Christian Bourdarias
email: bourdarias@univ-savoie.fr

Anupam Pal Choudhury
email: anupampcmath@gmail.com

Billel Guelmame
email: billel.guelmame@univ-cotedazur.fr

Stéphane Junca
email: stephane.junca@univ-cotedazur.fr

Entropy solutions in BV s for a class of triangular systems involving a transport equation

Keywords: AMS Classification: 35L65, 35L67, 35Q35, 26A45, 76N10 conservation laws, triangular systems, entropy solution, fractional BV spaces, linear transport, chromatography, Cauchy problem, non convex flux, regularity

published or not. The documents may come     Entropy solutions in BV s for a class of triangular systems involving a transport equation Christian Bourdarias, Anupam Pal Choudhury, Billel Guelmame,

Introduction

We consider triangular systems of the form, t > 0, x ∈ R,

∂ t u + ∂ x f (u) = 0, (1.1) ∂ t v + ∂ x (a(u)v) = 0. (1.2)
Here f is the scalar flux function (which we shall henceforth refer to as the flux) for the equation (1.1) and the function a denotes the velocity of the linear equation with respect to v (1.2). The system has a decoupled nonlinear conservation law and a coupled "linear" transport equation with a discontinuous velocity. The above system is complemented by a set of initial data, x ∈ R, u(x, 0) = u 0 (x), (1.3) v(x, 0) = v 0 (x).

(1.4)

The Pressure Swing Adsorption process (PSA) in chemistry [START_REF] Bourdarias | Hyperbolic models in gas-solid chromatography[END_REF] has such a triangular structure after a change of variables from Euler to Lagrange [START_REF] Bourdarias | Eulerian and Lagrangian formulations in BV s for gas-solid chromatography[END_REF][START_REF] Peng | Euler-Lagrange change of variables in conservation laws[END_REF]. Such systems are already of mathematical interest due to the coupling of the theory of scalar conservation laws with the theory of transport equations. This system was studied in [START_REF] Lefloch | An existence and uniqueness result for two nonstrictly hyperbolic systems[END_REF] in a non hyperbolic setting, f = a, with measure solutions for v. That even a strictly hyperbolic setting is not enough to avoid measure solutions was shown in [START_REF] Danilov | Delta shock wave formation in the case of triangular hyperbolic system of conservation laws[END_REF][START_REF] Hayes | Measure solutions to a strictly hyperbolic system of conservation laws[END_REF]. Here, strengthening the hyperbolicity of the system by a stronger transversality condition, global weak bounded entropy solutions are provided with an optimal fractional BV regularity for the initial data u 0 . At the first sight, such a system seems easy to solve in the "triangular" manner, that is solving the first equation (1.1) to get u and then solving the linear equation (1.2) keeping u fixed. This method works well for smooth solutions [START_REF] Ghoshal | Optimal BV s regularity for all time for entropy solutions of conservation laws[END_REF], but, when a shock wave appears in u the velocity a(u) becomes discontinuous. The theory of linear transport equations with discontinuous velocity is a delicate topic yielding measure solutions and a loss of uniqueness [START_REF] Bouchut | One-dimensional transport equations with discontinuous coefficients[END_REF][START_REF] Bouchut | Duality solutions for pressureless gases, monotone scalar conservations laws and uniqueness[END_REF][START_REF] Poupaud | Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients[END_REF].

In this paper, we propose a different approach to obtain global weak solutions in L ∞ . A main idea is to consider the system (1.1), (1.2) not as a triangular system, but, as a 2 × 2 hyperbolic system as in [START_REF] Baiti | Well-posedness for a class of 2 × 2 conservation laws with L ∞ data[END_REF]. If f is nonlinear, one field is nonlinear and the other one is linearly degenerate. When f is uniformly convex it is a particular case of 2 × 2 systems with one genuinely nonlinear field and a linearly degenerate one [START_REF] Haspot | BV s solutions for 2 × 2 systems of conservation laws with a linearly degenerate field[END_REF]. An example of such a system when the flux f is piecewise convex or concave arises from gas-liquid chromatography [START_REF] Bourdarias | Eulerian and Lagrangian formulations in BV s for gas-solid chromatography[END_REF]. The presence of linearly degenerate fields can simplify the study of solutions [START_REF] Peng | Explicit solutions for 2 × 2 linearly degenerate systems[END_REF][START_REF] Peng | Euler-Lagrange change of variables in conservation laws[END_REF], but can also produce blow-up behaviours [START_REF] Bourdarias | Eulerian and Lagrangian formulations in BV s for gas-solid chromatography[END_REF][START_REF] Neves | Ill-posedness of the Cauchy problem for totally degenerate system of conservation laws[END_REF].

We show that the behaviour of the proposed entropy solutions is linked to the fractional BV regularity of u, indeed, u 0 ∈ BV s , 0 < s ≤ 1. The BV s framework is optimal to study the regularity of entropy solutions of scalar conservation laws [START_REF] Castelli | Oscillating waves and the maximal smoothing effect for one dimensional nonlinear conservation laws[END_REF][START_REF] Marconi | Regularity estimates for scalar conservation laws in one space dimension[END_REF]. This framework is recalled in Section 1.3. For the scalar case, the theory works well for all s > 0. For the triangular system (1.1), (1.2), we prove that the regularity s = 1/3 of u 0 is critical for the existence of L ∞ entropy solutions. The exponent s = 1/3 is directly linked to the cubic estimate on the Lax curves [START_REF] Lax | Hyperbolic systems of conservation laws II[END_REF]. For nonconvex fields, it is known that the Lax curves are less regular [START_REF] Bianchini | On the Riemann problem for nonconservative hyperbolic systems[END_REF][START_REF] Ancona | Well-posedness for general 2 × 2 systems of conservation laws[END_REF][START_REF] Lefloch | Hyperbolic Systems of Conservation Laws: the theory of classical and nonclassical shock waves[END_REF]. However, for the triangular system (1.1), (1.2) with the flux f ∈ C 4 , the velocity a ∈ C 3 , and satisfying a uniformly strict hyperbolicity assumption, we prove that the exponent s = 1/3 is optimal for the existence theory of bounded entropy solutions. Our proof is based on a new cubic estimate that holds even when the flux f is non-convex. Our estimate generalizes the well known cubic estimates for genuinely nonlinear (convex) systems by Lax [START_REF] Lax | Hyperbolic systems of conservation laws II[END_REF].

The paper is organized as follows. The hyperbolicity of the triangular system and the key assumptions are given in Section 1.1. The definitions of weak and entropy solutions are stated in Section 1.2. The BV s framework is recalled in Section 1.3. In Section 2, the two main results are stated: existence for s ≥ 1/3 and blow-up for s < 1/3. In Section 3, the Riemann invariants and the Riemann problem are studied. The Lax curves and the key cubic estimates are studied in Section 4. The proof of existence is discussed in Section 5 and the blow-up is studied in Section 6. In the appendices A and B, we discuss on the uniqueness of the Riemann problem in the class of bounded weak entropy solutions.

The hyperbolic triangular system

The system (1.1), (1.2) of conservation laws is hyperbolic when f = a and can be rewritten, using the vectorial flux F, as ∂ t U + ∂ x F(U) = 0, U = (u, v) . The matrix of the linearized system has a triangular structure,

DF(U) = f (u) 0 a (u)v a(u)
.

This sytem has the unbounded invariant region

[-M, M ] u × R v , where M = u 0 ∞ .
In this paper, the system is assumed to be strictly hyperbolic through the condition

∀u ∈ [-M, M ], f (u) > a(u). (1.5)
Of course, the symmetric asumption: ∀u ∈ [-M, M ], f (u) < a(u), yields a similar study. For large data, the strict hyperbolicity condition (1.5) has to be strengthened on the set [-M, M ], by the following uniformly strict hyperbolicity (USH) condition:

inf |u|≤M f (u) > sup |u|≤M a(u). (1.6)
An interesting case is already when f is convex, f > 0 everywhere, or concave f < 0. This case occurs for a chromatography system with a convex isotherm written in appropriate Lagrangian coordinates [START_REF] Bourdarias | Eulerian and Lagrangian formulations in BV s for gas-solid chromatography[END_REF]. In this paper, the flux f (•) belongs to C 4 and the velocity a(•) belongs to C 3 . Few times in the paper, to present the nonconvex case, the flux is assumed to have a finite number of inflection points,

Z = {u, f (u) = 0, |u| ≤ M } is finite. (1.7)
Thus, Riemann problems are solved by a finite number of elementary waves, the so called composite wave [START_REF] Holden | Front tracking for hyperbolic conservation laws[END_REF]. The finiteness of the number of inflection points is not mandatory for the existence Theorem 2.1 below, because the proof uses approximate piecewise linear flux f ν . The important point is that f ν is also locally piecewise convex or concave so the solution of the Riemann problem has only a finite number of wavefronts [START_REF] Dafermos | Polygonal approximations of solutions of the initial value problem for a conservation law[END_REF].

Weak and entropy solutions

A weak solution of the system (1.

, ψ ∈ C 1 c (R × [0, +∞[, R
), the following integral identities hold:

+∞ 0 R (u ∂ t ϕ + f (u) ∂ x ϕ) dx dt + R u 0 (x) ϕ(x, 0)dx = 0, (1.8)
+∞ 0 R (v ∂ t ψ + a(u) v ∂ x ψ) dx dt + R v 0 (x) ψ(x, 0)dx = 0. (1.9)
The following regularity is usually required for (u, v): u ∈ L ∞ loc , v ∈ L 1 loc or v is a measure. In the case when v is a measure, there are some issues in defining the product a(u) × v [START_REF] Bouchut | One-dimensional transport equations with discontinuous coefficients[END_REF]. But our main focus in this paper is on bounded weak solutions u, v ∈ L ∞ .

We propose below a notion of entropy solutions for the system (1.1)-(1.2). As in [START_REF] Bourdarias | Existence of entropy solutions for gas chromatography with one or two actives species and non convex isotherms[END_REF][START_REF] Junca | Analysis of a Sugimoto's model of nonlinear acoustics in an array of Helmholt resonators[END_REF] the entropy condition is only tested on the nonlinear component u. Contact discontinuity waves linked to a linearly degenerate field are well known not to affect the entropy inequality which, therefore, remains an equality [START_REF] Dafermos | Hyperbolic Conservation Laws in Continuum Physics[END_REF]. Definition 1.2 (entropy solution). The pair (u, v) is an entropy solution of the system (1.1)-(1.2) with initial data (1.3)-(1.4) if it is a weak solution and for all convex function η and all non-negative test functions ϕ ∈ C 1 c (R×]0, +∞[, R), with q = η f , u satisfies the following inequality:

+∞ 0 R (η(u) ∂ t ϕ + q(u) ∂ x ϕ) dx dt + R η(u 0 (x)) ϕ(x, 0)dx ≥ 0. (1.10)
Thus, it suffices only to have a weak solution of the system and the entropy solution of (1.1).

