Strong stability with respect to weak limits for a hyperbolic system arising from gas chromatography

Du Lac Bourdarias

Description

Niveau: Supérieur, Licence, Bac+1
Strong stability with respect to weak limits for a hyperbolic system arising from gas chromatography C. Bourdarias ?, M. Gisclon †and S. Junca ‡ November 23, 2009 Abstract We investigate a system related to a particular isothermal gas-solid chromatography process, called “Pressure Swing Adsorption”, with two species and instantaneous exchange kinetics. This system presents the particularity to have a linearly degenerate eigenvalue: this allows the velocity of the gaseous mixture to propagate high frequency waves. In the case of smooth concentrations with a general isotherm, we prove L∞ stability for concentrations with respect to weak limits of the inlet boundary velocity. Using the Front Tracking Algorithm (FTA), we prove a L1 stability for concentrations with bounded variation (BV) under some convex assumptions on the isotherms. In both cases we show that high frequency oscillations with large amplitude of the inlet velocity can propagate without affecting the concentrations. Key words: systems of conservation laws, boundary conditions, BV estimates, entropy solu- tions, linearly degenerate fields, convex isotherms, Front Tracking Algorithm, waves interaction, geometric optics. MSC Numbers: 35L65, 35L67, 35Q35. 1 Introduction “Pressure Swing Adsorption (PSA) is a technology used to separate some species from a gas under pressure according to the molecular characteristics and affinity of the species for an adsorbent material.

• propagate high


• convex isotherm


• velocity without


• any equilibrium


• velocity


• velocity can


• isotherm related


• l∞


• system


• strictly convex

Sujets

• Tracking
• Jean Gisclon
• Velocity
• System

Informations

• Publié par: profil-ontoe-2012
• Nombre de lectures: 6
• Langue: English

Extrait

Strong stability with respect to weak limits for a hyperbolic system arising from gas chromatography C. Bourdarias , ∗ M. Gisclon † and S. Junca ‡

November 23, 2009

Abstract

We investigate a system related to a particular isothermal gas-solid chromatography process, called “Pressure Swing Adsorption”, with two species and instantaneous exchange kinetics. This system presents the particularity to have a linearly degenerate eigenvalue: this allows the velocity of the gaseous mixture to propagate high frequency waves. In the case of smooth concentrations with a general isotherm, we proveL ∞ stability for concentrations with respect to weak limits of the inlet boundary velocity. Using the Front Tracking Algorithm (FTA), we prove aL 1 batstiliy for concentrations with bounded variation (BV) under some convex assumptions on the isotherms. In both cases we show that high frequency oscillations with large amplitude of the inlet velocity can propagate without affecting the concentrations.

Key words:systems of conservation laws, boundary conditions, BV estimates, entropy solu- tions, linearly degenerate fields, convex isotherms, Front Tracking Algorithm, waves interaction, geometric optics. MSC Numbers:35L65, 35L67, 35Q35.

1 Introduction

“Pressure Swing Adsorption (PSA) is a technology used to separate some species from a gas under pressure according to the molecular characteristics and affinity of the species for an adsorbent material. Special adsorptive materials (e.g. zeolites) are used as a molecular sieve, preferentially adsorbing the undesired gases at high pressure. The process then swings to low pressure to desorb the adsorbent material” (source: Wikipedia). A typical PSA system involves a cyclic process where a number of connected vessels containing adsorbent material undergo successive pressurization and depressurization steps in order to produce a continuous stream of purified product gas. We focus here on a step of the cyclic process, restricted to isothermal behavior. As in general fixed bed chromatography, each of thedspecies (d≥2) simultaneously exists under two phases, a gaseous and movable one with velocityu(t, x) and concentrationc i (t, x) or a solid (adsorbed) with concentrationq i (t, x), 1≤i≤d. We assume that mass exchanges between the mobile and the stationary phases are infinitely fast, thus the two phases are constantly at composition equilibrium: the concentrations in the solid phase are given by some relationsq i = q i ∗ (c 1 , ..., c d ) where the functionsq i ∗ are the so-called equilibrium isotherms. A theoretical study of a model with finite exchange kinetics was presented in [6] and a numerical approach was developed in [7]. ∗ UMA,NRRC12S5737,[e-mail],LievoSade´eitsrevinU † eiovrf.inu@as-vnUt´sierivoiaveSedU,AMAL,e15SRNCRM337627,7urgeLeBoL-ca-tudlcnog.si ‡ erivt´siFMIUUnetaLobJ.DAdeNeci,e6621,Par,CNRSUMRN,80160,[e-mail]

