϕ 0 Y 0 g 0 .
Then we multiply the equation (44) by the opposite of the velocity u and we add it to (43). With the definition (8) of the thermodynamic flux, we have, taking into account the "trigonometrical relations" ( 16) and the expression (27) of the velocity:

0 = η * c - ε u 2 c + ϕ Y 0 g = η * c - ε c S 2 θ C 2 θ + 1 C θ ϕ 0 Y 0 g 0 = c η * C 2 θ + C θ ϕ 0 Y 0 g 0  François Dubois
and the relation (53) is established. We report now the expression (28) of a current state W inside the equation (44):

0 = ε c u η * + ϕ Y 0 Y θ • C θ W 0 + ε c S θ g 0 = C θ ϕ Y θ Y 0 W 0 + ε c u η * + S θ ϕ Y θ Y 0 g 0 = C θ ϕ Y θ Y 0 W 0 + ε c u η * -S θ c C θ η * = C θ ϕ Y θ Y 0 W 0 + ε η * u c - S θ C θ = C θ ϕ Y θ Y 0 W 0
and the relation (54) follows from (50).

•

Proposition 11. Dual entropy Under the hypotheses proposed at Proposition 9, we denote by σ * the dual of the entropy σ restricted on the manifold of null velocity:

(55) σ * ≡ ϕ 0 • W 0 -σ(W 0 ) on Ω 0 , with ϕ 0 = σ (W 0 ) .
Then we have (56)

η * = C θ σ *
and the relation (53) takes the form

(57) c σ * + ϕ 0 Y 0 g 0 = 0 on Ω 0 .
Proof of Proposition 11.

We have, thanks to (5), (31), ( 38) and (50) :

η * = ϕ • W -η(W ) = ϕ 0 • Y -θ Y θ C θ W 0 + ε c S θ g 0 -C θ σ = C θ ϕ 0 • W 0 -σ
due to Hypothesis 3 (relation (51)). The end of the proof is clear.

6) Hyperbolic systems of order two

• If we suppose N = 2, the matrix Y θ is a two by two matrix. We suppose here that this matrix has the same algebaic form that the form [START_REF] Kruzkov | First Order Quasi-Linear Systems in Several Independant Variables[END_REF] suggested for G θ . We set for this contribution (58)

Y θ =   C θ ε a S θ a S θ C θ   , θ ∈ IR , a > 0 .
In the relation (58), the functions S θ and C θ are analogous to the ones proposed in ( 13), ( 14), ( 15) and ( 16) for S θ and C θ . We have in particular

(59) C 2 θ -ε S 2 θ ≡ 1 .
The nilpotent representation of the group G corresponds to ε = 0, the hyperbolic representation to ε = 1 and the elliptic one to ε = -1. The derivative Y θ of the matrix Y θ proposed in (58) at θ = 0 is given by (60)

Y 0 =   0 ε a a 0   .
• We particularize the system by a simple choice for the null-velocity manifold. As in our previous work [START_REF] Dubois | Conservation laws invariants for Galileo group; CEMRACS preliminary results[END_REF], we suppose (61)

W 0 = ρ 0 0 ∈ Ω 0 and (62) g 0 (W 0 ) = 0 p 0 for W 0 ∈ Ω 0 .
With this choice, the hypotheses 2 and 3 are satisfied. The variable p 0 in the right hand side of the relation (62) is the thermodynamic pressure.

• Proposition 12. Pressure as a thermodynamic variable With the above hypotheses, we have the relation

(63) σ * + ε a c σ p 0 = 0 .
Proof of Proposition 12.

Just write the relation (57) with the help of ( 60) and (61) :

c σ * + ϕ 0 Y 0 g 0 = c σ * + σ 0   0 ε a a 0   0 p 0 = c σ * + ε a σ p 0
and the relation ( 63) is established.