Uniqueness of entropy solutions with a fixed initial data (u 0 , v 0 ) is a delicate matter [START_REF] Bressan | Hyperbolic Systems of Conservation Laws, The One-Dimensional Cauchy Problem[END_REF]. It is due to the lack of uniqueness of weak solutions for the linear transport equation with a discontinuous velocity ( [START_REF] Bouchut | One-dimensional transport equations with discontinuous coefficients[END_REF][START_REF] Bouchut | Duality solutions for pressureless gases, monotone scalar conservations laws and uniqueness[END_REF]).

For triangular systems as discussed in this paper, one can have measure solutions. A prototypical example of such triangular systems [START_REF] Hayes | Measure solutions to a strictly hyperbolic system of conservation laws[END_REF] is given by

∂ t u + ∂ x u 2 2 = 0, ∂ t v + ∂ x [(u -1)v] = 0, (1.11) with f (u) = u 2 2 and a(u) = u -1.
Clearly this satisfies the strict hyperbolicity condition (1.5) but not the more restrictive uniform strict hyperbolicity condition (1.6) for too large data. Similar phenomenon is well known for transport equations [START_REF] Bouchut | One-dimensional transport equations with discontinuous coefficients[END_REF] and is also widely observed in the case of non-strictly hyperbolic systems [START_REF] Bouchut | Duality solutions for pressureless gases, monotone scalar conservations laws and uniqueness[END_REF].

The main approach in solving the system (1.11) in [START_REF] Hayes | Measure solutions to a strictly hyperbolic system of conservation laws[END_REF] was to observe that the first equation in u can be solved independently. The second equation can then be considered as a transport equation in v with a discontinuous coefficient u (see also [START_REF] Bouchut | One-dimensional transport equations with discontinuous coefficients[END_REF]), but this idea of looking at the two equations separately might not be a good one. Our aim in this article is to prove the existence of solutions to systems of type (1.11) in the class of fractional BV functions and hence one need not appeal to δ-shock wave type solutions. Moreover, to obtain L ∞ solution with a wave front algorithm, we suppose that the system (1.1), (1.2) is uniformly strictly hyperbolic (1.6). Note that we already need the uniform strict hyperbolicity condition (1.6) to solve the Riemann problem appropriately.

BV s functions

For one dimensional scalar conservation laws, the spaces BV s are known to give optimal results for weak entropy solutions [START_REF] Bourdarias | Fractional BV spaces and first applications to conservation laws[END_REF][START_REF] Castelli | Oscillating waves and the maximal smoothing effect for one dimensional nonlinear conservation laws[END_REF][START_REF] Marconi | Regularity estimates for scalar conservation laws in one space dimension[END_REF][START_REF] Marconi | Structure and regularity of solutions to 1D scalar conservation laws[END_REF], first on the fractional Sobolev regularity, and second on the structure of such solutions. Such results on the maximal regularity have been extended for some systems and the multidimensional case [START_REF] Castelli | On the maximal smoothing effect for multidimensional scalar conservation laws[END_REF][START_REF] Ghoshal | Optimal BV s regularity for all time for entropy solutions of conservation laws[END_REF]. In this section, basic facts on BV s functions are recalled.

Definition 1.3. [39] A function u is said to be in BV s (R) with 0 < s ≤ 1 if T V s u < +∞, where T V s u := sup n∈N, x 1 <•••<xn n i=1 |u(x i+1 ) -u(x i )| 1/s .
The BV s semi-norm is defined by

|u| BV s := (T V s u) s ,
and a norm on this space is defined by

u BV s := u L ∞ + |u| BV s .
Fractional BV functions have traces like BV functions. This is a fundamental property to define the Rankine-Hugoniot condition for shock waves. Morerover, this property is not true for the Sobolev functions in W s,1/s , the Sobolev space nearest to BV s [START_REF] Bourdarias | Fractional BV spaces and first applications to conservation laws[END_REF].

Theorem 1.1. [START_REF] Musielak | On generalized variations[END_REF] For all s ∈]0, 1[, BV s functions are regulated functions.

The fractional total variation only depends on the local extrema of the function and the order of this extrema.

Lemma 1.1. [START_REF] Bourdarias | Fractional BV spaces and first applications to conservation laws[END_REF][START_REF] Haspot | BV s solutions for 2 × 2 systems of conservation laws with a linearly degenerate field[END_REF] If u is a piecewise monotonic function and if its local extrema are located in the increasing sequence (x i ) i∈I , then T V s u only depends on the sequence (u(x i )) i∈I . Moreover, there exists an ordered subset J of I such that

T V s u = j∈J,j =max J |u(x suc(j) ) -u(x j )| 1/s ,
where suc(j) denotes the successor of j in J, suc(j) = min{k ∈ J, k > j}.

Moreorever, it can be dangerous to refine the mesh to compute the fractional total variation [11, example 2.1], [START_REF] Bruneau | La variation totale d'une fonction[END_REF]. Consider u(x)

≡ x on [0, R], p = 1/s > 1. Then T V s u[0, R] = R p but, when n → +∞, n i=1 |u(iR/n) -u((i -1)R/n)| p = n(R/n) p = R p /n p-1 → 0.
This property, which is not true in BV , is used later to prove the existence of weak solutions in BV 1/3 for the triangular system (1.1), (1.2) when the flux f is convex.

Main results

The main results are stated in the BV s framework. The basic facts on this setting were recalled in Section 1.3. The critical space for the existence theory is BV 1/3 . Theorem 2.1 (Existence in L ∞ with (u 0 , v 0 ) ∈ BV 1/3 × L ∞ ). Suppose that the flux f (•) and the transport velocity a(•) satisfy the following assumptions on [-M, M ] where M = u 0 ∞ :

1. f (•) and a(•) belong to C 3 ([-M, M ], R),
2. f (u) > a(u) and satisfy the uniform strict hyperbolicity assumption (USH) (1.6), Then, if (u 0 , v 0 ) belongs to BV s × L ∞ and s ≥ 1/3, there exists an entropy solution (u, v)

of the system (1.1), (1.2), u ∈ L ∞ ([0, +∞) t , BV s (R x , R)), v ∈ L ∞ ((0, +∞) t × R x , R).
In addition, the positivity of the initial data is preserved, that is, if inf v 0 > 0 then inf v > 0.

Two mandatory conditions are required to avoid singular phenomena at t = 0+. The strong transversality condition (USH) (1.6) is the key assumption to avoid instantaneous δ-shock. The BV s regularity of u 0 is the other key condition to avoid instantaneous blow-up in L ∞ .

The positivity of v is the key tool to get the uniform strict hyperbolicity of the (PSA) system [START_REF] Bourdarias | Existence of entropy solutions for gas chromatography with one or two actives species and non convex isotherms[END_REF]. The complete proof of the positivity of v for the (PSA) system is obtained in [START_REF] Bourdarias | Kinetic formulation of a 2Ö2 hyperbolic system arising in gas chromatography[END_REF] without using the triangular structure of the (PSA) system [START_REF] Bourdarias | Eulerian and Lagrangian formulations in BV s for gas-solid chromatography[END_REF].

Notice that the sign of v is preserved a.e. by Theorem 2.1. Of course, by linearity of the equation (1.2) with respect to v, the negativity of the initial data is also preserved, that is, if sup v 0 < 0 then sup v < 0.

The triangular approach, that means solving first (1.1) with the unique Krushkov entropy solutions [START_REF] Kruzkov | First order quasilinear equations with several independent variables[END_REF] and then (1.2), usually yields non-unique measure solutions v [START_REF] Bouchut | One-dimensional transport equations with discontinuous coefficients[END_REF][START_REF] Poupaud | Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients[END_REF]. It is the reason why the uniqueness of entropy solutions for the whole triangular system is an open problem. Of course, we have uniqueness for u but the problem of uniqueness remains for v. For a weakly coupled system with some linearly degenerate fields, the entropy condition only on the nonlinear fields is enough to ensure uniqueness in [START_REF] Junca | Analysis of a Sugimoto's model of nonlinear acoustics in an array of Helmholt resonators[END_REF]. For the triangular system, the coupling by the transport velocity is too nonlinear to achieve uniqueness in the same way.

For the existence result, a Wave Front Tracking algorithm (WFT) is proposed as in [START_REF] Baiti | Well-posedness for a class of 2 × 2 conservation laws with L ∞ data[END_REF] for the whole system. As a matter of fact the approximate solution u of the scalar equation satisfies the BV s uniform bounds as in [START_REF] Bourdarias | Fractional BV spaces and first applications to conservation laws[END_REF][START_REF] Jenssen | One-dimensional scalar conservation laws with regulated data[END_REF]. The difficult point is to bound v in L ∞ . Since the system is linear with respect to v, an L ∞ bound for v is enough to get an existence theorem as in [START_REF] Bourdarias | Eulerian and Lagrangian formulations in BV s for gas-solid chromatography[END_REF].

Theorem (2.1) is optimal and, in general, we cannot reduce the BV 1/3 regularity of u 0 , else a blow-up can occur. For this purpose, we build a sharp example in a space very near to BV 1/3 , namely, BV 1/3-0

BV 1/3 BV 1/3-0 := s<1/3
BV s .

The simple example (1.11) from [START_REF] Hayes | Measure solutions to a strictly hyperbolic system of conservation laws[END_REF] with a Burgers' flux f and a linear velocity a is enough to provide a blow-up.

Theorem 2.2 (Blow-up in L ∞ at t = 0+ for u 0 ∈ BV 1/3-0 ). For the system (1.11) with the Burgers' flux f (u) = u 2 /2 and the velocity a(u) = u -1, there exists u 0 ∈ BV 1/3-0 , v 0 ∈ L ∞ such that the system (1.1), (1.
2) is uniformly strictly hyperbolic (USH) (1.6), and there doesn't exist any bounded entropy solution of the system (1.1), (1.2) with the initial data u 0 , v 0 .

Of course u 0 / ∈ BV 1/3 , otherwise the existence theorem 2.1 gives a bounded entropy solution. There is no blow-up for u since the entropy solution u of the scalar conservation law satisfies the maximum principle. Indeed, only the function v has a blow-up at time t = 0 + . Notice also that the blow-up depends only on the regularity of u 0 and not of v 0 .

The example presented below follows the construction of particular explicit solutions [START_REF] Adimurthi | Structure of entropy solutions to scalar conservation laws with strictly convex flux[END_REF] which provide optimal example in BV s for convex scalar conservation laws [START_REF] Ghoshal | Optimal BV s regularity for all time for entropy solutions of conservation laws[END_REF]. On any compact set avoiding the blow-up point, the entropy solution u is locally Lipschitz except on a finite number of lines. For initial data u 0 leading to such an entropy solution u, we show that a uniqueness result for the entropy solution (u, v) holds at least upto the time of first interaction of waves in u. Thus there is no way to avoid the blow-up. In general, a possible dense jump set in plane (t, x) is possible for the entropy solution u of a scalar conservation law [START_REF] Ghoshal | Optimal jump set in hyperbolic conservation laws[END_REF].

