Wassim Nasserddine

Harmonic analysis/ Analyse harmonique On the Hausdorff-Young theorem

Let G mn = ax + b be the matricial group of a local field. The Hausdorff-Young theorem for G 11 was proved by Eymard-Terp [3] in 1978. We will establish here the Hausdorff-Young theorem for G nn for all n ∈ N.

Résumé

Soit G mn = ax + b un groupe matriciel d'un corps local. En 1978 Eymard-Terp [3] ont prouvé le théorème de Hausdorff-Young sur G 11 . On établit ici le théorème de Hausdorff-Young sur G nn pour tout n ∈ N.

Introduction

The Hausdorff-Young theorem was generalized by several authors as they are passing over from a locally compact Abelian group to a locally compact group which is not Abelian but which is unimodular. In this paper we will tackle G mn = ax + b the matricial group of a local field, to be defined below, which is not unimodular and the theory will involve unbounded operators, as the case was before if m = n = 1.

Let K be a local field, n, m ∈ N, m ≥ n ≥ 1. Let M nm be the space of all n × m-matrices with elements from K and GL n be the multiplicative group of all n × n-inversible matrices with elements from K. G nm denotes the group of pairs (b, a), where b ∈ M nm and a ∈ GL n , with multiplication given by (b, a)(b , a ) = (b + ab , aa ). It is the semi-direct product M nm GL n .

Let H be the Hilbert space L 2 (GL n , du |det(u)| n ). For all λ in M mn , the formula [π λ (b, a)ξ](u) = τ (T r(bλu))ξ(ua), [START_REF] Bakali | Analyse harmonique sur les groupes ax + b matriciels[END_REF] defines a unitary representation of G nm in H, where τ is a fixed additive unitary nontrivial character on K (see [START_REF] Gel'fand | Representation theory and Automorphic Function[END_REF], p.224 and p.142). We define L q (H), 1 ≤ q < ∞, to be the Banach space of bounded linear operators on H with A q = {T r(|A| q )} 1 q < ∞, where for a bounded operator A, |A| denotes the operator (A * A) 1 2 and A ∞ will denote the operator norm of A. δ q denotes the unbounded operator in H defined by δ q ξ(u) = |det(u)| mq ξ(u).

Let G ess be the "essential" dual of G = G nm , 1 < p < 2,1 p + 1 q = 1. We will prove that if f ∈ D(G), the space of regular functions with compact support on G, then

F p (f ) ∈ L q ( G ess , L q (H)),
where F p (f )λ := Ff (λ) • δ 1/q := π λ (f ) • δ 1/q , and that if n = m, f ∈ L 1 ∩ L p (G), then the map f -→ F p (f ) extends uniquely to a linear map of L p (G) into L q ( G ess , L q (H)) with norm ≤ 1.

PRELIMINARY

In 1912 Young proved the Hausdorff-Young theorem for G = the circle group when p = 2k/(2k-1), 2 ≤ k ∈ N. In 1923 Hausdorff generalized this result for all p, 1 < p < 2. In 1940 Weil established this theorem for any locally compact Abelian groups. For groups which are not Abelian but which are unimodular, Kunze [START_REF] Kunze | Lp Fourier transforms on locally compact unimodular groups[END_REF] defined and proved the Hausdorff-Young theorem by using the theory of non-commutative integration. In 1974 Lipsman [START_REF] Lipsman | Non-Abelian Fourier analysis[END_REF] gave a concrete realization if those unimodular groups are separable and of type I. In this last case, let G be the set of unitary equivalence classes of continuous irreducible unitary representations of G and L p ( G) = {F is a measurable field of bounded operators on G :

F (π) p < ∞, µ almost all π ∈ G, b G F (π) p p dµ(π) < ∞, 1 ≤ p < ∞} If we write F 1 ∼ F 2 when F 1 (π) = F 2 (π) for µ-almost all π, then L p ( G) = L p ( G)/ ∼ is a Banach space under the norm F p = b G F (π) p p dµ(π) 1 p , 1 ≤ p < ∞,
and the Hausdorff-Young theorem ( [START_REF] Lipsman | Non-Abelian Fourier analysis[END_REF], th.2.2, p.214) asserts that if

1 < p < 2, 1 p + 1 q = 1, f ∈ L 1 ∩ L p (G), then F(f ) := f ∈ L q ( G) and f -→ f, where f (π) := G f (g)π(g)dg, extends uniquely to a map of L p (G) into L q ( G) with norm ≤ 1.
If the group G is not unimodular, new problemes appear to thoroughly deal with the Fourier transformation in L p (G). In our case, that is G = G nm , like Eymard-Terp's case [START_REF] Eymard | La transformation de Fourier et son inverse sur le groupe des ax + b d'un corps local. (French) Analyse harmonique sur les groupes de Lie[END_REF] which helped us as a model, we are led to temper the Fourier transformation of L 1 (G) by the unbounded operators δ 1 q . G nm is a locally compact group on which we fix the left Haar measure to be d(b, a) = dbda |det(a)| n+m , where db (resp. da) is the canonical measure on M nm K nm (resp. on GL n ⊂ M nn ) deduced from the Haar measure on K, and |.| is the module in K (for a ∈ K, |a| is defined by d(ax) = |a|dx, where dx is a Haar measure on (K, +) (see [START_REF] Weil | Basic number theory[END_REF], p.4) ). G nm is not unimodular ( dbda |det(a)| n is a right Haar measure) and its modular function is ∆ nm (b, a) =

1 |det(a)| m .
Letting Grass(m, k) (0 ≤ k ≤ min(n, m)) denote the set of k-finite dimensional vector subspaces in K m , M mk (k) denote the set of all m × k-matrices with rank k and S mk ([1], p.63) denote the canonical realization of Grass(m, k) in M mk (k). S mk is a compact subspace of M mk (k). There exists ( [START_REF] Bakali | Analyse harmonique sur les groupes ax + b matriciels[END_REF], prop.3.1.2, p.85) only one measure s on S mk (m, k ≥ 1, k ≤ m) such that, for all f ∈ L 1 (M mk , dλ), we have:

M mk f (λ)dλ = S mk GL k f (µa)|deta| m da |det(a)| k ds(µ) (2)
s is called the canonical measure of S mk . From now on, G and S denote respectively G nm and S mn . The representation π λ defined by (1) is only irreducible if rang(λ) = n, and the role of essential dual of G is played by S via the connection S λ -→ π λ . Let us recall two important results (which are in [START_REF] Bakali | Analyse harmonique sur les groupes ax + b matriciels[END_REF], p.96, p.118) for the groups G = G nm , m ≥ n:

Theorem 1 ( Plancherel Theorem). Let f ∈ D(G), then F(f ) • δ 1/2 ∈ L 2 (S, L 2 (H)) and f -→ F(f ) • δ 1/2 extends uniquely to an isometric, isomorphism of L 2 onto L 2 (S, L 2 (H)).
Theorem 2 ( Fourier Inversion Theorem). Let A(G) be the Fourier algebra of G and

f ∈ L 1 ∩ A(G), then F(f ) • δ 1 ∈ L 1 (S, L 1 (H))
and , for all x ∈ G, we have:

f (x) = S T r(π λ (x -1 )F(f ) • δ 1 )ds(λ).
2. The Hausdorff-Young theorem for the matricial groups

G nn = ax + b Lemma 3 ([1], p.94). Let K be a disconnected local field. If f ∈ D(G), then the function (λ, a) -→ f (., a)λ vanishes off a compact subset of M mn × GL n , where f(., a)(λ) = Mnm f (b, a)τ (T r(bλ))db.
Definition. We call the Fourier transform of f ∈ L 1 (G) and we denote by Ff , the function

λ -→ Ff (λ) = π λ (f ) defined in S with values in L ∞ (H) the space of bounded operators on H. Remark. If f ∈ L 1 (G), then [Ff (λ)]ξ(u) = Mnm GLn τ (T r(bλu))ξ(ua)f (b, a) dbda |det(a)| n+m , ξ ∈ H, (3)
in fact, let ξ and η in H. Then

| < Ff (λ)ξ, η > | = | < π λ (f )ξ, η > | = | G < π λ (g)ξ, η > f (g)dg| ≤ G GLn |τ (T r(bλu))ξ(ua)η(u)| du |det(u)| n |f (b, a)|d(b, a) ≤ G |f (b, a)|d(b, a) GLn |ξ(ua)||η(u)| du |det(u)| n < ∞, by Fubini's theorem, the integral G τ (T r(bλu))ξ(ua)η(u)f (b, a)d(b, a) exists, for almost all u ∈ GL n ,

and we have

GLn G τ (T r(bλu))ξ(ua)f (b, a)d(b, a) η(u) du |det(u)| n =< Ff (λ)ξ, η > = GLn Ff (λ)ξ(u)η(u) du |det(u)| n , since η is arbitrary and d(b, a) = dbda |det(a)| n+m , we see that (3) is valid. Lemma 4. Let K be a local field, and let 1 ≤ p ≤ 2, 1/p + 1/q = 1, f ∈ D(G). Then, for all λ ∈ M mn (n), the operator F p (f )λ := Ff (λ) • δ 1/q := π λ (f ) • δ 1/q extends to a Hilbert-Schmidt operator on H that we denote again Ff (λ) • δ 1/q . Proof. Let ξ ∈ dom(δ 1/q ), then [F p (f )λ]ξ(u) = [π λ (f ) • δ 1/q ]ξ(u) = Mnm GLn π λ (b, a)(δ 1/q ξ)(u)f (b, a) dbda |det(a)| n+m = Mnm GLn τ (T r(bλu))δ 1/q ξ(ua)f (b, a) dbda |det(a)| n+m = τ (T r(bλu))|det(a)| m q ξ(a)f (b, u -1 a) dbd(u -1 a) |det(au -1 )| n+m (a → u -1 a) = GLn f (., u -1 a)(λu) |det(a)| m p-1 p |det(au -1 )| m ξ(a) da |det(a)| n = GLn K λ f,p (u, a)ξ(a) da |det(a)| n Hence F p (f )λ acts in H with the kernel K λ f,p (u, a) = f(., u -1 a)(λu) |det(a)| m p-1 p |det(au -1 )| m
This kernel is Hilbert-Schmidt, in fact

GLn GLn |K λ f,p (u, a)| 2 duda |det(u)| n |det(a)| n = Mnn GLn | f (., u -1 a)(λu)| 2 |det(a)| 2m p-1 p |det(au -1 )| 2m duda |det(u)| n |det(a)| n = | f (., a)(λu)| 2 |det(ua)| 2m p-1 p |det(a)| 2m dud(au) |det(u)| n |det(au)| n (a → ua) = | f (., a)(λu)| 2 |det(a)| 2m p-1 p -n-2m |det(u)| 2m p-1 p -n duda
If K is disconnected, this last integral is finite since by lemma 3 (µ, a) -→ f (., a)µ vanishes off a compact subset of M mn × GL n . If K = R or C, let K n be the projection of suppf on GL n . For all u, if a / ∈ K n , we have f (b, a) = 0, ∀b, from which f(., a)(λu) = Mnm f (b, a)τ (T r(bλu))db = 0 follows. On the other hand, the functions µ -→ f(., a)µ are, uniformly on a, rapidly decreasing on M mn , hence

|det(µ * µ)| m(p-1) p -n 2 | f (., a)(µ)| 2 ≤ C 0 1 + µ 4n , for all µ ∈ M mn , where the constant C 0 is independent of a. Setting µ = λu, then |det(u)| 2 m(p-1) p -n 2 = | det(µ * µ) det(λ * λ) | m(p-1) p -n 2 It follows that GLn GLn |K λ f,p (u, a)| 2 duda |det(u)| n |det(a)| n ≤ C 0 det(λ * λ) m(p-1) p -n 2 Kn |det(a)| 2m(p-1) p -n-2m da Mnn du 1 + λu 4n < ∞ Lemma 5. Let us define f * p by f * p (g) = |det(a)| m-n+ m p f (g -1 )
, where g = (b, a). Then, for all f in D(G), we have:

F p (f * p ) = [F p (f )] * Proof. One only has to show that [F p (f )λ] * acts in H = L 2 (GL n , du |det(u)| n ) with the kernel K λ f * p ,p (u, a) = K λ f,p (a, u)
which is Hilbert-Schmidt by the previous lemma.

