Quang-Huy Nguyen

A pseudo-local property of gravity water waves system

By proving a weighted contraction estimate in uniformly local Sobolev spaces for the flow of gravity water waves, we show that this nonlocal system is in fact pseudo-local in the following sense: locally in time, the dynamic far away from a given bounded region has a small effect on that region (again, in a sense that we will make precise in the article). Our estimate on the flow also implies a new spatial decay property of the waves. To prove this result, we establish a paradifferential calculus theory in uniformly local Sobolev spaces with weights.

1. Introduction 1.1. The problem. We consider an incompressible, irrotational, inviscid fluid moving in a domain Ω underneath a free surface described by η and above a bottom described by a given function η * , which is assumed to be bounded and continuous. Namely, (1.1) Ω = {(t, x, y) ∈ [0, T ] × R d × R : η * (x) < y < η(t, x)}.

We also denote by Σ the free surface and by Γ the bottom, Σ = {(t, x, y) ∈ [0, T ] × R d × R : y = η(t, x))},

Γ = {(x, y) ∈ R d × R : y = η * (x)}.
The velocity filed v admits a potential φ : Ω → R such that v = ∇ x,y φ and ∆ x,y φ = 0 in Ω. We introduce the trace of the potential on the surface ψ(t, x) = φ(t, x, η(t, x))

and the Dirichlet-Neumann operator

(1.2) G(η)ψ = 1 + |∇ x η| 2 ∂φ ∂n    Σ = (∂ y φ)(t, x, η(t, x)) -∇ x η(t, x) • (∇ x φ)(t, x, η(t, x)).
Then (see [START_REF] Lannes | Water waves: mathematical analysis and asymptotics[END_REF]) the gravity water waves system in the Zakharov/Craig-Sulem formulation reads as follows

(1.3)    ∂ t η = G(η)ψ, ∂ t ψ = - 1 2 |∇ x ψ| 2 + 1 2 (∇ x η • ∇ x ψ + G(η)ψ) 2 1 + |∇ x η| 2 -gη
where g is the acceleration of gravity.

Following [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF] we shall consider the vertical and horizontal components of the velocity

The author was supported in part by Agence Nationale de la Recherche project ANA É ANR-13-BS01-0010-03.

on the free surface as unknowns which can be expressed in terms of η and ψ:

(1.4) B = (v y )| Σ = ∇ x η • ∇ x ψ + G(η)ψ 1 + |∇ x η| 2 , V = (v x )| Σ = ∇ x ψ -B∇ x η.
Recall also that the Taylor coefficient a = -∂P ∂y   Σ can be defined in terms of η, ψ, B, V only (see §4.2 in [START_REF] Alazard | On the Cauchy problem for gravity water waves[END_REF] and §4.3.1 in [START_REF] Lannes | Water waves: mathematical analysis and asymptotics[END_REF]).

The (local) well-posedness theory for gravity water waves (under the formulation (1.3) or the others) in Sobolev spaces H s (R d ) has been studied by many authors, for example Yosihara [START_REF] Yosihara | Gravity waves on the free surface of an incompressible perfect fluid of finite depth[END_REF], Wu [START_REF] Wu | Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent[END_REF][START_REF] Wu | Well-posedness in Sobolev spaces of the full water wave problem in 3-D[END_REF], Lannes [START_REF] Lannes | Well-posedness of the water waves equations[END_REF]; we refer to the recent book of Lannes [START_REF] Lannes | Water waves: mathematical analysis and asymptotics[END_REF] for a comprehensive survey of the subject. In these works, the waves were assumed to be of infinite extend (and vanish at infinity), that is, there is no restriction on the horizontal direction. However, in reality water waves always propagate in some bounded container (a lake, an ocean, etc) and hence there will be contacts at the "vertical boundary" of the container. A natural question then arises: (Q) can we justify the R d -approximation? More precisely, if (1.3) is a good model then it has to satisfy in certain sense the following property: the dynamic at "infinity" has a small effect on bounded regions. Since (1.3) appears to be nonlocal (due to the presence of the Dirichlet-Neumann operator) it is not clear that the above replacement at "infinity" is harmless. We should mention that in the special case of a canal or a rectangle basin where the walls are right vertical, the local theory was considered by Alazard-Burq-Zuily [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF], Kinsey-Wu [START_REF] Kinsey | A Priori Estimates for Two-Dimensional Water Waves with Angled Crests[END_REF], Wu [START_REF] Wu | A blow-up criteria and the existence of 2d gravity water waves with angled crests[END_REF]. Our goal in the present paper is to give the following answer to question (Q). Considering a bounded reference domain, we shall prove that in some sense, far away from this reference domain, the dynamic there has a small effect on the reference domain, and the farther it is the smaller the effect is. In other words, this proves that the gravity water waves system enjoys the "pseudo-local property" (the terminology "pseudo" will be clear in our explanation below).

Main results.

We recall first the definition of uniformly local Sobolev spaces (or Kato's spaces) introduced by Kato in [START_REF] Kato | The Cauchy problem for quasi-linear symmetric hyperbolic systems[END_REF].

Definition 1.1. Let χ ∈ C ∞ (R d ) with supp χ ⊂ [-1, 1] d , χ = 1 near [-1 4 , 1 4 ] d such that (1.5) q∈Z d χ q (x) = 1, ∀x ∈ R d , χ q (x) = χ(x -q). For s ∈ R define H s ul (R d ) the space of distributions u ∈ H s loc (R d ) such that u H s ul (R d ) := sup q∈Z d χ q u H s (R d ) < +∞.
This definition is independent of the choice of χ ∈ C ∞ 0 (R d ) satisfying (1.5) (see Lemma 7.1 in [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF]). Let us now define the classes of weights that we will consider. Definition 1.2. 1. We define the class W of acceptable weights to be the class of all functions w : R d → (0, ∞) satisfying the following conditions:

(i) r 1 := ∇w -1
w -1 and r 1 := ∇w w belong to C ∞ b (R d ), where w -1 (x) = 1/w(x),

(ii) for any C 1 > 0, there exists C 2 > 0 such that for any x 0 ∈ R d , there hold w(x) ≤ C 2 w(x 0 ) and w(x

) -1 ≤ C 2 w(x 0 ) -1 ∀x ∈ R d , |x -x 0 | ≤ C 1 ,
2. If w ∈ W and there exist ≥ 0, C > 0 such that for any x, y ∈ R d we have w(x)w -1 (y) ≤ C x -y then we say that w ∈ W po ( ).

3. If w ∈ W and there exist ≥ 0, C > 0 such that for any x, y ∈ R d we have w(x)w -1 (y) ≤ C exp(ρ x -y ) then we say that w ∈ W ex ( ).

Example 1.3. For any t, s ∈ R, C > 1, the functions x s , ln(C + |x| 2 ) belong to W po , and the functions e t x , e t x x s belong to W ex but not to any class W po if t = 0. See Remark A.4 for further remarks. Notation 1.4. 1. χ denotes a function in C ∞ 0 (R d ) such that χ = 1 on the support of χ in definition 1.1. For every k ∈ Z d , we also Define for x ∈ R d , χ k (x) := χ(x -k). 2. We set for all σ ∈ R,

H σ ul = H σ+ 1 2 ul (R d ) × H σ+ 1 2 ul (R d ) × H σ ul (R d ) × H σ ul (R d ), W σ = W σ+ 1 2 ,∞ (R d ) × W σ+ 1 2 ,∞ (R d ) × W σ,∞ (R d ) × W σ,∞ (R d ).
Denote also by U = (η, ψ, B, V ) the unknown of system (1.3) and by U 0 = (η 0 , ψ 0 , B 0 , V 0 ) its initial value.

The Cauchy theory proved in [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF] reads as follows Theorem 1.5. Let s > 1 + d 2 and U 0 ∈ H s ul with

(1.6) inf x∈R d (η 0 (x) -η * (x)) ≥ 2h > 0, inf x∈R d a(0, x) ≥ 2c > 0.
Then there exists T > 0 such that the Cauchy problem for (1.3) with datum U 0 has a unique solution

U ∈ L ∞ ([0, T ], H s ul ) ∩ C 0 ([0, T ], H r ul ), ∀r < s and (1.7) inf t∈[0,T ] inf x∈R d [η(t, x) -η * (x)] ≥ h, inf t∈[0,T ] inf x∈R d a(t, x) ≥ c.
Moreover, for given h, c > 0 the existence time T can be chosen uniformly for data belonging to a bounded set of H s ul . Conditions (1.6) mean that initially, the free surface is away from the bottom and the Taylor coefficient is positively away from 0. Then the conclusion (1.7) asserts that these properties are propagated by the waves, locally in time. We shall always consider in the sequel solutions of (1.3) obeying these properties, which for the sake of simplicity is denoted by (1.8) P s,T (h, c) := {U = (η, ψ, B, V ) ∈ L ∞ ([0, T ], H s ul ) solution to (1.3), satisfying (1.7) and U | t=0 satisfies (1.6)}.

Our main result concerning the solution map of the gravity water waves is stated in the following theorem.

Theorem 1.6. Let s > 1 + d 2 , T > 0 and two positive constants h, c. Then for every w ∈ W po ( ), ≥ 0 there exists a function K : R + × R + → R + nondecreasing in each argument, such that

(1.9) w(U 1 -U 2 ) C([0,T ],H s-1 ul ) ≤ K(M 1 , M 2 ) w(U 1 -U 2 )| t=0 H s-1
ul for all U 1 , U 2 ∈ P s,T (h, c), provided that the right-hand side is finite, where

M j := U j L ∞ ([0,T ],H s ul ) < +∞, j = 1, 2.
As a consequence, we have Corollary 1.7. Let s > 1 + d 2 ; h, c > 0 and A be a bounded set in H s ul . Denote by T the uniform existence time of solutions to (1.3) in P s,T (h, c) with data in A. Then there exists 0 < T 1 ≤ T such that the following property holds: for every w ∈ W po ( ), ≥ 0 one can find a constant C > 0 such that

(1.10) w(U 1 -U 2 ) C([0,T 1 ],H s-1 ul ) ≤ C w(U 1 -U 2 )| t=0 H s-1 ul ,
for all U j ∈ P s,T (h, c) with U j | t=0 ∈ A and provided that the right-hand side is finite.

In Corollary 1.7 if we take U 2 | t=0 = 0 and use the Sobolev embeddings (see Proposition 2.2, [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF])

H r ul (R d ) → W r-d 2 ,∞ (R d ), r > d 2 , r - d 2 / ∈ N,
we derive Corollary 1.8. Let s > 1 + d 2 and h, c > 0. Then for any bounded set A in H s ul , there exists a time T > 0 such that: for every w ∈ W po ( ), ≥ 0 one can find a constant C > 0 such that

(1.11) wU C([0,T ],H s-1 ul ) ≤ C wU | t=0 H s-1 ul for all U ∈ P s,T (h, c) with U | t=0 ∈ A and provided that the right-hand side is finite. Moreover, if s ≥ r > 1 + d 2 and r -d 2 / ∈ N it follows that (1.12) wU C([0,T ],W r-1-d 2 ) ≤ C wU | t=0 H s-1 ul . Remark 1.9. If w ∈ C ∞ b (R d
) then the right-hand sides of (1.9), (1.10), (1.11), (1.12) are automatically finite.

Remark 1.10. Persistence properties in weighted spaces have been studied extensively for asymptotic models of water waves in different regimes: (generalized) Korteweg-de Vries equation equations, Schrödinger equations, Benjamin-Ono equation, Camassa-Holm equation,...We refer to the works of Brandolese [START_REF] Brandolese | Breakdown for the Camassa-Holm Equation Using Decay Criteria and Persistence in Weighted Spaces[END_REF], Bona-Saut [START_REF] Lloyd | Dispersive Blowup of Solutions of Generalized Korteweg-de Vries Equations[END_REF], Fonseca-Linares-Ponce [START_REF] Fonseca | On persistence properties in fractional weighted spaces[END_REF], Nahas-Ponce [START_REF] Nahas | On the persistence properties of solutions of nonlinear dispersive equations in weighted Sobolev spaces. Harmonic analysis and nonlinear partial differential equations[END_REF], Ni-Zhou [START_REF] Ni | A new asymptotic behavior of solutions to the Camassa-Holm equation[END_REF].

Remark 1.11. It is natural to ask if the results in Theorem 1.6, Corollary 1.7, Corollary 1.8 hold for weights with exponential growth. For example, Theorem 1.6 with w = e -λ|x| , λ > 0 would give a strong pseudo-local property of gravity water waves. As we shall explain, the proof of our results can be divided into two parts: first, a study of the Dirichlet-Neumann operator in weighted spaces, and the second part makes use of a paradifferential machinery in weighted Sobolev spaces to paralinearize and symmetrize the system. For the first part, we are able to prove bound estimates for the Dirichlet-Neumann operator in the presence of "exponential weights" in the class W ex (see Proposition 2.14 below). However, for the (pseudo-) para-differential calculus, we have to restrict to "polynomial weights" in the class W po due to the fact that, in general, the kernel of a pseudodifferential operator only decays polynomially (see paragraph 1.3 3. and the proof of Proposition A.5).

Remark 1.12. It would of course be more satisfactory if the results could be formulated in terms of the derivatives of ψ since ψ is the trace of the velocity potential (on the free surface) and hence is determined up to additive constants. It should be possible to do so, modulo more technical complications; in particular, a Cauchy theory in Kato's spaces involving only regularity of ψ in homogeneous spaces. We refer to a relating result of Lannes in [START_REF] Lannes | Well-posedness of the water waves equations[END_REF] and the references therein.

Remark 1.13. Our proofs rely on the known Cauchy theories of Alazard-Burq-Zuily in [START_REF] Alazard | On the Cauchy problem for gravity water waves[END_REF], [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF]. To avoid a loss of 1 2 derivatives, the authors assumed that initially, the trace of velocity (B, V ) are 1 2 derivatives smoother than the natural threshold suggested by formula (1.4) (see also Remark 1.4,[START_REF] Alazard | On the Cauchy problem for gravity water waves[END_REF]) . Another way of avoiding this loss of derivatives can be found in Theorem 4.16, [START_REF] Lannes | Water waves: mathematical analysis and asymptotics[END_REF] where instead of directly imposing regularity condition on ψ, the author works with the "good unknown" ψ ≈ ψ -Bη.

