Yue-Jun Peng
email: peng@math.univ-bpclermont.fr

EULER-LAGRANGE CHANGE OF VARIABLES IN CONSERVATION LAWS

Keywords: Conservation laws, change of variables, explicit entropy solutions, uniqueness AMS Subject Classification (2000): 35L65, 76N15, 35Q35

We introduce a new method for studying the Cauchy problem for systems of conservation laws in one space dimension. This method is based on the equivalence of the Cauchy problems in Eulerian and Lagrangian coordinates, as regards the existence and uniqueness of entropy solutions. The main idea is to solve the problem in Lagrangian coordinates and determine the transformation linking the two coordinates. The main contributions are the uniqueness and explicit entropy solutions. Applications include the Keyfitz-Kranzer system, the Born-Infeld equations and linear Lagrangian systems which are linear in Lagrangian coordinates.

For these examples, the existence and uniqueness of solutions in L ∞ are obtained in explicit expressions. The linear Lagrangian system contains examples such as the equations of pressureless gas dynamics, all 2 × 2 linearly degenerate systems and the augmented Born-Infeld equations. In particular, we deduce the existence and uniqueness of entropy solutions of the Cauchy problem for the Born-Infeld equations. An explicit formula of its entropy solution is also provided.

Introduction

Both Eulerian and Lagrangian coordinates are used for the description of the evolution of one-dimensional flow of a compressible and inviscid gas. The equations of gas dynamics have a simple form in Lagrangian coordinates and are often treated in those in mathematical analysis instead of in Eulerian coordinates. See for instance [START_REF] Nishida | Global solutions for an initial boundary value problem of a quasilinear hyperbolic system[END_REF][START_REF] Nishida | Solutions in the large for some nonlinear hyperbolic conservation laws[END_REF][START_REF] Liu | Initial-boundary-value problems for gas dynamics[END_REF]. These two coordinate systems are linked by a nonlinear transformation depending on unknown variables [START_REF] Courant | Supersonic Flow and Shock Waves[END_REF][START_REF] Gel'fand | Some problems in the theory of quasilinear equations[END_REF]. The transformation is abbreviated by E-L. It possesses many interesting properties (see [START_REF] Sévennec | Géométrie des Systèmes Hyperboliques de Lois de Conservation[END_REF][START_REF] Wagner | Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions[END_REF]).

For a class of systems of conservation laws, after an E-L transformation we obtain a system in Lagrangian coordinates. It is easy to see that smooth solutions of the systems in the two coordinate systems are equivalent. In 1987, Wagner gave a rigorous justification of the equivalence of entropy solutions in L ∞ of the systems in the two coordinate systems [START_REF] Wagner | Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions[END_REF]. His result shows that the choice of coordinates is not important and one may work with either of them. However, in his work the initial data have not been taken into account and the equivalence of the Cauchy problem in the two systems has not been investigated.

In this paper, we prove that the Cauchy problems in the two coordinate systems are still equivalent as regards the existence and uniqueness of entropy solutions in L ∞ . An explicit relation between the entropy solutions in the two coordinate systems is exhibited. These results are given in Theorem 2.1 in the general case. They reveal that when the transformation is known, the entropy solution in Eulerian coordinates is obtained explicitly from the entropy solution in Lagrangian coordinates and vice versa. Note that the E-L transformation is uniquely determined by the entropy solution and its initial data. Thus, if the system in Lagrangian coordinates has a simple form, it is possible to solve the Cauchy problem for the original system by determining the transformation.

The E-L transformation is induced by one of the conservation equations. For the linearly degenerate system, entropy inequalities for all convex entropies become entropy equalities for all entropies. This provides many choices of conservation laws for making transformations. This idea is used throughout the applications in Sections 3-6.

The above considerations have some consequences in applications. The first application concerns the Keyfitz-Kranzer system of two equations for which one characteristic field is genuinely nonlinear and the other is linearly degenerate [START_REF] Keyfitz | A system of strictly hyperbolic conservation laws arising in elasticity theory[END_REF]. In a very simple way we establish the existence and uniqueness of entropy solutions for the Cauchy problem. For such a system with convex flux functions, an explicit formula of the entropy solution is provided with the help of the Lax formula for scalar conservation law.

The second application concerns linear Lagrangian systems, a class of systems which are linear in Lagrangian coordinates (see Definition 3.1). For such a hyperbolic system, we give a precise description of its mathematical structure on the eigenvalues, linear degeneracy of the characteristic fields, classical Riemann invariants and entropy-entropy flux pairs. Strictly convex entropies are also constructed. From the explicit entropy solution of the Cauchy problem for the linear system in Lagrangian coordinates, we deduce a unique E-L transformation which yields a unique entropy solution of its Cauchy problem with an explicit expression. This expression is of the form : w i (t, x) = w 0 i (α i (t, x)), where w i is an i-th classical Riemann invariant of the system, w 0 i is its initial value and α i is determined explicitly by the system and the initial data. Therefore, a maximum principle can be applied to each w i . Furthermore, we show that the entropy solution is weakly stable in L ∞ (R + × R) and satisfies the entropy equalities for all entropy-entropy flux pairs of the system.

We present three physical examples of linear Lagrangian systems. They are the system of pressureless gas dynamics, all 2 × 2 linearly degenerate systems and the augmented Born-Infeld (abbreviated to ABI) equations introduced by Brenier [START_REF] Brenier | Hydrodynamic structure of the augmented Born-Infeld equations[END_REF]. These systems are not necessarily strictly hyperbolic. For the system of pressureless gas dynamics which is not even hyperbolic, the local existence of solutions in L ∞ with explicit expressions is shown under a partial regularity assumption on the initial velocity. The solution is global if the initial velocity is a non-decreasing function. The 2 × 2 linearly degenerate system and the ABI equations are two hyperbolic linear Lagrangian systems. Thus our results can be immediately applied to them. The main application is the existence and uniqueness of entropy solutions of the Cauchy problem for the Born-Infeld (abbreviated by BI) system. The later is not a linear Lagrangian system. The results for this system are obtained from the entropy solution and a further study on the entropy-entropy flux pairs of the ABI equations. An explicit formula of the entropy solution of the BI equations is also given.

Finally, we remark that the equivalence of the two coordinate systems in several space dimensions is discussed in [START_REF] Dafermos | Equivalence of referential and spatial field equations in continuum physics[END_REF][START_REF] Dafermos | Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der Mathematischen Wissenschaften[END_REF][START_REF] Wagner | Conservation laws, coordinate transformations, and differential forms, Hyperbolic problems: theory, numerics, applications[END_REF]. See also [START_REF] Després | Lagrangian gas dynamics in two dimensions and Lagrangian systems[END_REF] for a numerical investigation in two dimensional equations of gas dynamics and the references therein. However, due to the essential difference between the systems of conservation laws in one dimension and several dimensions, it is not so obvious to obtain the results of this paper in several dimensions. A further study on the problem will be given in a forthcoming work.

This paper is organized as follows. In the next section, we consider the change of variables and establish the equivalence of entropy solutions of the Cauchy problems in the two coordinate systems. The main result of this section is Theorem 2.1. In Section 3 we apply Theorem 2.1 to the Keyfitz-Kranzer system. Section 4 is devoted to the study of the linear Lagrangian system with its first example : pressureless gas dynamics. Sections 5 and 6 contain another two examples of linear Lagrangian systems : the 2 × 2 linearly degenerate system and the ABI equations. At this stage the BI equations are studied. In the Appendix, we give the proof of Lemma 6.2 which is a key step to obtain the main results for the BI equations.

2. Equivalence of entropy solutions in two coordinate systems 2.1. Definitions. Consider the Cauchy problem for a system of conservation laws :

(2.1)

∂ t u + ∂ x f (u) = 0, t > 0, x ∈ R,
subject to an initial condition :

(2.2) t = 0 : u = u 0 (x), x ∈ R.

Here u = (u 1 , u 2 , • • •, u n ) t , u 0 = (u 0 1 , u 0 2 , • • •, u 0 n ) t ∈ L ∞ (R), and f = (f 1 , f 2 , • • •, f n ) t is a smooth function from a domain U ⊂ R n to R n . We denote by λ 1 (u), λ 2 (u), • • •, λ n (u)
the eigenvalues of the system (2.1), i.e., the eigenvalues of the matrix A(u) = f (u) and by r 1 (u), r 2 (u), • • •, r n (u) the corresponding right eigenvectors. The following notions are fundamental, see [START_REF] Lax | Hyperbolic systems of conservation laws II[END_REF] for instance. We say that the system (2.1) is hyperbolic in

U if λ 1 (u), λ 2 (u), • • •, λ n (u) are real and A(u) is diagonalizable for all u ∈ U . It is strictly hyperbolic if λ 1 (u), λ 2 (u), • • •, λ n (u) are all distinct. The i-th characteristic λ i (u)
is genuinely nonlinear in the sense of P.D.Lax if ∇λ i (u).r i (u) = 0 and is linearly degenerate if ∇λ i (u).r i (u) = 0 for all u ∈ U . If all the characteristics are linearly degenerate, the system (2.1) is said to be linearly degenerate.

It is known that solutions of the system (2.1) develop singularities in a finite time even if the initial data are smooth. As a consequence one has to seek global existence of weak solutions in the sense of distributions. In order to guarantee the uniqueness of weak solutions an entropy condition is imposed.

A function u ∈ L ∞ loc (R + × R) is called entropy solution if for all entropy-entropy flux pairs (E, F ) with E convex it satisfies (2.3) ∂ t E(u) + ∂ x F (u) ≤ 0,
in the sense of distributions.

2.2. Equivalence of systems. Let u ∈ L ∞ (R + ×R) be an entropy solution of the system (2.1) with u 1 (t, x) ≥ u 1 > 0. The variable u 1 plays a special role in the study. In physical models it may stand for the density, energy density etc. Using the first equation in (2.1) for u 1 , we may take an E-L transformation from (t, x) to (s, y) :

(2.4) (s, y) = (t, Y (t, x)) and dy = u 1 dx -f 1 (u)dt.
This change of variables can also be written as :

s = t and y = x X 1 (t) u 1 (t, ξ)dξ with X 1 (t) = f 1 (u(t, X 1 (t))) u 1 (t, X 1 (t)) .
It is unique if Y (0, x) or X 1 (0) is given. Then in Lagrangian coordinates (s, y), the system (2.1) reads (see [START_REF] Wagner | Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions[END_REF]) :

(2.5)

         ∂ s 1 ũ1 -∂ y f 1 (ũ) ũ1 = 0, ∂ s ũi ũ1 + ∂ y f i (ũ) - f 1 (ũ)ũ i ũ1 = 0 (2 ≤ i ≤ n),
for s > 0 and y ∈ R. Here and in what follows we denote by ũ the variable u in Lagrangian coordinates, i.e., ũ(s, y) = u(t, x). More generally, for an entropy-entropy flux pair (E, F ) of the system (2.1), let us define

(2.6) Ẽ(v) = E(u)/u 1 , F (v) = F (u) -f 1 (u)E(u)/u 1 , with (2.7) v 1 = 1 u 1 , v i = u i u 1 (2 ≤ i ≤ n), v = (v 1 , v 2 , • • •, v n ) t .
For

u 1 > 0 the function ϕ : u -→ v is a C ∞ -diffeomorphism with the property ϕ • ϕ = I d .
From [START_REF] Wagner | Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions[END_REF] we know that (E, F ) is an entropy-entropy flux pair of (2.1) if and only if ( Ẽ, F ) is an entropy-entropy flux pair for (2.5), and E is convex with respect to u if and only if Ẽ is convex with respect to v. Hence, in Lagrangian coordinates the entropy condition (2.3) is equivalent to

(2.8) ∂ s Ẽ(ṽ) + ∂ y F (ṽ) ≤ 0, Ẽ convex.
For linearly degenerate systems the Rankine-Hugoniot conditions are independent of the choice of the conservation laws among all entropy-entropy flux pairs. Therefore, it is expected that the entropy solution satisfies the entropy equality (2.9)

∂ t E(u) + ∂ x F (u) = 0
for all entropy-entropy flux pairs (E, F ). This condition is equivalent to

(2.10) ∂ s Ẽ(ṽ) + ∂ y F (ṽ) = 0. Definition 2.1. A function u ∈ L ∞ loc (R + × R)
is called an entropy solution of the linear degenerate system (2.1) if (2.9) holds for all entropy-entropy flux pairs (E, F ).

As we will see in Sections 4-6, condition (2.9) is satisfied for L ∞ weak solutions of the linear Lagrangian system and the BI equations. When only a part of the characteristic fields is linearly degenerate, we require the entropy equality (2.9) for all entropy-entropy flux pairs corresponding to the linearly degenerate fields. This is the case of the Keyfitz-Kranzer system (see section 3).

