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

J Ér

Émy Ruiz
email: ruiz@math.univ-bpclermont.fr

TWO LIMIT CASES OF BORN-INFELD EQUATIONS

Keywords: Born-Infeld equations, high and low field limits, classical Maxwell equations, pressureless magnetohydrodynamics system Mathematics Subject Classification 2000 : 35B40, 35L60, 35Q35

We study two limit cases λ → ∞ and λ → 0 in Born-Infeld equations.

Here the parameter λ > 0 is interpreted as the maximal electric field in the electromagnetic theory and the case λ = 0 corresponds to the string theory. Formal limits are governed by the classical Maxwell equations and pressureless magnetohydrodynamics system, respectively. For studying the limit λ → ∞, a new scaling is introduced. We give the relations between these limits and Brenier high and low field limits. Finally, using compensated compactness arguments, the limits are rigorously justified for global entropy solutions in L ∞ in one space dimension, based on derived uniform estimates and techniques for linear Lagrangian systems.

Introduction

The Born-Infeld system is a nonlinear version of classical Maxwell equations. It was introduced by Born and Infeld in 1930's to describe physical phenomena of electromagnetism [START_REF] Born | Foundation of the new field theory[END_REF]. Recently, the Born-Infeld system has attracted considerable attention because of its new applications in the string theory and high energy physics. See Boillat [START_REF] Boillat | Non linear hyperbolic fields and waves[END_REF], Brenier [START_REF] Brenier | Hydrodynamic structure of the augmented Born-Infeld equations[END_REF][START_REF] Brenier | A note on deformations of 2D fluid motions using 3D Born-Infeld equations[END_REF] and Gibbons [START_REF] Gibbons | Aspects of Born-Infeld theory and string / M-theory[END_REF].

In the electromagnetic theory, E and B stand for the electric and magnetic fields in R 3 . The Born-Infeld system is defined through the Born-Infeld Lagrangian L λ and the Faraday law. The expression of L λ is (see [START_REF] Brenier | A note on deformations of 2D fluid motions using 3D Born-Infeld equations[END_REF]) :

(1.1) L λ (B, E) = -λ 2 + |B| 2 -|E| 2 - (B • E) 2 λ 2 ,
where "•" stands for the inner product and | • | the Euclidean norm. The single parameter λ > 0 can be interpreted as the maximal electric field.

Introduce the electric induction D = ∂L λ ∂E , the Poynting vector P and the Born-Infeld energy density h [START_REF] Born | Foundation of the new field theory[END_REF][START_REF] Brenier | Hydrodynamic structure of the augmented Born-Infeld equations[END_REF][START_REF] Gibbons | Born-Infeld theory and stringy causality[END_REF] :

(1.2)

P = D × B, h = λ 2 + |B| 2 + λ 2 |D| 2 + |D × B| 2 .
1 Then (1.

3) E = ∂h(D, B) ∂D = λ 2 D + B × P h and the Born-Infeld equations, depending on the parameter λ > 0, read :

(1.4)

∂ t D + ∇ × -B + D × P h = 0, (1.5) ∂ t B + ∇ × λ 2 D + B × P h = 0,
The differential constraints are :

(1.6)

∇ • D = ∇ • B = 0,
which are compatible with equations (1.4)- (1.5). Furthermore, h and P satisfy the additional conservation laws :

(1.7) ∂ t h + ∇ • P = 0, (1.8)

∂ t P + ∇ • P ⊗ P -B ⊗ B -λ 2 D ⊗ D h -∇ λ 2 h = 0.
System (1.4)-(1.6) of 6 equations is abbreviated to λ-BI system. The case λ = 1 has been intensively investigated. In [START_REF] Brenier | Hydrodynamic structure of the augmented Born-Infeld equations[END_REF], Brenier enlarged the 1-BI system to an augmented Born-Infeld system of 10 equations by considering h, D, B and P as independent variables. He found many interesting properties of this last system, like fully linear degeneracy, Galilian invariance and existence of a strictly convex entropy which implies the hyperbolicity of the system in the sense of Friedrichs. Obviously, the augmentation of the λ-BI system can be carried out for any λ > 0 in a same way. The resulting system (1.4)- (1.8) is referred to λ-ABI system. It is a hyperbolic system which admits a strictly convex entropy (1.9) S λ (D, B, P, h) =

λ 2 + |B| 2 + λ 2 |D| 2 + |P | 2 2h , h > 0.
Note that a different enlargement of the Born-Infeld system was proposed by Serre in [START_REF] Serre | Hyperbolicity of the nonlinear models of Maxwell's equations[END_REF], in which only P is added as independent variable. Some extensions of the works [START_REF] Brenier | Hydrodynamic structure of the augmented Born-Infeld equations[END_REF][START_REF] Serre | Hyperbolicity of the nonlinear models of Maxwell's equations[END_REF] to more general equations derived from the variational principle were given by Boillat [START_REF] Boillat | Euler's variational equations with independent momentum[END_REF].

For smooth solutions to the 1-BI and 1-ABI systems, the reader is referred to Brenier [START_REF] Brenier | Hydrodynamic structure of the augmented Born-Infeld equations[END_REF] and Serre [START_REF] Serre | Hyperbolicity of the nonlinear models of Maxwell's equations[END_REF] for local existence by an argument for first order symmetrizable hyperbolic systems, to Chae-Huh [START_REF] Chae | Global existence for small initial data in the Born-Infeld equations[END_REF] for global existence with small initial data, and also to [START_REF] Brenier | Order preserving vibrating strings and applications to Electrodynamics and Magnetohydrodynamics[END_REF] for the solutions beyond singularities. Entropy solutions in one space dimension are studied in [START_REF] Peng | Entropy solutions of Born-Infeld systems in one space dimension[END_REF][START_REF] Peng | Euler-Lagrange change of variables in conservation laws[END_REF] by the change of variables of Euler-Lagrange type (see [START_REF] Wagner | Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions[END_REF]). For general theory of hyperbolic systems of conservation laws, we refer to [START_REF] Serre | Systèmes de Lois de Conservation[END_REF][START_REF] Dafermos | Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der Mathematischen Wissenschaften[END_REF][START_REF] Bressan | Hyperbolic Systems of Conservation Laws : The One Dimensional Cauchy Problem[END_REF][START_REF] Ph | Hyperbolic Systems of Conservation Laws[END_REF].

Following Brenier, (1.2) is called BI-manifold. It was shown that the BI-manifold of the 1-ABI system is time invariant for smooth solutions (see [START_REF] Serre | Hyperbolicity of the nonlinear models of Maxwell's equations[END_REF][START_REF] Boillat | Euler's variational equations with independent momentum[END_REF][START_REF] Brenier | Derivation of particle, string, and membrane motions from the Born-Infeld electromagnetism[END_REF]) and also for entropy solutions in one space dimension (see [START_REF] Peng | Entropy solutions of Born-Infeld systems in one space dimension[END_REF][START_REF] Peng | Euler-Lagrange change of variables in conservation laws[END_REF]). The invariance of the BI-manifold implies the equivalence of the corresponding solutions between the 1-BI system and 1-ABI system. As consequences, it yields the existence of local smooth solutions and global entropy solutions in one space dimension of the 1-BI system. Similar results hold for the λ-BI and λ-ABI systems for all λ > 0. Thus, the enlargement of the λ-BI system provides a useful technique to study its Cauchy problem and asymptotic analysis. Indeed, it suffices to study the λ-ABI system and then take the initial data in the BI-manifold to get the corresponding results for the λ-BI system.

In this work, we are concerned with two limits λ → ∞ and λ → 0 in the λ-BI system. In classical Born-Infeld theory, λ is a large parameter which controls the electric field, namely, |E| ≤ λ. However, in Tachyonic condensate equations of high energy physics and the classical string theory, the corresponding value of λ is zero (see [START_REF] Brenier | A note on deformations of 2D fluid motions using 3D Born-Infeld equations[END_REF][START_REF] Gibbons | Aspects of Born-Infeld theory and string / M-theory[END_REF]). Then it is important to consider both the limits λ → ∞ and λ → 0 in the λ-BI system to recover the link with the above physical situations.