1.3. Interpretation of the results. 1. The Zakharov system (1.3) appears to be nonlocal, which comes from the fact that the Dirichlet-Neumann operator defined by (1.2) is nonlocal. This can be seen more concretely by considering the case of fluid domain with infinite depth (i.e. Γ = ∅) and free surface at rest (i.e. η = 0). Then, the Dirichlet-Neumann operators is G(0) = |D x |. However, Corollary 1.7 shows that the system is in fact still weakly local as explained below.

Take

s > 1 + d 2 .
Let's restrict ourselves to a bounded set A of H s ul and suppose that we are observing a bounded domain, which by translation can be assumed to be centered at the origin, say O = B(0, 1). Let U 0,1 , U 0,2 be two data in A such that they are identical in a ball B(0, R) and have difference in H s-1 ul of size 1 outside this ball, where R > 1 is a given distance. Take a "window" φ around our observation region O, that is, φ ∈ C ∞ 0 (B(0, 3/2)) and φ ≡ 1 in O. Then by the estimate (1.10) we have for some T = T (A) > 0 and any N > 0

φ(U 1 -U 2 ) C([0,T ],H s-1 ul ) ≤ C N • -N (U 1 -U 2 ) C([0,T ],H s-1 ul ) ≤ C N,A • -N (U 0,1 -U 0,2 ) H s-1 ul ≤ C N,A R -N . -1 R -R 1 1 U 0,1 t = 0 U 0,2
1 1 R N U 2 -1 R N U 1 1 0 < t < T -1
Therefore, under the dynamic governed by system (1.3), a difference of size 1 outside the ball B(0, R) of initial data leads to a difference of size R -N of two solutions in the bounded domain B(0, 1) (see the figures above). When R → +∞, the difference of two solutions tends to 0 at a rate faster than any polynomial. In other words, to some extent, what happens far away has small effect on a given bounded region; moreover this effect becomes smaller and smaller when the distance increases to +∞. This gives us a weakly local property of gravity water waves. This property is indeed dictated by the polynomial decay off the diagonal of the kernel of differential operators in suitable classes, as we shall explain in point 3. below. 2. As a consequence of Corollary 1.8, the estimate (1.12) with λ > 0 provides a spatial decay property for solutions. In classical Sobolev spaces, solutions always vanish at infinity. On the other hand, Theorem 1.5 gives the existence of solutions in Kato's spaces which can be neither decaying nor periodic. The estimate (1.12) however gives a conclusion for the intermediate case: as long as the datum decays at some rate which is at most algebraic in Kato's space, the solution decays also, and moreover, at the same rate.

3. Let us explain why the class W po of weight that are at most polynomial growth is a reasonable choice in our results. For this purpose, a good way is to look at the linearization of system (1.3) around the rest state (η, ψ) = (0, 0) (take g = 1 and the flat bottom {y = -h, h > 0})

(1.13) ∂ t η -|D x | tanh(h|D x |)ψ = 0, ∂ t ψ + η = 0 or equivalently, with u := η + i|D x | 1 2 tanh(h|D x |)ψ, ∂ t u + i|D x | 1 2 tanh(h|D x |)u = 0.
Given a datum u 0 at time t = 0, this linearized equation has the explicit solution

u(t, x) = p(t, D x )u 0 ,
where the symbol p reads p(t, ξ) = e -it|ξ| √ tanh(h|ξ|) . Then for w ∈ W po ( ), ≥ 0 and 0 < T < ∞, we seek for the following estimate (1.14) wu C([0,T ];H s ul ) ≤ C wu 0 H s ul . Due to the presence of tanh(h|ξ|) we have that p satisfies

(1.15) |∂ α ξ p(t, ξ)| ≤ C α (1 + ξ) -1 2 |α| , ∀α ∈ N d , (t, ξ) ∈ [0, T ] × R d ,
which is usually denoted by p ∈ S 0 1 2 ,0 . An adaptation of the proof of Proposition A.5 then implies the estimate (1.14). Indeed, for simplicity let us consider s = 0. By writing χ q = χ q χ q (recall Notation 1.4) we need to show for any fixed k ∈ Z d (1.16)

A k := q wχ k p(t, D x )χ q w -1 : L 2 (R d ) → L 2 (R d )
with norm bounded uniformly in k. Using the classical pseudo-differential theory, it suffices to prove (1.16) for q satisfying |q -k| ≥ M for fixed M > 0 (cf estimate (A.2)). Due to the presence of χ k it suffices to prove

A k : L 2 (R d ) → L ∞ (R d ).
To this end, we call

(1.17) K(x, y) = (2π) -d R d
e i(x-y)ξ p(t, ξ)dξ the kernel of the pseudo-differential operator p(t, D x ) then the kernel of A k reads

H k (x, y) = |q-k|≥M w(x)χ k (x)K(x, y)χ q (y)w(y) -1 .
The Cauchy-Schwartz inequality implies

A k v L ∞ ≤ sup x H k (x, •) L 2 v L 2 .
By writing χ q = χ q χ q it suffices to show that H k L ∞ x,y is bounded by some constant independent of k. Indeed, remark that by choosing M large enough, on the support of χ k (x)χ q (y) (in the expression of H k ) we have |x -y| ≥ δ|k -q| for some δ > 0. Therefore, one can multiply both side of (1.17) by (x -y) γ , γ ∈ N d , integrate by parts and take into account the decay property (in ξ) (1.15) of p to derive

(1.18) |H k (x, y)| ≤ C N |q-k|≥M |χ k (x)w(x)χ q (y)w(y) -1 | k -q N , ∀N ∈ N.
Observe that for any w ∈ W the absolute value of the numerator of each term in the above series is bounded by

Cw(k)w(q) -1 .
Consequently, for the series in (1.18) to be convergent, it is reasonable to choose the weights that satisfy for some λ > 0

w(k)w(q) -1 k -q λ , ∀k, q ∈ Z d .
Then by choosing N large enough the series in (1.18) converges to some constant independent of k as desired. This argument suggests heuristically that the finiteness of the fluid depth is likely to be necessary since otherwise, the symbol p become p(t, ξ) = |ξ| 1/2 which is singular at 0 and does not belong to a good class of symbols. The local property of the system is closely related to the finite propagation speed property. Indeed, the plane waves u(t, x) = e i(x•k-ω(k)t) , ω(k) = |k| 1/2 are solutions to u t + i|D x | 1/2 u = 0. These plane waves propagate at (group) velocity ∇ k ω(k), which is unbounded when the wavenumber |k| tends to 0. In contrast, for the finite depth case (say h), the dispersion relation reads ω(k) = |k| tanh(h|k|) and thus the speed of propagation is bounded over all wavenumber k.

Remark 1.14. In the theory of pseudo-differential calculus, the terminology pseudo-local refers to the following property: if T is a pseudo-differential operator then the singular support of T u is contained in the singular support of u. The proof of this result makes use of the fact that: the kernel of T is C ∞ off the diagonal (x, x) in R d × R d . This in turn stems from the decay property of the symbol of T . The "pseudo-local property" in our result as explained above, also stems from the decay of the kernel of a pseudo-differential operator. However, such a decay is then translated not into the regularity (in term of the singular support) but the persistence in weighted spaces.

1.4. Plan of the proof. To prove Theorem 1.6 we follow essentially the scheme in [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF]. The first task is to adapt the paradifferential machinery to Kato's spaces with weights. This is done in Appendix A, which can be of independent interest for other studies in this framework. Having this in hand, compare to [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF] (and also [START_REF] Alazard | On the Cauchy problem for gravity water waves[END_REF]) the main ingredient for the proof of Theorem 1.6 reduces to the study of bound estimates, paralinearization and contraction estimate for the Dirichlet-Neumann operator. These are done in section 2 and 3 below, respectively.

Burq for his great guidance with many fruitful discussions and constant encouragement during this work. I sincerely thanks Prof. Claude Zuily for interesting discussions. Finally, I thanks the referees for proposing many valuable suggestions that helped improve both the content and the presentation of the manuscript. (2.1)

Ω = {(x, y) ∈ R d+1 : η * (x) < y < η(x)}
where η * is a fixed bounded continuous function on R d and η ∈ W 1,∞ (R d ). We assume that Ω contains a fixed strip

(2.2) Ω h := {(x, y) ∈ R d+1 : η(x) -h ≤ y < η(x)}.
2.1.1. Straightening the boundary. We recall here the change of variables introduced in [START_REF] Alazard | On the Cauchy problem for gravity water waves[END_REF] (see section 3.1.1) to flatten the domain with free boundary (which is in turn inspired by Lannes [START_REF] Lannes | Well-posedness of the water waves equations[END_REF]). Consider the map (x, z)

→ (x, ρ(x, z)) from Ω := R d × (-1, 0) to Ω h determined by (2.3) ρ(x, z) = (1 + z)e δz Dx η(x) -z e -(1+z)δ Dx η(x) -h if (x, z) ∈ Ω. If η ∈ W 1,∞ (R d
) and δ is small enough this map is a Lipschitz-diffeomorphism from Ω to Ω h and moreover, ∂ z ρ ≥ c 0 > 0 (see Lemma 3.6, [START_REF] Alazard | On the Cauchy problem for gravity water waves[END_REF]).

Notation 2.2. For any function f defined on Ω, we set

(2.4) f (x, z) = f (x, ρ(x, z)) then (2.5)        ∂f ∂y (x, ρ(x, z)) = 1 ∂ z ρ ∂ z f (x, z) := Λ 1 f (x, z) ∇ x f (x, ρ(x, z)) = ∇ x f - ∇ x ρ ∂ z ρ ∂ z f (x, z) := Λ 2 f (x, z). 2.1.2. Definition of the Dirichlet-Neumann operator G(η). Let ψ ∈ H 1 2 ul (R d ), we recall how G(η)ψ is defined (section 3.1, [1]). For every q ∈ Z d , set ψ q = χ q ψ ∈ H 1 2 (R d ) then one can find ψ q ∈ H 1 (Ω) such that ψ q | y=η(x) = ψ q (x) and for some F : R + → R + , (i) supp ψ q ⊂ {(x, y) : |x -q| ≤ 2, η(x) -h ≤ y ≤ η(x)} (ii) ψ q H 1 (Ω) ≤ F( η W 1,∞ (R d ) ) ψ q H 1 2 (R d ) . Let u q ∈ H 1,0 (Ω) := v ∈ H 1 (Ω)
, v| Σ = 0 be the unique variational solution, to equation ∆ x,y u q = -∆ x,y ψ q , which is characterized by

(2.6) Ω ∇ x,y u q (x, y) • ∇ x,y θ(x, y)dxdy = - Ω ∇ x,y ψ q (x, y) • ∇ x,y θ(x, y)dxdy for all θ ∈ H 1,0 (Ω). The series u := q∈Z d u q is then convergent in H 1,0 ul (Ω) := v : sup q∈Z d χ q v H 1 (Ω) < +∞ and v| Σ = 0 .
Finally, Φ := u + ψ := q∈Z u q + q∈Z ψ q solves uniquely the elliptic problem

(2.7) ∆ x,y Φ = 0 in Ω, Φ| Σ = ψ, ∂Φ ∂ν | Γ = 0,
in the variational sense and moreover, there exists

F : R + → R + such that Φ H 1 ul (Ω) ≤ F( η W 1,∞ (R d ) ) ψ H 1 2 ul (R d )
.

(see Proposition 3.

3, [1])

The Dirichlet-Neumann operator is defined by

(2.8) G(η)ψ(x) = (1 + |∇ x η| 2 ) 1 2 ∂Φ ∂n | Σ = ∂Φ ∂y -∇ x η • ∇ x Φ | Σ = Λ 1 Φ -∇ x η • Λ 2 Φ | z=0 = Λ 1 Φ -∇ x ρ • Λ 2 Φ | z=0 .
2.2. Elliptic regularity with weights. We observe that if u is a solution of the elliptic equation ∆u = 0 on Ω and u is its image via the diffeomorphism

(2.3) then (Λ 2 1 + Λ 2 2 ) u = 0, which is equivalent to (see equation (3.16), [2]) (2.9) (∂ 2 z + α∆ x + β • ∇ x ∂ z -γ∂ z ) u = 0, where (2.10) α := (∂ z ρ) 2 1 + |∇ρ| 2 , β := -2 ∂ z ρ∇ x ρ 1 + |∇ x ρ| 2 , γ := 1 ∂ z ρ ∂ 2 z ρ + α∆ x ρ + β • ∇ x ∂ z ρ .
These coefficients are estimated by

Lemma 2.3 ([1, Lemma 3.17]). Let J = (-1, 0) and s > 1 + d 2 .
There exists F : R + → R + non decreasing such that (see Definition A.1 for the definition of

X µ ul ) α X s-1 2 ul (J) + β X s-1 2 ul (J) + γ X s-3 2 ul (J) ≤ F η H s+ 1 2 ul .
Let us denote by L the linear differential operator

(2.11) L = ∂ 2 z + α∆ x + β • ∇ x ∂ z and consider the following inhomogeneous initial value problem (2.12) (L -γ∂ z ) u = F in R d × J, u| z=0 = ψ.
Recall Definition 1.2 for the definitions of weight classes W, W po , W ex . It is clear that W po ( ) ⊂ W ex ( ) for all ≥ 0. For any w ∈ W, defining (2.13)

r 2 := ∆w -1 w -1 , r 2 := ∆w w , we have that r 2 , r 2 ∈ C ∞ b (R d ). Now we fix a weight w ∈ W and set v = w u. A simple computation shows that v satisfies L v + (β • r 1 -γ)∂ z v + αr 2 v + 2αr 1 • ∇ x v = wF. Next, set v k = χ k v, then (2.14) L v k = χ k wF + F 0 + F 1
where

F 0 = α∆χ k v + 2α∇χ k • ∇ x v + β • ∇ x χ k ∂ z v -χ k β • r 1 ∂ z v -χ k αr 2 v -2χ k αr 1 • ∇ x v, F 1 = χ k γ∂ z v.
Estimates for F j 's are given in the next lemma.