The E-L transformation possesses many remarkable properties. In particular, if we denote by λi (ũ) the i-th eigenvalue of the system (2.5), from the second equation in (2.4), we obtain

dy dt = u 1 dx dt -f 1 (u), which gives, (2.11) λi (u) = u 1 λ i (u) -f 1 (u).
Moreover, let A(u) = f (u). The non-conservative equations

∂ t u + A(u)∂ x u = 0 are equivalent to ∂ s ũ + A(ũ)∂ y ũ = 0 with A(u) = u 1 A(u) -f 1 (u)I n .
Therefore, A(u) and A(u) have the same eigenvectors.

In the following proposition are listed the main equivalence properties between the systems (2.1) and (2.5). The proof can be found in [START_REF] Sévennec | Géométrie des Systèmes Hyperboliques de Lois de Conservation[END_REF][START_REF] Wagner | Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions[END_REF].

Proposition 2.1. The E-L transformation preserves the (strict) hyperbolicity, convexity of entropies with relation (2.6), entropy solutions, and for each i = 1, 2, •••, n, the existence of an i-th classical Riemann invariant and genuine nonlinearity or linear degeneracy of the i-th characteristic field of systems. Finally, the relation of the eigenvalues of the systems (2.1) and (2.5) in the two coordinate systems is given by (2.11).

Remark 2.1. If u 1 is a function of (u 2 , •••, u n ) and the first equation ∂ t u 1 +∂ x f 1 (u) = 0 in (2.1
) is exact and is derived from the sub-system composed of the n -1 equations in (2.1) with variables (u 2 , • • •, u n ), then this sub-system is equivalent to the sub-system without the first equation in (2.5), in the sense of Proposition 2.1.

2.3.

Equivalence of the Cauchy problems. From the definition (2.4) and ũ(s, y) = u(t, x), we have ũ(t, Y (t, x)) = u(t, x). If we set (2.12)

Y (0, x) = Y 0 (x) def = x 0 u 0 1 (ξ)dξ,
then for each given u ∈ L ∞ (R + × R), (2.4) and (2.12) define a unique E-L transformation. Note that the choice of Y 0 by (2.12) is compatible with (2.4). Obviously, for u 0 1 (x) ≥ u 1 > 0, Y 0 is strictly increasing, Lipschitzian and bijective from R to R. So is its inverse function X 0 . Similarly, for all t > 0, x -→ Y (t, x) is bijective from R to R and (t, x) -→ Y (t, x) is Lipschitzian. Concerning the inverse function of Y (t, •), we have the following result. ,x) and x = X(t, y) be the unique solution of

Lemma 2.1. Assume u 0 ∈ L ∞ (R) with u 0 1 (x) ≥ u 1 > 0, a.e. x ∈ R. Let u ∈ L ∞ (R + × R) be an entropy solution of (2.1)-(2.2) with u 1 (t, x) ≥ u 1 > 0, a.e. (t, x) ∈ R + × R. Let ũ(t, y) = u(t
(2.13) dx = 1 ũ1 dy + f 1 (ũ) ũ1 dt, X(0, y) = X 0 (y). Then X(t, •) = Y -1 (t,
= -f 1 (ũ) + ũ1 • f 1 (ũ) ũ1 (t, X(t, y)) = 0.
Since Y (0, X(0, y)) = Y 0 (X 0 (y)) = y, we obtain Y (t, X(t, y)) = y for all t ≥ 0. Similarly, X(t, Y (t, x)) = x. Hence, X(t, •) = Y -1 (t, •) for all t ≥ 0.

Taking t = 0 in ũ(t, y) = u(t, X(t, y)) gives ũ(0, y) = u 0 (X 0 (y)). Thus by Wagner's result [START_REF] Wagner | Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions[END_REF] which uses the E-L transformation (2.4) in weak formulations of the systems, we obtain the following equivalence result for the Cauchy problems.

Theorem 2.1. Let the assumptions of Lemma 2.1 hold and (t, x) -→ (s, y) be the unique E-L transformation defined by (2.4) and (2.12). Then ũ(t, y) = u(t, X(t, y)) is an entropy solution of (2.5) satisfying the initial condition ũ| s=0 = u 0 (X 0 ), where X 0 and X(t, •) are the inverse functions of Y 0 and Y (t, •), respectively.

Furthermore, if the entropy solution of the Cauchy problem for the system (2.5) is unique, then the entropy solution u

∈ L ∞ (R + × R) of (2.1)-(2.2) with u 1 (t, x) ≥ u 1 > 0, a.e. (t, x) ∈ R + × R is unique.
Proof. The proof of the first part of Theorem 2.1 is clear. It suffices to give the proof of its second part.

Let u ∈ L ∞ (R + × R) be an entropy solution of (2.1)-(2.2) with u 1 (t, x) ≥ u 1 > 0, a.e. (t, x) ∈ R + × R and (t, x) -→ (t, Y u (t, x)) be the unique E-L transformation with y = Y u (t, x) being the unique Lipschitzian solution of (2.4) and (2.12). From Lemma 2.1, the inverse function X u (t, •) of Y u (t, •) is given by (2.13). Then the first part of Theorem 2.1 shows that ũ(t, y) = u(t, X u (t, y)) is an entropy solution of (2.5) with initial value ũ0 = u 0 (X 0 ). From the uniqueness assumption of the Cauchy problem to the system (2.5), ũ depends only on u 0 and f but is independent of u for t > 0. It follows from (2.13) that X u is independent of u. So is Y u and we may write Y u = Y . Thus, any entropy solution u of (2.1)-(2.2) with u 1 (t, x) ≥ u 1 > 0 has the same expression u(t, x) = ũ(t, Y (t, x)). This proves the uniqueness of entropy solutions to the Cauchy problem (2.1)-(2.2). By Lemma 2.1, Theorem 2.1 shows the equivalence on the existence and uniqueness of entropy solutions for the Cauchy problems in the two coordinate systems. This result can be immediately applied to the Cauchy problem for the equations of gas dynamics in Eulerian coordinates :

(2.14)              ∂ t ρ + ∂ x (ρu) = 0, ∂ t (ρu) + ∂ x (ρu 2 + p) = 0, ∂ t ρ e + 1 2 u 2 + ∂ x ρ e + 1 2 u 2 + p u = 0, t > 0, x ∈ R, t = 0 : (ρ, u, e) = (ρ 0 , u 0 , e 0 )(x), x ∈ R
and in Lagrangian coordinates :

(2.15)

               ∂ s ṽ -∂ y ũ = 0, ∂ s ũ + ∂ y p(ṽ, ẽ) = 0, ∂ s ẽ + 1 2 ũ2 + ∂ y (p(ṽ, ẽ)ũ) = 0, s > 0, y ∈ R, s = 0 : (ṽ, ũ, ẽ) = (ṽ 0 , ũ0 , ẽ0 )(y), y ∈ R,
through the change of variables (2.4) and (2.12) with

u 1 = ρ, f 1 (ρ, u) = ρu, p(v, e) = p(1/v, e
) and (ṽ, ũ, ẽ)(s, y) = (1/ρ, u, e)(t, x). In the above equations, ρ > 0, u, e > 0 and p = p(ρ, e) are the density, velocity, specific internal energy and pressure, respectively. Thus, (ρ, u, e) ∈ L ∞ (R + × R) with ρ ≥ ρ > 0 is an entropy solution of (2.14) if and only if (ṽ, ũ, ẽ) is an entropy solution of (2.15) for which the initial data are linked by (ṽ 0 , ũ0 , ẽ0 ) = (1/ρ 0 , u 0 , e 0 ) • X 0 or (ρ 0 , u 0 , e 0 ) = (1/ṽ 0 , ũ0 , ẽ0 ) • Y 0 . The entropy solution of (2.14) is unique if and only if that of (2.15) is unique. In particular, the L ∞ entropy solutions of the Cauchy problems for the isentropic gas dynamics equations are equivalent in Eulerian and Lagrangian coordinate systems. Notice that the above equivalence result is also valid for systems of conservation laws with source terms provided that the source term does not appear in the conservation law for u 1 > 0. Hence, Theorem 2.1 can be applied to one-dimensional Navier-Stokes equations and Shallow Water equations. Further applications are given in the next sections.

A Keyfitz-Kranzer model

In mathematical modeling of enhanced oil recovery, the Keyfitz-Kranzer system is often used [START_REF] Keyfitz | A system of strictly hyperbolic conservation laws arising in elasticity theory[END_REF]. A simplified version of this system reads :

(3.1) ∂ t u + ∂ x (uφ(r)) = 0, t > 0, x ∈ R,
subject to an initial condition :

(3.2) t = 0 : u = u 0 (x), x ∈ R, where u = (u 1 , u 2 ) t , r = |u| = u 2 1 + u 2 2
, and φ : R + -→ R is a smooth function. The two eigenvalues of the system are :

(3.3) λ 1 (u) = (rφ(r)) , λ 2 (u) = φ(r).
We check easily that the second characteristic field is linearly degenerate. When (rφ(r)) ≡ 0, i.e., φ(r) = C 1 /r + C 2 for r > 0, with C 1 , C 2 ∈ R, the first characteristic field is also linearly degenerate. In this case, the problem is solved in Section 5. In what follows, suppose that (rφ(r)) = 0 for all r > 0. Then the first characteristic field is genuinely nonlinear. Let u = rθ with θ = (θ 1 , θ 2 ) t ∈ S 1 , the unit circle. For smooth solutions, the system (3.1) can be put in diagonal form :

∂ t r + (rφ(r)) ∂ x r = 0, ∂ t θ + φ(r)∂ x θ = 0, t > 0, x ∈ R.
Hence, it is hyperbolic but not strictly hyperbolic since it may occur φ(r) = (rφ(r)) , i.e., φ (r) = 0 for some r > 0. For the Cauchy problem (3.1)-(3.2) we refer the reader to [START_REF] Chen | Hyperbolic systems of conservation laws with a symmetry[END_REF][START_REF] Chen | The method of quasidecoupling for discontinuous solutions to conservation laws[END_REF][START_REF] Freistühler | Linear degeneracy and shock waves[END_REF][START_REF] Keyfitz | A system of strictly hyperbolic conservation laws arising in elasticity theory[END_REF][START_REF] Liu | On a nonstrictly hyperbolic system of conservation laws[END_REF][START_REF] Serre | Systèmes de Lois de Conservation II[END_REF] and the references therein. When the initial data satisfy u 0 ∈ BV (R), the existence of entropy solutions can be proved by using the Glimm scheme [START_REF] Glimm | Solutions in the large for nonlinear hyperbolic systems of equations[END_REF]. This is partially due to the decoupling of the system which compensates the loss of non strict hyperbolicity. The entropy solutions in L ∞ (R) is investigated in [START_REF] Chen | The method of quasidecoupling for discontinuous solutions to conservation laws[END_REF].

The two families of entropy-entropy flux pairs

(E i (u), F i (u)) 1≤i≤2 are E 1 (u) = g 1 (r), F 1 (u) = g 1 (r)(rφ(r)) dr and E 2 (u) = rg 2 (θ), F 2 (u) = rφ(r)g 2 (θ),
where g 1 and g 2 are smooth functions. By checking the Rankine-Hugoniot conditions of the system, we see that the entropy-entropy flux pairs (E 1 , F 1 ) and (E 2 , F 2 ) correspond to the genuinely nonlinear field λ 1 (u) and the linearly degenerate field λ 2 (u), respectively. Therefore, we require that any entropy solution u of (3.1) satisfies :

(3.4) ∂ t (rg 2 (θ)) + ∂ x (rφ(r)g 2 (θ)) = 0,
in the sense of distributions for any continuous function g 2 . Let g 2 (θ) = 1 and g 2 (θ) = θ i (i = 1, 2). We obtain

(3.5) ∂ t r + ∂ x (rφ(r)) = 0, ∂ t (rθ) + ∂ x (rφ(r)θ) = 0, t > 0, x ∈ R.
It is clear that (3.1) is not a linear Lagrangian system (see definition 4.1). However, it is equivalent to (3.5) for weak solutions satisfying (3.4). Since the system (3.5) is decoupled, we may first solve the scalar conservation law for r then find θ by an E-L transformation. More precisely, assume u 0 ∈ L ∞ (R) and inf x∈R r 0 (x) ≥ r > 0 with r 0 = |u 0 |. Let r be the unique entropy solution of the first equation in (3.5) with initial datum r 0 . It satisfies the entropy inequality

(3.6) ∂ t g 1 (r) + ∂ x g 1 (r)(rφ(r)) dr ≤ 0, ∀ g 1 convex,
in the sense of distributions. From the maximal principle, we know that r

∈ L ∞ (R + × R) and r(t, x) ≥ r, a.e., (t, x) ∈ R + × R. Let (t, x) -→ (s, y) = (t, Y (t, x)) be the unique E-L transformation defined by (3.7) dy = rdx -rφ(r)dt, Y (0, x) = Y 0 (x) def = x 0 r 0 (ξ)dξ.
Then