This paper is organized as follows. In the next section, we first show that the Born-Infeld Lagrangian L λ is well defined for all λ > 0. Then we perform formal asymptotic analysis in the λ-ABI system to recover the limit equations. They are governed by the classical Maxwell equations as λ → ∞ and the pressureless magnetohydrodynamics system as λ → 0. We show that these equations correspond to the limit cases of the Lagrangian and a new scaling is necessary to make the limit λ → ∞ rigorously. Furthermore, we give the relations between these limits and Brenier high and low field limits in the 1-ABI system. From section 3, we consider the problem in one space dimension. We show the hyperbolic structure of the λ-ABI system for all λ ≥ 0, which yields the existence of global entropy solutions of its Cauchy problem in L ∞ . These results are extensions of the corresponding results for the 1-ABI system. In particular, we give an explicit expression of the entropy solution and its dependence on λ. This yields uniform estimates for the sequences of the entropy solutions. The rigorous justification of the limits λ → ∞ and λ → 0 in the λ-ABI system is completed in section 4 and section 5 by using the compensated compactness arguments and techniques for linear Lagrangian systems. Finally, we remark that both convergence results hold globally in time because the limit systems are linearly hyperbolic.

Preliminary

2.1. Remarks on the definition of L λ . In the literature there exist another expressions for the Born-Infeld Lagrangian and energy density. They are given by (see [START_REF] Boillat | Euler's variational equations with independent momentum[END_REF]) :

(2.1) L k ( B, E) = -k 2 + k| B| 2 -k| E| 2 -( B • E) 2 , (2.2) h( D, B) = k 2 + k(| B| 2 + | D| 2 ) + ( D × B) 2 ,
with k ≥ 0 and

E = ∂h( D, B) ∂ D = k D + B × P h , P = D × B.
One may check that the relations between (1.1)-(1.2) and (2.1)-(2.2) are

k = λ, B = B √ k , D = √ kD.
Moreover, both expressions L λ and L k are well defined for sufficiently large λ and k, for instance, for |E| ≤ λ and | E| ≤ k, respectively. It is clear that L k is not well defined when k = 0 unless B • E = 0. However, in the limit case λ → 0, L λ corresponds to the Tachyonic condensate equations of high energy physics and the classical string theory (see [START_REF] Brenier | A note on deformations of 2D fluid motions using 3D Born-Infeld equations[END_REF]). Therefore, we should show that the Lagrangian L λ (B, E) given by (1.1) is well defined for all λ > 0. This fact is the advantage of the expression L λ compared to that of L k . A more precise result can be stated as follows.

Proposition 2.1. For all h > 0 and λ > 0, we have

(2.3) L λ (B, E) = - 1 h λ 2 + |B| 2 . Proof. Set ∆ λ (B, E) = λ 2 + |B| 2 -|E| 2 - (B • E) 2 λ 2 , with E = λ 2 D + B × P h , P = D × B, h = λ 2 + |B| 2 + λ 2 |D| 2 + |D × B| 2 . Since B • E = λ 2 (B • D) h + B • (B × P ) h = λ 2 (B • D) h and |E| 2 = |λ 2 D + B × P | 2 h 2 = 1 h 2 λ 4 |D| 2 + |B × P | 2 + 2λ 2 D • (B × P ) , we get ∆ λ (B, E) = 1 h 2 (λ 2 + |B| 2 )(λ 2 + |B| 2 + λ 2 |D| 2 + |D × B| 2 ) -(λ 4 |D| 2 + |B × P | 2 + 2λ 2 D • (B × P )) -λ 2 (D • B) 2 = 1 h 2 (λ 2 + |B| 2 ) 2 + λ 2 (|D × B| 2 + |D| 2 |B| 2 -(D • B) 2 ) +|B| 2 |D × B| 2 -|B × P | 2 -2λ 2 D • (B × P ) .
Using the relation

|ξ| 2 |η| 2 = |ξ • η| 2 + (ξ × η) 2 , ∀ ξ, η ∈ R 3 , we have ∆ λ (B, E) = 1 h 2 (λ 2 + |B| 2 ) 2 + 2λ 2 |D × B| 2 + |B • P | 2 -2λ 2 D • (B × P ) .
But,

B • P = B • (D × B) = 0, D • (B × P ) = (D × B) • P = |D × B| 2 . Then ∆ λ (B, E) = 1 h 2 λ 2 + |B| 2 2 ≥ 0, ∀ λ > 0.
This shows (2.3).

Formal asymptotic analysis.

As shown in the Introduction, the value of λ is large in the electromagnetic theory and zero in the classical string theory. Then, it is important to study the limit cases λ → 0 and λ → +∞ in the λ-ABI system. The detailed derivation of the formal asymptotic limits can be done as follows.

First, it is clear that the formal limit λ → 0 in the λ-ABI system (1.4)-(1.8) is given by (2.4)

                         ∂ t B + ∇ × B × P h = 0, ∂ t h + ∇ • P = 0, ∂ t P + ∇ • P ⊗ P -B ⊗ B h = 0, ∂ t D + ∇ × -B + D × P h = 0, ∇ • B = ∇ • D = 0.
In the limit of the λ-BI system (1.4)-(1.6), according to the definition (1.2)-(1.3), (h, P, E) should be given by (2.5)

P = D × B, h = |B| 2 + |D × B| 2 , E = B × P h .
Then, from (2.3), it is easy to check that the Born-Infeld Lagrangian satisfies

lim λ→0 L λ (B, E) = - |B| 2 |B| 2 + |D × B| 2 .
A straightforward computation using (2.5) shows that

lim λ→0 L λ (B, E) = -|B| 2 -|E| 2 ,
which was obtained in [START_REF] Brenier | A note on deformations of 2D fluid motions using 3D Born-Infeld equations[END_REF]. Here by (2.5), the conditions |B| 2 -|E| 2 ≥ 0 and B • E = 0 are always satisfied. Note that the first three equations in (2.4) for (B, h, P ), independent of the last one for D, are called pressureless magnetohydrodynamics (PMHD) system [START_REF] Brenier | Hydrodynamic structure of the augmented Born-Infeld equations[END_REF]. As we will see in section 5, in one space dimension the PMHD system is hyperbolic except some critical cases. This is different from the pressureless gas dynamics. Now consider the limit λ → +∞. To this end, let α ∈ [0, 1) be a constant and introduce the scaling

(2.6) B = λ α B λ , D = λ α-1 D λ .
Then from (1.2), we have

P = D × B = λ 2α-1 D λ × B λ , (2.7) h = λ 1 + ε(|B λ | 2 + |D λ | 2 ) + ε 2 |D λ × B λ | 2 ,
where ε = λ 2(α-1) → 0. If we choose the scaling

(2.8) h = λh λ , P = λ 2α-1 P λ ,
the λ-ABI system (1.4)-(1.8) becomes :

(2.9)

                         ∂ t D λ + ∇ × -B λ + εD λ × P λ h λ = 0, ∂ t B λ + ∇ × D λ + εB λ × P λ h λ = 0, ∂ t h λ + ε∇ • P λ = 0, ∂ t P λ + ∇ • εP λ ⊗ P λ -B λ ⊗ B λ -D λ ⊗ D λ h λ = 1 ε ∇ 1 h λ , ∇ • D λ = ∇ • B λ = 0. As ε → 0, i.e. λ → ∞, from (2.7)-(2.8
), we should have formally h λ → 1. Then from (2.9) the formal limit (B, D) = lim λ→∞ (B λ , D λ ) satisfies the linear Maxwell equations :

(2.10)

       ∂ t B + ∇ × D = 0, ∂ t D -∇ × B = 0, ∇ • B = ∇ • D = 0.
For the rest of the limit equations, we set h 1λ = (h λ -1)/ε. Since

1 h λ = 1 1 + εh 1λ = 1 -εh 1λ + O(ε 2 ), then 1 ε ∇ 1 h λ = -∇h 1λ + O(ε).
From h λ → 1 and P λ = O(1), we have

∇ • εP λ ⊗ P λ h λ -→ 0, as λ → +∞.
Thus, using the equations for h λ and P λ in (2.9), the formal limit (h 1 , P ) = lim λ→∞ (h 1λ , P λ ) satisfies the linear hyperbolic system :

(2.11)

∂ t h 1 + ∇ • P = 0, ∂ t P + ∇h 1 = ∇ • (B ⊗ B + D ⊗ D),
when D and B are solved by (2.10).

