Ferruccio Colombini

Daniele Del Santo

Francesco Fanelli

Guy Métivier

A note on complete hyperbolic operators with log-Zygmund coefficients

Keywords: Mathematics Subject Classification (2000). Primary 35L15; Secondary 35B65, 35S50, 35B45 Hyperbolic operators, non-Lipschitz coefficient, log-Zygmund regularity, energy estimates, well-posedness

, and it is devoted to strictly hyperbolic operators with non-regular coefficients. We focus here on the case of complete operators whose second order coefficients are log-Zygmund continuous in time, and we investigate the H ∞ well-posedness of the associate Cauchy problem.

Introduction

The present paper is, in a certain sense, the continuation of the recent work [START_REF] Colombini | Time-dependent loss of derivatives for hyperbolic operators with non-regular coefficients[END_REF]: it is devoted to the study of strictly hyperbolic operators with log-Zygmund in time coefficients. In particular, we want to prove the H ∞ well-posedness of the Cauchy problem related to a complete second order operator,

Lu = ∂ 2 t u - N j,k=1 ∂ j (a jk (t, x) ∂ k u) + N j=0 b j (t, x) ∂ j u + c(t, x) u , (1.1)
(with ∂ 0 = ∂ t ) whose highest order coefficients satisfy such a regularity assumption (see also relation (2.3) below). The core of the proof is to establish suitable energy estimates for L.

It is well-known (see [START_REF] Hurd | Questions of existence and uniqueness for hyperbolic equations with discontinuous coefficients[END_REF]; see also e.g. [START_REF] Hörmander | Linear partial differential operators[END_REF]Ch. IX] or [START_REF] Mizohata | The Theory of Partial Differential Equations[END_REF]Ch. 6] for analogous results) that, if the coefficients a jk are Lipschitz continuous with respect to t and only measurable in x, then the Cauchy problem for L is well-posed in H 1 × L 2 . If the a jk 's are Lipschitz continuous with respect to t and C ∞ b (i.e. C ∞ and bounded with all their derivatives) with respect to the space variables, one can recover the well-posedness in H s+1 × H s for all s ∈ R. Moreover, in the latter case, one gets, for all s ∈ R and for a constant C s depending only on it, the following energy estimate:

sup 0≤t≤T u(t, •) H s+1 + ∂ t u(t, •) H s ≤ (1.2) ≤ C s u(0, •) H s+1 + ∂ t u(0, •) H s + T 0 Lu(t, •) H s dt for all u ∈ C([0, T ]; H s+1 (R N )) ∩ C 1 ([0, T ]; H s (R N ))
for which Lu belongs to L 1 ([0, T ]; H s (R N )). Let us explicitly remark that previous inequality involves no loss of regularity for the function u: estimate (1.2) holds for every u of class C 2 ([0, T ]; H ∞ (R N )) and the Cauchy problem for L is well-posed in H ∞ with no loss of derivatives.

If the Lipschitz continuity (in time) hypothesis is not fulfilled, then (1.2) is no more true. Nevertheless, one can still try to recover H ∞ -well-posedness, with a loss of derivatives in the energy estimate.

The first case to consider is the case of the coefficients a jk depending only on t:

Lu = ∂ 2 t u - N j,k=1
a jk (t) ∂ j ∂ k u .

In [START_REF] Colombini | Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps[END_REF], Colombini, De Giorgi and Spagnolo assumed the coefficients to satisfy an integral log-Lipschitz condition:

T -ε 0 |a jk (t + ε) -a jk (t)| dt ≤ C ε log 1 + 1 ε , (1.3)
for some constant C > 0 and all ε ∈ ]0, T ]. They obtained an energy estimate with a fixed loss of derivatives: there exists a constant δ > 0 such that, for all s ∈ R, the inequality

sup 0≤t≤T u(t, •) H s+1-δ + ∂ t u(t, •) H s-δ ≤ (1.4) ≤ C s u(0, •) H s+1 + ∂ t u(0, •) H s + T 0 Lu(t, •) H s dt
holds true for all u ∈ C 2 ([0, T ]; H ∞ (R N )), for some constant C s depending only on s.

Let us explain briefly the main ideas of the proof, which are actually the basis to establish also the next results we are going to quote. The authors first smoothed out the coefficients using a mollifier kernel (ρ ε ) ε . Then, by Fourier transform, they defined an approximated energy E ε (ξ, t) in the phase space, where the problem becomes a family of ordinary differential equations. At that point, the key step was to perform a different approximation of the coefficients in different zones of the phase space, which was realized setting ε = |ξ| -1 . This particular choice allowed them to obtain the result.

Let us remark that if the coefficients a jk are not Lipschitz continuous then a loss of regularity cannot be avoided. In fact, in paper [START_REF] Cicognani | Modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic Cauchy problem[END_REF], Cicognani and Colombini proved that, if the regularity of the coefficients a jk is measured by a modulus of continuity, any intermediate modulus of continuity between the Lipschitz and the log-Lipschitz ones entails necessarily a loss of regularity, which, however, can be made arbitrarily small.

Recently Tarama (see paper [START_REF] Tarama | Energy estimate for wave equations with coefficients in some Besov type class[END_REF]) was able to prove an analogous result, and the consequent H ∞ well-posedness of the related Cauchy problem, when the coefficients satisfy an integral log-Zygmund condition: there exists a constant C > 0 such that, for all j, k and all ε ∈ ]0, T /2[, one has

T -ε ε |a jk (t + ε) + a jk (t -ε) -2 a jk (t)| dt ≤ C ε log 1 + 1 ε . (1.5)
It's obvious that, if the a jk 's satisfy (1.3), then they satisfy also (1.5): so, a more general class of functions is considered. The improvement with respect to paper [START_REF] Colombini | Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps[END_REF], however, was obtained defining a new energy, which involved (by derivation in time) second derivatives of the approximated coefficients.

In paper [START_REF] Colombini | Hyperbolic operators with non-Lipschitz coefficients[END_REF], Colombini and Lerner considered instead the case when the coefficients a jk depend both on time and space variables. In particular, they assumed an isotropic punctual log-Lipschitz condition, i.e. there exists a constant

C > 0 such that, for all ζ = (τ, ξ) ∈ R × R N , ζ = 0, one has sup z=(t,x)∈R×R N |a jk (z + ζ) -a jk (z)| ≤ C |ζ| log 1 + 1 |ζ| .
The analysis was based on a broad use of the Littlewood-Paley decomposition and paradifferential calculus. As a matter of facts, one cannot use the Fourier transform, due to the dependence of a jk on x. Also in this case, a different approximation of the coefficients in different zones of the phase space was required: this was obtained setting ε = 2 -ν , where 2 ν is the "size" of the frequencies in the ν-th ring of the Littlewood-Paley decomposition (see also section 3). In the end, they got the following statement: for all s ∈ ]0, 1/4], there exist positive constants β and C s and a time

T * ∈ ]0, T ] such that sup 0≤t≤T * u(t, •) H -s+1-βt + ∂ t u(t, •) H -s-βt ≤ (1.6) ≤ C s u(0, •) H -s+1 + ∂ t u(0, •) H -s + T * 0 Lu(t, •) H -s-βt dt for all u ∈ C 2 ([0, T ]; H ∞ (R N )).
Let us point out that the bound on s was due to this reason: the product by a log-Lipschitz function is well-defined in H s if and only if |s| < 1. Note also that this fact gives us a bound on the lifespan of the solution: the regularity index -s + 1 -βT * has to be strictly positive, so one can expect only local in time existence of a solution. Moreover in the case the coefficients a jk are C ∞ b in space, the authors proved inequality (1.6) for all s: so, they got well-posedness in H ∞ with a loss of derivatives increasing in time.

The case of a complete strictly hyperbolic second order operator,

Lu = N j,k=0 ∂ yj (a jk ∂ y k u) + N j=0 b j ∂ yj u + ∂ yj (c j u) + d u (here we set y = (t, x) ∈ R t × R N x )
, was considered by Colombini and Métivier in [START_REF] Colombini | The Cauchy problem for wave equations with non-Lipschitz coefficients; application to continuation of solutions of some nonlinear wave equations[END_REF]. They assumed the same isotropic log-Lipschitz condition of [START_REF] Colombini | Hyperbolic operators with non-Lipschitz coefficients[END_REF] on the coefficients of the second order part of L, while b j and c j were supposed to be α-Hölder continuous (for some α ∈ ]1/2, 1[ ) and d to be only bounded. The authors headed towards questions such as local-in-space existence and uniqueness, and finite propagation speed. Note that the restriction on α was due to the local-inspace study the authors performed (in particular, in order to have traces on the boundary well-defined): see for instance papers [START_REF] Colombini | Hyperbolic operators with non-Lipschitz coefficients[END_REF] and [START_REF] Colombini | A note on non-homogeneous hyperbolic operators with low regularity coefficients[END_REF], were no bounds on the Hölder regularity were required.

Trying to mix up up a Tarama-like hypothesis (concerning the dependence on the time variable) with the one of Colombini and Lerner (with respect to x) is at this point a natural step. More precisely, we want to consider a pointwise log-Zygmund condition in time and a pointwise log-Lipschitz condition in space, uniformly with respect to the other variable (see conditions (2.3) and (2.4) below). Under this assumption, it's possible to prove an energy estimate analogous to (1.6). The first result in this direction was due to Colombini and Del Santo, in paper [START_REF] Colombini | A note on hyperbolic operators with log-Zygmund coefficients[END_REF] (see also [START_REF] Santo | The Cauchy problem for a hyperbolic operator with Log-Zygmund coefficients, Further Progress in Analysis[END_REF], where smoothness in space was however required). Nevertheless, they had to restrict themselves to the case of space dimension N = 1 (see also [START_REF] Colombini | A note on non-homogeneous hyperbolic operators with low regularity coefficients[END_REF] for the case of a complete operator). The result in the general instance N ≥ 1 was only recently proved in paper [START_REF] Colombini | Time-dependent loss of derivatives for hyperbolic operators with non-regular coefficients[END_REF]. Even if the difficulty of the additional space variables seems to be just technical, the approach assumed in [START_REF] Colombini | Time-dependent loss of derivatives for hyperbolic operators with non-regular coefficients[END_REF], i.e. of paradifferential calculus depending on parameters, actually gives a more general and complete point of view of the problem.

The aim of the present note is to extend the result of [START_REF] Colombini | Time-dependent loss of derivatives for hyperbolic operators with non-regular coefficients[END_REF] for homogeneous second order operators to the case where also lower order terms come into play. In particular, according with the results of [START_REF] Colombini | Hyperbolic operators with non-Lipschitz coefficients[END_REF], [START_REF] Colombini | The Cauchy problem for wave equations with non-Lipschitz coefficients; application to continuation of solutions of some nonlinear wave equations[END_REF] and [START_REF] Colombini | A note on non-homogeneous hyperbolic operators with low regularity coefficients[END_REF], we will assume the first order coefficients to be Hölder continuous in the space variable, and the zero order coefficient to be just bounded. From an energy estimate analogous to (1.6), we will infer, for smooth in space coefficients, the H ∞ well-posedness of the related Cauchy problem with a finite loss of derivatives.

To complete the general picture of the state of the art, let us recall that in the already quoted paper [START_REF] Tarama | Energy estimate for wave equations with coefficients in some Besov type class[END_REF], Tarama proved also an energy estimate with no loss of derivatives for coefficients a jk (t) which fulfill a (integral) Zygmund hypothesis. This result has been recently generalized in [START_REF] Colombini | A well-posedness result for hyperbolic operators with Zygmund coefficients[END_REF] to the case of a jk 's Zygmund continuous both in t and x (assuming also Hölder first order coefficients and bounded zero order coefficient), but only in the space H 1/2 ×H -1/2 . If an isotropic Zygmund assumption implies or not loss of regularity for general initial data in H s × H s-1 is still an open problem.