Y (t, x) = Y 0 (x) - t 0 (rφ(r))(τ, x)dτ.
In Lagrangian coordinates (s, y), the second equation in (3.5) becomes :

∂ s θ(s, y) = 0, s > 0, y ∈ R.
From Theorem 2.1 we obtain a unique solution θ with initial datum θ 0 = u 0 /r 0 , given by

(3.8) θ(t, x) = θ 0 (X 0 (Y (t, x))), with X 0 = Y -1 0 .
When the function ψ(r) = rφ(r) is strictly convex, Y can be expressed explicitly by using the conjugated function due to the Lax formula [START_REF] Lax | Hyperbolic systems of conservation laws II[END_REF] : As we will see in the next sections, the class of linear Lagrangian systems contains many interesting examples such as the equations of pressureless gas dynamics, all 2 × 2 linear degenerate systems of conservation laws and the augmented Born-Infeld equations. By Proposition 2.1, it is easy to see that if a linear Lagrangian system is hyperbolic then it is necessarily linearly degenerate and admits a complete set of classical Riemann invariants. Moreover, since λi is a constant, from (2.11) the multiplicity of each eigenvalue of a linear Lagrangian system is constant. Now let us look at the structure of this system. The conservative variables of the system (2.7). Therefore, the system (2.5) is linear if and only if there are real constants ãij (1

(3.9) Y (t, x) = inf ξ∈R Y 0 (ξ) + tψ * ( x -ξ t ) , ψ * (y) = sup η∈R ηy -ψ(η) . Theorem 3.1. Assume u 0 ∈ L ∞ (R) with inf x∈R r 0 (x) > 0.
(2.5) is v = (v 1 , v 2 , • • •, v n ) t defined in
≤ i, j ≤ n) such that - f 1 (u) u 1 = n j=1 ã1j v j , f i (u) - f 1 (u)u i u 1 = n j=1 ãij v j (2 ≤ i ≤ n), which yield (4.1)
           f 1 (u) = -ã 11 - n j=2 ã1j u j , f i (u) = 1 u 1 ãi1 + u i f 1 (u) + n j=2 ãij u j (2 ≤ i ≤ n).
Hence, a linear Lagrangian system is necessarily of the form :

(4.2)            ∂ t u 1 -∂ x n j=2 ã1j u j = 0, ∂ t u i + ∂ x 1 u 1 ãi1 + u i f 1 (u) + n j=2 ãij u j = 0 (2 ≤ i ≤ n).
It can be written as a linear system in Lagrangian coordinates :

(4.3) ∂ s ṽ + Ã∂ y ṽ = 0,
where à = (ã ij ) 1≤i,j≤n is a constant matrix. Thus a linear Lagrangian system is completely determined by a constant matrix. We remark that f 1 is an affine function of u and the system (4.2) can be rewritten in an equivalent form with the variables (u

1 , v 2 , • • •, v n ) t : (4.4)            ∂ t u 1 -∂ x u 1 n j=2 ã1j v j = 0, ∂ t (u 1 v i ) + ∂ x ãi1 u 1 -ã11 v i + n j=2 (ã ij -ã1j u 1 v i )v j = 0 (2 ≤ i ≤ n).
In what follows we suppose that the linear Lagrangian system (4.2) is hyperbolic. By Proposition 2.1, the linear system (4.3) is also hyperbolic. Then ( l1 , l2 , • • •, ln ) is a basis of R n with li being an i-th left eigenvector of à associated to λi . It follows that w i = li v is an i-th classical Riemann invariant of (4.2) and satisfies (4.5)

∂ s wi + λi ∂ y wi = 0 (1 ≤ i ≤ n).
Therefore, each entropy-entropy flux pair ( Ẽ, F ) of the system (4.3) has the form :

(4.6)        Ẽ(v) = n j=1 g j (w j ) = n j=1 g j ( lj v), F (v) = n j=1 λj g j (w j ) = n j=1 λj g j ( lj v),
where g j (1 ≤ j ≤ n) are arbitrary continuous functions. From the equivalence relation (2.6), we obtain an explicit expression of each entropy-entropy flux pair of the linear Lagrangian system as :

(4.7) (E(u), F (u)) = n j=1 (u 1 , λj + f 1 (u))g j 1 u 1 lj (1, u 2 , • • •, u n ) t .
Moreover, let lij be the i-th component of lj , we have

∂ 2 Ẽ(v) ∂v i ∂v k = n j=1 g j (w j ) lji ljk def = α g ik (w), with w = (w 1 , • • • , w n ) t .
Therefore, by the equivalence of the convexity of entropies given in Proposition 2.1, E(u) defined by (4.7) is convex if and only if the symmetric matrix ∆ g (w) def = (α g ik (w)) 1≤i,k≤n is positively definite for all w ∈ R n . In particular, since ( l1 , l2 , •••, ln ) is linearly independent, the constant matrix (< li , lk >) 1≤i,k≤n is positively definite, where < •, • > denotes the inner product of R n . Thus a strictly convex entropy E(u) is constructed by choosing g j (ξ) = ξ 2 for all j = 1, 2, • • •, n.

We collect the above discussion in the following result.

Proposition 4.1. Suppose that the linear Lagrangian system (4.2) is hyperbolic. Then it is linearly degenerate, admits a complete set of classical Riemann invariants defined by

w i = li (1, u 2 , • • •, u n ) t /u 1 (1 ≤ i ≤ n)
and each of its eigenvalues has constant multiplicity, where ( l1 , l2 , • • •, ln ) is composed of the left eigenvectors of the corresponding linear system in Lagrangian coordinates. Moreover, each entropy-entropy flux pair (E(u), F (u)) of the system has an explicit expression (4.7), E(u) is convex if and only if the matrix ∆ g (w) is positively definite for all w ∈ R n and a strictly convex entropy E(u) is given when

g j (ξ) = ξ 2 for all j = 1, 2, • • •, n.
Finally, we mention that a similar notion was introduced in [START_REF] Després | Lagrangian systems of conservation laws[END_REF] where a system of conservation laws is called a Lagrangian system if in Lagrangian coordinates it admits a strictly convex entropy with zero flux function. The Lagrangian system is an extension of the equations for gas dynamics. Then the nature of the linear Lagrangian system and Lagrangian system is different.

4.2.

Explicit solutions of the Cauchy problem. Consider the Cauchy problem for the linear Lagrangian system (4.2) with the initial condition (2.2). Let

v 0 = (1, u 0 2 , •••, u 0 n ) t /u 0 1 and w = (w 1 , w 2 , • • •, w n ) t . By Theorem 2.1 we have (4.8) s = 0 : wi = w 0 i (X 0 (y)) = li v 0 (X 0 (y)), y ∈ R.
Therefore, the unique solution of the Cauchy problem (4.3) and (4.8) is given by wi (s, y) = w 0 i (X 0 (y -λi s)). It follows from Theorem 2.1 that the Cauchy problem (4.2) and (2.2) admits a unique entropy solution u. For seeking an explicit expression of u, we denote by ri the right eigenvector of à associated to λi normalized as li rj = δ ij , where δ ij stands for the Kronecker's symbol. Then we may write (4.9) ṽ(s, y)

= n i=1 ( li v 0 )(X 0 (y -λi s))r i .
Now let us determine the unique function y = Y (t, x) in the E-L transformation. From (2.4), we have dx = ṽ1 (t, y)dy -n j=1 ã1j ṽj (t, y)dt, which defines a unique function x = X(t, y) together with X(0, y) = X 0 (y). A straightforward calculation using the first equation in (4.3) gives

X(t, y) = X 0 (y) - t 0 n j=1 ã1j ṽj (τ, y)dτ.
Let rji be the j-th component of ri . Since n j=1 ã1j rji = λi r1i , it follows from (4.9) that (4.10) X(t, y) = X 0 (y) -

t 0 n i=1 λi r1i ( li v 0 )(X 0 (y -λi τ ))dτ.
Note that the function in the above integral is given by the initial data. To guarantee that the function y -→ X(t, y) is invertible for all t ≥ 0, we need the conditions v ∈

L ∞ (R + × R) and v 1 (t, x) ≥ v 1 > 0, or equivalently u ∈ L ∞ (R + × R) and u 1 (t, x) ≥ u 1 > 0, a.e. (t, x) ∈ R + × R.
That is why we have made the assumptions

(4.11) u 0 ∈ L ∞ (R), u 0 1 (x) ≥ u 1 > 0, a.e.
x ∈ R and furthermore, (4.12)

n i=1 ( li v 0 )(X 0 (y -λi t))r 1i ≥ v 1 > 0, a.e. (t, y) ∈ R + × R.
It is clear that under the conditions (4.11)-(4.12), the function y -→ X(t, y) is invertible and bi-Lipschitzian from R to R for all t ≥ 0. Therefore, from Lemma 2.1 we obtain y = Y (t, x) as the inverse function of y -→ X(t, y) satisfying Y (0, x) = Y 0 (x). Finally, (4.9) gives

(4.13) v(t, x) = n i=1 ( li v 0 )(X 0 (Y (t, x) -λi t))r i ,
which implies, together with (4.12), that v 1 (t, x) ≥ v 1 > 0, a.e. (t, x) ∈ R + × R. In particular, (4.14) w i (t, x) = w 0 i (X 0 (Y (t, x) -λi t)), which shows that the maximum principle holds for each w i .

On the other hand, the expression wi (s, y) = w0 i (y -λi s) shows that the entropy equality (2.10) is satisfied for all entropy-entropy flux pairs ( Ẽ(v), F (v)) given in (4.6). This equality is equivalent to (2.9). Thus we obtain the following conclusion. (2.7), where X 0 and Y (t, •) are the inverse functions of Y 0 and X(t, •) for all t ≥ 0 defined in (2.12) and (4.10), respectively. Remark 4.1. Formula (4.13) shows that the solution of the Cauchy problem (4.2) and (2.2) has the same regularity as its initial data. In particular, if

u 0 ∈ BV (R) then u(t, •) ∈ BV (R) for all t ≥ 0; and if u 0 ∈ C m (R) with m ∈ N * then X 0 , Y 0 ∈ C m (R) and X, Y ∈ C m (R + × R). Hence, u ∈ C m (R + × R).
Remark 4.2. From (4.10), we have

∂ tt X(t, y) = n i=1 λ2 i r1i li (v 0 • X 0 ) (y -λi t)
and

∂ yy X(t, y) = n i=1 r1i li (v 0 • X 0 ) (y -λi t).
Hence, when | λi | = λ for all i = 1, 2, • • •, n, X satisfies a linear wave equation

∂ tt X -λ2 ∂ yy X = 0.
As we will see in the next section, this situation occurs for all 2 × 2 linearly degenerate systems of conservation laws.

4.3.

Weak stability. Consider a hyperbolic linear Lagrangian system (4.2). Let (u 0 ε ) ε>0 be a bounded sequence in L ∞ (R) satisfying (4.11)-(4.12) uniformly with respect to ε. We denote by (u ε ) ε>0 the corresponding sequence of entropy solutions of the system (4.2) with the initial data (u 0 ε ) ε>0 . From (4.11)-(4.12) and the explicit formulas (4.13)-(4.14), we know that the sequence (u

ε ) ε>0 is bounded in L ∞ (R + × R) and there is a constant u > 0, independent of ε → 0, such that u 1ε (t, x) ≥ u, a.e. (t, x) ∈ R + × R.
Therefore, up to subsequences, still denoted by (u ε ) ε>0 and (u 0 ε ) ε>0 , we have

u ε --u, in L ∞ (R + × R) weakly- * and u 0 ε --u 0 , in L ∞ (R) weakly- * , with u ∈ L ∞ (R + × R) and u 0 ∈ L ∞ (R) satisfying u 0 1 (x) ≥ u 1 > 0 and u 1 (t, x) ≥ u, a.e. (t, x) ∈ R + × R.
The weak stability of the linear Lagrangian system means that the weak limit u is a unique entropy solution of (4.2) with initial datum u 0 . Indeed, since f 1 is an affine function, we obtain

f 1 (u ε ) --f 1 (u), in L ∞ (R + × R) weakly- * .
Then applying the div-curl lemma of compensated compactness (see [START_REF] Tartar | Compensated compactness and applications to partial differential equations[END_REF]) to the first and i-th (2 ≤ i ≤ n) equations in (4.2), we obtain, in the sense of distributions,

lim ε→0 (u 1ε f i (u ε ) -u iε f 1 (u ε )) = u 1 lim ε→0 f i (u ε ) -u i lim ε→0 f 1 (u ε ).
It follows from the definition of f i that

lim ε→0 ãi1 + n j=2 ãij u jε = u 1 lim ε→0 f i (u ε ) -u i f 1 (u).
Hence, for all 2 ≤ i ≤ n,

lim ε→0 f i (u ε ) = 1 u 1 ãi1 + u i f 1 (u) + n j=2 ãij u j = f i (u).
This allows to pass to the limits in the weak formulation of the Cauchy problem for u ε and deduce that u is a weak solution of (4.2) and the initial datum of u is just u 0 . To prove that the weak limit u is a unique entropy solution of (4.2), it remains to show that u fulfills the condition (4.12). To this end, we notice that the sequence (Y 0ε ) ε>0 defined by

Y 0ε (x) = x 0 u 0 1ε (ξ) dξ is bounded in W 1,∞ loc (R).
Then up to a subsequence, still denoted by (Y 0ε ) ε>0 , we have Y 0ε -→ Y 0 in L p loc (R) strongly for any p ≥ 1, where Y 0 is given in (2.12). Let φ ∈ C ∞ 0 (R) and

X 0ε = Y -1 0ε . Then φ(Y 0ε ) -→ φ(Y 0 ) in L p loc (R) strongly. It follows that R (v 0 ε • X 0ε )(y)φ(y)dy = R v 0 ε (x)φ(Y 0ε (x))Y 0ε (x)dx = R v 0 ε (x)u 0 1ε (x)φ(Y 0ε (x))dx = R (1, u 0 2ε , • • •, u 0 nε ) t (x)φ(Y 0ε (x))dx --→ R (1, u 0 2 , • • •, u 0 n ) t (x)φ(Y 0 (x))dx = R (v 0 • X 0 )(y)φ(y)dy, with v 0 = (1, u 0 2 , • • •, u 0 n ) t /u 0 1 .
Here we have used the changes of variables y = Y 0ε (x) and y = Y 0 (x) in the first and last equalities, respectively. This shows that

v 0 ε • X 0ε --v 0 • X 0 , in L ∞ (R) weakly- * . Since u 0
ε satisfies (4.12) uniformly with respect to ε, i.e., n i=1

( li v 0 ε )(X 0ε (y -λi t))r 1i ≥ v 1 > 0, a.e. (t, y) ∈ R + × R,
we deduce that u satisfies (4.12). Thus, u is a unique entropy solution of the Cauchy problem to the linear Lagrangian system (4.2).

