Antoine Benoit
email: antoine.benoit@univ-nantes.fr

Geometric optics expansions for linear hyperbolic boundary value problems and optimality of energy estimates for surface waves

Keywords:

In this article we are interested in energy estimates for initial boundary value problem when surface waves occur that is to say when the uniform Kreiss Lopatinskii condition fails in the elliptic region or in the mixed region. More precisely we construct rigorous geometric optics expansions for elliptic and mixed frequencies and we show using those expansions that the instability phenomenon is higher in the case of mixed frequencies even if the uniform Kreiss Lopatinskii condition does not fail on hyperbolic modes. As a consequence this result allow us to give a classification of weakly well posed initial boundary value problems according to the region where the uniform Kreiss Lopatinskii condition degenerates.

Introduction

In this article we will construct rigorous geometric optics expansions for solutions to a linear hyperbolic initial boundary value problem (ibvp in short) with constant coefficients. The uniform Kreiss-Lopatinskii condition (UKL in short) is the main point in the study of initial boundary value problem. Indeed it is a necessary and sufficient condition for strong well-possedness of the ibvp. This condition due to Kreiss in [START_REF] Kreiss | Initial boundary value problems for hyperbolic systems[END_REF] means that in the normal modes analysis no stable mode satisfies the homogeneous boundary conditions. In this article we are interested in ibvp for which this condition is not satisfied and fails for some "boundary" frequencies. The description of those frequencies will be made precise in section 2.2.

Geometric optics expansions for the ibvp are also highly linked to the structure of the resolvent matrix of the ibvp for boundary frequencies. Let us recall that thanks to the block structure of the resolvent equation (see theorem 2.1, [START_REF] Chazarain | Caractérisation des problèmes mixtes hyperboliques bien posés[END_REF], [START_REF] Benzoni-Gavage | Multidimensional hyperbolic partial differential equations[END_REF]) we know that there are four different kinds of frequencies, namely : elliptic, hyperbolic, mixed or glancing. There are already many articles about geometric optics expansions for ibvp according to the frequency of oscillations in the source term, the fact that the ibvp is characteristic or not, the stability assumption on the boundary ... For instance, on the one hand, under the uniform Kreiss-Lopatinskii condition (UKL), Williams treats in [START_REF] Williams | Nonlinear geometric optics for hyperbolic boundary problems[END_REF] the case of a noncharacteristic ibvp for all possible frequencies, and Lescarret [START_REF] Lescarret | Wave transmission in dispersive media[END_REF] deals with characteristic problems for a mixed-frequency. On the other hand, Marcou [START_REF] Marcou | Rigorous weakly nonlinear geometric optics for surface waves[END_REF] and Coulombel-Guès [START_REF] Coulombel | Geometric optics expansions with amplification for hyperbolic boundary value problems: linear problems[END_REF] build expansions without UKL but for an elliptic frequency and a hyperbolic frequency respectively. In this paper we are interested in noncharacteristic ibvp, when UKL fails in the elliptic region or in the mixed region of the boundary of the frequencies set (see section 2.1 for more details). So this paper will generalize Marcou's work to non homogenous problems and generalize Lescarret's work to problems for which UKL is violated. To some extent this paper will conclude the construction of geometric optics expansion for weakly well-posed ibvp. The main consequence of the geometric optics expansions will be to describe the influence of the region in which UKL fails on the energy estimate that we have on the solution of the ibvp. Indeed, in [START_REF] Kreiss | Initial boundary value problems for hyperbolic systems[END_REF] Kreiss shows that UKL is a necessary and sufficient condition for the strong well-possedness of the corresponding ibvp in L 2 . The corresponding energy estimate associated to the solution u is of the form :

γ u 2 L 2 γ + u |x d =0 2 L 2 γ ≤ C 1 γ f 2 L 2 γ + g 2 L 2 γ ,
where f is the source term in the interior and g the source term on the boundary. Then, Sablé-Tougeron showed in [START_REF] Sablé-Tougeron | Existence pour un problème de l'élastodynamique Neumann non linéaire en dimension 2[END_REF] that if UKL fails at the order one in the elliptic region of the boundary of the space of frequencies, it is possible to consider solutions in a weaker sense. That is to say that there exists a unique solution of the ibvp that satisfies an energy estimate with a loss of one derivative on the boundary. At last Coulombel treated in a more recent work cases where UKL is violated in the mixed region and in the hyperbolic region and showed two different energy estimates ([6] [START_REF] Coulombel | Well-posedness of hyperbolic initial boundary value problems[END_REF] ). In the mixed region he proved that there is a loss of one derivative on the boundary (as in Sablé-Tougeron's work) and also a loss of half a derivative in the interior. Whereas when UKL fails in the hyperbolic region, there is a loss of one derivative both on the boundary and in the interior.

The question is to know whether those energy estimates are sharp. In 2010, using geometric optics expansion for an ibvp with a failure of UKL in the hyperbolic region Coulombel-Guès [START_REF] Coulombel | Geometric optics expansions with amplification for hyperbolic boundary value problems: linear problems[END_REF] proved that the corresponding energy estimate is sharp. This paper will treat two of the three remaining behaviours for the failure of UKL, that is to say elliptic and mixed case. And we will be able to show the optimality of the energy estimates found by Sablé-Tougeron and Coulombel. In particular, our work shows that away from glancing points, the only possible failure of the UKL that allows for a homogeneous ibvp to be well-posed is in the elliptic region. This makes rigorous the discussion in ([3] , chapter 7) see also [START_REF] Ohkubo | On structures of certain L 2 -well-posed mixed problems for hyperbolic systems of first order[END_REF].

2 Notations, assumptions and main results.

2.1 Notations.

In this article we will consider ibvps in the half-space

R d + := x = (x , x d ) ∈ R d \ x d ≥ 0
and will we also note for T > 0, Ω T := ]-∞; T ] × R d + and finally the spatial boundary of Ω T will be denoted by ω T . Our typical ibvp will read :

     L(∂)u ε := ∂ t u ε + d j=1 A j ∂ j u ε = f ε in Ω T , Bu ε |x d =0 = g ε in ω T , u ε |t≤0 = 0, (1)
where the A j are square matrices of dimension N with real coefficients and B is a real matrix of dimension p × N . The value of p will be made precise in assumption 2.3. The superscript ε is sometimes used in order to highlight the main frequency of oscillations of the source terms and the solution in [START_REF] Benzoni-Gavage | Stability of multi-dimensional phase transitions in a van der Waals fluid[END_REF]. Energy estimates require the introduction of weighted Sobolev spaces. For s real and γ > 0 we will denote by H s γ (Ω T ) the weighted Sobolev space defined as follows :

H s γ (Ω T ) = u ∈ D (Ω T ) \ e -γt u ∈ H s (Ω T
) , and we also define H s γ (ω T ) and L 2

x d (H s γ (ω T )) in the same spirit. We introduce a partition of the frequencies space :

Ξ = ζ = (σ = γ + iτ, η) ∈ C × R d-1 , γ ≥ 0 , Ξ 0 = {ζ ∈ Ξ, γ = 0} .
Let A(ζ) be the resolvent matrix obtained after Fourier-Laplace transformation in the evolution equation of (1) :

A(ζ) = -A -1 d   σI + i d-1 j=1 η j A j   . (2)
We denote by E -(ζ) the stable subspace of A(ζ), and E + (ζ) the unstable subspace.

Thanks to Hersh's lemma (see [START_REF] Benzoni-Gavage | Multidimensional hyperbolic partial differential equations[END_REF] page 125) we know that for all σ such that Re σ > 0, A(ζ) does not have any purely imaginary eigenvalue and that dimE -(ζ) = p. However if Reσ is zero, Hersh's lemma is not true anymore and has to be substituted by the following result which is due to Kreiss [START_REF] Kreiss | Initial boundary value problems for hyperbolic systems[END_REF] and adapted by Métivier [START_REF] Métivier | The block structure condition for symmetric hyperbolic systems[END_REF] for constantly hyperbolic operators that is to say the following assumption is satisfied.

Assumption 2.1 There exist an integer q ≥ 1, real valued analytic functions λ 1 , ..., λ q on R d \ {0}and positive integers ν 1 , ..., ν q such that :

∀ξ ∈ S d-1 , det   τ + d j=1 ξ j A j   = q k=1 (τ + λ k (ξ)) ν k ,
with λ 1 (ξ) < ... < λ q (ξ) and the eigenvalues λ k (ξ) of

d k=1 A k are semi-simple. Theorem 2.1 [Block structure] If the ibvp (1) satisfies assumption 2.1 then for all ζ in Ξ there is a neighborhood V of ζ in Ξ, an integer L ≥ 1, a partition N = ν 1 + ... + ν L such that all ν i ≥ 1
, and a regular invertible matrix T defined on V such that we have :

∀ζ ∈ V, T (ζ) -1 A(ζ)T (ζ) = diag (A 1 (ζ), ..., A L (ζ)) ,
where the size of A i (ζ) is ν i , and A i (ζ) satisfies one of the following properties : i) All elements in the spectrum of A i (ζ) have a strictly negative real part .

ii) All elements in the spectrum of A i (ζ) have a strictly positive real part .

iii

) ν i = 1, A i (ζ) ∈ iR, ∂ γ A i (ζ) ∈ R \ {0} iv) ν i > 1, ∃k i ∈ iR such that A i (ζ) =    k i i 0 . . . i 0 k i    ,
and the coefficient in the down left corner of

∂ γ A i (ζ) ∈ R \ {0}.
Thanks to this theorem we are able to describe the four kinds of frequencies ζ in Ξ 0 .

Definition 2.1 Let :

• E be the set of elliptic frequencies, that is to say the set of ζ in Ξ 0 such that theorem 2.1 is satisfied with one block of type i) and consequently also one of type ii) only.

• H be the set of hyperbolic frequencies, that is to say the set of ζ in Ξ 0 such that theorem 2.1 is satisfied with blocks of type iii) only.

• EH be the set of mixed frequencies, that is to say the set of ζ in Ξ 0 such that theorem 2.1 is satisfied with one block of each type i) and consequently also one of type ii) and at least one of type iii), but no block of type iv)

• G be set the of glancing frequencies, that is to say the set of ζ in Ξ 0 such that theorem 2.1 is satisfied with at least one block of type iv).

We have the following partition of Ξ 0 :

Ξ 0 = E ∪ EH ∪ H ∪ G.
The analysis in [ [START_REF] Kreiss | Initial boundary value problems for hyperbolic systems[END_REF], [START_REF] Métivier | The block structure condition for symmetric hyperbolic systems[END_REF]] shows that E ± (ζ) admit a continous extension for the frequencies in Ξ 0 . Moreover if, ζ ∈ Ξ 0 \ G then we can write

C N = E -(ζ) ⊕ E + (ζ),
and

E ± (ζ) = E e ± (ζ) ⊕ E h ± (ζ), (3)
where

E h -(ζ) = ⊕ m∈C kerL(τ , η, ω m ), E h + (ζ) = ⊕ m∈N C kerL(τ , η, ω m ),
where for all ξ in R d+1 , L(ξ 0 , ..., ξ d ) = ξ 0 I + d j=1 ξ j A j .

We refer to [START_REF] Coulombel | Geometric optics expansions with amplification for hyperbolic boundary value problems: linear problems[END_REF] for the proof.