Our paper is organized in the following way. In the next section, we will state our main results. A brief presentation of the tools we need in order to prove them will follow, in Section 3: we refer to [START_REF] Colombini | Time-dependent loss of derivatives for hyperbolic operators with non-regular coefficients[END_REF] for a complete description. These tools mostly come from Fourier analysis. In particular, we will recall some basic facts on Littlewood-Paley decomposition and paradifferential calculus, both in its classical version (see e.g. [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF], [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF] and [START_REF] Métivier | Para-differential calculus and applications to the Cauchy problem for nonlinear systems, Centro di Ricerca Matematica "Ennio De Giorgi[END_REF]) and in the one depending on a parameter (see [START_REF] Métivier | Interactions de deux chocs pour un système de deux lois de conservation, en dimension deux d'espace[END_REF] and [START_REF] Métivier | Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems[END_REF]). We will also recall some basic facts on functions with low regularity modulus of continuity.

Section 4 is devoted to the proof of energy estimates for our strictly hyperbolic operator with non regular (both in t and x) coefficients: this is the most involving part. We will just give a sketch of the analysis of the higher order terms (already treated in [START_REF] Colombini | Time-dependent loss of derivatives for hyperbolic operators with non-regular coefficients[END_REF]), while we will give the most of the details about the lower order ones.

Finally, in Section 5 we will show how the just proved energy estimates allow to recover the H ∞ well-posedness of the related Cauchy problem, in the case of coefficients which are smooth in the space variables.

Main results

Let us consider the operator over [0, T 0 ] × R N (for some T 0 > 0 and N ≥ 1)

Lu = ∂ 2 t u - N j,k=1 ∂ j (a jk (t, x) ∂ k u) + N j=0 b j (t, x) ∂ j u + c(t, x) u (2.1)
(with the notation ∂ 0 = ∂ t ) and let us suppose L to be strictly hyperbolic with bounded coefficients, i.e. there exist two positive constants 0 < λ 0 ≤ Λ 0 such that, for all (t, x) ∈ R t × R N x and all ξ ∈ R N , one has

λ 0 |ξ| 2 ≤ N j,k=1 a jk (t, x) ξ j ξ k ≤ Λ 0 |ξ| 2 . (2.2)
Moreover, assume that the a jk 's are log-Zygmund continuous in the time variable, uniformly with respect to x: there exists a constant K 0 > 0 such that, for all τ > 0, one has

sup (t,x) |a jk (t + τ, x) + a jk (t -τ, x) -2a jk (t, x)| ≤ K 0 τ log 1 + 1 τ . (2.
3)

The goal of the present paper is to prove the following H ∞ well-posedness result.

Theorem 2.1. Consider the operator L defined in (2.1), under the strictly hyperbolicity condition (2.2). Let us suppose the following conditions on the coefficients of L:

(i) the a jk 's are log-Zygmund continuous in time, i.e. (2.3) holds;

(ii) the a jk 's are also of class C ∞ b with respect to x; (iii) b j , for all 0 ≤ j ≤ N , and c belong to

L ∞ ([0, T 0 ]; C ∞ b (R N )).
Then the Cauchy problem for L,

(CP ) Lu = f u |t=0 = u 0 , ∂ t u |t=0 = u 1 , is well-posed in H ∞ (R N ), globally in time.
Recall that the notation f ∈ C ∞ b means that f is of class C ∞ , and it is bounded with all its derivatives.

The proof of the previous result is essentially based on finding energy estimates for L: we will establish them when the coefficients satisfy minimal regularity assumptions also with respect to the space variable. This means that we will focus on log-Lipschitz continuous a jk 's and Hölder continuous b j 's, while the 0-th order coefficient c can be taken just bounded (see also [START_REF] Colombini | Hyperbolic operators with non-Lipschitz coefficients[END_REF], [START_REF] Colombini | The Cauchy problem for wave equations with non-Lipschitz coefficients; application to continuation of solutions of some nonlinear wave equations[END_REF] and [START_REF] Colombini | A note on non-homogeneous hyperbolic operators with low regularity coefficients[END_REF] about these hypothesis).

So, assume that there exists a constant

K 1 > 0 such that sup (t,x) |a jk (t, x + y) -a jk (t, x)| ≤ K 1 |y| log 1 + 1 |y| (2.4)
for all y ∈ R N \ {0}. Suppose also the following on the lower order terms:

b j ∈ L ∞ ([0, T 0 ]; C ω (R N )) , 0 ≤ j ≤ N , and c ∈ L ∞ ([0, T 0 ] × R N ) , (2.5)
for some real number 0 < ω ≤ 1 (with

C ω * := B ω ∞,∞ if ω = 1).
Theorem 2.2. Let L be the operator defined in (2.1), whose coefficients fulfill conditions (2.2), (2.3), (2.4) and (2.5).

Then, for all fixed θ ∈ ]0, min{ω, 1}[ , there exist a constant β * > 0, a time T > 0 and a constant C > 0 such that the estimate

sup 0≤t≤T u(t, •) H -θ+1-β * t + ∂ t u(t, •) H -θ-β * t ≤ (2.6) ≤ C u(0, •) H -θ+1 + ∂ t u(0, •) H -θ + T 0 Lu(t, •) H -θ-β * t dt holds true for all u ∈ C 2 ([0, T ]; H ∞ (R N )).
We will explain in Section 5 how one can recover the H ∞ well-posedness for (CP ) from the energy estimates of Theorem 2.2.

Remark 2.3. It's natural to wonder if a similar result still holds true under an isotropic log-Zygmund condition. Let's make some considerations. Firstly, in our proof, we will use the fact that the high frequencies of a log-Lipschitz function behave as the ones of a log-Zygmund one (see Propositions 3.10 and 3.11). The low frequencies of a log-Zygmund function, instead, have an additional logarithmic loss with respect a log-Lipschitz function: the idea, however, is that this loss can be absorbed as in our proof. Therefore, we expect to find a positive answer to the previous question, which will be matter of next studies.

Functional toolbox

The present section aims to give an overview of the tools, based on Fourier analysis techniques, we need to prove our results. We refer to paper [START_REF] Colombini | Time-dependent loss of derivatives for hyperbolic operators with non-regular coefficients[END_REF] (in particular Section 3) and the references therein for a more complete and detailed treatment.

Let us start with a definition. We set1 Π(D) := log(2 + |D|), i.e. its symbol is π(ξ) := log(2 + |ξ|). Definition 3.1. For all α ∈ R, we define the space H s+α log as Π -α H s , i.e.

f ∈ H s+α log ⇔ Π α f ∈ H s ⇔ π α (ξ) 1 + |ξ| 2 s/2 f (ξ) ∈ L 2 .
These logarithmic Sobolev spaces were introduced in [START_REF] Colombini | The Cauchy problem for wave equations with non-Lipschitz coefficients; application to continuation of solutions of some nonlinear wave equations[END_REF]. Due to the low regularity of the coefficients of L, this kind of spaces will come into play in our computations.

Littlewood-Paley theory and paradifferential calculus

Let us now define the so called Littlewood-Paley decomposition, based on a nonhomogeneous dyadic partition of unity with respect to the Fourier variable. We refer to [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF], [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF] and [START_REF] Métivier | Para-differential calculus and applications to the Cauchy problem for nonlinear systems, Centro di Ricerca Matematica "Ennio De Giorgi[END_REF] for the details.

So, fix a smooth radial function χ supported in the ball B(0, 2), equal to 1 in a neighborhood of B(0, 1) and such that r → χ(r e) is nonincreasing over R + for all unitary vectors e ∈ R N . Set also ϕ (ξ) = χ (ξ) -χ (2ξ) .

The dyadic blocks (∆ j ) j∈Z are defined by

∆ j := 0 if j ≤ -1, ∆ 0 := χ(D) and ∆ j := ϕ(2 -j D) if j ≥ 1.
We also introduce the following low frequency cut-off operator:

S j u := χ(2 -j D) = k≤j ∆ k for j ≥ 0.
The following classical properties will be used freely throughout the paper:

• for any u ∈ S , the equality u = j ∆ j u holds true in S ;

• for all u and v in S , the sequence (S j-3 u ∆ j v) j∈N is spectrally supported in dyadic annuli.

Let us also mention a fundamental result, which explains, by the so-called Bernstein's inequalities, the way in which derivatives act on spectrally localized functions.

Lemma 3.2. Let 0 < r < R. A constant C exists so that, for any nonnegative integer k, any couple (p, q) in [1, +∞] 2 with p ≤ q and any function u ∈ L p , we have, for all λ > 0,

(i) supp u ⊂ B(0, λR) =⇒ ∇ k u L q ≤ C k+1 λ k+N ( 1 p -1 q ) u L p ; (ii) supp u ⊂ {ξ ∈ R N | rλ ≤ |ξ| ≤ Rλ} =⇒ C -k-1 λ k u L p ≤ ∇ k u L p ≤ C k+1 λ k u L p .
By the Littlewood-Paley decomposition, we can characterize the new Sobolev class introduced in Definition 3.1 (see [START_REF] Métivier | Para-differential calculus and applications to the Cauchy problem for nonlinear systems, Centro di Ricerca Matematica "Ennio De Giorgi[END_REF]Prop. 4.1.11]). Proposition 3.3. Let s, α ∈ R. A tempered distribution u belongs to the space H s+α log if and only if:

(i) for all k ∈ N, ∆ k u ∈ L 2 (R N ); (ii) setting δ k := 2 ks (1 + k) α ∆ k u L 2 for all k ∈ N, then (δ k ) k ∈ 2 (N). Moreover, u H s+α log is equivalent to (δ k ) k 2 .
Littlewood-Paley decomposition allows also to define operators associated to low regularity symbols, via the so-called paradifferential calculus. For reasons which will appear clear in the sequel, we have to resort not to the classical version, as introduced by Bony in [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF], but to the one depending on a parameter γ ≥ 1 (see e.g. [START_REF] Métivier | Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems[END_REF] and [START_REF] Métivier | Interactions de deux chocs pour un système de deux lois de conservation, en dimension deux d'espace[END_REF]). Let us recall here the main ideas of the construction.

Hence, fix γ ≥ 1 and take a cut-off function ψ γ ∈ C ∞ (R N ×R N ) which verifies the following properties:

• there exist 0 < ε 1 < ε 2 < 1 such that ψ γ (η, ξ) = 1 for |η| ≤ ε 1 (γ + |ξ|) 0 for |η| ≥ ε 2 (γ + |ξ|) ;
• for all (β, α) ∈ N N × N N , there exists a constant C β,α such that

∂ β η ∂ α ξ ψ γ (η, ξ) ≤ C β,α (γ + |ξ|) -|α|-|β| .
Remark 3.4. We remark that ε 1 , ε 2 and the different C β,α 's occurring in the previous estimates must not depend on γ.