Lemma 2.4. Let J = (-1, 0) and s > 1 + d 2 . There exists F : R + → R + non decreasing such that for - 1 2 ≤ σ ≤ s -1 we have

(2.15) 1 j=0 F j Y σ+ 1 2 (J) ≤ F η H s+ 1 2 ul wψ H σ ul + ∇ x,z v X σ ul (J) .
(see Definition A.1 for the definition of Y µ ), where F depends on w only through the semi-norms of

r i , r i , i = 1, 2 in C ∞ b (R d ).
Proof. 1. It was proved in Lemma 3.20, [START_REF] Alazard | On the Cauchy problem for gravity water waves[END_REF] (applied with ε = 1 2 ) that under the conditions of this lemma,

γ∂ z v Y σ+ 1 2 (J) ≤ C γ L 2 (J,H s-1 ) ∂ z v X σ (J) ,
whose proof uses only the regularity of γ and

∂ z v. By writing χ k γ∂ z v = (χ k γ)( χ k ∂ z v)
and using the proof of preceding estimate we obtain

F 1 Y σ+ 1 2 (J) = χ k γ∂ z v Y σ+ 1 2 (J) ≤ C γ L 2 (J,H s-1 ul ) ∂ z v X σ ul (J) ≤ F η H s+ 1 2 ul ∂ z v X σ ul (J) . 2.
We turn to estimate F 0 . All the terms containing either ∇ x v or ∂ z v can be handled by the same method (remark that r 1 ∈ C ∞ b (R d )). Let us consider for example α∇χ k •∇ x v. There exists M > 0 such that if |k -j| > M then supp χ k ∩supp χ j = ∅. Therefore, it suffices to estimate A = χ j α∇χ k • ∇ x v for |j -k| ≤ M . We have the following product rule in Sobolev spaces (see for instance, Corollary 2.11 (i), [START_REF] Alazard | On the Cauchy problem for gravity water waves[END_REF]

): if s 0 ≤ s 1 , s 0 ≤ s 2 , s 1 + s 2 > 0 and s 0 < s 1 + s 2 -d 2 then there exists C > 0 such that for all u 1 ∈ H s 1 , u 2 ∈ H s 2 there holds (2.16) u 1 u 2 H s 0 ≤ C u 1 H s 1 u 2 H s 2 .
The preceding result applied with s 0 = σ, s 1 = s, s 2 = σ together with Lemma 2.3 leads to

A L 2 (J,H σ ) ≤ χ k α L 2 (J,H s ) ∇χ k • ∇ x v L ∞ (J,H σ ) ≤ F η H s+ 1 2 ul ∇ x v X σ ul .
3. We are left with two terms α∆χ k v and χ k αr 2 v, which can be treated in the same way (remark that

r 2 ∈ C ∞ b (R d ))
. Let us consider for example α∆χ k v. As in 2., one only need to estimate χ j α∆χ k v for j close to k. The product rule (2.16) gives

χ j α∆χ k v Y s+ 1 2 (J) ≤ χ j α∆χ k v L 2 (J,H σ ) ≤ C α∆χ k L 2 (J,H s ) χ j v L ∞ (J,H σ ) ≤ C α L 2 (J,H s ul ) χ j v L ∞ (J,H σ ) . Now, by writing (2.17) χ j v(x, z) = χ j v(x, 0) + z 0 χ j ∂ z v(x, τ )dτ = χ j wψ(x) + z 0 χ j ∂ z v(x, τ )dτ, we obtain χ j v L ∞ (J,H σ ) ≤ wψ H σ ul + ∂ z v L ∞ (J,H σ ) ul ≤ wψ H σ ul + ∂ z v X σ ul (J) . Consequently, χ j α∆χ k v Y σ+ 1 2 (J) ≤ F( η H s+ 1 2 ul ) wψ H σ ul + ∂ z v X σ ul (J) .
Remark 2.5. Lemma 2.4 is in the same spirit of Lemma 3.18, [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF]. However, in Lemma 3.18, [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF] the authors considered two cases corresponding to two ranges of σ:

-1 2 ≤ σ < s -3 2 and s -3 2 ≤ σ ≤ s -1.
This aimed to keep in the estimate (2.15) the function F depending only on η

H s 0 + 1 2 ul
for any 1 + d 2 < s 0 ≤ s, which appeared in their finial a priori estimate (see Proposition 4.7, [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF]). Here, however, for our contraction estimates we do not need this tame estimate. In fact, our contraction estimates shall be established in 1-derivative lower Sobolev spaces, hence we even do not use weighted bounds in the highest norms.

Next, we prove an elliptic regularity theorem with weights for ∇ x,z v:

Theorem 2.6. Let J = (-1, 0), s > 1 + d 2 and w ∈ W ex ( ), ≥ 0. Let u be a solution of the problem (2.12) and set v = w u. For -1 2 ≤ σ ≤ s-1 2 let η ∈ H s+ 1 2 ul (R d ) satisfying (2.2), wψ ∈ H σ+1 ul (R d ), F ∈ Y σ ul (J) and (2.18) ∇ x,z v X -1 2 ul (J) < +∞.
Then for every z 0 ∈] -1, 0[ there exists F : R + → R + non decreasing, depending only on (s, d) and the semi-norms of r i , r

i , i = 1, 2 (in C ∞ b (R d )) such that (2.19) ∇ x,z v X σ ul (z 0 ,0) ≤ F η H s+ 1 2 ul wψ H σ+1 ul + wF Y σ ul (J) + ∇ x,z v X -1 2 ul (J)
.

Consequently,

(2.20) w∇ x,z u X σ ul (z 0 ,0) ≤ F η H s+ 1 2 ul wψ H σ+1 ul + wF Y σ ul (J) + w∇ x,z u X -1 2 ul (J)
.

Proof. Estimate (2. [START_REF] Wu | Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent[END_REF]) is a consequence of Proposition 3.19, Proposition 3.20 and the proof of Proposition 3.16 in [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF], tacking into account Remark 2.5. We now derive (2.20) using (2.19). Remark first that ∂ z v = w∂ z u. Next, we write

w∇ x u = ∇ x v -u∇ x w, where u(x, z)∇ x w(x) = r 1 (x)w(x) u(x, z) = r 1 (x)w(x) u(x, 0) + z 0 ∂ z u(x, τ )dτ , which implies (using again r 1 ∈ C ∞ b (R d )) u∇ x w X σ ul ≤ C wψ H σ+ 1 2 ul + w∂ z u X σ ul ≤ C wψ H σ+ 1 2 ul + ∂ z v X σ ul .
We have proved that

w∇ x,z u X σ ul ≤ C wψ H σ+ 1 2 ul + ∇ x,z v X σ ul . Likewise, it holds that ∇ x,z v X -1 2 ul ≤ C wψ H 0 ul + w∇ x,z u X -1 2 ul .
The two inequalities above show that (2.20) is a consequence of (2.19) (notice that σ + 1 ≥ 0). Remark 2.7. We remark that in all the results stated below, the function F depends on w only through the semi-norms of r i and r

i , i = 1, 2 in C ∞ b (R d ).
To apply Theorem 2.6 we need a base estimate in the low norm X -1 2 ul . For the proof of this, let us recall a classical interpolation result.

Lemma 2.8 ([13, Theorem 3.1]). Let J = (-1, 0) and σ ∈ R. Let f ∈ L 2 z (J, H σ+ 1 2 (R d )) be such that ∂ z f ∈ L 2 z (J, H σ-1 2 (R d )). Then f ∈ C 0 z ([-1, 0], H σ (R d ))
and there exists an absolute constant C > 0 such that

f C 0 z ([-1,0],H σ (R d )) ≤ C f L 2 z (J,H σ+ 1 2 (R d )) + C ∂ z f L 2 z (J,H σ-1 2 (R d ))
.

Recall also here the Poincaré inequality proved in [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF] (cf. Remark 3.2) for fluid domains with finite depth of type Ω (cf. (1.1)).

Lemma 2.9. Let θ, θ * ∈ C 0 b (R d ) satisfying θ -θ * L ∞ (R d ) > 0. Define O = {(x, y) ∈ R d × R : θ * (x) ≤ y ≤ θ(x)} and H 1,0 (O) = {u ∈ L 2 (O) : ∇ x,y u ∈ L 2 (O) and u| y=θ(x) = 0}. Then for all u ∈ H 1,0 (O), α ∈ C ∞ b (R d ), α ≥ 0, there holds (2.21) O α(x)|u(x, y)| 2 dxdy ≤ θ -θ * 2 L ∞ (R d ) O α(x)|∂ y u(x, y)| 2 dxdy.
Remark 2.10. In Remark 3.2, [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF] the constant appearing in the Poincaré inequality is stated to be dependent only on θ L ∞ + θ * L ∞ . However, it is easy to track the proof to derive the explicit constant

θ -θ * 2 L ∞ (R d ) in (2.21). Proposition 2.11. Let J = (-1, 0), s > 1 + d 2 .
Let Φ be the unique solution to (2.7). Then the following statements hold true. (i) For every w ∈ W po ( ), ≥ 0, one can find a non decreasing function

F : R + → R + such that w∇ x,z Φ X -1 2 ul (J) ≤ F( η H s+ 1 2 ul (R d )
) wψ

H 1 2 ul (R d )
.

(ii) There exists an absolute constant C * = C * (d) > 0 such that for all w ∈ W ex ( ) and ≤ * with

(2.22) * C * η -η * 2 L ∞ (R d ) = 1 2 ,
one can find a non decreasing function

F : R + → R + such that w∇ x,z Φ X -1 2 ul (J) ≤ F( η H s+ 1 2 ul (R d )
) wψ

H 1 2 ul (R d )
.

Proof. We proceed in two steps.

Step 1. By Lemma 3.6, [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF] one can find an absolute constant C * > 0 such that for all µ > 0 satisfying

(2.23) µC * η -η * 2 L ∞ (R d )
≤ 1 there exists F : R + → R + non decreasing such that for all q ∈ Z d we have

(2.24) e µ x-q ∇ x,y u q L 2 (Ω) ≤ F( η W 1,∞ (R d ) ) ψ q H 1 2 (R d )
.

Using properties (i) and (ii) above of ψ q (see section 2.1.2), we see that (2.24) also holds for u q replaced by ψ q for any µ > 0 and thus (2.24) is true for u q replaced by Φ q = u q + ψ q , i.e.,

e µ x-q ∇ x,z Φ q L 2 (Ω) ≤ F( η W 1,∞ (R d ) ) ψ q H 1 2 (R d )
for any µ > 0 satisfying (2.23).

Using the diffeomorphism (2.3) we deduce that (2.25)

e µ x-q ∇ x,z Φ q L 2 (J,L 2 (R d )) ≤ F( η W 1,∞ (R d ) ) ψ q H 1 2 (R d )
.

Consider a weight w that is a priori in W. On the support of χ k , we have w(x)e µ x-q ∼ w(k)e µ k-q . Hence by the product rule (2.16) we have

(2.26) χ k w∇ x,z Φ q L 2 (J,L 2 (R d )) ≤ Cw(k)e -µ k-q F( η W 1,∞ (R d ) ) ψ q H 1 2 (R d )
, from which it follows that (2.27)

χ k w∇ x,z Φ L 2 (J,L 2 (R d )) ≤ q χ k w∇ x,z Φ q L 2 (J,L 2 (R d )) ≤ C q w(k)e -µ k-q F( η W 1,∞ (R d ) ) ψ q H 1 2 (R d ) ≤ C q w(k)w -1 (q)e -µ k-q F( η W 1,∞ (R d ) ) wψ q H 1 2 (R d )
.

Now we distinguish two cases: (i) w ∈ W po ( ), ≥ 0. By definition, w(k)w -1 (q) ≤ C k -q and thus the final sum in (2.27) converges for any ρ ≥ 0, which leads to (2.28)

χ k w∇ x,z Φ L 2 (J,L 2 (R d )) ≤ F( η W 1,∞ (R d ) ) wψ H 1 2 ul (R d )
.

(ii) w ∈ W ex ( ), ≥ 0. Choosing µ, * such that

µC * η -η * 2 L ∞ (R d ) = 1, * C * η -η * 2 L ∞ (R d ) =
1 2 then the final sum in (2.27) converges for all ≤ * and one also ends up with (2.28).

Step 2. Let us fix a weight w as in (i) or (ii). To complete the proof of this lemma, it remains to show for any k ∈ Z d that (2.29)

χ k w∇ x,z Φ L ∞ (J,H -1 2 (R d )) ≤ F( η H s+ 1 2 ul ) wψ H 1 2 ul (R d )
.

By the interpolation Lemma 2.8

wχ k ∇ x Φ L ∞ (J,H -1 2 ) ≤ wχ k ∇ x Φ L 2 (J,L 2 ) + wχ k ∂ z ∇ x Φ L 2 (J,H -1 )
.

The first term on the right-hand side is estimated by (2.28), so we need to estimate

M := wχ k ∇ x ∂ z Φ L 2 (J,H -1 ) .
Notice that for any acceptable weight ω ∈ W, there holds with χ ∈ C ∞ 0 (R d ) and χ = 1 on supp χ that

ωχ k ∇ x f H s (R d ) ≤ ∇(ωχ k f ) H s (R d ) + ω∇ x χ k f H s (R) d + ∇ x ωχ k f H s (R d ) ≤ χ k ωf H s+1 (R d ) + ω∇ x χ k f H s (R d ) + r 1 χ k H s χ k ωf H s (R d )
where r 1 = ∇ω ω as in Definition 1.2. This implies (2.30)

ωχ k ∇ x f H s (R d ) ≤ C ωf H s+1 ul (R d ) , ∀ω ∈ W, s ∈ R.