Q m L(ζ, ω m ) = I -Π m , Π m Q m = Q m Π m = 0.

Assumptions.

Let us deal with non characteristic ibvps with constant multiplicity that is assumption 2.1 and the following assumption are checked.

Assumption 2.2

The matrix A d is invertible.

The assumptions on the boundary condition are summarized in the following one : Assumption 2.3 B is a matrix of maximal rank p, with p ≥ 1 the number of positive eigenvalues of A d counted with multiplicity.

Our last assumption explains how the uniform Kreiss-Lopatinskii condition degenerates. There are two distincts cases depending on the region of degeneration : where

Assumption 2.4 • The Kreiss-Lopatinskii condition is satisfied, that is to say for all ζ ∈ Ξ \ Ξ 0 , kerB ∩ E -(ζ) = {0}. • Let ζ ∈ Ξ 0 such that kerB ∩ E -(ζ) = {0} then ζ ∈ E
∀ζ ∈ V, B [E 1 , ..., E p ] (ζ) = P (ζ)diag(γ + iθ(ζ), 1, ..., 1). Assumption 2.5 • The Kreiss-Lopatinskii condition is satisfied. • Let ζ ∈ Ξ 0 such that kerB ∩ E -(ζ) = {0} then ζ ∈ EH and such frequencies exist. • Let ζ ∈ EH such that kerB ∩ E -(ζ) = {0} then : E -(ζ) ∩ kerB = E e -(ζ) ∩ kerB. There is a neighborhood V of ζ in Ξ,
(E h 1 , ..., E h r )(ζ) is a regular basis of E h -(ζ).
Assumptions 2.4 and 2.5 correspond to situations where the uniform Kreiss-Lopatinskii condition breaks down because of surface waves that have exponential decay with respect to the normal variable x d . It is equivalent to assume that a Lopatinskii determinant vanishes at the order one for elliptic or mixed frequencies (see [START_REF] Benzoni-Gavage | Generic types and transitions in hyperbolic initial-boundary-value problems[END_REF], [START_REF] Sablé-Tougeron | Existence pour un problème de l'élastodynamique Neumann non linéaire en dimension 2[END_REF] ). However, assumption 2.5 considers a situation where hyperbolic modes also occur, even though they are not responsible for the breakdown of the uniform Kreiss-Lopatinskii condition. One example of an ibvp which satisfies the assumption 2.4 is the onset of Rayleigh waves for the equations of elastodynamics. That is to say that when UKL fails in the elliptic area one can see waves which are localized along the boundary.

One example of physical interest we know of an ibvp which satisfies the assumption 2.5 is the liquid-vapor phase transition model especially studied by Benzoni-Gavage in [START_REF] Benzoni-Gavage | Stability of multi-dimensional phase transitions in a van der Waals fluid[END_REF] and later on by Coulombel in [START_REF] Coulombel | Stabilité multidimensionnelle d'interfaces dynamiques[END_REF]. We provide in section 5 with another such example arising from the linearization of the Euler equations on a fixed domain.

Until the end we will work under one and only one of the two previous assumptions.

Now thanks to one of these assumptions we can describe the form of the source terms we are interested in.

Definition 2.4 • Let P os be the set of oscillating profiles ie functions u(t, x, X d ) in C ∞ (Ω T × R + ) which can be written :

u(t, x, X d ) = M m=1 e iωmX d u m (t, x), with (ω 1 , ..., ω M ) ∈ R M , (u 1 , ..., u M ) ∈ (H +∞ (Ω T )) M . • The set of functions U (t, x, X d ) in H +∞ (Ω T ×R + ) for which there is a positive δ such that e δX d U (t, x, X d ) is in H +∞ (Ω T × R + )
, will be denoted by P ev and will be the set of evanescent profiles.

• To conclude, the set of profiles P is defined as follows P = P os ⊕ P ev .

Source terms used in the geometric optics expansion will not be the same under assumption 2.4 or under assumption 2.5. We shall be more specific later on.

To conclude this section we define the useful following vectors :

Definition 2.5 Under assumption 2.4 or 2.5 there exists :

• a vector e ∈ C N \ {0} such that E -(ζ) ∩ kerB = E e -(ζ) ∩ kerB = vect(e). • A vector b ∈ C p \ {0} such that b.Bw = 0, for all w ∈ E -(ζ). • E -(ζ) = vect(e) ⊕ Ȇ-(ζ) and B is an isomorphism from Ȇ-(ζ) to b ⊥ .

Main results.

As mentioned in the introduction energy estimates depend on the region in which the uniform Kreiss-Lopatinskii condition fails. In the case of a frequency ζ that satisfies the assumption 2.5 we will need the extra assumption :

Assumption 2.6 If ζ satisfies 2.5, then E h + (ζ) is non trivial. Let us recall the different energy estimates : Theorem 2.2 [16] [Sablé-Tougeron] Under assumptions 2.1-2.2-2.3-2.4, for all f in L 2 (Ω T ), g ∈ H 1 2
(ω T ) that vanish for t ≤ 0, then there is a unique solution u ∈ L 2 (Ω T ) of the ibvp (1) which moreover admits the following energy estimate :

u 2 L 2 γ (Ω T ) + u |x d =0 2 H -1 2 γ (ω T ) ≤ C γ 2 f 2 L 2 γ (Ω T ) + g 2 H 1 2 γ (ω T ) , (5)
with γ sufficiently large and where C is independent of γ.

[6] [Coulombel] Under assumptions 2.1-2.2-2.3-2.5-2.6, for all f in L 2 x d (H 1 2 (ω T )), g ∈ H 1 2
(ω T ) that vanish for t ≤ 0, then there is a unique solution u ∈ L 2 (Ω T ) of the ibvp (1) which moreover admits the following energy estimate :

u 2 L 2 γ (Ω T ) + u |x d =0 2 H -1 2 γ (ω T ) ≤ C γ 2 1 γ f 2 L 2 x d (H 1 2 γ (ω T )) + g 2 H 1 2 γ (ω T ) , (6)
with γ sufficiently large and where C is independent of γ.

The estimates ( 5)-( 6) are true for γ above a threshold γ 0 , and it is possible to avoid the dependency in γ by multiplying C by e γ0T . So we can rewrite ( 5)-( 6) under the more pleasant form :

u 2 L 2 (Ω T ) + u |x d =0 2 H -1 2 (ω T ) ≤ C T f 2 L 2 x d (H s (ω T )) + g 2 H 1 2 (ω T )
, with s = 0 for (5) and s = 1 2 for (6), f, g, u zero for t < 0. Our first result is that one can make a geometric optics expansion if ζ satisfies assumption 2.4 or 2.5. The oscillating source terms f ε , g ε are made precise in (9) under assumption 2.4 and (9) under assumption (24). Theorem 2.3 Under assumptions 2.1-2.2-2.3-2.4, one can find a geometric optics expansion (U n ) n∈N of the ibvp (1) which satisfy equations [START_REF] Lescarret | Wave transmission in dispersive media[END_REF] (see below). Moreover, setting u ε N0,app the truncated geometric optics expansion defined by :

u ε N0,app = N0 n=0 e i ϕ ε ε n U n , (7)
then the error beetween u ε N0,app and the exact solution

u ε of (1) is an O(ε N0+ 3 2 ) in H ∞ ε (see 4.1
). Under assumptions 2.1-2.2-2.3-2.5, one can find a geometric optics expansion (U n ) n∈N of the ibvp (1) which satisfy equations ( 26)-(27) (see below). Such that the error beetween u ε N0,app and the exact solution u ε of ( 1) is an

O(ε N0+1 ) in H ∞ ε (see 4.1).
The two following theorems show that energy estimates ( 5) and ( 6) are sharp :

Theorem 2.4 Under assumptions 2.1-2.2-2.3-2.4, let s > 0.
We assume that for all f in L 2 (Ω T ), g ∈ H s (ω T ) vanishing for t ≤ 0, there is a unique solution u in L 2 (Ω T ) of the ibvp (1) with the energy estimate :

u 2 L 2 (Ω T ) ≤ C T f 2 L 2 (Ω T ) + g 2 H s (ω T ) ,
then necessarily s ≥ 1 2 . Theorem 2.5 Under assumptions 2.1-2.2-2.3-2.5-2.6, let s 1 , s 2 > 0. We assume that for all f in L 2

x d (H s1 (ω T )), g ∈ H s2 (ω T ) vanishing for t ≤ 0, there is a unique solution u in L 2 (Ω T ) of the ibvp (1) with the energy estimate :

u 2 L 2 (Ω T ) ≤ C T f 2 L 2 x d (H s 1 (ω T )) + g 2 H s 2 (ω T ) , (8)
then necessarily s 1 ≥ 1 2 and s 2 ≥ 1 2 . Theorem 2.5 indicates that the existence of hyperbolic outgoing modes has a serious impact on the stability properties of (1). In particular, strong wellpossedness of (1) for g ≡ 0, as considered in [[3], chapter 7] can not occur if such modes are present. This result gives a rigorous justification to the discussion in [ [START_REF] Benzoni-Gavage | Multidimensional hyperbolic partial differential equations[END_REF] pages 205-207]. We refer to section 5 for further comments and consequences.

3 Geometric optics expansions.

Let ϕ(t, x ) = τ t + η.x , with (τ , η) a frequency that satisfies assumption 2.4 or assumption 2.5.

Geometric optics expansions under assumption 2.4.

This section gives a rigorous construction of geometric optics expansions when UKL fails in the elliptic area. It is a generalization of Marcou's work in the case of a non-homogenous boundary condition. Indeed, in this paper we are interested in the influence of a non-homogenous boundary condition on the energy estimate. Moreover, the case of an elliptic frequency will be a good preparatory work for the case of a mixed frequency in the next paragraph.

Remark Hereinafter as no confusion is possible we will set Π e ± = Π ± . From now to the end of the paragraph, we will deal with source terms f ε and g ε of the form,

f ε (t, x) = e i ϕ ε f ev t, x, x d ε , (9)
g ε (t, x ) = εe i ϕ ε g(t, x ),
with amplitudes f ev in P ev and g in H +∞ (ω T ),both zero for negative times. We also take an ansatz of the form,

u ε (t, x) = n≥0 ε n e i ϕ ε U n t, x, x d ε , (10)
where for all n > 0, U n is an element of P ev .

Taking such ansatz allows us to treat elliptic modes in one piece as in Lescarret's work [START_REF] Lescarret | Wave transmission in dispersive media[END_REF]. Thus, in comparison to Williams's method in [START_REF] Williams | Nonlinear geometric optics for hyperbolic boundary problems[END_REF] see also [START_REF] Hernandez | Resonant leading term geometric optics expansions with boundary layers for quasilinear hyperbolic boundary problems[END_REF] it is unnecesary to assume that the stable part of A(ζ) is diagonalizable, moreover it does not require to solve in an approximate way transport equations with complex coefficients. Plugging the ansatz [START_REF] Lax | Asymptotic solutions of oscillatory initial value problems[END_REF] in the ibvp (1), the following cascade of equations appears :

       L(∂ X )U 0 = 0, L(∂ X )U n + L(∂)U n-1 = δ n,1 f, ∀n ≥ 1, BU n |x d =X d =0 = δ n,1 g, ∀n, U n |t≤0 = 0, ∀n (11)
where δ n,p is the Kronecker's symbol and :

L(∂ X ) := A d ∂ X d -A(ζ) . Lemma 3.1 Let P ev U (X d ) := e X d A(ζ) Π -U (0), (12)
Q ev F (X d ) := X d 0 e (X d -s)A(ζ) Π -A -1 d F (s)ds - +∞ X d e (X d -s)A(ζ) Π + A -1 d F (s)ds. ( 13
)
Then for all F in P ev , the equation

L(∂ X )U = F,
has a solution U in P ev . Moreover, one can write any such solution U under the following form :

U = P ev U + Q ev F.
proof : P ev and Q ev defined in ( 12)-( 13) are explicitly given by Duhamel's formula.