For instance, if γ = 1, one can take

ψ γ (η, ξ) ≡ ψ 1 (η, ξ) := +∞ l=0 χ l-3 (η) ϕ l (ξ) ,
where χ and ϕ are the localization (in the phase space) functions associated to a Littlewood-Paley decomposition. Similarly, if γ > 1 it is possible to find a suitable integer µ ≥ 0 such that

ψ γ (η, ξ) := χ µ (η) χ µ+2 (ξ) + +∞ l=µ+3 χ l-3 (η) ϕ l (ξ) (3.1)
is a function with the properties just described. Define now G ψγ as the inverse Fourier transform of ψ γ with respect to the variable η:

G ψγ (x, ξ) := F -1 η ψ γ (x, ξ) . Lemma 3.5. For all (β, α) ∈ N N × N N , there exist constants C β,α , independent of γ, such that:

∂ β x ∂ α ξ G ψγ (•, ξ) L 1 (R N x ) ≤ C β,α (γ + |ξ|) -|α|+|β| , (3.2)
| • | log 2 + 1 | • | ∂ β x ∂ α ξ G ψγ (•, ξ) L 1 (R N x ) ≤ C β,α (γ + |ξ|) -|α|+|β|-1 • (3.3) • log(1 + γ + |ξ|) .
Thanks to G ψγ , we can smooth out a symbol a in the x variable and then define the paradifferential operator associated to a as the pseudodifferential operator related to this smooth function. We set the classical symbol associated to a to be

σ a (x, ξ) := ( ψ γ (D x , ξ) a ) (x, ξ) = G ψγ (•, ξ) * x a(•, ξ) (x)
, and then the paradifferential operator associated to a:

T a := σ a (x, D x ) ,
where we have omitted ψ γ because the definition is independent of it, up to lower order terms (see [START_REF] Métivier | Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems[END_REF]).

Remark 3.6. Let us note that if a = a(x) ∈ L ∞ and if we take the cut-off function ψ 1 , then T a is actually the usual paraproduct operator. If we take ψ γ as defined in (3.1), instead, we get a paraproduct operator which starts from high enough frequencies, which will be indicated with T µ a (see [START_REF] Colombini | The Cauchy problem for wave equations with non-Lipschitz coefficients; application to continuation of solutions of some nonlinear wave equations[END_REF]Par. 3.3]). Let us point out that we can also define a γ-dyadic decomposition. First of all, we set Λ(ξ, γ) := γ 2 + |ξ| 2 1/2 . Then, taken the usual smooth function χ associated to a Littlewood-Paley decomposition, we define

χ ν (ξ, γ) := χ 2 -ν Λ(ξ, γ) , S γ ν := χ ν (D x , γ) , ∆ γ ν := S γ ν+1 -S γ ν .
The usual properties of the support of the localization functions still hold, and for all fixed γ ≥ 1 and all tempered distributions u, we have

u = +∞ ν=0 ∆ γ ν u in S .
Moreover, with natural modifications in definitions, we can introduce the space H s+α log γ as the set of tempered distributions for which

u 2 H s+α log γ := R N ξ Λ 2s (ξ, γ) log 2α (1 + γ + |ξ|) | u(ξ)| 2 dξ < +∞ .
For the details see [START_REF] Métivier | Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems[END_REF]Appendix B]. What is important to retain is that, once we fix γ ≥ 1 (for example, to obtain positivity of paradifferential operators involved in our computations), all the previous construction is equivalent to the classical one; in particular, the space H s+α log γ coincides with H s+α log , the respective norms are equivalent and the characterization given by Proposition 3.3 still holds true.

On log-Lipschitz and log-Zygmund functions

Let us now recall some definitions and properties about the low regularity conditions we assume on the second order coefficients of L. If not otherwise specified, one can find all the proofs to next results in [7, Section 3].

Definition 3.7. A function f ∈ L ∞ (R N
) is said to be log-Lipschitz, and we write

f ∈ LL(R N ), if the quantity |f | LL := sup x,y∈R N , |y|<1   |f (x + y) -f (x)| |y| log 1 + 1 |y|   < +∞ .
We set

f LL := f L ∞ + |f | LL .
Let us define also the space of log-Zygmund functions. We will give the general definition in R N , even if the one dimensional case will be the only relevant one for our purposes. Definition 3.8. A function g ∈ L ∞ (R N ) is said to be log-Zygmund, and we write g ∈ LZ(R N ), if the quantity

|g| LZ := sup x,y∈R N , |y|<1   |g(x + y) + g(x -y) -2 g(x)| |y| log 1 + 1 |y|   < +∞ .
We set g LZ := g L ∞ + |g| LZ .

Remark 3.9. Let us immediately point out that, by monotonicity of logarithmic function, we can replace the factor log (1 + 1/|y|) in previous definitions with log (1 + γ + 1/|y|), for all parameters γ ≥ 1. As paradifferential calculus with parameters will play a fundamental role in our computations, it's convenient to perform such a change, and so does also in hypothesis (2.3) and (2.4) of section 2.

Let us recall also the following characterization of the space LZ.

Proposition 3.10. The space LZ(R N ) coincides with the logarithmic Besov space B 1-log ∞,∞ , i.e. the space of tempered distributions u such that

sup ν≥0 2 ν (ν + 1) -1 ∆ ν u L ∞ < +∞ . (3.4)
On the one hand, from definitions 3.7 and 3.8, it's obvious that LL(R N ) → LZ(R N ): Proposition 3.3 of [START_REF] Colombini | Hyperbolic operators with non-Lipschitz coefficients[END_REF] explains this property in terms of dyadic decomposition.

Proposition 3.11. There exists a constant C such that, for all a ∈ LL(R N ) and all integers k ≥ 0, we have

∆ k a L ∞ ≤ C (k + 1) 2 -k a LL . (3.5)
Moreover, for all k ∈ N we have

a -S k a L ∞ ≤ C (k + 1) 2 -k a LL (3.6) S k a C 0,1 ≤ C (k + 1) a LL . (3.7)
Recall that, again from Proposition 3.3 of [START_REF] Colombini | Hyperbolic operators with non-Lipschitz coefficients[END_REF], property (3.7) is a characterization of the space LL.

On the other hand, we have also B 1 ∞,∞ → LL: the Zygmund condition entails a logarithmic loss in the first variation. As the next lemma states, this instance actually occurs also for log-Zygmund functions. For our purposes, it's enough to consider a log-Zygmund function a depending only on the time variable t, but the same reasoning holds also in higher dimensions. Lemma 3.12. For all a ∈ LZ(R), there exists a constant C, depending only on the LZ norm of a, such that, for all γ ≥ 1 and all 0 < |τ | < 1 one has

sup t∈R |a(t + τ ) -a(t)| ≤ C |τ | log 2 1 + γ + 1 |τ | . (3.8)
Now, given a log-Zygmund function a(t), we can regularize it by convolution. So, take an even function ρ ∈ C ∞ 0 (R t ), 0 ≤ ρ ≤ 1, whose support is contained in the interval [-1, 1] and such that ρ(t)dt = 1, and define the mollifier kernel

ρ ε (t) := 1 ε ρ t ε ∀ ε ∈ ]0, 1] .
We smooth out the function a setting, for all ε ∈ ]0, 1],

a ε (t) := (ρ ε * a) (t) = Rs ρ ε (t -s) a(s) ds . (3.9)
The following proposition holds.

Proposition 3.13. Let a ∈ LZ(R). There exist constants C such that, for all γ ≥ 1,

|a ε (t) -a(t)| ≤ C a LZ ε log 1 + γ + 1 ε (3.10) |∂ t a ε (t)| ≤ C a LZ log 2 1 + γ + 1 ε (3.11) ∂ 2 t a ε (t) ≤ C a LZ 1 ε log 1 + γ + 1 ε .
(3.12)

Low regularity symbols and calculus

Let us now consider symbols a(t, x, ξ), which are LZ t -LL x , and investigate properties of the corresponding paradifferential operators, built as explained above. As we will have to make use of the paradifferential calculus with parameters, we will allow the symbols to depend also on some γ ≥ 1.

We point out that in our calculus the time can be treated as an additional parameter, while ξ represents, as usual, the dual variable to x. Definition 3.14. A symbol of order m+δ log (for m, δ ∈ R) is a function a(t, x, ξ, γ) which is locally bounded on [0, T 0 ]×R N ×R N ×[1, +∞[ , of class C ∞ with respect to ξ and which satisfy the following property: for all α ∈ N N , there exists a constant

C α > 0 such that ∂ α ξ a(t, x, ξ, γ) ≤ C α (γ + |ξ|) m-|α| log δ (1 + γ + |ξ|) (3.13)
for all (t, x, ξ, γ).

So, take a symbol a(t, x, ξ, γ) of order m ≥ 0, which is log-Zygmund in t and log-Lipschitz in x, uniformly with respect to the other variables. Now we smooth a out with respect to time, as done in (3.9). Next lemma, whose proof can be found in [START_REF] Colombini | Time-dependent loss of derivatives for hyperbolic operators with non-regular coefficients[END_REF], provides us estimates on classical symbols associated to a ε and its time derivatives.

Lemma 3.15. The classical symbols associated to a ε and its time derivatives satisfy:

∂ α ξ σ aε ≤ C α (γ + |ξ|) m-|α| ∂ β x ∂ α ξ σ aε ≤ C β,α (γ + |ξ|) m-|α|+|β|-1 log (1 + γ + |ξ|) ∂ α ξ σ ∂taε ≤ C α (γ + |ξ|) m-|α| log 2 1 + γ + 1 ε ∂ β x ∂ α ξ σ ∂taε ≤ C β,α (γ + |ξ|) m-|α|+|β|-1 log (1 + γ + |ξ|) 1 ε ∂ α ξ σ ∂ 2 t aε ≤ C α (γ + |ξ|) m-|α| log 1 + γ + 1 ε 1 ε ∂ β x ∂ α ξ σ ∂ 2 t aε ≤ C β,α (γ + |ξ|) m-|α|+|β|-1 log (1 + γ + |ξ|) 1 ε 2
, where all the constants which occur here don't depend on the parameter γ.

Note that first and second inequalities are satisfied also by the symbol a (not smoothed in time). Now let us quote some basic facts on symbolic calculus, which follow from previous lemma. Before doing this, we recall a definition (see also [START_REF] Métivier | Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems[END_REF]Def. B.8]). Definition 3.16. We say that an operator P is of order m + δ log if, for every (s, α) ∈ R 2 and every γ ≥ 1, P maps

H s+α log γ into H (s-m)+(α-δ) log γ continuously.
Proposition 3.17. (i) Let a be a symbol of order m which is L ∞ in the x variable.

Then T a is of order m, i.e. it maps H s+α log γ into H s-m+α log γ .

(ii) Let us take two symbols a, b of order m and m respectively. Suppose that a, b are LL in the x variable. The composition of the associated operators can be approximated by the symbol associated to the product of these symbols, up to a remainder term:

T a • T b = T ab + R . R has order m + m -1 -log: it maps H s+α log γ into H s-m-m +1+(α+1) log γ . (iii)
Let a be a symbol of order m which is LL in the x variable. The adjoint (over L 2 ) operator of T a is, up to a remainder operator, T a . The remainder operator is of order m -1 + log, and it maps

H s+α log γ into H s-m+1+(α+1) log γ .
Let us end this subsection stating a basic positivity estimate. In this situation, paradifferential calculus with parameters comes into play. Proposition 3.18. Let a(t, x, ξ, γ) be a symbol of order m, which is LL in the x variable and such that Re a(t, x, ξ, γ) ≥ λ 0 (γ + |ξ|) m .

Then, there exists a constant λ 1 , depending only on |a| LL and on λ 0 (so not on γ), such that, for γ large enough, one has

Re (T a u, u) L 2 (R N x ) ≥ λ 1 u 2 H m/2 γ . Remark 3.19.
If the positive symbol a has low regularity in time and we smooth it out by convolution with respect to this variable, we obtain a family (a ε ) ε of positive symbols, with same constant λ 0 . Now, all the paradifferential operators associated to these symbols will be positive operators, uniformly in ε: the constant λ 1 of the previous inequality can be chosen independently of ε.

Then we have the following corollary, which turns out to be fundamental in our energy estimates.

Corollary 3.20. Let a be a positive symbol of order 1 and suppose that a is LL in the x variable.

Then there exists γ ≥ 1, depending only on the symbol a, such that

T a u L 2 ∼ u H 1 γ for all u ∈ H 1 γ (R N ).