Applying this estimate and (2.28) leads to

M ≤ C w∂ z Φ L 2 (J,L 2 ) ul ≤ F( η H s+ 1 2 ul ) wψ H 1 2 ul .
Finally, to obtain (2.29) we shall prove

(2.31) w∂ z Φ L ∞ (J,H -1 2 ) ul ≤ F( η H s+ 1 2 ul ) wψ H 1 2 ul .
Again, by interpolation,

wχ k ∂ z Φ L ∞ (J,H -1 2 ) ≤ wχ k ∂ z Φ L 2 (J,L 2 ) + wχ k ∂ 2 z Φ L 2 (J,H -1 )
.

It remains to estimate

A := wχ k ∂ 2 z Φ L 2 (J,H -1 )
. Taking into account the fact that Φ q satisfies equation (2.9), we have

A ≤ q A 1,q + A 2,q + A 3,q ,
where by the product rule (2.16) (remark that s > 1 + d 2 is sufficient), Lemma 2.3 and (2.28),

A 1,q = χ k wα∆ Φ q L 2 (J,H -1 ) ≤ α L ∞ (J,H s-1 2 ) ul w∆ Φ q L 2 (J,H -1 ) ul , A 2,q = χ k wβ∂ z ∇ x Φ q L 2 (J,H -1 ) ≤ β L ∞ (J,H s-1 2 ) ul w∂ z ∇ x Φ q L 2 (J,H -1 ) ul , A 3,q = χ k wγ∂ z Φ q L 2 (J,H -1 ) ≤ γ L ∞ (J,H s-3 2 ) ul w∂ z Φ q L 2 (J,L 2 ) ul .
Finally, to sum w∇ x,z Φ q L 2 (J,L 2 ) ul over q ∈ Z d , one makes use of (2.26) and argues as in (2.27). The proof of Proposition 2.11 is complete.

Remark 2.12. In statement (ii) above, the function F depends on , which is in turn bounded from above by C η -η * -2

L ∞ . Therefore, F is really increasing in η H s+ 1 2 ul
if the fluid depth η -η * L ∞ is bounded from below by some positive constant.

Using Proposition 2.11 as the ground step for the regularity Theorem 2.6 we now prove a weighted estimate for Φ and its gradient.

Corollary 2.13. Let J = (-1, 0), s > 1 + d 2 .
Let Φ be the unique solution to (2.7). Then the following statements hold true. (i) For every w ∈ W po ( ), ≥ 0 and -1 2 ≤ σ ≤ s -1 2 , one can find a non decreasing function F : R + → R + such that for any z 0 ∈ (-1, 0),

w Φ X σ+1 ul (z 0 ,0) + w∇ x,z Φ X σ ul (z 0 ,0) ≤ F η H s+ 1 2 ul wψ H σ+1 ul .
(ii) For every w ∈ W ex ( ) with ≤ * (defined by (2.22)) and - 1 2 ≤ σ ≤ s -1 2 , one can find a non decreasing function F : R + → R + such that for any z 0 ∈ (-1, 0),

w Φ X σ+1 ul (z 0 ,0) + w∇ x,z Φ X σ ul (z 0 ,0) ≤ F η H s+ 1 2 ul wψ H σ+1 ul .
Proof. Observe that Φ satisfies (2.12) with F = 0. According to Proposition 2.11, with the weight w given either in (i) or (ii) we have

w∇ x,z Φ X -1 2 ul (z 0 ,0) ≤ F( η H s 0 + 1 2 ul ) wψ H 1 2 ul < +∞.
Theorem 2.6 then leads to the desired estimate for w∇ x,z Φ X σ ul (z 0 ,0) . Consequently, the argument in (2.17) leads to w Φ X σ ul (z 0 ,0) ≤ wψ

H σ+ 1 2 + w∂ z Φ X σ ul (z 0 ,0) ≤ F η H s+ 1 2 ul wψ H σ+1 ul .
Finally, using the fact that for any W ∈ W,

W u H σ+1 ul ≤ C W ∇u H σ ul + W u H σ ul we derive w Φ X σ+1 ul (z 0 ,0) ≤ F η H s+ 1
→ R + such that wG(η)ψ H σ ul ≤ F η H s+ 1 2 ul wψ H σ+1 ul .
Proof. Let w be the weight as in (i) or (ii). By (2.8),

G(η)ψ = Λ 1 Φ -∇ x ρ • Λ 2 Φ | z=0 =: H| z=0 .
Owing to Lemma 2.8, we have for any J = (z 0 , 0) ⊂ (-1, 0)

χ q wG(η)ψ H σ ≤ C χ q wH L 2 (J,H σ+ 1 2 ) + χ q w∂ z H L 2 (J,H σ-1 2 )
.

For the term χ q w∂ z H

L 2 (J,H σ-1 2 )
we make use of the following identity (see (3.21), [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF])

∂ z H = -∇ (∂ z ρ)Λ 2 Φ to have χ q w∂ z H L 2 (J,H σ-1 2 ) ≤ w(∂ z ρ)Λ 2 Φ L 2 (J,H σ+ 1 2 ) ul .
On the other hand, we observe by definition (2.5) of Λ 1,2 that the terms in H have the same structure as (∂ z ρ)Λ 2 Φ and thus, it suffices to prove, for example, that

w∇ x ρ • ∇ x Φ L 2 (J,H σ+ 1 2 ) ul ≤ F η H s+ 1 2 ul wψ H σ+1 ul .
By virtue of Corollary 2.13, this reduces to proving (2.32)

w∇ x ρ • ∇ x Φ L 2 (J,H σ+ 1 2 ) ul ≤ F η H s+ 1 2 ul w∇ x,z Φ X σ ul .
We consider two cases:

Case 1: -1 2 ≤ σ ≤ s -1.
We apply the product rule (2.16) with

s 0 = σ + 1 2 , s 1 = s -1 2 , s 2 = σ + 1 2 to obtain χ k w∇ x ρ • ∇ x Φ L 2 (J,H σ+ 1 2 ) ≤ C χ k ∇ x ρ L ∞ (J,H s-1 2 ) χ k w∇ x Φ L 2 (J,H σ+ 1 2 )
, from which (2.32) follows in view of Lemma 2.3.

Case 2: s -1 ≤ σ ≤ s -1 2 . Since σ + 1 2 > d 2
we have the following well-known inequality ab

H σ+ 1 2 a L ∞ b H σ+ 1 2 + a H σ+ 1 2 b L ∞ .
On the other hand, σ

+ 1 2 ≤ s and σ ≥ s -1 > d 2 so ab H σ+ 1 2 a H s-1 2 b H σ+ 1 2 + a H s b H σ .
Applying the preceding inequality to a = χ k ∇ x ρ and b = χ k w∇ x Φ yields

χ k w∇ x ρ • ∇ x Φ L 2 (J,H σ+ 1 2 ) ≤ χ k ∇ x ρ L ∞ (J,H s-1 2 ) χ k w∇ x Φ L 2 (J,H σ+ 1 2 ) + χ k ∇ x ρ L 2 (J,H s ) χ k w∇ x Φ L ∞ (J,H σ ) .
Lemma 2.3 then implies the desired estimate (2.32). The proof is complete.

∃λ > 0, ∀j ∈ N, j ≥ 1, ∃C j > 0, ∀x ∈ R, |∂ j x ψ(x)| ≤ C j e -λ|x| .
Then for η ∈ H ∞ (R) with η -H ≥ h > 0, G(η)ψ also has an exponential decay, that is, there exist 0 < ε < λ such that for any j ∈ N we can find a constant C j > 0 such that

|∂ j x G(η)ψ)(x)| ≤ C j e -ε|x| .
1. The advantage of Proposition 2.16 is that it does not assume decay on ψ itself but its derivatives, which is compatible with the solitons studied there. The authors were not interested in the way the estimates depend on the regularity of the surface η. 2. Proposition 2.16 is asymmetric in the sense that the exponential decay of G(η)ψ is lower than the decay of ψ. 3. Proposition 2.14 assumes also that ψ is decay (choosing for example w(x) = e x ). However, compare to Proposition 2. [START_REF] Nahas | On the persistence properties of solutions of nonlinear dispersive equations in weighted Sobolev spaces. Harmonic analysis and nonlinear partial differential equations[END_REF], it has the following advantages:

(i) Proposition 2.14 holds in any dimension, with varying bottom (η * is only assumed to be in C 0 b ) and allows domains with non smooth surfaces (η ∈ C 2+ε ).

(ii) The decay rate of G(η)ψ is preserved, i.e., the same as the rate of ψ.

Moreover, the exponential rate of ψ can be chosen as

(2.33) ≤ 1 2C * η -η * 2 L ∞
(for some absolute constant C * = C * (d)) which is decreasing in the square of the fluid depth. Such a rate is therefore higher for shallow water but deteriorates when the depth tends to infinity.

κ = (1 + |∇ x η| 2 )|ξ| 2 -(∇ x η • ξ) 2 1 2
and define the remainder

(2.34) R(η)ψ := G(η)ψ -T κ ψ.
Our aim in this section is to prove the following weighted version of Theorem 3.11 in [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF].

Theorem 2.17. Let s > 1 + d 2 and w ∈ W po ( ), ≥ 0. Then there exists F :

R + → R + non decreasing such that for 0 ≤ t ≤ s -1 2 , η ∈ H s+ 1 2 ul (R d ) satisfying (2.2) we have wR(η)ψ H t ul ≤ F η H s+ 1 2 ul wψ H t+ 1 2 ul provided that wψ ∈ H t+ 1 2 ul (R d ).
Proof. Let us fix a real number t ∈ [0, s - 1 2 ]. By definition of the Dirichlet-Neumann operator, one has

G(η)ψ = h 1 ∂ z Φ -h 2 • ∇ x Φ   z=0 , h 1 = 1 + |∇ x ρ| 2 ∂ z ρ , h 2 = ∇ x ρ.
Let A be the symbol of class Γ 1 (R d × J) given in Lemma 3.20, [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF]. We set

g k = (∂ z -T A )(χ k w Φ), h j | z=0 = h 0 j , j = 1, 2, A| z=0 = A 0 . Then we can write (2.35) χ k wG(η)ψ = h 0 1 (∂ z (χ k w Φ))| z=0 -χ k wh 0 2 • ∇ x ψ = h 0 1 g k | z=0 + h 0 1 T A 0 (χ k χ k wψ) -χ k χ k wh 0 2 • ∇ x ψ = h 0 1 g k | z=0 + h 0 1 [T A 0 , χ k ]( χ k wψ) + χ k h 0 1 T A 0 -h 0 2 • ∇ ( χ k wψ) + χ k h 0 2 • ∇( χ k wψ) -χ k χ k wh 0 2 • ∇ x ψ. Therefore, χ k wG(η)ψ = B 1 + B 2 , where B 1 = h 0 1 g k | z=0 + h 0 1 [T A 0 , χ k ]( χ k wψ) + χ k h 0 1 T A 0 -h 0 2 • ∇ ( χ k wψ), B 2 = χ k h 0 2 • (∇ χ k )wψ + χ k h 0 2 • (∇w)ψ = χ k h 0 2 • (∇w)ψ.
The proof of Theorem 3.11, [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF] shows that

B 1 = χ k T κ ( χ k wψ) + S
with the remainder S satisfies

S H t ≤ F η H s+ 1 2 ul ) wψ H t+ 1 2 ul . Writing ∇w = wr 1 with r 1 ∈ C ∞ b (R d ). Since h 0 2 ∈ H s-1 2 (R d ) ul with norm bounded by F( η H s+ 1 2 ul
) and t ≤ s -1 2 the product rule yields

B 2 H t = χ k h 0 2 • r 1 wψ H t = χ k h 0 2 • r 1 H s-1 2 χ k wψ H t ≤ F η H s+ 1 2 ul wψ H t ul .
We have proved the following result

χ k wG(η)ψ = χ k T κ ( χ k wψ) + S, S H t ≤ F η H s+ 1 2 ul wψ H t+ 1 2 ul .
The proof of Theorem 2.17 boils down to showing that the commutator

T := χ k [T κ , χ k w]ψ satisfies (2.36) T H t ≤ F η H s+ 1 2 ul wψ H t+ 1 2 ul . Indeed, introduce χ ∈ C ∞ c (R d
), χ = 1 on the support of χ and define also χ q (•) = χ(• -q) for all q ∈ Z d . Denoting w k = χ k w and noticing that w k = w k χ k we can write

T = χ k T κ (w k χ k ψ) -χ k w k χ k T κ ψ = χ k T κ (w k χ k ψ) -χ k w k [χ k , T κ ]ψ -χ k w k T κ (χ k ψ) = χ k [T κ , w k ](χ k ψ) -χ k w k [χ k , T κ ]ψ =: R 1 -R 2 .
1. R 1 can be written as

R 1 = χ k [T κ , T w k ](χ k ψ) + χ k T κ [(w k -T w k )(χ k ψ)] -χ k (w k -T w k )(T κ (χ k ψ)) =: R 1,a + R 1,b -R 1,c .
a) For R 1,a we apply the symbolic calculus for paradifferential operators in Kato's spaces (cf. Theorem 7.16 (ii), [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF])

to κ ∈ Γ 1 1 , w k ∈ Γ 0 1 (see also Theorem A.11 (ii) in the Appendix) to have R 1,a H t ≤ CF η H s+ 1 2 ul w k W 1,∞ χ k ψ H t ul .
Now by properties (i), (ii) in Definition 1.2, the weight w satisfies

w k W 1,∞ ≤ Cw(k) and w(k) χ k ψ H t ul ≤ C wψ H t ul . Consequently, R 1,a H t ≤ F η H s+ 1 2 ul wψ H t ul . b) For R 1,b one first uses the boundedness of T κ from H t+1 ul to H t ul (see Theorem 7.16 (i), [1]) to have R 1,b H t ≤ F η H s+ 1 2 ul (w k -T w k )(χ k ψ) H t+1 ul
Next, the paraproduct rule in Proposition 7.18, [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF] gives for n large enough

(w k -T w k )(χ k ψ) H t+1 ul ≤ C w k H n ul χ k ψ H t ul
As in a) we remark that w k H n ul ≤ Cw(k) and hence obtain the desired estimate for R 1,b . The term R 1,c can be handled using exactly the same method. In summary, we have proved an estimate better than needed:

R 1 H t ≤ F η H s+ 1 2 ul wψ H t ul . 2. To study R 2 we decompose R 2 = w k [T χ k , T κ ]ψ + w k (χ k -T χ k )T κ ψ -w k T κ [(χ k -T χ k )ψ]
By the same arguments as in the the study of R 1 but using instead the symbolic calculus in Kato's spaces with weights in Theorem A.11 together with inequality (A.21) one ends up with

R 2 H t ≤ F η H s+ 1 2 ul wψ H t ul .
The proof of Proposition 2.17 is complete.