Concerning the convergence of the Born-Infeld Lagrangian, we use (1.3), (2.6) and (2.8) to obtain

E = λ 2 λ α-1 D λ + λ α λ 2α-1 B λ × P λ λh λ . It follows that (B • E) 2 λ 2 = λ 2(2α-1) (B λ • D λ ) 2 . Writing E = λ α (D λ + O(ε))
and noting that 2(2α -1) < 2α, from (1.1) we obtain

L λ (B, E) + λ = λ -λ 2 + |B| 2 -|E| 2 -λ -2 (B • E) 2 = |E| 2 -|B| 2 -λ -2 (B • E) 2 λ + λ 1 + λ -2 |B| 2 -|E| 2 -λ -4 (B • E) 2 = λ 2α-1 |D λ | 2 -|B λ | 2 + O(ε) 1 + 1 + ε |B λ | 2 -|D λ | 2 + O(ε 2 ) = λ 2α-1 1 2 (|D λ | 2 -|B λ | 2 ) + O(ε) .
Thus, as λ → +∞, we have

lim λ→∞ λ 1-2α (L λ (B, E) + λ) = 1 2 |D| 2 -|B| 2 .
This shows that the classical Maxwell theory corresponds to the limit of the Lagrangian

λ 1-2α (L λ (B, E) + λ).
In particular, it corresponds to the limit of L λ (B, E) + λ when α = 1/2 and to the limit of λ(L λ (B, E) + λ) when α = 0. The latter case was given by Brenier (see [START_REF] Brenier | A note on deformations of 2D fluid motions using 3D Born-Infeld equations[END_REF]).

2.3.

Relations with Brenier high and low field limits. In [START_REF] Brenier | Hydrodynamic structure of the augmented Born-Infeld equations[END_REF], Brenier introduced the high and low field limits in the 1-ABI system. The low field limit is defined by the scaling (2.12)

B = τ B τ , D = τ D τ , P = τ 2 P τ , h = h τ = 1 + O(τ 2 ),
where (B, D, h, P ) is a solution of the 1-ABI system and τ > 0 is an artificial small parameter. Put the expression (2.12) into the 1-ABI system and let τ → 0, he found that the formal limit of (B τ , D τ ) satisfies the linear Maxwell equations (2.10). It is easy to check that the low field limit corresponds to the limit λ → ∞ in the λ-ABI system with the choice τ = λ α-1 . Indeed, with the above choice and ε = λ 2(α-1) , (B τ , D τ , h τ , P τ ) defined by (2.12) satisfies exactly the system (2.9). Similarly, the high field limit is defined by the following scaling in the 1-ABI system :

(2.13)

B = τ B τ , D = D τ , P = τ P τ , h = τ h τ ,
where (B, D, h, P ) is a solution of the 1-ABI system and τ > 0 is an artificial large parameter. Put the expression (2.13) into the 1-ABI system and let τ → ∞, then the formal limit of (B τ , h τ , P τ ) satisfies the PMHD system. We may check easily that this high field limit corresponds to the limit λ → 0 in the λ-ABI system with the choice τ = λ -1 . The high and low field limits have been rigorously justified for smooth solutions, in a recent paper of Brenier and Yong [START_REF] Brenier | Derivation of particle, string, and membrane motions from the Born-Infeld electromagnetism[END_REF], in which the physical regimes of these limits are explained. The convergence holds essentially locally in time for the high field limit. Whereas it is global in time for the low field limit because its limit equations are linear hyperbolic and admit global smooth solutions. Using the relations between the high and low field limits and those of λ → ∞ and λ → 0 discussed above, we may apply their results to yield the convergence of the limits λ → ∞ and λ → 0 for smooth solutions in the λ-ABI system. On the other hand, the results of this paper justify the high and low field limits for global entropy solutions in one space dimension. For simplify the presentation of the paper, we will not write these results in details.

3. Born-Infeld systems in one space dimension 3.1. Hyperbolic structure of the λ-ABI system. From now on we consider the Born-Infeld systems in one space dimension. As in [START_REF] Brenier | Hydrodynamic structure of the augmented Born-Infeld equations[END_REF], we denote x = x 1 and consider the problems independent of x 2 and x 3 . Then B 1 and D 1 are constants from (1.4)- (1.6). It follows that the λ-BI system (1.4)-(1.6) and λ-ABI system (1.4)-(1.8) are reduced to

(3.1)                              ∂ t D 2 + ∂ x B 3 + D 2 P 1 -D 1 P 2 h = 0, ∂ t D 3 + ∂ x -B 2 + D 3 P 1 -D 1 P 3 h = 0, ∂ t B 2 + ∂ x -λ 2 D 3 + B 2 P 1 -B 1 P 2 h = 0, ∂ t B 3 + ∂ x λ 2 D 2 + B 3 P 1 -B 1 P 3 h = 0, P = D × B, h = λ 2 + |B| 2 + λ 2 |D| 2 + |D × B| 2 , (3.2)
                                                     ∂ t h + ∂ x P 1 = 0, ∂ t P 1 + ∂ x P 2 1 -(λ 2 + B 2 1 + λ 2 D 2 1 ) h = 0, ∂ t D 2 + ∂ x B 3 + D 2 P 1 -D 1 P 2 h = 0, ∂ t D 3 + ∂ x -B 2 + D 3 P 1 -D 1 P 3 h = 0, ∂ t B 2 + ∂ x -λ 2 D 3 + B 2 P 1 -B 1 P 2 h = 0, ∂ t B 3 + ∂ x λ 2 D 2 + B 3 P 1 -B 1 P 3 h = 0, ∂ t P 2 + ∂ x P 1 P 2 -λ 2 D 1 D 2 -B 1 B 2 h = 0, ∂ t P 3 + ∂ x P 1 P 3 -λ 2 D 1 D 3 -B 1 B 3 h = 0,
respectively. Here we have used the notations

B = (B 1 , B 2 , B 3 ) t , D = (D 1 , D 2 , D 3 ) t , P = (P 1 , P 2 , P 3 ) t .
Now let us rewrite the λ-ABI system (3.2) by introducing :

P 1 = hv 2 , D 2 = hv 3 , D 3 = hv 4 , B 2 = hv 5 , B 3 = hv 6 , P 2 = hv 7 , P 3 = hv 8 .
Then the λ-ABI system (3.2) becomes :

(3.3)                                  ∂ t h + ∂ x (hv 2 ) = 0, ∂ t (hv 2 ) + ∂ x (hv 2 2 -a 2 h -1 ) = 0, ∂ t (hv 3 ) + ∂ x (hv 2 v 3 -D 1 v 7 + v 6 ) = 0, ∂ t (hv 4 ) + ∂ x (hv 2 v 4 -D 1 v 8 -v 5 ) = 0, ∂ t (hv 5 ) + ∂ x (hv 2 v 5 -B 1 v 7 -λ 2 v 4 ) = 0, ∂ t (hv 6 ) + ∂ x (hv 2 v 6 -B 1 v 8 + λ 2 v 3 ) = 0, ∂ t (hv 7 ) + ∂ x (hv 2 v 7 -λ 2 D 1 v 3 -B 1 v 5 ) = 0, ∂ t (hv 8 ) + ∂ x (hv 2 v 8 -λ 2 D 1 v 4 -B 1 v 6 ) = 0,
where the constant a, depending on λ, is defined by

a = λ 2 + B 2 1 + λ 2 D 2 1 .
Remark that the first two equations in (3.3) form the system for Chaplygin gas dynamics on which some results are obtained in [START_REF] Serre | Un système hyperbolique non linéaire avec des données oscillantes[END_REF][START_REF] Peng | Explicit solutions for 2 × 2 linearly degenerate systems[END_REF][START_REF] Brenier | Hydrodynamic structure of the augmented Born-Infeld equations[END_REF].

Suppose h, v 2 ∈ L ∞ (R + × R) with h > 0.
Using the first equations in (3.3), we may introduce a change of variables (t, x) -→ (s, y) from R + × R to R + × R, with (see [START_REF] Wagner | Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions[END_REF][START_REF] Peng | Euler-Lagrange change of variables in conservation laws[END_REF])

s = t, dy = hdx -hv 2 dt.
For simplicity, denote by v(s, y) = v(t, x). Then in Lagrangian coordinates (s, y), the system (3.3) is written as :

(3.4) ∂ s v + A∂ y v = 0, t > 0, y ∈ R, where v = (h -1 , v 2 , . . . , v 8 ) t and A =               0 -1 0 0 0 0 0 0 -a 2 0 0 0 0 0 0 0 0 0 0 0 0 1 -D 1 0 0 0 0 0 -1 0 0 -D 1 0 0 0 -λ 2 0 0 -B 1 0 0 0 λ 2 0 0 0 0 -B 1 0 0 -λ 2 D 1 0 -B 1 0 0 0 0 0 0 -λ 2 D 1 0 -B 1 0 0              
Recall that a linear Lagrangian system is a linear system in Lagrangian coordinates via a transformation of Euler-Lagrangian type [START_REF] Peng | Euler-Lagrange change of variables in conservation laws[END_REF]. Since A is a constant matrix, the λ-ABI system (3.3) is a linear Lagrangian system and the results in [START_REF] Peng | Euler-Lagrange change of variables in conservation laws[END_REF] can be applied. A straightforward computation gives the eigenvalues of A :