Proof of the energy estimates

Let us now tackle the proof of Theorem 2.2. The analysis of the higher order terms, which is the most involving part, was already performed in [START_REF] Colombini | Time-dependent loss of derivatives for hyperbolic operators with non-regular coefficients[END_REF]: so, we will give just a sketch of it.

We will give the most of the details, instead, when treating the lower order terms. In doing this, we will follow a different approach than the one used in [START_REF] Colombini | A note on non-homogeneous hyperbolic operators with low regularity coefficients[END_REF], based on the Bony's paraproduct decomposition (as done in the quoted work [START_REF] Colombini | Time-dependent loss of derivatives for hyperbolic operators with non-regular coefficients[END_REF] for the second order part).

Approximate and total energy

We first regularize coefficients a jk in the time variable by convolution, as done in (3.9), and define the 0-th order symbol

α ε (t, x, ξ) := γ 2 + |ξ| 2 -1/2   γ 2 + j,k a jk,ε (t, x) ξ j ξ k   1/2 .
We take ε = 2 -ν (see also [START_REF] Colombini | Hyperbolic operators with non-Lipschitz coefficients[END_REF] and [START_REF] Colombini | A note on hyperbolic operators with log-Zygmund coefficients[END_REF]), and (for convenience) we will miss out the ε in the notation.

We immediately fix a real number γ ≥ 1, which will depend only on λ 0 and on the sup j,k |a jk | LLx , such that (see Corollary 3.20)

T α -1/2 w L 2 ≥ λ 0 2 w L 2 and T α 1/2 (γ 2 +|ξ| 2 ) 1/2 w L 2 ≥ λ 0 2 w H 1 (4.1)
for all w ∈ H ∞ . Recall that the choice of γ is equivalent to the choice of the parameter µ in (3.1) (see Remark 3.6): hence, from now on we will consider paraproducts starting from this µ, according to definition (3.1), even if we will omit it in the notations. Consider in (2.1) a function u ∈ C 2 ([0, T 0 ]; H ∞ ). We want to get energy estimates for u. We rewrite the equation using paraproduct operators by the coefficients a jk :

∂ 2 t u = Lu + j,k ∂ j (a jk ∂ k u) - N j=0 b j (t, x) ∂ j u -c(t, x) u = Lu + j,k ∂ j T a jk + A jk ∂ k u - N j=0 T bj + B j ∂ j u -T c + C u ,
where A jk ∂ k u = (a jk -T a jk )∂ k u and analogous for B j and C. Let us apply the operator ∆ ν : we get

∂ 2 t u ν = j,k ∂ j T a jk ∂ k u ν - j T bj ∂ j u ν -T c u ν + (4.2) + (Lu) ν + j,k ∂ j A jk ∂ k u ν - j B j ∂ j u ν -Cu ν + + j,k ∂ j ∆ ν , T a jk ∂ k u - j ∆ ν , T bj ∂ j u -[∆ ν , T c ] u ,
where we have set f ν = ∆ ν f and [∆ ν , T g ] is the commutator operator between ∆ ν and the paramultiplication by g. Now, we define

v ν (t, x) := T α -1/2 ∂ t u ν -T ∂t(α -1/2 ) u ν w ν (t, x) := T α 1/2 (γ 2 +|ξ| 2 ) 1/2 u ν z ν (t, x) := u ν
and the approximate energy associated to u ν :

e ν (t) := v ν (t) 2 L 2 + w ν (t) 2 L 2 + z ν (t) 2 L 2 . (4.3) Remark 4.1. Note that w ν (t) 2 L 2 ∼ ∇u ν 2 L 2 ∼ 2 2ν u ν 2 L 2
, thanks to hypothesis (2.2) and the choice of the frequency µ from which the paraproduct starts, recall also (4.1). Now, we fix a θ ∈ ]0, 1[ , as required in hypothesis, and we take a β > 0 to be chosen later; we can define the total energy associated to the solution u as

E(t) := ν≥0 e -2β(ν+1)t 2 -2νθ e ν (t) . (4.4)
It's not difficult to prove (see also inequality (4.8) below) that there exist constants C θ and C θ , depending only on the fixed θ, for which one has:

(E(0)) 1/2 ≤ C θ ( ∂ t u(0) H -θ + u(0) H -θ+1 ) (4.5) (E(t)) 1/2 ≥ C θ ( ∂ t u(t) H -θ-β * t + u(t) H -θ+1-β * t ) , (4.6)
where we have set β * = β (log 2) -1 .

Time derivative of the approximate energy

Let's find an estimate on the time derivative of the energy. We start analysing each term of (4.3).

z ν term.

For the third term we have:

d dt z ν (t) 2 L 2 = 2 Re (u ν , ∂ t u ν ) L 2 . (4.7)
Now, we have to control the term ∂ t u ν : using positivity of operator T α -1/2 , we get

∂ t u ν L 2 ≤ C T α -1/2 ∂ t u ν L 2 (4.8) ≤ C v ν L 2 + T ∂t(α -1/2 ) u ν L 2 ≤ C (e ν ) 1/2 .
So, we find the estimate:

d dt z ν (t) 2 L 2 ≤ C e ν (t) . (4.9) 4.2.2. v ν term. Straightforward computations show that ∂ t v ν (t, x) = T α -1/2 ∂ 2 t u ν -T ∂ 2 t (α -1/2
) u ν . Therefore, keeping in mind relation (4.2), we get:

d dt v ν (t) 2 L 2 = -2Re v ν , T ∂ 2 t (α -1/2 ) u ν L 2 + (4.10) +2 j,k Re v ν , T α -1/2 ∂ j T a jk ∂ k u ν -T bj ∂ j u ν -T c u ν L 2 + +2Re v ν , T α -1/2 Lu ν L 2 + 2Re (v ν , T α -1/2 X u) L 2 ,
where we have defined

Lu := Lu + j,k ∂ j A jk ∂ k u - j B j ∂ j u -Cu X u := j,k ∂ j ∆ ν , T a jk ∂ k u + j ∆ ν , T bj ∂ j u + [∆ ν , T c ] u .
Obviously, we have

2 Re v ν , T α -1/2 Lu ν L 2 ≤ C (e ν ) 1/2 Lu ν L 2 , ( 4.11)
while from Lemma 3.15 and the choice ε = 2 -ν one immediately recovers

2 Re v ν , T ∂ 2 t (α -1/2 ) u ν L 2 ≤ C (ν + 1) e ν .
(4.12)

The other terms (paraproducts, commutators) of (4.10) will be treated later.

d dt w ν 2 L 2 = 2 Re v ν , T α -1/2 T α 2 (γ 2 +|ξ| 2 ) u ν L 2 + R ν , (4.13)
where R ν is a suitable remainder term which can be bounded by the localized energy, at the cost of a logarithmic loss:

|R ν | ≤ C (ν + 1) e ν . (4.14)
Remark 4.2. The analysis of the remainder term R ν is performed by use of the symbolic calculus, introduced in Section 3. Let us point out that, concerning the regularity in x of the symbols, only the first order derivatives come into play. This fact will be relevant in Section 5, in particular in Lemma 5.1.

Let us now consider the paraproduct terms. First of all, we pair up the second term of (4.10) with the first one of (4.13). As proved in [START_REF] Colombini | Time-dependent loss of derivatives for hyperbolic operators with non-regular coefficients[END_REF], we have

2Re   v ν , T α -1/2   j,k ∂ j T a jk ∂ k u ν + T α 2 (γ 2 +|ξ| 2 ) u ν     L 2 ≤ C (ν + 1) e ν , (4.15)
where the constant C depends on the log-Lipschitz norm (with respect to space) of the coefficients a jk and on the fixed parameters µ and γ.

For the lower order terms, instead, we just use continuity results for the paraproduct operator (and (4.8) for j = 0 below):

|2 Re (v ν , T α -1/2 T c u ν ) L 2 | ≤ C e ν (4.16) N j=0 2 Re v ν , T α -1/2 T bj ∂ j u ν L 2 ≤ C e ν , (4.17)
where the constants C depend on c L ∞ ([0,T0]×R N ) and sup j b j L ∞ ([0,T0];C ω ) .

To sum up, from inequalities (4.9), (4.11), (4.12), from the analysis of the paradifferential operators in (4.15), (4.16) and (4.17), and from the estimate for the remainder term (4.14), we can conclude that

d dt e ν (t) ≤ C 1 (ν + 1) e ν (t) + C 2 (e ν (t)) 1/2 Lu ν (t) L 2 + (4.18) + 2 Re (v ν , T α -1/2 X u) L 2 .

Commutator terms

Let us now focus on the commutator terms. We will recall also the most of the details concerning the analysis of the higher order terms, because conditions over β * and T in Theorem 2.2 will follow from it.

So, first of all let us consider the quantity

j,k Re v ν , T α -1/2 ∂ j ∆ ν , T a jk ∂ k u L 2 .
We start remarking that

[∆ ν , T a jk ]w = [∆ ν , S µ a jk ]S µ+2 w + +∞ l=µ+3 [∆ ν , S l-3 a jk ] ∆ l w,
where µ is fixed, as usual (see Remark 3.6). Consequently,

∂ j [∆ ν , T a jk ] ∂ k u = ∂ j ([∆ ν , S µ a jk ] ∂ k (S µ+2 u)) + + ∂ j   +∞ l=µ+3 [∆ ν , S l-3 a jk ] ∂ k (∆ l u)   .
The support of the Fourier transform of

[∆ ν , S µ a jk ] ∂ k (S µ+2 u) is contained in {|ξ| ≤ 2 µ+4 } and moreover [∆ ν , S µ a jk ] ∂ k (S µ+2 u) is identically 0 if ν ≥ µ + 5.
Then, from Bernstein's inequality and [4, Th. 35] we have

∂ j ([∆ ν , S µ a jk ] ∂ k (S µ+2 u)) L 2 ≤ C µ sup j,k a jk LLx S µ+2 u L 2 ;
hence, putting all these facts together, we find

+∞ ν=0 e -2β(ν+1)t 2 -2νθ j,k 2Re v ν , T α -1/2 ∂ j ([∆ ν , S µ a jk ]∂ k (S µ+2 u)) L 2 ≤ (4.19) ≤ C µ sup j,k a jk LLx µ+4 ν=0 e -2β(ν+1)t 2 -2νθ v ν L 2 µ+2 h=0 u h L 2 ≤ C µ sup j,k a jk LLx e β(µ+5)T 2 (µ+4)θ × × µ+4 ν=0 e -β(ν+1)t 2 -νθ v ν L 2 µ+4 h=0 e -β(h+1)t 2 -hθ u h L 2 ≤ C µ sup j,k a jk LLx e β(µ+5)T 2 (µ+4)θ µ+4 ν=0 e -2β(ν+1)t 2 -2νθ e ν (t) .
Next, let's consider

∂ j   +∞ l=µ+3 [∆ ν , S l-3 a jk ] ∂ k (∆ l u)   .
Looking at the support of the Fourier transform, it is possible to see that

[∆ ν , S l-3 a jk ] ∂ k (∆ l u)
is identically 0 if |l -ν| ≥ 3. Consequently the sum over l is reduced to at most 5 terms:

∂ j ([∆ ν , S ν-5 a jk ] ∂ k (∆ ν-2 u)) + • • • + ∂ j ([∆ ν , S ν-1 a jk ] ∂ k (∆ ν+2 u))
, each of them having the support of the Fourier transform contained in {|ξ| ≤ 2 ν+1 }. Let's consider one of these terms, e.g.

