New interaction estimates for the Baiti-Jenssen system

We establish new interaction estimates for a system introduced by Baiti and Jenssen. These estimates are pivotal to the analysis of the wave front-tracking approximation. In a companion paper we use them to construct a counter-example which shows that Schaeffer's Regularity Theorem for scalar conservation laws does not extend to systems. The counter-example we construct shows, furthermore, that a wave-pattern containing infinitely many shocks can be robust with respect to perturbations of the initial data. The proof of the interaction estimates is based on the explicit computation of the wave fan curves and on a perturbation argument.


Introduction
We deal with the system of conservation laws (1) ∂ t U + ∂ x F η (U ) = 0.
The unknown U = U (t, x) attains values in R 3 : and the flux function F η : R 3 → R 3 is defined as In the previous expression, the parameter η attains values in the interval [0, 1/4[ and to simplify the exposition we fix the functions p 1 and p 3 by setting Note, however, that for some of the results discussed in the following the precise expression of the functions p 1 and p 3 is irrelevant.
System (1), (2) was introduced by Baiti and Jenssen in [3,19] and it was used to construct an example of a Cauchy problem where the initial data have finite, but large, total variation and the L ∞ -norm of the admissible solution blows up in finite time. More recently, the authors of the present paper used the Baiti-Jenssen system (1) to exhibit an explicit counter-example which shows that Schaeffer's regularity result for scalar conservation laws does not extend to systems, see [11]. The counter-example we construct shows, furthermore, that a wave-pattern containing infinitely many shocks can be robust with respect to perturbations of the initial data. We refer to § 2.1 in the present paper for a brief overview of these counterexamples. See also [10].
This note aims at establishing new quantitative interaction estimates for the Baiti-Jenssen systems (1), (2). The estimates we obtain are pivotal to the analysis of the so-called wave front-tracking approximation of the Cauchy problem obtained by coupling (1) with an initial datum U (0, ·) = U 0 . We refer to [5,14,18] for an extended discussion on the wave front-tracking approximation. Here we only mention that the wave front-tracking algorithm is based on the construction of a piecewise constant approximation of the Cauchy problem. Under suitable conditions on the initial datum U 0 and on the flux function F η , one can show that the wave front-tracking approximation converges to an admissible solution of the Cauchy problem, see in particular the analysis in [5]. In [11] we construct wave fronttracking approximations of the Cauchy problems obtained by coupling (1) with suitable initial data. We then rely on the wave front-tracking approximation to establish qualitative properties of the limit solutions. In the following we do not consider all the possible interactions one has to handle when constructing the wave front-tracking approximation. We only discuss those that we encounter in [11] and that cannot be handled by relying on straightforward considerations on the structure of the flux F η .
Before going into the technical details, we make some further remarks. First, in the present note we fix a very specific system in the wider class considered in [3]. The motivation for this choice is twofold: i) it simplifies the notation and ii) the analysis in the present note is sufficient for the applications in [11]. Note, moreover, that in the proof of Lemma 1.1 we use (although not in an essential way) the exact expression of the function F η evaluated at η = 0. However, we are confident that our results can be extended to wider classes of systems of the type considered in [3].
Second, in this note the only occurrences where we explicitly use the precise expression of the functions p 1 and p 3 is in the results discussed in § 2.3. More precisely, the proofs of Lemmas 1.1 and 1.2 both rely on a perturbation argument: we first show that our system verifies the statement of the lemmas in the case η = 0 and then we show that the same holds provided η is sufficiently small. The proof of Lemma 1.2 is completely independent of the specific expression of p 1 and p 3 . In the perturbation argument in the proof of Lemma 1.1 we use some results from § 2.3, but we never directly use the specific expression of p 1 and p 2 .