Another consequence of this blow-up example is that for general nonlinear 2×2 strictly hyperbolic systems with a genuinely nonlinear eigenvalue and a linearly degenerate one, the existence result proven in [START_REF] Haspot | BV s solutions for 2 × 2 systems of conservation laws with a linearly degenerate field[END_REF] is optimal.

3 The uniformly strictly hyperbolic system

In this section, we study the Riemann invariants and the solution of the Riemann problem for the triangular system (1.1)-(1.2).

Riemann invariants

A 2 × 2 strictly hyperbolic system admits, at least locally, a set of coordinates which diagonalizes the hyperbolic system for smooth solutions [START_REF] Smoller | Shock waves and reaction-diffusion equations[END_REF]. The knowledge of this coordinate system, given by the Riemann invariants, is often useful in understanding the structure of the system. Next we study the Riemann invariants for the system (1.1)-(1.2).

The eigenvalues of the system (1.1)-(1.2) are

λ 1 = f (u) > a(u) = λ 2 . (3.1)
Notice that the eigenvalues are functions of u only. Let r 1 and r 2 denote the corresponding right eigenvectors. Clearly, u is a 2-Riemann invariant associated to the right eigenvector r 2 = (0, 1) and satisfies

∂ t u + f (u)∂ x u = 0.
A 1-Riemann invariant, which we denote as z(u,v), corresponding to the right eigenvector r 1 , can be computed in the following manner. We note that a right eigenvector of the matrix DF(u) corresponding to the eigenvalue f (u) is given by

r 1 =   1 a (u)v f (u) -a(u)   . Then z satisfies 1 v ∂ u z = a (u) a(u) -f (u) ∂ v z.
This can be solved using a separation of variables.

∂ u z = a (u) a(u) -f (u) , ∂ v z = 1 v .
For instance, a Riemann invariant is given by

z(u, v) = A(u) + ln(|v|), A (u) = a (u) a(u) -f (u)
.

To avoid the singularity at v = 0, it suffices to take the exponential

Z = exp(z) = v exp(A(u)), (3.2)
which satisfies the transport equation

∂ t Z + a(u)∂ x Z = 0. (3.3)

The Riemann problem

We study the Riemann problem for the system (1.1)-(1.2) with initial data:

u 0 (x) = u ± , ± x > 0, (3.4) v 0 (x) = v ± , ± x > 0. ( 3.5)
A direct and somewhat naive approach is to solve the conservation law (1.1) first and then the second equation (1.2) using the solution u. In such an approach, one faces the difficulty of solving the linear transport equation with a discontinuous coefficient.

Instead, we consider the two equations together as a system. This is a key point as in [START_REF] Baiti | Well-posedness for a class of 2 × 2 conservation laws with L ∞ data[END_REF]. The solution of the Riemann problem consists of two waves separated by an intermediary state (u m , v m ) where u m = u -and v m is unknown. In accordance with the labeling of the eigenvalues (3.1), a wave associated to u is called a 1-wave and a wave associated to Z is called a 2-wave.

1. A wave associated to the eigenvalue λ 1 = f is a shock wave or a rarefaction wave if f is convex or concave. For a non-convex flux f , this wave is a composite wave.

2. A linearly degenerate wave associated to the eigenvalue λ 2 = a is called a contact discontinuity. The speed of this 2-wave is a(u -).

The intermediate value v m has to be computed through the 1-wave. Now, the various 1-waves that can occur are detailed. For this purpose, we consider Riemann problems yielding only a 1-wave.

Shock waves

Let us denote U = (u, v) and Ũ = (u, Z).

The Rankine-Hugoniot condition gives

s[u] = [f (u)], s[v] = [a(u) v], (3.6)
where s denotes the speed of the discontinuity (or the slope of the jump). Thus, the slope of the jump is determined by u ± and the flux f ,

s = [f (u)] [u] . (3.7)
Since entropy solutions are considered, the Oleinik-entropy condition [START_REF] Bressan | Hyperbolic Systems of Conservation Laws, The One-Dimensional Cauchy Problem[END_REF] for (1.1) enforces the Lax-entropy conditions,

f (u -) ≥ s ≥ f (u + ), (3.8)
whence

s ≥ f (u + ) > a(u + ). (3.9)
Using the second equation of (3.6) yields

(s -a(u + ))v + = (s -a(u -))v -, (3.10)
v + = v - s -a(u -) s -a(u + ) := S -(u + ; U -). (3.11)
The equation (3.11) can be interpreted in terms of the Lax shock curve. For a fixed U -= (u -, v -), the right hand side of (3.11) is only a function of u + as s is given by (3.7) as a function of u + . In the plane U = (u, v), (3.11) describes the set of U + such that the Riemann problem with initial data U ± is solved by a shock. On this curve, only U + satisfying the Oleinik condition (3.8) are considered to allow an entropic shock. Conversely, if U + is fixed, the Lax shock curve is parametrized by u -and reads

v -= v + s -a(u + ) s -a(u -) := S + (u -; U + ). (3.12)
Proceeding similarly with Z = v exp(A(u)), Ũ-= (u -, Z -) and keeping the notation S in coordinates (u, Z) yields,

Z + = Z - s -a(u -) s -a(u + ) exp [A(u + ) -A(u -)] := S -(u + ; Ũ-),
and

Z -= Z + s -a(u + ) s -a(u -) exp [A(u -) -A(u + )] := S + (u -; Ũ+ ).

1-Contact discontinuity

This is a limiting case of the preceding one, when f is constant on the interval [u -, u + ]. The same formula for the shock wave follows.

s = f (u ± ) = f (u) > a(u ± ), u ∈ [u -, u + ], (3.13)
v + = v - f (u -) -a(u -) f (u + ) -a(u + ) = S -(u + ; U -), (3.14)
Z + = Z - f (u -) -a(u -) f (u + ) -a(u + ) exp [A(u + ) -A(u -)] := S -(u + ; Ũ-). (3.15)
Such waves arise in the wave-front tracking (WFT) method when the flux is approximated by piecewise linear functions.

Rarefaction waves

Since Z is a 1-Riemann invariant, the Lax rarefaction curve R -( Ũ-) is simply,

Z + = Z -. (3.16)
Using the defiinition of Z, this implies v + exp(A(u + )) = v -exp(A(u -)) and thus the Lax rarefaction curve can be written explicitly,

v + = R -(u + ; U -) = v -exp(A(u -) -A(u + )), (3.17)
v -= R + (u -; U + ) = v + exp(A(u + ) -A(u -)). (3.18)
Notice that v + and v -have the same signs. In particular, if v -= 0, v = 0 is constant through the rarefaction wave.

1-Composite waves

The composite waves only occur for the waves associated to the first eigenvalue f (u). In general, the entropy solution u is a composite wave [START_REF] Holden | Front tracking for hyperbolic conservation laws[END_REF]. If f has a finite number N of inflection points, then there are at most N contact-shock waves [START_REF] Dafermos | Hyperbolic Conservation Laws in Continuum Physics[END_REF][START_REF] Lefloch | Hyperbolic Systems of Conservation Laws: the theory of classical and nonclassical shock waves[END_REF]. The Lax curves associated to such waves are studied below in Section 4.

2-Contact discontinuity

u is a 2-Riemann invariant and hence is constant along the 2-contact discontinuity. Thus the Lax curve is simply a vertical line in the plane U = (u, v) or the plane Ũ = (u, Z),

u -= u + . (3.19)
4 The Lax curves

A fundamental theorem due to Lax [START_REF] Lax | Hyperbolic systems of conservation laws II[END_REF] states that the shock curve and the rarefaction curve emanating from a constant state U -in the plane (u, Z) or (u, v) have a contact of the second order for a genuinely nonlinear eigenvalue. This means that the shock curve can be replaced by the rarefaction curve up to an error of order [u] 3 where [u] = u +u - [START_REF] Bressan | Hyperbolic Systems of Conservation Laws, The One-Dimensional Cauchy Problem[END_REF][START_REF] Dafermos | Hyperbolic Conservation Laws in Continuum Physics[END_REF][START_REF] Serre | Systems of conservation laws. I: Hyperbolicity, entropy, shock waves[END_REF]. For the triangular system, a genuinely nonlinear eigenvalue means f > 0 (or f < 0) everywhere. For nonconvex cases, typically f locally has a finite number of roots where f changes its sign and the Lax curves are less regular due to the occurrence of contact-shocks. Under a concave-convex assymption, which means here that f does not vanish, the regularity of the Lax curves is only piecewise C 2 [START_REF] Lefloch | Hyperbolic Systems of Conservation Laws: the theory of classical and nonclassical shock waves[END_REF], see also [START_REF] Ancona | Well-posedness for general 2 × 2 systems of conservation laws[END_REF]. As a consequence, the error becomes of order [u] 2 for the variation of Z through a contact-shock [START_REF] Lefloch | Hyperbolic Systems of Conservation Laws: the theory of classical and nonclassical shock waves[END_REF]. The situation is worse in general, the Lax curves are only Lipschitz [START_REF] Bianchini | On the Riemann problem for nonconservative hyperbolic systems[END_REF]. However, we prove that cubic estimates are still valid for the triangular system (1.1), (1.2). It is mainly due to the existence of Riemann invariant coordinates.

In this section, cubic estimates on the Lax curves for the triangular system in the plane (u, Z) are proved.

Let (u -, v -), (u + , v + ) be two constant states connected by a rarefaction or a shock wave. It is more convenient to use the Riemann invariant coordinates (u, Z). The rarefaction curves in the plane (u, Z) are simply,

Z -= Z + , (4.1)
which means that in the (u, v) plane

v -exp(A(u -)) = v + exp(A(u + )). (4.2)
For the shock curve, the Rankine-Hugoniot condition is written in the conservative variables (u, v),

s = [f (u)] [u] = f (u + ) -f (u -) u + -u - , (4.3)
v -(s -a(u -)) = v + (s -a(u + )). (4.4)
Since f ∈ C 4 , s is a C 3 function of its arguments. Moreover, fixing (u + , v + ) and considering u -as a variable, the Lax shock curve is C 3 with respect to u -,

v -= v + s -a(u + ) s -a(u -) . (4.5)
Indeed, the denominator never vanishes due to the uniformly strict hyperbolicity assumption (USH) (1.6). The same regularity of the shock curve holds in the variables (u, Z)

Z -= Z + s -a(u + ) s -a(u -) exp(A(u -) -A(u + )). (4.6)
Of course, the Lax rarefaction curve and the Lax shock curve has to be restricted on the subset satisfying entropy conditions. Nevertheless, we use these curves for all range of u -in R (at least for -M ≤ u -≤ M ) to obtain a generalized Lax cubic estimate for the nonconvex case.