Theorem 4.2. The linear Lagrangian system is weakly stable in L ∞ (R + × R).

Similar results were obtained by Serre [START_REF] Serre | Un système hyperbolique non linéaire avec des données oscillantes[END_REF] for 2 × 2 linearly degenerate systems and by Brenier [START_REF] Brenier | Hydrodynamic structure of the augmented Born-Infeld equations[END_REF] for a simplified ABI system. Since these two systems are included in the class of linear Lagrangian systems, Theorem 4.2 can be viewed as an extension of their results. However, the entropy equality and uniqueness of entropy solutions were not discussed in their papers. Finally, we point out that the above weak stability result is obtained in a very simple way with the help of a good choice of conservation laws to apply the div-curl lemma among all entropy-entropy flux pairs given in (4.7). 4.4. An example of non-hyperbolic linear Lagrangian systems. Consider the Cauchy problem for the system of pressureless gas dynamics :

(4.15) ∂ t ρ + ∂ x (ρu) = 0, ∂ t (ρu) + ∂ x (ρu 2 ) = 0, t > 0, x ∈ R,
with initial conditions :

(4.16) t = 0 : ρ = ρ 0 (x), u = u 0 (x), x ∈ R.
Let ρ > 0 and (t, x) -→ (s, y) = (t, Y (t, x)) be the E-L transformation defined by (4.17)

dy = ρdx -ρudt, Y (0, x) = Y 0 (x) def = x 0 ρ 0 (ξ)dξ.
Then in Lagrangian coordinates (s, y), the system (4.15) becomes :

(4.18) ∂ s ṽ -∂ y ũ = 0, ∂ s ũ = 0, s > 0, y ∈ R.
It is clear that the equations of pressureless gas dynamics are linear Lagrangian system but not hyperbolic. The initial conditions (4.16) correspond to :

(4.19) s = 0 : ṽ = ṽ0 (y), ũ = ũ0 (y), y ∈ R, where (ṽ

0 , ũ0 ) = 1 ρ 0 , u 0 (X 0 ), X 0 = Y -1 0 .
We deduce from (4.18)-(4.19) that ṽ(t, y) = ṽ0 (y) + t(ũ 0 ) (y), ũ(t, y) = ũ0 (y).

To guarantee the change of variables (4.17), we assume that (4.20)

ρ 0 ∈ L ∞ (R), u 0 ∈ W 1,∞ (R) with ρ 0 (x) ≥ ρ > 0, a.e. x ∈ R.
Then ṽ(t, y) > 0 for small t > 0. It follows that x = X(t, y) satisfies dx = (ṽ 0 (y) + t(ũ 0 ) (y))dy + ũ0 (y)dt.

The unique function x = X(t, y) satisfying X(0, y) = X 0 (y) = Y -1 0 (y) is X(t, y) = X 0 (y) + tũ 0 (y). Since X 0 (y) = ṽ0 (y), the function y -→ X(t, y) is bijective and bi-Lipschitzian for small t ≥ 0. From Lemma 2.1, we have Y (t, •) = X -1 (t, •) for small t ≥ 0. Hence,

(4.21) ρ(t, x) = ρ 0 (X 0 (Y (t, x))) 1 + t(ρ 0 (u 0 ) )(X 0 (Y (t, x)))
, u(t, x) = u 0 (X 0 (Y (t, x))).

Proposition 4.2. Under the assumption (4.20), there exists t 1 > 0 such that the Cauchy problem (4.15)-(4.16) admits a unique solution (ρ, u) ∈ L ∞ ((0, t 1 )×R)∩W 1,∞ ((0, t 1 )×R), defined by (4.21). This solution is global if u 0 is a non-decreasing function.

Beyond the interval [0, t 1 ), ρ becomes a measure. For the study of measure-valued solutions of (4.15), we refer to [START_REF] Bouchut | On zero pressure gas dynamics[END_REF][START_REF] Bouchut | Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness[END_REF][START_REF] Brenier | Sticky particles and scalar conservation laws[END_REF][START_REF] Rykov | Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics[END_REF] and the references therein. See also [START_REF] Ya | Gravitational instability : an approximate theory for large density perturbations[END_REF] from a physical point of view.

The 2 × 2 linearly degenerate system

Consider the Cauchy problem for the 2 × 2 linearly degenerate system. Since the characteristic fields are both linearly degenerate, the eigenvalues λ 1 and λ 2 are two classical Riemann invariants of the system associated to the eigenvalues λ 2 and λ 1 , respectively. Set z = λ 1 and w = λ 2 , then for smooth solutions all 2 × 2 linearly degenerate systems can be written in a canonical form :

(5.1)

∂ t w + z ∂ x w = 0, ∂ t z + w ∂ x z = 0, t > 0, x ∈ R.
It is supplemented by initial conditions :

(5.2) t = 0 : w = w 0 (x), z = z 0 (x), x ∈ R.

It is easy to check that each entropy-entropy flux pair (E, F ) of the system (5.1) is of the form (see [START_REF] Serre | Un système hyperbolique non linéaire avec des données oscillantes[END_REF]) :

(5.

3)

E(w, z) = g 1 (w) + g 2 (z) w -z , F (w, z) = zg 1 (w) + wg 2 (z) w -z ,
where g 1 and g 2 are two arbitrary continuous functions. For w > z and any a > 0, let

(5.4) ρ = 2a w -z , u = w + z 2 .
Then we obtain the equations for Chaplygin gas dynamics [START_REF] Whitham | Linear and Nonlinear Waves[END_REF] :

(5.5)

∂ t ρ + ∂ x (ρu) = 0, ∂ t (ρu) + ∂ x (ρu 2 + p(ρ)) = 0,
in which the pressure is given by the state law of Von Kármán-Tsien : p(ρ) = p -a 2 ρ -1 , where p > 0 is a constant such that p(ρ) > 0 (see [START_REF] Courant | Supersonic Flow and Shock Waves[END_REF], p.10 and [START_REF] Boillat | Non linear hyperbolic fields and waves[END_REF], p.7).

Comparing the systems (4.4) and (5.5), we see that (5.5) is a linear Lagrangian system with n = 2, u 1 = ρ and u 2 = ρu. According to Definition 2.1, (w, z) ∈ L ∞ (R + × R) with w > z is an entropy solution of the Cauchy problem (5.1)-(5.2) if all entropy-entropy flux pairs (E(w, z), F (w, z)) defined by (5.3) satisfy (5.6) ∂ t E(w, z) + ∂ x F (w, z) = 0, in the sense of distributions. This makes sense of the system (5.1) for weak solutions.

The system (5.1) was studied by Serre for oscillatory solutions (see [START_REF] Serre | Un système hyperbolique non linéaire avec des données oscillantes[END_REF] or [START_REF] Serre | Systèmes de Lois de Conservation II[END_REF], p.110-113), and by E and Kohn for measure-valued solutions [START_REF] Kohn | The initial-value problem for measure-valued solutions of a canonical 2 × 2 system with linearly degenerate fields[END_REF]. When the initial data satisfy (5.7)

w 0 , z 0 ∈ L ∞ (R), inf x∈R w 0 (x) > sup x∈R z 0 (x),
the existence and uniqueness of entropy solutions were established by Chen by using the compensated compactness and quasidecoupling method [START_REF] Chen | The method of quasidecoupling for discontinuous solutions to conservation laws[END_REF]. The derivation of explicit solutions of the Cauchy problem (5.1)-(5.2) was investigated by Serre [START_REF] Serre | Intégrabilité d'une classe de système de lois de conservation[END_REF] and by Brenier [START_REF] Brenier | Hydrodynamic structure of the augmented Born-Infeld equations[END_REF][START_REF] Brenier | A note on deformations of 2D fluid motions using 3D Born-Infeld equations[END_REF] in the case of periodic smooth solutions. Under the conditions in (5.7), the explicit entropy solution in L ∞ of the Cauchy problem was constructed by the author [START_REF] Peng | Explicit solutions for 2 × 2 linearly degenerate systems[END_REF] by means of the method presented in Sections 2-4. The resulting formula is the same as that in [START_REF] Brenier | Hydrodynamic structure of the augmented Born-Infeld equations[END_REF][START_REF] Brenier | A note on deformations of 2D fluid motions using 3D Born-Infeld equations[END_REF] but is different from that of [START_REF] Serre | Intégrabilité d'une classe de système de lois de conservation[END_REF]. Indeed, it is a particular case of the linear Lagrangian system for which the second condition in (5.7) corresponds to (4.12). This condition implies that w(t, x) > z(t, x) and then the system (5.1) is strictly hyperbolic for almost all (t, x) ∈ R + × R.

The E-L transformation can be defined explicitly by using the first equation in (5.5) and (5.4) with a = 1. It is given by (s, y) = (t, Y (t, x)) with (5.8)

dy = 2 w -z dx - w + z w -z dt, Y (0, x) = Y 0 (x) def = x 0 2d ξ w 0 (ξ) -z 0 (ξ)
.

The corresponding linear system in Lagrangian coordinates (s, y) is (5.9)

∂ s w -∂ y w = 0, ∂ s z + ∂ y z = 0.
Let X 0 = Y -1 0 . Then w(s, y) = w 0 (X 0 (y + s)) and z(s, y) = z 0 (X 0 (y -s)).

Hence, a unique function x = X(t, y) satisfying X(0, y) = X 0 (y) is given by (5.10)

X(t, y) = 1 2 y+t 0 w 0 (X 0 (ξ))d ξ - 1 2 y-t 0 z 0 (X 0 (ξ))d ξ.
Obviously, X(t, •) is bijective and Lipschitzian from R to R for all t ≥ 0. From Lemma 2.1, we have Y (t, •) = X -1 (t, •). Finally, since the eigenvalues of the linear system (5.9) are λ1 = -1 and λ2 = 1. From Remark 4.2, X satisfies the linear wave equation :

∂ tt X -∂ yy X = 0.
As an application of Theorem 4.1, we obtain the following result already stated in [START_REF] Peng | Explicit solutions for 2 × 2 linearly degenerate systems[END_REF].

Theorem 5.1. Let the conditions in (5.7) hold. Then the Cauchy problem (5.1)-( 5.2) has a unique solution (w, z) ∈ L ∞ (R + × R) satisfying the entropy equality (5.6) for all entropy-entropy flux pairs (E, F ) defined by (5.3). This solution is given by

(5.11) w(t, x) = w 0 (X 0 (Y (t, x) + t)), z(t, x) = z 0 (X 0 (Y (t, x) -t)), t > 0, x ∈ R.
Remark 5.1. In view of the expressions (5.8) and (5.11), the second condition in (5.7) can be replaced by the following weak condition :

inf t>0, y∈IR
w 0 (X 0 (y + t)) -z 0 (X 0 ((y -t)) > 0.

By the expression (5.3) on the entropy-entropy flux pairs, it is easy to see that this weak condition is necessary and sufficient to define any weak solution.