Letting g = (b, a), then g -1 = (-a -1 b, a -1 ), so that

[f * p ] ∧ (., au -1 )(λu) = Mnm f * p (b, u -1 a)τ (T r(bλu))db = |det(au -1 )| m-n+ m p f (- b u -1 a , 1 u -1 a )τ (T r(bλu))db = |det(au -1 )| m-n+ m p |det(u -1 a)| n f (-b, a -1 u)τ (T r(bλa))db (b → u -1 ab) = |det(au -1 )| m+ m p f (-b, a -1 u)τ (T r(bλa))db Thus K λ f * p ,p (u, a) = [f * p ] ∧ (., u -1 a)(λu) |det(a)| m p-1 p |det(au -1 )| m = |det(au -1 )| m+ m p f (., a -1 u)(λa) |det(a)| m p-1 p |det(au -1 )| m = |det(u)| m-m p -m |det(a)| m+ m p + m(p-1) p -m f (., a -1 u)(λa) = f (., a -1 u)(λa) |det(u)| m p-1 p |det(ua -1 )| m = K λ f,p (a, u) Theorem 6 ([7], p.177). Let 1 < p < 2, 1/p + 1/q = 1, and let k ∈ L 2 (X × X)
, where X is a σ-finite measure space. If K is the integral operator with kernel k, then

K q ≤ ( k p,q k * p,q ) 1 2 ,
where k p,q = ( ( |k(x, y)| p dx) q p dy)

1 q et k * (x, y) = k(y, x).
The main result of this paper is theorem 7 which follows. If m = n in its last statement, the annouced result is obtained. Let us give some simple formulae to use afterwards.

Let λ ∈ M mn and C(λ) := inf { λx , x ∈ M nn , x = 1}, then ([1] lemme3.1.1, p.84) we have

|C(λ) -C(µ)| ≤ λ -µ , (4)
for all λ et µ in M mn . In fact, for all x ∈ M nn with x = 1, we have

C(λ) ≤ λx ≤ λx -µx + µx =⇒ C(λ) ≤ λ -µ + C(µ), similarly, C(µ) ≤ λ -µ + C(λ), from which (4) follows.
Let λ 0 be fixed in M mn , by the definition of C(λ), we have, for all 0

= u ∈ M nn , C(λ) ≤ λ u u and λu ≥ C(λ) u If λ is very close to λ 0 , then λ -λ 0 ≤ C(λ0) 2 , it follows from (4) that |C(λ) -C(λ 0 )| ≤ C(λ0) 2 and C(λ) ≥ C(λ0) 2 . Therefore λu ≥ C(λ 0 ) 2 u (5) λ 0 u ≥ C(λ 0 ) 2 u (6)
Theorem 7. Let f ∈ D(G), and let K n be the projection of suppf on GL n , then (i) For all λ ∈ M mn (n), the operator F p (f )λ is in L q (H).

(ii) The function λ -→ F p (f )λ is continuous of M mn (n) into the Banach space L q (H). In particular that F p is a linear map of D(G) into L q (S, L q (H)).

(iii) F p (f ) q ≤ C 1 2 f f p , where C f = sup a∈Kn |det(a -1 )| (m-n)p 1 p .
Proof. Let us show that F p (f )λ is in L q (H), in fact, since by lemma 4 F p (f )λ acts in H with the kernel of Hilbert-Schmidt

K λ f,p (u, a) = f (., u -1 a)(λu) |det(a)| m p-1 p |det(au -1 )| m ,
it follows from theorem 6 that

F p (f )λ q ≤ K λ f,p p,q (K λ f,p ) * p,q 1 2
,

where K λ f,p p,q = GLn GLn |K λ f,p (u, a)| p du |det(u)| n q p da |det(a)| n 1 q and (K λ f,p ) * (u, a) = K λ f,p (a, u) Now (K λ f,p ) * q p,q = GLn GLn |(K λ f,p ) * (a, u)| p da |det(a)| n q p du |det(u)| n = GLn GLn |K λ f,p (u, a)| p da |det(a)| n q p du |det(u)| n = | f (., u -1 a)(λu)| p |det(a)| m(p-1) |det(au -1 )| mp da |det(a)| n q p du |det(u)| n = | f (., a)(λu)| p |det(au)| m(p-1) d(ua) |det(a)| mp |det(ua)| n q p du |det(u)| n (a → ua) = Mnn GLn | f(., a)(λu)| p |det(a)| -m-n |det(u)| m(p-1)-np q da q p du (7)
If K is disconnected, then this last integral is finite by lemma 3 since f ∈ D(G) and λ is fixed. If K = R or C, let K n be the projection of suppf on GL n and µ = λu. Then

|det(µ * µ)| m(p-1) 2 -np 2q | f (., a)(µ)| ≤ C 0 1 + µ 4n ,
and consequently

(K λ f,p ) * q p,q ≤ C 0 |det(λ * λ )| m(p-1) 2 -np 2q q p |det(a)| -m-n da 1 + λu 4n q p du ≤ C 0 |det(λ * λ )| m(p-1) 2 -np 2q q p Kn |det(a)| -m-n da q p Mnn du (1 + λu 4n ) q p < ∞ because q p > 1)
By the proof of lemma 5, we have

K λ f * p ,p (u, a) = K λ f,p (a, u), hence K λ f,p q p,q = GLn GLn |K λ f,p (a, u)| p da |det(a)| n q p du |det(u)| n = GLn GLn |K λ f * p ,p (u, a)| p da |det(a)| n q p du |det(u)| n
On the other hand, the function (b, a) -→ |det(a)| m-n+ m p is, infinitely differentiable if K = R or C, locally constant when K is disconnected since the module |.| in K is locally constant. Therefore f * p ∈ D(G) and the previous reasoning implies that K λ f,p q p,q < ∞, and thus F p (f )λ ∈ L q (H).

(ii) Let λ 0 be fixed in M mn (n), and let λ be very close to λ 0 in M mn (n). We define K λ,λ0 f,p to be the kernel of the operator F p (f )λ -F p (f )λ 0 . Then (repeat the same computation which gave ( 7))

(K λ,λ0 f,p ) * q p,q = Mnn GLn | f(., a)(λu) -f(., a)(λ 0 u)| p |det(a)| -m-n |det(u)| m(p-1)-np q da q p du
If K is disconnected, then, by lemma 3, and by ( 5),( 6), we can integrate, in a neighborhood of λ 0 , on a compact K 1 × K 2 of M n × GL n which is independent of λ, so | f(., a)(λu)| ≤ 1 K1×K2 (u, a)2| f(., a)(λ 0 u) for all λ very close to λ 0 , and by Lebesgue's dominated convergence theorem, we have (K λ,λ0 f,p ) * q p,q -→ 0 when λ -→ λ 0 .

If K = R or C, there exists an independent constant C 1 of a such that, for all µ and ν in M mn , we have

| f (., a)(µ) -f(., a)(ν)| ≤ C 1 µ -ν , since the functions b -→ f (b, a) vanish off a compact part of M nm which is independent of a.
Let K n be the projection of suppf on GL n . For any > 0, we choose a compact subset K of M nn such that

Mnn\K 2 1 + ( C(λ0) 2 ) 4n u 4n q p du ≤ Then (K λ,λ0 f,p ) * q p,q = K Kn | f (., a)(λu) -f (., a)(λ 0 u)| p |det(a)| -m-n |det(u)| m(p-1)-np q da q p du + Mnn\K Kn | f (., a)(λu) -f (., a)(λ 0 u)| p |det(a)| -m-n |det(u)| m(p-1)-np q da q p du ≤ [(C 1 λ -λ 0 ) p ] q p K Kn u p |det(a)| -m-n |det(u)| m(p-1)-np q da q p du + C q p 0 C λ0 Mnn\K Kn ( |det(a)| -m-n 1 + λu 4n + |det(a)| -m-n 1 + λ 0 u 4n )da q p du where C λ0 is a constant depending on λ 0 ≤ C q p 1 λ -λ 0 q C K ( where C K is a constant depending on K ) + C q p 0 C λ0 Kn |det(a)| -m-n da q p Mnn\K 1 1 + λu 4n + 1 1 + λ 0 u 4n q p
du Since, by ( 5) and ( 6), we have

Mnn\K 1 1 + λu 4n + 1 1 + λ 0 u 4n q p du ≤ Mnn\K 2 1 + ( C(λ0) 2 ) 4n u 4n q p du ≤ ,
it follows that (K λ,λ0 f,p ) * p,q -→ 0 when λ -→ λ 0 . Similarly, we prove that K λ,λ0 f,p p,q -→ 0 when λ -→ λ 0 , and the inequality of theorem 6 completes the proof of (ii).

(iii) In fact, we have

S F p (f )λ q q ds(λ) ≤ S K λ f,p q 2 p,q (K λ f,p ) * q 2
p,q ds(λ) by theorem 6

≤ S K λ f,p q ds(λ) 1 2

S

(K λ f,p ) * q ds(λ)

1 2
by Hölder inequality On the other hand, from the calculation which gave [START_REF] Russo | Recent advances in the Hausdorff-Young theorem[END_REF], it follows that

S (K λ f,p ) * q ds(λ) = S GLn GLn | f (., a)(λu)| p |det(a)| -m-n |det(u)| m(p-1)-np q da q p duds(λ) ≤ GLn S GLn | f(., a)(λu)| q det(u)| m-n duds(λ) p q da |det(a)| m+n q p
We used the Minkowski's generalized inequality (α ≥ 1):

( |φ(a, λ, u)|d µ (a)) α d ν (u)ds(λ) = f q p , by applying to [START_REF] Weil | Basic number theory[END_REF] the Hausdorff-Young theorem for the additive group M nm since the formula τ λ (b) = τ (T r(bλ)) defines a unitary character of the additive group M nm and λ -→ τ λ is an isomorphism of the additive group M mn onto the dual of (M nm , +).

Similarly, since K λ f * p ,p (u, a) = K λ f,p (a, u) by the proof of lemma 5, and f * p ∈ D, we show that

S K λ f,p q ds(λ) ≤ f * p q p Now G |f * p | p dg = G |det(a)| (m-n)p+m | f (g -1 )| p dg = G |det(a -1 )| (m-n)p+m | f(g)| p ∆(g -1 )dg (g → g -1 ) = G |det(a -1 )| (m-n)p | f(g)| p dg ≤ (C f ) p f p p Hence S K λ f,p q ds(λ) ≤ f * p q p ≤ (C f f p ) q ,
and therefore

F p (f ) q = S F p (f )λ q q ds(λ) 1 q ≤ (C f ) 1 2 f p
Remarks. If m < n, the theorems 1, 2 are already true, but in this case you have to alter the definitions of S, H, π λ , δ 1/q (see [START_REF] Bakali | Analyse harmonique sur les groupes ax + b matriciels[END_REF]); and it may not be difficult to prove an analogy of theorem 7 with another C f in (iii). We tackled the case where m ≥ n to be able to reach the value m = n and obtain the Hausdorff-Young theorem.

Theorem 8 ( Hausdorff-Young Theorem).

If n = m, 1 < p < 2, 1 p + 1 q = 1, f ∈ L 1 ∩ L p (G),
then F p (f ) ∈ L q (S, L q (H)) and f -→ F p (f ) extends uniquely to a, norm-decreasing, linear map of L p (G) into L q (S, L q (H)) with norm ≤ 1.

Proof. In fact, by theorem 7 (even if m ≥ n) we have F p (f ) ∈ L q (S, L q (H)) for any f ∈ D(G).

If m = n, we get (C f ) 1 2 = 1, hence f -→ F p (f ) is continuous of (D(G), . p ) into L q (S, L q (H)) with norm ≤ 1, it extends uniquely to L p (G), in particular, if f ∈ L 1 ∩ L p (G), we take a sequence (f k ) ∈ D(G) which tends to f in norm of L 1 (G) and in norm of L p (G). Corollary 9. Setting 1 < p < 2, 1 p + 1 q = 1, x ∈ R. Then (i) The domain of δ x F p (f ) is dense in H for any f ∈ D(G). (ii) If n = m, f ∈ L p (G) and ∆ x f ∈ L p (G), then Dom(δ x F p (f ) is also dense in H. Proof. (i) Let f ∈ D(G), since ∆ x f is also ∈ D(G)
, from the proof of lemma 4 it follows that F p (∆ x f )λ acts in H with the kernel of Hilbert-Schmidt

K λ ∆ x f,p (u, a) = [∆ x f ] ∧ (., u -1 a)(λu) |det(a)| m p-1 p |det(au -1 )| m = Mnm ∆ x f (b, u -1 a)τ (T r(bλu))db = |det(u -1 a)| -mx K λ f,p (u, a) Let ξ ∈ Dom(δ x ), then [F p (∆ x f )λ]δ x ξ(u) = GLn K λ ∆ x f,p (u, a)δ x ξ(a) da |det(a)| n = GLn |det(u -1 a)| -mx K λ f,p (

Hence

F p (∆ x f )δ x ⊆ δ x F p (f ), and consequently the domain of δ x is stable by F p (f ) and thus, since Dom(δ x ) is dense in H, (i) follows.