The Lax cubic estimate on the Rankine-Hugoniot curve

The Lax cubic estimate [START_REF] Lax | Hyperbolic systems of conservation laws II[END_REF] can be written as follows for a shock wave connecting (u -, Z -) to (u + , Z + ) for the triangular system (1.1),(1.2), as soon as Z is bounded,

[Z] = O([u]) 3 , [Z] = Z + -Z -, [u] = u + -u -. (4.7)
The Riemann invariant Z is constant along the rarefaction curves. The Lax cubic estimate means that the shock curve and the rarefaction curve have a contact of order 2. The Lax cubic estimate was written in a genuinely nonlinear framework [START_REF] Lax | Hyperbolic systems of conservation laws II[END_REF]. This means that f does not vanish. Indeed, the Lax computations are still valid without this convex assumption and without only considering the entropic part of the Rankine-Hugoniot curve.

Here, we also use the non-entropic part of the Rankine-Hugoniot curve. It is a useful tool used many times in this paper, first in the next section 4.2 to get a cubic estimate for the entropy solution of the Riemann problem for the triangular system with a non-convex flux f . For the triangular system, the Rankine-Hugoniot curve is global, explicit and well defined thanks to the uniformly strict hyperbolicity assumption (1.6). For 2 × 2 systems, such a global curve does not always exist [START_REF] Keyfitz | A strictly hyperbolic system of conservation laws admitting singular shocks[END_REF]. Now, to prove cubic estimates, we have to write the Rankine-Hugoniot curve. Here, we choose to write Z -as a function of u -when (u + , Z + ) are fixed for the following reasons.

1. To solve the Riemann problem and compute the intermediary state v m which corresponds to Z m and here Z -.

2. To obtain the cubic estimate on the global Rankine-Hugoniot curve below.

3. To obtain the existence result, bounding Z along the 2-characteristics.

4. To build a blow-up, again computing Z from the right to the left on the 2-characteristics.

The Rankine-Hugoniot curve RH + when Ũ+ = (u + , Z + ) is fixed and Z -is a function of u -, is given by

Z -= Z + s -a(u + ) s -a(u -) exp [A(u -) -A(u + )] = Z + r(u -, u + ) := RH + (u -; u + , Z + ). (4.8)
Note that the Rankine-Hugoniot curve RH + is a property of (u, v), which means this curve is always computed in the conservative variables (u, v) [START_REF] Dafermos | Hyperbolic Conservation Laws in Continuum Physics[END_REF]. For our convenience, here we write it in terms of (u, Z) due to the fundamental role of the line Z = constant in the Lax cubic estimate [START_REF] Lax | Hyperbolic systems of conservation laws II[END_REF]. This classic cubic estimate on the shock curve is generalized on the global Rankine-Hugoniot curve.

s = s(u -, u + ) = [f ] [u] = f (u + ) -f (u -) u + -u - ∈ C 3 ([-M, M ] 2 , R), r = r(u -, u + ) = s -a(u + ) s -a(u -) exp [A(u -) -A(u + )] ∈ C 3 ([-M, M ] 2 , R), r = 1 + O(1) [u] 3 > 0, ∀(u -, u + ) ∈ [-M, M ] 2 , (4.9) Z -= Z + (1 + O(1)[u] 3 ), ∀(u -, u + , Z + ) ∈ [-M, M ] 2 × R, (4.10)
where the constant O(1) depends only on the derivatives of f and a on [-M, M ]. Moreover, Z -has the same sign as Z + , more precisely,

Z + = 0 ⇒ Z -= 0, Z + = 0 ⇒ Z -Z + > 0.
The cubic flatness of the Rankine-Hugoniot curve RH + is expressed in (4.10). It does not depend on the convexity of the fields, although in the classical textbooks [START_REF] Bressan | Hyperbolic Systems of Conservation Laws, The One-Dimensional Cauchy Problem[END_REF][START_REF] Dafermos | Hyperbolic Conservation Laws in Continuum Physics[END_REF][START_REF] Holden | Front tracking for hyperbolic conservation laws[END_REF][START_REF] Lefloch | Hyperbolic Systems of Conservation Laws: the theory of classical and nonclassical shock waves[END_REF][START_REF] Serre | Systems of conservation laws. I: Hyperbolicity, entropy, shock waves[END_REF][START_REF] Smoller | Shock waves and reaction-diffusion equations[END_REF] some nonlinearity assumptions on the fields are given. The reason of these nonlinear assumptions is usually to introduce the rarefaction curve and the shock curve. Here, the global Rankine-Hugoniot curve defined for all u -∈ [-M, M ] is the main subject without looking at the entropic parts of this curve. A careful reading of the classical proof of the cubic estimate in textbooks shows that it is a geometric property of the Rankine-Hugoniot curve itself, without referring at the entropic or nonlinearity assumptions. This geometric property is a consequence of the symmetry of the Rankine-Hugoniot condition with respect to U -and U + as explained in [START_REF] Serre | Systems of conservation laws. I: Hyperbolicity, entropy, shock waves[END_REF]. It is very important in this paper to prove the cubic estimate for the non-entropic part of the Rankine-Hugoniot curve for two reasons.

1. The cubic estimates on the Lax curve (which is not piecewise C 3 for nonconvex f [START_REF] Lefloch | Hyperbolic Systems of Conservation Laws: the theory of classical and nonclassical shock waves[END_REF]) uses the global Rankine-Hugoniot curve, as discussed in the next section 4.2.

2. Rarefaction wave fans are replaced by weak non-entropic jumps in the wave front tracking algorithm [START_REF] Bressan | Hyperbolic Systems of Conservation Laws, The One-Dimensional Cauchy Problem[END_REF]. The error in the weak formulation is controlled by the cubic estimate to pass to the limit and get a weak solution of (1.1), (1.2).

The proof appears as a direct consequence of (4.9). An elementary and self-contained proof using only Taylor's expansions is proposed. A more tedious computation can give the more precise result

r(u -, u + ) = 1 + E[u] 3 + O([u] 3 ), (4.11)
where E depends in a quite complicated way on the derivatives of f and a at u = u + . The computation of E is quite intricate and not useful here, except in the last section 6 on the blow-up where a direct computation of E at u + = 0 is given when f is quadratic and a is linear. Now, Lemma 4.1 is proven.

Proof. The positivity on r is a consequence of the assumption (1.6). This positivity implies that Z -has the same sign as Z + .

To obtain (4.9), we use Taylor expansion. Here u + is fixed and u -is the variable near u + . The notations a -= a(u -), a + = a(u + ) and so on are used to shorten the expressions.

a -= a + -a + [u] + (a + /2)[u] 2 + O([u] 3 ), s = [f ] [u] = f --f + -[u] = f + -(f + /2)[u] + (f + /6)[u] 2 + O([u] 3 ).
The hyperbolic quantity h is used,

h := f -a > 0. (4.12)
The sign of h is the consequence of the strict hyperbolicity assumption (1.6). We start with the fraction part of r and h

+ = f + -a + = 0. s -a + s -a - = (f + -a + ) -(f + /2)[u] + (f + /6)[u] 2 (f + -a + ) -(f + /2 -a + )[u] + (f + /6 -a + /2)[u] 2 + O([u] 3 ) (4.13) = 1 - a + h + [u] + h + a + -a + f + + 2a 2 + 2h 2 + [u] 2 + O([u] 3 ). (4.14)
For the term exp(-[A]) in r, the Taylor expansion of A -is used at the first order.

-

A -= a - h - = a + -a + [u] h + -h + [u] + O([u] 2 ) = a + h + + a + h + -a + h + h 2 + [u] + O([u] 2 ).
Integrating with respect to u -yields,

A --A + = a + h + [u] + a + h + -a + h + 2h 2 + [u] 2 + O([u] 3 ). Since exp(x) = 1 + x + x 2 /2 + O(x) 3 , it yields exp [A --A + ] = 1 + a + h + [u] + a + h + -a + h + + a 2 + 2h 2 + [u] 2 + O([u] 3 ). ( 4

Cubic estimates for the Riemann problem

Now, the intermediary state Z m of a Riemann problem has to be estimated. For this purpose, the variation of Z along a composite wave is studied. When the flux is convex, Lax proved that the variation of Z is a cubic order of the variation of u [START_REF] Lax | Hyperbolic systems of conservation laws II[END_REF]. For a nonconvex flux, it is well known that the Lax curve is less regular, piecewise C 2 [START_REF] Lefloch | Hyperbolic Systems of Conservation Laws: the theory of classical and nonclassical shock waves[END_REF] or only Lipschitz [START_REF] Bianchini | On the Riemann problem for nonconservative hyperbolic systems[END_REF]. However, we prove that for our triangular system we are able to keep a cubic order. This is mainly due to the existence of Riemann coordinates for 2 × 2 systems and the cubic estimate for the global Rankine-Hugoniot locus, Lemma 4.1. As a consequence, we can prove a similar estimate for the variation of Z over a composite 1-wave. This improves the well known square root estimate for concave-convex eigenvalues [START_REF] Lefloch | Hyperbolic Systems of Conservation Laws: the theory of classical and nonclassical shock waves[END_REF], which correspond to cubic degeneracies for f . That means that for the triangular system (1.1), (1.2), the estimate is as precise as for the convex case [START_REF] Lax | Hyperbolic systems of conservation laws II[END_REF].

Proposition 4.1 (Variation of Z through a composite wave).

Assume that the flux has N inf l points.

Let the states Ũi = (u i , Z i ), i = 1, . . . , m, where

u 0 < u 1 < • • • < u m (m ≤ N inf l + 1)
comprise a composite 1-wave, and Z -= Z 0 and Z + = Z m . Then the total variation of Z through a 1-wave is,

Z ∞ ≤ |Z + | exp O(|u + -u -| 3 ) , (4.16) T V Z ≤ O(1)|Z + ||u + -u -| 3 .
(4.17)

Later, as the consequence of the existence proof, the assumption of the finite number of inflection points can be removed and the same inequalities (4.16), (4.17) hold.

Proof. We note that Z is a 1-Riemann invariant and therefore it remains constant over a rarefaction wave. Moreover, if the states Ũi and Ũi+1 are joined by a jump, by Lemma 4.1 and the classical inequality 1 + x ≤ exp(x), we have the estimate

|Z i | = |Z i+1 r(u i , u i+1 )| = |Z i+1 | 1 + O(|u i -u i+1 | 3 ) ≤ |Z i+1 | exp O(|u i -u i+1 | 3 ) .
Summing up and noting that u 0 , u 1 , . . . , u m are ordered, the estimate (4.16) follows,

max i |Z i | ≤ |Z + | exp O(|u + -u -| 3 ) .
Now, the BV bound for Z through the composite wave is computed.

Z i -Z i+1 = Z i+1 (r(u i , u i+1 ) -1) = Z i+1 O(|u i -u i+1 | 3 ), T V Z ≤ max i |Z i | i O(|u i -u i+1 | 3 ) ≤ |Z + |O(|u + -u -| 3 ) exp O(|u + -u -| 3 ) .

Since u satisfies the maximum principle, we further have exp O(|u

+ -u -| 3 ) = O(1)
which only depends on the L ∞ bound of the initial data u 0 . Hence the inequality (4.17) follows.