∂ j ([∆ ν , S ν-3 a jk ] ∂ k (∆ ν u))
, the computation for the other ones being similar. We have, from Bernstein's inequality,

∂ j ([∆ ν , S ν-3 a jk ] ∂ k (∆ ν u)) L 2 ≤ C 2 ν [∆ ν , S ν-3 a jk ] ∂ k (∆ ν u) L 2 .
On the other hand, using [START_REF] Coifman | Au delà des opérateurs pseudo-différentiels[END_REF]Th. 35] again, we get

[∆ ν , S ν-3 a jk ]∂ k (∆ ν u) L 2 ≤ C ∇S ν-3 a jk L ∞ ∆ ν u L 2 ,
where C does not depend on ν. Consequently, using also (3.7), we deduce

∂ j ([∆ ν , S ν-3 a jk ] ∂ k (∆ ν u)) L 2 ≤ C 2 ν (ν + 1) sup j,k a jk LLx ∆ ν u L 2 .
From this last inequality and similar ones for the other terms, we infer

j,k Re   v ν , T α -1/2 ∂ j   +∞ l=µ+3 [∆ ν , S l-3 a jk ]∂ k (∆ l u)     L 2 ≤ ≤ C sup j,k
a jk LLx (ν + 1)e ν (t)

and then, keeping in mind also (4.19),

+∞ ν=0 e -2β(ν+1)t 2 -2νθ j,k 2 Re v ν , T α -1/2 ∂ j ∆ ν , T a jk ∂ k u L 2 ≤ (4.20) ≤ C +∞ ν=0 (ν + 1) e -2β(ν+1)t 2 -2νθ e ν (t) ,
where C depends on µ, sup j,k a jk LLx , on θ and on the product β T .

Let us now deal with the commutators coming from the lower order parts. As the analysis is analogous to just performed one, we will only sketch the line of the argument. Moreover, let us consider only the b j 's terms, which are actually worse than the commutator

[∆ ν , T c ].
First of all, we focus on the case of space derivatives: so, we take 1 ≤ j ≤ N . As before, we can write

∆ ν , T bj ∂ j u = [∆ ν , S µ b j ] ∂ j S µ+2 u + +∞ l=µ+3 [∆ ν , S l-3 b j ] ∂ j ∆ l u .
From Bernstein's inequality and [START_REF] Coifman | Au delà des opérateurs pseudo-différentiels[END_REF]Th. 35], one gets

[∆ ν , S µ b j ] ∂ j S µ+2 u L 2 ≤ C ∇S µ b j L ∞ S µ+2 u L 2 ≤ C µ b j C ω S µ+2 u L 2 ;
hence, exactly as done in estimate (4.19), we finally have

+∞ ν=0 e -2β(ν+1)t 2 -2νθ N j=1 2Re (v ν , T α -1/2 [∆ ν , S µ b j ] ∂ j S µ+2 u) L 2 ≤ ≤ C µ sup j b j C ω e β(µ+5)T 2 (µ+4)θ E(t) .
Moreover, also in this case, the series over l reduces to a finite sum, due to spectral localization of operators S l and ∆ l . As before, it's enough to consider only the term for l = ν: [4, Th. 35] tells us, once again, that

[∆ ν , S ν-3 b j ] ∂ j ∆ ν u L 2 ≤ C ∇S ν-3 b j L ∞ ∆ ν u L 2 ≤ C 2 ν b j C ω ∆ ν u L 2 ,
and therefore we have

+∞ ν=0 e -2β(ν+1)t 2 -2νθ N j=1 2Re   v ν , T α -1/2 +∞ l=µ+3 [∆ ν , S l-3 b j ] ∂ j ∆ k u   L 2 ≤ ≤ C sup j b j C ω E(t) .
The term involving the time derivative of u, instead, requires us a little bit harder work because we can no more shift the derivative on the coefficient b 0 (as done before thanks to [START_REF] Coifman | Au delà des opérateurs pseudo-différentiels[END_REF]Th. 35]). It's not difficult to see that the kernel of the operator

[∆ ν , S κ b 0 ] ,
for all κ ≥ 0, is the function

k(x, y) = 2 κN φ(2 κ (x -y) ) (S κ b 0 (x) -S κ b 0 (y)) ,
where we have set φ to be the inverse Fourier transform of the localization function ϕ (defined in subsection 3.1). So, by Schur's Lemma the operator norm of the previous commutator over L 2 is controlled by

[∆ ν , S κ b 0 ] L(L 2 ) ≤ C b 0 C ω 2 -ωκ .
This having been established, we consider now the term [∆ ν , S µ b 0 ] S µ+2 ∂ t u. From the previous estimate we get

[∆ ν , S µ b 0 ] S µ+2 ∂ t u L 2 ≤ C b 0 C ω 2 -ωµ S µ+2 ∂ t u L 2 ,
and arguing as for the (4.19) leads us to

+∞ ν=0 e -2β(ν+1)t 2 -2νθ 2Re (v ν , T α -1/2 [∆ ν , S µ b 0 ] S µ+2 ∂ t u) L 2 ≤ ≤ C b 0 C ω e β(µ+5)T 2 (µ+4)(θ-ω) E(t) .
The same arguments give an analogous estimate for [∆ ν , S ν-3 b 0 ] ∆ ν ∂ t u. Therefore the commutator term with first order coefficients can be bounded by

+∞ ν=0 e -2β(ν+1)t 2 -2νθ N j=0 2Re v ν , T α -1/2 ∆ ν , T bj ∂ j u L 2 ≤ C E(t) , (4.21)
where C depends also on sup j b j C ω . As already said, the commutator term [∆ ν , T c ] is actually better, even if c is only L ∞ , because there are no derivatives acting on u. The estimate is finally Applying Hölder inequality for series again implies (see also [START_REF] Colombini | Time-dependent loss of derivatives for hyperbolic operators with non-regular coefficients[END_REF]Sect. 4.4] for the details)

+∞ ν=0 e -2β(ν+1)t 2 -2νθ 2 Re (v ν , T α -1/2 [∆ ν , T c ] u) L 2 ≤ C c L ∞ E(t) . ( 4
+∞ ν=0 e -2β(ν+1)t 2 -2νθ (e ν (t))

1/2   j,k ∂ j A jk ∂ k u   ν L 2 ≤ ≤ +∞ ν=0 (ν + 1) e -2β(ν+1)t 2 -2νθ e ν (t) 1/2 j,k ∂ j A jk ∂ k u H -θ-β * t-1 2 log
, and from [START_REF] Colombini | The Cauchy problem for wave equations with non-Lipschitz coefficients; application to continuation of solutions of some nonlinear wave equations[END_REF]Prop. 3.4] we have

j,k ∂ j A jk ∂ k u H -s-1 2 log ≤ C sup j,k a jk LLx u H 1-s+ 1 2 log , (4.26)
with C uniformly bounded for s in a compact set of ]0, 1[. Consequently,

j,k ∂ j A jk ∂ k u H -θ-β * t-1 2 log ≤ C sup j,k a jk LLx u H 1-θ-β * t+ 1 2 log ≤ C +∞ ν=0 (ν + 1) e -2β(ν+1)t 2 -2νθ e ν (t) 1/2
, and finally

+∞ ν=0 e -2β(ν+1)t 2 -2νθ (e ν (t)) 1/2   j,k ∂ j A jk ∂ k u   ν L 2 ≤ (4.27) ≤ C 4 +∞ ν=0 (ν + 1)e -2β (
F u := f u -T f u . (i) If f ∈ C ω (R N ) for some ω > 0 (with f ∈ C ω * := B ω ∞,∞ if ω ∈ N), then F maps H -s (R N ) into H ω-s (R N ) continuously for all s ∈ ]0, ω[. (ii) If f is only L ∞ (R N ), then F maps continuously H s (R N ) into L 2 (R N ) for all s ≥ 0.
Moreover, the operator norms of F depend on the norm of f in the respective functional space, and they are uniformly bounded for s varying in a compact subset of ]0, ω[ and [0, +∞[ respectively.

Proof. (i) We can write

F u = κ≥µ-3 S κ+2 u ∆ κ f = κ T κ .
As each term T κ is spectrally supported in a ball of size 2 κ and ω -s > 0, by [1, Lemma 2.84] it's enough to estimate its L 2 norm. By [1, Prop. 2.79] and dyadic characterization of Hölder spaces (see e.g. [START_REF] Métivier | Para-differential calculus and applications to the Cauchy problem for nonlinear systems, Centro di Ricerca Matematica "Ennio De Giorgi[END_REF]), we have

S κ+2 u ∆ κ f L 2 ≤ S κ+2 u L 2 ∆ κ f L ∞ ≤ C f C ω u H -s 2 (s-ω)κ λ κ ,
for a sequence (λ κ ) κ ∈ 2 . Therefore we immediately get

F u H ω-s ≤ C f C ω u H -s .
(ii) Thanks to continuity properties of the paraproduct operator, we have

T f u H s ≤ C f L ∞ u H s
for any s ∈ R: in particular, if s ≥ 0 the L 2 norm can be bounded by the previous quantity. Moreover, if s ≥ 0, obviously

f u L 2 ≤ f L ∞ u L 2 ≤ f L ∞ u H s .
The lemma is completely proved.

Let us come back to our estimates. As done before, we find

+∞ ν=0 e -2β(ν+1)t 2 -2νθ N j=0 2Re v ν , T α -1/2 B j ∂ j u ν L 2 ≤ ≤ +∞ ν=0 e -2β(ν+1)t 2 -2νθ 2 -2νω e ν 1/2 N j=0 B j ∂ j u H ω-θ-β * t . At this point, Lemma 4.3 immediately tells us +∞ ν=0 e -2β(ν+1)t 2 -2νθ N j=0 2 Re v ν , T α -1/2 B j ∂ j u ν L 2 ≤ C 5 E(t) , (4.28)
where the constant C 5 depends also on sup j b j C ω .

For Cu we argue as before: by Hölder inequality we have

+∞ ν=0 e -2β(ν+1)t 2 -2νθ N j=0 2Re v ν , T α -1/2 Cu ν L 2 ≤ C E 1/2 Cu H -θ-β * t ,
and so, applying Lemma 4.3 again,

+∞ ν=0 e -2β(ν+1)t 2 -2νθ N j=0 2 Re v ν , T α -1/2 Cu ν L 2 ≤ C 6 E(t) , (4.29)
for C 6 depending on c L ∞ ([0,T0]×R N ) .

Final estimates

Now we take β > 0 (recall that β * = β(log 2) -1 ) and T ∈ ]0, T 0 ] such that

β * T = δ < min {1 , ω} -θ ; (4.30)
in particular, we have 0

< θ ≤ θ + β * t ≤ θ + δ < 1.
Therefore, putting inequalities (4.25), (4.27), (4.28) and (4.29) into (4.24) we finally find

d dt E(t) ≤ (K -2β) +∞ ν=0 (ν + 1) e -2β(ν+1)t 2 -2νθ e ν (t) + + C 2 +∞ ν=0 e -2β(ν+1)t 2 -2νθ (e ν (t)) 1/2 (Lu(t)) ν L 2 ,
where we have set

K = C 1 + C 4 C 2 + C 3 + C 5 C 2 + C 6 C 2 .
Now we fix β large enough, such that K -2β ≤ 0: this corresponds to take T > 0 small enough in (4.30). Hence, we finally get the estimate

d dt E(t) ≤ C 2 (E(t))

The well-posedness in H ∞

In this section we want to sketch how recovering the thesis of Theorem 2.1 from Theorem 2.2. We will follow the outline of the proof in [START_REF] Colombini | Hyperbolic operators with non-Lipschitz coefficients[END_REF] for isotropic log-Lipschitz coefficients. So, let us impose more regularity with respect to x on the coefficients. A preliminary lemma is in order. Lemma 5.1. Consider the operator L defined by (2.1), with coefficients a jk fulfilling conditions (2.2) and (2.3). Assume that a jk ∈ L ∞ ([0, T 0 ]; C 2 b (R N ) ), while b j and c ∈ L ∞ ([0, T 0 ]; C 1 b (R N ) ). Then there exist a β * > 0, a time T > 0 and a constant C > 0 such that estimate (2.6) holds true also for θ = 0.