Remark 5.2. In the literature there exists an another way to integrate the system (5.1). Consider the second order nonlinear partial differential equation

(5.12) (∂ x Φ) 2 + 1 ∂ tt Φ -2∂ t Φ ∂ x Φ ∂ tx Φ + (∂ t Φ) 2 -1 ∂ xx Φ = 0, t > 0, x ∈ R with initial conditions (5.13) t = 0 : Φ = Φ 0 (x), ∂ t Φ = Φ 1 (x), x ∈ R.
Under the hyperbolic condition 1 + (∂ x Φ) 2 -(∂ t Φ) 2 > 0, Lochak obtained the following explicit formulas for smooth solutions of (5.12)-(5.13), see [START_REF] Lochak | Sur la théorie non linéaire des ondes[END_REF][START_REF] Lochak | Un modèle d'équation non linéaire conservant certaines ondes planes[END_REF],

(5.14)

t = 1 2 α β 1 + (Φ 0 (λ)) 2 1 + (Φ 0 (λ)) 2 -Φ 2 1 (λ)
dλ,

(5.15)

x = α + β 2 + 1 2 α β Φ 0 (λ)Φ 1 (λ) 1 + (Φ 0 (λ)) 2 -Φ 2 1 (λ) dλ, (5.16) Φ = Φ 0 (α) + Φ 1 (β) 2 - 1 2 α β Φ 1 (λ) 1 + (Φ 0 (λ)) 2 -Φ 2 1 (λ)
dλ.

If we introduce u = ∂ t Φ and v = ∂ x Φ, then (5.12) can be rewritten as a non conservative system :

(v 2 + 1)∂ t u -2uv∂ x u + (u 2 -1)∂ x v = 0, ∂ t v -∂ x u = 0.
This system is equivalent to (5.1) for smooth solutions in which w and z are defined by

w = 1 v 2 + 1 √ 1 + v 2 -u 2 -uv , z = - 1 v 2 + 1 √ 1 + v 2 -u 2 + uv .
It is clear that Lochak formulas (5.14)-(5.16) are different from (5.11).

6. The Born-Infeld equations 6.1. Preliminary. The Born-Infeld system is one of the nonlinear models of Maxwell equations and related to the Born-Infeld Lagrangian (See Born-Infeld [START_REF] Born | Foundation of the new field theory[END_REF], Boillat [START_REF] Boillat | Non linear hyperbolic fields and waves[END_REF], Boillat-Ruggeri [START_REF] Boillat | Energy momentum, wave velocities and characteristic shocks in Euler's variational equations with application to the Born-Infeld theory[END_REF], Brenier [START_REF] Brenier | Hydrodynamic structure of the augmented Born-Infeld equations[END_REF], Gibbons [START_REF] Gibbons | Aspects of Born-Infeld theory and string / M-theory[END_REF][START_REF] Gibbons | Born-Infeld theory and stringy causality[END_REF] and Serre [START_REF] Serre | Hyperbolicity of the nonlinear models of Maxwell's equations[END_REF])

(6.1) L(E, B) = -1 + |B| 2 -|E| 2 -< E, B > 2 ,
where E and B are the electric and magnetic fields in R 3 , < •, • > stands for the inner product and |B| =< B, B >

P = D × B, h = 1 + |B| 2 + |D| 2 + |D × B| 2 ,
then the BI equations read :

(6.3)      ∂ t D + ∇ × -B + D × P h = 0, ∂ t B + ∇ × D + B × P h = 0, Y.J. Peng with differential constraints (6.4) ∇ • D = ∇ • B = 0.
Remark that in the Born-Infeld model, the electric field E ∈ R 3 is given by

E = ∂h(D, B) ∂D = 1 h (D + B × P ).
It is well known that the Born-Infeld energy h and Poynting vector P satisfy the additional conservation laws (6.5)

∂ t h + ∇ • P = 0, (6.6)
∂ t P + ∇ • P ⊗ P -B ⊗ B -D ⊗ D h -∇ 1 h = 0,
which mean that h and P are just mathematical entropies of the system (6.3). In [START_REF] Brenier | Hydrodynamic structure of the augmented Born-Infeld equations[END_REF] Brenier lifted the BI system of 6 equations (6.2)-(6.4) to the ABI system of 10 equations (6.3)-(6.6) by considering h, D, B and P as independent variables. He revealed the hydrodynamic structure of the ABI system similar to that of the Magnetohydrodynamics equations. An another enlargement of the BI system of 6 equations to 9 equations was described by Serre [START_REF] Serre | Hyperbolicity of the nonlinear models of Maxwell's equations[END_REF]. Note that the idea of enlargement of systems was developed by Qin [START_REF] Qin | Symmetrizing nonlinear elastodynamic system[END_REF], by Dafermos [START_REF] Dafermos | Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der Mathematischen Wissenschaften[END_REF] and by Demoulini-Stuart-Tzavaras [START_REF] Demoulini | A variational approximation scheme for threedimensional elastodynamics with polyconvex energy[END_REF] for nonlinear elasticity.

For smooth solutions to the BI and ABI systems, the reader is referred to Brenier [START_REF] Brenier | Hydrodynamic structure of the augmented Born-Infeld equations[END_REF] and Serre [START_REF] Serre | Hyperbolicity of the nonlinear models of Maxwell's equations[END_REF] for local existence of smooth solutions by an argument for first order symmetrizable hyperbolic systems, to Chae-Huh [START_REF] Chae | Global existence for small initial data in the Born-Infeld equations[END_REF] for global existence with small initial data, and also to [START_REF] Brenier | Order preserving vibrating strings and applications to Electrodynamics and Magnetohydrodynamics[END_REF] for the solutions beyond singularities. In this section, we consider the entropy solutions for both BI and ABI systems in one-dimensional case. In a simple way we show that both systems are hyperbolic and linearly degenerate. Therefore, L ∞ entropy solutions of the BI system should satisfy the conservation laws (6.5)-(6.6) according to Definition 2.1. Moreover, the ABI system is a linear Lagrangian one thus allowing to apply Theorem 4.1. Although the BI system is not a linear Lagrangian one, the explicit entropy solutions can still be constructed via the solutions of the ABI system.

As in [START_REF] Brenier | Hydrodynamic structure of the augmented Born-Infeld equations[END_REF] we denote by x = x 1 and consider the problems independent of x 2 and x 3 . Then B 1 and D 1 are constants from (6.3)-(6.4). It follows that the BI system (6.3)-(6.4) and the ABI system (6.3)-(6.6) are reduced to (6.7)

                           ∂ t D 2 + ∂ x B 3 + D 2 P 1 -D 1 P 2 h = 0, ∂ t D 3 + ∂ x -B 2 + D 3 P 1 -D 1 P 3 h = 0, ∂ t B 2 + ∂ x -D 3 + B 2 P 1 -B 1 P 2 h = 0, ∂ t B 3 + ∂ x D 2 + B 3 P 1 -B 1 P 3 h = 0, P = D × B, h = 1 + |B| 2 + |D| 2 + |D × B| 2 , (6.8)
                                                     ∂ t h + ∂ x P 1 = 0, ∂ t P 1 + ∂ x P 2 1 -(1 + D 2 1 + B 2 1 ) h = 0, ∂ t D 2 + ∂ x B 3 + D 2 P 1 -D 1 P 2 h = 0, ∂ t D 3 + ∂ x -B 2 + D 3 P 1 -D 1 P 3 h = 0, ∂ t B 2 + ∂ x -D 3 + B 2 P 1 -B 1 P 2 h = 0, ∂ t B 3 + ∂ x D 2 + B 3 P 1 -B 1 P 3 h = 0, ∂ t P 2 + ∂ x P 1 P 2 -D 1 D 2 -B 1 B 2 h = 0,
∂ t P 3 + ∂ x P 1 P 3 -D 1 D 3 -B 1 B 3 h = 0,
respectively. Here we have used the notations

B = (B 1 , B 2 , B 3 ) t , D = (D 1 , D 2 , D 3 ) t , P = (P 1 , P 2 , P 3 ) t .
6.2. Entropy solutions of the ABI system. Obviously, the ABI system is decoupled since the first two equations in (6.8) are nothing but the equations for Chaplygin gas (5.5). Therefore, we may first solve these equations to get (h, P 1 ), and then determine the remainder unknowns by a linear system of six equations. This is possible for smooth solutions. For weak solutions, we will not do so since the linear system for unknown (D 2 , D 3 , B 2 , B 3 , P 2 , P 3 ) contains discontinuous coefficients which makes the problem difficult to be treated. Now let us rewrite the system (6.8) by introducing (6.9)

P 1 = hv 2 , D 2 = hv 3 , D 3 = hv 4 , B 2 = hv 5 , B 3 = hv 6 , P 2 = hv 7 , P 3 = hv 8 .
Then the ABI system (6.8) becomes :

(6.10)

                                       ∂ t h + ∂ x (hv 2 ) = 0, ∂ t (hv 2 ) + ∂ x hv 2 2 -a 2 h -1 = 0, ∂ t (hv 3 ) + ∂ x (hv 2 v 3 -D 1 v 7 + v 6 ) = 0, ∂ t (hv 4 ) + ∂ x (hv 2 v 4 -D 1 v 8 -v 5 ) = 0, ∂ t (hv 5 ) + ∂ x (hv 2 v 5 -B 1 v 7 -v 4 ) = 0, ∂ t (hv 6 ) + ∂ x (hv 2 v 6 -B 1 v 8 + v 3 ) = 0, ∂ t (hv 7 ) + ∂ x (hv 2 v 7 -D 1 v 3 -B 1 v 5 ) = 0, ∂ t (hv 8 ) + ∂ x (hv 2 v 8 -D 1 v 4 -B 1 v 6 ) = 0,
where a = 1 + B 2 1 + D 2 1 > 0 is a constant. We associate to the system (6.10) the following initial conditions :

(6.11) t = 0 : h = h 0 (x), v i = v 0 i (x) (2 ≤ i ≤ 8), x ∈ R.
In view of (4.4), we see that (6.10) is a linear Lagrangian system with n = 8 and u 1 = h. Moreover, let (t, x) -→ (s, y) = (t, Y (t, x)) be the change of variables given by (6.12)

dy = hdx -hv 2 dt, Y (0, x) = Y 0 (x) def = x 0 h 0 (ξ)dξ.
Then in Lagrangian coordinates (s, y) the system (6.10) is written as (6.13) ∂ s ṽ + Ã∂ y ṽ = 0,

where v 1 = 1/h, v = (v 1 , v 2 , • • •, v 8 ) t and (6.14) Ã =               0 -1 0 0 0 0 0 0 -a 2 0 0 0 0 0 0 0 0 0 0 0 0 1 -D 1 0 0 0 0 0 -1 0 0 -D 1 0 0 0 -1 0 0 -B 1 0 0 0 1 0 0 0 0 -B 1 0 0 -D 1 0 -B 1 0 0 0 0 0 0 -D 1 0 -B 1 0 0              
.

Notice that à is composed of two block matrix à = diag( Ã1 , Ã2 ), with Ã1 ∈ M 2 (R) being obviously diagonalizable and Ã2 ∈ M 6 (R) being symmetric. Hence, à is diagonalizable and the linear system (6.13) is hyperbolic. Furthermore, a straightforward computation gives the eigenvalues of à : (6.15) λ1 = λ2 = λ3 = -a, λ4 = λ5 = 0, λ6 = λ7 = λ8 = a.

Thus applying Proposition 4.1, we obtain Proposition 6.1. For h > 0 and any given B 1 , D 1 ∈ R, the ABI system (6.10) is a linear Lagrangian one and hyperbolic. Let

u = h(1, v 2 , • • •, v 8 ) t .
The eigenvalues λ i = λ i (u) (1 ≤ i ≤ 8) of the system (6.10) are :

(6.16) λ 1 = λ 2 = λ 3 = u 2 -a h , λ 4 = λ 5 = v 2 , λ 6 = λ 7 = λ 8 = u 2 + a h .
It admits a complete set of classical Riemann invariants w j = lj v (1 ≤ j ≤ 8), where lj is the left eigenvector of à associated to the eigenvalue λj . Moreover, each entropy-entropy flux pair (E, F ) of the system is expressed as

(6.17) (E(u), F (u)) = h 8 j=1 (1, λ j (u))g j 1 h lj (1, u 2 , • • •, u 8 ) t ,
with arbitrary continuous functions g j (1 ≤ j ≤ 8), and a strictly convex entropy E(u) is given when g j (ξ) = ξ 2 for all j = 1, 2, • • •, 8.