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In gas chromatography, velocity variations accompany changes in gas composition, especially in the case of high concentration solute: it is known as the sorption effect. In the present model, the sorption effect is taken into account through a constraint on the pressure (or on the density in this isothermal case). See [33] and [37] for a precise description of the process and [11] for a survey on various related models. The system for two species (d= 2) with three unknowns (u, c 1 , c 2 ) is: ∂ t (c 1 +q 1 ∗ (c 1 , c 2 )) +∂ x (u c 1 ) = 0,(1) ∂ t (c 2 +q ∗ 2 (c 1 , c 2 )) +∂ x (u c 2 ) = 0,(2) c 1 +c 2 =ρ(t),(3) with suitable initial and boundary data. The functionρrepresents thegiventotal density of the mixture. The experimental device is realized so that it is a given function depending only upon time and in the sequel we assume that

ρ≡1,

(4)

(which is not really restrictive from a theoretical point of view). Notice that we seek positive solutions (c 1 , c 2 ), thus, in view of (3) with (4),c 1 ,c 2 must satisfy 0≤c 1 ≤1,0≤c 2 ≤1.(5) Let us rewrite system (1), (2), (3) as a 2× For this purpose, we use the2 system as in [9, 11]. following notations, introduced in [10]: we setc=c 1 ∈[0,1], thenc 2 =ρ−c 1 = 1−c, and we set q i (c) =q i ∗ (c,1−c), i= 1,2, h(c) =q 1 (c) +q 2 (c), I(c) =c+q 1 (c).

Adding (1) and (2) we get, thanks to (3):

∂ t (q 1 (c) +q 2 (c)) +∂ x u= 0, thus our purpose is to study the following system: ( ∂ t I ∂ t ( h c ( ) c+ + ) ∂ x ∂ ( x cu u 0)= =0 , , (6) supplemented by initial and boundary data:  c(0, x) =c 0 (x)∈[0,1] >, x0,   (7) c(t,0) =c b (t)∈[0,1] >, t0,   u(t,0) =u b (t)>0, t >0. Notice that we assume in (7) an incoming flux at the boundary, i.e.∀t >0, u b (t)>0. In the case where the first species is inert, that isq 1 = 0, theIfunction reduces to identity. First existence results of solutions, with large data, which satisfy some entropy criterium in the case of two chemical species were obtained in [9, 10] for system (6) with initial and boundary dataBVfor the concentration and onlyL ∞ boundary data for the velocityu b  the. Furthermore, velocityubecomesBVwith respect toxin the quartert >0 andx > we have But,0, see [9, 10]. no gain of regularity with respect to time for the velocity. Indeed the following simple example is illuminating. Letcbe a constant andu b anyL ∞ positive function, then (c(t, x), u(t, x)) = (c, u b (t))

2

is a weak entropy solution of (6). So strong singularities with respect to time for the velocityuare expected, and these singularities do not seem affect the concentration. In this paper, we study more precisely the specific structure of the velocity in two cases, namely the case with smooth concentrations and the more complicated case withBVconcentrations. The specific structure of velocity has an interesting application, the stability in strong topologies of concentrations with respect to weak ∗  Forlimit for the incoming velocity. instance, physically, a high oscillating velocity can be replaced by its mean to compute the concentration. It is well known in the chromatography field that there are models with constant velocity ([37, 34, 11]). In the first case (smooth concentration), if the velocityu(., .) is also smooth, we have already proven in [9] that the system (6) reduces to a nonlinear scalar conservation law for the concentration. Indeed, for weak entropy solution with lipschitz concentration but onlyL ∞ velocity, the reduction to one equation forc(., . Furthermore) remains true, see Section 3. the velocity isedstratifiin the following sense:  u v ( ( ,xt .,.assia)rgearlsu )=u b (t) c( ×v .,. ( ) t,x . ), (8)

There is no restriction on the isothermsq 1 , q 2 for such phenomenons. only limitations are The the local existence of smooth concentration before shocks. Thus with regular concentration (Lipschitz or with bounded variation) the principal singularity of u(., .) comes from its boundary value and it is a temporal singularity. This decomposition for the velocity is a key ingredient to pass to the weak limit foruand to the strong limit for the concen- tration. This also allows to propagate high oscillations with large amplitudes for velocity without affecting the concentration.