Remark 2.18. In the proof above, it is for the study of the remainder R 2 that we need to restrict the weight w to the class of "polynomial weights" W po ( ).

2.4.

A weighted estimate for Φ. We use the elliptic regularity theorem 2.6 to prove a weighted estimate for Φ-solution to (2.7), which will be used later in proving a contraction estimate for Dirichlet-Neumann operator.

Lemma 2.19. Let s > 1+ d 2 and w ∈ W po ( ), ≥ 0. With µ > 0 satisfying (2.23) and Φ q , ψ q as in section 2.1.2 there exists a non-decreasing function F independent of q such that

k∈Z d we µ 2 x-q χ k ∇ x,z Φ q L ∞ (J×R d ) ≤ F( η H s+ 1 2 ul
) wψ q H s .

Proof. We remark that wψ q ∈ H s (R d ) for every q ∈ Z d provided that ψ ∈

H s ul (R d ). It is clear that we µ 2 x-q χ k ∇ x,z Φ q L ∞ (J×R d ) ≤ e -µ 4 k-q we 3µ/4 x-q ∇ x,z Φ q L ∞ (J×R d ) .
Consider the weight we 3µ/4 x-q ∈ W which has semi-norms independent of q. Applying Theorem 2.6 to Φ q (with σ = s -1) and taking into account Remark 2.7 , we may estimate

k∈Z d we µ 2 x-q χ k ∇ x,z Φ q L ∞ (J×R d ) ≤ C k∈Z d e -µ 4 k-q we 3µ/4 x-q ∇ x,z Φ q L ∞ (J×R d ) ≤ C we 3µ/4 x-q ∇ x,z Φ q L ∞ (J,H s-1 (R d )) ul ≤ F( η H s+ 1 2 ul
) we 3µ/4 x-q ψ q H s ul + we 3µ/4 x-q ∇ x,z Φ q X -1 2

ul (J) ≤ F( η H s+ 1 2 ul
) wψ q H s + we 3µ/4 x-q ∇ x,z Φ q X -1 2 ul (J)

.

Remark that in the first inequality, we have used the trivial fact that k∈Z d e -µ 4 k-q is finite and independent of q.

To complete the proof we need to prove that (2.37)

we 3µ/4 x-q ∇ x,z Φ q X -1 2 ul (J) ≤ F( η H s+ 1 2 ul ) wψ q H 1 2 .
However, using interpolation inequality as in step 2 of the proof of Proposition 2.11, it suffices to show that (2.38)

we 3µ/4 x-q ∇ x,z Φ q L 2 (J,L 2 ) ul ≤ F( η H s+ 1 2 ul ) wψ q H 1 2 .
Indeed, by virtue of (2.25) one can estimate

χ p we 3µ 4 x-q ∇ x,z Φ q L 2 (J,L 2 ) e -µ 4 p-q w(p) χ p e µ x-q ∇ x,z Φ q L 2 (J,L 2 ) e -µ 4 p-q w(p)F( η W 1,∞ ) χ q ψ H 1 2 e -µ 4 p-q w(p)w(q) -1 F( η W 1,∞ ) wχ q ψ H 1 2 e -µ 4 p-q p -q ( η W 1,∞ ) wχ q ψ H 1 2 F( η W 1,∞ ) wχ q ψ H 1 2
, which is the desired bound.

Remark 2.20. As in Proposition 2.11, Lemma 2.17 can be formulated for weights in the class W ex ( ) with sufficiently small.

Weighted contraction for the Dirichlet-Neumann operator

The main ingredient in proving the contraction for the Dirichlet-Neumann operator is the contraction estimate for solutions to the elliptic problem (2.7). The key idea then is to compare the two variational solutions Φ j after changing the variables Φ j to Φ j as in (2.4). However, after straightening the fluid domains by the diffeomorphism (2.3), the new domains will depend on their upper surface. To overcome this, we use a slightly different diffeomorphism as follows.

Given

η * ∈ C 0 b (R d ) and h > 0, there exists η ∈ C ∞ b (R d ) such that (3.1) η * (x) < η(x) < η * (x) + h 3 , ∀x ∈ R d .
Then, because η j > η * + h we set

         Ω 1,j = {(x, y) : x ∈ R d , η j (x) -h 3 < y < η j (x)}, Ω 2,j = {(x, y) : x ∈ R d , η(x) ≤ y ≤ η j (x) -h 3 }, Ω 3,j = {(x, y) : x ∈ R d , η * (x) < y < η(x)}, Ω j = Ω 1,j ∪ Ω 2,j ∪ Ω 3,j , and
         Ω 1 = R d x × (-1, 0) z , Ω 2 = R d x × [-2, -1] z , Ω 3 = {(x, z) ∈ R d × (-∞ -2) : z + 2 + η(x) > η * (x)}, Ω = Ω 1 ∪ Ω 2 ∪ Ω 3 .
Remark that Ω depends on η * , h but not on η j . Thus, we can define (3.2)

ρ j (x, z) =      ρ 1,j (x, z) = (1 + z)e δz Dx η j (x) -z e -(1+z)δ Dx η j (x) -h 3 , in Ω 1 , ρ 2,j (x, z) = (2 + z) e δ(z+1) Dx η j (x) -h 3 -(1 + z) η, in Ω 2 , ρ 3,j (x, z) = z + 2 + η(x), in Ω 3 . Lemma 3.1. Assume that η j ∈ W 1,∞ (R d ), j = 1, 2. There exists an absolute constant C > 0 such that if Cδ η j W 1,∞ (R d ) ≤ h, j = 1, 2
then the mappings (x, z) → (x, ρ j (x, z)) are Lipschitz diffeomorphisms from Ω to Ω j and there exists a constant c 0 > 0 such that ∂ z ρ j ≥ c 0 a.e. in Ω.

Proof. Observe first that ρ k,j are Lipschitz for k = 1, 2, 3; j = 1, 2. Clearly, (x, z) → (x, ρ 3,j (x, z)) are diffeomorphisms from Ω 3 to Ω 3,j and ∂ z ρ 3,j = 1 ≥ c 0 > 0. The same properties hold for ρ 1,j as in (2.3). We now prove it for ρ 2,j . Notice first that

ρ 2,j (-1, x) = η j - h 3 , ρ 2,j (-2, x) = η.

Compute now

∂ z ρ 2,j = e δ(z+1) Dx η j (x) - h 3 -(2 + z)δe δ(z+1) Dx D x η j -η = e δ(z+1) Dx η j (x) -η j (x) -(2 + z)δe δ(z+1) Dx D x η j + η j (x) -η - h 3 .
By writing e δ(z+1) Dx η j -η j = δ(z + 1)

1 0 e rδ(z+1) Dx D x η j dr we deduce that e δ(z+1) Dx η j -η j

L ∞ (R d ) + (2 + z)δe δ(z+1) Dx D x η j L ∞ (R d ) ≤ Cδ η j W 1,∞ (R d ) ≤
h 6 for δ > 0 small enough. On the other hand, thanks to (2.2) and (3.1) it holds that

η j -η - h 3 = (η j -η * ) + (η * -η) - h 3 > h - h 3 - h 3 = h 3 and thus ∂ z ρ 2,j ≥ h 3 -h 6 = h 6 in Ω 2 . Therefore, we can conclude that (x, z) → (x, ρ 2,j (x, z)) are diffeomorphisms from Ω 2 to Ω 2,j .
With the functions ρ j above we denote for every f : Ω → R

(3.3) f j (x, z) = f (x, ρ j (x, z))
and as in (2.5) we define the differential operators Λ j = (Λ j 1 , Λ j 2 ). Hereafter, we denote J = (-2, 0) and assume that (3.4)

η j ∈ H s+ 1 2 ul (R d ), s > 1 + d 2 , j = 1, 2. Lemma 3.2. Let w ∈ W po ( ), ≥ 0. We have Λ 1 -Λ 2 = ℘∂ z = (℘ 1 , ℘ 2 )∂ z with ℘ = 0 for z < -2 and (3.5) w℘ L 2 (J,L 2 (R d )) ul ≤ F( (η 1 , η 2 ) H s+ 1 2 ul ×H s+ 1 2 ul ) w(η 1 -η 2 ) H 1 2 ul .
Proof. By definition, one gets

℘ 1 = ∂ z (ρ 2 -ρ 1 ) ∂ z ρ 1 ∂ z ρ 2 , ℘ 2 = - ∇ x (ρ 2 -ρ 1 ) ∂ z ρ 1 -∇ x ρ 2 ∂ z (ρ 2 -ρ 1 ) ∂ z ρ 1 ∂ z ρ 2
so in Ω 3 , ℘ = 0. To obtain (3.5) one writes

w℘ L 2 (J,L 2 (R d )) ul ≤ w℘ L 2 ((-1,0),L 2 (R d )) ul + w℘ L 2 ((-2,-1),L 2 (R d )) ul to use definition (3.2), the fact that η ∈ C ∞ b (R d
) and the 1 2 -smoothing effect of the Poisson kernel, which is Proposition A.7 applied with r = 1.

Theorem 3.3. Let ψ j ∈ H 1 2
ul (R d ) and Φ j , j = 1, 2 be the unique solution in H 1 ul (Ω j ) of the problem

(3.6) ∆ x,y Φ j = 0 in Ω, Φ j | Σ = ψ j , ∂Φ j ∂ν | Γ = 0. Set η = η 1 -η 2 , ψ = ψ 1 -ψ 2 , Φ = Φ 1 -Φ 2 where Φ j is the image of Φ j as in (3.3).
Then for every w ∈ W po ( ), ≥ 0 there exists a nonnegative function F such that (3.7)

w∇ x,z Φ X -1 2 ul (J) ≤ F( (η 1 , η 2 ) H s+ 1 2 ul ×H s+ 1 2 ul ) wη H s-1 2 ul ψ 2 H s ul + wψ H 1 2 ul
.

For the proof of this result, we shall apply Lemma 2.19 for Φ q . However, Φ q here is the image of Φ q via the diffeomorphism corresponding to one of ρ j defined by (3.2) instead of (2.3). We want the same result as Lemma 2.19 in this situation.

To have this, we notice that on J = (-2, 0), ρ j is comprised of two functions ρ 1,j for z ∈ (-1, 0) and ρ 2,j for z ∈ (-2, -1]. The function ρ 1,j possesses the same properties as ρ does and so does ρ 2,j since η ∈ C ∞ b ⊂ H ∞ ul . Therefore, we obtain Lemma 3.4. Let w ∈ W po ( ), ≥ 0 and Φ j,q , ψ j,q , j = 1, 2, q ∈ Z as in section 2.1.2. There exists F 1 non-decreasing such that: if 0 < µF 1 ( η j H s ul ) ≤ 1 then one can find a non-decreasing function F independent of q such that

k∈Z d we µ 2 x-q χ k ∇ x,z Φ j,q L ∞ (J×R d ) ≤ F( η j H s+ 1 2 ul
) wψ j,q H s .

Proof. (of Theorem 3.3) For simplicity in notations we shall denote

F = F( η 1 H s+ 1 2 ul , η 2 H s+ 1 2 ul
) which may change from line to line. We proceed in the following steps.

Step 1. Let Φ j,q = u j,q + ψ j,q where u j,q is the variational solution characterized by (2.6). After changing the variables, (2.6) becomes Ω Λ j Φ j,q Λ j θJ j dX = 0, ∀θ ∈ H 1,0 ( Ω), j = 1, 2 with the Jacobian J j = |∂ z ρ j | = ∂ z ρ j ≥ c 0 > 0 a.e. in Ω (by Lemma 3.1).

Set Φ q = Φ 1,q -Φ 2,q , ψ q = ψ 1,q -ψ 2,q and choose θ = e 2δgε ( Φ q -ψ q ) ∈ H 1,0 ( Ω)

where g ε = x-q 1+ε x-q . It follows that

Ω Λ 1 Φ q Λ 1 θJ 1 dX ≤ 3 j=1 A j ,      A 1 = Ω |(Λ 1 -Λ 2 ) Φ 2,q Λ 1 θJ 1 |dX, A 2 = Ω |Λ 2 Φ 2,q (Λ 1 -Λ 2 )θJ 1 |dX, A 3 = Ω |Λ 2 Φ 2,q Λ 2 θ(J 1 -J 2 )|dX. By Lemma (3.2) we know that Λ 1 -Λ 2 = 0 in Ω 3 . Likewise, J 1 -J 2 = ∂ z ρ 1 -∂ z ρ 2 = 0 in Ω 3 .
Consequently, with Ω 0 = R d × J we have A j , j = 1, 2, 3 are equal to the corresponding integrals over Ω 0 .