Using lemma 3.1, the cascade of equations ( 11) can be rewritten equivalently as follows :

   P ev U 0 = U 0 , BU 0 |x d =X d =0 = 0 , U 0 |t≤0 = 0 (14)
and for higher order terms :

   (I -P ev )U n = Q ev (δ n,1 f ev -L(∂ x )U n-1 ) , BU n |x d =X d =0 = δ n,1 g , U n |t≤0 = 0 , ∀n ≥ 1. ( 15
)
3.1.1 Determining the leading order term U 0 .

Using P ev U 0 = U 0 in the boundary condition of ( 14) one has to solve the following system for the principal term :

   P ev U 0 = U 0 , BΠ -U 0 (t, x , 0, 0) = 0 , U 0 |t≤0 = 0 . ( 16
)
Thanks to equation ( 12), the first equation implies that in order to know U 0 it is sufficient to know its trace on {X d = 0}. Unfortunately, the boundary condition does not determine this trace but a double trace on {X d = x d = 0}. Moreover, this equation can not be solved easily due to the degeneracy of UKL on E -(ζ). We will start by determining the double trace U 0 (t, x , 0, 0). Let

Π -U 0 (t, x , 0, 0) = α 0 (t, x )e, (17)
with e given in definition 2.5, α 0 a scalar function.

The determination of α 0 follows Marcou's method described in [START_REF] Marcou | Rigorous weakly nonlinear geometric optics for surface waves[END_REF], which is here adapted to a non homogenous boundary condition. Equations satisfied by the term U 1 are (see [START_REF] Ohkubo | On structures of certain L 2 -well-posed mixed problems for hyperbolic systems of first order[END_REF]) :

   (I -P ev )U 1 = Q ev (f ev -L(∂ x )U 0 ) , BU 1 |x d =X d =0 = g , U 1 |t≤0 = 0 . ( 18
)
Injecting, the first equation written for x d = X d = 0 in the boundary condition and after multiplying by the vector b (given in definition 2.5), we are led to :

b.BQ ev (f ev -L(∂ x )U 0 ) |x d =X d =0 = b.g .
However by definition of Q ev (see ( 13)), we can develop the left-hand side of the previous equation as follows :

b.BQ ev (f ev -L(∂ x )U 0 ) |x d =X d =0 = b.B +∞ 0 e -sA(ζ) Π + A -1 d (-f ev +L(∂ x )U 0 (s))ds.
We thus obtain

b.B +∞ 0 e -sA(ζ) Π + A -1 d L(∂ x )U 0 (s)ds = b. g + B +∞ 0 e -sA(ζ) Π + A -1 d f ev (s)ds := g0 .
Once more developing the left-hand side and using the equality [START_REF] Williams | Nonlinear geometric optics for hyperbolic boundary problems[END_REF] we have the final equation :

b.B +∞ 0 e -sA(ζ) Π + A -1 d L(∂ x )U 0 (s)ds = b.BI t ∂ t α 0 + d-1 j=1 b.BI j ∂ j α 0 ,
where we set :

I t = +∞ 0 e -sA(ζ) Π + A -1 d e sA(ζ) e ds, I j = +∞ 0 e -sA(ζ) Π + A -1 d A j e sA(ζ) e ds. ( 19
)
So it is clear that α 0 satisfies the transport equation

  b.BI t ∂ t + d-1 j=1 b.BI j ∂ j   α 0 = g0 .
It does not seem obvious that the coefficients of this equation are real. So we can not conclude that this equation is well-posed yet. To do that, we will need to make the I t , I j more explicit which is possible thanks to the following lemma.

Lemma 3.2 [Marcou] Set I t = I 0 , ∂ η0 = ∂ t then for all j ∈ {0, ..., d -1},

I j = Π + ∂ ηj Π -(ζ)e,
where we use the notation η 0 = τ .

proof : This proof is the same as in [ [START_REF] Marcou | Rigorous weakly nonlinear geometric optics for surface waves[END_REF], lemma 6.3] . We recall it here for the sake of completeness.

Let µ be an eigenvalue of A(ζ)Π -, we have Reµ < 0 and moreover e sµ is an eigenvalue of e sA(ζ) Π -; applying Dunford's formula we can write :

e sA(ζ) Π -= 1 2iπ Γ- e sz (A(ζ) -z) -1 dz,
where Γ -is a closed simple curve included in the complex half-plane with negative real part, surronding the eigenvalues of A(ζ)Π -.

As Π -e = e, we have

I j = 1 2iπ Γ- +∞ 0 e s(z-A(ζ)) Π + ds A -1 d A j (A(ζ) -z) -1 dz e.
On the other hand we can explicitely compute

+∞ 0 e s(z-A(ζ)) Π + ds = e s(z-A(ζ)) Π + (z -A(ζ)) -1 +∞ 0 = Π + (A(ζ) -z) -1 .
Setting A 0 = I we have,

I j = Π + 1 2iπ Γ- (A(ζ) -z) -1 A -1 d A j (A(ζ) -z) -1 dz e = Π + 1 2iπ Γ- ∂ ηj (A(ζ) -z) -1 dz e = Π + ∂ ηj Π -(ζ)e
Thanks to the stability assumption 2.4 one can write in a neighborhood of ζ the equality :

Be(ζ) = β(ζ) (γ + iθ(ζ)) e(ζ), (20)
with e, σ, β smooth such that

   e(ζ) = e , θ(ζ) = 0 ; θ(ζ) ∈ R , β (ζ)
B∂e(ζ) = iβ(ζ)∂θ(ζ)e.
Therefore we have proved the following proposition :

Proposition 3.1 The function α 0 defined in ( 17) is a solution of the real transport equation

iβ(ζ)b.e ∂ τ θ(ζ)∂ t α 0 + ∇ η θ(ζ).∇ x α 0 = g0 , α 0 |t≤0 = 0 , (21)
thus α 0 is uniquely determined and it is nonzero if and only if g0 is. Moreover α 0 has the same regularity as g0 (ie H +∞ (ω T )). By the equation ( 17), the same properties are also true for Π -U 0 (t, x , 0, 0).

We readily observe that g0 is nonzero if we choose a nonzero g and f ev ≡ 0. This explains the scaling of g ε in order to have an O(1) solution u ε in L ∞ (Ω T ).

The double trace Π -U 0 (t, x , 0, 0) is now fully determined. To conclude the construction of U 0 it is sufficient to note that in U 0 (t, x, X d ) = P ev U 0 (t, x, X d ), x d is a parameter. Consequently we can arbitrary lift the double trace Π -U 0 (t, x , 0, 0) in a simple one Π -U 0 (t, x, 0). For instance, set as in [START_REF] Lescarret | Wave transmission in dispersive media[END_REF] :

U 0 (t, x, X d ) = α 0 (t, x )χ(x d )e X d A(ζ) e, where χ ∈ C ∞ c (R) checking χ(0) = 1.
Remark We stress that from now on the geometric optics expansion is not uniquely determined. Indeed it highly depends on the function χ.

But following Métivier's work [START_REF] Métivier | Small viscosity and boundary layer methods[END_REF] we can show that this dependency is only an error term of upper order in ε. Indeed, by Taylor's expansion in x d :

U 0 (t, x, X d ) = U 0 (t, x , 0, X d ) + x d V (t, x, X d ),
where V is a function in P ev , as well as

W := X d V . For X d = x d ε : U 0 (t, x, X d ) = U 0 (t, x , 0, x d ε ) + εW (t, x, x d ε ).
Consequently the difference beetwen two functions U 0 lifted by two different functions χ is at most a term of order one in ε.

Second term in the WKB expansion.

The construction of the second term of the expansion is very similar to that of the first one. Indeed the only modifications come from the fact that U 1 is not polarized so that we have to consider the unpolarized part and that there is a little change in the source term of the transport equation ( 21).

Let us recall that the second term satisfies equation [START_REF] Ohkubo | On structures of certain L 2 -well-posed mixed problems for hyperbolic systems of first order[END_REF]. But thanks to the previous section from now U 0 is known, so the first equation of (15) allows to determine the unpolarized part of U 1 . Thus to complete the construct of U 1 it is sufficient to determine is polarized part that is to say P ev U 1 . But reiterating the arguments of the previous section, to do that it is sufficient to know the trace Π -U 1 (t, x, 0). Let us start by determining the double trace Π -U 1 (t, x , 0, 0). Plugging this double trace in the boundary condition of ( 15) we obtain :

BΠ -U 1 (t, x , 0, 0) = g -Q ev (f ev -L(∂ x )U 0 ) |x d =X d =0 . ( 22
)
As noticed in the introduction of this paragraph there is a little difference with the first term. Indeed, "a priori" the left side of the previous equation is not zero so we can not deduce that Π

-U 1 (t, x , 0, 0) is in kerB ∩ E -(ζ). So let us decompose, Π -U 1 (t, x , 0, 0) = α 1 (t, x )e + v 1 (t, x ), (23)
where α 1 is a scalar function and v 1 (t, x ) is in Ȇ-(ζ) (notation of definition 2.5). The boundary condition (22) can now be rewritten as follows :

v 1 (t, x ) = B -1 | Ȇ-(ζ) g -Q ev (f ev -L(∂ x )U 0 ) |x d =X d =0 ,
which determines in a unique way v 1 in H +∞ (ω T ), zero for negative times. At this moment it is sufficient to determine α 1 to know the double trace Π -U 1 (t, x , 0, 0). In order to do so, one can reiterate the construction described in paragraph 3.1.1 to exhibit a transport equation on α 1 . We end up with :

Proposition 3.2 The function α 1 defined in (23) is solution of the transport equation iβ(ζ)b.e ∂ τ θ(ζ)∂ t α 1 + ∇ η θ(ζ).∇ x α 1 = g1 , α 1 |t≤0 = 0 , with g1 := -b.BQ ev L(∂ x ) (I -Π -)U 1 |x d =X d =0 + v 1
Mimicking the end of paragraph 3.1.1 we have to extend Π -U 1 (t, x , 0, 0) to all positive x d . Thus we can set

U 1 (t, x, X d ) = χ(x d )e X d A(ζ) (α 1 (t, x )e + v(t, x ))+Q ev (f ev -L(∂ x )U 0 )(t, x d , X d ).

Higher order terms and summary.

It is clear that the process of the previous sections allows us to determine the (n + 1) th term if we already know the terms uppon the n th . Indeed let us summarize this method : If we know the n th term, equation ( 15) determines the unpolarized part of the (n + 1) th term. [START_REF] Marcou | Rigorous weakly nonlinear geometric optics for surface waves[END_REF] implies that to know the polarized part it is sufficient to know Π -U n+1 (t, x, 0). We begin by computing the double trace Π -U n+1 (t, x , 0, 0).