Third, the Baiti-Jenssen (1) system in not physical, in the sense that it does not admit strictly convex entropies, see [3] for a proof. It is natural to wonder whether or not the results established in the present note can be extended to physical systems. Very loosely speaking, by combining Lemmas 1.2 and 1.1 below with the analysis in [11, §3.1-3.2] we get the following statement. Under suitable conditions, the only waves generated at the interactions between two shocks are shock waves, or, more precisely, no rarefaction waves are generated at the interaction between two shocks. There are actually several physical systems that share this property: for instance, one can consider the 2 × 2 example discussed by DiPerna in [15, §5] and assume that the data have sufficiently small total variation. We refer to [6, §4] for the analysis of shock interactions for this system. On the other hand, a much more challenging question is whether or not there is any physical system that exhibit the same behaviors as those discussed in [3,11]. In other words, one can wonder whether or not a physical system can i) exhibit finite time blow up or ii) violate the regularity prescribed, for scalar conservation laws, by Schaeffer's Theorem. To the best of the authors' knowledge, the answers to the above questions is presently open.
We now give some technical details about the estimates we establish. First, we point out that the Baiti-Jenssen system (1) is strictly hyperbolic in the unit ball, which amounts to say that the Jacobian matrix DF η admits three real and distinct eigenvalues for every U such that |U | < 1. Also, if η > 0 every characteristic field is genuinely nonlinear. In other words, let r 1 , . . . , r 3 denote the right smooth eigenvectors associated to the eigenvalues λ 1 , λ 2 , λ 3 . Then for some suitable constant c > 0 and for every i = 1, 2, 3 and |U | < 1. In the following, we distinguish three families of shocks: we term a given shock 1-, 2-or 3-shock depending on whether the speed of the shock is close to λ 1 , λ 2 or λ 3 . We also point out that establishing interaction estimates for system (1) boils down to the following. Consider the so-called Riemann problem, namely the Cauchy problem obtained by coupling (1) with an initial datum in the form where U , U r ∈ R 3 are constant states. The above problem admits, in general, infinitely many distributional solutions: we term admissible the solution constructed by Lax in the pioneering work [21], see § 2.2 for a brief overview. Establishing interaction estimates for (1) amounts to establish estimates on the admissible solution of the Riemann problem (1)-(7) in the case when U and U r satisfy suitable structural assumptions.
The first case we consider is the case of the interaction of two 2-shocks, see Figure 1, left part. In other words, we assume that there is a state U m ∈ R 3 such that • U and U m are the left and the right states of a Lax admissible 2-shock, • U m and U r are the left and the right states of a Lax admissible 2-shock and • the shock between U and U m has higher speed than the shock between U m and U r . We now give an heuristic formulation of our interaction estimate and we refer to § 3 for the rigorous statement, which requires some technical notation. Here we only point out that the strength of a shock is a quantity defined in § 2.2 which is proportional to the modulus of the difference between the left and the right state of the shock.
Fix a constant a such that 0 < a < 1/2 and set U := (a, 0, −a). Consider the interaction between two 2-shocks and assume that the states U and U r are sufficiently close to U . If the strengths of the interacting 2-shocks are sufficiently small, then the admissible solution of the Riemann problem (1)-(7) is obtained by patching together a 1-shock, a 2-shock and a 3-shock.
We remark that the relevant point in the above result is that the solution of the Riemann problem that we consider in the statement contains no rarefaction wave.
The second case we consider is the case of the interaction between a 1-shock and a 2-shock, see Figure 1, right part. In other words, we assume that there is a state U m ∈ R 3 such that • U and U m are the left and the right states of a Lax admissible 2-shock, • U m and U r are the left and the right states of a Lax admissible 1-shock. The case of the interaction of a 3-shock with a 2-shock is analogous. We now give an heuristic formulation of our result and we refer to § 4 for the rigorous statement. Lemma 1.2. Consider the interaction between a 1-shock and a 2-shock and assume both shocks have sufficiently small strength. Then the admissible solution of the Riemann problem (1)-(7) is obtained by patching together a 1-shock, a 2-shock and a 3-shock. Also, we establish quantitative bounds from above and from below on the strength of the outgoing shocks, see formulas (35).
Note that the fact that the three outgoing waves are shocks follows from the analysis in [3]. Also, the bound from above on the strength of the outgoing 3shocks follows from by now classical interaction estimates, see [5, Page 133, (7.31)]: the main novelty in Lemma 1.2 is that we have a new bound from below on the strength of the outgoing 3-shock, see the left hand side of formula (35). This estimate is important for the analysis in [11].
This note is organized as follows. In § 2 we go over some previous results. In particular, in § 2.1 we provide some motivation for studying the Baiti-Jenssen system (1) by describing two counter-examples that use it. In § 2.2 we recall some results from [21] and in § 2.3 we apply these results to the Baiti-Jenssen system. In § 3 we discuss the interaction of two 2-shocks and in § 4 the interaction of a 1-shock and a 2-shock.