Step 2. (Estimate for A 1 ) First of all, we remark that

(3.8) Λ j (e 2δgε U ) = e 2δgε Λ j U + (0, U )2δe 2δgε ∇g ε .
Using Lemma 3.2 and formula (3.8) with j = 1, U = Φ q -ψ q one can write

A 1 = Ω 0 e 2δgε |℘∂ z Φ 2,q Λ 1 ( Φ q -ψ q )J 1 |dX + 2δ Ω 0 e 2δgε |∇g ε ℘ 2 ∂ z Φ 2,q ( Φ q -ψ q )J 1 |dX := A 1,1 + A 1,2 . Since J j L ∞ x,z
≤ F, we may estimate

A 1,1 ≤ F Ω 0 e 2δgε |℘∂ z Φ 2,q Λ 1 ( Φ q -ψ q )|dX ≤ F ℘e δgε ∂ z Φ 2,q L 2 ( Ω 0 ) e δgε Λ 1 ( Φ q -ψ q ) L 2 ( Ω 0 ) .
On the other hand, there holds (3.9)

f 1 f 2 L 2 (J,L 2 (R d )) ≤ k χ k f 1 χ k f 2 L 2 (J,L 2 (R d )) ≤ f 1 L 2 (J,L 2 (R d ) ul ) k χ k f 2 L ∞ (J×R d ) .
Now we choose δ > 0 such that

(3.10) δF 1 ( η 2 H s+ 1 2 ul ) ≤ 1 2
then the condition of Lemma 3.4 is fulfilled with µ = 2δ. It then follows from (3.9) and Lemma 3.2 that (3.11)

℘e δgε ∂ z Φ 2,q L 2 ( Ω 0 ) ≤ w℘w -1 e δ x-q ∂ z Φ 2,q L 2 (J,L 2 (R d )) ≤ w℘ L 2 (J,L 2 (R d ) ul ) k χ k w -1 e δ x-q ∂ z Φ 2,q L ∞ (J×R d ) ≤ F wη H 1 2 ul w -1 ψ 2,q H s .
Therefore,

A 1,1 ≤ F e δgε Λ 1 ( Φ q -ψ q ) L 2 ( Ω 0 ) wη H 1 2 ul w -1 ψ 2,q H s .
For A 1,2 we have

A 1,2 ≤ 2δF e δgε ℘ 2 ∂ z Φ 2,q L 2 ( Ω 0 ) e δgε ( Φ q -ψ q ) L 2 ( Ω 0 ) .
The first L 2 -norm on the right-hand side is already estimated by (3.11). For the second term on the right-hand side, one applies the Poincaré inequality in Lemma 2.9 (with

O = Ω 1,1 ∪ Ω 2,1 , which is diffeomorphic to Ω 0 ) to the image of ( Φ q -ψ q )
under the inverse of ρ 1 (x, z), (x, z) ∈ Ω 0 and then changes the variables back to Ω 0 to derive (3.12)

e δgε ( Φ q -ψ q ) L 2 ( Ω 0 ) ≤ F e δgε ∂ z ( Φ q -ψ q ) L 2 ( Ω 0 ) ≤ F e δgε Λ 1 ( Φ q -ψ q ) L 2 ( Ω 0 )
from which we deduce that A 1,2 satisfies the same estimate as A 1,1 does and hence, so does A 1 , i.e., (3.13)

A 1 ≤ F e δgε Λ 1 ( Φ q -ψ q ) L 2 ( Ω 0 ) wη H 1 2 ul
w -1 ψ 2,q H s .

Step 3. (Estimates for A 2 , A 3 ) By Lemma 3.5 we have

(Λ 1 -Λ 2 )θ = ℘e 2δgε ∂ z ( Φ q -ψ q ).
It follows that

A 2 ≤ F ℘e δgε Λ 2 Φ 2,q L 2 ( Ω 0 ) e δgε ∂ z ( Φ q -ψ q ) L 2 ( Ω 0 ) .
Using the definition of Λ 2 and the same method as in (3.11) one obtains that the first term is also bounded by the right-hand side of (3.11). On the other hand, it is easy to see the second term is bounded by F e δgε Λ 1 ( Φ q -ψ q ) L 2 ( Ω 0 ) . Therefore, A 2 also satisfies the bound (3.13).

For A 3 one uses the formula (3.8) to get

A 3 ≤ A 3,1 + A 3,2 with A 3,1 = Ω 0 e 2δgε |Λ 2 Φ 2,q Λ 2 ( Φ q -ψ q )(J 1 -J 2 )|dX, A 3,2 = δ Ω 0 e 2δgε |∇g ε Λ 2 2 Φ 2,q ( Φ q -ψ q )(J 1 -J 2 )|dX. First, A 3,2 is estimated by (J 1 -J 2 )e δgε Λ 2 Φ 2,q L 2 ( Ω 0 ) e δgε ( Φ q -ψ q ) L 2 ( Ω 0 )
. The second term is estimated by (3.12) and the first term is estimated as in (3.11) with

℘ replaced by J 1 -J 2 which satisfies w(J 1 -J 2 ) L 2 (J,L 2 ul ) ≤ F wη H 1 2 ul
. Similarly,

A 3,1 ≤ (J 1 -J 2 )e δgε Λ 2 Φ 2,q L 2 ( Ω 0 ) e δgε Λ 2 ( Φ q -ψ q ) L 2 ( Ω 0 ) .
We only need to study the second term on the right-hand side. With u := Φ q -ψ q one has Λ

2 1 u = ∂zρ 1 ∂zρ 2 Λ 1 1 u which implies e δgε Λ 2 1 u L 2 ( Ω 0 ) ≤ F e δgε Λ 1 1 u L 2 ( Ω 0 ) . On the other hand, Λ 2 2 u = ∇ x u - ∇ x ρ 2 ∂ z ρ 2 ∂ z u = Λ 1 2 u + ∇ x ρ 1 ∂ z ρ 1 - ∇ x ρ 2 ∂ z ρ 2 ∂ z ρ 1 1 ∂ z ρ 1 ∂ z u .
Hence,

e δgε Λ 2 2 u L 2 ( Ω 0 ) ≤ F e δgε Λ 1 u L 2 ( Ω 0 ) and e δgε Λ 2 u L 2 ( Ω 0 ) ≤ F e δgε Λ 1 u L 2 ( Ω 0 ) .
In conclusion, we have proved that: for any (small) δ > 0 satisfying (3.10), there holds

Ω Λ 1 Φ q Λ 1 θJ 1 dX ≤ F e δgε Λ 1 ( Φ q -ψ q ) L 2 ( Ω 0 ) wη H 1 2 ul w -1 ψ 2,q H s . (3.14)
Step 4. Next, in view of (3.8) we write

(3.15) Ω Λ 1 Φ q Λ 1 θJ 1 dX = Ω e 2δgε Λ 1 Φ q Λ 1 ( Φ q -ψ q )J 1 dX + 2δ Ω Λ 1 2 Φ q .( Φ q -ψ q )e 2δgε ∇g ε J 1 dX = Ω e 2δgε |Λ 1 ( Φ q -ψ q )| 2 J 1 dX + Ω e 2δgε Λ 1 ψ q Λ 1 ( Φ q -ψ q )J 1 dX + 2δ Ω Λ 1 2 ( Φ q -ψ q )( Φ q -ψ q )e 2δgε ∇g ε J 1 dX + 2δ Ω Λ 1 2 ψ q ( Φ q -ψ q )e 2δgε ∇g ε J 1 dX := B 1 + B 2 + B 3 + B 4 .
Owing to the Poincaré inequality in Lemma 2.21 (applied with O = Ω 1 ) and a change of variables one has

(3.16) |B 3 | ≤ δF 2 ( η 1 H s+ 1 2 ul ) e δgε Λ 1 ( Φ q -ψ q ) 2 L 2 ( Ω)
where F 2 : R + → R + is a non decreasing function. Likewise, (3.17)

|B 4 | ≤ δF 2 ( η 1 H s+ 1 2 ul ) e δgε Λ 1 ψ q L 2 ( Ω) Λ 1 ( Φ q -ψ q )e δgε L 2 ( Ω) .
Finally, it is clear that

(3.18) |B 2 | ≤ F 2 ( η 1 H s+ 1 2 ul ) e δgε Λ 1 ψ q L 2 ( Ω) Λ 1 ( Φ q -ψ q )e δgε L 2 ( Ω) .
Now, remark that there exists a constant c 0 depending only on h such that

|J 1 | ≥ c 0 . Choose δ > 0 satisfying (3.19) δ F 1 ( η 2 H s+ 1 2 ul ) + F 2 ( η 1 H s+ 1 2 ul ) = min( c 0 2 , 1 2 ).
A combination of (3.14)-(3.18) yields

e δgε Λ 1 ( Φ q -ψ q ) L 2 ( Ω) ≤ F wη H 1 2 ul w -1 ψ 2,q H s + e δgε Λ 1 ψ q L 2 ( Ω) .
Step 5. Now, letting ε → 0 and taking into account properties (i), (ii) of ψ q in section 2.1.2 lead to

(3.20) e δ x-q Λ 1 Φ q L 2 ( Ω) ≤ F wη H 1 ul w -1 ψ 2,q H s + e δ x-q Λ 1 ψ q L 2 ( Ω) ≤ F wη H 1 ul w -1 ψ 2,q H s + ψ q H 1 2
.

Hence e δ x-q ∇ x,z Φ q L 2 ( Ω) ≤ F wη H 1 ul w -1 ψ 2,q H s + ψ q H 1 2
.

Consequently, (

χ p w∇ x,z Φ q L 2 (J,L 2 (R d )) ≤ e -δ p-q w(p)F wη H 1 2 ul w -1 ψ 2,q H s + ψ q H 1 2 (Ω) 3.21)
≤ e -δ p-q w(p)w(q) -1 F wη

H 1 2 ul ψ 2,q H s + wψ q H 1 2
.

Finally, we get (3.22)

χ p w∇ x,z Φ L 2 (J,L 2 (R d )) ≤ q χ p w∇ x,z Φ q L 2 (J,L 2 (R d )) ≤ F wη H 1 2 ul ψ 2 H s ul + wψ H 1 2 ul .
Step 6. It remains to prove that

χ p w∇ x,z Φ L ∞ (J,H -1 2 (R d ))
is bounded by the right hand side of (3.7). The estimate of

χ p w∇ x Φ L ∞ (J,H -1 2 (R d ))
follows from (3.22) and the interpolation Lemma 2.8. By the same lemma, for χ p w∂ z Φ

L ∞ (J,H -1 2 (R d ))
it remains to

estimate χ p w∂ 2 z Φ L 2 (J,H -1 (R d ))
. For this purpose we use equation (3.26) below, satisfied by Φ to have (3.23)

χ p w∂ 2 z Φ L 2 (J,H -1 (R d )) ≤ χ p wα 1 ∆ Φ L 2 (J,H -1 (R d )) + χ p wβ 1 .∇∂ z Φ L 2 (J,H -1 (R d )) + χ p wγ 1 ∂ z Φ L 2 (J,H -1 (R d )) + χ p wF L 2 (J,H -1 (R d )) .
In the above inequality, α 1 , β 1 , γ 1 are defined as in (2.10), for the diffeomorphism ρ defined in term of η 1 .

Because -1 < s -2, the estimate (3.28) applied with f = ψ 2 implies the desired estimate for

χ p wF L 2 (J,H -1 (R d ))
. Concerning the other terms, the product rule (2.16) gives (3.24)

χ p wα 1 ∆ Φ L 2 (J,H -1 ) ≤ C χ p α 1 L ∞ (J,H s-1 2 ) χ p w∆ Φ L 2 (J,H -1 ) χ p wβ 1 • ∇∂ z Φ L 2 (J,H -1 ) ≤ C χ p β 1 L ∞ (J,H s-1 2 ) χ p w∇∂ z Φ L 2 (J,H -1 ) χ p wγ 1 ∂ z Φ L 2 (J,H -1 ) ≤ C χ p γ 1 L ∞ (J,H s-3 2 ) χ p w∂ z Φ L 2 (J,L 2 ) .
Owing to (3.22) we are left with the estimates for the first term on the right-hand side of the above inequalities. Again, this is done along the same line as in the proof of Lemma 2.

3 noticing that η ∈ C ∞ b ⊂ H ∞ ul .
This completes the proof. We are now in position to derive the weighted estimate for the Dirichlet-Neumann operator:

Theorem 3.5. Assume that s > 1 + d 2 .
Then for every w ∈ W po ( ), ≥ 0 there exists F : R + → R + non decreasing such that for all η 1 , η 2 ∈ H

s+ 1 2 ul (R d ) and f ∈ H s ul (R d ) we have w[G(η 1 ) -G(η 2 )]f H s-3 2 ul ≤ F( (η 1 , η 2 ) H s+ 1 2 ul ×H s+ 1 2 ul ) w(η 1 -η 2 ) H s-1 2 ul f H s ul .
Proof. Let Φ j defined as in (3.6) (with ψ j = f , j = 1, 2) and Φ j be its image via the diffeomorphism (x, z) → (x, ρ j (x, z)) given by (3.2). We have from definition (2.8) of the Dirichlet-Neumann operator

(3.25) G(η j )f = 1 + |∇ x ρ j | 2 ∂ z ρ j ∂ z Φ j -∇ x ρ j ∇ x Φ j    z=0 .
Recall from (2.9) that each Φ j satisfies the elliptic problem

(∂ 2 z + α j ∆ x + β j • ∇ x ∂ z -γ j ∂ z ) Φ j = 0
, where α j , β j , γ j are defined as in (2.10) with η replaced by η j .