To do so, we decompose Π -U n+1 (t, x , 0, 0) in a component on kerB which depends on a scalar function α n+1 and a component on Ȇ-(ζ) which is determined algebraically by the boundary condition.

For α n+1 we write the equations for the (n + 2) th term and Marcou's work allows to show that α n+1 is solution of a real transport equation. So at this stage Π -U n+1 (t, x , 0, 0) is known.

To conclude it is sufficient to lift the double trace Π -U n+1 (t, x , 0, 0) in a "simple" trace Π -U n+1 (t, x, 0). Thus we have shown the following proposition : Proposition 3.3 Under assumptions 2.1-2.2-2.3-2.4, one can find a geometric optics expansion (U n ) n∈N of the ibvp (1) which satisfy equations [START_REF] Lescarret | Wave transmission in dispersive media[END_REF], where all U n belong to P ev .

Geometric optics expansion under assumption 2.5.

From now on the frequency ζ will satisfy assumption 2.5 raher than assumption 2.4. So we will have to include highly oscillating terms in the geometric optics expansion. The philosophy of the construction will be the same but we will have to be more careful about the resolution of boundary conditions. We choose source terms f ε and g ε under the form,

f ε (t, x, X d ) = e i ϕ ε (εf os (t, x, X d ) + f ev (t, x, X d )) , (24)
g ε (t, x ) = εe i ϕ ε g(t, x ), with f os = M m=1 e iω m X d f os m in P os , f ev in P ev and g in H +∞ (ω T ), both zero for the negative times. Let us consider the ansatz

u ε (t, x) = n≥0 ε n e i ϕ ε U n t, x, x d ε , (25)
where for all integer n, U n ∈ P is decomposed as

U n (t, x, X d ) := U os n + U ev n = M m=1 e iω m X d u n,m + U ev n .
Injecting the ansatz (25) in the ibvp (1) we obtain the cascade of equations :

   L(∂ X )U 0 = 0, BU 0 |x d =X d =0 = 0, U 0 |t≤0 = 0 , (26)
for the principal term. For higher order terms :

   L(∂ X )U n + L(∂ x )U n-1 = δ n,1 f ev + δ n,2 f os , BU n |x d =X d =0 = δ n,1 g , U n |t≤0 = 0 , ∀n ≥ 1. ( 27
)
The following lemma which is just a generalization of lemma 3.1, gives us a solution of the equation L(∂ X )U = F with F in P .

Lemma 3.3 [START_REF] Lescarret | Wave transmission in dispersive media[END_REF] There are projectors P, P i , a partial inverse Q such that for all profiles F ∈ P , the equation :

L(∂ X )U = F ( 28
)
admits a solution U in P if the following compatibility condition P i F = 0 is satisfied. In that case, this solution U can be written :

U = QF + PU.
In particular on P ev , P = P ev , Q = Q ev and P i is zero. On P os , P i = P i os = P os .

We refer to [START_REF] Lescarret | Wave transmission in dispersive media[END_REF] for a complete proof but since it will be useful we give below the expression of P os , and Q os : 26) and ( 27) can be rewritten as follows :

P os V = P i os V = M m=1 e iX d ω m Π m V m , (29)
Q os V = M m=1 e iX d ω m Q m V m , where V = M m=1 e iω m X d V m . Using lemma 3.3, (
       PU 0 = U 0 , P i L(∂ x )PU 0 = 0, BU 0 |x d =X d =0 = 0 , U 0 |t≤0 = 0 . (30) and        (I -P)U n = Q(δ n,2 f os + δ n,1 f ev -L(∂ x )U n-1 ) , P i L(∂ x )U n = P i δ n,1 f os + P i δ n,1 f ev , BU n |x d =X d =0 = δ n,1 g , U n |t≤0 = 0 . ∀n ≥ 1. ( 31
)
The first equations in (30)-( 31) are due to the particular form of the solution of (28), and the second come from the compatibility condition P i F = 0. We see that the first term U 0 has to satisfy the conditions (30) of polarization and propagation together with some boundary conditions. Decomposing on the profiles spaces P ev and P os it follows:

                 P ev U ev 0 = U ev 0 , P os U os 0 = U os 0 , P i os L(∂ x )P os U os 0 = 0, P i ev L(∂ x )P ev U ev 0 = 0, B U os 0 + U ev 0 |X d =0 |x d =0 = 0 , U os 0 |t≤0 = U ev 0 |t≤0 = 0 . ( 32
)
However, thanks to the relation P i ev = 0 given by lemma 3.3 the fourth equation of ( 32) is a tautology. Since this lemma also gives the relation P i os = P os one can reformulate the third equation of (32) :

P os L(∂ x )P os U os 0 = 0.
So we see that the evolution equations on the oscillating terms and on the evanescent terms are just linked by the boundary condition. Consequently one can hope to solve the evolution equations independently for the oscillating and the evanescent terms. Let us start by determining the principal oscillating term.

Leading order oscillating term.

Thanks to the decomposition given in lemma 2.1 and (29) we will identify each term in the expansion of U os 0 :

         Π m u 0,m = u 0,m , Π m L(∂ x )u 0,m = 0 , B M m=1 u 0,m + U ev 0 |X d =0 |x d =0 = 0 , u 0,m |t≤0 = 0 , ∀m ∈ {1, M } . ( 33
)
Let us recall the classical lemma due to Lax :

Lemma 3.4 [START_REF] Lax | Asymptotic solutions of oscillatory initial value problems[END_REF] Under assumption 2.1, the following equality is true

Π m L(∂ x )Π m u 0,m = (∂ t + v m .∇ x )u 0,m ,
where v m = ∇λ km (η, ω m ) is the group velocity associated to λ km defined in (4) .

Distinguish two cases according to the set in which m lies.

First case : m ∈ N C

The group velocity v m is outgoing thus the transport equation associated to (33) is :

(∂ t + v m .∇ x )u 0,m = 0 , u 0,m |t≤0 = 0 ,
and can be explicitely solved by integration along the characteristics. Consequently u 0,m as solution of a homogeneous transport equation is zero and the same remains true for its trace on {x d = 0}.

Second case : m ∈ C Conversely when m is causal the phenomenon is incoming so the transport equation has to take the boundary condition into account; thus it reads :

     (∂ t + v m .∇ x )u 0,m = 0 , B m∈C Π m u 0,m + (P ev U ev 0 ) |X d =0 |x d =0 = -B m∈N C Π m u 0,m = 0 , u 0,m |t≤0 = 0 .
Focusing on the left-hand side of the boundary condition, we obtain :

m∈C Π m u 0,m |x d =0 + Π e -U ev 0 (t, x , 0, 0) ∈ vect{e.}
So in order to satisfy the boundary condition, we need

Π m u 0,m |x d =0 = 0 , ∀m ∈ C Π e -U ev 0 (t, x , 0, 0) = α 0 (t, x )e , (34)
where α 0 is a scalar function.

Consequently the transport equation for a causal term reads :

   (∂ t + v m .∇ x )u 0,m = 0 , u 0,m |x d =0 = 0 , u 0,m |t≤0 = 0 ;
one more time, one can solve it explicitly and show that u 0,m and the trace on {x d = 0} are zero.

Principal term for the evanescent part.

For the principal evanescent term we have to solve the same cascade of equations as in the previous paragraph (see [START_REF] Sablé-Tougeron | Existence pour un problème de l'élastodynamique Neumann non linéaire en dimension 2[END_REF]), that is to say :

   P ev U ev 0 = U ev 0 , BΠ e -U ev 0 (t, x , 0, 0) = 0 , U ev 0 |t≤0 = 0 . (35)
So we will try to apply the method described in paragraph 3.1.1 up to a few modifications.

As before, equations for the higher order evanescent term U ev 1 read :

     (I -P ev )U ev 1 = Q ev (f ev -L(∂ x )U ev 0 ), B U os 1 + U ev 1 |X d =0 |x d =0 = g, U ev 1 |t≤0 = 0. (36)
The only change is that now oscillating terms of order one appear in the boundary condition. Since we do not know those terms, Marcou's method can not be applied at this stage. However, in paragraph 3.2.1 we managed to determine oscillating terms without any consideration on the evanescent ones, so we will try to do that once more.

Second oscillating terms in the WKB expansion.

Using the cascade of equations (31) for the oscillating part, we obtain

       (I -P os )U os 1 = -Q os (L(∂ x )U os 0 ), P os L(∂ x )U os 1 = f os , BU 1 |x d =X d =0 = g, U os 1 |t≤0 = 0 . ( 37
)
Since from now we know that U os 0 is zero, the first equation allows to determine the unpolarized part of U os 1 and it is zero. So it is sufficient to determine the polarized part of U os 1 , to do that combining the first and the second equations of (37) it follows :

P os L(∂ x )P os U os 1 = f os .
Reiterating the arguments of paragraph 3.2.1, we exhibit the following transport equation on Π m u 1,m :

(∂ t + v m .∇ x )Π m u 1,m = f os m , Π m u 1,m |t≤0 = 0 , , m ∈ N C. (38)
And for m ∈ C :

     (∂ t + v m .∇ x )Π m u 1,m = f os m , B m∈C Π m u 1,m |x d =0 + U ev 1 (t, x , 0, 0) = g -B m∈N C Π m u 1,m |x d =0 Π m u 1,m |t≤0 = 0 , (39)
(38) is a transport equation which determines in a unique way Π m u 1,m = u 1,m and its trace on {x d = 0} for all m in N C. We stress that u 1,m is not zero unless the source terms meet specific requirements. We have to be more careful with causal terms. Indeed, as said at the end of paragraph 3.2.2, evanescent and oscillating terms both appear in the boundary condition. Let us decompose the evanescent term in its polarized and unpolarized parts. The boundary condition in (39) now reads :

B m∈C Π m u 1,m |x d =0 + Π e -U ev 1 (t, x , 0, 0) = g -B m∈N C Π m u 1,m |x d =0 -BQ ev (f ev -L(∂ x )U ev 0 ), (40)
where we used :

(I -P ev )U ev 1 = Q ev (f ev -L(∂ x )U ev 0 )
. The main point is that one more time the vector

m∈C Π m u 1,m |x d =0 + Π e -U ev 1 (t, x , 0, 0),
which appears on the left-hand side of ( 40) is an element of E -(ζ). Thus after multiplying by the vector b defined in definition 2.5 we obtain the following compatibility condition :

b.BQ ev (f ev -L(∂ x )U ev 0 ) = b. g -B m∈N C Π m u 1,m |x d =0 . (41)
Moreover thanks to the definition 2.5 one can find w 1 in Ȇ-(ζ) such that the following decomposition is satisfied :

m∈C Π m u 1,m |x d =0 + Π e -U ev 1 (t, x , 0, 0) = α1 (t, x )e + w 1 (t, x ). ( 42
)
Applying B, (39) implies that w 1 is given by the formula :

w 1 (t, x ) = B -1 | Ȇ-(ζ) g -B m∈N C Π m u 1,m |x d =0 -BQ ev (f ev -L(∂ x )U ev 0 ) , (43)
showing that w 1 is known if and only if U ev 0 is determined.

3.2.4

End of the construction of the evanescent term.