(ii) Let us take a sequence (f k ) ∈ D(G) such that f k -→ f and ∆ x f k -→ ∆ x f in L p (G). Then, for any ξ ∈ Dom(δ x ), and for almost all u, we have Thus F p (∆ x f )δ x ⊆ δ x F p (f ), which completes the proof of corollary 9.

F p (∆ x f )δ x ξ(u) = lim k F p (∆ x f k )δ x
If n divides m, the essential dual of G amounts to a single point by ([1], p.80) which always bears the Plancherel measure and the Fourier inverse theorem (see th.1 and th.2). Whether the essential dual is very large or a single point, the proof of theorem 7 asserts that the constant C f only depends on f, m and n. So let us raise the following problem Problem 1. What happens to the last two statements if m > n?

Remarks. In Eymard-Terp's article ( [START_REF] Eymard | La transformation de Fourier et son inverse sur le groupe des ax + b d'un corps local. (French) Analyse harmonique sur les groupes de Lie[END_REF], p.227), if m = n = 1, the following result appears

F p (f ) q ≤ A p f p , A p = p 1 2p q -1 2q , f ∈ L 1 ∩ L p (G).
Their proof relies on Babenko-Beckner's inequality ( [START_REF] Beckner | Inequalities in Fourier analysis[END_REF], th.1, p.162) which is valid in the case where K = R (or K = R n , A p → A n p ). If K is disconnected, this inequality fails by ( [START_REF] Russo | Recent advances in the Hausdorff-Young theorem[END_REF], th.B, p.180).

If K is connected, in theorem 8, the norm of F p is strictly < 1, F p ≤ A nm p if K = R, and F p ≤ A 2nm p when K = C, in fact, you just have to apply ( [START_REF] Beckner | Inequalities in Fourier analysis[END_REF], th.1, p.162) to [START_REF] Weil | Basic number theory[END_REF].

  , λ, u)| α d ν (u)ds(λ) 1 α d µ (a), with φ(a, λ, u) = | f (., a)(λu)| p |det(u)| m(p-1)-np q , and α = q p , d µ (a) = da |det(a)| m+n , d ν (u) = du. From (2), for k = n, we have GLn S GLn | f (., a)(λu)| q det(u)| m-n duds(λ) |f (b, a)| p dbda |det(a)| m+n q p

  u, a)|det(a)| mx ξ(a) da |det(a)| n = |det(u)| mx GLn K λ f,p (u, a)ξ(a) da |det(a)| n = δ x F p (f )λξ(u).

8 =

 8 )| mx F p (f k )ξ(u) = |det(u)| mx F p (f )ξ(u)by theorem δ x F p (f )ξ(u).

Institut de Recherche Mathématique Avancée, Université Louis Pasteur et CNRS, 7, Rue René Descartes, 67084 Strasbourg Cedex, France. E-mail : nasserdd@math.u-strasbg.fr 1


Ta-Tsien Li

Rao Bopeng

Exact boundary controllability of unsteady flows in a tree-like network of open canals

Keywords: Exact boundary controllability, Saint-Venant system, tree-like network of open canals AMS Subject Classification : 93B05, 35B37, 93C20, 35L50

In this paper we establish the exact boundary controllability of unsteady flows in a tree-like network of open canals with general topology.

§1. Introduction.

The goal of this paper is to establish the exact boundary controllability of unsteady flows in a tree-like network of open canals.

The one-dimensional mathematical model of unsteady flows in an open canal was given by Saint-Venant system [START_REF] De Saint-Venant | Théorie du mouvement non permanent des eaux, avec application aux crues des rivières et l'introduction des marées dans leur lit[END_REF], which has been frequently used by hydraulic engineers in their practice (see also [START_REF] Leugering | On the modelling and stabilization of flows in networks of open canals[END_REF]- [START_REF] De Halleux | Boundary feedback control in networks of open channels[END_REF]). Using the theory on the semi-global C 1 solution and the exact boundary controllability for quasilinear hyperbolic systems (cf [START_REF] Li | Semi-global C 1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems[END_REF][START_REF] Li | Solution C 1 semi-globale et contrôlabilité exacte frontière de systèmes hyperboliques quasi linéaires[END_REF][START_REF] Lii | Exact boundary controllability for quasilinear hyperbolic systems[END_REF][START_REF] Li | Local exact boundary controllability for a class of quasilinear hyperbolic systems[END_REF][START_REF] Li | Exact controllability for first order quasilinear hyperbolic systems with zero eigenvalues[END_REF][START_REF] Li | Boundary Value Problems for Quasilinear Hyperbolic Systems[END_REF]), the exact boundary controllability of unsteady flows in both a single open canal and a star-like network of open canals was obtained in [START_REF] Li | Exact boundary controllability of unsteady flows in a network of open canals[END_REF]. Later on, the exact boundary controllability of unsteady flows in a string-like network of open canals was established in [START_REF] Li | Exact boundary controllability for quasilinear hyperbolic systems and its application to unsteady flows in a network of open canals[END_REF].

In this paper we will generalize the previous results to a tree-like network of open canals with general topology. For this purpose, we will serve the total flux at the multiple node as an interface control. By means of a method different from that in [START_REF] Li | Exact boundary controllability of unsteady flows in a network of open canals[END_REF], we establish another type of result on the exact boundary controllability of unsteady flows in a star-like network of open canals. Based on this new result and the results obtained in [START_REF] Li | Exact boundary controllability of unsteady flows in a network of open canals[END_REF][START_REF] Li | Exact boundary controllability for quasilinear hyperbolic systems and its application to unsteady flows in a network of open canals[END_REF] on the exact boundary controllability of unsteady flows for a single open canal, a star-like network and a string-like network of open canals, we establish the exact boundary controllability of unsteady flows in a tree-like network of open canals with general topology.

This paper is organized as follows. In §2 we give the statement of the exact boundary controllability of unsteady flows in a network of open canals and show the main idea of realizing it. In §3, we recall some results given in [START_REF] Li | Exact boundary controllability of unsteady flows in a network of open canals[END_REF][START_REF] Li | Exact boundary controllability for quasilinear hyperbolic systems and its application to unsteady flows in a network of open canals[END_REF] on the exact boundary controllability of unsteady flows in a network of open canals with simpler topology. In §4, a fundamental Lemma on the continuation of piecewise C 1 solution in a star-like network of open canals is established. Then, combining the results in § §3-4, we establish in §5 the exact boundary controllability of unsteady flows in a general tree-like network of open canals. Finally, some remarks are given in §6. §2 Statement on the exact boundary controllability of unsteady flows in a network of open canals.

We consider unsteady flows in a network composed of N open canals. Let a i0 and a i1 be the x-coordinates of two ends of the i-canal and L i = a i1a i0 its length. Suppose that there is no friction and the canal is horizontal and cylindrical, the corresponding Saint-Venant system can be written as (cf. [START_REF] Leugering | On the modelling and stabilization of flows in networks of open canals[END_REF], [START_REF] Li | Exact boundary controllability of unsteady flows in a network of open canals[END_REF]

-[9]) (2.1)          ∂A i ∂t + ∂(A i V i ) ∂x = 0, t ≥ 0, a i0 ≤ x ≤ a i1 (i = 1, • • • , N ), ∂V i ∂t + ∂S i ∂x = 0,
where, for the i-th canal, A i = A i (t, x) stands for the area of the cross section at x occupied by the water at time t, V i = V i (t, x) the average velocity over the cross section and

(2.2)

S i = 1 2 V 2 i + gh i (A i ) + gY ib ,
where g is the gravity constant, constant Y ib denotes the altitude of the bed and

(2.3)

h i = h i (A i )
is the depth of the water, h i (A i ) being a suitably smooth function of A i such that

(2.4) h i (A i ) > 0.
The initial condition is

(2.5) t = 0 : (A i , V i ) = (A i0 (x), V i0 (x)) , a i0 ≤ x ≤ a i1 (i = 1, • • • , N ).
When a i0 (resp. a i1 )is a simple node, we have the flux boundary condition :

(2.6) x = a i0 : A i V i = q i0 (t) (resp. x = a i1 : A i V i = q i1 (t)).
While, when a i0 (resp. a i1 ) is a multiple node, we denote by J i0 (resp. J i1 ) the set of indices corresponding to all the canals jointed at a i0 (resp. a i1 ). At a i0 (resp. a i1 ) we have (cf. [START_REF] Leugering | On the modelling and stabilization of flows in networks of open canals[END_REF], [START_REF] Li | Exact boundary controllability of unsteady flows in a network of open canals[END_REF])the energy-type interface conditions (2.7)

S i = S j , ∀j ∈ J i0 (resp. ∀j ∈ J i1 )
and the total flux interface condition (2.8)

j∈J i0 ±A j V j = Q i0 (t) resp. j∈J i1 ±A j V j = Q i1 (t) . An equilibrium state (A i , V i ) = (A i , V i ) with A i > 0 (i = 1, • • • N ) is subcritical if (2.9)        λ i0 = V i -gA i h i (A i ) < 0, (i = 1, • • • , N ). µ i0 = V i + gA i h i (A i ) > 0,
For the purpose of the exact boundary controllability of unsteady flows in a network of open canals, we want to find a time T > 0 such that for any given initial data

(A i , V i ) = (A i0 (x), V i0 (x)) and final data (A i , V i ) = (A iT (x), V iT (x)) (i = 1, • • • N ) in a C 1 neighbourhood of a given subcritical equilibrium state (A i , V i ) (i = 1, • • • N ),
there exist suitable flux boundary controls at some simple nodes and suitable total flux interface controls at some multiples nodes, such that the corresponding mixed initial-boundary value problem for Saint-Venant system (2.1) with the initial data (2.5), the corresponding boundary conditions (2.6) and the interface conditions (2.7)-(2.8) admits a unique piecewise C 1 solution on the time interval 0 ≤ t ≤ T , which satisfies exactly the final condition (2.10)

t = T : (A i , V i ) = (A iT (x), V iT (x)) , a i0 ≤ x ≤ a i1 (i = 1, • • • , N ).
To this end, it suffices to construct a piecewise C 1 solution to Saint-Venant system (2.1) on the time interval 0 ≤ t ≤ T , which satisfies the initial condition (2.5), the final condition (2.10), all the energy-type interface conditions (2.7) and a part of flux boundary conditions (2.6) and total flux interface conditions (2.8). Once this solution is obtained, we put it into the rest of flux boundary conditions (2.6) and total flux interface conditions (2.8) to get the desired flux boundary controls q i0 (t) (resp. q i1 (t)) and total flux interface controls Q i0 (t) (resp. Q i1 (t)), which realize the exact boundary controllability for the Saint-Venant system.

To construct this kind of piecewise C 1 solution is a nonstandard problem which can be regarded as an inverse problem. There is no uniqueness in general. However we can get this solution in a stable way, i.e., by solving several well-posed problems. The main principle in the resolution consists in making the number of controls or the controllability time T as small as possible. However, for a smaller number of controls, we need a larger controllability time, and vise versa. §3 Construction of piecewise C 1 solution in a network of canals with simpler topology.

In this section, we recall some results given in [START_REF] Li | Exact boundary controllability of unsteady flows in a network of open canals[END_REF][START_REF] Li | Exact boundary controllability for quasilinear hyperbolic systems and its application to unsteady flows in a network of open canals[END_REF], which will be useful in what follows.

1). Consider a single canal with flux boundary conditions at one end. If

(3.1) T > 2L max - 1 λ 0 , 1 µ 0 ,
where λ 0 and µ 0 are given by (2.9) and L = a 1a 0 is the length of the canal, we can construct a C 1 solution on the single canal, such that it satisfies any given initial and final conditions in a C 1 neighbourhood of a given subcritical equilibrium state(A, V ) as well as the the flux boundary condition on one end (see [START_REF] Li | Exact boundary controllability of unsteady flows in a network of open canals[END_REF] for the detail). We illustrate this procedure by the following figure : Here a bold point "•" denotes the starting end on which the flux boundary condition is satisfied, while a hollow point "•" denotes the arriving end, on which the value of solution is determined by the procedure of resolution, and the arrow denotes the direction along which we solve the problem.