(3.5) µ 1 = µ 2 = µ 3 = -a, µ 4 = µ 5 = 0, µ 6 = µ 7 = µ 8 = a.
Moreover, set

β = B 2 1 + λ 2 D 2 1 , φ ± = aD 1 ± B 1 , ψ ± = aB 1 ± λ 2 D 1 .
For β = 0 (then a = 0) we choose the left eigenvectors l iλ and right eigenvectors r jλ of A as follows :

(3.6)                                  l 1λ = (0, 0, λ 2 φ -, λ 2 φ + , ψ + , ψ -, β, β), l 2λ = (0, 0, λ 2 φ + , -λ 2 φ -, ψ -, -ψ + , β, -β),
l 3λ = (a, 1, 0, 0, 0, 0, 0, 0),

l 4λ = 1 2a 2 (0, 0, B 1 , -B 1 , -D 1 , D 1 , 1, 1), l 5λ = 1 2a 2 (0, 0, -B 1 , -B 1 , D 1 , D 1 , 1, -1), l 6λ = (0, 0, λ 2 φ + , λ 2 φ -, ψ -, ψ + , -β, -β), l 7λ = (0, 0, λ 2 φ -, -λ 2 φ + , ψ + , -ψ -, -β, β), l 8λ = (a, -1, 0, 0, 0, 0, 0, 0), (3.7)                                                r 1λ = 1 4a 2 β (0, 0, φ -, φ + , ψ + , ψ -, β, β) t , r 2λ = 1 4a 2 β (0, 0, φ + , -φ -, ψ -, -ψ + , β, -β) t , r 3λ = 1 2a
(1, a, 0, 0, 0, 0, 0, 0) t ,

r 4λ = (0, 0, B 1 , -B 1 , -λ 2 D 1 , λ 2 D 1 , λ 2 , λ 2 ) t , r 5λ = (0, 0, -B 1 , -B 1 , λ 2 D 1 , λ 2 D 1 , λ 2 , -λ 2 ) t , r 6λ = 1 4a 2 β (0, 0, φ + , φ -, ψ -, ψ + , -β, -β) t , r 7λ = 1 4a 2 β (0, 0, φ -, -φ + , ψ + , -ψ -, -β, β) t , r 8λ = 1 2a
(1, -a, 0, 0, 0, 0, 0, 0) t .

We see that l iλ and r jλ are defined for all λ ≥ 0 and satisfy l iλ r jλ = δ ij for all 1 ≤ i, j ≤ 8. Hence, the matrix A is diagonalizable and the linear system (3.4) is hyperbolic. When β = 0 and a = 0, i.e., B 1 = D 1 = 0 and λ = 0, we may write A = diag(A 1 , A 2 , 0), with

A 1 = 0 -1 -a 2 0 , A 2 =      0 0 0 1 0 0 -1 0 0 -λ 2 0 0 λ 2 0 0 0      .
It is easy to check that A 1 and A 2 are diagonalizable. So is A and then the λ-ABI system is hyperbolic for all a = 0 and all λ ≥ 0. The above computations are elementary but the expressions of l iλ and r jλ are important in the study of the limit λ → 0. That is why we keep the subscript λ in their expressions to see uniform estimates with respect to λ. Now applying the results in [START_REF] Peng | Euler-Lagrange change of variables in conservation laws[END_REF], we have

w i = l iλ v (1 ≤ i ≤ 8). The eigenvalues µ i = µ i (v) (1 ≤ i ≤ 8) of the system (3.
3) are :

µ 1 = µ 2 = µ 3 = v 2 -ah -1 , µ 4 = µ 5 = v 2 , µ 6 = µ 7 = µ 8 = v 2 + ah -1 .
Moreover, all entropy-entropy flux pairs (E, F ) of the system are expressed as

(3.8) (E(v), F (v)) = h 8 i=1 (1, µ i (v))g i (l iλ v),
with arbitrary continuous functions g i (1 ≤ i ≤ 8).

Remark 3.1. When a = 0, the system for Chaplygin gas dynamics becomes the pressureless gas dynamics which is not hyperbolic. This means that a = 0 is also a necessary condition to keep the hyperbolicity of the λ-ABI system.

3.2.

Entropy solutions of the Born-Infeld systems. Consider the Cauchy problem to the λ-ABI system (3.3) associated to the initial conditions :

(3.9)

t = 0 : h = h 0 (x), v i = v 0 i (x) (2 ≤ i ≤ 8), x ∈ R.
Following [START_REF] Peng | Explicit solutions for 2 × 2 linearly degenerate systems[END_REF], in order to guarantee the strict hyperbolicity of the system for Chaplygin gas for all time t ≥ 0 we need to assume that

(3.10) h 0 , v 0 i ∈ L ∞ (R) (2 ≤ i ≤ 8), inf x∈R v 0 2 (x) + a h 0 (x) > sup x∈R v 0 2 (x) - a h 0 (x) ,
which imply that h(t, x) > 0 for almost all (t, x) ∈ R + × R. Note that conditions (3.10) appeared first time in [START_REF] Serre | Un système hyperbolique non linéaire avec des données oscillantes[END_REF] in the study of BV weak solutions and their oscillations. The Cauchy problems to the 1-ABI system and 1-BI system have been studied in [START_REF] Peng | Entropy solutions of Born-Infeld systems in one space dimension[END_REF][START_REF] Peng | Euler-Lagrange change of variables in conservation laws[END_REF] through an Euler-Lagrange change of variables. It was shown that the Cauchy problems to these two systems admit each a unique entropy solution with an explicit expression. Since the λ-ABI system is hyperbolic linear Lagrange system, the mentioned results can be extended easily to the λ-ABI and λ-BI systems.

Recall that by entropy solution of the λ-ABI system we mean weak solution satisfying the entropy equality ∂ t E(v) + ∂ x F (v) = 0 for all entropy-entropy flux pairs given in (3.8). Now let us define

(3.11) Y 0 (x) = x 0 h 0 (ξ)dξ, (3.12) X(t, y) = 1 2 y+t 0 v 0 2 + a h 0 (X 0 (ξ))d ξ - 1 2 y-t 0 v 0 2 - a h 0 (X 0 (ξ))d ξ.
It was shown that Y 0 and y -→ X(t, y) are Lipschitzian and bijective for all t ≥ 0. We denote their inverse functions by (3.13)

X 0 = (Y 0 ) -1 , Y (t, •) = X -1 (t, •).
The results of the existence of entropy solutions to the λ-ABI system can be stated as follows. Their proofs will be omitted since they are similar to the cases for 1-ABI system [START_REF] Peng | Euler-Lagrange change of variables in conservation laws[END_REF]. Proposition 3.2. Let the assumption (3.10) hold and a = 0. Then the Cauchy problem (3.3) and (3.9) for the λ-ABI system admits a unique entropy solution h, v ∈ L ∞ (R + ×R). The solution satisfies the maximum principle :

(3.14) inf ξ∈R v 0 2 (ξ) + a h 0 (ξ) ≤ v 2 (t, x) + a h(t, x) ≤ sup ξ∈R v 0 2 (ξ) + a h 0 (ξ) , (3.15) inf ξ∈R v 0 2 (ξ) - a h 0 (ξ) ≤ v 2 (t, x) - a h(t, x) ≤ sup ξ∈R v 0 2 (ξ) - a h 0 (ξ) ,
and has an explicit expression :

(3.16) v(t, x) = 8 i=1 (l iλ v 0 )(X 0 (Y (t, x) -µ i t))r iλ ,
where

v 0 = ((h 0 ) -1 , v 0 2 , • • • , v 0 8
) t , µ i , l iλ and r iλ are defined in (3.5)-(3.7), respectively. Remark that the estimates (3.14)-(3.15) depend only on the system of Chaplygin gas dynamics and independent of the rest equations in the λ-ABI system. They are consequences of the fact that the system of Chaplygin gas dynamics is fully linearly degenerate. Applying Proposition 3.2 together with the time invariance of the BI-manifold proved in [START_REF] Peng | Euler-Lagrange change of variables in conservation laws[END_REF] yields a result of the existence and uniqueness of entropy solutions to the λ-BI system.

Proposition 3.3. Let (B 0 2 , B 0 3 , D 0 2 , D 0 3 ) ∈ L ∞ (R)
and (h 0 , P 0 ) be defined by the BImanifold (1.2) satisfying the second condition in (3.10). Assume a = 0. Then there exists a unique entropy solution

(B 2 , B 3 , D 2 , D 3 ) ∈ L ∞ (R + × R) to the λ-BI system (3.1) with the initial data (B 0 2 , B 0 3 , D 0 2 , D 0 3 ).