Existence in BV s

In this section, Theorem 2.1 is proved using a simplified Wave Front Tracking (WFT) [START_REF] Bressan | Hyperbolic Systems of Conservation Laws, The One-Dimensional Cauchy Problem[END_REF][START_REF] Holden | Front tracking for hyperbolic conservation laws[END_REF] algorithm for such a triangular system (1.1), (1.2). The BV s estimates for u are a consequence of such estimates for scalar conservation laws [START_REF] Bourdarias | Fractional BV spaces and first applications to conservation laws[END_REF][START_REF] Jenssen | On Φ-variation for 1-d scalar conservation laws[END_REF][START_REF] Jenssen | One-dimensional scalar conservation laws with regulated data[END_REF]. The L ∞ bound for v and the proof of existence of a weak solution for the triangular system are based on an approach using the BV 1/3 regularity for u.

The Wave Front Tracking algorithm

The WFT depends on an integer parameter ν > 0. The approximate solutions will be denoted by u ν , v ν , Z ν . We shall mostly use the Riemann invariant coordinates (u ν , Z ν ) except when passing to the limit in the weak formulation.

This algorithm is explained in many books [START_REF] Bressan | Hyperbolic Systems of Conservation Laws, The One-Dimensional Cauchy Problem[END_REF][START_REF] Dafermos | Hyperbolic Conservation Laws in Continuum Physics[END_REF][START_REF] Holden | Front tracking for hyperbolic conservation laws[END_REF] on hyperbolic systems. Taking advantage of the structure of the triangular system (1.1), (1.2), we will mix the WFT for the scalar case [START_REF] Dafermos | Polygonal approximations of solutions of the initial value problem for a conservation law[END_REF] and for systems [START_REF] Bressan | Hyperbolic Systems of Conservation Laws, The One-Dimensional Cauchy Problem[END_REF][START_REF] Holden | Front tracking for hyperbolic conservation laws[END_REF]. The main principle is to work with piecewise constant approximations.

As in the scalar case [START_REF] Bressan | Hyperbolic Systems of Conservation Laws, The One-Dimensional Cauchy Problem[END_REF][START_REF] Holden | Front tracking for hyperbolic conservation laws[END_REF][START_REF] Jenssen | On Φ-variation for 1-d scalar conservation laws[END_REF][START_REF] Jenssen | One-dimensional scalar conservation laws with regulated data[END_REF], the values of u ν are taken on a uniform grid parametrized by the integer ν, that is,

u ν ∈ ν -1 Z.
(5.1)

On the other hand, v ν , or equivalently Z ν , is not required to stay on the uniform grid, that is,

v ν , Z ν ∈ R, (5.2)
as we solve the Riemann problem with the exact flux f to compute v ν , or equivalently Z ν . In this way, we can use the cubic estimate on the global Rankine-Hugoniot curves of Lemma 4.1. A similar approach for a convex case was used in [START_REF] Haspot | BV s solutions for 2 × 2 systems of conservation laws with a linearly degenerate field[END_REF].

The initial data are approximated as follows. Let N 0 + 1 be the number of constant states for the approximate initial data u 0,ν , v 0,ν and the corresponding Z 0,ν . Here u 0,ν takes values on the grid and N 0 is a function of ν. To ensure that (u 0,ν , v 0,ν ) converges towards the initial data (u 0 , v 0 ), we need the condition that N 0 goes to infinity as ν tends to infinity.

The approximate initial data can be chosen to satisfy the following uniform estimates with respect to ν [START_REF] Jenssen | On Φ-variation for 1-d scalar conservation laws[END_REF][START_REF] Jenssen | One-dimensional scalar conservation laws with regulated data[END_REF]:

u 0,ν ∞ ≤ u 0 ∞ , (5.3) T V s u 0,ν ≤ T V s u 0 , (5.4) v 0,ν ∞ ≤ v 0 ∞ .
(5.5)

Moreover, (u 0,ν , v 0,ν ) converges pointwise almost everywhere to (u 0 , v 0 ) when ν → +∞ and therefore, the previous inequalities become equalities at the limit ν → +∞.

The flux f is replaced by a piecewise linear continuous flux f ν [START_REF] Dafermos | Polygonal approximations of solutions of the initial value problem for a conservation law[END_REF] that coincides with f on the uniform grid [START_REF] Bressan | Hyperbolic Systems of Conservation Laws, The One-Dimensional Cauchy Problem[END_REF]Ch. 6], [27, p. 70],

f ν (k/ν) = f (k/ν), ∀k ∈ Z.
(5.6)

At t = 0+, N 0 Riemann problem are solved. The approximate solution u ν is the weak entropy solution of

∂ t u ν + ∂ x f ν (u ν ) = 0, u ν (x, 0) = u 0,ν (x). (5.7)
With strong compactness on u ν , the Kruzkov entropy solution is recovered [START_REF] Bressan | Hyperbolic Systems of Conservation Laws, The One-Dimensional Cauchy Problem[END_REF][START_REF] Holden | Front tracking for hyperbolic conservation laws[END_REF].

For (u ν , v ν ) the approximate vectorial flux F ν (u, v) = (f ν (u), a(u)v) is used, as detailed below. The reason for this choice is twofold. First, to have a simpler wave front tracking for the scalar equation for u with a non-convex flux following the seminal paper [START_REF] Dafermos | Polygonal approximations of solutions of the initial value problem for a conservation law[END_REF] (this case is also very well detailed in the textbook [START_REF] Holden | Front tracking for hyperbolic conservation laws[END_REF]). This wave front tracking algorithm does not lead to any consistency error in the weak formulation [START_REF] Dafermos | Polygonal approximations of solutions of the initial value problem for a conservation law[END_REF]. Secondly, to use the new generalized cubic estimates for Z ν which also hold for the solution (u ν , v ν ) of the system with the approximate flux F ν , as

f ν = f on the grid {(u, v) ∈ (ν -1 Z) × R}.
After some time, some of the Riemann problems interact and hence, new Riemann problems have to be solved. The non-linear interactions for the triangular system are described in detail below. The process continues until the second time of interactions and so on.

This process can be continued for all time because there is only a finite number of interactions for all time. For u ν this follows from the existing results for the scalar case. An explicit bound on the number of interactions is given in [27, p. 71-72]. The 2-waves, associated to the linearly degenerate eigenvalue a(u ν ), never interact together since they are contact discontinuities. Due to the transversality assumption (USH), a 2-wave can interact only once with a 1-wave and creates a new 2-wave (and does not modify the 1-wave). Thus, the number of such interactions and of 2-waves is finite. This proves that the WFT is well defined for all time. Now, the approximate Riemann solver and the nonlinear interactions of the waves are detailed.

Approximate Riemann solver

In this short section, the approximate Riemann solver is detailed. Let (u -, Z -), (u + , Z + ) denote the left and right states in the initial data. For u ν the solution is a series of entropic jumps u -= u 0 < u 1 < . . . < u m = u + for the piecewise linear flux f ν . For Z ν , there are many possibilities. We want to approximate the exact solution of the Riemann problem and keep the cubic estimates (Proposition 4.1) which generalize the Lax cubic estimates for genuinely nonlinear waves. So, we use the Rankine-Hugoniot curve to determine Z m-1 , . . . , Z 1 . The states Z i , i = m, m -1, . . . , 2 are built as follows:

If there is a jump between u i-1 and u i corresponding to a shock or a contact discontinuity for f ν , then Z i-1 is given by the Rankine-Hugoniot curve (4.8),

Z i-1 = RH + (u i-1 ; u i , Z i ).
(5.8)

Note that RH ν + = RH + on the grid {(u, v) ∈ (ν -1 Z) × R}, where RH ν + denotes the Rankine-Hugoniot curve corresponding to the approximate flux F ν . Thus, the jump between (u i-1 , Z i-1 ) and (u i , Z i ) satisfies the Rankine-Hugoniot condition and there is no error in the weak formulation of the linear equation satisfied by v and hence, no error in the weak formulation of the approximate triangular system (5.7), (1.2), where the exact flux f is replaced by the piecewise-linear flux f ν in (1.1).

If there is a 2-contact discontinuity (along which u is constant and there is a jump between v ± ), then the contact discontinuity is solved with the exact speed a(u). Thus, again, there is no error of consistency for the approximate system (5.7), (1.2).

Nonlinear wave interactions

We briefly describe the different possible nonlinear wave interactions and the details of the interactions are given in the (u, Z) plane. Here Z denotes the 1-Riemann invariant with respect to the exact flux F . Lemma 5.1. The cubic estimates for Z proved in Proposition 4.1 also hold true across 1-waves (shock and 1-contact discontinuity) corresponding to the approximate flux F ν .

Proof. This immediately follows from (5.8), the fact that RH ν + = RH + on the grid and the cubic estimates for Z across 1-waves corresponding to f (Proposition 4.1).

In all the cases, we consider three states Ũ-, Ũ0 , Ũ+ before the interaction. Here Ũ0 denotes the intermediary state which disappears after the interaction. The states Ũ-, Ũ0 are connected by an elementary wave and similarly the states Ũ0 , Ũ+ are connected by an elementary wave. An elementary wave is either a 1-wave: shock or a small jump, or a 2-wave: a contact discontinuity. At a time of interaction t interact , the Riemann problem is solved with a new intermediary state Ũm (see Figure 1). We shall use the following notations.

S or S 1 stands for a shock wave which is always a 1-wave.

C 1 or C 2 stands for a contact discontinuity associated to λ 1 or λ 2 , a 1-wave or a 2-wave. C 1 is considered as a degenerate shock S 1 .

The key point here is to understand the effect of the L ∞ norm of Z ν after the interaction. First of all, we note that there is no self interaction for the second family since there are only C 2 waves. Also the case of interactions between the 1-waves of the first family have already been well-studied [START_REF] Bressan | Hyperbolic Systems of Conservation Laws, The One-Dimensional Cauchy Problem[END_REF], [START_REF] Dafermos | Hyperbolic Conservation Laws in Continuum Physics[END_REF], [START_REF] Holden | Front tracking for hyperbolic conservation laws[END_REF] for the component u ν and are not presented here. The new feature is the effect of this interaction on the second component Z ν .We have seen in lemma 5.1 that the change in L ∞ norm of Z ν is of the order of the cube of the change in u (or u ν ).

Finally, we consider the interaction of a 1-wave with a 2-wave.

1. In the case of an interaction of the form S 1 -C 2 (which means that a 1-shock interacts with a 2-contact discontinuity), the outgoing wave is of the form C 2 -S 1 .

The shock continues with the same slope and the same value u -, u + and still satisfies the entropy condition (3.8). Thus, for u there is no change before and after the interaction. Roughly speaking, the interaction of a 1-shock with a 2-contact discontinuity is transparent for u. On the contrary, there is a change for Z ν and following [START_REF] Haspot | BV s solutions for 2 × 2 systems of conservation laws with a linearly degenerate field[END_REF], it can be shown that the change in L ∞ norm of Z ν is of the order of cube of the change in u ν .