2). By rescaling and folding all corresponding x-interval to the interval 0 ≤ x ≤ 1, the previous method can be applied to a star-like network of N open canals (see [START_REF] Li | Exact boundary controllability of unsteady flows in a network of open canals[END_REF] for the detail). Let T satisfy

(3.2) T > 2 max 1≤i≤N L i max - 1 λ i0 , 1 µ i0 ,
where λ i0 and µ i0 are given by (2.9) and

L i = a i1 -a i0 is the length of the i-th canal (i = 1, • • • N )).
Then, by solving the problem from the joint end to all simple nodes, we can get a piecewise C 1 solution which satisfies any given initial and final conditions in a piecewiseC 1 neighbourhood of a given subcritical equilibrium

(A i , V i ) (i = 1, • • • N )
as well as the interface conditions (2.7)-(2.8) at the joint end (the multiple node). We illustrate this procedure by the following figure : Here all simple nodes are the arriving ends "•" on which the solution is determined by the procedure of resolution. The joint node is the starting end "•" on which all the interface conditions (2.7)-(2.8) are satisfied. The procedure of resolution is indicated by the arrows : from "•" to "•".

3). For a string-like network of N open canals, if

(3.4) T > 2 N i=1 L i max - 1 λ i0 , 1 µ i0 ,
where λ i0 and µ i0 are given by (2.9) and

L i = a i1 -a i0 is the length of i-th canal (i = 1, • • • N ))
, by solving the problem from one simple node "•" to another simple node "•", indicated by Figure 3.3, we can still get a piecewise C 1 solution which satisfies any given initial and final conditions in a piecewise C 1 neighbourhood of a given subcritical equilibrium (A i , V i ) (i = 1, • • • N ) as well as the flux boundary condition on one simple node and the interface conditions (2.7)-(2.8) on all the multiple nodes denoted by " " (see [START_REF] Li | Exact boundary controllability for quasilinear hyperbolic systems and its application to unsteady flows in a network of open canals[END_REF] for the detail). This situation is similar to the case shown by Figure 3.1 for a single canal, however the controllability time should be much larger. 4). Based on the results given in 2) and 3), if T is large enough (bigger than the maximum of the exact controllability time corresponding to each string-like sub-network), similarly we can get a piecewise C 1 solution for a star-like network of string-like sub-networks, which satisfies any given initial and final conditions in a piecewise C 1 neighbourhood of a given subcritical equilibrium as well as the interface conditions (2.7)-(2.8) on each multiple node. We illustrate this procedure by the following figure : Here the joint node of all the string-like sub-networks is the starting end "•", all the simple nodes are the arriving end "•", while all the joint nodes in the string-like sub-networks are denoted by " " on which the interface conditions (2.7)-(2.8)are satisfied. §4. Fundamental Lemma on the continuation of piecewise C 1 solution.

In order to get a desired piecewise C 1 solution for a network of N open canals with general topology, we should extend the piecewise C 1 solution on a network of open canals with simpler topology shown in the previous section. To this end, we now give another way to construct a piecewise C 1 solution for a star-like network of N open canals by assuming that a C 1 solution satisfying any given initial and final conditions in a C 1 neighbourhood of a corresponding subcritical equilibrium is obtained on one canal, and then solving the problem from the joint end to other simple ends respectively, which is illustrated in the following figure : Let (A 1 , V 1 ) be a known C 1 solution on the first canal denoted by " ". In order to construct the desired C 1 solution on other canals in the star-like network, we serve the energy-like interface condition as a boundary condition at the joint node x = 0 for each corresponding canal. Thus the piecewise C 1 solution constructed satisfies only the energy-like interface conditions (2.7), while the total flux interface condition (2.8) is no longer satisfied at the joint node which is then denoted by " ". More precisely, we have the following result.

Lemma 4.1 Let (A i , V i ) = (A i , V i ) (i = 1, • • • , N ) be a subcritical equilibrium state. Suppose that (4.1) T > 2 max i=2,•••,N L i max - 1 λ i0 , 1 µ i0 ,
where λ i0 and µ i0 (i = 2, • • • , N ) are still given by (2.9). Let (A i0 (x), V i0 (x)) and (A iT (x), V iT (x)) (i = 1, • • • , N ) be any given initial state and final state with small C 1

norms n i=1 (A i0 (•)-A i , V i0 (•)-V i ) C 1 [0,L i ] and n i=1 (A iT (•)-A i , V iT (•)-V i ) C 1 [0,L i ] ,
which satisfy the conditions of piecewise C 1 compatibility with the energy-like interface conditions (2.7) at the points (t, x) = (0, 0) and (T, 0) respectively. Let

(4.2) R i (T ) = {(t, x)| 0 ≤ t ≤ T, 0 ≤ x ≤ L i } (i = 1, • • • , N ).
Assume that we have already determined the solution

(A 1 , V 1 ) with small C 1 norm (A 1 -A 1 , V 1 -V 1 ) C 1 [R 1 (T )]
, which satisfies the initial condition (2.5) and the final condition (2.10). Then there exists a piecewise

C 1 solution (A i , V i )(i = 1, • • • , N ) on the domain ∪ N i=1 R i (T ) with small piecewise C 1 norm n i=1 (A i -A i , V i -V i ) C 1 [R i (T )] ,
which satisfies the initial condition (2.5), the final condition (2.10) and the energy-like interface conditions (2.7) at the joint node x = 0.

Proof. For i = 1, • • • , N , introducing the Riemann invariants r i and s i as follows :

(4.3) 2r i = V i -V i -G i (A i ), 2s i = V i -V i + G i (A i ),
where

(4.4) G i (A i ) = A i A i gh i (A i ) A i dA i .
Let H i be the inverse function of G i (A i ). We have (4.5)

V i = r i + s i + V i , A i = H i (s i -r i ) and (4.6) H i (s i -r i ) = A i gh i (A i ) > 0. Since (A i , V i ) = (A i , V i ) correspond to (r i , s i ) = (0, 0), we get (4.7) H i (0) = A i , H i (0) = A i gh i (A i ) > 0.
Taking (r i , s i ) (i = 1, • • • , N ) as new unknown variables, system (2.1) can be reduced to the following system of diagonal form :

(4.8)          ∂r i ∂t + λ i ∂r i ∂x = 0, (i = 1, • • • N ), ∂s i ∂t + µ i ∂s i ∂x = 0,
where (4.9)

   λ i = V i -gA i h i (A i ) < 0, (i = 1, • • • N ). µ i = V i + gA i h i (A i ) > 0,
The energy-type interface conditions (2.7) now become

x = 0 : 1 2 (r i + s i + V i ) 2 + gh i (H i (s i -r i )) + gy bi (4.10) = 1 2 (r 1 + s 1 + V i ) 2 + gh 1 (H 1 (s 1 -r 1 )) + gy b1 (i = 2, • • • N ).
For each i = 2, • • • , N , let

P i (r i , s i , r 1 , s 1 ) = 1 2 (r i + s i + V i ) 2 + gh i (H i (s i -r i )) + gy bi (4.11) - 1 2 (r 1 + s 1 + V 1 ) 2 + gh i (H 1 (s 1 -r 1 )) + gy b1 . Since the state (A k , V k ) = (A k , V k ) corresponds to (r k , s k ) = (0, 0) for k = 1, • • • N , condition (
4.10) implies (4.12) P i (0, 0, 0, 0) = 0.

On the other hand, a straightforward computation gives (4.13)

∂P i ∂s i (0, 0, 0, 0) = V i + gA i h i (A i ) > 0.
Then, by the implicit function theorem, we can find a C 1 function f i in a neighbourhood of (r i , r 1 , s 1 ) = (0, 0, 0) such that (4.10) can be equivalently rewritten as (4.14)

s i = f i (r i , r 1 , s 1 )
with f i (0, 0, 0) = 0. Since the solution (r 1 , s 1 ) is known, putting the values r 1 (t, 0) and s 1 (t, 0) into (4.14) and noting f i (0, 0, 0) = 0, (4.14) can be equivalently rewritten as (4.15)

s i = G i (t, r i ) + H i (t),
where G i and H i are C 1 functions such that (4.16) G i (t, 0) ≡ 0 and (4.17)

H i C 1 [0,T ] is small enough. Similarly, for each i = 2, • • • , N , we have (4.18) ∂P i ∂r i (0, 0, 0, 0) = V i -gA i h i (A i ) < 0.
Then, in a neighbourhood of (r i , s i , r 1 , s 1 ) = (0, 0, 0, 0), condition (4.10) can be equivalently rewritten as

(4.19) r i = G i (t, s i ) + H i (t),
where G i and H i are C 1 functions such that (4.20) G i (t, 0) ≡ 0 and (4.21)

H i C 1 [0,T ] is small enough.
Moreover, under the assumption of Lemma 4.1, it is easy to see that the conditions of C 1 compatibility with (4.15)(resp.(4.19)) are still satisfied at the point (t, x) = (0, 0) (resp. (T, 0)). Then, applying the results given in [START_REF] Li | Local exact boundary controllability for a class of quasilinear hyperbolic systems[END_REF] and [START_REF] Li | Exact boundary controllability of unsteady flows in a network of open canals[END_REF] to system (4.8) for the i-th canal, we get the solution (r i , s i ), i.e., (A i , V i ) for each i = 2, • • • , N , which, together with (A 1 , V 1 ), satisfy the initial condition (2.5), the final condition (2.10) and the energy-type interface conditions (2.7) at x = 0. The proof is then complete. Remark 4.1 Lemma 4.1 is still valid for a star-like network of string-like subnetworks, when a piecewise C 1 solution is known on one string-like sub-network and T > 0 is suitably large. A network is called to have a tree-like configuration, if any two nodes in the network can be connected by a unique path of canals. In other words, a tree-like network is a connected network without loop. In what follows, for T > 0 suitably large, taking the results given in §3 as the first step, we will successively use Lemma 4.1 or Remark 4.1 to get a desired piecewise C 1 solution on the whole network, which implies the exact boundary controllability for a tree-like network of open canals.

Arbitrarily choosing a node as the starting end, we construct a desired piecewise C 1 solution as follows.

Case A. When the starting end O k is a simple node, on which the boundary flux

q k with small C 1 norm q k (•) -A k V k C 1 [0,T ]
is given, we assume that the initial data and the final data satisfy the conditions of C 1 compatibility with the flux boundary condition (2.6) at O k . For T > 0 suitably large, by the procedure illustrated in Figure 3.1, we can get a C 1 solution of system (2.1) on the canal with the end O k (another end of this canal is denoted by O l ), which satisfies the corresponding initial condition (2.5) and the final condition (2.10) on the canal and the flux boundary condition (2.6) at the simple node O k . This procedure can be shown in the following figure : Then, repeating the procedure given by Lemma 4.1, we can construct a piecewise C 1 solution on the whole network of canals, which satisfies the initial condition (2.5) and the final condition (2.10) on the whole network, the boundary condition (2.6) at the starting end O k and the energy-like interface conditions (2.7) at all multiple nodes. Finally, we substitute this solution in (2.8) to get the total flux interface controls Q i (t) for all multiple nodes, and in (2.6) to get the flux boundary controls q i (t) for all simple nodes except the starting end O k . We illustrate this procedure by the following figure : Case B. When the starting end O l is a multiple node, on which the total flux Q l with small norm Q l (•) -j∈J l ±A j V j C 1 [0,T ] is given, we assume furthermore that the initial data and the final data also satisfy the conditions of piecewise C 1 compatibility with the total flux interface condition (2.8) at the multiple node O l . For T > 0 suitably large, by the procedure illustrated in Figure 3.2, we get a piecewise C 1 solution for the local star-like network jointed at the multiple node O l , which satisfies the corresponding initial condition (2.5) and the final condition (2.10) on the whole local star-like network, the interface conditions (2.7) and (2.8) at multiple node O l . We illustrate this procedure by the following figure. Then repeating the procedure given by Lemma 4.1 as shown in the previous situation, we finally get a piecewise C 1 solution for the whole network of canals, which satisfies the initial condition (2.5) and the final condition (2.10) on the whole network, the total flux interface condition (2.8) at the starting multiple node O l and the energy-type interface conditions (2.7) at all multiple nodes. Thus, in a similar way, we can define the flux boundary controls q i (t) for all simple nodes, and the total flux controls Q i (t) for all multiple nodes except the starting end O l , respectively. We illustrate this procedure by the following figure :   O l starting end 2). The flux boundary condition (2.6) can be replaced by the water level boundary condition (6.1) x = a i0 : A i = q i0 (t).