Overview of previous results
For the reader's convenience, in this section we go over some previous results. More precisely: § 2.1: we discuss two counter-examples based on the Baiti-Jenssen system (1): the original one in [3] and a more recent one devised in [11]. § 2.2: we follow the famous work by Lax [21] and we outline the construction of the solution of the Riemann problem. § 2.3: we apply Lax's construction to the Baiti-Jenssen system.

2.1.
Counter-examples based on the Baiti-Jenssen system. This paragraph is organized as follows: § 2.1.1: we discuss the counter-example in [3] § 2.1.2: we discuss the counter-example in [11]. Before dealing with the specific examples, we recall two main features of the Baiti-Jenssen system: first, it is strictly hyperbolic, namely (5) holds. Note that strict hyperbolicity is a standard hypothesis for results concerning systems of conservation laws, see [14]. Also, if η > 0 every characteristic field is genuinely nonlinear, which means that (6) is satisfied for every i = 1, 2, 3. This is a remarkable property because loosely speaking systems where all the characteristic field are genuinely nonlinear are usually better behaved than general systems. For instance, the celebrated decay estimate by Oleȋnik [23], which applies to scalar conservation laws with convex fluxes, has been extended to systems of conservation laws where all the characteristic field are genuinely nonlinear, see for instance the works by Glimm and Lax [17], by Liu [22] and, more recently, by Bressan and Colombo [7], Bressan and Goatin [8] and Bressan and Yang [9], while for balance laws we refer to Christoforou and Trivisa [12].
2.1.1. Finite time blow up of admissible solutions with large total variation. Consider the general system of conservation laws where the unknown U (t, x) attains values in R N , the variables (t, x) ∈ [0, +∞[×R and the flux function F : R N → R N is smooth and strictly hyperbolic (5). Consider furthermore the Cauchy problem obtained by coupling (8) with the initial condition Under some further technical assumption on the structure of the flux, Glimm [16] established existence of a global in time solution of the Cauchy problem provided that TotVar U 0 , the total variation of the initial datum, is sufficiently small. Under the same assumptions, Bressan and several collaborators established uniqueness results, see [5] for a detailed exposition. The requirement that the total variation TotVar U 0 is small is highly restrictive, but necessary to obtain well-posedness results unless additional assumptions are imposed on the flux function F . Indeed, explicit examples have been constructed of systems and data U 0 where TotVar U 0 is finite, but large, and the admissible solution blows up in finite time. In particular, in [3] Baiti and Jenssen constructed an initial datum for system (1) such that the L ∞ -norm of the admissible solution blows up in finite time. The solution is admissible in the sense that it is piecewise constant and every shock is Lax admissible. For further examples of finite time blow up, see the references in [3] and [14].