• The nilpotent case ( ε = 0 ) has no interest when N = 2 because the dual of the entropy is necessary null, as we have remarked in [START_REF] Dubois | Conservation laws invariants for Galileo group; CEMRACS preliminary results[END_REF] in the particular case of the Galileo group (ε = 0). We can explicit the first relations of (28). With the notation

(64) W ≡ ρ J we have (65) ρ = C θ C θ ρ 0 + ε ε a c S θ S θ p 0 , J a = C θ S θ ρ 0 + ε a c S θ C θ p 0 .
• In the case of Galileo group, we have ε = 0, C θ = 1, S θ = θ and u = c θ. The explicitation of the parameter θ and the state of null velocity ρ 0 is simple by resolution of the system (65). We have

ρ = C θ ρ 0 , J a = S θ ρ 0 , ε = 0 , Galileo group.
In the hyperbolic case, we have

tanh θ = J a ρ , u = c θ , ε = 0 , ε = 1
and in the elliptic one,

tg θ = J a ρ , u = c θ , ε = 0 , ε = -1
as we have explained in [START_REF] Dubois | Conservation laws invariants for Galileo group; CEMRACS preliminary results[END_REF].

• In the following, we focus on hyperbolic ( ε = 1) and elliptic ( ε = -1) representations of the Lorentz (ε = 1) and circular (ε = -1) groups. With other words, (66)

ε 2 = ε 2 = 1 .
We can replace the pressure p 0 with an appropriate thermodynamic quantity because

σ * σ = ρ 0 σ -σ σ = ρ 0 - σ σ .
Then the relations (65) take the form

(67) ρ = C θ C θ -ε S θ S θ ρ 0 + ε S θ S θ σ σ , J a = C θ S θ -ε ε S θ C θ ρ 0 + ε ε S θ C θ σ σ .
For the Lorentz (ε = 1) or the circular (ε = -1) group, the resolution of the system (67) with unknowns θ and ρ 0 is absolutly non trivial. In order to be able to differentiate the solution (θ, ρ 0 ) relatively to (ρ , J), we have the following result.

• Proposition 13. Jacobian We denote by ζ 0 the partial derivative of the ratio σ/σ relatively to the dentity ρ 0 at rest :

(68) ζ 0 ≡ ∂ ∂ρ 0 σ σ .
If the hypothesis (66) is satisfied and if the following jacobian determinant

(69) ∆ ≡ ∂ (ρ, J) ∂ (θ, ρ 0 ) = a σ σ (ε ε -1) C 2 θ σ * σ + ε S 2 θ ζ 0 -(C 2 θ + ε S 2 θ )
is not null, the solution (θ, ρ 0 ) of the system (67) is locally unique and we can differentiate the parameters θ and ρ 0 relatively to the state W introduced in (64).

Proof of Proposition 13.

We have

                         ∂ρ ∂θ = ( ε -ε) C θ S θ ρ 0 + ε C θ S θ + S θ C θ σ σ ∂ρ ∂ρ 0 = C θ C θ -ε S θ S θ + ε S θ S θ ζ 0 1 a ∂J ∂θ = (1 -ε ε) C θ C θ ρ 0 + ε ε C θ C θ + ε S θ S θ σ σ 1 a ∂J ∂ρ 0 = C θ S θ -ε ε S θ C θ + ε ε S θ C θ ζ 0 .
Then the determinant ∆ ≡ ∂ (ρ, J) ∂ (θ, ρ 0 ) can be evaluated as follows :

∆ = ∂ρ ∂θ ∂J ∂ρ 0 - ∂ρ ∂ρ 0 ∂J ∂θ = a ( ε -ε) C θ S θ ρ 0 + ε C θ S θ + S θ C θ σ σ C θ S θ -ε ε S θ C θ + ε ε S θ C θ ζ 0 -a C θ C θ -ε S θ S θ + ε S θ S θ ζ 0 (1 -ε ε) C θ C θ ρ 0 + ε ε C θ C θ + ε S θ S θ σ σ -(C 2 θ + ε S 2 θ ) + ε S 2
θ ζ 0 and the expression (69) is established.

• After determining the parameters θ and ρ 0 it is possible to evaluate the mathematical entropy η thanks to the relation (31). Then we can differentiate this mathematical entropy relatively the density ρ and the momentum J. We have the following remarquable coherence property.

•

Proposition 14. Entropy variables If the jacobian determinant (69) is not null, then the entropy variables ϕ ≡ (α, β) can be determined according to the relations (50). In other terms, (71)

α ≡ ∂η ∂ρ = C θ σ (ρ 0 ) , β ≡ ∂η ∂J = - ε a S θ σ (ρ 0 ) .
Proof of Proposition 14.