O k O l
Accordingly, the energy-like interface conditions (2.7) can be also replaced by the water level interface conditions (6.2)

h i (A i ) + Y ib = h j (A j ) + Y jb , ∀j ∈ J i0 .
3). The flux boundary condition (2.6) can be replaced by the following boundary condition of underflow gate (cf. [START_REF] De Halleux | Boundary feedback control in networks of open channels[END_REF]) (6.3) x = a i0 :

A i V i |A i V i | = 2gu 2 i0 (y i0 -h i (A i )),
where y i0 is the water level outside the i-th canal and u i0 is the height of gate opening at x = a i0 . Accordingly, on each intermediate joint end a i0 of a string-like sub-network, the interface conditions (2.7)-(2.8) can be also replaced by the interface conditions of underflow gate (cf. [START_REF] De Halleux | Boundary feedback control in networks of open channels[END_REF])

(6.3) A i-1 V i-1 |A i-1 V i-1 | = A i V i |A i V i | = 2gu 2 i0 (h i-1 (A i-1 ) -h i (A i )).
4). When one considers the friction in a network of open canals, Saint-Venant system (2.1) becomes (6.4)

         ∂A i ∂t + ∂(A i V i ) ∂x = 0, t ≥ 0, a i0 ≤ x ≤ a i1 (i = 1, • • • , N ) ∂V i ∂t + ∂S i ∂x + F i (A i , V i ) = 0
with F i (A i , 0) ≡ 0 and V i F i (A i , V i ) ≥ 0. For any given equilibrium state (A i , V i ) = (A i , V i ) with A i > 0(i = 1, • • • , N ), we still have the exact boundary controllability.

5

). Since both the initial and final states belong to a piecewise C 1 neighbourhood of an arbitrarily given subcritical state, our results give only the local exact boundary controllability for unsteady flows in a tree-like network of open canals, however, that is enough to get the corresponding global exact boundary controllability. In fact, in order to drive any initial state given in a piecewise C 1 neighbourhood of a subcritical state to any final state given in a piecewise C 1 neighbourhood of another subcritical state, it suffices to use the previous results to drive the initial state to the first equilibrium state, then successively transfer the first equilibrium state to the second equilibrium state, and finally reach the final state. Thus we get the corresponding global exact boundary controllability, however, the controllability time must be enlarged many times.

Figure 3 . 1 .

 31 Figure 3.1. A single canal

Figure 3 . 2 .

 32 Figure 3.2. A star-like network of canals

Figure 3 . 3 .

 33 Figure 3.3. A string-like network of canals

Figure 3 .

 3 Figure 3.4. A star-like network of string-like sub-networks

Figure 4 . 1 A

 41 Figure 4.1 A star-like network of canals

Figure 4 . 2 .

 42 Figure 4.2. A star-like network of string-like sub-networks

Figure 5 . 1 .Figure 5 . 2 .

 5152 Figure 5.1. Starting from a simple node Still for T > 0 suitably large, applying Lemma 4.1 illustrated in Figure 4.1 to the local star-like network of open canals jointed at the multiple node O l , we get a piecewise C 1 solution which satisfies the corresponding initial condition (2.5) and the final condition (2.10) on this local star-like network, the flux boundary condition (2.6)

Figure 5 . 3 .

 53 Figure 5.3. Exact controllability starting from a simple node (23 controls for 23 canals)

Figure 5 . 4 .

 54 Figure 5.4. Starting from a multiple node

Figure 5 . 5 .Figure 5 . 6 .Figure 5 . 7 .

 555657 Figure 5.5. Exact controllability starting from a multiple node (23 controls for 23 canals) Case C. If the tree-like network contains string-like sub-networks, we can use the procedure shown in Figure 3.3 or Figure 3.4as the first step and then apply repeatedly Remark 4.1 to get a desired C 1 piecewise solution for the whole network of the canals (seeFigures 5.6 and 5.7). In this way we can reduce the number of controls, however, the controllability time should be correspondingly enlarged (cf.(3.4)).

Acknowledgments. LI Ta-Tsien is supported by the Special Funds for Major State Basic Research Projets of China. The paper was written during Rao Bopeng's visit in the Institut Sino-Français de Mathématiques Appliquées (ISFMA) at Fudan University on 2003-2004.


Ph. D D Gialis

On the Signature of Highest Energy Particles in Gamma-Ray Bursts

Keywords: gamma rays, bursts -radiation mechanisms, non-thermal -magnetic fields

The launch of Fermi observatory opens new possibilities to develop diagnosis of Gamma-Ray Burst parameters, especially during their prompt emission. In this paper, first, we focus on the issue of the observability of the electron distribution cut-off, which is very sensitive to magnetic field parameters (intensity, turbulence level...). Its observability mainly depends on the importance of the Self-Synchro-Compton emission and, of γ-γ absorption process, that we calculate by taking account of the important stratification of the medium. Actually we found wide possibilities to observe or deduce the electron distribution cut-off and thus constrain magnetic parameters in the outflow of the burst. Then, we also focus on the possible gamma signature of the cosmic ray production during the prompt emission, through their synchrotron emission around a few GeV. Again we look at the γ-γ absorption process and found nevertheless favorable conditions (particularly if B ∝ 1/r) to observe this signature for some range of magnetic parameters, when the Compton parameter is sufficiently small (Y < 10 -2 ).

Introduction

The interpretation of the Gamma-Ray Burst (hereafter GRB) prompt emission is one of the most important challenge for the "standard" internal shock model (see e.g. Piran (2005)). Also, a key ingredient, on the way of understanding, is the magnetic field; its intensity, its turbulence level and its temporal evolution in the outflow are crucial for both the particle acceleration processes and the resulting synchrotron and Self-Synchro-Compton (hereafter SSC) high energy emissions. We think that future Fermi observations of GRBs will provide us with some very interesting elements of diagnosis, that we want to emphasize here. In our investigation, and according to previous work (Gialis & Pelletier (2005)), we first intend to clarify and to develop some constraints on the particle acceleration efficiency in the internal shocks, following the characteristics of the magnetic field. In Sec. 2, we determine the local cut-off energy for both the accelerated particle energy distribution and their synchrotron emission and we show that the possible observation of a global energy cut-off, in the electronic synchrotron spectrum, would give us some indications on the magnetic field. In another hand, we have to take into account opacity effects, which could occur in GRB outflows and, which could prevent any observational signature relating to distribution cut-off. In Sec. 3, we then discuss about the γ-γ absorption between the highest energy photons produced by the electronic population, and the lower energy ones, and we look at which conditions it could play an important role, especially in hiding the electron distribution cut-off. In last section, we focus on the acceleration of hadronic particles, like protons: because the usual Fermi acceleration process, at internal and external shocks, seems to be inefficient to reach the UHECR energy range (see e.g. Lemoine et al. (2006), and Gialis & Pelletier (2003)), we first examine some properties of a secondary acceleration process by scattering off relativistic hydro-magnetic fronts, as previously proposed (Gialis & Pelletier (2004)). In this context, we conclude about the possibility to observe a synchrotron signature for UHECRs, depending on the magnetic parameters, against the γ-γ absorption and for a sufficiently small Compton parameter (Y < 10 -2 ) corresponding to a magnetic dominated outflow.

Evolution of the electronic distribution cut-off during the prompt emission

Theoretical framework and magnetic parameters

The internal shock model was first introduced (see e.g. Rees & Mészáros (1994)) to describe the evolution of an unsteady ultra-relativistic outflow produced by an inner engine (black hole surrounded by a magnetized torus) during a time ∆t w . The simplest version of this model assumes that the collimated outflow is constituted by a set of layers ejected at different times, with different Lorentz factors of the order of 10 2 to 10 3 , in order to solve the well-known "compactness problem". The average Lorentz factor Γ can reach the value of the baryonic loading parameter η = E/M b c 2 , where E is the total energy of the burst and M b the baryonic mass carried into the outflow.

Send offprint requests to: denis gialis@yahoo.fr Internal shocks occur when slower layers catch up with faster ones. These shocks are mildly relativistic, with a relative Lorentz factor in the co-moving frame 2. For a typical size r 0 = 10 7 cm of the inner engine, and an average Lorentz factor Γ in the observer frame, shocks occur at a distance r b Γ2 r 0 or η 2 r 0 , η being the baryonic parameter (E/M b c 2 ). In this internal shock phase, the width, ∆R, of the layers is proportional to the distance r from the black hole. Assuming that the layers are optically thin to e --γ interactions beyond r b , particles can be accelerated in the layers via first and second order Fermi processes, which constitute the origin of the GRB prompt emission. The internal shock stage lasts until the deceleration radius is reached, i.e. the distance where the interaction with the surrounding medium produces an external and a reverse shock.

In our model, the profile of the mean magnetic field follows a power-law shape, B ∝ r -α (see Appendix 1). The parameter that we consider as the less constrained is the intensity of the magnetic field at some point, either at the origin scale r 0 of a few gravitational radii, or at the beginning of the acceleration stage in the internal shocks at r b . The other important index that controls the particle acceleration efficiency as a function of the particle energy is the index of the turbulent spectrum β (e.g. β = 5/3 for Kolmogorov theory). The efficiency of the Fermi acceleration process is directly related to the efficiency of particle scattering off magnetic irregularities. The most efficient acceleration is obtained with the so-called Bohm scaling, which corresponds to β = 1. This efficiency also scales with the level of magnetic irregularities η t ≡ < δB 2 > / < B 2 >. In the following, we argue that the three parameters α, β and η t can be determined by Fermi observations of GRB prompt emission, in particular through the measurement of the position of E peak (i.e the maximum of the ν F ν spectrum) and its evolution with time (see Appendix 1). More generally, the evolution of the synchrotron spectrum and the value of the maximum synchrotron emitted energy all simply depend on these three parameters.

Acceleration processes and characteristic time

The acceleration processes are all collisionless, and they mainly depend on the local magnetic field irregularities. In this paper, we do not hypothesize on the geometrical configuration of the magnetic field. We only assume an evolution profile for the magnetic field intensity, B ∝ r -α with α ∈ [1, 2], and the existence of a turbulent component. Following previous works (Gialis & Pelletier (2003)), we introduce a Kolmogorov turbulence spectrum such that

< δB 2 >∝ k max k min k -5/3 dk , ( 1
)
where k min 2π/∆R. The upper bound k max is given by the cyclotron pulsation of the non-relativistic protons and the Alfvèn speed: k max ω cp /V A with ω cp = e B/m p c and 0.1 c V A < c. A particle of charge q, with an energy , can be scattered off by Alfvèn waves only if its Larmor's radius, r L = /|q| B, is greater than ∼ 2π/k max , thus a particle of mass m is accelerated only if its Lorentz factor is γ min = (|q|/e)(m p /m) (V A /c) (Lacombe (1977)). We want to emphasize this theoretical estimate which is based on the physical description of the acceleration process in a turbulent magnetic field. A similar result is usually obtained by invoking the fraction of energy which is injected in the electronic population, though it is not well known (see e.g. Pe'er & Waxman (2004)). Also, thermal protons can be accelerated while electrons are injected in Fermi acceleration processes only if their Lorentz factors 10 2 to 10 3 . The lower bound of the synchrotron energy spectrum, in the co-moving frame, corresponding to a minimal electronic Lorentz factor γ e,min close to a few 10 2 , is thus given by low syn 0.29 γ e,min 400

2 B b 10 5 G r r b -α keV.
(2)

However, in a mildly relativistic internal shock (with a Lorentz factor γ 1.5), electronic and hadronic temperatures, respectively T e and T p , are of the same order: T e T p γ m p c 2 . Since γ e,min < m p /m e , a large fraction of the electronic population is injected in the acceleration process when a slow cooling regime occurs (see Appendix 2). According to the Diffusive Shock Acceleration (DSA) theory (see e.g. Drury (1983), Blandford & Ostriker (1978), or Bell (1978)), we will assume that mildly relativistic internal shocks produce a power law energy distribution for the accelerated particles1 , with a density of particles per unit of energy dn p /d ∝ -p , with p ∈ [2, 3[. Then, a thermal electronic distribution, with a maximum close to e min = γ e,min m e c 2 , is extended by such a power law distribution for > e min .