Does this structure (8) remain true after shock waves? In the second case, we try to generalize the structure (8) and its consequences for concentrationsc(., .) inBV general entropy, i.e. for solutions (c, u the realistic case with shock-waves, we restrict ourselves to the For) of system (6). classical treatment of hyperbolic systems. That is to say we assume thateigenvalues are linearly de- generate or genuinely nonlinear. This restriction implies some convex assumptions on the isotherms q 1 , q 2 . Nevertheless, this limitation allows us to use the classical Front Tracking Algorithm (FTA see [13]) which gives new estimates for the velocity. For instance, if the boundary velocity belongs toBVwith respect  we obtain betterto time, the velocity has the same smoothness. Furthermore interaction estimates when the shock and rarefaction curves are monotonic in coordinates (c,lnu). It is the case for instance for an inert gas and an active gas with the Langmuir isotherm. Above all we prove thatu/u b isBVboth in time and space with only ln(u b )∈L ∞ andc 0 , c b ∈BV. With, this new estimate for velocity, we again get the stratified structure (8). This decomposition for the velocity yields a strong stability inL 1 for concentration. general weak entropy solu- For tions of quasilinear system (6) high oscillations with large amplitude for velocity can propagate as in semilinear systems, see for instance [25, 26, 28], with strong profile foruwith double scale as in [27].

We conjecture that our result is still valid for general isotherms without convex restrictions.

Why does velocity of system (6) has such simple structure (8)? There are two fundamental reasons to explain it. First the system (6) has a constant zero eigenvalue which is linearly degenerate (since it is constant). Second the nonlinear system is linear with respect tou. The system expounded in [4] also has a null eigenvalue, an evident stratified component, but, instead [4], we cannot reduce system (6) to a single equation for solutions with shocks. In [3] is studied another interesting system with linearly degenerate eigenvalues which modelises some traf- fic flows, but it is a Temple system ([40, 41]). As in [18, 19, 17, 16, 31], the zero eigenvalue makes the existence of stratified solutions possible or the propagation of large-amplitude high frequency waves. For genuinely nonlinear conservation laws, it is the converse since only high oscillating solutions with small amplitude can propagate: see for instance [23, 15]. Moreover, the linearity of the system (6) with respect to the velocity helps to pass to the weak limit foruand to obtain a strong stability result for the concentration. Furthermore, entropies are also linear with respect to

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u, see section 2.

This system has strong features. Is it a Temple system? Generally it is not a Temple system, see section 2.2 for some comments and [12] for a detailled study about the intersection beetween the Temple class and systems (6). In [12] we construct an entropy solution withL ∞ data exhibiting a blow up. Obviously from [5], such blow up is impossible for Temple system.

The paper is organized as follows: In Section 2 we recall some basics results from [10] concerning hyperbolicity, entropies, weak entropy solutions of System (6). In Section 3, we study the case where the concentration is smooth and velocity is onlyL ∞ . In the remainder of the paper we study the case with onlyBV short sectionconcentrations. In 4 we briefly expound the Front Tracking Algorithm (FTA) for System (6). Section 5 is devoted to the study of both shock and rarefaction curves. We state the assumptions that we need to perform estimates with the Front Tracking Algorithm. These assumptions restrict us to convex (or concave) isotherms and we give some examples from chemistry. We obtain the fundamental interaction estimates in Section 6 andBVestimates forv Finally,in Section 7. we obtain strong stability for concentration with respect to weak limit on the boundary velocity in Section 8.

2 Hyperbolicity and entropies

In order for this paper to be self contained, we recall without any proof some results expounded in [10]. It is well known that it is possible to analyze the system of Chromatography, and thus System (6), in terms of hyperbolic system of P.D.E. provided we exchange the time and space variables and u >0: see [34] and also [36] for instance. In this framework the vector state will beU=  m u  wherem=u cis the flow rate of the first species. In this vector state,umust be understood as u ρ, that is the total flow rate. In the sequel, we will make use of the functionf=q 1 c 2 −q 2 c 1 introduced by Douglas andal. in [29], written here under the form

f(c) =q 1 c 2 −q 2 c 1 =q 1 (c)−c h(c).(9) Any equilibrium isotherm related to a given species is always increasing with respect to the corre- ∗ sponding concentration (see [29]) i.e. c∂ q∂ i ≥0. Sincec=c 1 andc 2 = 1−c, it follows: i q 1 0 ≥0≥q 0 2 .(10)

Let us define the functionHby

H(c) = 1 + (1−c)q 0 1 (c)−c q 2 0 (c) = 1 +q 1 0 (c)−ch 0 (c). From (10),HsatisfiesH≥and we have the following relation between1 f,Handh: f 00 (c) =H 0 (c)−h 0 (c).