Set Φ = Φ 1 -Φ 2 then (3.26) ∂ 2 z + α 1 ∆ x + β 1 • ∇∂ z -γ 1 ∂ z Φ = F with F = {(α 2 -α 1 )∆ x + (β 2 -β 1 ) • ∇∂ z -(γ 2 -γ 1 )∂ z } Φ 2 .
We fix z 0 ∈ (-1, 0) and set I 0 = (z 0 , 0), I = (-1, 0). We first prove that

(3.27) w∇ x,z Φ X s-3 2 ul (I 0 ) ≤ F( (η 1 , η 2 ) H s+ 1 2 ul ×H s+ 1 2 ul ) w(η 1 -η 2 ) H s-1 2 ul f H s ul .
To prove the preceding estimate, we claim that

(3.28) wF L 2 (I,H s-2 ) ul ≤ F( (η 1 , η 2 ) H s+ 1 2 ul ×H s+ 1 2 ul ) w(η 1 -η 2 ) H s-1 2 ul f H s ul .
Indeed, the H s ul version of the product rule (2.16) (see Proposition 7.3 (i), [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF]) applied with s 0 = s -2, s 1 = s -1, s 2 = s -2 yields (3.29)

wF L 2 (I,H s-2 ) ul w(α 2 -α 1 ) L 2 (I,H s-1 ) ul ∆ x Φ 2 L ∞ (I,H s-2 ) ul + w(β 2 -β 1 ) L 2 (I,H s-1 ) ul ∇∂ z Φ 2 L ∞ (I,H s-2 ) ul + w(γ 2 -γ 1 ) L 2 (I,H s-2 ) ul ∂ z Φ 2 L ∞ (I,H s-1 ) ul .
On the other hand, applying Proposition A.7 for the 1 2 -smoothing effect of the Poisson kernel in weighted spaces gives

(3.30) w(α 2 -α 1 ) L 2 (I,H s-1 ) ul + w(β 2 -β 1 ) L 2 (I,H s-1 ) ul + w(γ 2 -γ 1 ) L 2 (I,H s-2 ) ul ≤ F( (η 1 , η 2 ) H s+ 1 2 ul ×H s+ 1 2 ul ) w(η 1 -η 2 ) H s-1 2 ul .
Remark that in Corollary 2.13 one can replace the assumption z 0 ∈ (-1, 0) by z 0 ∈ J = (-2, 0) because on (-2, 1] the diffeomorphism ρ j satisfies the same bounds as the diffeomorphism defined in (2.3) does (again, this is true because η ∈ C ∞ b (R d )). This remark applied with σ = s -1 and w ≡ 1 leads to

(3.31) ∇ x,z Φ 2 L ∞ (I,H s-1 ul ) ≤ F( η 2 H s+ 1 2 ul ) f H s ul .
Putting together (3.29), (3.30), (3.31) one obtains the claim (3.28).

Since Φ| z=0 = 0, with the aid of Theorem 2.6 (which is applicable since ρ 1,j and ρ in (2.3) have exactly the same form), the proof of (3.27) now reduces to estimate

w∇ x,z Φ X -1 2 ul (I)
. This is a consequence of Theorem 3.3 applied with ψ 1 = ψ 2 = f (and the fact that I ⊂ J).

In view of (3.25), to obtain the bound for w

[G(η 1 )f -G(η 2 )]f in H s-3 2 ul
it is necessary to bound w∇ x,z Φ H s-3 2 ul at z = 0. More precisely, we shall prove that

w∇ x,z Φ| z=0 H s-3 2 ul ≤ F( (η 1 , η 2 ) H s+ 1 2 ul ×H s+ 1 2 ul ) w(η 1 -η 2 ) H s-1 2 ul f H s ul .
To this end, we use the argument in step 6 of the proof of Theorem 3.3. By virtue of Lemma 2.8 and (3.27), we then only need to estimate w∂ 2 z Φ L 2 (I 0 ,H s-2 ) ul , which in turn follows by using equation (3.26) together with the estimate (3.28) and the product rule (A.22). Finally, using (3.25) and the product rule (A.22) once again, we conclude the proof of Theorem 3.5.

Remark 3.6. Theorem 3.3 is also a crucial ingredient in proving contraction of the remainder R appearing in the reformulation of water waves system-equation (4.4) in Proposition 4.2, [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF]. Notice that our estimate (3.7) is sufficient for this purpose because w(ψ 1 -ψ 2 )

H 1 2 ul ≤ w(ψ 1 -ψ 2 ) H s-1 ul owing to the fact that s > 1 + d 2 .
4. Proof of the main results 4.1. Proof of Theorem 1.6. The contraction estimate in Theorem 1.6 was proved in [START_REF] Alazard | On the Cauchy problem for gravity water waves[END_REF] (see Theorem 5.1) for classical Sobolev spaces and then in [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF] for Kato's spaces. Both use the following scheme: 1) study the Dirichlet-Neumann operator: bound estimates and paralinearization 2) contraction estimate for the Dirichlet-Neumann operator 3) paralinearization of the difference equations (after reformulation) 4) estimates for the good unknown 5) back to the original unknowns.

Here, we shall follow the same scheme as above. The first two items are the real new points in our problem and have been studied in Section 2 and 3. For the last three items we need a para-differential machinery in Kato's spaces with weights and this is established in Appendix A. The key point in this machinery is that: whenever we estimate S(u, v) in weighted norms, where S is an operator of two variables, we are always able to shift the weight to u or v. Having this in hand, items 3), 4), 5) follow line by line those in [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF] and [START_REF] Alazard | On the Cauchy problem for gravity water waves[END_REF]: one only need to replace • H σ ul or • H σ by w • H σ ul in the relevant estimates (w is the weight). We conclude the proof. 4.2. Proof of Corollary 1.7. We need to show how (1.9) implies (1.10). To this end, it suffices to prove that there exist 0 < T 1 ≤ T and N > 0 (both are independent of U j ) such that (4.1)

U j L ∞ ([0,T 1 ],H s ul ) ≤ N, j = 1, 2. Define the Sobolev norms of the solutions as

M j σ (τ ) = U j L ∞ ([0,τ ],H σ ul )
, ∀τ ∈ [0, T ], ∀j ≥ 0. Let us recall the a priori estimate derived in [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF]: for any 1 + d 2 < σ ≤ s and T > 0 one can find a non decreasing function F : R + → R + such that (4.2)

M j s (T ) ≤ F M j σ (0) + T M j σ (T ) M j s (0) + T M j s (T ) , ∀j ≥ 0.

Fix s 0 ∈ 1 + d 2 , s . Since each U j is a solution to the gravity waters system in C 0 ([0, T ], H s 0 ul ), the estimate (4.2) gives for some non decreasing F 1 : R + → R + (independent of U ) M j s 0 (τ ) ≤ F 1 (M j s 0 (0) + τ M j s 0 (τ )), ∀τ ∈ [0, T ], ∀j ≥ 0. According to Theorem 1.5, U j is continuous in time with value in H s 0 ul since s 0 < s. Consequently, M j s 0 (τ ) is continuous in τ . In addition, M j s 0 (0) can be bounded by some constant independent of j, say A. The standard argument then gives the existence of T 0 ∈ (0, T ] and N > 0, both are independent of U j (but depend on A), such that (4.3)

M j s 0 (τ ) ≤ N, ∀τ ∈ [0, T 0 ], ∀j ≥ 0. Applying again the estimate (4.2) with σ = s 0 < s we get for some non-decreasing function F : R + → R + (independent of U )

M j s (τ ) ≤ F(M j s 0 (0) + T 0 M j s 0 (τ )) M j s (0) + τ M j s (τ ) , ∀τ ∈ [0, T 0 ], ∀j ≥ 0. By (4.3), this implies M j s (τ ) ≤ F(N (1 + T 0 )) M j s (0) + τ M j s (τ ) , ∀τ ∈ [0, T 0 ]. Now, let T 1 ∈ (0, T 0 ] satisfying T 1 F(N (1 + T 0 )) ≤ 1 2 one deduces
M j s (T 1 ) ≤ 2F(N (1 + T 0 ))M j s (0), ∀j ≥ 0 which concludes the proof.

Appendix A. Paradifferential calculus in Kato's spaces with weights

In this section, we adapt the paradifferential machinery for the presence of weights which can be of independent interest. The proofs of these results follow those in [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF] but we need to take some care (so we only present the proof whenever it is necessary). We recall first various spaces which will be used in the sequel.

Definition A.1. Let p ∈ [1, +∞], J = (z 0 , 0), z 0 < 0 and σ ∈ R. 1. The space L p (J, H σ (R d )) ul is defined as the space of measurable functions u from J z × R d x to C such that u L p (J,H σ (R d )) ul := sup q∈Z d χ q u L p (J,H σ (R d )) < +∞. 2. We set X σ ul (J) = L ∞ (J, H σ (R d )) ul ∩ L 2 (J, H σ+ 1 2 (R d )) ul Y σ ul (J) = L 1 (J, H σ (R d )) ul + L 2 (J, H σ-1 2 (R d )) ul
endowed with their natural norms. The same spaces without subscript "ul" are defined for classical Sobolev spaces.

Notice that L ∞ (J, H σ (R d )) ul = L ∞ (J, H σ ul (R d )).
Notation A.2. For t ∈ R, we denote t the smallest integer strictly greater than or equal t.

A.1. Weighted continuity of pseudo-differential operators. In [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF], the authors proved the continuity of pseudo-differential operators on the framework of L 2 based uniformly local Sobolev spaces. Here, we perform similar results with the presence of weights in classes W po ( ), > 0 (see Definition 1.2), which are composed of functions that are at most polynomial growth. For the sake of clarity, let us redefine this class.

Definition A.3. For every ≥ 0, we define W po ( ) to be the class of all functions w : R d → (0, ∞) satisfying the following conditions:

(i) r 1 := ∇w -1
w -1 and r 1 := ∇w w belong to C ∞ b (R d ), where w -1 (x) = 1/w(x),

(ii) for any C 1 > 0, there exists C 2 > 0 such that for any x 0 ∈ R d , there hold

w(x) ≤ C 2 w(x 0 ) and w(x) -1 ≤ C 2 w(x 0 ) -1 ∀x ∈ R d , |x -x 0 | ≤ C 1 , (iii
) there exists C > 0 such that for any x, y ∈ R d we have w(x)w -1 (y) ≤ C x -y .

Remark A.4. 1. For all λ ∈ R and C > 1, the functions x λ , ln(C + |x| 2 ) belong to W po (|λ|) and W po (1) respectively. For every t ∈ R \ {0}, the function e t x does not belong to any class W pol ( ).

If w

j ∈ W po ( j ), j = 1, 2 then w 1 w 2 ∈ W po ( 1 + 2 ). 3. If w ∈ W po ( ) and w > 0 then w λ ∈ W po ( |λ|) for any λ ∈ R. 4. Condition (iii) in Definition A.3 is equivalent to (iii ) there exist M 1 , M 2 > 0 and λ 1 , λ 2 ∈ R such that M 1 x λ 1 ≤ w(x) ≤ M 2 x λ 2 , ∀x ∈ R d .
Although (iii ) seems to be more natural than (iii), in the following proofs, condition (iii) is more convenient.

We denote by S m 1,0 the standard Hörmander's class of symbols p

∈ C ∞ (R d × R d ) satisfying |D α ξ D β x p(x, ξ)| ≤ C α,β (1 + |ξ|) m-|α| ∀α, β ∈ N d , ∀(x, ξ) ∈ R d × R d .
Proposition A.5. Let P be a pseudo-differential operator whose symbol p belongs to S m 1,0 and let w be a weight in W po ( ), ≥ 0. Then for any s ∈ R, there exists C > 0 such that wP u H s ul ≤ C wu H s+m ul , provided that the right hand side is finite.

Proof. We write (A.1)

wχ k P u = |k-q|≤2 wχ k P χ q u + |k-q|>3 wχ k P χ q u := A + |k-q|>3 B k,q .
Since χ q u = (χ q w(x)u)( χ q w(x) -1 ) ∈ H s+m (R d ), we have by properties (i), (ii), (iii) of the weight w, the product rule (2.16) and the classical pseudo-differential calculus that

wχ k P χ q u H s ≤ Cw(k) P χ q u H s ≤ Cw(k) χ q u H s+m ≤ Cw(k)w(q) -1 wχ q u H s+m ≤ C k -q wχ q u H s+m ≤ C wu H s+m ul , provided |k -q| ≤ 2. Thus, (A.2) A ≤ C wu H s+m ul .
To bound the second part, we fix n 0 ∈ N, n 0 ≥ s. We shall prove

(A.3) D α x B k,q L 2 (R d ) ≤ C k -q d+1 wu H s+m ul , |α| ≤ n 0 which implies the desired estimate for |k-q|>3 B k,q . By the presence of χ k , D α x B k,q L 2 (R d ) ≤ C D α x B k,q L ∞ (R d ) . We have D α x B k,q (x) = D α x K(x, •), χ q u with (A.4) K(x, y) = (2π) -d R d e i(x-y)ξ p(x, ξ)dξχ k (x)w(x) χ q (y). Fix n 1 ∈ N, n 1 ≥ -(s + m) and β ∈ N d , |β| ≤ n 1 . Let γ ∈ N d be such that |γ| = N with (A.5) N ≥ max(m + n 0 + n 1 + d + 1, + d + 1).
Multiplying D α x D β y K(x, y) by (x -y) γ and integrating by parts with a remark that |x -y| ≥ δ|k -q| (for some δ > 0) on the support of χ k (x)w(x) χ q (y), we obtain

D α x D β y K(x, y) ≤ C β,d,λ k -q N w(k) |β 1 |≤|β| ∂ β 1 χ q (y) . It follows that (A.6) |D α x B k,q (x)| ≤ D α x K(x, •) H -(s+m) χ q u H s+m ≤ C k -q N w(k) χ q u H s+m ≤ C k -q N w(k)w(q) -1 χ q wu H s+m ≤ C k -q N k -q wu H s+m ul ≤ C k -q d+1 wu H s+m ul which proves (A.3).
In a particular case the proof above gives the following more precise result.

Proposition A.6. Let m ∈ R, and w ∈ W po ( ), ≥ 0. Let h(ξ) = h ξ |ξ| |ξ| m ψ(ξ) with h ∈ C ∞ (S d-1 ), and
(A.7) ψ ∈ C ∞ (R d ), ψ(ξ) = 1 if |ξ| ≥ 1, ψ(ξ) = 0 if |ξ| ≤ 1 2 .
Then for every s ∈ R and

(A.8) r > m + s + -(m + s) + + 3d 2 + 1,
there exists a constant C such that

wh(D x )u H s ul (R d ) ≤ C h H r (S d-1 ) wu H s+m ul (R d )
, provided that the right-hand side is finite.