Limit toward linear Maxwell equations in one dimension

4.1. Uniform estimates of the entropy solutions. In this section, we study the asymptotic limit λ → ∞ in the λ-ABI system. We want to prove the convergence of the system (2.9) to the system (2.10)-(2.11) in one space dimension. First, in one space dimension the system (2.9) reads :

(4.1)

                                                     ∂ t h λ + ∂ x (εP 1λ ) = 0, ∂ t (εP 1λ ) + ∂ x ε 2 P 2 1λ -ρ 2 λ h λ = 0, ∂ t D 2λ + ∂ x B 3λ + ε(D 2λ P 1λ -D 1 P 2λ ) h λ = 0, ∂ t D 3λ + ∂ x -B 2λ + ε(D 3λ P 1λ -D 1 P 3λ ) h λ = 0, ∂ t B 2λ + ∂ x -D 3λ + ε(B 2λ P 1λ -B 1 P 2λ ) h λ = 0, ∂ t B 3λ + ∂ x D 2λ + ε(B 3λ P 1λ -B 1 P 3λ ) h λ = 0, ∂ t P 2λ + ∂ x εP 1λ P 2λ -D 1 D 2λ -B 1 B 2λ h λ = 0, ∂ t P 3λ + ∂ x εP 1λ P 3λ -D 1 D 3λ -B 1 B 3λ h λ = 0, where ε = λ 2(α-1) , α ∈ [0, 1), ρ λ = √ 1 + εσ, σ = B 2 1 + D 2 1
, with B 1 , D 1 being two real constants. A more general case for which (B 1λ ) λ and (D 1λ ) λ are bounded sequences in R (then admitting convergent subsequences) can be dealt with in a similar way.

Let us recall the definition :

(4.2) P 1λ = h λ v 2λ , D 2λ = h λ v 3λ , D 3λ = h λ v 4λ , B 2λ = h λ v 5λ , B 3λ = h λ v 6λ , P 2λ = h λ v 7λ , P 3 = h λ v 8λ .
According to the scaling (2.7)-(2.8), we associate to (4.1) the following initial conditions :

(4.3) t = 0 : h λ = h 0 λ (x) def = 1 + εh 0 1λ (x), v iλ = v 0 iλ (x) (2 ≤ i ≤ 8), x ∈ R.
Observe that the first two equations in (4.1) are decoupled with the rest. Then uniform estimates for the variables (h λ , P 1λ ) and (D 2λ , D 3λ , B 2λ , B 3λ , P 2λ , P 3λ ) can be treated separately.

Lemma 4.1. Assume that the sequences of the initial data

(h 0 1λ ) λ , (v 0 iλ ) λ (2 ≤ i ≤ 8) are bounded in L ∞ (R).
Then the Cauchy problem (4.1)-( 4.3) has a unique entropy solution (h λ , P 1λ , D 2λ , D 3λ , B 2λ , B 3λ , P 2λ , P 3λ ), and for sufficiently large λ > 0, we have

(4.4) h λ -1 L ∞ (R + ×R) ≤ Cε, P 1λ L ∞ (R + ×R) ≤ C,
where C > 0 is a constant independent of λ.

Proof. Let us notice that under the assumption of Lemma 4.1, Proposition 3.2 can be applied to the Cauchy problem (4.1) and (4.3). Indeed, the hyperbolicity of the system is invariant under the change of variables (2.6) and (2.8). Moreover, for sufficiently large λ, we have

ρ λ = 1 + O(ε), h λ (0, x) = 1 + εh 0 1λ (x), which yield ρ λ > 0, w λ def = inf ξ∈R εv 0 2λ (ξ) + ρ λ 1 + εh 0 1λ (ξ) = 1 + O(ε), and
z λ def = sup ξ∈R εv 0 2λ (ξ) - ρ λ 1 + εh 0 1λ (ξ) = -1 + O(ε).
Thus, the condition (3.10) is trivially fulfilled and there is a unique entropy solution to the Cauchy problem (4.1) and (4.3). Similarly,

w λ def = sup ξ∈R εv 0 2λ (ξ) + ρ λ 1 + εh 0 1λ (ξ) = 1 + O(ε), z λ def = inf ξ∈R εv 0 2λ (ξ) - ρ λ 1 + εh 0 1λ (ξ) = -1 + O(ε).
Applying the maximum principle (3.14)- (3.15) to the solution, we obtain

w λ ≤ εv 2λ (t, x) + ρ λ h λ (t, x) ≤ w λ , z λ ≤ εv 2λ (t, x) - ρ λ h λ (t, x) ≤ z λ .
It follows that h λ (t, x) = 1 + O(ε) and the sequence (v 2λ ) λ is bounded in L ∞ (R + × R), which implies that the sequence (P 1λ ) λ is bounded in L ∞ (R + × R).

Lemma 4.2. Under the assumptions of Lemma 4.1, the sequence (D 2λ , D 3λ , B 2λ , B 3λ , P 2λ , P 3λ ) λ is bounded in L ∞ (R + × R).

Proof. Since h λ > 0 for all λ > 0 large, by the first equation in (4.1), we may make the change of variables : (t, x) -→ (s, y) with

s = t, dy = h λ dx -εP 1λ dt.
Then in Lagrangian coordinates (s, y), v λ (s, y) = (v 3λ , • • • , v 8λ ) t (t, x) satisfies an equivalent system

∂ s v λ + A λ ∂ y v λ = 0,
where

A λ =          0 0 0 1 -εD 1 0 0 0 -1 0 0 -εD 1 0 -1 0 0 -εB 1 0 1 0 0 0 0 -εB 1 -D 1 0 -B 1 0 0 0 0 -D 1 0 -B 1 0 0         
.

A straightforward computation shows that

ν 1 = ν 2 = -ρ λ , ν 3 = ν 4 = 0, ν 5 = ν 6 = ρ λ
are the eigenvalues of A λ . For σ = 0, i.e., B 2 1 + D 2 1 = 0, we choose the corresponding left eigenvectors l iλ and right eigenvectors r iλ of A λ as follows :

                                       l 1λ = 1 2 (ρ λ D 1 -B 1 , ρ λ D 1 + B 1 , ρ λ B 1 + D 1 , ρ λ B 1 -D 1 , εσ, εσ), l 2λ = 1 2 (ρ λ D 1 + B 1 , -ρ λ D 1 + B 1 , ρ λ B 1 -D 1 , -(ρ λ B 1 + D 1 ), εσ, -εσ), l 3λ = 1 2ρ 2 λ (B 1 , -B 1 , -D 1 , D 1 , 1, 1), l 4λ = 1 2ρ 2 λ (-B 1 , -B 1 , D 1 , D 1 , 1, -1), l 5λ = 1 2 (ρ λ D 1 + B 1 , ρ λ D 1 -B 1 , ρ λ B 1 -D 1 , ρ λ B 1 + D 1 , -εσ, -εσ), l 6λ = 1 2 (ρ λ D 1 -B 1 , -(ρ λ D 1 + B 1 ), ρ λ B 1 + D 1 , -ρ λ B 1 + D 1 , -εσ, εσ),                                    r 1λ = 1 2ρ 2 λ σ (ρ λ D 1 -B 1 , ρ λ D 1 + B 1 , ρ λ B 1 + D 1 , ρ λ B 1 -D 1 , σ, σ) t , r 2λ = 1 2ρ 2 λ σ (ρ λ D 1 + B 1 , -ρ λ D 1 + B 1 , ρ λ B 1 -D 1 , -(ρ λ B 1 + D 1 ), σ, -σ) t , r 3λ = (εB 1 , -εB 1 , -εD 1 , εD 1 , 1, 1) t , r 4λ = (-εB 1 , -εB 1 , εD 1 , εD 1 , 1, -1) t , r 5λ = 1 2ρ 2 λ σ (ρ λ D 1 + B 1 , ρ λ D 1 -B 1 , ρ λ B 1 -D 1 , ρ λ B 1 + D 1 , -σ, -σ) t , r 6λ = 1 2ρ 2 λ σ (ρ λ D 1 -B 1 , -(ρ λ D 1 + B 1 ), ρ λ B 1 + D 1 , -ρ λ B 1 + D 1 , -σ, σ) t .
In this case, the constant sequences ( l iλ ) λ and ( r iλ ) λ are bounded and l iλ r jλ = δ ij for all 1 ≤ i, j ≤ 6. On the other hand, when σ = 0, we have B 1 = D 1 = 0, and the matrix A λ is symmetric and independent of λ. Then we may choose arbitrarily its left and right eigenvectors l i and r j such that l i r j = δ ij for all 1 ≤ i, j ≤ 6.