Each eigenvalue of the system (6.10) being multiple, then its linear degeneracy can alos be obtained from the result in [START_REF] Boillat | Chocs caractéristiques[END_REF]. To solve the Cauchy problem (6.10)-(6.11), we translate the conditions in (5.7) into (6.18)

h 0 , v 0 i ∈ L ∞ (R) (2 ≤ i ≤ 8), inf x∈R v 0 2 (x) + a h 0 (x) > sup x∈R v 0 2 (x) - a h 0 (x) ,
which imply that h(t, x) > 0 for almost all (t, x) ∈ R + × R. As above, for t ≥ 0, define (6. [START_REF] Després | Lagrangian systems of conservation laws[END_REF])

X 0 = Y -1 0 , Y (t, •) = X -1 (t, •), with (6.20) X(t, y) = 1 2 y+t 0 v 0 2 + a h 0 (X 0 (ξ))d ξ - 1 2 y-t 0 v 0 2 - a h 0 (X 0 (ξ))d ξ.
Then applying Theorems 4.1-4.2, we obtain the following result. Theorem 6.1. Under the condition (6.18), the Cauchy problem (6.10)-(6.11) admits a unique solution h, v ∈ L ∞ (R + × R) satisfying the entropy equality ∂ t E(u) + ∂ x F (u) = 0 in the sense of distributions for all entropy-entropy flux pairs defined by (6.17). This solution is weakly stable in L ∞ (R + × R) and is given by

(6.21) v j (t, x) = 8 i=1 ( li v 0 )(X 0 (Y (t, x) -λi t))r ji (1 ≤ j ≤ 8),
where ri is the right eigenvector of à associated to λi such that li rj = δ ij (1 ≤ i, j ≤ 8) and rji is the j-th component of ri .

6.3. Entropy solutions of the BI system. It is clear that (E(u), F (u)) defined in (6.17) is an entropy-entropy flux pair of the BI system (6.7) when h and v lie in the 4-dimensional sub-manifold (6.2) of R 8 (BI manifold following Brenier [START_REF] Brenier | Hydrodynamic structure of the augmented Born-Infeld equations[END_REF]). By a further computation, we obtain the following result.

Proposition 6.2. For h > 0 and any given B 1 , D 1 ∈ R, the BI system (6.7) is hyperbolic and linearly degenerate with the eigenvalues

(6.22) µ 1 (U ) = µ 2 (U ) = P 1 -a h , µ 3 (U ) = µ 4 (U ) = P 1 + a h ,
where U = (B 2 , B 3 , D 2 , D 3 ) t , P 1 and h are given by (6.2). It admits a complete set of classical Riemann invariants W i = L i U/h (1 ≤ i ≤ 4) where L i (1 ≤ i ≤ 4) are linearly independent constant vectors given by (6.23)

L 1 = (a, β 1 , 0, -β 3 ), L 2 = (-β 1 , a, β 3 , 0), L 3 = (0, -β 2 , a, β 1 ), L 4 = (β 2 , 0, -β 1 , a), with β 1 = B 1 D 1 , β 2 = 1 + B 2 1 and β 3 = 1 + D 2 1 .
Moreover, each entropy-entropy flux pair has an explicit expression :

(6.24) (E(U ), F (U )) = h 4 i=1 (1, µ i (U ))g i 1 h L i U ,
where h is defined in (6.2) and g i (1 ≤ i ≤ 4) are arbitrary continuous functions.

Proof. First, replacing P 2 and P 3 from the expression P = D × B and using the notation (6.9), we obtain (6.25)

v 7 = B 1 v 4 -D 1 v 6 , v 8 = D 1 v 5 -B 1 v 3 .
Then in Lagrangian coordinates (s, y) with (6.26) s = t, dy = hdx -P 1 dt, h > 0, the BI system (6.7) becomes (6.27)

             ∂ s ṽ3 -∂ y B 1 D 1 ṽ4 -(1 + D 2 1 )ṽ 6 = 0, ∂ s ṽ4 + ∂ y B 1 D 1 ṽ3 -(1 + D 2 1 )ṽ 5 = 0, ∂ s ṽ5 -∂ y (1 + B 2 1 )ṽ 4 -B 1 D 1 ṽ6 = 0, ∂ s ṽ6 + ∂ y (1 + B 2
1 )ṽ 3 -B 1 D 1 ṽ5 = 0. From Remark 2.1, the BI system (6.7) is equivalent to the linear system (6.27) via the E-L transformation. Now the eigenvalues of ( 6

i = L i U/h = L i (v 3 , v 4 , v 5 , v 6 ) t (1 ≤ i ≤ 4).
Similarly to the proof of (6.17), we may check that each entropy-entropy flux pair of the BI system has the expression (6.24).

Obviously, all entropy solutions (B, D) of the BI system (6.7) satisfy the ABI system (6.8). Conversely, if (h, B, D, P ) is a smooth solution of (6.8), with an initial datum in the BI manifold (6.2), then (B, D) is also a smooth solution of (6.7) (see [START_REF] Boillat | Euler's variational equations with independent momentum[END_REF][START_REF] Serre | Hyperbolicity of the nonlinear models of Maxwell's equations[END_REF]). This means that the BI-manifold is invariant for smooth solutions. Here we show that it is still invariant for the entropy solution. It implies that if the initial data lie in the BI manifold (6.2), then (B, D) is an entropy solution of (6.7) if and only if (h, B, D, P ) is an entropy solution of (6.8). Note finally that the BI system is not strictly hyperbolic. For strictly hyperbolic systems having all linearly degenerate characteristics, we refer to [START_REF] Chen | The method of quasidecoupling for discontinuous solutions to conservation laws[END_REF] on the existence of entropy solutions. Theorem 6.2. Let the condition (6.18) be fulfilled and (h, B, D, P ) ∈ L ∞ (R + × R) be an entropy solution of the ABI system (6.8) with initial data (h 0 , B 0 , D 0 , P 0 ). If (h 0 , B 0 , D 0 , P 0 ) lies in the BI manifold (6.2), then (B, D) is an entropy solution of the BI system (6.7).

We first show the following lemmas. Lemma 6.1. For the ABI system (6.8), under the assumptions of Theorem 6.2, we have (6.29)

P 2 = B 1 D 3 -D 1 B 3 , P 3 = D 1 B 2 -B 1 D 2 . Proof. Let e 2 = P 2 -(B 1 D 3 -D 1 B 3 ) and e 3 = P 3 -(D 1 B 2 -B 1 D 2 )
. We want to show e 2 = 0 and e 3 = 0. By the definition (6.9), we have Therefore, using the system (6.10) we get

∂ t e 2 + ∂ x (v 2 e 2 ) = 0 and ∂ t e 3 + ∂ x (v 2 e 3 ) = 0.
In the coordinates (s, y) they are equivalent to

∂ s ẽ2 h = 0 and ∂ s ẽ3 h = 0.
Since the initial data (h 0 , B 0 , D 0 , P 0 ) lie in the BI manifold (6.2), we deduce that e 2 (0, x) = e 3 (0, x) = 0. Hence, ẽ2 = ẽ3 = 0 and then e 2 = e 3 = 0. Lemma 6.2. For the ABI system (6.8), let us introduce

(6.30) E 1 (u) = 1 + |B| 2 + |D| 2 + |P | 2 h , F 1 (u) = hP 1 E 1 (u) -2a 2 P 1 + 2a 2 hE 2 (u) h 2 , (6.31) E 2 (u) = B 3 D 2 -B 2 D 3 h , F 2 (u) = P 1 E 2 (u) h - P 2 1 + a 2 -hE 1 (u) 2h 2 .
If (6.29) holds, then (E 1 , F 1 ) and (E 2 , F 2 ) are two entropy-entropy flux pairs of the ABI system (6.8).

As the proof of Lemma 6.2 involves only elementary calculations, it is postponed to Appendix.

Proof of Theorem 6.2. Let us denote by

P = D × B, h = 1 + |B| 2 + |D| 2 + |P | 2 .
Lemma 6.1 shows that P 2 = P 2 and P 3 = P 3 . Now we prove that h = h and P 1 = P 1 , i.e., E 1 (u) = h and E 2 (u) = h -1 P 1 . On one hand, since the solution of the ABI system satisfies the entropy equality for all entropy-entropy flux pairs, Lemma 6.2 together with the notation h -1 P 1 = v 2 gives (6.32)

       ∂ t E 1 + ∂ x v 2 E 1 + 2a 2 (E 2 -v 2 ) h = 0, ∂ t E 2 + ∂ x v 2 E 2 - 1 2 v 2 2 + a 2 h 2 - E 1 h = 0.
On the other hand, from the first two equations in (6.10), we have (6.33)

     ∂ t h + ∂ x (hv 2 ) = 0, ∂ t v 2 + 1 2 ∂ x v 2 2 - a 2 h 2 = 0. Let W = E 1 -h and Z = 2(E 2 -v 2 )
. Subtracting (6.32) and (6.33) and using h

-1 E 1 = h -1 W + 1, we obtain      ∂ t W + ∂ x v 2 W + a 2 Z h = 0, ∂ t Z + ∂ x v 2 Z + W h = 0.
In Lagrangian coordinates (s, y), this system is equivalent to

       ∂ s W h + a 2 ∂ y Z h = 0, ∂ s Z h + ∂ y W h = 0.
Since (h 0 , B 0 , D 0 , P 0 ) lies in the BI manifold (6.2) which implies that W = Z = 0 at s = 0, we deduce from the uniqueness of solutions to the linear system that W = Z = 0.

Hence, E 1 (u) = h and E 2 (u) = v 2 . This shows that h = h and P 1 = P 1 . Then (h, B, D, P ) lies in the BI manifold (6.2) and (B, D) is a weak solution of (6.7). Besides, since (h, P 1 ) satisfies the conservation law ∂ t h + ∂ x P 1 = 0, we may use the Lagrangian coordinates (s, y) defined in (6.26). In these coordinates the BI system is equivalent to the linear system (6.27) which yields the entropy equality for all entropy-entropy flux pairs ( Ẽ, F ). Therefore, Theorem 2.1 implies that (B, D) satisfies the entropy equality for all entropyentropy flux pairs given in (6.24). Thus, (B, D) is an entropy solution of the BI system (6.7).

Since the entropy solution (B, D) of the BI system (6.7) satisfies the ABI system (6.8), from the uniqueness of the entropy solution of the Cauchy problem for the ABI system, we obtain a result of the uniqueness of the entropy solution to the BI system. Moreover, by solving the equations of Chaplygin gas dynamics and the linear system (6.27), we obtain as above an explicit expression of the entropy solution. Theorem 6.3. Let U 0 = (B 0 2 , B 0 3 , D 0 2 , D 0 3 ) t ∈ L ∞ (R) and (h 0 , P 0 ) be defined by the BI manifold (6.2) satisfying the second condition in (6.18). Then there exists a unique entropy solution U = (B 2 , B 3 , D 2 , D 3 ) t ∈ L ∞ (R + × R) to the system (6.7) with the initial data U 0 . The solution is weakly stable in L ∞ (R + × R) and is given by the expressions :

(6.34) U (t, x) h(t, x) = 4 i=1
L i U 0 h 0 (X 0 (Y (t, x) -μi t))R i , (6.35) h(t, x) = 2a w 0 (X 0 (Y (t, x) + at)) -z 0 (X 0 (Y (t, x) -at)) ,

where X 0 , Y , L i and μi are defined by (6. [START_REF] Després | Lagrangian systems of conservation laws[END_REF]), (6.20), (6.23) and (6.28), respectively,

z 0 = v 0 2 - a h 0 , w 0 = v 0 2 + a h 0 , v 0 2 =
P 0 1 h 0 , and R i is defined by L i R j = δ ij for all i, j = 1, 2, 3, 4. Remark 6.1. In one space dimension there is an another way to obtain the existence and uniqueness of entropy solutions of the BI system. Indeed, it suffices to enlarge the BI system by adding the equations for h and P 1 instead of h and P . Then we obtain a linear Lagrangian system of 6 equations of independent variables h, P 1 , (B 2 , B 3 ) and (D 2 , D 3 ). We may use this system (which is smaller than the ABI system) to replace the ABI system. Here the essential thing is that the relations in (6.25) are linear. Appendix : Proof of Lemma 6.2

Step 1 : Equation for (E 1 (u), F 1 (u)).

Let v 2 = h -1 P 1 . Following Brenier [START_REF] Brenier | Hydrodynamic structure of the augmented Born-Infeld equations[END_REF], E 1 satisfies

∂ t E 1 + ∂ x (v 2 E 1 ) = 2∂ x P 1 -hE 2 + B 1 < B, P > +D 1 < D, P > h 2 .
Using Lemma 6.1, we get

∂ t E 1 + ∂ x (v 2 E 1 ) = 2a 2 ∂ x P 1 -hE 2 h 2 , namely, ∂ t E 1 (u) + ∂ x F 1 (u) = 0.
Step 2 : Derivation of the equation for V = hE 2 .