2.1 Hyperbolicity

Concerning hyperbolicity, we refer to [21, 38, 39]. System (6) takes the form u ∂ x U+∂ t Φ(U) = 0 withU=  m  and Φ(U) =   h I ( ( /um /um) )   .

4

(11)

(12)

The eigenvalues are:

0 andλ= H(c) , u thus in view of (11) the system is strictly hyperbolic. The zero eigenvalue is of course linearly degenerate, moreover the right eigenvectorr=  1 + h 0 q (c 1 0 ) (c)  associated toλsatisfiesdλ∙r= H u ( 2 c) f 00 (c), thusλis genuinely nonlinear in each domain wheref 00 6= 0.

Proposition 2.1 ([10] Riemann invariants) System (6) admits the two Riemann invariants:

candw= ln(u) +g(c) =L+g(c),whereg 0 (c) = −h H 0 ( ( c c ) ) andL= ln(u).

Furthermore this system can be rewritten for smooth solutions as:

2.2 Entropies

. ∂ x c+ H(c u ) ∂ t c= 0, ∂ x (ln(u) +g(c)) =∂ x w= 0

Dealing with entropies, it is more convenient, as shown in [10], to work with the functions

(13)

G(c) = exp(g(c)), W= exp(w) =u G(c). Notice thatGis a positive solution ofH G 0 +h 0 G= 0. DenoteE(c, u) any smooth entropy andQ=Q(c, u Then, for smooth) any associated entropy flux. solutions,∂ x E+∂ t Q= 0. Moreover: Proposition 2.2 ([10] Representation of all smooth entropies) The smooth entropy functions for System (6) are given by

E(c, u) =φ(w) +u ψ(c)

whereφandψare any smooth real functions. The corresponding entropy fluxes satisfy Q 0 (c) =h 0 (c)ψ(c) +H(c)ψ 0 (c).

Moreover, in [8], the authors looked for convex entropies for System (12) (i.e. System (6) written in the (u, m) variables) in order to get a kinetic formulation. The next proposition gives us a family of degenerate convex entropies independently of convex assumption on the functionfor on the isotherms.

Proposition 2.3 ([10] Existence of degenerate convex entropies) Ifψis convex or degenerate convex, i.e.ψ 00 ≥0, thenE=u ψ(c)is a degenerate convex entropy.

There are some few cases (water vapor or ammonia for instance) where the isotherm is convex. There is also the important case with an inert carrier gas and an active gas with a concave or convex isotherm (see [9, 10, 11]). In these cases, the next proposition ensures the existence of λ In such cases,-Riemann invariants which are also strictly convex entropies.wis monotonic with respect toxfor any entropy solution.

Proposition 2.4 ([10] Whenλ-Riemann invariant is a convex entropy) There are strictly convex entropy of the formE=φ(w)if and only ifG 00 does not vanish. More precisely, forα >0,E α (c, u) =u α G α (c)is an increasing entropy with respect to the Riemann invariantW convex for. It is strictlyα >1ifG 00 >0and forα <1ifG 00 <0.

5

Unfortunately, whenGhas an inflexion point such system does not admit any strictly convex entropy. When one gas is inert, it is always the case if the sign of the second derivative of the isotherm changes. See for instance [10] for the BET isotherm. wis a Riemann invariant associated with a linearly degenerate eigenvalue, namely 0, then through a contact discontinuity∂ x w= 0, see for instance Theorem 8.5.2 page 223 from [21] (the reader can also directly check it). Thus through a contact discontinuity∂ x φ(w) = 0 for anyφ without any convex assumption. Does this equality remains valid through aλ−shock? Whenλ is genuinely nonlinear (f 00 does not vanish), the answer is affirmative if System (6) is a Temple System ([40, 41, 12]). We recall that a 2×2 strictly hyperbolic system is a Temple system if there exists a system of coordinates consisting of Riemann invariants and if shock and rarefaction curves coincide. For instance, System (6) with two linear isotherms is a Temple system. More generally System (6) is a Temple system if∂ x w= 0 through anyλ−shock and any entropy solution. Indeed wlevel curves are the rarefaction curves, and∂ x w= 0 through anyλ−shock means that rarefaction and shock curves coincide. The reader interested by this subject could consult [12], more precisely Lemma 3.1 therein. But unfortunately we have the following remark proven in [12]. Remark 2.1 ([12])In general, System (6) is not a Temple system.