Using the explicit formula (3.16), we have

v λ (t, x) = 6 i=1 ( l iλ v 0 λ )(X 0 λ (Y λ (t, x) -ν i t)) r iλ , where v 0 λ = (v 0 3λ , • • • , v 0 8λ ) t , X 0 λ = (Y 0 λ ) -1 , Y λ (t, •) = X -1 λ (t, •), with Y 0 λ (x) = x 0 h 0 λ (ξ)dξ, X λ (t, y) = 1 2 y+t 0 v 0 2λ + ρ λ h 0 λ (X 0 λ (ξ))d ξ - 1 2 y-t 0 v 0 2λ - ρ λ h 0 λ (X 0 λ (ξ))d ξ.
Since the sequence ( v 0 λ ) λ is assumed to be bounded in L ∞ (R), we deduce that ( v λ ) λ is bounded in L ∞ (R + × R). This proves Lemma 4.2 together with the definition (4.2) and h λ = 1 + O(ε).

4.2.

Convergence. From Lemmas 4.1-4.2, we have

(4.5) h λ -→ 1, in L ∞ (R + × R) strongly,
and up to subsequences (not relabeled),

(4.6) (D 2λ , D 3λ , B 2λ , B 3λ , P 2λ , P 3λ ) --(D 2 , D 3 , B 2 , B 3 , P 2 , P 3 ), in L ∞ (R + × R) weakly- * . Recall ε = λ 2(α-1) → 0 as λ → ∞.
Since the strong convergence of h λ is involved in all its nonlinear terms, it is easy to pass to the limit in the system (4.1) in the sense of distributions. The weak limit (D 2 , D 3 , B 2 , B 3 ) satisfies the linear Maxwell equations :

(4.7)            ∂ t D 2 + ∂ x B 3 = 0, ∂ t D 3 -∂ x B 2 = 0, ∂ t B 2 -∂ x D 3 = 0, ∂ t B 3 + ∂ x D 2 = 0,
and (P 2 , P 3 ) satisfies (4.8)

∂ t P 2 -∂ x (D 1 D 2 + B 1 B 2 ) = 0, ∂ t P 3 -∂ x (D 1 D 3 + B 1 B 3 ) = 0.
Now consider the limit of h 1λ = (h λ -1)/ε and P 1λ . Using (4.1), we have (4.9)

     ∂ t h 1λ + ∂ x P 1λ = 0, ∂ t P 1λ + ∂ x ε 2 P 1λ -ρ 2 λ ε(1 + εh 1λ ) = 0.
From Lemma 4.1, the sequences (h 1λ ) λ and (P 1λ ) λ are bounded in L ∞ (R + × R). Then

ε 2 P 1λ ε(1 + εh 1λ ) -→ 0 in L ∞ (R + × R) strongly,
and up to subsequences, (4.10)

h 1λ --h 1 , P 1λ --P 1 , in L ∞ (R + × R) weakly- * . Since ρ 2 λ = 1 + εσ, 1 1 + εh 1λ = 1 -εh 1λ + O(ε 2 ), we obtain -∂ x ρ 2 λ ε(1 + εh 1λ ) = ∂ x h 1λ + e λ ,
with lim λ→∞ e λ = 0 in L ∞ (R + ; W -1,∞ (R)). Obviously, we can pass to the limit in (4.9) in the sense of distributions to obtain the linear equations :

(4.11) ∂ t h 1 + ∂ x P 1 = 0, ∂ t P 1 + ∂ x h 1 = 0.
On the other hand, the sequences (h 0 1λ ) λ and (v 0 iλ ) λ (2 ≤ i ≤ 8) being bounded in L ∞ (R), then up to subsequences, we have (4.12)

h 0 1λ --h 0 1 , v 0 iλ --v 0 i , in L ∞ (R) weakly- * , with (h 0 1 , v 0 2 , • • • , v 0 8 ) ∈ L ∞ (R).
Together with (4.3) for the expression of h 0 λ , they imply that

(P 0 1λ , D 0 2λ , D 0 3λ , B 0 2λ , B 0 3λ , P 0 2λ , P 0 3λ ) --(v 0 2 , • • • , v 0 8 ), in L ∞ (R)
∂ t (P 2 + D 1 B 3 -B 1 D 3 ) = ∂ t (P 3 + B 1 D 2 -D 1 B 2 ) = 0,

Limit toward PMHD equations in one dimension

In this last section, we justify the limit λ → 0 in the λ-ABI system (3.2) with the initial conditions : . By Proposition 3.1, to guarantee the hyperbolicity of the λ-ABI system for all λ ≥ 0, we require a λ = 0. Therefore, we assume here B 1 = 0.

Applying Proposition 3.2 to the Cauchy problem (3.2) and (5.1), we obtain a unique entropy solution

v λ = (h -1 λ , v 2λ , • • • , v 8λ ) t ∈ L ∞ (R + × R).
This solution satisfies the maximum principle (3.14)- (3.15). Together with the assumption (H2), we deduce that for sufficiently small λ > 0, (5.2) 0 < h ≤ h λ (t, x) ≤ h, a.e. (t, x) ∈ R + × R, where h and h are two constants independent of λ. From (3.16), v λ can be expressed as :

(5. ), it is easy to see that (l iλ ) λ and (r iλ ) λ are bounded sequences. We conclude from (5.2)-(5.3) that the sequence U λ = (h λ , P 1λ , D 2λ , D 3λ , B 2λ , B 3λ , P 2λ , P 3λ ) is bounded in L ∞ (R + × R). Then, up to subsequences, we have U λ --(h, P 1 , D 2 , D 3 , B 2 , B 3 , P 2 , P 3 ), in L ∞ (R + × R) weakly- * .

In order to pass to the limit in the nonlinear terms of (3.2), we proceed as in [START_REF] Peng | Euler-Lagrange change of variables in conservation laws[END_REF] using compensated compactness arguments. For instance, let

V λ = (h λ , P 1λ ) t , W λ = P 2 1λ -a 2 λ h λ , -P 1λ t .
Then (V λ ) λ and (W λ ) λ are two bounded sequences in L ∞ (R + × R) then in L 2 loc (R + × R). Moreover, they satisfy div t,x V λ = 0, rot t,x W λ = 0.

Proposition 3 . 1 .

 31 Let B 1 , D 1 ∈ R and h > 0. Assume a = 0. Then the λ-ABI system (3.3) is a linear Lagrangian one and hyperbolic for all λ ≥ 0. It admits a complete set of classical Riemann invariants

  which is in accordance with P = D × B. Then its Cauchy problem with the initial conditions :(4.13) t = 0 : (h 1 , P 1 , D 2 , D 3 , B 2 , B 3 , P 2 , P 3 ) = (h 0 1 , v 0 2 , • • • , v 0 8 )(x), x ∈ R admits a unique solution (h 1 , P 1 , D 2 , D 3 , B 2 , B 3 , P 2 , P 3 ) ∈ L ∞ (R + × R).This implies the convergence of the whole sequence.

Theorem 4 . 1 .

 41 Under the assumptions of Lemma 4.1, as λ → ∞, the solution of the Cauchy problem (4.1) and (4.3) is convergent in the sense of (4.5)-(4.6), (4.10) and (4.12), where the limit (h 1 , P 1 , D 2 , D 3 , B 2 , B 3 , P 2 , P 3 ) ∈ L ∞ (R + ×R) is the unique solution of the problem (4.7)-(4.8), (4.11) and (4.13).

(5. 1 )b 1

 11 t = 0 : h = h 0 λ (x), v i = v 0 iλ (x) (2 ≤ i ≤ 8), x ∈ R.Similarly to (3.10), we assume that the sequences of the initial data satisfy :(H1) (h 0 λ ) λ and (v 0 iλ ) λ (2 ≤ i ≤ 8) are bounded in L ∞ (and b 2 are two constants independent of λ, a λ = λ 2 + B 2 1 + λ 2 D 2 1

8 i=1(

 8 3) v λ (t, x) = l iλ v 0 λ )(X 0 λ (Y λ (t, x) -µ i t))r iλ ,where µ i , l iλ and r iλ are defined in (3.5)-(3.7), respectively, X 0 λ and Y λ are given by the formulas (3.11)-(3.13) in which (h 0 , v 0 2 ) are replaced by (h 0 λ , v 0 2λ ). Sincea λ = |B 1 | + O(λ 2 ) and β = B 2 1 + λ 2 D2 1 , from (3.6)-(3.7

  weakly- * .