Proof. First of all, we note that, thanks to the additional regularity on the principal part of L, it makes sense to consider the problem in H 1 × L 2 . Now, it's enough to retake the same steps of the previous proof. The fundamental fact is that, taking into account the additional regularity of the coefficients of L, the logarithmic loss disappears whenever space derivatives of the coefficients are involved. Nevertheless, this is not the case when the time derivative ∂ t acts on the symbol α: the loss in the energy estimates comes from these terms.

Taking into account Remark 4.4, note that the lifespan T depends only on the constants λ 0 and Λ 0 (recall (2.2)) and on the norms of the coefficients of L in their respective functional spaces.

Let us now analyse what happens to the space derivatives of u.

Lemma 5.2. Consider the operator L defined by (2.1), with coefficients a jk fulfilling conditions (2.2) and (2.3).

Assume that coefficients a jk , b j and c all belong to the space L ∞ ([0, T 0 ]; C ∞ b (R N ) ). Then, for all m ≥ 0, there exists a constant C m > 0 such that, for all, u ∈ C 2 ([0, T ]; H ∞ (R N )), one has the following estimate:

sup 0≤t≤T u(t, •) H m+1-β * t + ∂ t u(t, •) H m-β * t ≤
(5.1)

≤ C 1 + e CmT u(0, •) H m+1 + ∂ t u(0, •) H m + T 0 Lu(t, •) H m-β * t dt .
Here, the "loss index" β * , the time T and the constant C are the same of those in Lemma 5.1.

Proof. The proof is analogous to that of [10, Th. Let us just point out that the global in time issue can be obtained remarking that the constants which come into play in the previous proof depend only on the operator L (or better, on its coefficients), and not on the initial data.

Guy Métivier

Institut de Mathématiques de Bordeaux, UMR 5251 Univeristé de Bordeaux 1 351, Cours de la Libération F33405 Talence France e-mail: guy.metivier@math.u-bordeaux1.fr

0

  2.3]. Hence, let us just give a sketch of it.For notation convenience, we defineX m (t) := t 0 u(τ ) H m+1-β * τ + ∂ t u(τ ) H m-β * τ dτand the energy of the dataU(t) := u(0) H m+1 + ∂ t u(0) H m + t Lu(τ ) H m-β * τ dτ .We set moreover Λ(D x ) := 1 + |D x | 2 1/2 . From Lemma 5.1, for any m ≥ 0 we immediately getX m (t) ≤ C 0 U(t) + t 0 L, Λ m (D x ) u(τ ) H -β * τ dτ .Now, differentiating equation (2.1) with respect to x, for any multi-index η such that |η| = m we findL(D η x u) -D η x (Lu) = N j,k=1 1≤| |≤m ∂ j D x a jk ∂ k D η- Lemma 5.1.Second, third and fourth terms on the right-hand side of the previous equality are obviously of lower order. Hence, we finally getX m (t) ≤ C 0 U(t) + C m X m (t) ,for a constant C m depending only on the L ∞ norms of the coefficients of L and their space derivatives.Therefore, by Gronwall's inequality we inferX m (t) ≤ C 0 C m e Cmt U(t) ,which immediately implies estimate (5.1). Now, Theorem 2.1 easily follows from Lemma 5.2.

  where C 3 depends on µ, on the norms of the coefficients on the respective functional spaces, on θ and on the product β T .

	In the end, putting inequalities (4.20), (4.21) and (4.22) all together, we get
	+∞
		e -2β(ν+1)t 2 -2νθ 2Re (v ν , T α -1/2 X u) L 2 ≤	(4.23)
	ν=0
							+∞
							≤ C 3	(ν + 1)e -2β(ν+1)t 2 -2νθ e ν (t) ,
							ν=0
	4.4. The analysis of Lu
	From (4.18) and (4.23) we get
	d dt	E(t) ≤ (C 1 + C 3 -2β)	+∞ ν=0	(ν + 1) e -2β(ν+1)t 2 -2νθ e ν (t) +	(4.24)
				+∞
		+ C 2	ν=0	e -2β(ν+1)t 2 -2νθ (e ν (t)) 1/2	Lu(t)	ν L 2	.
	Let us now consider each term of Lu separately. From Hölder inequality we
	immediately gather
	+∞
	e -2β(ν+1)t 2 -2νθ (e ν (t))	1/2 (Lu) ν L 2 ≤ C (E(t))
	ν=0
							.22)

1/2 Lu(t) H -θ-β * t . (4.25)

  ν+1)t 2 -2νθ e ν (t) , with C 4 uniformly bounded for β * t + θ in a compact set of ]0, 1[ . For the lower order terms, let us state a lemma, which is the analogous of [11, Prop. 3.4]. Lemma 4.3. Given a function f , we define the remainder operator

  Remark 4.4. Let us point out that condition (4.30) gives a bound on the lifespan T of a solution to the Cauchy problem for (2.1). It depends on θ ∈ ]0, 1[ and on β * > 0, hence on constants C 1 . . . C 6 . Going after the guideline of the proof, one can see that, in the end, T depends only on the index θ, on the parameter µ defined by conditions (4.1), on constants λ 0 and Λ 0 defined by (2.2), and on the quantities sup j,k |a jk | LZt , sup j,k |a jk | LLx , sup j b j L ∞ ([0,T0];C ω ) and c L ∞ ([0,T0]×R N ) .

	1/2 Lu(t) H -θ-β * t ;
	by Gronwall's Lemma and relations (4.5) and (4.6) we infer estimate (2.6).

Throughout we agree that f (D) stands for the pseudo-differential operator u → F -1 (f F u).

Acknowledgment

The third author was partially supported by Grant MTM2011-29306-C02-00, MICINN, Spain, ERC Advanced Grant FP7-246775 NUMERIWAVES, ESF Research Networking Programme OPTPDE and Grant PI2010-04 of the Basque Government.


Joseph Gril¹

Patrick Perré
email: perre@nancy-engref.inra.fr

HYGROMECHANICAL BEHAVIOUR OF PAINTED WOODEN PANELS FROM THE CULTURAL HERITAGE

Keywords: Wood, hygromechanical behaviour, panel painting, cultural heritage

Provided it has been properly processed and implemented in structural applications, including protecting against biological attacks and accounting for moisture induced movements, wood is able to withstand considerable duration of loading thanks to its good resistance to fatigue and sophisticated multi-scale composite structure. For that reason, it is still present as the structural element of various types of ancient objects belonging to our cultural heritage, from buildings to painted panels or musical instruments. The requests by conservators to contribute to the maintenance of these old objects offer stimulating challenges to the wood scientists. Long-term creep predictions, essential for the design of wooden structures, would be safely validated when they manage to describe the present state of timber in an old building, while the better understanding of the ageing processes responsible for the sound quality of an old instrument could inspire novel approaches for wood drying and improvement by hygrothermal or chemical modifications.

This paper will focus on the case of wooden panels serving as the support of paintings, that usually require preventive conservation in order to tackle with microclimatic aggressions consecutive to visitors, transportation and generalization of heating and air conditioning. Moisture induced movements of the painted surface might have damaged the painting and tend to detach it for the support, while restrictions imposed by the frames to prevent these movements often provoked major cracks in the wood and the paint. One of the challenges is the explanation of the present state of the observed objects, based on the knowledge of the long term response of a wood part submitted to hypothesized boundary conditions. Another challenge is the prediction of the consequences of a restoration, considered as a modification of these boundary conditions. These questions can be formulated for specific cases, such as highly valuable objects, or in more general terms to evaluate the long term impact of conservation choices.

The wooden support of Mona Lisa, made of a single poplar plate with a pre-existing crack starting from the upper edge, has been previously modelled based on data gathered in October 2004 (fig. 1). The instantaneous response of the panel to the application of the crossbars on the back frame was rather correctly predicted, which allowed to calculate a release energy rate low enough to guarantee against a short-term risk of propagation of the crack [START_REF] Dureisseix | The Mona Lisa's wooden support. Mechanical modelling of the activity of the flexible frame[END_REF]. On the other hand, a separate simulation of the effect of hygrothermal fluctuations considerably overestimated the predicted deflections [START_REF] Perré | The Mona Lisa's wooden support[END_REF] compared to the measured values using a continuous monitoring of the panel [START_REF] Uzielli | The Mona Lisa's wooden support. Physical and mechanical characterization of the support[END_REF]. Several explanations could be given to that discrepancy: partial permeability of the painted face, that would reduce the asymmetry of the moisture exchanged responsible for the curvature variations; adsorption/desorption hysteresis reducing the moisture content variations resulting from the microclimatic fluctuations; damage heterogeneity of the panel due to thickness gradients of hygromechanical loading history; dominance of longitudinal stresses induced by the frame restraint at the upper and lower edges; ageing of the material resulting in slower transfer or significantly smaller hygroexpansion ratios -or a combination of these various factors. Using additional data gathered on the same and other historical panels located in various museums or churches (fig. 2), as well as from tests on mock-up panels or clear wood specimens, some new analysis will be proposed with the objective of validating a predictive model that could be used to optimise the boundary conditions to apply to a painting and assess the risk associated with microclimatic changes for given panel configuration.

Fig. 1 :Fig 2 .

 12 Fig. 1: 2D Finite elements mesh of the Mona Lisa wooden support and applied boundary conditions


Frederic Gruau
email: frederic.gruau@lri.fr

Luidnel Maignan

Distributed execution of automata networks on a computing medium: introducing IfAny machines

Keywords: Distributed computing, spatial computing, simulation, embedding, automata network, computing medium, cellular automata

A computing medium is a set of Processing Elements (PE) homogeneously distributed in space, with connections local in space. PEs are fine grain, and are therefore modeled as Finite State Machine (FSM). In this elementary framework, the interaction between PEs can be defined by a set of instructions, which return a value depending on the neighbor's state. That value is then used as an input to the FSM. This paper studies an instruction set reduced to a single instruction called "IfAny q" that tests IfAny of the neighbors has a given state q. This instruction puts a minimal requirement on hardware: there is no need for addressing channels, communication can be done by local radio broadcasting. An IfAny machine A running on a network tailored for a specific computational task can be executed in parallel on an IfAny medium whose network is fixed and reflects the locality in space. The execution involves an embedding of A's network, and a transformation of A's FSM, adding a 3 states register. We analyse the example of A realizing the addition of n binary numbers. With a carefully chosen network embedding, the resulting parallel execution is optimal in time and space with respect to VLSI complexity. This work demonstrates that IfAny machines can be seen as a rudimentary programming method for computing media. It represents a first step of our long term project which is to realize general purpose parallel computation on a computing medium.