Schaeffer's Regularity Theorem does not extend to systems. In [24]
Schaeffer established a regularity result which can be loosely speaking formulated as follows. Consider a scalar conservation law with strictly convex flux, namely equation (8) in the case when U (t, x) attains real values and F : R → R is uniformly convex, i.e. F ≥ c > 0 for some constant c > 0. The work by Kružkov [20] establishes existence and uniqueness of the so-called entropy admissible solution of the Cauchy problem posed by coupling (8) and (9). It is known that, even if U 0 is smooth, the entropy admissible solution can develop shocks, namely discontinuities that propagate in the (t, x)-plane. Schaeffer's Theorem states that, for a generic smooth initial datum, the number of shocks of the entropy admissible solution is locally finite. The word "generic" is here to be interpreted in a suitable technical sense, which is related to the Baire Category Theorem, see [24] for the precise statement.
In [11] we discuss whether or not Schaeffer's Theorem extends to systems of conservation laws where every characteristic field is genuinely nonlinear, namely (6) holds. Note that the assumption that every characteristic field is genuinely nonlinear can be loosely speaking regarded as the analogous for systems of the condition (which applies to scalar equations) that the flux is strictly convex. Indeed, regularity results for scalar equations with strictly convex fluxes have been extended to systems where every characteristic field is genuinely nonlinear: as we mentioned before, this is the case of Oleȋnik's [23] decay estimate, see for instance [7,8,9,12,17,22] for possible extensions to systems. Also, the SBV regularity result by Ambrosio and De Lellis [1], which applies to scalar conservation laws with strictly convex fluxes, has been extended to systems where every characteristic field is genuinely nonlinear, see [2,4,13].
Despite the above considerations, in [11] we exhibit an explicit example which rules out the possibility of extending Schaeffer's Theorem to systems of conservation laws where every characteristic field is genuinely nonlinear. More precisely, we construct a "big" set of initial data such that the corresponding solutions of the Cauchy problems for the Baiti-Jenssen system (1) develop infinitely many shocks on a given compact set of the (t, x)-plane. The term "big" is to be again interpreted in a suitable technical sense, which is related to the Baire Category Theorem, see [11] for the technical details.
2.2. The Lax solution of the Riemann problem. We consider a system of conservation laws (8) and we assume that F : R 3 → R 3 is strictly hyperbolic (5) and that every characteristic field is genuinely nonlinear, namely (6) holds for i = 1, 2, 3. Lemma 2.1 below states that the Baiti-Jenssen system satisfies these conditions. The Riemann problem is posed by coupling (8) with an initial datum in the form where U + and U − are given states in R 3 . In [21], Lax constructed a solution of the Riemann problem (8)-(10) under the assumptions that the states U + and U − are sufficiently close: we now briefly recall the key steps of the analysis in [21]. We fix i = 1, 2, 3 andŪ ∈ R 3 and we define the i-wave fan curve throughŪ by setting In the previous expression, R i is the i-rarefaction curve throughŪ and S i is the i-Hugoniot locus throughŪ . The i-rarefaction curve R i is the integral curve of the vector field r i , namely the solution of the Cauchy problem The i-th Hugoniot locus S i is the set of states that can be joined toŪ by a shock with speed close to λ i (Ū ). The i-Hugoniot locus S i is determined by imposing the Rankine-Hugoniot conditions. We term the value |s i | strength of the i-wave connecting the statesŪ (on the left) and D i [s,Ū ] (on the right). Note that, owing to (11), when s i > 0 the i-wave is a i-th rarefaction wave, when s i < 0 the i-wave is an i-shock satisfying the so-called Lax admissibility criterion. The solution of the Riemann problem (8)-(10) is computed by imposing and by using the Local Invertibility Theorem to solve for (s 1 , s 2 , s 3 ). From the value of (s 1 , s 2 , s 3 ) one can reconstruct a solution of the Riemann problem (8)-(10), see [21] for the precise construction. This solution is obtained by patching together rarefaction waves and shocks that satisfy the Lax admissibility criterion. In the following, we refer to this solution as the Lax solution of the Riemann problem (8)-(10).

2.3.
The wave fan curves of the Baiti-Jenssen system. We collect in this paragraph some features of the Baiti-Jenssen system. For the proof, we refer to [3,11].
The first result states that in the unit ball the Baiti-Jenssen system is strictly hyperbolic whenever 0 ≤ η < 1/4. Also, when η > 0 all the characteristic fields are genuinely nonlinear. Note that when η = 0 this last condition is lost because two characteristic fields became linearly degenerate. See [3] or [11] for the explicit computations.
We now discuss the structure of the wave fan curves. We start by giving the explicit expression of the 1-and the 3-wave fan curve. In the statement of the following result, we denote by (ū,v,w) the components of the stateŪ ∈ R 3 . Lemma 2.2. Consider the flux function (2), assume that 0 < η < 1/4 and fix U ∈ R 3 such that |Ū | < 1. Then the following properties hold true.
i) The 1-wave fan curve D 1 [σ,Ū ] is a straight line in the plane v =v, more precisely where Note that r 1 (Ū ) is the first eigenvector of the Jacobian matrix DF (Ū ). Also, the statesŪ (on the left) and D 1 [σ,Ū ] (on the right) are connected by a wave which is a 1-rarefaction wave when σ > 0, a Lax admissible 1-shock when σ < 0. ii) The 3-wave fan curve D 3 [τ,Ū ] is a straight line in the plane v =v, more precisely where The vector r 3 (Ū ) is the third eigenvector of the Jacobian matrix DF (Ū ). Also, the statesŪ (on the left) and D 3 [τ,Ū ] (on the right) are connected by a wave which is a 3-rarefaction wave when τ < 0, a Lax admissible 3-shock when τ > 0.
Note that, for the 3-wave fan curve, the positive values of τ correspond to shocks, the negative values to rarefaction waves. This is the contrary with respect to (11) and it is a consequence of the fact that we use the same notation as in [3,11] and we choose the orientation of r 3 in such a way that when η > 0 condition (6) is replaced by the opposite inequality ∇λ 3 · r 3 < 0. We now turn to the structure of the 2-wave fan curve. In the following statement, we use the notation Also, we consider entropy admissible solutions of scalar conservation laws, in the Kružkov [20] sense.