	It is clear that the linear system (4.7)-(4.8) and (4.11) is hyperbolic. In particular, (4.7)-
	(4.8) imply that

Hence, by Tartar equation [START_REF] Tartar | Compensated compactness and applications to partial differential equations[END_REF] and up to subsequences, we have, in the sense of distributions lim

where χ is the weak limit of the sequence ((P 2 1λ -a 2 λ )/h λ ) λ . Since lim λ→0 a 2 λ = -B 2 1 , we obtain

Thus, (h, P 1 ) is a solution of the system :

(5.4)

Similarly, using the first and the rest equations in the λ-ABI system (3.2) and the compensated compactness arguments, we can pass to the limit in (3.2) in the sense of distributions to obtain :

(5.5)

Furthermore, as λ → 0, we have obviously (up to subsequences),

From Proposition 3.1 with B 1 = 0, we know that the system (5.4)-(5.5) is a linear Lagrangian one and hyperbolic. Then its Cauchy problem with the initial conditions :

(5.6) t = 0 : (h, P 1 , D 2 , D 3 , B 2 , B 3 , P 2 , P 3 ) = (h 0 , P 0 1 , D 0 2 , D 0 3 , B 0 2 , B 0 3 , P 0 2 , P 0 3 )(x), x ∈ R admits a unique entropy solution (h, P 1 , D 2 , D 3 , B 2 , B 3 , P 2 , P 3 ) ∈ L ∞ (R + × R). This implies the convergence of the whole sequence.


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

Ingrid Violet
email: violet@math.univ-bpclermont.fr

Example of supersonic solutions to a steady state Euler-Poisson system

Keywords: Euler-Poisson equations, steady state flow, supersonic solutions, existence, uniqueness. 2000 Mathematics Subject Classification. 35J25, 35Q35, 76J20

We give an example of supersonic solutions to a one-dimensional steady state Euler-Poisson system arising in the modeling of plasmas and semiconductors. The existence of the supersonic solutions which correspond to large current density is proved by Schauder's fixed point theorem. We show also the uniqueness of solutions in the supersonic region.

Introduction

The Euler-Poisson system plays an important role in the mathematical modeling and numerical simulation for plasmas and semiconductors [START_REF] Chen | Introduction to Plasma Physics and Controlled Fusion[END_REF][START_REF] Gardner | Numerical methods for the hydrodynamic device model : subsonic flow[END_REF][START_REF] Markowich | The Stationary Semiconductor Device Equations[END_REF]. In the steady state isentropic case the existence and uniqueness of smooth solutions are obtained in the subsonic region for a one-dimensional flow [START_REF] Degond | On a one-dimensional steady-state hydrodynamic model for semiconductors[END_REF] or potential flows [START_REF] Degond | A steady state potential flow model for semiconductors[END_REF]. See also [START_REF] Amster | Subsonic solutions to a onedimensional non-isentropic hydrodynamic model for semiconductors[END_REF] for the subsonic solutions to a one-dimensional non-isentropic model. In [START_REF] Gamba | Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductors[END_REF][START_REF] Gamba | A viscous approximation for a 2-D steady semiconductor or transonic gas dynamic flow : existence theorem for potential flow[END_REF], the stationary transonic solutions are studied by an artificial viscosity approximation. The existence of the transonic solutions is proved by passing to the limit in the approximate Euler-Poisson system as the viscosity coefficient goes to zero. However, the existence of the purely supersonic solutions has not been discussed yet.

In this paper, we give an example of the supersonic solutions in a one-dimensional steady state Euler-Poisson system : ∂ x j = 0, (1.1)

∂ x j 2 n + p(n) = n∂ x φ -j/τ, (1.2) 1 -∂ xx φ = b -n.
(1.3) Equation (1.1) implies that j is a constant. Here, n, j and φ are the electron density, the current density and the electric potential, respectively. The parameter τ > 0 stands for the momentum relaxation time depending on n and j in general. For simplicity, we assume that τ is a constant. The given function b = b(x) is the doping profile for the semiconductors. The pressure function p = p(n) is assumed to be smooth and strictly increasing for n > 0. As in [START_REF] Degond | On a one-dimensional steady-state hydrodynamic model for semiconductors[END_REF], we consider equations (1.1)-(1.3) in the interval (0, 1) subject to the following Dirichlet boundary conditions :

n(0) = n 0 , n(1) = n 1 , φ(0) = φ 0 , φ(1) = φ 1 , (1.4)
where n 0 > 0, n 1 > 0 and φ 0 ,

φ 1 ∈ IR are given data. If n > 0 is a smooth function, after eliminating φ in (1.2)-(1.
3), we obtain a Dirichlet problem for n :

-∂ xx F j (n) - 1 j ∂ x 1 τ n + 1 j 2 (n -b) = 0 in (0, 1), (1.5
)

n(0) = n 0 , n(1) = n 1 , (1.6)
where

F j (n) = 1 2n 2 + h(n) j 2 with h(n) = n 1 p (y) y dy.
Once n is solved, from (1.2) φ is given explicitly by :

φ(x) = φ 0 + j 2 (F j (n(x)) -F j (n 0 )) + x 0 j τ n(y)
dy.

(1.7)

Then φ 1 is linked with j by the following relation

φ 1 = φ 0 + j 2 (F j (n 1 ) -F j (n 0 )) + 1 0 j τ n(y) dy. (1.8)
It is easy to see that (n, φ) with n > 0 is a smooth solution of (1.2)-(1.4) if and only if (n, φ) is a smooth solution of (1.5)-(1.7). Therefore, we may first solve n to the Dirichlet problem (1.5)-(1.6) and then determine φ by (1.7). Now the equation (1.5) is elliptic if and only if F j (n) = 0. Since p is strictly increasing, there is a unique n c (j) such that F j (n c (j)) = 0, or equivalently

p (n c (j)) = |j| n c (j) .
Here the quantities c = p (n) and j/n stand for the speed of sound and the electron velocity, respectively. If n -→ n 2 p (n) is strictly increasing, we obtain the following alternative :

subsonic flow ⇐⇒ F (n) > 0 ⇐⇒ n > n c (j) =⇒ (1.5) is elliptic, (1.9) supersonic flow ⇐⇒ F (n) < 0 ⇐⇒ n < n c (j) =⇒ (1.5) is elliptic.
(1.10)

Note that the linear term n/j 2 in (1.5) has not a good sign. Nevertheless, it is small as j is large and then can be controlled by the L 2 (0, 1) norm of ∂ x n by Poincaré's inequality. Similar argument holds for the term ∂ x (1/jτ n). This is the main feature of the problem to yield the existence and uniqueness of solutions.

Existence of solutions

Assume b ∈ L ∞ (0, 1). In view of (1.9), the subsonic solutions to (1.2)-(1.4) correspond to the small value of j. They have been considered in [START_REF] Degond | On a one-dimensional steady-state hydrodynamic model for semiconductors[END_REF]. We study here the supersonic solutions which correspond to the case (1.10). To this end, let M 1 and M 2 be any two constants satisfying

0 < M 1 < min(n 0 , n 1 ), max(n 0 , n 1 ) < M 2 .
(2.1)

Choosing j such that n c (j) > M 2 , then (1.10) and (2.1) imply that the boundary data n 0 and n 1 are in the supersonic region. Since the maximum principle can not be applied to (1.5) in the supersonic region, the solutions of (1.5)-(1.6) may not be supersonic flow. To seek for a supersonic solution, we define a smooth and strictly decreasing function Fj on

IR + such that Fj (+∞) = 0, Fj (n) = F j (n) for all n ≤ M 2 .
Then we study the following problem instead of (1.5)-(1.6) :

-∂ xx Fj (n) - 1 j ∂ x 1 τ n + 1 j 2 (n -b) = 0 in (0, 1), (2.2
)

n(0) = n 0 , n(1) = n 1 . (2.3)
Our strategy is to prove the existence of a smooth solution n to (2.2)-( 2.3) such that 0 < n ≤ M 2 . Then n is a supersonic solution of (1.5)-(1.6) by the definition of Fj . Since Fj is smooth and strictly decreasing from IR + to IR + , we may make a change of variable v = Fj (n) for n > 0. Let G j be the inverse of Fj , which is also smooth and strictly decreasing from

IR + to IR + . Then the problem (2.2)-(2.3) is equivalent to -∂ xx v - 1 j ∂ x 1 τ G j (v) + 1 j 2 (G j (v) -b) = 0 in (0, 1), (2.4
)

v(0) = v 0j = F j (n 0 ), v(1) = v 1j = F j (n 1 ). (2.5)
To study the problem (2.4)-(2.5), we will apply Schauder's fixed point theorem. For this purpose, let's define a closed convex set

S = {v ∈ C([0, 1]); F j (M 2 ) ≤ v ≤ F j (M 1 )},
and a map T by v = T (σ) for σ ∈ S, where v solves the linear problem :

-∂ xx v + 1 jτ α j (σ)∂ x v + 1 j 2 β j (x, σ) = 0 in (0, 1), (2.6) v(0) = v 0j , v(1) = v 1j , (2.7)
with

α j (σ) = G j (σ) G 2 j (σ) = 1 G 2 j (σ) F j (G j (σ)) , β j (x, σ) = G j (σ) -b(x).
We observe that σ ∈ S implies that

F j (M 2 ) ≤ σ ≤ F j (M 1 ).
From Fj (σ) = F j (σ) for σ ≤ M 2 , we have

M 1 ≤ G j (σ) ≤ M 2 .
Therefore, from the definition of F j , there is a j 1 > 0 depending only on M 1 and M 2 such that α j and β j are two bounded functions with bounds depending on M 1 and M 2 but independent of j and σ for any j ∈ IR satisfying |j| ≥ j 1 .