We first invert the jacobian matrix :

    ∂ρ ∂θ ∂ρ ∂ρ 0 ∂J ∂θ ∂J ∂ρ 0     -1 =     ∂θ ∂ρ ∂θ ∂J ∂ρ 0 ∂ρ ∂ρ 0 ∂J     = 1 ∆     ∂J ∂ρ 0 - ∂ρ ∂ρ 0 - ∂J ∂θ ∂ρ ∂θ     . Then                        ∂θ ∂ρ = 1 ∆ ∂J ∂ρ 0 = a ∆ (C θ S θ -ε ε S θ C θ ) + ε ε S θ C θ ζ 0 ∂ρ 0 ∂ρ = - 1 ∆ ∂J ∂θ = - a ∆ (1 -ε ε) C θ C θ ρ 0 + ε ε C θ C θ + ε S θ S θ σ σ ∂θ ∂J = - 1 ∆ ∂ρ ∂ρ 0 = - 1 ∆ (C θ C θ -ε S θ S θ ) + ε S θ S θ ζ 0 ∂ρ 0 ∂J = 1 ∆ ∂ρ ∂θ = 1 ∆ ( ε -ε) C θ S θ ρ 0 + ε (C θ S θ + S θ C θ ) σ σ .
We first observe that

∂η ∂θ = ∂ ∂θ C θ σ(ρ 0 ) = ε S θ σ , ∂η ∂ρ 0 = ∂ ∂ρ 0 C θ σ(ρ 0 ) = C θ σ .
Then with the chain rule, we have

α = ∂η ∂ρ = ∂η ∂θ ∂θ ∂ρ + ∂η ∂ρ 0 ∂ρ 0 ∂ρ = a ∆ ε S θ σ (C θ S θ -ε ε S θ C θ ) + ε ε S θ C θ ζ 0 -C θ σ (1 -ε ε) C θ C θ ρ 0 + (ε ε C θ C θ + ε S θ S θ σ σ = a ∆ (ε ε -1) C 2 θ C θ ρ 0 σ + ε S 2 θ C θ ζ 0 σ -ε C θ (S 2 θ + ε C 2 θ ) σ = a ∆ C θ σ (ε ε -1) C 2 θ ρ 0 + ε S 2 θ ζ 0 σ σ -ε (S 2 θ + ε C 2 θ ) σ σ = C θ σ
due to (70). In an analogous way, we have

β = ∂η ∂J = ∂η ∂θ ∂θ ∂J + ∂η ∂ρ 0 ∂ρ 0 ∂J = 1 ∆ -ε S θ σ (C θ C θ -ε S θ S θ ) + ε S θ S θ ζ 0 + C θ σ ( ε -ε) C θ S θ ρ 0 + ε (C θ S θ + S θ C θ ) σ σ = ε ∆ (1 -ε ε) C 2 θ S θ ρ 0 σ + ε S 2 θ + ε C 2 θ S θ σ -ε S 2 θ S θ ζ 0 σ = ε ∆ S θ σ (1 -ε ε) C 2 θ ρ 0 + ε S 2 θ + ε C 2 θ σ σ -ε S 2 θ ζ 0 σ σ = - ε a S θ σ
due to (70). The proof is completed.

• We particularize our study and wish to construct nontrivial hyperbolic systems with entropy velocity covariant under the Lorentz group and -or the circular group. It seems essential to maintain a constant sign for the determinant ∆ presented at the relation (69). Moreover, we have to enforce the strict convexity of the mathematical entropy (ρ, J) -→ η(ρ, J). In this contribution, we only consider the particular case (72) ε = ε .

We focus on "circular elliptic" and "Lorentz hyperbolic" two by two systems of conservation laws. Observe first that due to the condition (72), the expression (67) of the conserved variables simplify into

(73) ρ = ρ 0 + ε S 2 θ σ σ , J = a S θ C θ σ σ .
Then the mathematical entropy η constrained to the condition (31) is strictly convex if and only if the determinant of its hessian det (d2 η) is stricly positive and if the second derivative ∂ 2 η ∂ρ 2 is also strictly positive.