Casse et al. (2002) (see also Gialis & Pelletier (2003)) showed that the characteristic acceleration time, which is proportional to the particle mean free path, mainly depends on both the particle rigidity and the magnetic turbulence level. For a Kolmogorov spectrum, it is given by

t acc 6.6 × 10 -2 ¯ c , ( 3
)
where the mean free path, ¯ , of a particle is given by

¯ = r L η t r L c -2/3 . ( 4
)
In this last expression, we define c ∆R, as the magnetic correlation length and, η t ≡ <δB 2 > <B 2 > , the magnetic turbulence level. At this point, it is important to stress the importance of the magnetic field intensity (via particle Larmor's radius), its evolution with the distance (B ∝ r -α ) and its turbulence level: these parameters are not only essential in the definition of the characteristic acceleration time t acc which controls the efficiency of the acceleration process for both leptons and hadrons, but they also determine the magnetic confinement capabilities of the medium and, consequently, they play a key role in the problem of Ultra-High Energy Cosmic Ray generation.

Cut-off energy and electron cooling regimes

The upper bound of the energy distribution is obtained by comparing the local acceleration time t acc with both the synchrotron cooling time, t syn , and the co-moving dynamical time t dyn r/η c. Here, we assume that other cooling processes, like e.g. the inverse Compton cooling for electrons, are negligible. Thus, according to the magnetic field evolution with the distance r, if t syn > t dyn at r b , the energy of particles is only limited by the expansion, during all the internal shock stage. Now, if t syn < t dyn at r b , the energy limitation follows two stages of evolution. For usual magnetic parameters, this last case seems to apply to the electronic population, as we will demonstrate below. The electron energy distribution cut-off (see Appendix 1), due to the synchrotron cooling, can be written (in c.g.s units) e max 1.9 × 10 7 κ 0 10

-3/4 η 1/2 (B b r α b ) -5/4 r s 1 , ( 5
)
where B b 10 5 G is the magnetic field intensity at r b ( η 2 r 0 ), the index s 1 = (5α -2)/4, and we define the turbulence parameter κ 0 2.2×10 -1 /η t . At the very beginning of the internal shock phase, we obtain an electronic maximal Lorentz factor close to 400. The corresponding cut-off energy of synchrotron photons at r b , namely e b,syn , does not depend on the index α of the magnetic field intensity: e b,syn 0.31 κ 0 10

-3/2 η 100 -1 B b 10 5 G -3/2 r 0 10 7 cm -1 keV, ( 6
)
in the co-moving frame. The observational signature of such a cut-off energy could be observed at the very beginning of the prompt emission where the instantaneous electronic synchrotron spectrum could lie in a narrow energy band (10 -100 keV) for a terrestrial observer. It is noteworthy that this cut-off energy is also independent of η in the observer frame. Finally, because low syn at r b has to be ≤ e b,syn , we obtain a strong constraint on the upper bound of the magnetic field intensity B b :

B b 1.0 × 10 5 κ 0 10 -3/5
r 0 10 7 cm -2/5 γ e,min 400

-4/5 η 100 -2/5 G, (7)
or, for the turbulence parameter,

κ 0 10 B b 10 5 G -5/3 r 0 10 7 cm -2/3 γ e,min 400
-4/3 η 100 -2/3 . ( 8
)
From Eq. ( 7), we deduce that only high magnetic field intensity (B b 10 5 G) requires an high turbulence level (i.e κ 0 10, see above). Thus, because the magnetic field at r b can vary from 10 3 to 10 7 G depending on the magnetic properties of the central engine, one has to consider different levels of turbulence when studying particle acceleration processes in the internal shocks (see Sec. 3). The evolution of the cut-off energy of synchrotron photons, with the distance r and beyond r b , is given by

max syn = e b,syn r r b 2s 1 -α . ( 9
)
Eq. ( 9) is valid until what we could name the "cooling radius" r c , where the co-moving dynamical time becomes shorter than the cooling synchrotron time. The cooling radius is such that

r c 1.0 × 10 -24 κ 0 10 -3 η -2 (B b r α b ) 3 1 3α-2 cm. ( 10
)
The variations of this radius with the magnetic parameters α, B b , and κ 0 are strong important (see Fig. 1). For example, on one hand, no transition occurs for an index α close to 1 and for a high turbulence level (κ 0 10): the synchrotron limitation prevails and the cooling radius is moved away beyond the deceleration radius, namely r d . On the other hand, for a low turbulence level (κ 0 ≥ 100) with α > 1.5, the cooling radius is smaller than r b and the limitation is only due to expansion.

Beyond the cooling radius, and until the deceleration radius, the cut-off energy of the electronic distribution is (in c.g.s units) e max 1.9 × 10 -11 κ 0 10

-3 η -1 B b r α b r s 2 , ( 11
)
with the index s 2 = 1 -α. Thus, we obtain a new evolution for the cut-off energy with the distance r, and beyond r c :

max syn = e c,syn r r c 2s 2 -α . ( 12
)
In Appendix 2, we detail the calculation of the electronic synchrotron spectrum which is locally produced at the distance r, by taking into account both the previous cut-off energy and the variation of the electronic cooling regime.

In the co-moving frame, and at the cooling radius r c , the cut-off energy of the corresponding synchrotron spectrum only depends on the turbulence level. Indeed, Eq. ( 9) and ( 10) yield e c,syn

10 κ 0 10 -3
MeV.

(13)

The energy e c,syn is now considered as the cut-off energy of the whole electronic synchrotron spectrum (Gialis & Pelletier 2005) when one neglects absorption processes like e.g. γ-γ interactions (see Sec. 3). A direct observation of its spectral signature, in the GeV range, would provide a very good estimate of the turbulence level. Moreover, as suggested by Eq. ( 10) and by the Fig. 1, the determination of the duration separating the very beginning of the prompt emission and the observation of the highest energy synchrotron photons, i.e ∆t (r c -r b )/η 2 c for a terrestrial observer, could constitute a very interesting observational test to constrain the index α of the magnetic field evolution law in the outflow. Sec. 3 will examine the possible absorption of high energy photons in γ-γ interactions with the lower energy photons.

Pair production process and high energy cut-off in synchrotron spectrum

In the internal shock stage of GRBs, the highest energy synchrotron photons can interact with lower energy photons to create some e + -e -pairs. The resulting absorption can prevent any observational signature, in the synchrotron spectrum, of an high energy cut-off in the electronic distribution. In this section, we will assume that effects from a possible Compton Inverse process are negligible, which holds for magnetic dominated GRB outflows.

General considerations

Many studies of pair production process have been made (see e.g. Granot et al. (2008)) in GRB internal shock model, but the evolution of the magnetic field during the expansion is not generally taking into account. Indeed, the threshold energy of this process is very dependent on the magnetic field parameters. We propose here an analytical estimation of pair production effects by using our previous results concerning the synchrotron spectrum and its evolution with time. We consider a possible interaction of a photon having a high energy h with a photon having a lower energy . These energies will be expressed in the co-moving frame. The cross section depends on the angle θ between their directions of propagation. Then, the cross section is given by (Gould & Schréder (1967))

σ γγ ( h , , µ) 3 16 σ T (1 -β 2 ) (3 -β 4 ) ln 1 + β 1 -β -2β (2 -β 2 ) , ( 14
)
where

β = 1 - 2(m e c 2 ) 2 (1 -µ) h 1/2 , ( 15
)
and µ = cos θ.

The threshold energy for this interaction is such that

th = 2(m e c 2 ) 2 (1 -µ) h , ( 16
)
and σ γγ is maximal for β = √ 2/2: σ max γγ 0.26 σ T . In our case, the co-moving threshold energy depends on r and, for the highest energy photons (see Eq. ( 9) and ( 12)), we have

th (r) =      2(m e c 2 ) 2 (1-µ) e b,syn r r b α-2s 1 for r b ≤ r ≤ r c 2(m e c 2 ) 2 (1-µ) e c,syn r r c α-2s2 for r ≥ r c . ( 17
)
At r c , for κ 0 = 10 and for a head-on interaction (µ = -1), th 26 keV in the co-moving frame, and whatever α. For a sufficient low energy photon density, the highest energy photons can create pairs only when th (r) ≤ max syn i.e between the radii r th,min = m e c 2 e b,syn

1 2s 1 -α r b , ( 18
)
and

r th,max = m e c 2 e c,syn 1 2s 2 -α r c , ( 19
)
with r c ∈ [r th,min , r th,max ]. As illustrated in the Fig. 2, these radii are relevant only for a set of magnetic parameters: for example, for α close to 1, there is no possibility of γ-γ absorption. This is a specific case where the cut-off in the electronic distribution could have an observational signature (see Fig. 3), only if the turbulence is weak enough (κ 0 10). For a higher turbulence level (κ 0 10), this cut-off could only be observable for α 1.2 (i.e when r c < r d ). When the γ-γ absorption is possible, i.e when r th,min < deceleration radius, we have to examine the cut-off energy in the synchrotron spectrum. For a high energy photon, the mean free path, γγ ( h ), is such that where n γ ( ) is the number of photons per cm -3 at the energy . Following Gould & Schréder (1967), we introduce s = / th = s 0 Θ, with Θ = 1 2 (1 -µ) and, s 0 = h /(m e c2 ) 2 . Thus, we can write

-1 γγ ( h ) = 1 2 1 -1 dµ(1 -µ) max syn th σ γγ ( h , , µ) dn γ ( ) d d , (20)
-1 γγ ( h ) = 3 8 σ T m 2 e c 4 h 2 ∞ m 2 e c 4 h -2 dn γ ( ) d d s 0 ( ) 1 s σ(s) ds , ( 21
)
with σ(s)

16 σ γγ 3 σ T 1 if s 1
, and σ(s) ln(s)/s if s 1. Also, we can define the internal optical depth, namely τ γγ ( h , r), as the ratio between the co-moving dynamical time, t dyn , and the mean time between two pair production interactions, i.e γγ ( h )/c.

Absorption estimate in GRB outflows

Considering now that the resulting photon distribution, in the outflow and expressed in the co-moving frame, is well described by a Band's function (Band et al. (1993)), we have 2 :

dn γ d K η L iso γ ∆t w 4π r 2 ∆R E 2 peak F η E peak , ( 22
)
where E peak ( 100 keV) is the energy for which the observed ν F ν spectrum is maximum, L iso γ 10 51 erg•s -1 is the observed isotropic γ-ray luminosity and,

F (x) = x α B exp (-(2 + α B ) x) for x ≤ x 0 x β B x α B -β B 0 exp (-(2 + α B ) x 0 ) for x ≥ x 0 , ( 23
) with x 0 = (α B -β B )/(2 + α B )
, and the normalization constant K = ( x F (x)dx) -1 . The Band's parameters, α B and β B , depend on the GRB and, their mean values are close to -1 and -2.2 respectively (Preece et al. (2000)). Moreover, following the previous section, the resulting photon distribution could exhibit a cut-off energy at e c,syn . Assuming that σ γγ σ max γγ , and that γ-γ interactions occur only at the threshold energy:

-1

γγ ( h ) 0.26 σ T e c,syn 0 δ( -th ) dn γ d d , ( 24
)
where th (m e c 2 ) 2 / h , and δ, Dirac's function. Then, in the co-moving frame, we deduce the internal optical depth

τ γγ ( h , r) K 0.26 σ T η L iso γ ∆t w (m e c 2 ) 2 4π r 2 E 2 peak h F η (m e c 2 ) 2 E peak h . ( 25
)
At the distance r, this expression is valid only for photons with an energy h > m m 2 e c 4 / max syn , the minimal energy threshold. The energy of cut-off, namely cut , is defined by τ γγ ( cut , r) = 1 for cut > m .

The possibility to have an observational signature, in the synchrotron spectrum, of the cut-off energy relating to the electronic distribution, is determined by both the choice of the magnetic parameters (turbulence level, index α and intensity B b ) and the duration ∆t w and/or the luminosity of the bursts. As it is illustrated, for example, on Fig. 4 and5, for high turbulence level, the γ-γ absorption could only occur during a short while around r c comparing to the internal shock phase duration. We could thus distinguish two situations: first, when cut > e c,syn at the distance r c , we could directly observe the photons with the highest cut-off energy, e c,syn , and obtain the occuring date of this maximum (see Sec. 2.3). When cut < e c,syn around r c , we could deduce e c,syn by determining the slopes (2s 1 -α and 2s 2 -α, see Eq. ( 9) and ( 12)) of the evolution, with time, of the cut-off energy in the photon spectrum when there is no γ-γ absorption i.e during the largest part of the internal shock stage. It is important to note that the original observational test that we propose here, is only based on the observation of the cut-off energy of the resulting synchrotron spectrum which evolves during all the internal shock phase: in fact, we don't need to observe the precise shape of the spectrum. In both cases, observations by Fermi observatory could easily constrain magnetic parameters.