2. The interaction C 1 -C 2 (that is, when a 1-contact discontinuity interacts with a 2-contact discontinuity) generates outgoing waves of the form C 2 -C 1 . This case can be dealt in a similar manner as in the case of S 1 -C 2 and it follows that the change in L ∞ norm of Z ν is of the order of cube of the change in u ν .

The reader interested by all the cases S 1 - [START_REF] Haspot | BV s solutions for 2 × 2 systems of conservation laws with a linearly degenerate field[END_REF] (and different notations where the 1 and 2 waves have to be exchanged) in the plane of Riemann invariant, a technique used in [START_REF] Smoller | Shock waves and reaction-diffusion equations[END_REF] for the isentropic Euler system. Here we present the last two cases for the triangular system, the case S 1 -C 2 in figure 2 and the case C 1 -C 2 in figure 3.

C 1 , C 1 -S 1 , S 1 -C 2 , C 1 -C 2 can consult

Uniform estimates

The BV s estimates for u ν are already known since u is the entropy solution of the scalar conservation law (1.1). These estimates are recalled briefly in the first paragraph. The only difficulty in this section is to obtain the L ∞ estimates for v. For this purpose, we generalize the approach first used in [START_REF] Bourdarias | Eulerian and Lagrangian formulations in BV s for gas-solid chromatography[END_REF] and recently in [START_REF] Haspot | BV s solutions for 2 × 2 systems of conservation laws with a linearly degenerate field[END_REF]. The approach consists of bounding v ν , indeed Z ν , along the 2-characteristics. For the chromatography system [START_REF] Bourdarias | Eulerian and Lagrangian formulations in BV s for gas-solid chromatography[END_REF], the 2-characteristics are simply straight lines. In general, here, the 2-characteristics are piecewise linear. They are uniquely defined since the second eigenvalue a(u) is linearly degenerate and its integral curves are transverse to the discontinuity lines of the first field (assumption (USH)). The precise definition of such characteristics is given in the second paragraph. Then the estimate on Z ν along 2-characteristics, as in [START_REF] Bourdarias | Eulerian and Lagrangian formulations in BV s for gas-solid chromatography[END_REF][START_REF] Haspot | BV s solutions for 2 × 2 systems of conservation laws with a linearly degenerate field[END_REF], is given in the last paragraph using our generalized estimates on the Lax curves. The interacting waves are represented by dotted lines, a 1-wave in black (horizontal line) followed by a 2-wave in red (vertical line). The full lines represent the resulting waves.

BV s estimates for u The T V s decay is known for the Glimm scheme, the Godunov scheme and the Wave Front Tracking algorithm [START_REF] Bourdarias | Fractional BV spaces and first applications to conservation laws[END_REF][START_REF] Bourdarias | Eulerian and Lagrangian formulations in BV s for gas-solid chromatography[END_REF]. Another argument is that u ν is also the exact entropy solution of the scalar conservation law (5.7) with the piecewise-constant initial data u 0,ν , and the decay of T V s u gives the uniform estimates with respect to ν, T V s u ν (•, t) ≤ T V s u 0,ν ≤ T V s u 0 .

(5.9)

The approximate 2-characteristics

Essentially, for the WFT, an approximate i-characteristic is a continuous curve which is piecewise linear following the velocity λ i , i = 1, 2. Since the eigenvalues depend only on u, there is a problem to define an i-characteristic where u is not defined. For i = 2, there is no problem of uniqueness, since a 2-characteristic is always transverse to the discontinuity lines of u. Thus, the 2-characteristic crosses the u discontinuity with a kink. Let γ ν (x 0 , t) be the forward generalized 2-characteristic starting at the point x 0 , that is γ ν (x 0 , t) is a solution of the differential inclusion

d dt γ ν (x 0 , t) ∈ [a (u ν (γ ν (x 0 , t) -0, t)) , a (u ν (γ ν (x 0 , t) + 0, t))] , γ ν (x 0 , 0) = x 0 .
For the wave front tracking, these 2-characteristics are uniquely determined and are piecewise linear continuous curves (thanks to the transversality assumption (USH)) and satisfy the differential equation

d dt γ ν (x 0 , t) = a (u ν (γ ν (x 0 , t), t)) , γ ν (x 0 , 0) = x 0 , (5.10)
except for a finite number of times which correspond to a jump of the piecewise constant function u ν .

L ∞ estimates for v

The L ∞ estimate on v ν is first obtained on the approximate Riemann invariant Z ν . The key point is then to bound Z ν in L ∞ . Z ν is not constant through a 1-wave: a shock wave or a 1-contact discontinuity. But, we know that the variation of Z ν is of order of the cube of the variation of u ν , by lemma 5.1. When there is no 1-wave, the simple curve to bound Z ν is the 2-characteristic. In this case, Z ν (γ ν (x 0 ), t) = Z ν (x 0 , 0). Note that the 2-characteristics starting at time t = 0 span the whole (t, x) half-plane but Z ν is only defined outside a finite number of 2-characteristics.

For example, Z ν is not well-defined on a 2-characteristic that contains a 2-wave (contact discontinuity for the second family). But these are only finite in number as the number of 2-wave fronts is finite. Also Z ν is not well-defined on 2-characteristics passing through the points of intersections of waves corresponding to the first family. Again there are only finite number of such curves.

Z ν is also not well-defined on a finite number of 2-characteristics originating from the points on t = 0 where the Riemann problems are solved initially. Figure 4: Wave front tracking algorithm. For the picture, f (u) > 0 > a(u). Thus the 1-waves in blue go to the right and the 2-waves in black to the left. Notice that these 1-waves are not affected by the interaction with the 2-waves, but the 2-waves are affected by the interaction.

But since there are only finite number of such 2-characteristics, this is enough to estimate Z ν ∞ . Now, due to the transversality conditions, a 2-characteristic meets many 1-waves. Thanks to lemma 5.1, we have a cubic estimate for the L ∞ norm and the total variation of Z ν through a 1-wave. More precisely, if t 1 is a time just before the 2-characteristic meets a 1-wave and t 2 is the time just after the 2-characteristic crosses the 1-wave, we have the following estimate,

Z ν ∞,(γν (x 0 ,•),•)[t 1 ,t 2 ] ≤ |Z ν (t 1 )| exp O T V 1/3 u ν (γ ν (x 0 , t 2 ), t 2 ) , T V Z ν (γ ν (x 0 , •), •)[t 1 , t 2 ] ≤ Z ν ∞,(γν (x 0 ,•),•)[t 1 ,t 2 ] O(T V 1/3 u ν (γ ν (x 0 , •), •)[t 1 , t 2 ]) .
That means that the L ∞ norm and the total variation of Z ν along the piece of curve {(γ ν (x 0 , t), t), t ∈ [t 1 , t 2 ]} is controlled by T V 1/3 u ν along the same piece of curve. Notice also that the sign of Z ν is constant along a 2-characteristic by lemma 4.1.

First, the L ∞ norm of Z is bounded on the 2-characteristic and, second, the estimate on T V Z ν follows. The total variation is additive and the fractional total variation is sub-additive [START_REF] Bruneau | La variation totale d'une fonction[END_REF]. So adding all these estimates, on the whole 2-characteristic Γ(x 0 ) = {(γ ν (x 0 , t), t), t > 0} starting at x = x 0 , we have the estimates

Z ∞,Γ(x 0 ) ≤ |Z 0,ν (x 0 )| exp O T V 1/3 u ν [Γ(x 0 )] , ≤ Z 0 ∞ exp O T V 1/3 u ν [Γ(x 0 )] , T V Z ν [Γ(x 0 )] ≤ Z 0 ∞ Ψ O T V 1/3 u ν [Γ(x 0 )] , Ψ(x) = x exp(x).
Now note that Γ(x 0 ) is a space like curve for the 1-characteristics of (5.7) ( [START_REF] Dafermos | Hyperbolic Conservation Laws in Continuum Physics[END_REF]). Hence

T V 1/3 u ν [Γ(x 0 )] ≤ T V 1/3 u 0,ν ≤ T V 1/3 u 0 .
This follows from the fact the wave-fans corresponding to u ν are monotone with respect to the left and right end states. Therefore no extremal values are added when we measure the fractional variation along Γ(x 0 ). Together these yield L ∞ and BV bounds for Z ν (uniform with respect to ν) along the 2-characteristics,

Z ν ∞ ≤ Z 0 ∞ exp O(1)T V 1/3 u 0 , (5.11)
T V Z ν [Γ(x 0 )] ≤ Z 0 ∞ Ψ O T V 1/3 u 0 .
(5.12)

Notice also that the positivity is preserved. If inf v 0 > 0, then inf Z 0 > 0 and by similar arguments presented above, we have inf Z ν > inf Z 0 exp O(1)T V 1/3 u 0 > 0, where the constant O(1) is negative. Finally, a L ∞ bound (respectively a positivity) for Z ν yields a L ∞ bound (respectively a positivity) for v ν . For the triangular system, the L ∞ bound of Z ν and hence of v ν is enough to pass to the weak limit in (1.2) since the left hand side is linear with respect to v ν . The uniform BV bound on Z ν along the 2-characteristics can be used to recover a strong trace at t = 0, like in [START_REF] Bourdarias | Existence of entropy solutions for gas chromatography with one or two actives species and non convex isotherms[END_REF] (at x = 0).

Passage to the limit in the weak formulation

Passing to the strong-weak limit in the equation (1.2) which is linear with respect to v allows us to get a weak solution. But to do so, we need to understand the error of consistency of the scheme. There is no error of consistency in the independent scalar equation for u ν (5.7) [START_REF] Dafermos | Polygonal approximations of solutions of the initial value problem for a conservation law[END_REF]. By construction, our approximate solver does not produce error for the linear equation (1.2). That means that the approximate solution (u ν , v ν ) is an exact solution of the approximate system (5.7),(1.2):

∂ t u ν + ∂ x f ν (u ν ) = 0, ∂ t v ν + ∂ x (a(u ν )v ν ) = 0.
Therefore as in [START_REF] Dafermos | Polygonal approximations of solutions of the initial value problem for a conservation law[END_REF], one can pass to the limit and obtain a weak solution of the exact system (1.1),(1.2).

6 Blow-up of an entropy solution with u 0 / ∈ BV 1/3 In this section, we provide a proof of Theorem 2.2. For this purpose, we construct initial data u 0 , Z 0 that satisfy u 0 ∈ BV 1/3-0 (R), Z 0 ∈ L ∞ (R), and exhibit blow-up in L ∞ norm of Z at t = 0+ for the system (1.1)-(1.2).

Recall that the notation u 0 ∈ BV 1/3-0 (R) means that u 0 is in BV s for all s ∈ [0, 1/3). Our idea of construction is motivated by similar examples studied in [START_REF] Adimurthi | Structure of entropy solutions to scalar conservation laws with strictly convex flux[END_REF][START_REF] Castelli | On the maximal smoothing effect for multidimensional scalar conservation laws[END_REF][START_REF] Lellis | On the optimality of velocity averaging lemmas[END_REF][START_REF] Ghoshal | Optimal BV s regularity for all time for entropy solutions of conservation laws[END_REF].