•

Proposition 15. Technical lemma for convexity conditions Under the hypotheses (66) and (72), the expression of the jacobian determinant ∆ is given according to

(74) ∆ = a δ σ σ with (75) δ = ε S 2 θ ζ 0 -(C 2 θ + ε S 2 θ
) . The two signed expressions are given according to (76)

det d 2 η = ε σ σ a ∆ = ε a 2 σ and (77) ∂ 2 η ∂ρ 2 = ∂α ∂ρ = C θ δ ε S 2 θ ζ 0 σ 2 σ -C 2 θ + ε S 2 θ σ .
Proof of Proposition 15.

The expression δ in (75) is just a specific name for the big braket in right hand side of the expression (69) and the relation (74) is satisfied. Under the condition (72), the expression δ is clearly reduced to the expression (75). We establish now the expression (76) of the determinant of the hessian matrix. From the chain rule, we have

det d 2 η = ∂ (α, β) ∂ (ρ, J) = ∂(α, β) ∂(θ, ρ 0 ) ∂(θ, ρ 0 ) ∂(ρ, J) = 1 ∆ det     ∂α ∂θ ∂α ∂ρ 0 ∂β ∂θ ∂β ∂ρ 0     = 1 ∆ det ε S θ σ C θ σ - ε a C θ σ - ε a S θ σ = - 1 ∆ ε a σ σ det ε S θ C θ C θ S θ
due to (71) and (72). Then det d 2 η = 1 a δ σ σ ε a σ σ is a consequence of the trigonometrical property [START_REF] Noether | Invariante Variationsprobleme[END_REF] and the expression (76) follows. The evaluation of (77) is also elementary. From the chain rule, we have

∂α ∂ρ = ∂α ∂θ ∂θ ∂ρ + ∂α ∂ρ 0 ∂ρ 0 ∂ρ = a ∆ ε S θ σ S θ C θ ζ 0 -C θ σ C 2 θ + ε S 2 θ σ σ = 1 δ σ σ C θ ε S 2 θ σ ζ 0 -σ C 2 θ + ε S 2
θ σ σ and the expression (77) is clear.

• We detail now two explicit examples of hyperbolic system of order two. The first one is invariant under the action of the circular group and the second under the Lorentz group.

•

Proposition 16. Exponential entropy for a "circular-elliptic" system We introduce two strictly positive constants ρ * and σ and an exponential entropy on the manifold Ω 0 thanks to

(78) σ(ρ 0 ) = σ exp ρ 0 ρ * .
We set (79)

         Ψ = 2 J a ρ * , ρ 0 = ρ + 1 2 1 - √ 1 -Ψ 2 ρ * , θ = 1 2 Arcsin Ψ , u = c J | J | 1 - √ 1 -Ψ 2 1 + √ 1 -Ψ 2 = c sgn(J) tan θ , p 0 = a c ρ 0 -ρ * .
The system of conservation laws (80)

     ∂ρ ∂t + ∂ ∂x c a J = 0 ∂J ∂t + ∂ ∂x u J + p 0 = 0  François Dubois is hyperbolic if we suppose (81) | Ψ | ≤ 1 , | θ | ≤ π 4 .
It admits a mathematical entropy given by the expression

(82) η(ρ, J) ≡ 1 2 1 + √ 1 -Ψ 2 σ ρ + ρ * 2 1 - √ 1 -Ψ 2
and the dissipation of entropy of entropy solutions is given by the inequality

(83) ∂η ∂t + ∂ ∂x u η ≤ 0 .
Proof of Proposition 16.

With the choice (78) of an exponential entropy, the parameter ζ 0 introduced at the relation ( 68) is null. Because σ is supposed strictly positive, σ is a convex function and the condition det d 2 η > 0 is reduced to δ < 0 if we look to a "circular-elliptic" hyperbolic system, id est if we suppose ε = -1 (see ( 76)). Then with the notation Ψ introduced in (79), the relations (73) can be written

sin (2 θ) = Ψ , ρ = ρ 0 -sin 2 θ ρ * .
and the calculus of θ proposed in (79) is natural. We have the same remark for the expression of ρ 0 in (79) as a consequence of the identity sin 2 θ = 1 2 1 -1 -sin 2 (2 θ) . The second condition in (81) is a consequence of the expression of θ proposed in (79). This last condition is also necessary when we impose δ < 0 because the relation (75) can be now written δ = -(cos 2 θ -sin 2 θ). The velocity in (79) is simply the actual formulation of the relation (30) because

tan θ = sgn(θ) 1 -1 -sin 2 (2 θ) 1 + 1 + sin 2 (2 θ
) .