Lemma 2.3. Assume that U is a Lax solution of the Riemann problem (8)-(10).
Then the second component v is an entropy admissible solution of the Cauchy problem Also, we can choose the eigenvector r 2 and the parametrization of the 2-wave fan curve D 2 [s,Ū ] in such a way that the second component of D 2 [s,Ū ] is exactlyv + s.

Interaction of two 2-shocks
We first rigorously state Lemma 1.1 Lemma 3.1. There is a sufficiently small constant ε > 0 such that the following holds. Fix a constant a such that 0 < a < 1/2 and set U := (a, 0, −a). Assume that Assume furthermore that Then there are σ < 0 and τ > 0 such that Note that by combining (17) with the inequalities σ < 0, τ > 0 and s 1 + s 2 < 0 we get that the three outgoing waves are all shocks. The proof of Lemma 3.1 is organized as follows: § 3.1: by relying on a perturbation argument, we show that the proof of Lemma 3.1 boils down to the proof of the Taylor expansion (21). § 3.2: we complete the proof by establishing (21). 3.1. Proof of Lemma 3.1: first step. We start with some preliminary considerations. Assume that the states U and U r satisfy (16). Next, solve the Riemann problem between U (on the left) and U r (on the right): owing to [21], this amounts to determine by relying on the Local Invertibility Theorem the real numbers σ, s and τ such that Establishing the proof of Lemma 3.1 amounts to prove that s = s 1 + s 2 < 0 and that σ < 0, τ > 0.
To prove that s = s 1 + s 2 we recall Lemma 2.3 and the fact that the v component is constant along the 1-st and the 3-rd wave fan curves D 1 and D 3 . We conclude that s = v r − v = s 1 + s 2 < 0 . Note that v r and v are the second component of U r and U .
We are left to prove that σ < 0 and τ > 0. We first introduce some notation: we regard σ and τ as functions of η, s 1 and s 2 and U and we write σ η (s 1 , s 2 , U ) and τ η (s 1 , s 2 , U ) to express this dependence. Note that σ and τ depend on η because the wave fan curve D 2 depends on η.
Owing to the Implicit Function Theorem, the regularity of σ η (s 1 , s 2 , U ) and σ η (s 1 , s 2 , U ) is at least the same as the regularity of the functions D 1 , D 2 and D 3 . Also, note that the Lax Theorem [21] (see also [5, p.101]) states that the wave fan curves D 1 , D 2 and D 3 are C 2 . The reason why we can achieve C ∞ regularity is because we are actually considering the wave fan curves in regions where they are C ∞ . To see this, we first point out that, owing to (13) and (14), the wave fan curves D 1 , D 3 are straight lines and hence they are C ∞ . Next, we point out that we are only interested in negative values of s 1 + s 2 . Hence, we can replace the 2-wave fan curve D 2 defined as in (11) with the 2-Hugoniot locus S 2 . We recall that the 2-Hugoniot locus S 2 [s,Ū ] contains all the states that can be connected toŪ by a shock, namely all the states such that the couple (Ū , S 2 [s,Ū ]) satisfies the Rankine-Hugoniot conditions. The 2-Hugoniot locus S 2 [s,Ū ] is C ∞ and by combining all the previous observations we can conclude that σ η (s 1 , s 2 , U ) and τ η (s 1 , s 2 , U ) are both C ∞ with respect to the variables (η, s 1 , s 2 , U ).
Next, we discuss the partial derivatives of σ η (s 1 , s 2 , U ) and τ η (s 1 , s 2 , U ) with respect to (s 1 , s 2 ) at the point (η, 0, 0, U ). By arguing as in the proof of estimate (7.32) in [5, p.133] we conclude that • for every U , for every η > 0 and every integer k ≥ 1 we have the following equalities: • For every U and for every η > 0 we also have the following equality concerning the derivatives of second order: This implies that σ η and τ η admit the following Taylor expansions In § 3.2 we prove that when η = 0 and U = U the functions σ and τ admit the Taylor expansions Next, we use the Lipschitz continuous dependence of the derivatives of third order with respect to η and U and we conclude that provided that 0 ≤ η ≤ εa and |U − U | ≤ εa. In the above expression, C denotes a universal constant. By plugging the above expressions into (20) and recalling that s 1 , s 2 < 0 we can eventually conclude that, if ε is sufficiently small, then The proof of the lemma is complete.