We would like to emphasize that the solution proposed below is not based on the Wave-Front Tracking algorithm discussed earlier, rather it is an exact solution.

Notice that when v 0 ≡ 0, i.e. Z 0 ≡ 0, no blow-up occurs since (u, v) ≡ (u, 0) gives a global entropy solution where u the entropy solution associated to the L ∞ initial data u 0 . Thus, in the following construction, we have to avoid the value 0 for Z.

Let us consider the system (1.1)-(1.2) with flux f (u) = u 2 2 , a(u) = u -1 and initial data u 0 as described below and Z 0 ≡ 1.

Let x 0 = 0 and x n = 1 -1 2 n , n ≥ 1. Let B n be chosen such that Note that u 0 ∞ ≤ 1 3 and therefore, the condition of uniform strict hyperbolicity is satisfied. The first interaction times T n for the Riemann problems for the equation

x n = x n-1 + 2B n , that is, B n = 1 2 n+1 . Let b n = 1 (n + 26)
∂ t u + ∂ x f (u) = 0
with initial data u 0 is given by The initial data u 0 , described above, clearly does not belong to BV 1 3 (R), but, as in [START_REF] Bourdarias | Eulerian and Lagrangian formulations in BV s for gas-solid chromatography[END_REF][START_REF] Castelli | On the maximal smoothing effect for multidimensional scalar conservation laws[END_REF] we can conclude that u 0 ∈ BV 1/3-0 (R). We use the forward generalized characteristic for Z. Since a(u) = u -1 and the first interaction times of the Riemann problems for u satisfy (6.1), it follows that the forward generalized characteristic for Z starting at the point x ∞ := 1 crosses infinitely many shocks before the first interaction of the waves in u. As we have already seen from the nonlinear wave interactions, the L ∞ norm of Z does not change when it interacts with a 1-rarefaction wave.

T n = B n b n = (n + 26)

Lemma 4 . 1 (

 41 Cubic flatness of the global Rankine-Hugoniot curve). If f and a belong to C 3 (R, R) and satisfy the uniform strict hyperbolicity condition (1.6), then

. 15 )

 15 Now, multiplying (4.14) and (4.15) yields r = 1 + O([u] 3 ).

Figure 1 :

 1 Figure1: Nonlinear interaction of two waves. The 1-wave on the left crosses the interaction point with the same speed and the same value u -on the left of the 1-wave (u m = u -) and u + on the right. On the other hand, the second wave is affected by the interaction: the speed of the 2-wave and a new value Z m appears.

Figure 2 :

 2 Figure 2: Interaction of a shock with a 2-contact discontinuity. The interacting waves are represented by dotted lines, a 1-wave in black followed by a 2-wave in red. The full lines represent the resulting waves.

Figure 3 :

 3 Figure 3: Interaction between a 1-contact discontinuity and a 2-contact discontinuity.The interacting waves are represented by dotted lines, a 1-wave in black (horizontal line) followed by a 2-wave in red (vertical line). The full lines represent the resulting waves.

  U 0 (x, b, B) = b1 [0,B) (x) -b1 [B,2B) (x).Using this we define the initial data u 0 asu 0 (x) = n≥1 U 0 (xx n-1 , b n , B n ).

  Note that the first interaction time T n satisfies the relationT n > 1x n . (6.1) x , b , B)

Figure 5 :

 5 Figure 5: A typical building-block

2 < 1

 21 Now, let us consider a left state u -and a right state u + connected by a 1-shock wave. In our example, we have u -= b > 0 and u + = -b < 0. Hence, the speed of the shock s = 0. Also by construction 0 < b n < 1 and hence we assume that the prototype b satisfies the same. NowZ - Z + = sa(u + ) sa(u -) exp [A(u -) -A(u + )] = a(u + ) a(u -) exp [A(u -) -A(u + )] = u + -1 u --1 exp [A(u -) -A(u + )] = 1 + b 1b exp [A(u -) -A(u + )] . (6.2) Also A (u) = a (u) a(u)f (u) = 1 (u -1)u = -1 < 0,

Acknowledgments

A part of this work was done when the last author was visiting NISER, Bhubaneswar, India. We also thank the IFCAM project "BV s , control & interfaces" which allowed the interaction between the Indian hyperbolic community with the French one. We also thank Boris Haspot for fruitful discussions.

and hence

Therefore from (6.2), we have

We show that for b positive and small enough,

and therefore Z -> Z + . Thus, Z increases in strength as the forward 2-generalized characteristic crosses a shock (from right to left).

Notice that we simply need that Z increases when b ∼ 0. It is for very small oscillations that Z blows-up. Inequality (6.4) follows from a Taylor expansion up to the third order:

for b sufficiently small and hence Z -> Z + > 0.

Thus, due to an interaction with a 1-shock wave, there is a change of order

= +∞, we find that the L ∞ norm for Z blows up.

Remark 6.1. Let Γ(x 0 ) = {(γ(x 0 , t), t), t ≥ 0} be the 2-characteristic issued from x = x 0 at t = 0. The solution is well defined under Γ(1) that is on the set {(x, t), x < γ(1, t), t ≥ 0}. But, over Γ(1), {(x, t), 1 > x > γ(1, t), t > 0}, Z and v blow up, v = +∞. Remark 6.2. This example does not contradict the Lax-Oleinik smoothing effect ( [START_REF] Lax | Hyperbolic systems of conservation laws II[END_REF]) as the blow-up for Z occurs only at t = 0, that is, there is an immediate blow-up. Such a blow-up is not possible for a time t 0 > 0 due to the BV smoothing of u.

Now suppose that there exists a bounded weak solution for the system (1.1)-(1.2) with the above initial data. Then since the solution of the Riemann problem in the class of bounded weak solutions is unique upto the time of first interaction of the waves (see appendix B), any bounded weak solution will satisfy the construction discussed above. This, in turn, implies that the bounded weak solution exhibits a blow-up in L ∞ norm, which contradicts its definition. Thus, there cannot exist a bounded weak solution for the system (1.1)-(1.2) with the above initial data, which proves Theorem 2.2 . This also shows that the existence Theorem 2.1 is optimal in the class of bounded weak solutions.

A On Lipschitz test functions

Here we briefly discuss the use of Lipschitz test functions instead of C 1 test functions for bounded weak entropy solutions for triangular systems (1.1)-(1.2). We show that if u, v ∈ L ∞ is a weak bounded solution (and therefore the solution constructed in Section 5) for the triangular system, then the C 1 test functions can be enlargerd to Lipschitz functions. This can be proved using the following observations. Note that (u, v) is a weak solution and hence satisfies the integral identities (1.8)-(1.9) for compactly supported smooth test functions. Now if we consider a compactly supported Lipschitz continuous test function ψ, we can construct (using a standard mollifier) a sequence of compactly supported smooth test functions {φ n } that converges pointwise almost everywhere to ψ. An application of dominated convergence theorem shows that ∂ t φ n and ∂ x φ n converge to ∂ t ψ and ∂ x ψ respectively at points of continuity (and therefore almost everywhere) of ψ. Since the integral identities (1.8)-(1.9) are satisfied for each φ n , we can use the above facts to pass to the limit in the identities to prove the integral identities for the Lipschitz continuous test function ψ.

B Uniqueness of Riemann problem in L ∞

We next show that in the case of strictly convex fluxes f , the solution of the Riemann problem is unique in the class of bounded entropy solutions at least upto the time of first interaction of waves. In this direction, we first note the following result.

in open sets where u is Lipschitz continuous.

Here

) denotes the dual of the space of compactly supported Lipschitz continuous functions.

Proof. Since u is Lipschitz continuous and A ∈ C 1 (R), we can write any ψ ∈ Lip c in the form ψ = φ exp(-A(u)) for a suitable φ ∈ Lip c .Therefore, for

we can write,

where we use the fact that v satisfies the integral identity (1.9) for any φ ∈ Lip c (R × [0, +∞[, R). Now ∂ t u and ∂ x f (u) belong to L ∞ and hence in L 1 loc , and also satisfy the integral identity (1.8) for all compactly supported smooth functions. Therefore ∂ t u = -∂ x f (u) almost everywhere. Also using the fact that f ∈ C 1 and u is continuous, using Volpert's formula [START_REF] Volpert | Spaces BV and quasilinear equations[END_REF], it follows that -∂ x f (u) = f (u)∂ x u almost everywhere. Using these facts and the definition of A in (B.2), it follows that

Now suppose that f is strictly convex and consider the Riemann problem with left and right states given by (u -, v -) and (u + , v + ) respectively. Let the corresponding states in (u, Z) coordinates be given by (u -, Z -) and (u + , Z + ) respectively. Without loss of generality, we assume that the Riemann problem is based at the point (0, 0).

First we consider the case when there is a rarefaction wave in u. Note that in this case u is a Lipschitz continuous function away from a ball B(0, r) of radius r around (0, 0), for any r > 0. From the previous lemma, we conclude that away from the ball B(0, r), Z (corresponding to any bounded weak solution (u, v)) satisfies the identity (B.1) and by unicity of u and the unicity of weak solutions of a transport equation with Lipschitz continuous velocity field, Z is uniquely determined. Since this is true for any r > 0, it follows that in the class of bounded weak solutions, (u, Z) (and therefore (u, v)) is unique away from the set {x = a(u -)t} of measure zero.

Next let us consider the case when there is a shock wave in u. In this case, using the previous lemma, we see that Z (corresponding to any bounded weak solution (u, v)) satisfies the transport equation (B.1) with velocity a(u -) in the region

and therefore is uniquely determined in this region and has a trace on the line x = st. Similarly, Z satisfies the transport equation (B.1) with velocity a(u + ) in the region (x, t) : s < x t < +∞ , and therefore is uniquely determined in this region as well.

But any bounded weak solution (u, v) satisfies the same Rankine-Hugoniot condition across the shock curve, and therefore, the left trace of Z on the line x = st is uniquely determined by the states u -, u + , v + , which implies the unicity of Z as well. Therefore, for a strictly convex flux f , at least upto the time of first interaction of waves, the solution of the Riemann problem is unique in the class of bounded entropy solutions. Note that the above proof holds for initial data u 0 which gives rise to entropy solutions u that are locally Lipschitz except for a finite number of lines, and hence applies to the initial data that we construct in Section 6. Uniqueness for the slightly smaller class of picewise C 1 entropy solutions are well known [START_REF] Lax | Hyperbolic partial differential equations[END_REF] and has been already obtained for the (PSA) system [START_REF] Bourdarias | Some mathematical results on a system of transport equations with an algebraic constraint describing fixed-bed adsorption of gases[END_REF].