• The static pressure p 0 is related to the entropy σ thanks to (63). The expressions ( 8) and (33) of the physical and thermodynamix fluxes, the algebraic forms (58) of the matrix Y θ and the choice (62) for g 0 show that the first component f 1 of f (W ) is given by

f 1 = ρ u - sin θ a cos θ p 0 = ρ u - 1 a c u a c (ρ 0 -ρ * ) = ρ * u (1 -sin 2 θ) = ρ * c sin θ cos θ = 1 2 ρ * c sin(2 θ) = 1 2 ρ * c Ψ = 1 2 ρ * c 2 J a ρ * = c a J .
Then the relations (80) are established. The expression (82) of the mathematical entropy is then a consequence of (31) : η = cos θ σ(ρ 0 ). A particular hyperbolic system with entropy velocity invariant under the action of the circular group is constructed.

• Proposition 17. Homographic entropy for a "Lorentz-hyperbolic" system We consider two strictly positive constants ρ * and σ and an homographic entropy σ on the manifold Ω 0 defined by the relation

(84) σ(ρ 0 ) = -σ ρ 0 ρ 0 + ρ * , ρ 0 > 0 . We set (85) Φ(ρ 0 ) ≡ 2 J a σ σ = 2 J a ρ * ρ 0 (ρ 0 + ρ * ) and we suppose (86) | J | a ≤ ρ ≤ 1 2 ρ * .
Then the equation ( 87)

F (ρ 0 ) ≡ ρ 0 + σ 2 σ 1 + Φ(ρ 0 ) 2 -1 = ρ has a unique solution ρ 0 such that (88) 0 ≤ ρ 0 ≤ 1 2 ρ * .
We consider the expressions

(89) p 0 = a c ρ 2 0 ρ * , θ = 1 2 Argsh Φ(ρ 0 ) , u = a tanh θ = c sgn(J) √ 1 + Φ 2 1 √ 1 + Φ 2 + 1 .
Then the system of conservation laws (90)

       ∂ρ ∂t + ∂ ∂x ρ + ρ 0 ρ * u = 0 ∂J ∂t + ∂ ∂x u J + p 0 = 0
is hyperbolic and admits a mathematical entropy given by the expression

(91) η(ρ, J) ≡ 1 2 1 + 1 + Φ(ρ) 2 σ(ρ 0 ) .
The dissipation of entropy of entropy solutions is still given by the inequality (83).

Proof of Proposition 17.

In this case of the Lorentz group, we have ε = 1, S θ ≡ sinh θ and C θ ≡ cosh θ. We have chosen with (84) a negative convex function σ. Then (c.f. ( 76)) the condition δ < 0 has to be satisfied to assume det d 2 η > 0. We have

σ σ = ρ 0 ρ * (ρ 0 + ρ * ) , ζ 0 ≡ ∂ ∂ρ 0 σ σ = 1 + 2 ρ 0 ρ * .
The relations (73) can now be written as

sinh (2 θ) = Φ(ρ 0 ) , ρ = ρ 0 + sinh 2 θ σ σ .
The identity sinh 2 θ ≡ 1 2 1 + sinh 2 (2 θ) -1 shows that the density ρ 0 at null velocity associated with an arbitrary state W ≡ (ρ, J) t has to solve the equation

ρ -ρ 0 = 1 2 1 + Φ(ρ 0 ) 2 -1 σ σ
 which is equivalent to (87). We remark that F (0) = | J | /a and the first condition of (88) is natural.

• We prove now that the equation (87) has a unique solution if the conditions (86) are satisfied. We have

∂ ∂ρ 0 σ 2 σ √ 1 + Φ 2 = 1 2 ∂ ∂ρ 0 σ σ 2 + 4 J 2 a 2 = 2 4 σ σ 2 + 4 J 2 a 2 σ σ ζ 0 > 0 if ρ 0 > 0. We have also ∂ ∂ρ 0 ρ 0 - σ 2 σ = 1 - ζ 0 2 = 1 2 - ρ 0 ρ * > 0 under the con- dition (
p 0 = -a c σ * σ = a c σ σ -ρ 0 = a c ρ 2 0
ρ * and the first relation of ( 89) is established. The two other relations of (89) are an elementary consequence of the previous considerations.