Motivation and review

Spatial computing [START_REF] Dehon | Computing Media and Languages for Space-Oriented Computation[END_REF] models hardware called "computing medium" made of fine grain Processing Elements (PEs), distributed homogeneously in space, and communicating locally between neighbors. Cellular Automata [START_REF] Ilachinski | Cellular Automata: A Discrete Universe[END_REF] (CAs) is an emblematic example. The locality of communication is nice because it enables arbitrary large medium size. However, it also makes it difficult to abstract space away: most of the problems considered usually involves space both as an input and as an output; A typical example is the computation of the discrete Voronoi diagram. Our goal is to achieve more general purpose computing, in order to better exploit the potentially enormous power brought by arbitrary scalable hardware. A first intermediate level of abstraction, is to program a virtual network of communicating agents. In particular, when the agents are restricted to logic gates, the programmed object is simply a circuit, and is often used to prove universality results: For example, for the so called "game of life" universal circuits are designed using gliders for signals [START_REF] Rennard | Implementation of Logical Functions in the Game of Life[END_REF], and for amorphous medium [START_REF] Coore | Botanical computing: a developmental approach to generating interconnect topologies on an amorphous computer[END_REF], Coore builds arbitrary circuits using gradients and particles. In our view, universality alone is not very meaningful, we consider computational qualities including programming expressiveness, space-time performance, and medium scalability. 1-Expressiveness: Our nodes are agents which are not executing a mere logic gate but an arbitrary Finite State Machine (FSM). 2-Time Performance: computation can be pipelined in space. Large distances between a producer and a consumer of a data augment the latency, which is unavoidable due to the locality of communication. But thanks to pipelining, it does not diminish the throughput. 3-Space Performance: our simulation consists in adding a 3-states register to each agents. In contrast universality demonstration for CA often requires so huge configurations that they are purely theoretical. 4-Scalability: we relax the constraint on cristal regularity and global synchronicity. We consider Amorphous medium [START_REF] Abelson | Amorphous computing[END_REF] defined by scattering PEs in space, and using local broadcast radio communication.

In Coore's work the network can be dynamically instantiated, and this is a fundamental requisite for programming. This work reports only purely static network: the medium is assumed to be initially configured by an external entity in order to execute a desired network which remains fixed. The dynamic instantiation will be modeled through self development [START_REF] Gruau | Self developing networks, part 1: the formal system[END_REF][START_REF] Gruau | Self developing networks, part 2: Universal machines[END_REF].

IfAny Distributed Finite State Machine

Computing medium as Quasi-Synchronous Distributed FSM

To force the computation to spread in space, we prefer fine grain over coarse grain medium. This means that the PE's behavior is adequately modeled by a Finite State Machine (FSM). A distributed machine needs a set of instructions I allowing interaction between PEs, it can be modeled as a Moore machine (Σ, Q, δ, I, γ) where the input alphabet Σ is the set of possible values returned by the instructions. Q is the set of states, δ : Q × Σ → Q is the next state function, γ : Q → I is the output function. A PE executes the instruction γ(q) of its current state q ∈ Q, and uses the returned value v ∈ Σ as the input determining the next state δ(q, v).

Unlike generic asynchronous execution in distributed system, the PEs of a computing medium are identical, and therefore run with the same average speed, although they do not necessarily share the same global clock. A simple way to model those two features is to consider the same discrete time for each PE, but skip the update with a small probability, at each time step for each PE. This schedule that we call quasi-synchronous obliges the designer of programs to engineer specific methods if he needs a behavior which is robust with respect to the non determinism introduced by the random skips.

IfAny Machine

The instructions send and receive are often used to exchange messages through point to point communication channels. However, since a computing medium's most important feature is its potential scalability, communicating by radio broadcasting in the local neighborhood is more relevant. IfAny machines takes into account the "rudimentary-ness" of such medium: neither PEs, nor communication channels need to be identified. We will see that a particular sub-class of these machine can indeed be implemented with radio broadcasting.

Definition 1 (IfAny Machine). It is a distributed FSM with a single type of instructions ∃Q which tests ifAny of its neighbors has its state in Q ⊂ Q.

The atomic instruction of an IfAny machine is a test, therefore it returns a boolean value, and Σ = {0, 1}. In this work, we use IfAny machines both for modeling a computing medium and for programming. For modeling hardware, the network is homogeneous, similar to a 2D grid, and each PE has a bounded small number of neighbors. For programming, the PEs are virtual, we call them agents. The structure of the network reflects a specific pattern of communication needed for a particular algorithm, each agent can have arbitrary many neighbors.

When programming agents interconnected in a static network, we test presence of states among the neihgbors for two purposes: 1-the agent is awaiting a particular source signal s ∈ S ⊂ Q to appear; 2-The agent is awaiting for inhibitory states i ∈ I ⊂ Q to disappear. Sources model signal propagation, while inhibitors can delay this propagation, in order to make it deterministic despite the random skips. To increase the expressiveness, we consider macro-instruction written ∃S¬∃I declaring both a set of sources S and inhibitors I. Let Q N be the set of states represented amongst the neighbors, the agent executing the macroinstruction is blocked in the same state until there are no inhibitors Q N ∩ I = ∅ and at least one source Q N ∩ S = ∅. The unblocking of a macro-instruction can be programmed using one state with instruction ∃I, a transition 1 leading to itself, and 0 leading to a states with instruction ∃S. However, this is true only if the sources in the neighborhood are stable. Fortunately, this will be the case for the subset of machines considered in this paper.

Macro instructions define a new machine where Σ = Q instead of {0, 1}. If a single source appears among the neighbors, Q N ∩ S = {s} the transition is called mono-source; and we decide that s is the value returned by the macro instruction, i.e. the source is propagated. For multi-source transitions, the returned value must be computed from Q N ∩ S. We want to keep this computation as simple as possible to be able to execute it in parallel with fine grain PEs. We consider ordered IfAny machines, where an order is defined on Q, and decided that the value returned is the smallest element of Q N ∩ S. A mono-source transition, implements pure communication, merely copying states from neighbors, all the computation has to be done within the PEs. In contrast, a multi-source transition needs to compute a minimum state over arbitrary many neighbors. As an illustrating example, we propose an IfAny machine using only monosource transitions for solving the computing task of adding p numbers using a binary tree of agents. The numbers are fed bits after bits at the leaves, least significant bits first, each branch agent adds two flows of bits representing integers that are intermediate sums. The result is available bits by bits from the agent at the root of the tree. For simplicity, the machine presented here just describes a permanent computing regime: we do not handle the initialisation nor the termination phase of the addition.

An

The agent's state is a 5-uplet (d, n, q, c, r) of registers. The registers c, r store bits used in the computations: and q is a control state. The transition graph has only two control states q = 0 and q = 1 and cycles between the two as indicated by fig. 1. The state q = 0 just collects the first input r = a, while q = 1 collects the second input b, it also updates the carry c ← c+a+b ≥ 2 and simultaneously computes the next output bit r = (c + a + b) modulo 2 .

The registers d, n store an address: d is the distance to the root, it can be stored modulo 3, in order to need only tree states, n is the child number: 0 or 1. Both control states use the same instruction parametrized by state registers d, n, q, which includes one set of sources : S rec(α) and two sets of inhibitors I child ∪ I father . The source is used to collect successively each of the two input bits: S rec (α) = {(d + 1, q, 0, -, α), α = 0, 1}, it reads the result α from child number q at distance d + 1. Inhibition from the father (at distance d -1) is I father = (d -1, -, (n + q) mod 2, -, -), it ensures that a result will be used at least once by preventing to go from q = 0 to q = 1 (resp. from q = 1 to q = 0) if the father is reading (resp. not reading) the agent's output. Inhibition from the children I child = (d + 1, ¬q, 0, -, -) ensures that a result is not used twice by preventing to go to q = i if the i th child still has an already used result. The inhibitions and source cause the pair of control states between a father f and its first child c 0 (resp. second child c 1 ) to follow the 4-cycle: (q c0 , q f ) = (0, 0) → (0, 1) → (1, 1) → (1, 0) (resp. (q c1 , q f ) = (0, 1) → (0, 0) → (1, 0) → (1, 1)). In both cases, a father never updates simultaneously with any of its children and the order of updates is deterministic. Recall that the set of sources present in the neighbors determines which transition is taken, while the inhibitions just delay it. Because of quasi-synchronous execution, the sources for a given transition in one run may come too late or leave too early to achieve their effect in another run. The preceding example shows how inhibitions can implement causal precedence preventing these happenings, leading to a deterministic behavior. More generally: Definition 2. An IfAny machine with given initial configuration, and FSM is source-deterministic, if whenever an agent in state a is source for a neighbor agent in state b , then prec a < next b < next a .

Here, prec a is the transition leading to a while next b is the next transition done after b. prec a < next b implies that a is present before b's transition, and next b < next a that it is still present after, together, it proves that a is present during b's transition. The adder is source-deterministic, the diagram 2 (a) clearly shows that the transitions preceding, (resp. following) the source state for a given transition e are causing, (resp are caused by) e. The definition implies that for each transition, the set of sources which is present at the time of the transition is deterministic, therefore the transition itself is deterministic, and the whole execution is deterministic.

Given a deterministic machine, we can build the execution graph defined as follows: For each agent a, it has one node for each of a's transition, indexed by the transition number. The node a t is labeled by the state taken after the t th transition. For each t, a t , is connected to a t+1 , and a t is connected to a t if a and a are neighbors, and there is a possible configuration of the system, in which agents a and a have done respectively t and t transitions. The execution graph of the addition machine is shown in fig. 2 (b). It compactly represents all the reachable configurations which are connected sub-graphs with exactly one node on each horizontal line.

Implementation of IfAny machines by radio broadcasting.

Our primary concern with medium is to take into account the locality, however, we should not ignore the specific robustness of the IfAny machine allowing to consider communicating by radio broadcast over an anonymous medium. In this case a PE cannot identify the sender of a message. Yet, this rudimentary framework can implement IfAny machines if they are source-deterministic. Each PE broadcasts its state at regular time-interval τ . When a message arrives, the date of arrival is stored together with the message. Messages that arrived more than τ + time ago are discarded. Here, accounts for slight difference in clock frequency. A PE can maintain an image of the set of state present in its neighborhood, and be sure that each neighbor will be represented, up to a timing precision. Source-determinism imposes that the sources which determine a transition are causally before the transition, and their removal is caused by the transition considered. The lack of time precision does not change the fact that every PEs will safely get its sources, and the transition made by each PE will be deterministic.

Execution of IfAny machines on a computing medium

We now want to execute a given programmed virtual IfAny machine P having a specific network of agents (such as a binary tree of adders), on a given computing medium M . Due to the locality of connections in M , the number of links per PEs is upper-bounded by a small constant like 6 for the hexagonal lattice. In contrast the network of P can have arbitrary large degree. The simulation of circuits usually lays out wires for transmitting signals over arbitrary long distances between an operator that produces the signal, and another operator that uses it. We propose a different view in which it is the agent's support which can be arbitrary large in space, so that the support of communicating agents can be directly adjacent in space. For example, if an agent has n neighbors, and has a support without holes, it needs a large support of area O(n 2 ), so that its perimeter has a length of O(n) and can offer enough border length for touching all the n neighbors. The transmission of signals over wires will be replaced by the propagation of new state within a support. Definition 3. An embedding of a network (V P , E P ) in another (V M , E M ), is a mapping φ : V P → P(V M ) such that (i) ∀a ∈ V P , φ(a) is connected, (ii) ∀a 1 , a 2 ∈ V P , φ(a 1 ) ∩ φ(a 2 ) = ∅ (iii) a 1 and a 2 are adjacent ⇔ φ(a 1 ) and φ(a 2 ) are adjacent

We will also consider "quasi-embedding" verifying only condition (ii) and (iii). The complement of φ(V P ) is called the background. Its role is to separate the different supports. Note that a completely connected network (cliques) are not good for targets, they can embed only cliques, because the supports will always be adjacent. Grids with 3 dimensions can embed any networks if they are large enough. Fig. 3 (a) represents an embedding of a binary tree reduced to three nodes, within a planar 2D networks. Adjacency in such a network is equivalent to adjacency in the underlying Euclidian 2D space, the underlying network can thus be omitted in the figure .  The state of an agent a of P is distributed on the PEs of a connected component φ(a) called its support. The coordination of those PEs needs some additional mechanisms. First a PE must be able to identify whether an adjacent PE belong to the same support or not. We define the equivalence relation ∼ by q 1 ∼ q 2 if q 1 and q 2 can be taken by the same agent. The class of q is noted [q]. By definition, all the states that can be taken by an agent belong to the same state class, which is the agent's state class. Definition 4. An IfAny machine is called state-separated, if any two neighbors always have distinct state class.