The compatibility condition (41) will allow to apply Marcou's method (see 3.1.1) indeed it reads :

b.BQ ev (f ev -L(∂ x )U ev 0 ) |x d =X d =0 = b. g -B m∈N C u 1,m |x d =0 := g0 (t, x ),
where all source terms on the right-hand side are known. So one can show the following proposition Proposition 3.4 The function α 0 defined in (34) is solution of the transport equation

iβ(ζ)b.e ∂ τ θ(ζ)∂ t α 0 + ∇ η θ(ζ).∇ x α 0 = g0 , α 0 |t≤0 = 0 , (44)
Thus α 0 is determined in a unique way. It is zero if and only if g0 is zero, and it has the same regularity as g0 . Thanks to (34) it is the same for Π e -U ev 0 (t, x , 0, 0). Reiterating the end of the analysis of paragraph 3.1.1, it is sufficient to lift Π e -U ev 0 (t, x , 0, 0) to obtain a solution of (35). A typical example is

U ev 0 (t, x, X d ) = α 0 (t, x )χ(x d )e X d A(ζ) e, (45)
with χ ∈ C ∞ c (R) such that χ(0) = 1.

End of the construction for causal terms.

After the choice of χ, the principal evanescent term is determined. So, it is the same for w 1 defined in (42) and it is explictly given by the formula (43). However, w 1 is in Ȇ-(ζ) so it can be decomposed as follows :

w 1 = v h 1 + v e 1 , with v h 1 an element of E h -(ζ), v e 1 in a supplementary space of E h -(ζ) in Ȇ-(ζ)
. We emphasize that v e 1 and v h 1 are known since w 1 is known from equation (43). By identification on those spaces we obtain the two equations :

m∈C u 1,m |x d =0 = v h 1 . Π e -U ev 1 (t, x , 0, 0) = α 1 (t, x )e + v e 1 (t, x )
.

terms are determined. The boundary condition for the causal oscillating and the evanescent terms reads as follows :

B m∈C Π m u n+1,m |x d =0 + Π e -U ev n+1 = -B m∈N C u n+1,m |x d =0 + ((I -P ev )U ev n+1 ) |x d =X d =0 - m∈C (I -Π m )u n+1,m |x d =0 .
Decomposing as in (42) allows to determine u n+1,m |x d =0 thanks to Lax's lemma.

To determine the transport coefficient on the boundary of U ev n+1 we write the boundary condition for U ev n+2 , after the determination of the noncausal oscillating terms of order n + 2 it reads : 4 Proof of the main theorems.

B m∈C Π m u n+2,m |x d =0 + Π e -U ev n+2 = G + B(Q ev L(∂ x )U ev n+1 |x d =X d =0

Justification of the expansion.

The aim of this section is to show that the two geometric optics expansions built in previous sections converge to the exact solution of (1). Since proofs are very similar in both cases we will just give it in the case of the geometric optics expansion under assumption 2.5. We give the following definition : Definition 4.1 For all integer N and for all (u ε ) ε of H +∞ (Ω T ), we say that

(u ε ) ε is O(ε N ) in H +∞ ε (Ω T ) if for all α ∈ N d+1 there is a positive constant C α such that ∀ε ∈ ]0, 1] , ε |α| ∂ α u ε L 2 (Ω T ) ≤ C α ε N .
We will also need the following proposition in view of justifying the geometric optics expansion :

Proposition 4.1 Let U be a fonction in P ev then U (t, x, x d ε ) and

(L(∂ x )U (t, x, X d )) |X d = x d ε are O(ε 1 2 ) in L 2 (Ω T ). proof : Since U is in P ev , there is δ such that e δX d U (t, x, X d ) is in H ∞ (Ω T ×R + ) so (x d , X d ) → U (•, •, x d , X d )e δX d ∈ L ∞ x d ,X d (L 2 (ω T )).
We have,

U |X d = x d ε 2 L 2 (Ω T ) = +∞ 0 ]-∞,T ]×R d-1 |e δ x d ε U (t, x, x d ε )e -δ x d ε | 2 dtdx dx d = +∞ 0 e δ x d ε U (•, •, x d , x d ε ) 2 L 2 (ω T ) e -2δ x d ε dx d ≤ U e δ• L ∞ x d ,X d (L 2 (ω T )) +∞ 0 e -2δ x d ε dx d , ≤ Cε.
Let ∂ y be a differential operator, y in {t, x 1 , ..., x d } we have :

(∂ y U (•, •, X d )) |X d = x d ε 2 L 2 (Ω T ) = +∞ 0 e -2δ x d ε e δ x d ε ∂ y U (•, •, x d , x d ε ) 2 L 2 (ω T ) dx d ,
as the assessment X d equals x d ε is made after the derivative. Since e δX d ∂ y U (t, x, X d ) is in H ∞ (Ω T ×R + ) we can work as for the first assertion to show that

(∂ y U (•, •, X d )) |X d = x d ε 2
L 2 (Ω T ) ≤ Cε, and we conclude by the triangle inequality.

We have to prove the following result to complete the proof of theorem 2.3.

Theorem 4.1 Under assumptions 2.1-2.2-2.3-2.5, u N0 app -u ε is an O(ε N0+1 ) in H +∞ ε (Ω T )
, where we recall that u N0 app is defined in [START_REF] Coulombel | Geometric optics expansions with amplification for hyperbolic boundary value problems: linear problems[END_REF] .

proof : The remainder u N0+2 app -u ε satisfies the system :

   L(∂) u N0+2 app -u ε = ε N0+2 e i ϕ ε L(∂ x )U os N0+2 + L(∂ x )U ev N0+2 B u N0+2 app -u ε = 0 u(t < 0) = 0
Thanks to proposition 4.1 and by interpolation between spaces L 2 (ω T ) and H 1 (ω T ) one can show that for all index N , L(∂

x )U ev N is O(1) in L 2 x d (H 1 2 (ω T )). The same kind of argument also shows that L(∂ x )U os N is O(ε -1 2 ) in L 2 x d (H 1
2 (ω T )). Thus in terms of powers of ε the limiting term is the oscillating one. Using the energy estimate (6), we obtain

u N0+2 app -u ε L 2 (Ω T ) ≤ Cε N0+ 3 2 ,
For all α, tangential derivatives ∂ α t,x are estimated directly by differentiation of the ibvp (1), so we have :

∂ α t,x u N0+2 app -u ε L 2 (Ω T ) ≤ ε N0+2 ∂ α t,x e i ϕ ε L(∂ x )U N0+2 L 2 (Ω T ) .
Thanks to Leibniz's formula it comes :

∂ α t,x e i ϕ ε L(∂ x )U N0+2 L 2 (Ω T ) ≤ 1 ε |α| L(∂ x )U N0+2 L 2 (Ω T ) + ∂ α t,x (L(∂ x )U N0+2 ) L 2 (Ω T ) ≤ C ε |α| . Consequently, ε |α| ∂ α t,x u N0+2 app -u ε L 2 (Ω T ) ≤ Cε N0+ 3 2 .
Using the evolution equation, we can write the x d -derivative of u N0+2 app -u ε as a linear combination the tangential derivatives. Thus the previous estimate still holds if we add derivatives in x d . Consequently we have shown that u

N0+2 app -u ε is O(ε N0+ 3 2 ) in H +∞ ε (Ω T ). It is now easy to show that u N0+2 app -u N0 app is O(ε N0+1 ) in H +∞ ε
(Ω T ) and we conclude by the triangle inequality.

Optimality of energy estimate, proof of theorem 2.5

Once again since proofs are very similar in both cases we just describe here the proof of theorem 2.5. We will need the following lemma : Lemma 4.1 Let U 0 be as in (45), then U 0 (t, x,

x d ε ) is O(ε 1
2 ) in L 2 (Ω T ). This property is sharp in the sense that if there is κ > 0 such that U 0 is an O(ε

1 2 +κ ) in L 2 (Ω T ), then U 0 is zero in L 2 (Ω T ).
proof : The first assertion has already been proved in proposition 4.1. To show the second assertion we will use the fact that for all w

∈ C k , M ∈ M k (C) we have |e M w| 2 ≥ e -2 M |w| 2 . So, if v ε = U 0 t, x, x d ε v ε L 2 (Ω T ) ≥ α 0 2 L 2 +∞ 0 χ(x d ) 2 e -2x d ε A(ζ) |e| 2 dx d 1 2 , by change of variable u = x d ε it comes v ε L 2 (Ω T ) ≥ ε 1 2 C α0,e +∞ 0 χ(εu) 2 e -2u A(ζ) du 1 2 , ≥ C α0,e,χ,A(ζ) ε 1 2 .
proof : (theorem 2.5) We argue by contradiction and assume s 1 = 1 2 -δ, δ ∈ 0, 1 2 . Let g ε ≡ 0, and consider source terms

f ε = εe i ϕ ε f os t, x, x d ε , f os t, x, x d ε = ψ(t, x)e iω m 0 x d ε e h + , (46)
where e h + ∈ kerL(iζ, iω m0 ) \ {0}, which is possible from assumption 2.6. So by interpolation between spaces L 2 (ω T ) and

H 1 2 (Ω T ), f ε L 2 x d (H s 1 (ω T )) ≤ Cε 1 2 +δ , the energy estimate (8) allows to show that u ε is O(ε 1 2 +δ ) in L 2 (Ω T ).
Using the geometric optics expansion given by theorem 2.3 and thanks to the triangle inequality, one shows that U 0 = U ev 0 the first term in the expansion is O(ε

1 2 +δ ) in L 2 (Ω T ). But, U 0 = U ev 0 t, x, x d ε = α 0 (t, x )χ(x d )e x d ε A(ζ) e,
implies that for α 0 nonzero, U ev 0 is O(ε

2 ) in L 2 (Ω T ). This result is sharp thanks to lemma 4.1. Therefore α 0 is zero. But, α 0 is solution of the transport equation (44) with :

g0 = -b.B m∈N C u 1,m |x d =0 .
The argument to conclude is the same as in [START_REF] Coulombel | Geometric optics expansions with amplification for hyperbolic boundary value problems: linear problems[END_REF]. For m ∈ N C, u 1,m is solution of the transport equation :

(∂ t + v m .∇ x ) u 1,m = δ m,m0 ψ(t, x)e h + , u 1
,m |t≤0 = 0 , so the only nonzero amplitude is u 1,m0 and it is computable by integration along the characteristics. Let us choose ψ in (46) such as the source term -b.Bu 1,m0 |x d =0 is nonzero. It implies that α 0 is nonzero and we have thus proved s 1 ≥ 1 2 . Once more we argue by contradiction so we suppose that s 2 = 1 2 -δ, δ ∈ 0, 1 2 . Let f ε ≡ 0 and choose now g ε (t, x ) := εe i ϕ ε ψ(t, x )b, with b as in definition 2.5 and ψ ∈ C ∞ c . Reiterating the same kind of interpolation arguments described in the first part of the proof and using the estimate for a mixed frequency, we see that a loss of half a derivative appears in the interior of the domain :

u 2 L 2 (Ω T ) + u |x d =0 2 H -1 2 (ω T ) ≤ C T f 2 L 2 x d (H 1 2 (ω T )) + g 2 H 1 2 (ω T )
.

At last, [START_REF] Coulombel | Geometric optics expansions with amplification for hyperbolic boundary value problems: linear problems[END_REF] show that the worst case occurs for hyperbolic frequencies for which we lose one derivative in the interior and one derivative on the boundary :

u 2 L 2 (Ω T ) + u |x d =0 2 L 2 (ω T ) ≤ C T f 2 L 2 x d (H 1 (ω T )) + g 2 H 1 (ω T ) . ( 47
)
5.2 Boundary conditions for linearized Euler equations.