Proposition 2.5 ([10] Non Existence of strictly convex entropy) If sign ofG 00 changes then System (6) does not admit strictly convex smooth entropy.

2.3 Definition of weak entropy solution

We have seen that there are two families of entropies:u ψ(c) andφ(u G(c)). The first family is degenerate convex (in variables (u, uc)) providedψ 00 ≥0. So, we seek after weak entropy solutions which satisfy∂ x (u ψ(c)) +∂ t Q(c)≤0 in the distribution sense. The second family is not always convex. There are only two interesting cases, namely±G 00 (c)>0 for allc∈[0,1]. WhenG 00 >0 andα >1, we expect to have∂ x (u G(c)) α ≤0 from Proposition 2.4. But, the mappingW7→W α is increasing onR + . So, the last inequality reduces to∂ x (u G(c))≤0. In the same way, ifG 00 <0, we get∂ x (u G(c))≥0. Now, we can state a mathematical definition of weak entropy solutions. Definition 2.1Let beT >0,X >0,u∈L ∞ ((0, T)×(0, X),R + ),0≤c(t, x)≤ρ≡1for almost all(t, x)∈(0, T)×(0, X). Then(c, u)is aweak entropy solutionof System (6)-(7) with respect to the family of entropiesu ψ(c)if, for all convex (or degenerate convex)ψ: ∂ ∂ x (u ψ(c)) +Q t∂ ∂ (c)≤0,(14) inD 0 ([0, T[×[0, X[), whereQ 0 =H ψ 0 +h 0 ψ, that is, for allφ∈ D([0, T[×[0, X[): Z 0 X Z 0 T (u ψ(c)∂ x φ+Q(c)∂ t φ)dt dx+ Z 0 T u b (t)ψ(c b (t))φ(t,0)dt+ Z X Q(c 0 (x))φ(0, x)dx≥0. 0 Remark 2.2If±G 00 ≥0thenu ψ=±u G(c)is a degenerate convex entropy, with entropy flux Q≡0, contained in the family of entropiesu ψ(c). So, ifG 00 keeps a constant sign on[0,1],(c, u) has to satisfy: ± x∂ ∂ (u G(c))≤0,if±G 00 ≥0on[0,1].(15) Notice that the entropiesu ψ(c)and the entropyu G(c)are linear with respect to the velocityu.

2.4 About the Riemann Problem

The implementation of the Front Tracking Algorithm used extensively from Section 4 requires some results about the solvability of the Riemann problem. We first recall the solution of the boundary Riemann Problem (6), (16). c(0, x) =c 0 ∈[0,1] >, x0,  u c( ( , t, t) 0) 0 = = u c + + ∈ > [0 0 , , 1]  > , t 0.(16)

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We restrict ourselves for the casef 00 6= 0, thus there is noλ−discontinuity wave and no composite wave as in [10]. The complete Riemann Problem (6), (17)  c u( (t t, ,= )= )0 0u c − − ∈ > 0[ 0, ,1]  < , t 0,  u c( ( , t, t = =) 0) 0c u + + ∈ > [ 0 0 , ,1]  > , t 0.(17) is simply solved by a 0-contact discontinuity wave followed by aλ−wave given by following propo- sitions.

For the boundary Riemann Problem (6), (16), we are classically looking for a selfsimilar solution, i.e.:c(t, x) =C(z),u(t, x) =U(z) withz= x t >0 (see Fig. 1). In the domaint >0, x >0, the boundary Riemann problem is solved with aλ−wave sinceλ is the only positive eigenvalue of the system. Let us recall the following results obtained in [10], wherefandHare given by (9, 11).