For v ∈ H 1 (0, 1) and z ∈ H 1 0 (0, 1), let

a(v, z) = 1 0 ∂ x v∂ x z + 1 jτ α j (σ)z∂ x v dx, l(z) = - 1 j 2 1 0 β j (x, σ)zdx.
It is clear that l(•) is linear and continuous on H 1 0 (0, 1), and a(•, •) is bilinear and continuous on H 1 0 (0, 1) × H 1 0 (0, 1). Moreover, by Poincaré's inequality,

a(z, z) = 1 0 (∂ x z) 2 + 1 jτ α j (σ)z∂ x z dx ≥ ||∂ x z|| 2 L 2 (0,1) - 1 |j|τ ||α j || L ∞ (0,1) ||z|| L 2 (0,1) ||∂ x z|| L 2 (0,1) ≥ 1 - C 1 |j|τ ||α j || L ∞ (0,1) ||∂ x z|| 2 L 2 (0,1) , ∀ z ∈ H 1 0 (0, 1),
where C 1 > 0 is the constant in Poincaré's inequality. Then there exists a

j 2 ≥ 2C 1 τ ||α j || L ∞ (0,1) depending only on M 1 and M 2 such that a(z, z) ≥ 1 2 ||∂ x z|| 2 L 2 (0,1) , ∀ |j| ≥ j 2 , ∀ z ∈ H 1 0 (0, 1). (2.8)
Therefore, a(•, •) is coercive. By Lax-Milgram's theorem, there exists a unique solution v ∈ H 1 (0, 1) to the variational problem a(v, z) = l(z), ∀ z ∈ H 1 0 (0, 1) and (2.7). This shows that the map T is well defined.

We prove now that T (S) is a compact set of C([0, 1]). Indeed, let v j = (1-x)v 0j +xv 1j . Then v -v j ∈ H 1 0 (0, 1). From the continuity of l(•) and a(•, •), the coercivity estimate (2.8) and

a(v -v j , v -v j ) = l(v -v j ) -a(v j , v -v j ),
it is easy to obtain

||∂ x (v -v j )|| L 2 (0,1) ≤ 2C 1 j 2 ||β j || L ∞ (0,1) + 2C 1 |j|τ ||α j || L ∞ (0,1) ||∂ x v j || L 2 (0,1) .
(2.9)

Recall that α j and β j are bounded independent of σ. We conclude from Poincaré's inequality and the compact imbedding from H 1 (0, 1) into C([0, 1]) that T (S) is a compact set of C([0, 1]). Moreover, there are constants C 2 > 0 and j 3 ≥ j 2 which depend only on M 1 and M 2 such that

|v(x) -v j (x)| ≤ C 2 |j| , ∀ |j| ≥ j 3 , ∀ x ∈ [0, 1]. Since F j (max(n 0 , n 1 )) ≤ v j (x) ≤ F j (min(n 0 , n 1 )), ∀ x ∈ [0, 1],
it follows that

F j (max(n 0 , n 1 )) - C 2 |j| ≤ v(x) ≤ F j (min(n 0 , n 1 )) + C 2 |j| , ∀ |j| ≥ j 3 , ∀ x ∈ [0, 1].
The function n -→ F j (n) being strictly decreasing for n ≤ M 2 , from (2.1) there is a j 4 ≥ j 3 depending only on M 1 and M 2 such that

F j (M 2 ) ≤ v(x) ≤ F j (M 1 ), ∀ |j| ≥ j 4 , ∀ x ∈ [0, 1]. (2.10)
Hence, v ∈ S and then T is a self map from S to S. Finally, the continuity of T follows from a standard argument. More precisely, for σ 1 , σ 2 ∈ S, we can prove that there is a constant C 3 > 0 depending only on M 1 and M 2 such that

1 - C 3 |j|τ ||T (σ 1 ) -T (σ 2 )|| C([0,1]) ≤ C 3 |j|τ ||σ 1 -σ 2 || C([0,1]) .
Thus, T is continuous for |j| > j 5 = max(j 4 , C 3 /τ ). We conclude from Schauder's fixed point theorem the existence of a solution v ∈ H 1 (0, 1) ∩ S of v = T (v). This shows the existence of a solution v ∈ H 1 (0, 1) ∩ S to the problem (2.4)-(2.5), and then the existence of a solution n = G j (v) ∈ H 1 (0, 1) to the problem (2.2)- (2.3)

. Since v = Fj (n) = F j (n) for n ≤ M 2 , from (2.10) we obtain M 1 ≤ n(x) ≤ M 2 , ∀ |j| ≥ j 5 , ∀ x ∈ [0, 1].
(2.11) Therefore, n ∈ H 1 (0, 1) is a supersonic solution to the problem (1.5)- (1.6). Thus, we have proved Theorem 1 Let n 0 > 0 and n 1 > 0. Let M 1 , M 2 be two constants satisfying (2.1) and b ∈ L ∞ (0, 1). Then there exists a j e > 0 depending only on M 1 and M 2 such that for any current density j satisfying |j| ≥ j e , the problem (1.2)-(1.4) admits a solution (n, φ) ∈ H 1 (0, 1) × H 1 (0, 1). This solution is located in the supersonic region and satisfies (2.11).

There doesn't exist a general result on the uniqueness of solutions when the boundary data are located in the supersonic region. Indeed, for large j the formation of shocks cannot be avoided and the transonic solutions should be investigated. We refer to [START_REF] Gamba | Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductors[END_REF][START_REF] Gamba | A viscous approximation for a 2-D steady semiconductor or transonic gas dynamic flow : existence theorem for potential flow[END_REF] for the analysis of the transonic solutions. Here we give a uniqueness result in the supersonic region for large j. This result can be stated as follows.

Theorem 2 Let M 1 and M 2 be two constants with 0 < M 1 < M 2 . Let (n (1) , φ (1) ) and (n (2) , φ (2) ) be two supersonic solutions of (1.2)-(1.3) in H 1 (0, 1) × H 1 (0, 1) with M 1 ≤ n (1) , n (2) ≤ M 2 . Then there exists a j u > 0 depending only on M 1 and M 2 such that for any current density j ∈ IR satisfying |j| ≥ j u , we have (n (1) , φ (1) ) = (n (2) , φ (2) ).

Proof. In view of (1.7), it suffices to show that n (1) = n (2) . Let w = n (2) -n (1) . By subtracting the equation (1.5) satisfied by n (1) and n (2) we obtain :

∂ xx (A j (x)w) + 1 jτ ∂ x (B(x)w) + 1 j 2 w = 0 in (0, 1), (

where A j (x) = -1 0 ∂F j ∂n n (1) (x) + s n (2) (x) -n (1) (x) ds,

1 M 2 2 ≤ B(x) = 1 n (1) n (2) ≤ 1 M 2 1 in (0, 1). From F j (n) = - 1 n 3 + h (n) j 2 ,
it is easy to check that there are constants C 4 > 0 and j 6 > 0 which depend only on M 1 and M 2 such that A j (x) ≥ C 4 , ∀ |j| ≥ j 6 , ∀ x ∈ [0, 1].

Multiplying (3.1) by A j w ∈ H 1 0 (0, 1) and integrating over (0, 1) give : It follows from Poincaré's inequality that :

||∂ x (A j w)|| 2 L 2 (0,1) ≤ 1 C 4 C 0 M 2 1 |j|τ + C 2 0 j 2 ||∂ x (A j w)|| 2 L 2 (0,1) .
This shows that A j w = 0 and then w = 0 provided that |j| ≥ j 7 for some large j 7 > 0 depending only on M 1 and M 2 .

1 0[

 1 ∂ x (A j (x)w)] 2 dx = )w∂ x (A j (x)w) + 1 j 2 A j (x)w 2 dx.

Acknowledgments :

The authors acknowledge partial support from the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282.