For example, the addition machine has six state classes, one for each value of d and n. Any two neighbors always have distinct value of d, so they are stateseparated. Now, let us consider a computing medium M also executing the macro IfAny instructions with sources and inhibitors. We look for a restricted form of universality of M that could be called "network universality": using the same fixed network M can simulate a machine P having an arbitrary embeddable network, but the FSM of M is built according to the FSM of P . Theorem 1. An Ifany medium can simulate any embeddable, source-deterministic, state-separated IfAny machine. It needs only 3 times more states.

Proof: Let P be the machine to simulate including an FSM A P and a network, and M the medium along with φ an embedding of P 's network, into M . The FSM A M governing the medium's evolution is obtained from A P by "enriching" the instructions and adding a fixed set of transitions. The proof run on two subsections, the first one considering the simpler case of mono-source transitions, and the second one dedicated to multi-source transitions

Mono-source transition

Because of locality, when simulating a transition for a PE p of P , the PEs of the support φ(p) cannot acquire the new state simultaneously. For a mono-source transition, only those PEs which are adjacent to the support of the source know what is the new state, and will be able to take it first. Once "born", the new state must then be propagated throughout the support. A PE of M in state q needs to identify whether a neighbor of the same support carries a new state that should be propagated. To this end, we add an "age" component to the state, which is simply the number of transitions already done by the agent, and assign the additional sources S {q,t} = [q]×{t+1}. The age of two PEs belonging to distinct support is not correlated, therefore, the original sources S (resp. inhibitors I), must be replaced by S × N (resp. I × N ) in order to take into account all the possible ages. The "death" of the previous state happens when the new state has invaded the whole support. Note that the presence of an inhibitor does not impede the new state to be born and propagate, but it does impede the previous state to die. The following summarizes the transition function δ M with two cases for birth and propagation. δ M ((q, t), (s, t )) = (δ P (q, s), t

+ 1) if ¬s ∈ [q] (birth) (s, t + 1) if s ∈ [q] ∧ t = t + 1 (propagation) (1)
Several state transition waves of increasing age can be pipelined through the support of an agent. We assign inhibitors [q] × {t -1} in order to isolates the wave aged t + 1 from the t -1, and bound by one the age variation between two neighbor PEs of the same support. The age is needed only for distinguishing t+1 from t -1, knowing the age modulo 3 is obviously sufficient for this purpose, therefore only t modulo 3 is stored. This explains why only three times as many states are necessary for the execution. In summary, the rule 2 shows how to rewrite a mono-source instruction q, ∃S¬∃I of

A P into an instruction of A M q, ∃S¬∃I ⇒ (q, t), ∃S {q,t} ∪ (S × N) ¬∃I {q,t} ∪ (I × N) (2)
In order to prove the correctness, we need to characterize what are the possible configurations on M . Due to pipelined state transition waves, a configuration can simultaneously embed several snapshots of P 's execution, as shown in fig. 3 (b). We prove by recurrence the following hypothesis H(k): After k time steps there exist a quasi-embedding φ k from a sub-graph G k of the execution graph, such that: (i) for each agent a, G k contains a single continuous segment a t , a t+1 , . . . , a t+k whose image by φ k partition the support φ(a). (ii) the state of PEs in φ k (a t ) is (q, t) where q is the label of a t .

The hypothesis H(0) is trivial at the start; If H(k -1) is true, consider the configuration after k -1 time steps, and a given PE, not in the background, with state (q, t), and let (q , t ) be the state of a neighbor p . Condition (iii) of def. 3 is equivalent to the following "neighbor-invariant property": either p is in a different agent's support, and q is a possible neighbor state in the simulated execution, or it is in the same support, and q ∼ q and |t-t | = 1. In all cases, the transition of rule 2 is defined and allow to define G k and φ k from G k-1 and φ k-1 by adding, or deleting nodes and connections; The birth (resp. death) event lead to the adding (resp. deleting) of nodes. Propagation events create (resp. deletes) connection when the border to a neighbor is reached, (resp, totally invaded). A border is usually not invaded in one time step, as a result, a connection moves from G k-1 to G k in a "worm like" manner, it is duplicated first, before being deleted. Fig. 4. Race condition: if the state (q = 1, t = 2) of the father propagates faster than its source in child0 (q = 0, t = 1), it may become adjacent to the previous state of the source (q = 1, t = 0) which is forbidden.

G k verifies (i) by "Reductio ad absurdum": if (i) was not true an entire intermediate component aged t would have updated, including PEs adjacent to the t -1 component, which is forbidden by the inhibition [q] × {t -1}. In order to fullfil condition (iii) of def. 3 for φ k we need to check the neighbor-invariant property for the new configuration. A problem happens if the next state (q , t+1) propagates faster than its source (s, t ) does, some PEs with state (q , t + 1) can become adjacent to PEs of the source's previous state (s , t -1), whereas s is never adjacent to q . This is illustrated in fig. 4. Due to source-determinism, the transition from s to s causes the transition from q to q . The solution is to apply a prior transformation of S by adding s as an additional inhibitor of q, which does not modify semantic of S, but will prevent (q , t + 1) to propagate to a PE adjacent to (s , t -1) in the transformed system.

The mapping φ k is a quasi-embedding, which means that it does not check condition (i) of definition 3. The support of a wave may be unconnected because birth events can happen independently throughout the boarder. assign the additional sources S min = {([q], r, t), r = 1 . . . k} to the sources S R1 of rule 2. The input of a branch PE will be the local minimum by using the ordering: ([q], r, -) < (s r , -) < ([q], r + 1, -). In fact, ([q], r, -) and (s r , -) are equivalent with respect to the next state, so δ M ((q, t), ([q], r, t)) = δ M ((q, t), (s r , -)). This next state will be ([q], r, t) for branch PE. However, for the PE at the root of T , since r = r min , the next state can directly be the final new state (δ P (q, s rmin ), t + 1). Because of its higher age, this new state will thereafter be propagated by PE throughout the support which are in in a rank state (c, r, t). We reuse the propagation case of rule 1 by regrouping the first two components of rank states: (c, r, t) = ((c, r), t).

A branch node PE should compute its minimum only if all its children have done so. We need to assign an additional inhibitors I not ready to prevent too early computation. In summary, the following rule 3 shows how to rewrite a multi-source instruction q, ∃s i=1...k ¬∃I of A P into an instruction of A M . q, ∃s i=1...k ¬∃I ⇒ (q, t)∃S R1 ∪ S min ¬∃I R1 ∪ I not ready

Let us now study how to implement the tree T , and compute I not ready . First, note that the operation a, b → min(a, b) is idempotent: min(x, x) = x, a value may be used more than once in the computation of the global minimum, and a spanning Direct Acyclic Graph (DAG) can be used instead of a spanning tree, as long as it has a single root. A simple way to implement a DAG is to designate a leader PE in each support, and compute the hop count d to the leader using a separate distance layer on the medium. The relation between father and children is determined by successive distance values. The tree root is the leader itself. The inhibitors I not ready are agents on the same support, at greater distance, who do not carry ranks.

The computation of the hop count can be done in bounded state using a separate distance layer on the medium, the method is presented in details in [START_REF] Maignan | Integer gradient for cellular automata: Principle and examples[END_REF], it also work in a dynamic environment, the resulting spanning DAG will then be automatically updated if the support and/or the leader move during execution. Note that however, the method assumes a synchronous update, it can nevertheless always be made to work on the quasi synchronous medium using standard synchronisation algorithms. The time cost will be bounded since the number of neighbors is upper bounded in a medium.

Optimality of the addition machine's execution on a 2D medium

We apply our transformation on the FSM of the addition machine which add n binary numbers, and execute it on a 2D computing medium. We evaluate the performance with respect to the VLSI complexity which states that moving a bit over one unit of space costs one unit of time The modified FSM simply needs an additional register to store the age modulo 3. The network to be simulated is a binary tree, where each node does an addition, bit by bit, and store the intermediate carry. The transition are mono-source and can be pipelined. The frequency at which the bits of the result are produced is the inverse of the the spatial distance between the support of two children. If that distance is constant, (independent of n), the frequency will also be constant. It is possible to map a binary tree with constant distance of only one PE between brothers, by encapsulating membranes, the membranes of the father contains the two membranes of the children as shown in fig. 6. Usually, the inputs are fed on the boarder of a circuit, but this optimised mapping forces the inputs to be fed directly at the right PEs within the medium. On the other hand, the output will be available throughout the whole boarder. The space needed is O(n) and the latency, which is the length of the smallest path from input to output (leave to root) is O( √ n). In summary, the transformation produces an asynchronous cellular automata rule that does a real computation in optimal time and space.

∃Fig. 1 .

 1 Fig. 1. Implementation of the IfAny adder, where α is a name used to bind the operand.

Fig. 2 .

 2 Fig. 2. The IfAny addition machine's execution shown on 3 agents (a) Space time diagram showing inhibitions (dotted arrows) and sources (plain arrows). (b) Execution graph with a reachable configuration in bold

Fig. 3 .

 3 Fig. 3. Mapping on the medium. (a) Embedding the agent network, (b) Embedding a slice of the execution graph.

Fig. 6 .

 6 Fig.6. 2D embedding of a binary tree of 2 k leaves, in O(2 k ) space, with constant euclidian distance between brother nodes. We color distinctly d = 0, d = 1 and d = 2. The bold path indicate the trajectory followed by one bit from leave to root.

If two sources are present, one could be suppressed without impeding the transition to take place since one source is sufficient for triggering a transition.

For the addition machine there is 6 state classes, but setting c = d is sufficient to distinguish supports. A practical use of multiple source is to consider only two sources encoding a 0 and a 1, and compute a logical AND of arbitrary many neighbors. In this case, kmax = 2. Thus 18 extra states are needed.

Figure 5 illustrates the propagation of state waves in the support of the root node of the addition machine. The spatial period p of the waves is equal to the distance between both children, and is inversely proportional to the throughput. If the host is not reading the output, the computation can still go on on the medium, the space behaves as a buffer that stores the bits of the result.

Multi-source transition.

Consider now an agent a of P in state q, age t, doing an instruction potentially having k multiple sources s 1 , . . . , s k , where the index r = 1 . . . k is the source rank i.e. s r < s r+1 . We note k max the maximal value of k, for all the possible multisource transition of any agents. The agents in the support of p must compute the minimum rank among the sources actually present, because it is this value that will determine the transition's input.

Consider a PE p on the border of the support of a, it can compute the ranks of the sources present in its immediate local neighborhood and find out what is the minimum rank r 0 , however this is only a local minimum. The computation of the global minimum over the ranks in the whole border needs a centralized processing: the ranks must be input at the leaves of a tree T and then propagated by letting each branch PE computes the minimum of its children. The global minimum r min will then be available at the PE corresponding to the root of T . We will first assume the existence of such a tree T embedded in the support, whose leaves span the whole border, and then show how to install it.

We need to introduce additional rank states in order to store the rank while they are being propagated . Those states have three components: (c, r, t) where c ranges over the possible state class, and is needed to identify the support of the simulated agent a. r ∈ {1 . . . k max } is a possible rank and t ∈ {0, 1, 2} is the age. If the network uses α state classes, the total supplementary number of rank states is α * k max * 3 2 .

Agents in the boarder memorise their local minimum r 0 by taking the state ([q], r 0 , t). The ranks are then propagated within the support through T , We