Theorems 2.4-2.5 allow to describe areas where one can impose a maximal dissipative boundary condition for linearized Euler equations. Indeed this system is both symmetric and constantly hyperbolic so we can study the question of dissipative boundary conditions. Let us recall that the boundary conditions are maximal dissipative if the following is true :

∀v ∈ kerB, A d v, v ≤ 0,
It is known (see [START_REF] Benzoni-Gavage | Multidimensional hyperbolic partial differential equations[END_REF] chapter 3) that a symetrizable ibvp with a maximal dissipative boundary condition admits an energy estimate without loss of derivative in the interior. Moreover for linearized Euler equations the elliptic area is empty so we have :

Ξ 0 = EH ∪ H ∪ G.
Consequently thanks to theorem 2.5 and the optimality of the energy estimate (47) proved in [START_REF] Coulombel | Geometric optics expansions with amplification for hyperbolic boundary value problems: linear problems[END_REF], the only possibility to have a maximal dissipative boundary conditions that does not satisfy UKL is that the Lopatinskii determinant vanishes at a glancing point. We deal with the linearized isentropic Euler equations in dimension 2 for an outgoing subsonic fluid. This correspond to the problem :

   ∂ t U + A 1 ∂ 1 U + A 2 ∂ 2 U = f , BU |x2=0 = 0 , U |t=0 = 0, (48)
with :

A 1 =   0 -1 0 -1 0 0 0 0 0   , A 2 =   M 0 -1 0 M 0 -1 0 M   ,
where M ∈ ]-1, 0[ denotes the Mach number. We suppose that B reads : such that UKL fails in the glancing region. G is easily computable (similar computations can be found in [ [START_REF] Benzoni-Gavage | Generic types and transitions in hyperbolic initial-boundary-value problems[END_REF],chapter 14]) and is given by :

B =
G = (b 2 , b 3 ) ∈ R 2 \ b 2 = ± 1 + M b 3 √ 1 -M 2 = Γ g + ∪ Γ g -.
We introduce ∆(σ, η) a Lopatinskii determinant associated to the ibvp (48).

The other possibilities of behaviour for a boundary condition (see [START_REF] Benzoni-Gavage | Generic types and transitions in hyperbolic initial-boundary-value problems[END_REF]) are a bit more difficult to compute, we enumerate all the possible cases below Remark The transitions described above are those predicted in [START_REF] Benzoni-Gavage | Generic types and transitions in hyperbolic initial-boundary-value problems[END_REF]. The set Γ w contain frequencies which satisfy the assumption 2.5 and assumption 2.6.

It has already been mentioned in the begining of this paragraph that

D ⊂ G ∪ SS.
But for (b 2 , b 3 ) ∈ G the matrix associated to the quadratic form A 2 X, X is

M =   M 1 + (1+M b3) 2 1-M 2 ± (1+M b3) 2 √ 1-M 2 ± (1+M b3) 2 √ 1-M 2 M (1 + b 2 3 ) + 2b 3   ,
and it is easy to see that det(M) < 0 which means that A 2 X, X is nonpositive.

In other words D ∩ G is empty. For (b 2 , b 3 ) ∈ SS the matrix associated to the quadratic form A 2 X, X is

Definition 2 . 3

 23 Set Π ± e := Π ± e (ζ) the projector on E e ± (ζ) associated to the decomposition (3). Set Π m := Π m (ζ) the projector on kerL(ζ, ω m ) and Q m := Q m (ζ) the partial inverse of L(ζ, ω m ) satisfying :

= 0 .

 0 Using lemma 3.2, we also have in a neighborhood of ζ, b.BI = b.BΠ + (ζ)∂Π -(ζ)e. Differentiating the identity Π -(ζ)e(ζ) = e(ζ) in a neighborhood of ζ then composing on the left by Π + (ζ) in order to avoid the second term we have : Π + (ζ)∂Π -(ζ)e = ∂e(ζ). Now differentiating (20) in ζ we obtain :

  with G a function depending on the traces of u n+2,m , (I -Π m )u n+2,m and possibly of the source terms on the boundary. Composing by b gives a compatibility condition on U ev n+1 |x d =X d =0 and Marcou's method allows to write it as a transport equation which permits to determine α n+1 . Thus we have shown the following proposition : Proposition 3.5 Under assumptions 2.1-2.2-2.3-2.5, one can find a geometric optics expansion (U n ) n∈N of the ibvp (1) which satisfy equations (26)-(27).

1 b 2 b 3 ,

 3 with b 2 , b 3 ∈ R. We denote by D the set of (b 2 , b 3 ) ∈ R 2 such that B defines a maximal dissipative boundary condition and by G the set of (b 2 , b 3 ) ∈ R 2

Γ 0 := (b 2 , b 3 )∪ (b 2 , b 3 )

 2323 ∈ R 2 \ ∆(1, 0) = 0 = (b 2 , b 3 ) ∈ R 2 \ b 3 = 1 Γ w := (b 2 , b 3 ) ∈ R 2 \ ∃ζ ∈ EH, ∆(ζ) = 0 = (b 2 , b 3 ) ∈ R 2 \ b 3 = -M, Γ g -< b 2 < Γ g + Γ s := (b 2 , b 3 ) ∈ R 2 \ ∃ζ ∈ H, ∆(ζ) = ∂ σ ∆(ζ) = 0 = (b 2 , b 3 ) ∈ R 2 \ b 3 > -M, b 2 2 + b 2 3 = 1 SU := (b 2 , b 3 ) ∈ R 2 \ ∃ζ ∈ Ξ \ Ξ 0 , ζ = (1, 0), ∆(ζ) = 0 = (b 2 , b 3 ) ∈ R 2 \ b 3 > -M, b 2 2 + b 2 3 < 1 SS := (b 2 , b 3 ) ∈ R 2 \ ∀ζ ∈ Ξ, ∆(ζ) = 0 = (b 2 , b 3 ) ∈ R 2 \ b 3 < -M, Γ g -< b 2 < Γ g + ∈ R 2 \ b 3 > -1 M , Γ g + < b 2 < Γ g -All other parameters (b 2 , b 3 )give rise to problems in the so-called W R class, that is for which the Lopatinskii determinant vanishes exactly at first order in the hyperbolic region H.

  The set of causal (resp. noncausal) indeces C (resp. N C) is defined by : m is in C (resp. N C) if and only if for λ km defined in (4) we have∂ η d λ km (η, ω m ) > 0 (resp. ∂ η d λ km (η, ω m ) < 0). v m := ∇λ km (η, ω m) is called the group velocity associated to the phase τ +η.x + ω m x d .

	E e -(ζ) (resp. E e + (ζ)) is the generalized space associated to the eigenvalues
	of A(ζ) of negative (resp. positive) real part, and E h ± (ζ) are sums of eigenspaces
	of A(ζ) associated to some purely imaginary eigenvalues of A(ζ).
	In fact, it is possible to give a precise decomposition of E h ± (ζ). Let iω m be a
	purely imaginary eigenvalue of A(ζ), then
	det(τ I + A(η, ω m )) = 0.
	Using the constant hyperbolicity of L(∂) (see assumption 2.1), there is an index
	k m such that
	τ + λ km (η, ω m ) = 0,	(4)
	with λ km (η, ω m ) smooth.
	Definition 2.2 With such notations, we have
	Lemma 2.1 For all ζ ∈ (EH ∪ H)

of size p and at last a regular real valued function θ such that we can write :

	and there exist such
	frequencies.
	• Moreover let ζ ∈ E such that kerB ∩E -(ζ) = {0}, then there is a neighborhood
	V of ζ in Ξ, a regular basis (E 1 , ..., E p )(ζ) of E -(ζ), a regular and invertible
	matrix P (ζ)

b 2 (1 + M b

) b 2 (1 + M b 3 ) M (1 + b 2 3 ) + 2b 3.

Consequently the transport equation (39) can be rewritten as follows :

and it is now possible to determine u 1,m for all causal m, by integration along the characteristics.

Second evanescent term.

The method to build the second evanescent term is similar to the method for the first one. Taking equation (36), since U ev 0 is now known it is sufficient to determine the polarized part that is to say Π e -U ev 1 (t, x, 0). Decomposing as follows :

where v e 1 (t, x ) ∈ Ȇ-(ζ) is given by the boundary condition. Consequently it is sufficient to know α 1 . To do that we will need to deal with equations on the third evanescent term and since it contains oscillating terms we will need to know them as well. As before, noncausal terms are easy to determine since they satisfy outgoing transport equations. For causal terms we have to decompose the boundary condition for n = 2 as in sections 3.2.3 and 3.2.4, one can show that α 1 is the solution of the transport equation :

So it is now sufficient to extend the double trace Π -U ev 1 (t, x , 0, 0) for the positive x d . For instance, we can set :

Higher order terms.

Let us summarize the construction of the higher order terms. The first equation of (31) gives the unpolarized parts of U os n+1 and U ev n+1 . So one more time it is sufficient to determine the polarized part.

For noncausal oscillating terms , the second equation of (31) and Lax's lemma lead to an outgoing transport equation :

where F m is a function depending on (I -Π m )u n+1,m and possibly on the oscillating source terms. Integrating along the characteristics, noncausal oscillating (8) one can show that u ε is O(ε

, consequently once again α 0 is zero. But, α 0 is the solution of the transport equation ( 44) with g0 = ψ|b| 2 , and we can choose ψ such that is not g0 zero.So, α 0 as a solution of the nontrivial transport equation ( 44) is not zero which is the desired contradiction. So we proove that necessarily s 2 ≥ 1 2 .

5 Consequences

Classification and well-posed homogeneous ibvp.

The first consequence of theorems 2.4-2.5 is to show rigorously the intuition in the beginning of [[3], chapter 7] and [START_REF] Ohkubo | On structures of certain L 2 -well-posed mixed problems for hyperbolic systems of first order[END_REF]. It says that the only well-posed homogeneous ibvp, that is ibvp of the form (1) with g ε ≡ 0 and that satisfy the estimate :

meet on of the following conditions : i) UKL is satisfied.

ii) UKL fails in the elliptic region E.

iii) UKL fails in the glancing region G under some dimension restriction. Indeed, if UKL fails in the mixed or in the hyperbolic area then necessarily there is a loss of at least half a derivative in the domain Ω T , so the homogeneous ibvp is not well-posed. If UKL fails in the elliptic region, thanks to theorem 2.4 the energy estimate is :

, and the only way to have strong well-posedness is that there is no loss of derivative on the boundary that is to say g = 0. Moreover theorems 2.4-2.5 and the optimality of energy estimate when UKL fails in the hyperbolic region given by [START_REF] Coulombel | Geometric optics expansions with amplification for hyperbolic boundary value problems: linear problems[END_REF] allow the following classification of weakly well-posed ibvp when UKL fails outside the glancing area.

When UKL fails for an elliptic frequency then the energy estimate is :

X, X is nonpositive for all X if and only if det(M ) ≥ 0 that is to say : The boundary of D is a hyperbola, included in SS, for which the corresponding boundary conditions are maximal dissipative though not stricly dissipative. As predicted by theorem 2.5, Γ w does not meet D. We summarize the above discussion in the following scheme.