From the system (6.8), we obtain

∂ t V = ∂ t (B 3 D 2 -B 2 D 3 ) = B 3 ∂ t D 2 + D 2 ∂ t B 3 -B 2 ∂ t D 3 -D 3 ∂ t B 2 = -B 3 ∂ x B 3 + D 2 P 1 -D 1 P 2 h -D 2 ∂ x D 2 + B 3 P 1 -B 1 P 3 h +B 2 ∂ x -B 2 + D 3 P 1 -D 1 P 3 h + D 3 ∂ x -D 3 + B 2 P 1 -B 1 P 2 h and ∂ x (v 2 V ) = ∂ x (v 2 (B 3 D 2 -B 2 D 3 )) .
Therefore,

∂ t V + ∂ x (v 2 V ) = ε 1 + ε 2 ,
with

ε 1 = B 2 ∂ x (v 2 D 3 ) -B 3 ∂ x (v 2 D 2 ) -D 2 ∂ x (v 2 B 3 ) + D 3 ∂ x (v 2 B 2 ) + ∂ x (v 2 (B 3 D 2 -B 2 D 3 )) , ε 2 = -B 2 ∂ x B 2 h -B 3 ∂ x B 3 h -D 2 ∂ x D 2 h -D 3 ∂ x D 3 h -B 2 ∂ x D 1 P 3 h + B 3 ∂ x D 1 P 2 h + D 2 ∂ x B 1 P 3 h -D 3 ∂ x B 1 P 2 h .
By developing ε 1 , ε 2 and using the definition of E 1 , we get

ε 1 = -V ∂ x v
h -E 1 .
Hence,

∂ t V + ∂ x v 2 V - a 2 h = (P 1 -V )∂ x v 2 + 1 2h ∂ x (hE 1 ) -∂ x E 1 .
Step 3 : Equation for (E 2 (u), F 2 (u)). By the definition of V , we obtain

∂ t (hE 2 ) + ∂ x hv 2 E 2 - a 2 h -h(v 2 -E 2 )∂ x v 2 = 1 2h ∂ x (hE 1 ) -∂ x E 1 .
Using the first equation in (6.10) and after simplification, we have

∂ t E 2 + ∂ x (v 2 E 2 ) - a 2 h ∂ x 1 h -v 2 ∂ x v 2 = 1 2h 2 ∂ x (hE 1 ) - 1 h ∂ x E 1 . Since 1 2h 2 ∂ x (hE 1 ) - 1 h ∂ x E 1 = - 1 2 ∂ x (h -1 E 1 ),
we obtain finally

∂ t E 2 (u) + ∂ x F 2 (u) = 0.
This ends the proof of Lemma 6.2.

Acknowledgment The author is very grateful to G.Boillat for informing me timely of references [START_REF] Brenier | Hydrodynamic structure of the augmented Born-Infeld equations[END_REF][START_REF] Serre | Hyperbolicity of the nonlinear models of Maxwell's equations[END_REF], and for helpful discussions on the Born-Infeld theory. He would also like to thank C.Dafermos for reading and revising the manuscript carefully, which considerably improves the presentation of the paper.

Theorem 4 . 1 .

 41 Let the assumptions (4.11)-(4.12) hold and the linear Lagrangian system (4.2) be hyperbolic. Then the Cauchy problem (4.2) and (2.2) admits a unique entropy solution in the sense of Definition 2.1. It is given by the explicit expression (4.13) and

1 2

 1 is the Euclidean norm. Introducing the electric induction D = ∂L/∂E, the Poynting vector P and the Born-Infeld energy density h [5, 26, 9]: (6.2)

e 2 =

 2 hv 7 -B 1 hv 4 + D 1 hv 6 and e 3 = hv 8 -D 1 hv 5 + B 1 hv 3 .

Remark 6 . 2 .( 1 +

 621 The Born-Infeld Lagrangian L(E, B) given in (6.1) is well defined if and only if∆(E, B) = 1 + |B| 2 -|E| 2 -< E, B > 2 ≥ 0, with E = 1 h (D + B × P ), P = D × B.Using the definition (6.2) on h,< B × P, B > = 0, < B, P > = < B, D × B > = 0and the formula|ξ| 2 |η| 2 = < ξ, η > 2 +|ξ × η| 2 , ∀ ξ, η ∈ R 3 , we have h 2 ∆(E, B) = (1 + |B| 2 ) 2 + 2|D × B| 2 -2 < D, B × P > . Since < D, B × P > = < P, D × B > = |D × B| 2 , we obtain finally h 2 ∆(E, B) = (1 + |B| 2 ) 2 > 0.Thus, the Born-Infeld Lagrangian L(E, B) is well defined for all B, D ∈ R 3 and all h > 0.In addition, we haveL(E, B) = -1 h |B| 2 ).

B 1 <

 1 B, P > +D 1 < D, P > = B 1 (B 1 P 1 + B 2 P 2 + B 3 P 3 ) + D 1 (D 1 P 1 + D 2 P 2 + D 3 P 3 ) = (B 2 1 + D 2 1 )P 1 + (B 1 B 2 + D 1 D 2 )(B 1 D 3 -B 3 D 1 ) + (B 1 B 3 + D 1 D 3 )(B 2 D 1 -B 1 D 2 ) = (B 2 1 + D 2 1 )(P 1 -hE 2 ). Hence,

  •) for all t ≥ 0.

	Proof. Let us just compute, for all t > 0,
	dY (t, X(t, y)) dt	=	∂Y (t, X(t, y)) ∂t	+	∂Y (t, X(t, y)) ∂x	•	∂X(t, y) ∂t

  Then the Cauchy problem (3.1)-(3.2) has a unique global entropy solution u = rθ satisfying the entropy equality (3.4) and entropy inequality (3.6) for all continuous functions g 1 and g 2 with g 1 convex. Definition and structure of the system. Let us begin with a definition.

	Definition 4.1. A system of conservation laws (2.1) is called a linear Lagrangian one if
	it is linear in Lagrangian coordinates (s, y) defined by the E-L transformation (2.4).

Moreover, if the function ψ(r) = rφ(r) is strictly convex for r > 0, this solution is given explicitly by r = ∂ x Y and (3.8), where Y is defined in (3.9).

4. Study of linear Lagrangian systems 4.1.

  with the corresponding left eigenvectors L 1 , L 2 , L 3 and L 4 given in(6.23). According to(2.11), the eigenvalues of the BI system (6.7) are given by(6.22). Since the vectors L 1 , L 2 , L 3 and L 4 are linearly independent, the linear system (6.27) is hyperbolic. It follows from Proposition 2.1 and Remark 2.1 that the BI system (6.7) is hyperbolic and linearly degenerate. It admits a complete set of classical Riemann invariants W

			.27) are
	(6.28)	μ1 = μ2 = -a,	μ3 = μ4 = a,


Gaëlle Ferré
email: gaelle.ferre@univ-nantes.fr

Gesture catchments and density in narratives of personal experience

 proposed a framework for the semantic and syntactic analysis of spontaneously produced narratives of personal experience in audio recordings. Labov later developed this first approach in two papers (1997, 2007). In the field of Conversation Analysis, the Goodwins (1982, 1984, 1986, 1992) analyzed the complexity of interaction frameworks in narratives which were video-recorded. They particularly considered participants' body orientation and actions to reveal how tellers managed through the production of their narratives or why they didn't. At the time, they however lacked the technical tools to go further into their analyses: they couldn't, for instance, add a prosodic analysis with fine phonetic detail which could have corroborated their views. Nor did they have video annotation tools, with which it is now so easy to align transcriptions and annotations with the visual signal. This has been made possible only quite recently, and some work has been started on narrative at McNeill's Lab at the University of Chicago. McNeill and colleagues (2001) video-recorded participants retelling the story of a film or cartoon they had watched before. This experimental procedure enabled them to establish links between prosody and discourse as well as between discourse organization and gestures. Namely, they found out that metaphoric gestures showed the structure of the different scenes recounted and that a narrative was given coherence through the use of gesture catchments, e.g. one or several recurring gesture features through the whole narrative. The aim of the present paper is to extend the studies presented above on spontaneously produced narratives, that is narratives of personal experience produced in a larger speech context (video-recorded conversations, TV shows, etc.). Although the work is based on a larger collection of narratives, for the sake of clarity, one complex narrative will be presented here and it will be shown that intonation units are probably more adapted than syntactic ones to determine the Labovian phases of a spoken narratives and that both modal density (Norris 2004) and gesture catchments adapt to the complexity of a narrative.

Introduction

Narrative analysis is a vast field of study -not only in linguistics -which is constantly enriched with the apparition of new technology. In the 1960s, recording devices made it possible to reconsider previous work, adding to the collection of existing studies the analysis of narratives of personal experience in which new emphasis could be given to the oral and visual dimensions of narratives. Two major series of papers were then written, on the internal structure of narratives by [START_REF] Labov | Narrative analysis: Oral versions of personal experience[END_REF] and later [START_REF] Labov | Some Further Steps in Narrative Analysis[END_REF][START_REF] Labov | Narrative pre-construction[END_REF], and on the occurrence of narratives in larger action-levels by the Goodwin couple (1982Goodwin couple ( , 1984Goodwin couple ( , 1986Goodwin couple ( , 1992)). Whereas the first series mentioned focused on the semantic and syntactic structure of narratives, the latter initiated observations on actions and the gestures made by the participants in the video-recordings. Besides, they based their analysis on spontaneously produced narratives which are also the object of this paper.

At the turn of the century, new technology again revived interest in narrative studies as it then became possible to annotate both prosodic and gestural phenomena with extreme precision and make new observations on co-occurring marks. Interesting observations were made on narratives recorded with an experimental procedure by McNeill et al. (2001, among other studies). They described how gestures take part in the structuring of narratives thanks to "gesture catchments", which will be described below.

The present paper is part of a larger analysis on a collection of narratives of personal experience, and shows how prosody and gestures help understand the structure of a complex narrative, spontaneously produced, and rather than giving general conclusions on several narratives, it will present one narrative in its complexity as an illustration of the type of process involved in the telling, focusing namely on gesture / modal density and gesture catchments.

Theoretical background

Discourse Analysis

This work has been carried out in the theoretical framework in Discourse Analysis proposed by [START_REF] Labov | Narrative analysis: Oral versions of personal experience[END_REF] and [START_REF] Labov | Some Further Steps in Narrative Analysis[END_REF][START_REF] Labov | Narrative pre-construction[END_REF] who analyzed the structure of narratives as being made of non narrative clauses (in syntactic terms) and narrative ones. Non narrative clauses are comments made by the participants which elaborate on or generalize from the narrative but do not recount events of the personal experience itself. For instance, in the narrative given as example in the appendix (section 5) and which will be fully described in the next section, the narrator starts one narrative with "this happened a long time ago" (l.18), which is part of the narrative itself (a narrative orientation clause) and then elaborates with "cos of course recently I've just turned a very useful forty" which is a comment or parenthesis (a non narrative clause) made to say that this story would not happen nowadays as the events recounted could only occur in her youth and that she would not behave in such a way now. Non narrative clauses may be produced at any time during the narrative (they may either be uttered spontaneously by the speaker or be triggered by audience feedback as is the case in the example lines 41-45) and do not follow any particular order.

Conversely, narrative clauses are temporally-sequentially ordered and are commonly grouped into the following narrative phases:

Abstract: A phase which shortly announces the punchline of the narrative. This phase is typically absent from narratives produced in the course of a larger action like a conversation or a TV interview. Orientation: A phase in which the teller presents the context of the narrative (time, place, participants…). Complication: A sequence of events temporally organized and leading to a climax or a punchline.

Resolution: According to the type of narrative, it is a phase in which the teller briefly recounts how the problem was solved or what happy conclusion was drawn from the complication phase. Evaluation: A phase in which the teller proposes a personal evaluation either of the events in the narrative or of the narrative itself as a genre (funny, sad…). Coda: A short phase in which the teller switches from the narrative to the larger action. Few of the narratives examined for this and other studies contain this phase though. The example presented in the appendix does not.

Prosody

As presented in the previous subsection, Labov and Waletsky's work has had a major impact on narrative studies and it will be shown in this paper that the structure they proposed for spoken narratives is still valid today, despite the slight fault in their analysis. Although they worked on spoken narratives, their analysis relies on a syntactic segmentation of speech, which is not what one would think the most appropriate initial segmentation of spontaneous speech which may contain false starts, hesitation marks, repeated items and abandoned constructions. For instance, look at the beginning of line 18 in the example given in the appendix "in fact I may even have to uh", an abandoned clause: this type of clause is often met in spoken narratives, and neither syntax nor semantics help us much in knowing whether this utterance was meant by the teller as the end of the previous evaluation or the beginning of the orientation clause. This is where prosody has something to add since the fundamental frequency (ƒ0) has a different shape in beginnings and in ends of units, as described by [START_REF] Wichmann | Intonation in Text and Discourse[END_REF]. There are two types of intonation units (IUs): minor and major ones [START_REF] Cruttenden | Intonation[END_REF]. Minor IUs have at least one nuclear stress, have their own prosodic contour, and are grouped into major IUs. This grouping of IUs is called Paragraph intonation; in each paragraph, pitch declines progressively from one group to another. Across paragraphs, it is on the contrary reset to mid values. In the example provided in the appendix, each line break in the narrative corresponds to a major unit and may contain several minor units, as in l.56 "so it was just a great end / to a great night" which constitutes one major unit containing two minor ones, and is congruent with any syntactic analysis one could have made on the script which is not always the case. With this segmentation made, it is then easier to classify each major unit into a narrative phase on semantic criteria. The whole procedure is described in section 3.