3.2.
Proof of formula (21). The proof of the Taylor expansion (21)  Note that in this paragraph we always assume η = 0 because formula (21) deals with this case.

The 2-Hugoniot locus.
Before giving the technical results, we introduce some notation. First, we recall that we term F 0 the flux function F η in (2) in the case when η = 0. In the following, we will mostly focus on the behavior of the first and the third component of U . Hence, it is convenient to termÛ andF 0 the vectors obtained by erasing the second components of U and F 0 , respectively. We have the relation where we have also introduced the 2 × 2 matrix J(v).
Finally, we recall that we term S 2 [s,Ū ] the 2-Hugoniot locus passing throughŪ , namely the set of states that can be connected toŪ by a (possibly not admissible) shock of the second family. Also, as usual we denote byū,v andw the first, second and third component ofŪ , respectively. We use the notation Ū = (ū,w).
Proof The first and the third equations in the Rankine-Hugoniot conditions can be written as where J is the same as in (22). Next, we introduce the 2 × 2 matrix and we rewrite (26) as which implies (25) provided that By recalling that γ = 2v + s we can compute the explicit expression of the above matrices: The determinant of the matrix A(v + s, 2v + s) is and hence the matrix is invertible when |2v + s| < 4. We can now complete the lemma by computing the explicit expression of E, namely

3.2.2.
Conclusion of the proof of formula (21). We are now ready to establish (21). We first recall some notation: we consider the system of conservation laws with flux F 0 , see (2). We consider the collision between two 2-shocks and we assume that U = (a, 0, −a), U m and U r are the left, middle and right states before the interaction. This means that for some s 1 < 0, s 2 < 0 we have In the above expression, S 2 represents the 2-Hugoniot locus. To establish the last equality we used the fact that s 1 and s 2 are both negative. We plug (23) into (27) and we use the equality v = 0: we arrive at Next, we focus on the states after the interaction. By arguing as at the beginning of § 3.1, we conclude that it suffices to determine σ = σ 0 (s 1 , s 2 , U ) and τ = τ 0 (s 1 , s 2 , U ) such that By the explicit expression of D 1 and D 3 and by applying Lemma 3.2 we infer that the above equality implies U r = U + σ r 1 (0) + E(0, s 1 + s 2 ) U + σ r 1 (0) + τ r 3 (s 1 + s 2 ) = U + E(0, s 1 + s 2 ) U + I + E(0, s 1 + s 2 ) σ r 1 (0) + τ r 3 (s 1 + s 2 ) In the previous expression we denote by r 1 and r 3 the vectors obtained from r 1 and r 2 by erasing the second component. Also, we introduced the matrix H: its first column is I + E(0, s 1 + s 2 ) r 1 (0), the second column is r 3 (s 1 + s 2 ). In the following, we will prove that H(s 1 + s 2 ) is invertible provided that s 1 and s 2 are both sufficiently close to 0. By comparing (28) and (29) we then obtain (30) U .
Assume that we have established the following asymptotic expansion for G: Then by plugging both (31) and U = (a, −a) into (30) we obtain the asymptotic expansion (21). Hence, to conclude the proof of (21) we are left to establish (31). First, we point out that, owing to the expression of E in the statement of Lemma 3.2, This implies that when s 1 = s 2 = 0, the matrix E(0, s 1 + s 2 ) vanishes and hence We compute now the asymptotic expansion of E(0, s 1 ) + E(0 + s 1 , s 2 ) − E(0, s 1 + s 2 ) + E(0 + s 1 , s 2 )E(0, s 1 ).

Interaction of a 1-shock and a 2-shock
We first rigorously state Lemma 1.2.
Furthermore, assume that |U | < 1/2. Then there are real numbers σ and τ such that Note that (35) implies σ < 0 and τ > 0. If we combine these inequalities with (34) and s < 0 we see that the three outgoing waves are all shocks.
Establishing the proof of Lemma 4.1 amounts to establish (35). Indeed, (1) by using Lax's construction (see § 2.2) we determine σ , s , τ such that (2) By combining (13), (14) and Lemma 2.3 we obtain that s = s. To establish (35) we proceed as follows: § 4.1: we establish (35) in the case when η = 0. § 4.2: we conclude the proof by relying on a perturbation argument. More precisely, by using the fact that the flux F η in (2) smoothly depends on η we show that (35) holds provided η is sufficiently small. Note that, as we have mentioned in the introduction, the precise expression of the function p 1 and p 3 plays no role in the proof of Lemma 4.1, what is actually relevant is that estimates (35) hold at η = 0 with strict inequalities and that η is sufficiently small.