• We now establish the algebraic form of the hyperbolic system [START_REF] Bargmann | On unitary ray representations of continuous groups[END_REF]. From ( 8) and (33), the particular expression (58) of the matrix Y θ and the choice g 0 ≡ (0, p 0 ) t proposed in (62) show that the first component f 1 of f (W ) is given by

f 1 = ρ u + sinh θ a cosh θ p 0 = ρ u + 1 a c u a c ρ 2 0 ρ * = ρ + ρ 2 0 ρ * u .
Then the fist line of the relations (90) is established. The second line is straightforword to evidence. The expression of the mathematical entropy is a direct consequence of (31) and the explitation of cosh θ as a function of Φ(ρ 0 ) as proposed in (89). This remark completes the construction of an hyperbolic system with entropy velocity (90) invariant under the action of the Lorentz group.

7) Conclusion

• In this contribution, we have extended our previous work [START_REF] Dubois | Conservation laws invariants for Galileo group; CEMRACS preliminary results[END_REF] relative to the construction of hyperbolic systems covariant under the Galileo group and space reflection. We have precised the notion of "entropy velocity" as an intrinsic velocity associated with the conservation of entropy for regular solutions. In particular, the usual velocity of gas dynamics system is an entropy velocity. An hyperbolic system of conservation laws and the associated conservation of entropy are covariant under the action of a group of space-time transformations if the algebraic form of the equations is not changed after the action of the corresponding group. We have proposed sufficient conditions in order to satisfy a group covariance property. These conditions introduce naturally a representation of the group in the space of states that can be chosen in multiple ways. We have also introduced a natural manifold of "null velocity". When thermodynamic data are given on this manifold, the natural question is the construction of the whole hyperbolic system from the covariance property. We have also natural constraints for the entropy variables.

• After the study of the particular case of Galileo covariance in [START_REF] Dubois | Conservation laws invariants for Galileo group; CEMRACS preliminary results[END_REF], we have particularized in this contribution our study for the two-dimensional case with the Lorentz and circular groups. We have proposed nontrivial hyperbolic systems covariant under each of these two groups. For the circular example, we find a condition (81) that shows that the associated hyperbolic system is not covariant under the action of the whole circular group. The question is set [START_REF] Vallée | Personal communication[END_REF] to know if there exists a mathematical obstruction to find hyperbolic systems covariant under the action of the whole circular group. A better knowledge of such systems of conservation laws is a possible next step. Last but not least, the link with relativistic gas dynamics (see e.g. Souriau [START_REF] Souriau | Thermodynamique relativiste des fluides[END_REF][START_REF] Souriau | Thermodynamique et géométrie[END_REF], Chiu [START_REF] Chiu | Relativistic gas dynamics[END_REF] and Vallée [START_REF] Vallée | Lois de comportement des milieux continus dissipatifs compatibles avec la physique relativiste[END_REF]) is also a natural question. The extension of this work to several space dimensions is, in principle, straightforward.

  88) for ρ 0 . Then we have F (ρ 0 ) > 0 and the equation (87) has a unique solution when (88) is satisfied. The maximum of the function F (•) on the interval [0,1 2 ρ * ] can be estimated as follows. We have first 2 and the second condition of (86) is natural. The pressure p 0 is given by the relation (63) :

											σ σ	ρ * 2	= 1 2	1 2 + 1 ρ * = 3 4 ρ * . Then
	F	ρ * 2	=	ρ * 2	-	3 8	ρ * +	ρ * 2	3 4	2	+		2 J a ρ *	2	≥	ρ * 2	-	3 8	ρ * +	3 8	ρ * =	ρ

*

σ σ 1 δ

Acknowledgments

The author thanks Géry de Saxcé and Claude Vallée for their kind invitation to participate to the 56th and 57th CITV in august 2012 and august 2013, and for stimulating discussions during these workshops. In november 2014, Claude Vallée passed away. Of course, this contribution is also dedicated to him.