Growth points and catchments

Elicited narratives have also been studied, namely by McNeill and colleagues, from the gesture perspective. [START_REF] Mcneill | Growth points, catchments and contexts[END_REF] and [START_REF] Mcneill | Growth points in thinking-for-speaking[END_REF] have launched a vast series of studies on elicited narratives, in which they show that speech in this context is organized in Growth Points.

Growth points are idea units, and therefore constitute units which are related to the cognitive process at work on the speaker's part. Growth points, according to the authors, are made visible through the gestural production of speakers, especially thanks to what they call "gesture catchments". A catchment consists of the repetition of one or several recurrent gesture features throughout a growth point. They give examples of different iconic gestures used by the speakers to differentiate between parts of speech (i.e. growth points). [START_REF] Quek | The Catchment Feature Model: A Device for Multimodal Fusion and a Bridge Between Signal and Sense[END_REF] shows in his paper that catchments may be expressed by very fine gesture features, such as holds, symmetry, etc. In the present paper, it will be shown that gesture plane and hand orientation may suffice to constitute a gesture catchment in a narrative where growth points need not be expressed with the same gesture types. Although Labov's analysis of narratives as described above does not quite correspond to McNeill's analysis in terms of growth points, both visions are useful for the understanding of how narratives are structured.

Methods and materials

This work is following a series of studies [START_REF] Ferré | Gesture, Intonation and the Pragmatic Structure of Narratives in British English Conversation[END_REF][START_REF] Ferré | Récits de femmes -Analyse multimodale du récit conversationnel en français : une étude de cas[END_REF]) on a collection of French and English narratives of personal experience. They were taken from two types of corpora: video-recordings of freely going dialogues and broadcast TV-shows. The annotation process was slightly different for French and English narratives, namely the annotation of prosody which was adapted for French, and therefore, only the annotation procedure for narratives in English, which will be the focus of this paper, will be presented below. Although only one example will be discussed at length, the other narratives of the collection have been treated in exactly the same fashion.

Sound

Narratives have been transcribed manually in current spelling with Praat, initially segmented into minor IUs following the British tradition of segmentation [START_REF] Cruttenden | Intonation[END_REF]. Once this first transcription was made, the algorithm Momel-Intsint [START_REF] Hirst | A Praat plugin for Momel and Intsint with improved algorithms for modelling and coding intonation[END_REF]) was used to automatically annotate the ƒ0 target points and ƒ0 values in the signal. The target points give a stylisation of the curve which is as close as possible to the raw ƒ0 curve calculated by Praat. After a manual correction of the target points, the mean ƒ0 was calculated for each minor IU as an approximation of the ƒ0 slope, and from these values, minor IUs were grouped into major units which constituted the basis for the semantic analysis. At last, nuclear stresses were annotated on a separate tier, and speech rate calculated in number of syllables per minute for each major IU. In personal recordings, intensity was measured and annoted, but not in broadcasted narratives, since we expect it to have been controlled during the recording by sound technicians, who level off any peak in loudness.

Major units, on semantic criteria, were then analyzed into narrative and non narrative clauses, and the narrative clauses grouped into the different phases presented in 2.1. All the phases were entered into a Praat tier.

From sound to gesture

Once the sound had been treated, the resulting Praat tiers were imported into Elan for the annotation of the gestures made by the tellers. Hand gestures were annotated following the typology proposed by [START_REF] Mcneill | Hand and Mind : What Gestures Reveal about Thought[END_REF], to which emblems and adaptors (auto-contact gestures) were added. Head movements (nods, shakes, tilts, jerks, etc.) as well as eyebrow movements (raising and frowning) were also transcribed. Having the prosodic, discursive and gesture annotations on a single Elan file enabled the cross-comparison of the various narratives and some characteristics in narrative structure were highlighted; they are now presented in the discussion, which concentrates on the analysis of a complex narrative.

Discussion and conclusion

The narrative presented in its whole in the appendix is an excerpt from a TV-show, broadcasted by ABC television, in which an Australian comedy actress and writer presents her new book. The two action levels -participating in the TV-show and telling a narrative -are alternatively foregrounded [START_REF] Norris | Analyzing Multimodal Interaction. A Methodological Framework[END_REF]) in this excerpt thanks to hand gestures which are either related to the narrative (metaphorics, beats, iconics) or to the construction of intersubjectivity as the actress points at the host, the audience or herself.

To concentrate on the narrative itself, it displays a complex semantic structure as it can be analyzed as one main narrative (description of the book content, drinking stories) in which a second narrative is embedded (the drinking story which was not inserted in the book). The two narratives are presented as narratives of personal experience, whether it be true or not. Besides this complex super-structure, the second narrative can also be seen as complex since it contains two powerful moments: how the actress got inebrieted at a party and what shameful consequence this had on her behaviour. It will be shown below how the two powerful moments are highlighted through gesture amplitude, speech rate and gesture catchments.

Modal density

The notion of modal density [START_REF] Norris | Analyzing Multimodal Interaction. A Methodological Framework[END_REF] has proved particularly valuable in the analysis of this narrative. Although it is used here in a much simpler way than in Norris' analyses, we can see it in two ways: first, at some points in the telling, there is higher gesture density (highlighted parts in the script). By this we mean that the teller produces a hand gesture, as well as a head and eyebrow movement. The two movements can be seen as reinforcing gestures described in [START_REF] Ferré | Intensive Gestures in French and their Multimodal Correlates[END_REF] which give more intensity to some parts of discourse. Now higher gesture density sometimes coincide with prosodic highlighting such as syllable lengthening as in "long time ago" (l.18) were both "long" and "-go" are lengthened and carry nuclear stress. The same happens at l.25 on the word "laudanum" which is thus made prominent and a key point in the narrative, through which the audience is guided with a succession of high modal density parts. The modal density is even greater in the apex of narrative 1 and apex 2 of narrative 2 with the sudden slackening of speech rate (accompanied with greater sound articulation), and in apex 2, with the larger amplitude of the iconic gesture (a gesture mimicking the throwing away of something, the largest gesture in the whole excerpt). Besides highlighting key moments in the telling, the actress produces hand gestures which reveal the underlying structure of her narrative.

Gesture catchments

The first observation to be made in terms of catchments is the orientation of the gestures made by the actress throughout the narrative, which reveals the underlying structure of the story: during Narrative 1, all the metaphoric gestures are made with hand palms turned upwards and as soon as she starts Narrative 2, the metaphoric gestures are made palms turned downwards, an orientation which switches again at the beginning of the complication phase (l.24: palms upwards). The next change (l.34: palms downwards) occurs immediately after the non narrative proposition (l.33) and again after the non narrative proposition (l.47) to the end of the narrative (palms upwards).

Whereas metaphorics are used by the teller to reveal the different phases of the narrative, the iconic gestures highlight the two powerful moments in Narrative 2 where two different planes are adopted by the teller. Both the metaphoric and the iconic gestures (l.32 & 34) of the first key moment are made with a movement in the horizontal plane whereas the succession of iconics (l.52-53) of the second key moment are made with a movement or gesture orientation in the vertical plane. Although probably unconsciously produced by the teller, the gestures in the two planes coincide with the two moments she highlights in the narrative: first that she had taken substances and thereafter woke up on the floor, and second that she threw her soiled underpants out of the window in a place where they would be visible from all.

Conclusion

As a short conclusion, we may say that both prosody and gesture participate in the structuring of a narrative which is an organized unit of speech even when it is part of a larger action-level as was the case in the example presented in this paper. Whereas pointing and the use of vocatives in speech reveal the larger action (the interview), metaphorics, iconics and prosodic devices such as speech rate, sound lengthening and intonation grouping structure the smaller action (telling a narrative). Inside a complex narrative, key moments are highlighted through gesture density (simultaneous production of several gestures) and modal density (gestures produced at times of prosodic emphasis). Gesture catchments may apply to different levels in the narrative. As seen in the example provided in this paper, one type of catchment (palm orientation) reveals the structure in terms of narrative phases (Labov, op. cit.) whereas the other type of catchment (plane) reveals the two key moments, so that the verbal complexity is mirrored by a gestural one.

Appendix 1 1

Host: was there it was # was there any story you thought I can't # I can't put this in here that it seems cos it covers so much and so much personal stuff good and bad

  (255) AND / {(laughter)} the NEXT morning / I've WOken I up / on the FLOOR of this PLACE (290.5) 35 FORtunately at least next to the people I had come with (358) M and A ROY / that's when I realized / that I needed a toilet very QUICKly # (285) {mhmh} AND / uh well basically I FOUND a toilet (291) and I was pretty BUsy / from both ENDS: (250) {(laughter)} 40 climax 1: and it SEEMS a way (+ laughter) / P esSENtially Roy / I've THROWN up / on my own underPANTS (203) {(laughter)} [NOW / YEAH (+ clapping and laughter) it's # {(laughter) 'kay okay} A AND you know (220) P COME with me # / come WITH me # (224.5) {I'm I'm drugged I want to wake up now # but at the same time not want to (+ laughter)} 45 M don't well # (117) {yeah yeah}] complication: WELL: / M hem NOW / cos the REAL problem was / that I M was at a comPLETE stranger's HOUSE: / and I'm THINKing this (300) {at your house it would have been okay} [ M oh that's XX {(laughter)}] why I thought this story is no ICE-breaker (264) 50 E HI / WE haven't met / BUT / (h) P uh so I didn't know what M to DO / and I thought / I did the most sensible thing I COULD do: (274) climax 2: I I # TOOK the underpants off: / I and THREW them out the window (242) {(laughter)} resolution: but it was just a bit of a SHAME / that I didn't realize / that the window FACED / I the FRONT: of the HOUSE: (325) {(laughter)] 55 and we actually ALL walked PAST: them / as we LEFT {(laughter)} (224) evaluation: so it was B just a GREAT: end / to a GREAT: night: # (206) BACK TO NARRATIVE 1 Host: XX # I look forward to chapter one of the next book already

  [in fact I MAY even have to uh /] M this happened a LONG: time aGO: / [uh P cos of course REcently / P I've just turned a very useful FORty (291) 20 (h) hem P and Roy may I just say to YOU: # / M well I don't care what peo P ple say FOR:ty B # / is the new FOR:ty # (299) {I'm please to hear that I've been worried} XXX WELL: / PEOple who say anything M else / are living a LOT] (420) A (h) but oKAY / M so YEARS ago I'm hem / I'm having a bit of a big DAY / with some BUddies # (318) complication: AND: / then one of them's M going HEY # / B I know a PARty / we can GO to (321) 25 and take some LAU:danum / [(h) M which was of COURSE / an Opiate based drug: (217) + laughter) then I'd gone OFF / this isn't DOing anything for M me / and I've done the senseful maTURE thing M then / and drank half a bottle of BOUR:bon (296.7) {(laughter) mhmh} [no I am a REAL model #]

	NARRATIVE 1 orientation: well there's ONly one story / that I really WANted to put in / that I COULDn't / beCAUSE hem: # (281 syll/min) complication: well BAsically / my VEry wonderful editor Sam / SAID uh (347) you M know the WHOLE thing / with you being fond of a DRINK: (351.5) climax: B (h) I think that's pretty CLEAR: # (241) 10 in the BOOK: # (206) 5 resolution: M you PRObably don't: need yet / another drinking STOry (290) M so I had to loose one of my FA:vourites # / shall I share it toNIGHT # (266.5){please do please do (laughter)} 15 evaluation: it's hem well look M I'd LOVE to / cos it's really one of my most sophisticated TALES: / TOO: {(laughter)} (324) EMBEDDED NARRATIVE 2 so (+ laughter) M I've gone SURE: / so we've gone to this PARty / M I've TAken the stuff (257) it wasn't LAUdanum / it was like someone had popped M NEUrofen / into a bottle of BAI:leys (369.6) 30 M I don't know WHAT it was / {you were right at the end?} M but I've given it a CRACK / reGARDless (312) orientation: from MAny years ago: / M at LEAST / you'd have LEARNED: children] (246.5) (h) uh (

Transcription conventions: [ ] non narrative clauses; { } feedback from host and/or audience; # silent pause; (h) audible in-breath; (347) speech rate in number of syllables/minute; capital letters: nuclear stress; : perceived syllable lengthening; / minor intonation boundary; XXX inaudible speech; highlighted parts: high gesture density; underline: hand gesture (Metaphoric, Iconic, Beat, Pointing, Adaptor, Emblem).

AcknowledgementsI am grateful to ABC Television, Australia, to broadcast some of their recordings and allow the public to use the videos and scripts in research papers for educational purposes. The script used in this study has been anonymized. I would also like to thank the reviewers of this paper for their useful commentaries on a previous version.