4.1.
Proof of Lemma 4.1: the case η = 0. We establish (35) in the case η = 0. This part of the proof is actually the same as in [3, p. 844-845], but for completeness we go over the main steps.
We term σ 0 , τ 0 the real numbers satisfying (34) when η = 0. Let v r and v denote the second components of U r and U , respectively. We term γ the speed of the incoming 2-shock (which is the same as the speed of the outgoing 2-shock), we recall Lemma 2.3 and the fact that the second component varies only across 2-shocks. We conclude that Since by assumption |U | < 1/2 and |s| < 1/4, then (36) |γ| < 3.
By imposing the Rankine-Hugoniot conditions on the incoming and outgoing 2shocks and by arguing as in [3, pp. 844-845], with the choice c = 4, we arrive at the following system: and (38) then the above linear system can be recast as AX 0 = Y . The explicit expression of the matrix A −1 is We solve for σ 0 and τ 0 and we obtain Next, we use [3, eq. (5.3)-(5.4)] and we recast the Rankine-Hugoniot conditions for 2-shocks as a nonlinear system in the form where A and Y are as in (37) and (38), respectively. Also, the vector X is defined by setting X := σ τ and the nonlinear term F(X, U , s, σ) is equal to In the above expression, the functions p 1 and p 3 are the same as in (3). Note, however, that the precise expression of p 1 and p 3 plays no role in the proof, the only relevant point is that p 1 and p 3 are both regular (say twice differentiable with Lipschitz continuous second derivatives). Note furthermore that we can regard F as a function of X, U , s and σ because, owing to (42) and (43), U m , U m and U m are functions of X, U , s and σ. Next, we rewrite equation (44) as where the vector X 0 = A −1 Y is given by (38) and (40). We now fix s, σ, η and |U | satisfying the assumptions of Lemma 4.1 and we define the closed ball (47) K := X = (σ , τ ) ∈ R 2 : |X − X 0 | ≤ kησs .
In the above expression, k > 0 is a universal constant that will be determined in the following and σ 0 and τ 0 are defined by (40). We also define the function T : R 2 → R 2 by setting (48) T (X) := X 0 − ηA −1 F(X, U , s, σ).
Assume that T is a strict contraction from K to K. Then the proof of Lemma 4.1 is complete: indeed, owing to (46) the fixed point X satisfies the Rankine-Hugoniot conditions (44). Also, owing to (41) and to (47) we infer that the inequalities (35) are satisfied provided that the parameter η is sufficiently small.

4.2.2.
Conclusion of the proof of Lemma 4.1. In this paragraph we prove that the map T defined by (48) is a strict contraction on the closed set K defined by (47). First, we make some remarks about notation. To simplify the exposition, in the following we denote by C a universal constant: its precise value can vary from occurrence to occurrence. Also, in the following we will determine the constant k in (47) and then choose the constant η in such a way that kη ≤ 1. This choice implies in particular that, when X belongs to the set K defined as in (47) and the hypotheses of Lemma 4.1 are satisfied, then the map F attains values on a bounded set and so F and all its derivatives are bounded by some constant C. Finally, note that, if X ∈ K and the hypotheses of Lemma 4.1 are satisfied, then |A −1 | ≤ C.
We now proceed according to the following steps.
Step 1: we point out that to show that the map T is a contraction it suffices to show that ≤ η|A −1 ||F(X 1 , U , s, σ) − F(X 2 , U , s, σ)| provided that the constant η is sufficiently small. This implies that T is a contraction and concludes the proof of Lemma 4.1.
By using an analogous argument, we control the second component of F and we arrive at (50) |F(X, U , s, σ)| ≤ C|σ|.
Next, we point out that when s = 0 we have U m = U m and U = U m and by using again the Lipschitz continuity of the functions p 1 and p 3 we conclude that (51) |F(X, U , s, σ)| ≤ C|s|.
Finally, we use the regularity of the function F and, by arguing as in the proof of [5, Lemma 2.5, p. 28], we combine (50) and (51) to obtain (49). This concludes the proof of Lemma 4.1.