Germain M Souche

Julien Huillery

H Pothier

Philippe Gandit

Jérôme I Mars

Sergey E Skipetrov

Olivier Bourgeois
email: olivier.bourgeois@grenoble.cnrs.fr

Searching for thermal signatures of persistent currents in normal metal rings

Keywords: numbers: 73.23.-b, 73.23.Ra, 68.65.-k, 64.70.Nd

We introduce a calorimetric approach to probe persistent currents in normal metal rings. The heat capacity of a large ensemble of silver rings is measured by nanocalorimetry under a varying magnetic field at different temperatures (60 mK, 100 mK and 150 mK). Periodic oscillations versus magnetic field are detected in the phase signal of the temperature oscillations, though not in the amplitude (both of them directly linked to the heat capacity). The period of these oscillations (Φ0/2, with Φ0 = h/e the magnetic flux quantum) and their evolution with temperature are in agreement with theoretical predictions. In contrast, the amplitude of the corresponding heat capacity oscillations (several kB) is two orders of magnitude larger than predicted by theory.

INTRODUCTION

At very low temperatures T , a small isolated metal ring carries a weak but non-dissipative current that is not destroyed even by the presence of disorder in the ring [1]. It follows from very general arguments that this persistent current I is periodic in the magnetic flux Φ threading the ring [START_REF] Imry | Introduction to Mesoscopic Physics[END_REF][START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF]:

I(Φ, T ) = ∞ m=1 I m (T ) sin 2πm Φ Φ 0 , (1)
where Φ 0 = h/e is the magnetic flux quantum. The typical magnitude of the persistent current in a ring of circumference L is I ∼ eE Th /h, where E Th = hD/L 2 is the Thouless energy and D is the diffusion coefficient determined by the scattering of electrons from the disorder in the ring. For L ∼ 1 µm, we find I ∼ 1 nA. The precise value of I and the direction in which this current is flowing are determined by the configuration of impurities and are thus random. Due to the weak magnitude of the persistent current, only the first two harmonics m = 1, 2 in Eq. (1) were observed experimentally [START_REF] Lévy | [END_REF][5][6][7][8][9][10]. They were also in the focus of numerous theoretical studies (see Refs. [11][12][13][14][15][16][17][18] for a representative selection of theoretical works). The ensemble of the existing literature indicates that the statistical properties of the first harmonics I 1 can be understood within the model of noninteracting electrons, whereas the second harmonics I 2 is dominated by interaction effects. In the experiments performed on single rings [5,9] or on small ensembles of identical rings [7,10], both the first and the second harmonics of I have been detected. Hence, the measured period of current oscillation with Φ is Φ 0 . The mean values of I 1 and I 2 , their variances, and possibly the full probability distributions can be measured [18]. On the contrary, if large ensembles of rings (up to 10 7 rings in Ref. [START_REF] Lévy | [END_REF]) are used [START_REF] Lévy | [END_REF]8], averaging over all the rings of the ensemble inevitably takes place and one only has access to the average current I . Because I 1 = 0 [START_REF]More precisely, I1 ∝ exp(-L/2 ), with the elastic mean free path due to disorder [3, 11[END_REF], the measured signal is mainly due to the second harmonics I 2 = 0. The observed current then oscillates with a period Φ 0 /2.

The properties of the first harmonics I 1 of the persistent current are nowadays relatively well understood. If early observations were somewhat contradictory [5,7], more recent works have demonstrated an impressive agreement between experiment and the noninteracting electron theory [9,10,18]. The second harmonics I 2 , on the contrary, still represents a challenge for both theory and experiment. Even though a consensus exists on the importance of electron-electron interactions to explain its properties, the calculation assuming repulsive interactions [12,14] yields the magnitude of I 2 which is significantly smaller than the one measured in the experiments. In addition, the paramagnetic response of the rings at small magnetic field predicted by the theory, disagrees with observations [START_REF] Lévy | [END_REF]8]. Diamagnetic response can be obtained in a theory assuming attractive interactions [13]. However, interactions which are sufficiently strong to reproduce the experimentally observed values of I 2 would induce a transition to the superconducting state at temperatures which are too high to be compatible with known properties of (some of the) metals used in the persistent current experiments: copper, gold, and silver. Indeed these metals do not exhibit super-conductivity even at the lowest temperatures accessible experimentally (down to 0.1 mK [16]). A possible solution to this problem has been recently proposed by Bary-Soroker et al. [17]: a tiny amount of magnetic impurities can destroy the superconductivity but has little impact on the persistent current. Two new parameters -the spin-scattering rate 1/τ s and the bare superconducting transition temperature T 0 c of the material without magnetic impurities -appear in the theory and allow for a reasonable explanation of the experiments reported in Refs. [START_REF] Lévy | [END_REF] and [7].

Motivated by the recent progress in the research on persistent currents in normal metal rings, we propose here a new way of detecting these currents. Our approach is radically different from the one employed in all previous experiments: if these anterior works relied on the measurement of the rings' magnetic moment (either using a more or less sophisticated version of a SQUID magnetometer [START_REF] Lévy | [END_REF]5,7,9], coupling the rings to a superconducting microresonator [8], or using an elegant micromechanical detector [10]), we propose to focus on the rings' heat capacity. Our idea stems from the basic principles of thermodynamics. On the one hand, the heat capacity at constant pressure is given by C p = -T (∂ 2 F/∂T 2 ) p , where F is the thermodynamic free energy. On the other hand, the persistent current is I = -∂F/∂Φ. Therefore, we find

∂C ∂Φ = T ∂ 2 I ∂T 2 . (2)
From here on we omit the subscript 'p' of the heat capacity to lighten the notation. Equations ( 1) and [START_REF] Imry | Introduction to Mesoscopic Physics[END_REF] imply that

C(Φ, T ) = C(0, T ) + ∞ m=1 C m (T ) 1 -cos 2πm Φ Φ 0 , (3)
where C m (T ) = (T Φ 0 /2πm)∂ 2 I m (T )/∂T 2 . The heat capacity C is therefore also a periodic function of the magnetic flux Φ. Equation [START_REF] Imry | Introduction to Mesoscopic Physics[END_REF] shows that the dependence of C on Φ is intimately related to the dependence of I on T . This link between C and I was already exploited by Yang and Zhou to investigate the impact of spin-orbit coupling on both the persistent current and the heat capacity in the model of noninteracting electrons [START_REF] Yang | [END_REF], as well as by (some of the) present authors to study the entrance of magnetic vortices into superconducting loops [21][22][23]. The experiments reported here are, however, much more involved than those performed in the superconducting state: not only we work at much lower temperatures (T ∼ 100 mK instead of T ∼ 1 K), but also the variations of the heat capacity to detect are much weaker (∆C ∼ k B instead of ∆C ∼ 10 3 k B ).

EXPERIMENTAL SETUP

Our sample (Fig. 1) is composed of N = 5 × 10 6 noninteracting silver rings (2r = 600 nm diameter, w 140 nm arm width, d = 34 nm thickness, total mass m = 330 ng). The silver rings are deposited by e-gun evaporation under 10 -6 mb vacuum. Thanks to the high purity of the silver the phase coherence length is large, of about 10 µm at the temperature of 100 mK, which is far above the diffusive ring circumference L [24]. The specific heat of silver at T = 0.1 K can be estimated to be of the order of 10 -6 J/gK, and hence the magnetic-field-independent part of the heat capacity of the rings is expected to be 3.3 × 10 -13 J/K; representing approximately 10% of the total heat capacity. The rings have been patterned by electron beam lithography on the thermal sensor made of a suspended silicon membrane (size 4 mm × 4 mm, thickness ∼ 5 µm). The transducers, a copper heater and a highly sensitive NbN thermometer are integrated on each side of the membrane (see Fig. 2), the silver rings being located between the two elements [25].

A sketch of the ac calorimetry technique that we use is given in Fig. 2 and described in detail in previous articles [26][27][28]. It consists in applying an ac current through the heater; here at a frequency f elec = 30 Hz. This induces oscillations T ac (t) of the temperature of the suspended membrane. These oscillations are detected by the thermometer. For a specific experimentally determined operating frequency, quasi-adiabatic conditions are fulfilled, allowing measurements of the specific heat. The typical signal obtained from the lock-in amplifier is composed of a modulus measuring a root mean square (RMS) voltage proportional to the temperature oscillation of the membrane (T RMS ac ∝ V RMS ac ) and of a phase ϕ, both related to the heat capacity through the equations:

T ac (t) = P 0 ωC cos(2ωt + ϕ), (4) tan
(∆ϕ) = 2ωC K , (5)
where P 0 is the power dissipated in the heater, ω = 2πf elec the electrical excitation frequency, C the heat capacity of the membrane and K the thermal conductance between the membrane and the heat bath (for more details see [21,26,27,29]). The setup is cooled down to very low temperatures using a dilution fridge, equipped with a superconducting coil supplying a magnetic field H normal to the plane of the rings. As compared to previous measurements [21][22][23], the thermometry has been adapted to work at very low temperature and the arm width has been increased from 40 µm to 150 µm to allow measurements at a higher frequency. Consequently, δT dc = T membrane -T cryostat decreased and we were able to obtain temperatures compatible with the purpose of our study (T 70 mK on the membrane for a regulated temperature of 50 mK of the sample holder) [29].

EXPERIMENTAL RESULTS

Figure 3 shows a scan of the heat capacity versus the temperature of the membrane. The heat capacity measured at the lowest temperature is around 10 pJ/K, with the error not exceeding 0.1 pJ/K. It must be noticed that this value corresponds to the heat capacity of the whole sample (rings + membrane). In the inset of Fig. 3 Our experimental method to detect persistent currents is based on scanning the heat capacity versus the applied magnetic field, at a constant temperature. During the measurement, two components of the temperature oscillation are recorded: the modulus and the phase. Both bring physical informations about the heat capacity variations (see equations ( 4) and ( 5)).

A typical scan is shown in Fig. 4. Starting at H = 0 mT, we slowly increase the field to 50 mT by small steps ( 1 mT). C is measured at each H for signal integration times of the order of 10 seconds. In order to improve the sensitivity and to detect the very weak oscillations, we realized a hundred of identical scans by carrying out cycles between H = -50 mT and H = 50 mT. These data are averaged during the signal processing described below.

The measurements were performed at three different temperatures: T = 60, 100 and 150 mK. For each of them, the temperature of the sensor membrane oscillates with an amplitude of ∼ 10% compared to the control temperature.

FIG. 4: Phase of the temperature oscillation (at the top) and modulus of the temperature oscillations, expressed in terms of heat capacity using Eq. ( 4)(at the bottom), of mesoscopic silver rings versus magnetic field measured at T = 150 mK.