Finally, for a low turbulence level (κ 0 > 100), the γ-γ absorption does not affect the synchrotron spectrum (see Fig. 6), but the highest cut-off energy of the electronic distribution, can yield an observational signature at r b : according to Eq. ( 11), it is noteworthy that this cut-off energy at r b is not dependent on the index α, and does not correspond to any transition between two different energy limitations (i.e r c ). However, because of a too strong adiabatic cooling (see Eq. ( 7) and ( 8)), this situation would be incompatible with the acceleration process in the internal shocks: indeed, this process could not simply start. The upper boundary of the grey zone is the energy of the highest synchrotron photons which are observable. For a high turbulence level, the γ-γ absorption is important, and the cut-off relating to the electronic distribution, at rc, is unobservable. The upper boundary of the grey zone is the energy of the highest synchrotron photons which are observable. For a high turbulence level, the γ-γ absorption is important only for very long bursts, and the cut-off relating to the electronic distribution, at rc, could be easily observable.

Synchrotron emission from UHECRs

In the previous section, we focused on the electronic population. Here, we intend to study the possibility of an observational signature of Ultra-High Energy Cosmic Rays (UHECRs), produced in the internal shocks, via their synchrotron emission. In fact, we want to determine on which conditions the synchrotron photons from accelerated protons and, more particularly, from UHECRs, could be distinguished from leptonic synchrotron and SSC components, without undergoing γ-γ absorption.

Efficiency of UHECR acceleration processes

Following our previous works (Gialis & Pelletier (2003), and Gialis & Pelletier (2004)), let us first consider the energy of the accelerated protons resulting from usual Fermi's process, as described at the very beginning of Sec. 2: because t syn > t dyn at r b , we deduce that the only energy limitation results from the expansion3 . Also, contrary to electrons, protons undergo a single cooling regime during all the expansion: the slow one (see Appendix 2). Then, the energy distribution of protons is such that dn p /d ∝ -p , with p ∈ [2, 3[, between 0 1 GeV and, p max , the highest energy that protons can achieve in the co-moving frame. According to Eq. ( 11) and with r b η 2 r 0 , we have

p max 1.2 × 10 6 κ 0 10 -3 η 100 r 0 10 7 cm B b 10 5 G r r b s 2 GeV, (26)
with the index s 2 = 1 -α. This is the same energy limitation as for electrons, because it depends only on the charge of the particle, and not on its mass. Whatever any reasonable magnetic parameters, it shows that protons cannot achieve the UHECR energy range4 , by considering a such scenario for Fermi's process, as it was suggested by Waxman (1995).

To produce UHECRs, a secondary acceleration process is required. As it has been proposed by Gialis & Pelletier (2004), the protons, pre-accelerated in the internal shocks, could be scattered off multiple magnetized fronts, the internal shocks themselves, and could undergo a kind of "second order" Fermi process, which is efficient in this mildly relativistic regime. The scenario is the following one: each internal shock can be described as a magnetized front in the resulting shocked layer. These magnetized fronts can propagate in both opposite directions in the co-moving frame. Thus, protons are not only accelerated by diffusion process on both sides of the internal shocks, but they can also undergo an acceleration mechanism directly by reflection or scattering off the mildly relativistic magnetized fronts. Also, at each scattering, the cosmic rays have an energy gain of order γ 2 * , where 1 < γ * 2 is the average relative Lorentz factor of a front in the co-moving frame. Because of the magnetic field strength and the relativistic expansion of the layer width, the accelerated protons could be confined in their initial layer, until they reach the local confinement energy limit, given by Hillas' criterium. Because of the anisotropy of their angular distribution function, the particles could mainly escape downstream or upstream of their layer, and consequently stay in the relativistic flow. Then, cosmic rays can be accelerated again, out of their initial layer, when they encounter other magnetized layers. As we have previously seen in a Monte-Carlo simulation (Gialis & Pelletier (2004)), the escaping probability out of the collimated wind flow is weak enough during the acceleration stage from r b to a few hundreds of r b . This secondary acceleration process could be efficient to produce UHECRs: the reason is that the characteristic acceleration time of this process is of the order of the time separating two layer collisions, which is much smaller than the synchrotron cooling time, t syn , and the dynamical time, t dyn , for a sufficient number of layers, namely N s . Let us write this acceleration time, in the co-moving frame, for a flow with N s 1;

t acc L com -N s ∆R c N s m t dyn , ( 27
)
where L com = c ∆t w r/η r 0 is the co-moving length of the flow, ∆R = r/η, the width of a layer, and

m = c ∆t w N s r 0 -1 . ( 28
)
Assuming that 0.5N max s < N s < N max s , with N max s = c ∆t w /r 0 , we deduce that 0 < m < 1. Writing ˙ /t acc , for a particle of energy , we obtain the time for which, a particle, with an initial energy 0 at t 0 , achieve an energy ;

t = t 0 0 m < t 0 0 . ( 29
)
Consequently, a proton with 0 10 6 GeV (see Eq. ( 26)), at a few r b , achieve the UHECR energy range ( = 10 8 GeV) at the time t 10 2 t b , in the co-moving frame, where t b = r b /η c. At last, it is interesting to mention that the characteristic time, t pγ , of p-γ interaction cooling is such that t pγ /t acc r/r b for r > r b : thus, we can neglect this cooling process for UHECRs, even if a resulting high energy neutrino emission may be possible (Gialis & Pelletier (2003)).

Numerical simulations showed that such an acceleration process extends the energy distribution of protons until the local confinement energy limit, i.e cl e B ∆R. Then, the corresponding cut-off energy for synchrotron photons, produced by these UHECRs, is

max p,syn 1.0 η 100 2 r 0 10 7 cm 2 B b 10 5 G 3 r r b 2s 2 -α GeV, ( 30
)
in the co-moving frame. However, according to the acceleration process, UHECRs cannot be produced before a few tens of r b . Also, because of the magnetic field decrease, the emitted synchrotron photons will have a lower energy, and the energy cut-off of these photons would be equal to a few tens of MeV in the co-moving frame, i.e a few GeV for a terrestrial observer, for α close to 1. In fact, the energy range of the hadronic synchrotron peak is very sensitive to the index α of the magnetic field decrease (see Fig. 7): from Eq. ( 9) and ( 30), and for a dominant leptonic emission (see Sec. 4.2), we deduce that synchrotron signature of UHECRs would be compromised for a magnetic field evolution very different from B ∝ r -1 . For this reason, hereafter, we will assume an index α very close to 1.

Comparison between hadronic and leptonic synchrotron emissions

First, we examine observability of an UHECR synchrotron component by comparing the hadronic and leptonic luminosities. Their ratio can be simply expressed by

L p syn L e syn = m e m p 2 N p N e < γ 2 p > < γ 2 e > , ( 31
)
where N p and N e are the numbers of emitting particles and, γ p and γ e their Lorentz factors. In the case of non-relativistic shocks, we could have N p N e , but no generation of UHECRs would be possible in our Fermi acceleration scenario. The situation is quite different in mildly relativistic internal shocks: in fact, because of an higher temperature (see Sec. 2.2), a large fraction of the electronic population is injected in the acceleration process and then, produce an higher synchrotron emission. Also, in Eq. ( 31), we can assume N p N e . For energy distributions ∝ γ -2 , inserting Eq.( 5), we obtain 1.8 × 10 -3 γ e,min 400

-2 r r b -s1 , ( 32
)
in the synchrotron cooling stage (r < r c ). An hadronic synchrotron component seems to be undetectable. However, for α close to 1, the energy range of hadronic and leptonic components are very different around the distance (∼ 100 r b ) where the UHECR population is formed: indeed, r c 100 r b and the cut-off energy for hadronic and leptonic synchrotron components are respectively around a few GeV and a few hundreds of keV for a terrestrial observer (see Fig. 7). Thus, because this emission would begin around a few hundreds of t b /η, i.e a few tens of ms after the very beginning of the prompt emission, in the observer frame, hadronic synchrotron photons could be detectable if one can observe the evolution of the spectrum with time as it was previously suggested in Sec. 3.2.

SSC and pair creation effects on observability

Concerning a possible pair creation process, the energy of threshold, for this synchrotron photons, is

p th (m e c 2 ) 2 max p,syn 2.6 × 10 -1 η 100 -2 r 0 10 7 cm -2 B b 10 5 G -3 r r b α-2s 2 keV, (33)
beyond ∼ 100 r b . Thus, for α close to 1, there is no γ-γ absorption, because p th is always max syn , the cut-off energy of the electronic synchrotron spectrum, whatever the distance r (see Fig. 8).

Let us now investigate the possible obstacle to the observation of the synchrotron emission of UHECRs due to the SSC emission. This depends on the Compton parameter Y ≡ U s /U B , i.e the ratio between the energy densities of the synchrotron radiation and of the magnetic field. Indeed, the synchrotron spectrum is Comptonized such that a second bump at few 100 MeV range is generated (see below); its luminosity peak is almost the synchrotron luminosity peak multiplied by Y . Actually the real parameter is weakened by the Klein-Nishina regime, which mostly amounts to multiply Y by the fraction of soft photons of energy below the threshold of pair creation. Anyway this does not change the order of magnitude. However, the synchrotron spectrum is not simply duplicate; its high energy part decays more rapidly (its index is more or less increased by 1). Nevertheless, this SSC emission can totally hide the high energy cut-off of the leptonic synchrotron spectrum for Y 10 -2 , unfortunately; and thus we cannot determine directly the electron distribution cut-off. But an indirect determination is not excluded, as explained previously when discussion about the γ-γ absorption (see 3.2).

The dominant electronic population has a Lorentz factor γe 10 3 during all the expansion, and, in the outflow, the most important photon field corresponds to the E peak i.e photons with energy peak few keV in co-moving frame. Thus, the SSC-peak is estimated around 100 MeV ( η γ2 e peak in terrestrial observer frame), and the hadronic synchrotron spectrum could be observed around a few GeV as predicted above, provided that the Compton parameter Y be sufficiently small (< 10 -2 ), which is consistent with the requirement of a magnetic dominated outflow suitable for UHECR generation (see also, e.g., Asano & Inoue (2007)). th , with the cut-off energy, max syn , of the electronic synchrotron spectrum, for α = 1, and in the co-moving frame. There is no possible γ-γ absorption.

Discussion and conclusion

The analysis we have made allows to draw several unexpected conclusions about a possible diagnosis of magnetic field parameters (turbulence level, index α and intensity B b ) and particle acceleration processes in the internal shock stage, by taking the observational limitations due to γ-γ absorption and SSC process into account. We proposed to estimate these magnetic field parameters by studying some observational consequences of their variations and, more particularly, by determining the evolution of the cut-off energy of the synchrotron part of the electronic spectrum. We then detailed on which conditions the signatures of accelerated leptonic and hadronic population of particles could be observed during the GRB prompt emission. Let us summarize our results:

-First, we show that the electronic synchrotron spectrum which can be observed at the very beginning of the internal shock phase by Fermi observatory could lie in a narrow energy band (10-100 keV) for a terrestrial observer. Moreover, the cut-off energy of this spectrum places constraints upon both magnetic field intensity and its turbulence level, that we determined (see Eq. ( 7) and ( 8)).

-For a magnetic field (B ∝ r -α ) with an index 1 ≤ α 1.2 and by considering an high level of magnetic field turbulence (η t 10 -2 ), we cannot observe any highest cut-off energy in the electronic synchrotron spectrum, corresponding to e c,syn , because the cooling radius, r c , would be too far from the inner engine. In another hand, this case would be the most favourable to observe an early high energy synchrotron signature from UHECRs, with a cut-off energy around a few GeV, on the condition that there is no too strong SSC component in this energy range (in fact, when Y < 10 -2 ), in the internal shocks at the same moment. It is noteworthy that the magnetic field decreases in 1/r in two important cases: one case corresponds to the existence of a large scale mean field anchored in a central rotator (this is then a toroidal field that dominates at large distance), the other case corresponds to the amplification of the magnetic field to equipartition at each shock (P mag ∼ P ∼ ρ γ c 2 with ρ ∝ 1/r 2 ).

-For 1.2 α 1.4, the electronic synchrotron spectrum could reveal an highest cut-off energy just before the end of the prompt emission, despite the pair creation process with gamma photons of lower energy and for a Compton parameter Y < 10 -2 . This cut-off energy, around 1 GeV for the terrestrial observer, could be directly observable for sufficient short GRBs (i.e ∆t w < a few tens of seconds) and for reasonable luminosity (i.e L iso γ 10 51 erg•s -1 ), by the new Fermi observatory.