Notice also that for the Kruzkov solution u a continuity in time with respect to L 1 loc is space is required [START_REF] Kruzkov | First order quasilinear equations with several independent variables[END_REF], but for a nonlinear flux this property is automatically fullfilled [START_REF] Chen | Initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws[END_REF]. Notice that since u 0 belongs to BV s , s > 0, the solution is continuous in time with values in L 1 loc in space [START_REF] Bourdarias | Fractional BV spaces and first applications to conservation laws[END_REF][START_REF] Jenssen | On Φ-variation for 1-d scalar conservation laws[END_REF]. Thus, the initial data is recovered.


J Pénarier

L Torres

Guillaume Varani

L Tohmé

S Blin

P Nouvel

A Pénarier

J Torres

L Varani

G Ducournau

P Artillan

S Bollaert

Y Roelens

D Coquillat

D But

W Knap

F Teppe

Signal-to-noise ratio in Terahertz wireless communication using field-effect-transistors as detectors

de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Significative increase of bit data-rate in present wireless communication systems requires an increase of the carrier frequency. Wireless communication systems based on terahertz (THz) carrier frequencies are interesting as allowing for datarates up to 100 Gbps. However, adequate THz sources and detectors are still under development [START_REF] Dragoman | Terahertz fields and applications[END_REF]. Preliminary wireless communication systems have been successfully demonstrated mainly using Schottky barrier diodes as detectors [START_REF] Song | Present and future of terahertz communications[END_REF]- [START_REF] Ishigaki | Direct intensity modulation and wireless data transmission characteristics of terahertz-oscillating resonant tunneling diodes[END_REF]. At the moment, Schottky barrier diodes are by far the best detectors for communication, due to their high sensitivity (10 kV/W at 0.3 THz) [START_REF] Jastrow | 300 GHz transmission system[END_REF]. More recently, field-effect transistors (FETs) have also been demonstrated to be an alternative for low cost THz detection in wireless communication [START_REF] Blin | Plasma-Wave Detectors for TerahertzWireless Communication[END_REF]. Detection by FETs can be very promising in comparison to existing detectors based on Schottky diodes as it is more easily compatible with integrated circuits such as demodulation circuits. The best reported performance of FETs detectors can achieve a responsivity up to few kV/W at 0.3 THz and a noise equivalent power in the range of a few pW/Hz 1/2 [START_REF] Schuster | Broadband terahertz imaging with highly sensitive silicon CMOS detectors[END_REF], [START_REF] Ojefors | A 0.65 THz focal-plane array in a quarter-micron CMOS process technology[END_REF]. These values are comparable to the best results reported for Schottky diode THz detectors, placing FETs as important competitors with the additional advantage of low resistance, easier integration as already mentioned, and relatively low price. In the real experimental conditions of THz wireless communication, signal-to-noise ratio is a combination of noises mainly due to amplifier and detrimental coupling signal at the modulation frequency.

In this paper, we propose to investigate the signal-tonoise ratio in communication systems based on FET detectors. Section 2 is devoted to the characterization of FET detectors, their integration, the determination of their sensitivity and bandwidth. In Section 3, we present experimental results of THz wireless communication with transmission of pseudorandom bit sequence using FET detector operating at room temperature, and show that the limitation is due to coupling through the gate and the source. Finally, we investigate some improvements and integration of the system for communication around 0.3 THz.

II. FET DETECTOR CHARACTERIZATION

As we mentioned earlier, our objective is to study the potential of commercial and low cost transistors operating at room temperature for the THz communication. For that purpose, we choose a pseudomorphic high-electron-mobility GaAs transistor (pHEMT) with a 200 nm gate length. Previous measurements have shown that these transistors are very sensitive to accidental external coupling through their gate and their source (ground coupling) [START_REF] Blin | Plasma-Wave Detectors for TerahertzWireless Communication[END_REF]. Indeed, coupling through the gate induces amplification of the coupled signal. In order to protect the transistor from external coupling, we decide to place it in a guiding horn.

The wave-guide, set with a cut-off frequency of 0.174 THz, permits to realize a Faraday cage in order to block the major part of accidental coupling signals. The additional horn placed in front of the wave-guide, focuses the incoming THz radiation on the transistor, thus increasing its effective sensitivity. The latter ensures theoretical enhancement of 10 times for the detected signal at 0.3 THz. In this paper, all measurements considered have been realized using the transistor into the horn.

Transistor

G D S

Gate voltage

Horn

Electronic Source

Electrical spectrum analyzer

Fig. 1. Sensitivity characterization setup of pHEMT transistor. The transistor is placed respectively in front of 0.3 THz and 0.65 THz electronic THz tunable sources in order to measure sensitivity.

1) Transistors sensitivity:

A first experiment consists in measuring the GaAs pHEMT transistor sensitivity as a function of the THz frequency around 0.3 THz and 0.65 THz. As shown in figure 1, the setup for this sensitivity characterization consists of an electronic THz source and the transistor in the horn. The transistor is biased with an optimum gate-source voltage to increase its sensitivity and the rectified signal of the ac current induced by the incoming radiation is measured between drain and source with an electrical spectrum analyzer (ESA) [START_REF] Dyakonov | Generation and detection of Terahertz radiation by field effect transistors[END_REF]. To cover the large frequency domain, we use two electronic THz tunable sources: the frequency of the first one ranges from 0.22 THz to 0.34 THz, while the second one ranges from 0.64 THz to 0.69 THz. The power of both sources is around 1 mW. As can be seen in figure 2, the transistor has a sensitivity of about 1 V/W at 0.3 THz and about 1 mV/W at 0.65 THz. We also point out that the response is quite flat over 30 GHz, between 0.24 THz and 0.27 THz. These results suggest to use these transistors in THz communication systems. 2) Detector bandwidth measurement: A second experiment consists in measuring the detector bandwidth. In this study, we use an electronic THz source whose frequency is fixed at 0.324 THz, with an output power of about 1 mW. We make the modulation frequency varying from 100 kHz to 20 GHz in order to determine the modulation bandwidth of the detector (figure 3). Again the transistor is biased with an optimum gatesource voltage.

Transistor

G D S

Gate voltage

Horn

Figure 4 illustrates the variation of the amplitude of the detected signal as a function of the modulation frequency. The red curve represents the signal and noise amplitude, and the black curve is the measurement of the noise alone, when an opaque screen is positioned between the source and the transistor. The noise is the result of the white noise of the ESA and the coupled signal via the transistor, and the measured one. Signal-to-noise ratio is about 20 to 30 dB. This ratio can be improved by 15 dB when using an amplifier between the transistor drain and the ESA. We also notice that at 0.324 THz, we get a 10 GHz bandwidth at detection, what is confirmed by the green curve, which represents a first-order low-pass fit using a 10 GHz cut-off frequency.

III. TERAHERTZ WIRELESS COMMUNICATION

In this section, we discuss THz communication using the commercial pHEMT transistor in the horn, as well as the first transmission of pseudo-random bit sequence (PRBS). The experimental setup is presented in figure 5. The setup consists in a 0.2 THz electronic source whose amplitude is modulated using a PRBS pattern generator. The output source power is around 2 mW (measured with calibrated pyroelectric sensor). The beam is collimated and focused by off-axis parabolic mirrors, after propagation of 20 cm and detected by the pHEMT transistor in the horn. The transistor is biased with a gate-source voltage of -0.4 V to increase its sensitivity. The detected source-drain signal is amplified and observed on a high-speed oscilloscope. To evaluate the detected signal quality, eye-patterns are used and provide useful visual information when the data-rate frequency is varying and ensure the different amplifiers are operating properly.

First, we use an amplifier with 200 MHz bandwidth, 40 dB gain and 50 Ohm input impedance. We make the datarate frequency varying and we observe eye-patterns with the oscilloscope. As shown in figure 6, the eye opening is still wide up to data-rates of 0.250 Gbps. Using a 18 GHz, 50 Ohms amplifier, the data-rate was increased up to 1.5 Gbps, where the noise begins to degrade the eye-pattern (figure 7). We are working on optimized bias supply to reduce the coupled noise. In order to avoid this coupling, we will change gate supply from a source meter into an isolated battery, along with electronic devices for voltage adjustment.

IV. CONCLUSION

As a summary, we have investigated the performances of a GaAs pHEMT commercial transistor as THz detector. This transistor has been integrated in a wave-guide module, and has a sensitivity of about 1 V/W at 0.3 THz and of about 1 mV/W at 0.65 THz, and a 10 GHz modulation bandwidth at 0.324 THz with a signal-to-noise ratio pf about 20 dB. We have shown the importance of the integration of the commercial transistor into the guided horn, and high-data rate wireless communication was demonstrated for the first time using FETs as detectors at 0.2 THz. Data-rates up to 0.250 Gbps were observed for the first time. As such data-rates are likely limited noise due to direct coupling, enhanced data-rates are expected in the very near future thanks to careful electromagnetic isolation.

Fig. 2 .

 2 Fig. 2. Variation of the sensitivity, respectively at 0.3 THz and 0.65 THz of pHEMT transistor operating at room temperature. The response is quite flat on 30 GHz between 0.24 THz and 0.27 THz.

Fig. 3 .

 3 Fig.3. Setup of detector bandwidth characterization. The pHEMT transistor with horn is placed in front of a 0.324 THz source whose modulation frequency varies from 100 kHz to 20 GHz, in order to determine the bandwidth of the detector.

Fig. 4 .

 4 Fig.[START_REF] Moeller | 2.5 Gb/s error-free transmission at 625 GHz using a narrow-bandwidth 1 mW THz source[END_REF]. Variation of the amplitude of the detected signal by the pHEMT transistor as a function of the modulation frequency. The first graph gives a signal-to-noise ratio of about 20 to 30 dB using a FSU 46 spectrum analyzer with a resolution bandwidth of 10 Hz. The second graph illustrates a 10 GHz bandwidth at 0.324 THz.

Fig. 5 .

 5 Fig. 5. Wireless communication setup. The 0.2 THz electronic source is modulated using a PRBS pattern generator. Eye-patterns are observed on a high-speed oscilloscope at different data-rate frequencies.

Fig. 6 .

 6 Fig. 6. Wide and high eye openings for three different data-rate frequencies of pseudo-random bit sequence: (a) 10 Mbps. (b) 50 Mbps. (c) 250 Mbps.

Fig. 7 .

 7 Fig. 7. Disrupted eye-pattern by direct coupling from the pattern generator through the gate supply for three different data-rate frequencies of pseudorandom bit sequence: (a) 500 Mbps. (b) 1 Gbps. (c) 1.5 Gbps.

ACKNOWLEDGMENT

This work was supported by ANR-JST project WITH "WIreless communication using TeraHertz plasmonic-nano ICT devices", by the GDR-I project "Semiconductor sources and detectors of THz frequencies" and by the Groupement d'Intérêt Scientifique "Teralab". Authors thank the Languedoc-Roussillon region through the "Terahertz Platform" and the "HERMES Platform". We also thank the IEMN facilities, partially supported by the "Contrat de Plan Etat Région" (CPER) and the Nord Pas-de-Calais region.