Proposition 2.6 (λ−rarefaction waves) Any smooth non-constant self-similar solution(C(z), U(z))of (6) in an open domain Ω ={0≤α < z < β}wheref 00 (C(z))does not vanish, satisfies: dC H(C) dz z f 00 (C) , U(z) = H( z C) . = In particular, d Cd has the same sign asf 00 (C). z Assume for instance that0≤a < c 0 < c + < b≤1andf 00 >0in]a, b[. Then the only smooth self-similar solution of (6) is such that :  C(z) =c 0 0< z < z 0 , ,   zd Cd = f H z 0 ( 0 ( C C ) ) , z 0 < z < z + , (18)   C(z) =c + , z + < z, wherez + = H(c u + + ) ,z 0 =z + e −Φ(c + ) withΦ(c) = Z c 0 c H f 00 ( ( ξ ξ ) ) dξ. Moreoveru 0 = z H(c 0 0 ) andUis given by:   U(z) =u 0 ,0< z < z 0 , U H(C(z)) , z 0 < z < z +    (z) =z , (19) U(z) =u + z + < z. Proposition 2.7 (λ−shock waves)If(c 0 , c + )satisfies the following admissibility condition equiv- alent to the Liu entropy-condition ([30]): f(c + )−f(c 0 ) (c 0 ) for allcbetweenc 0 andc + c , + −c 0 ≤ f( c c) − − c f 0 , then the boundary Riemann problem (6),(16)is solved by a shock wave defined as: C(z) =  c c + 0 fi fi0 s ,s<<z U, z< (z) =  u u 0 + if if 0 z<, <z<s, s (20) whereu 0 and the speedsof the shock are obtained through 1 +h + u [ 0 f [ ] c 1 + ]u + 0 h 0 =s= u [ + f [ ] c ] + + , u where[c] =c + −c 0 ,[f] =f + −f 0 =f(c + )−f(c 0 ),h + =h(c + ),h 0 =h(c 0 ).

A 0−wave always appears on the line{t= 0}. Proposition 2.8 (0−contact discontinuity waves)Two distinct statesU − andU 0 are nected by a0−contact discontinuity if and only ifc − =c 0 (with of courseu − 6=u 0 ).

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con-

3

Case with smooth concentration

System (6) has the strong property that there exist weak entropy solutions withsmoothconceartn- tionc(., .) on (0, T)×(0, X) but not necessarily smooth velocityu(., .), for some positive constants TandX. Furthermore,c(., .) is the solution of a scalar conservation law.

3.1 Existence of weak entropy solutions with smooth concentration

About existence of weak entropy solutions of some hyperbolic systems with one component less smooth than the others, we refer the reader to [18, 19, 31]. Here, we obtain by the classical method of characteristicsexistence and uniquenessof weak entropy solutions with smooth concentration andL ∞  have a similar result in [9] but only with smooth velocity.velocity. We

Theorem 3.1 (Unique weak entropy solution withsmoothcnoectnarit)on LetT 0 >0,X >0,c 0 ∈W 1,∞ ([0, X],[0,1]),c b ∈W 1,∞ ([0, T 0 ],[0,1]),lnu b ∈L ∞ ([0, T 0 ],R). Ifc 0 (0) =c b (0)then there existsT∈]0, T 0 ]such that System (6)-(7) admits auniqueweak entropy solution(c, u)on[0, T]×[0, X]with c∈W 1,∞ ([0, T]×[0, X],[0,1]),lnu∈L ∞ ([0, T], W 1,∞ ([0, X],R)). Furthermore, for anyψ∈C 1 ([0,1],R), setting F 0 (c) = (H(c)G(c)) −1 andQ 0 =H ψ 0 +h 0 ψ,

(c, u)satisfies:

∂ x (u ψ(c)) +∂ t Q(c) = 0, ∂ x (u G(c)) = 0 ,

(21)

∂ t c+u b (t)G(c b (t))∂ x F(c) = 0.(22) Equations (21) mean that entropy inequalities become equalities. This fact easily implies the strat- ified structure (8) of the velocity, see equation (23) below. Notice that system (6) degenerates into scalar equation (22).

Proof:We build a solution using the Riemann invariants and we check that such a solution is an entropy solution. Next, we prove uniqueness. Using the Riemann invariantW=u G(c) (∂ x W= 0) and the boundary data we defineuby: u(t, x) = u b (t G ) (c( G t (c x, b ( )) t)) ,(23) souis smooth with respect tox. Then, the first equation of (13) can be rewritten as follows: ∂ t c+µ ∂ x c= 0,withµ=λ −1 = u H(c ) = u b ( H t ( ) c G ) ( G c b ( ( t c )) ) =µ(t, c).(24) We solve (24) supplemented by initial-boundary value data (c 0 , c b ) by the standard characteristics method. Let us define, for a given (τ, x),X(∙, τ, x) as the solution of: dX( d ,x,τs s ) =µ(s, c(s, X(s, τ, x))), X(τ, τ, x) =x.

Since dc ds (s, X(s, τ, x)) = 0 from (24), we have

X(s, τ, x) =x−b(s, τ)F 0 (c(τ, x))

with

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b(s, τ) = Z τ s u b (σ)G(c b (σ))dσ.