-For 1.4 α 2, we cannot observe directly any highest cut-off energy in the electronic synchrotron spectrum, but this energy could be easily deduced from the observation of the instantaneous cut-off energy during the prompt emission stage and, more particularly, from its variations with time, as described in Sec. 3.2.

-For a low level of turbulence (η t < 10 -3 ), and whatever the magnetic field index α, there is no γ-γ absorption.

However, the highest energy cut-off of the electronic synchrotron spectrum does not correspond to the previous cut-off energy e c,syn , and it only results from the comparison between t acc and t dyn at the very beginning of the internal shock stage (i.e at r b ). Also, it is important to note that this early cut-off energy is not dependent on the magnetic field index. In this case, the electronic population would undergo a pure slow cooling regime, but the adiabatic cooling is too strong and the acceleration process cannot start.

-At last, for a low turbulence level, the UHECR generation could be also compromise, because of inefficiency of the primary acceleration process in the internal shocks (see Eq. ( 26) with κ 0 10) and, consequently, a too weak energy of injection in any secondary acceleration process.

We emphasize that any observability of the electron distribution cut-off or of the UHECR synchrotron emission would stem from a detailed analysis of the spectrum temporal evolution; no single zone modeling of the emissions will provide with this important diagnosis. The diagnosis of magnetic parameters through the spectrum evolution seems already at hand with Fermi observations.

1 β-3 η 1-β β-3 (B b r α b ) β β-3 r s 1 , ( 38
)
where the index s 1 can be written as

s 1 = 1 + β (α -1) 3 -β , ( 39
)
and the coefficient C 2 (β) as MeV.

C 2 (β) = 6π m 2 e c 3 σ T C 1 (β) 1 3-β . ( 40
(41)

Another limitation comes from adiabatic cooling due to the relativistic expansion of the layer. Comparing the characteristic dynamical time ( r/c η) with the acceleration time leads to a new limitation for the energy of the electrons. This expansion limitation, namely exp and calculated in the co-moving frame, is such that

exp = (c C 1 (β)) 1 β-2 κ 0 10 1 β-2 η -1 B b r α b r s2 , ( 42
)
where the index s 2 is equal to 1 -α. For α = 1, and β = 5/3, we have

exp = 1.2 × 10 9 κ 0 10 -3 η 100 B b 10 5 G r 0 10 7 cm MeV. ( 43
)
The transition radius r c between these two limitations can be easily obtained:

r c = Λ(α, β) κ 0 10 1 (β-2) (3α-2) η 2 2-3α (B b r α b ) 3 3α-2 , ( 44
)
where the coefficient is such that

Λ(α, β) = 20 (2π) 1-β 3 (β -1) 3-β (β-2) (3α-2) 9π m 2 e c 4 (β -1) 10 σ T (2π) 1-β e 1 2-3α . ( 45
)
For α = 1 and β = 5/3, the distance may be close to r d ;

r c = 1.0 × 10 20 κ 0 10 -3 η 100 4 B b 10 5 G 3 r 0 10 7 cm 3 cm . ( 46
)
Thus, during the expansion, the electron energy distribution undergoes a two stage self-similar evolution, that depends on magnetic parameters.

For an electron in a magnetic field B, the synchrotron emission reaches a maximum power close to the cut-off energy

max h e B 4 m 3 e c 5 2 , ( 47
)
where is the energy of the electron. According to Eq. ( 38), for r b ≤ r ≤ r c , this energy, in the observer frame, can be expressed as

syn max h e 4m 3 e c 5 C 2 2 (β) κ 0 10 2 β-3 η 1+β 3-β (B b r α b ) 3(1-β) 3-β r 2s 1 -α . ( 48
)
And Eq. ( 42) leads to the evolution for r ≥ r c , which is ∝ r 2-3α ;

syn max h e 4m 3 e c 5 (c C 1 (β)) 2 β-2 κ 0 10 2 β-2 η -1 (B b r α b ) 3 r 2-3α . ( 49
)
The energy c at the break radius r c is given by

c 4.4 × 10 -8 (c C 1 (β)) 2 β-2 20 (2π) 1-β 3 (β -1) β-3 β-2 9π m 2 e c 4 (β -1) 10 σ T (2π) 1-β e κ 0 10 1 β-2 η .
For r r b and according to Eq. ( 48), one can write the initial syn max , namely i peak , as

i peak ∝ η λ 3-β η 2 3-β t Ω c ∆t w r 0 3 2 β-1 3-β M 5/2 BH E -3/2 mag β-1 3-β , ( 50
)
where E mag is the magnetic energy released by the burst, M BH the mass of the inner black hole, and λ = 5-3β+6α (β-1).

Assuming that E mag ∝ E and M BH ∝ E, this relation yields to a kind of Amati's law involving β only:

i peak ∝ E β-1 3-β . ( 51
)
Appendix 2: Determination of the local electronic synchrotron spectrum

In this Appendix, we propose to determine the electronic distribution and the resulting synchrotron spectrum locally produced, which both depend on the distance from the central engine: we usually distinguish two cooling regimes for the electronic population (Sari et al. (1998), Gupta & Zhang (2007)) separating by the condition t syn = t dyn corresponding to a critical Lorentz factor: (53)

Obviously, whatever α ∈ [1, 2], r b < r t < r c . Then, because of the outflow evolution during the expansion, we show that accelerated electrons undergo these two cooling regimes (see Fig. 9). Indeed, let be f ( , t), the electronic distribution, at a time t in the co-moving frame. According to the acceleration process, the continuity equation, for the fast cooling regime, can be simply expressed as

∂f ∂t + ∂ ∂ ( ˙ f ) ∝ -p , ( 54
)
by assuming that t acc t syn , and only considering synchrotron losses, i.e ˙ ∝ 2 . In the stationary case (∂/∂t ≡ 0), we deduce that f ( , t) ∝ -(p+1) . In slow cooling regime, the distribution is mainly governed by the acceleration process, i.e f ( , t) ∝ -p . Thus, during the expansion, and in the co-moving frame, we derive the following results, without taking account of the thermal component for < e min : -When γ e,min > γ cr , which is always the case at the very beginning of the internal shock phase, (61)

The resulting synchrotron photon distribution can be numerically calculated by the integration of the previous local distributions, all along the outflow. The obtained result is really different from the previous works (see e.g. Gupta & Zhang (2007)) because it takes into account both the important stratification of the outflow and a precise cut-off energy, in the local electronic synchrotron spectrum, resulting from a detailed analysis of the acceleration process. This point will be fully detailled in a forthcoming paper. However, in this paper, we have just assumed that the resulting photon distribution is given by a phenomenological Band's function (Band et al. (1993)).

Fig. 1 .

 1 Fig. 1. Variation of the cooling radius rc with the magnetic field intensity, the turbulence level and the index α. The horizontal line corresponds to about the deceleration radius i.e the end of the internal shock stage.

Fig. 2 .

 2 Fig. 2. Left: Variation of the threshold radii with the index α for an usual turbulence level. The horizontal line corresponds to a distance of about the deceleration radius i.e the end of the internal shock stage. Right: Variation of the threshold radii with the turbulence level for α = 1.5.

Fig. 3 .

 3 Fig. 3. The case α = 1.0 (i.e B ∝ r -1 ) with B b = 10 5 G: comparison of the maximum energy (co-moving) of synchrotron photons with the corresponding threshold energy of the pair creation process, for two different turbulence levels. No pair creation process is possible.

Fig. 4 .

 4 Fig.4. The case α = 1.5 (i.e B ∝ r -3/2 ) with B b = 10 5 G, E peak = 100 keV, αB = -1 and βB = -2.2, for different burst durations ∆tw: comparison of the maximum energy (co-moving) of synchrotron photons with the corresponding threshold energy of the pair creation process ( th = m), and the cut-off energy cut. The upper boundary of the grey zone is the energy of the highest synchrotron photons which are observable. For a high turbulence level, the γ-γ absorption is important, and the cut-off relating to the electronic distribution, at rc, is unobservable.

Fig. 5 .

 5 Fig. 5. The case α = 1.3 (i.e B ∝ r -1.3 ) with B b = 10 5 G, E peak = 100 keV, αB = -1 and βB = -2.2, for different burst durations ∆tw: comparison of the maximum energy (co-moving) of synchrotron photons with the corresponding threshold energy of the pair creation process ( th = m), and the cut-off energy cut.The upper boundary of the grey zone is the energy of the highest synchrotron photons which are observable. For a high turbulence level, the γ-γ absorption is important only for very long bursts, and the cut-off relating to the electronic distribution, at rc, could be easily observable.

Fig. 6 .

 6 Fig. 6. Comparison of the maximum energy (co-moving) of synchrotron photons with the corresponding threshold energy of the pair creation process ( th = m ) for a low turbulence level, and for B b = 10 5 G: there is no γ-γ absorption.

  p,min γ p,max γ e,min γ e,max

Fig. 7 .

 7 Fig. 7. Comparison of the cut-off energy, max p,syn , of the UHECR synchrotron spectrum (left), with the cut-off energy, max syn , of the electronic synchrotron spectrum (right), for different magnetic index α, and in the co-moving frame.

Fig. 8 .

 8 Fig. 8. Comparison of the threshold energy, pth , with the cut-off energy, max syn , of the electronic synchrotron spectrum, for α = 1, and in the co-moving frame. There is no possible γ-γ absorption.

)

  For α = 1, and a Kolmogorov scaling with β = 5/3, this limitation can be expressed as

  with a Lorentz factor γ e , undergoes a fast cooling regime if γ e > γ cr and, a slow cooling regime otherwise. The transition radius, r t , for which γ cr = γ e,min , is

--

  = γ e,min m e c 2 and cr = γ cr m e c 2 . This is a "pure" fast cooling regime for all the accelerated electrons. The corresponding synchrotron photon distribution, in the co-moving frame, 4π m e c) for γ cr > 1. More precisely, the scaling is the following one When γ e,min < γ cr < γ e,max , which is always the case between r t and r c , the two cooling regimes for the accelerated electron population and, the synchrotron photon distribution, in the co-moving frame, When γ cr > γ e,max , which is always the case beyond the cooling radius r c , dn e d ∝ -p for e min < < e max .(60) This is a "pure" slow cooling regime for the accelerated electrons. Here, the synchrotron photon distribution is

However, we think that this power law could be modified by considering a continuous second order Fermi process in the shocked and/or unshocked layers.

We neglect corrections due to cosmological redshift.

We assume that the outflow is always transparent to p-p interactions beyond a few r b (seeGialis & Pelletier (2003)), and we neglect proton energy losses due to p-γ interactions.

Particles with an energy ≥ 10 8 GeV in the co-moving frame.

Appendix 1: Cut-off energy of the electronic population in the internal shock stage

The Fermi acceleration (first and second order) in the internal and midly relativistic shocks was first considered by Waxman (1995) with a characteristic acceleration time proportional to Larmor's time of accelerated particles (Bohm scaling). We showed in a previous paper (Gialis & Pelletier 2003) that, in a more realistic situation, Fermi's acceleration time in the internal shocks depends on the mean free path of the particle, namely ¯ , in an irregular magnetic field. Assuming a turbulence spectrum of magnetic perturbations that follows a power law of index β, on both sides of the shock, we have

for Larmor's time r L < c , with η t ≡ <δB 2 > <B 2 > the turbulence level, and c the magnetic correlation length, which is of the order of the layer thickness i.e r/η beyond the radius r b . Eq. ( 34) is known in weak turbulence theory and has been extended into the regime of strong turbulence (Casse et al. 2002). Then, the acceleration time t acc is given by

where µ is the cosine of the pitch angle.

We suppose that layers start to expand by carrying a magnetic field the intensity of which decays like r -α (1 ≤ α < 2), and is equal to B b 10 5 G at r b . According to the equation ( 35), the acceleration time, in the co-moving frame, for an electron is (c.g.s units)

where is the energy of the electron, e is elementary charge, κ 0 (≡ |µ| 1-β 2π η t ), the ratio of the acceleration time over Larmor's time for r L = c , and the coefficient

The acceleration process is clearly more efficient for a value of α close to 1 i.e for a slow decaying magnetic field.

In the internal shock stage, we obtain the electron maximal energy by comparing the acceleration time with the characteristic synchrotron loss time. This energy, namely syn and calculated in the co-moving frame, is the synchrotron limitation syn = C 2 (β) κ 0 10