DATA PROCESSING

In order to detect a periodicity in our data, a Fourier analysis of the phase of the temperature oscillation and of its amplitude, expressed in terms of heat capacity using Eq. ( 4), has been performed. As a first step, the low frequency trend observed in every scan has been removed with a third-order polynomial regression. As a second step, the power spectral densities have been calculated as follows. The autocorrelation function of the signal x(H) (where x is the modulus or the phase),

Γ x (H) = lim M →∞ 1 2M M -M x(H )x(H -H)dH , (6)
is first estimated for every scan. Averaging over a hundred of scans is then performed to reduce the noise. The power spectral density S x (ν) is calculated as the Fourier transform of the averaged autocorrelation function Γx (H). The power spectral densities (refered to PSD in the following) of both the phase of the temperature oscillation and the modulus of the heat capacity (extracted from equations 4) for the three temperatures 60, 100 and 150 mK are displayed in Fig. 5 . We first stress that the high peaks observed for all curves at a low frequency ν ∼ 0.05 mT -1 are a reminiscence of the trend which has not been completely removed by the polynomial regression. These peaks do not carry any useful signal. Unfortunately, they can mask any physically interesting signatures of persistent currents that might be present at low frequencies and would correspond to the first harmonics C 1 of the heat capacity.

In addition to the low-frequency peak, our spectral analysis of the phase of the temperature oscillation (upper panel of Fig. 5) reveals peaks at ν 0.11 mT -1 (the period is 9 mT). This value corresponds to a half of flux quantum through a ring of 264 nm in radius. Our rings have the inner radius of 220 nm and the outer radius of 360 nm. Thus, the observed spectral peak is in the range of frequencies in which signatures of the second harmonics of heat capacity oscillations, C 2 , are expected. However, as it can be seen from the lower panel of Fig. 5, no clear signature of spectral peaks at ν 0.11 mT -1 is observed in the modulus of heat capacity, although the phase and the modulus of the oscillating heat capacity signal are related through Eq. ( 5). The reason for observing a peak in the phase signal but not in the modulus signal most probably resides in a lower sensitivity on the modulus signal. At 100 mK, the sensitivity obtained for the modulus is ∼ 10 -14 J/K or ∼ 10 -20 J/K (∼ 100 k B ) per ring (a sensitivity of ∆C/C 5 × 10 -4 ) as compared to the sensitivity obtained for the phase estimated to be of the order of ∆ϕ/ϕ 3 × 10 -5 . In order to compare our measurements with theoretical predictions, we estimated the average amplitude of the second harmonics of heat capacity oscillations, | C 2 (T ) |, (upper panel). First, the area A of the spectral peak is estimated (second column). For this purpose, we fit the curves of the upper panel of Fig. 5 with smooth polynomials and evaluate the integral of the difference between the data and the fit taken between the vertical lines. Next, the associated heat capacity is calculated using Eq. ( 5) with ϕ = √ 2A (column 5). Finally, the heat capacity is divided by the number of rings N = 5 × 10 6 to obtain the signal per ring (last column).

T (mK) Area A (degree 2 ) Phase ϕ (degree) tan ϕ K(W/K) | C2 | (J/K) | C2 |/ring (k B )
from the area below the curves of Fig. 5 (see Table I).

COMPARISON WITH THEORY

We compare our results with the predictions of two theoretical models: the model of Ambegaokar and Eckern (AE) [12] and the model of Bary-Soroker, Entin-Wohlman and Imry (BEI) [17]. Both models rely on taking into account interactions between electrons in the rings, but the interactions are repulsive in the AE model and attractive in the BEI model. In order to suppress the superconductivity that strong attractive interactions may induce, the latter model includes scattering of electrons by magnetic impurities.

Even if the applicability of the AE model to realistic experiments is made questionable by the disagreement of the sign of the current that it predicts with measurements [8], it is still of great interest to describe experimental data [10]. Using Eq. ( 18) of Ref. [12] and our Eq. ( 2), we readily obtain:

C AE 2 (T ) = k B × 4 9π N (0) V k B T E Th exp - k B T 3E Th , (7)
where N (0) is the electronic density of states at Fermi energy and V is the mean value of the attractive interaction potential [12]. Using the data from Ref. [8], where silver rings similar to ours were studied, we estimate N (0) V 0.37 and E Th /k B 40 mK. The resulting dependence of C AE 2 on temperature is shown in Fig. 6 by a dashed line, with the scale given on the right. The theoretical Eq. ( 7) reproduces the trend of the temperature dependence of our data that has a maximum at

T 3E Th /k B ≈ 100 mK. But the values of | C AE 2
| are 2 orders of magnitude smaller than the data.

In the BEI model, assuming attractive interactions in combination with scattering on magnetic impurities [17], we find: where Ψ(x) and Ψ (x) are the digamma function and its derivative, respectively, τ s is the spin-scattering time, T 0 c is the bare superconducting transition temperature in the absence of magnetic impurities (i.e. at τ s → ∞), and

C BEI 2 (T ) = -k B × 4T E Th k B ∞ n=-∞ ∞ 0 dx sin(2πx) × ∂ 2 ∂T 2 Ψ [F n (x, T, T 0 c , τ s )] ln(T /T 0 c ) + Ψ[F n (x, T, T 0 c , τ s )] -Ψ(1/2) , (8)
F n (x, T, T 0 c , τ s ) = (1 + |n|)/2 + (1/2πτ s + πx 2 E Th /k B )/T .
The minus sign in Eq. ( 8) reflects the diamagnetic nature of the persistent current in this model, in contrast to the paramagnetic current leading to Eq. ( 7). It follows from the comparison of the theory with previous measurements of persistent currents [START_REF] Lévy | [END_REF]7] that it is reasonable to choose T 0 c = 0.1E Th /k B [17]. At the same time, s = 1/πT 0 c τ s should exceed 0.87 to suppress the superconductivity at all temperatures. We set s = 1 and show the absolute value of the resulting heat capacity | C BEI 2 | in Fig. 6 by a solid line.

DISCUSSION

Interestingly, the two theoretical approaches considered above yield very similar results both for the magnitude of heat capacity oscillations (| C 2 | ∼ 10 -2 k B ) and for its temperature dependence (maximum values of | C 2 | reached at T ≈ 100 mK for experimental parameters). Whereas the latter temperature dependence is in agreement with our measurements, the predicted magnitude of the heat capacity oscillations is far too small to explain the observed | C 2 | ∼ k B . This discrepancy may result from the insufficiencies of the theoretical models, as well as from the uncontrolled errors in the estimation of the absolute values of heat capacities in our experiment.

On the theory side, the two models considered here were previously shown to be compatible with experiments (see Refs. [12] and [17]). They provided reasonable results for the magnitude and the temperature dependence of persistent currents measured in Refs. [START_REF] Lévy | [END_REF] and [7]. It should be noted, however, that, on the one hand, the heat capacity is proportional to the second derivative of the persistent current I with respect to temperature [see Eq. ( 2)] and hence, it is sensitive to fine details of the temperature dependence of I that might not be captured by the theory. Equation (8), for example, changes sign for T ≈ 30 mK, leading to a peculiar low-temperature behavior of the heat capacity in Fig. 6. This is due to the change of the shape of I 2 (T ) curve from convex to concave. Also the surface states (evanescent states), when included in the theoretical model, may modify its predictions significantly [30]. In addition, it was noticed by several authors that other phenomena (such as, e.g., the fluctuations of electron spin density [31] or the ambient electromagnetic field [32]) may induce magnetic fluxperiodic currents in mesoscopic rings. These currents can be comparable or even larger than the persistent currents. Given the large signals measured in our experiments and taking into account that our calorimetric technique may be particularly sensitive to currents that dissipate heat, we believe that it is likely that these phenomena may be important in our setup. Because Eq. ( 2) does not hold for these non-equilibrium processes, it remains to be seen if and how they could be included into the theoretical model.

The disagreement between theory and experiment seen in Fig. 6 might also stem from the difficulties in obtaining quantitatively correct values of heat capacity, intrinsic to the extreme difficulty of the measurements: measuring very low thermal signals at extremely low temperatures. These difficulties are obvious already from the comparison of the heat capacities extracted from the absolute value and the phase of the voltage signal measured in our experiment (compare the two panels of Fig. 5).

CONCLUSION

We introduced a calorimetric approach to the study of persistent currents in mesoscopic rings made of normal metals. The approach relies on the measurement of periodic variations of heat capacity of a large ensemble of rings with magnetic field. Under the experimental conditions reported here, the approach was at the limit of its sensitivity, the signal being strongly masked by noise. Despite this, we estimated the amplitude of heat capacity oscillations to be of the order of several k B per ring at ∼ 100 mK. The amplitude is two orders of magnitude larger than expected from the existing theories which, however, correctly predict the range of temperatures where the heat capacity signal is maximum. Further experiments and theoretical investigations are necessary to elucidate the sources of this discrepancy.

Both experiment and theory suggest that, in contrast to the persistent current I, the heat capacity C of an ensemble of mesoscopic rings is not a monotonic function of temperature. In particular, the average value of its second harmonic C 2 vanishes for T → 0. More experiments having better sensitivity will be necessary to evidence the real position of this maximum.

FIG. 1 :

 1 FIG. 1: (a) Scanning electron microscope (SEM) image of a single silver mesoscopic ring, the scale bar represents 200 nm. (b) SEM image of the array of rings.

FIG. 2 :

 2 FIG.2: Schematic drawing of the experimental set-up of heat capacity measurement on a suspended silicon membrane. The internal oscillation signal of the lock-in 1 is used to pilot the voltage current convertor used to apply the current to the heater. The lock-in 2 is used to measure the resistance of the heater, and then estimate the power dissipated. The oscillation of temperature Tac is measured on the lock-in 1 at the second harmonic (2f). The scale bar represents 1 mm.

  Figure3shows a scan of the heat capacity versus the temperature of the membrane. The heat capacity measured at the lowest temperature is around 10 pJ/K, with the error not exceeding 0.1 pJ/K. It must be noticed that this value corresponds to the heat capacity of the whole sample (rings + membrane). In the inset of Fig.3, we show a histogram of the noise measurement obtained from the upper panel of this figure, indicating a noise of 0.8 nV/ √ Hz.

FIG. 3 :

 3 FIG. 3: (color online) Top panel: voltage at twice the excitation frequency versus time at T = 200 mK. Inset of the bottom panel: histogram of the noise extracted from the measurement presented in the top panel. Bottom panel: heat capacity C of silver rings versus temperature in the absence of magnetic field.

FIG. 5 :

 5 FIG. 5: (Color online) Power spectral densities of the average signal for phase (upper panel (a)) and in the lower panel (b), the PSD of the modulus of the heat capacity (extracted from the equation 4). In the upper panel the red and the blue curves have been shifted from the green for clarity, and in the medium panel only the red curve has been shifted by 10 -30 (J/K) 2 .mT. The vertical straight lines delimit the frequency range in which signatures of persistent currents are expected.

  -6 2.7×10 -8 3.5×10 -16 5 TABLE I: The amplitude of the second harmonics of heat capacity oscillations | C2 | is extracted from the curves of Fig.5

FIG. 6 :

 6 FIG. 6: (Color online) Average amplitude of the second harmonic C2 of the heat capacity oscillations with the magnetic field. Points show the experimental results from TableI, with the scale on the left. Lines show theoretical predictions: Eq. (7) (dashed blue line) and Eq. (8) (solid red line), scale on the right.

Acknowledgements

We acknowledge support from Nanofab and Capthercal. We would like to thank P. Brosse-Maron, J.-L. Garden, T. Fournier and U. Gennser for help and fruitful scientific exchanges. We are grateful to G. Montambaux, L. Saminadayar and G. Gaudin for discussions at early stages of this work and to R. Deblock for his comments on the manuscript. This research has been funded by the Région Rhône-Alpes (PhD fellowship of GS) and by the ANR through the Quantherm project.