Remark that the condition on r above comes from the choice of N in (A.5), plus d 2 + ε derivatives from Sobolev embeddings. Next, tracking the proof of Lemma 7.10 in [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF] and Proposition A.5 above, we easily obtain the following proposition.

Proposition A.7. Let r > 0, and m ∈ R and w ∈ W po ( ), ≥ 0 . Let p ∈ S r 1,0 (R d ), a ∈ S m 1,0 (R d ) be two symbols with constant coefficients. We assume that there exists c 0 > 0 such that for all ξ ∈ R d we have p(ξ) ≥ c 0 |ξ| r . Then for all s ∈ R and I = [0, T ], one can find a positive constant C such that

(A.9) we -tp(D) a(D)u L ∞ (I,H s ) ul + we -tp(D) a(D)u L 2 (I,H s+ r 2 ) ul ≤ C wu H s+m ul ,
provided that the right-hand side is finite.

A.2. Para-differential calculus with weights. Assuming the theory of paradifferential calculus for classical Sobolev spaces (see [START_REF] Metivier | Para-differential calculus and applications to the Cauchy problem for nonlinear systems[END_REF]) and for uniformly local Sobolev spaces (see [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF]), we present in this section such a theory with the presence of weights. Given m ∈ R, ρ ≥ 0 we denote by Γ m ρ (R d ) the class of symbols of order m and by T a the associated para-differential operator as in Definition 7.15, [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF]. In particular, Γm ρ (R d ) denotes the subspace of Γ m ρ (R d ) which consists of symbols a(x, ξ) homogeneous of degree m with respect to ξ. To deal with the weights in the class W po ( ), for any symbol a ∈ Γ m ρ and any real number s, let us define the semi-norm

(A.10) M m ρ (a, s) = sup |α|≤I(d,m,s) sup |ξ|≥ 1 2 (1 + |ξ|) |α|-m ∂ α ξ a(•, ξ) W ρ,∞ (R d ) ,
where I(d, m, s) is the smallest even integer strictly greater than

(A.11) m + s + -(m + s) + + 5d 2 + 2.
If a is a symbol independent of ξ, the associated operator T a is called a paraproduct and we have the formal decomposition of Bony au = T a u + T u a + R(a, u).

A.2.1. Symbolic calculus. The following technical lemmas will be used in proving results on symbolic calculus.

Lemma A.8. Let µ ∈ R, w ∈ W po ( ), ≥ 0 and N ≥ + d + 1. Then there exists C > 0 such that (A.12) sup

x∈R d w(x) x -• -N u H µ (R d ) ≤ C wu H µ ul (R d )
provided that the right hand side is finite.

Proof. We write w(x) x -y -N χ q (y)u(y) = w(x)w(y) -1 1 x -q N

x -q N x -y N χ q (y)w(y)χ q (y)u(y).

Since the function y → x-q N

x-y N χ q (y) belongs to W ∞,∞ (R d ) with semi-norms uniformly bounded (independently of x and q), we deduce that

w(x) x -• -N u H µ (R d ) ≤ q∈Z d w(x) x -• -N χ q u H µ ≤ C N q∈Z d w(x)w(q) -1
x -q N wu H µ ul ≤ C N q∈Z d

x -q ρ x -q N wu H µ ul ≤ C N wu H µ ul .

Combining this lemma and the proof of Lemma 7.13, [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF], we obtain Lemma A.9. Let w ∈ W po ( ), ≥ 0. Let χ ∈ C ∞ 0 (R d ) and χ ∈ C ∞ 0 (R d ) be equal to one on the support of χ. Let ψ, θ ∈ S(R d ). For every m, σ ∈ R there exists a constant C > 0 such that (A. [START_REF] Lions | Problèmes aux limites non homogènes[END_REF] j≥-1

wχ k ψ(2 -j D)((1 -χ k )u)θ(2 -j D)v H m (R d ) ≤ C wu H σ ul (R d ) v L ∞ (R d ) .
For every m, σ, t ∈ R one can find a constant C > 0 such that (A.14)

j≥-1

wχ k ψ(2 -j D)((1 -χ k )u)θ(2 -j D)v H m (R d ) ≤ C wu H σ ul (R d ) v H t ul (R d )
and (A.15)

j≥-1

wχ k ψ(2 -j D)((1 -χ k )u)θ(2 -j D) χ k v H m (R d ) ≤ C wu H σ ul (R d ) χ k v H t (R d ) .
Remark A.10. It follows easily from the proof of the above lemma that the same estimates as in (A.13), (A.14) and (A.15) hold if on the left-hand sides 2 -j is replaced by 2 -j-j 0 where j 0 ∈ Z is fixed. We shall use this remark to deal with paraproduct estimates.

It turns out that the symbolic calculus with weights possesses the same features as in the usual setting. (iii) Let a ∈ Γ m ρ (R d ) and denote by T * a the adjoint operator of T a and by a * the complex conjugate of a (in case a is a matrix, a * is its conjugate transpose). Then for all µ ∈ R there exists a constant C > 0 such that Proof. We give the proof for the first assertion only since these three points are proved along the same lines. For simplicity we shall consider symbols in Γm ρ (R d ).

Step 1. Consider first the case where a is a bounded function and write χ k wT a u = χ k wT a ( χ k u) + χ k wT a ((1 -χ k )u).

The classical theory gives

χ k wT a ( χ k u) H µ ≤ Cw(k) a L ∞ χ k u H µ ≤ Cw(k) a L ∞ w(k) -1 χ k wu H µ ≤ C a L ∞ wu H µ ul .
The estimate for the second term follows immediately from (A.13).

Step 2. Next we consider the case a(x, ξ) = b(x)h(ξ) where h(ξ) = |ξ| m h ξ |ξ| with h ∈ C ∞ (S d-1 ). Then directly from the definition we have T a = T b (ψh)(D x ), where the cut-off ψ is given by (A.7). The desired estimate in (i) follows from Step 1 and Proposition A. By the Weyl formula we know that λ ν ∼ cν 1 d . In addition, there exists a positive constant K such that for all ν ≥ 1 (A.17) h ν H r (S d-1 ) ≤ Kλ r ν .

Now using the steps above and Proposition A.6 we obtain (ψ is given by (A.7)) A.3. Transport equations. For the sake of completeness we present a weighted version of Lemma 7.19, [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF] on transport equations in uniformly local Sobolev spaces. The result holds for weights in a large class than W while its proof is a direct consequence of Lemma 7.19, [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF]. We denote by W 0 the class of all functions w : R d → (0, ∞) such that ∇w w ∈ C ∞ b (R d ).

wT a u H µ ul ≤ ν≥1 T bν (ψh ν )(D x )u H µ ul ≤ C ν≥1 b ν L ∞ (R d ) h ν H r (S d

2 .

 2 A weighted description for the Dirichlet-Neumann operator Notation 2.1. Throughout this paper, we denote C > 0 and F : R + → R + are multiplicative constants and functions that may change from line to line within a proof. The notation A B means that there exist C > 0 such that A ≤ CB. 2.1. Definition of the Dirichlet-Neumann operator. In this sections, we drop the time dependence of the domain and work on the domain of the form.

Remark 2 . 15 .

 215 Several comments are in order about Proposition 2.14. Let us recall a relating result on the exponential decay of the Dirichlet-Neumann operator (for d = 1) obtained by Ming-Rousset-Tzvetkov [15]: Proposition 2.16 ([15, Proposition 3.2]). Assume that η * = -H and ψ ∈ C ∞ b (R) having an exponential decay:

2. 3 .

 3 Paralinearization of the Dirichlet-Neumann operator. We denote by κ the principal symbol of the Dirichlet-Neumann operator:

Theorem A. 11 .

 11 Let m, m ∈ R, ρ ≥ 0 and w ∈ W po ( ), ≥ 0. (i) If a ∈ Γ m 0 (R d ), then for all µ ∈ R, there exist a constant C > 0 such that wT a u H µ ul (R d ) ≤ CM m 0 (a, µ) wu H µ+m ul (R d ) . (ii) If a ∈ Γ m ρ (R d ), b ∈ Γ m ρ (R d ) then, for all µ ∈ R, there exist a constant C > 0 such that w(T a T b -T a b )u H µ ul (R d ) ≤ C M m ρ (a, µ) M m 0 (b, µ) + M m 0 (a, µ) M m ρ (b, µ) wu H µ+m+m -ρ ul (R d )

  w(T * a -T b )u H µ ul (R d ) ≤ CM m ρ (a, µ) wu H µ+m-x a * (x, ξ).

6 . 3 .S d- 1 ∆ I 2 ω 2 + 1

 631221 Step Finally, for the general case we introduce ( h ν ) ν∈N * an orthonormal basis of L 2 (S d-1 ) consisting of eigenfunctions of the (self adjoint) Laplace Beltrami operator∆ ω = ∆ S d-1 on L 2 (S d-1 ) i.e. ∆ ω h ν = λ 2 ν h ν . Setting h ν = |ξ| m h(ω), ω = ξ |ξ| when ξ = 0, we can write a(x, ξ) = ν∈N * b ν (x)h ν (ξ) where b ν (x) = S d-1 a(x, ω) h ν (ω)dω. With I = I(d, m, µ) we have λ I ν b ν (x) = a(x, ω) h ν (ω)dω, which gives (A.16) b ν L ∞ (R d ) ≤ Cλ -I ν M m 0 (a, µ). By definition of I, we can find an integer r such that m + µ + -(m + µ) + + 3d < r < I -d.

Lemma A. 15 .

 15 Let I = [0, T ], s 0 > 1 + d 2 , µ > 0 and w ∈ W 0 . Then there exists F : R + → R + non decreasing such that for V j ∈ L ∞ (I,H s 0 (R d )) ul ∩ L ∞ (I, H µ (R d )) ul j = 1, d, wf ∈ L 1 (I, H µ (R d )) ul , wu 0 ∈ H µ ul (R d ) and any solution u with wu ∈ L ∞ (I, H s 0 (R d )) ul to the problem (A.23) (∂ t + V • ∇)u = f, u| t=0 = u 0we have(A.24) wu L ∞ (I,H µ ) ul ≤ F T V L ∞ (I,H s 0 ) ul wu 0 H µ ul + wf L 1 (I,H µ ) ul + sup k∈Z d T 0 wu(σ) H s 0 ul χ k V (σ) H µ dσ .Proof. If u is a solution to the transport problem (A.23) then wu is a solution to(∂ t + V • ∇)(wu) = g, wu| t=0 = u 0 with g := f + uV • ∇w = f + uwV • r, r = ∇w w ∈ C ∞ b (R d ).We are in position to apply Lemma 7.19,[START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF] to havewu L ∞ (I,H µ ) ul ≤ F T V L ∞ (I,H s 0 ) ul wu 0 H µ ul + wg L 1 (I,H µ ) ul + sup k∈Z d T 0 wu(σ) H s 0 ul χ k V (σ) H µ dσ .On the other hand, for k ∈ Z d using the product rule (2.16) one has with n large enoughχ k wu(σ)V (σ) • r H µ ≤ C χ k wu(σ)V (σ) H µ χ k r H n Since r ∈ C ∞ b (R d ), χ k r H n canbe bounded by a constant independent of k. Finally, applying once again the product rule (2.16) (remark that s 0 > 1 + d2 ) givesχ k wu(σ)V (σ) H µ ≤ |q-k|≤M χ q wu(σ) H s 0 χ k V (σ) H µ ≤ C wu(σ) H s 0 ul χ k V (σ) H µ . Consequently, wuV • r L 1 (I,H µ ) ul = supk∈Z d T 0 χ k wu(σ)V (σ) • r H µ dσ ≤ C sup k∈Z d T 0 wu(σ) H s 0 ul χ k V (σ) H µ dσ, from which the estimate (A.24) follows.

  Corollary 2.13 implies the following weighted estimate for the Dirichlet-Neumann operator, which is of independent interest. Proposition 2.14. Let s > 1 + d 2 and η ∈ H s+ 1 2 (R d ) ul . Then the following statements hold true. (i) For every w ∈ W po ( ), ≥ 0 and -1 2 ≤ σ ≤ s -1 2 , one can find a non decreasing function F : R + → R + such that

		ul	2	ul wψ H σ+1	.
	wG(η)ψ H σ ul ≤ F η	ul H s+ 1 2	wψ H σ+1

ul

.

(ii) For every w ∈ W ex ( ) with ≤ * (defined by (2.22)) and -

1

2 ≤ σ ≤ s -1 2 , one can find a non decreasing function F : R +

  -1 ) wu H µ+m Let s 0 , s 1 , s 2 ∈ R satisfying s 0 ≤ s 2 and s 0 < s 1 + s 2 -d 2 .Then there exists C > 0 such that wT a u H s 0

				ul
	M m 0 (a, µ) wu H µ+m ul	ν≥1	ν	-I+r d
	M m 0 (a, µ) wu H µ+m ul	.
	A.2.2. Paraproducts.

Proposition A.12. Let w ∈ W po ( ), ≥ 0. ul ≤ C min a H s 1 ul wu H s 2 ul , wa H s 1 ul u H s 2 ul .

Acknowledgment. This work was partially supported by the labex LMH through the grant no ANR-11-LABX-0056-LMH in the "Programme des Investissements d'Avenir". I would like to send my deepest thanks to my advisor, Prof. Nicolas

Proof. We write (A. [START_REF] Shatah | Local well-posedness for fluid interface problems[END_REF] χ k wT a u = χ k wT a (1 -χ k )u + χ k wT χ k a χ k u + χ k wT (1-χ k )a χ k u.

By the classical result, we have (A. [START_REF] Wu | Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent[END_REF])

On the other hand, applying (A. [START_REF] Metivier | Para-differential calculus and applications to the Cauchy problem for nonlinear systems[END_REF] gives

and it follows form (A.15) (applied with w ≡ 1) that

Consequently, we obtain

2), we decompose

The proof is complete. [START_REF] Wu | Well-posedness in Sobolev spaces of the full water wave problem in 3-D[END_REF] wR(a, u)

(ii) If s 1 + s 2 > 0, s 0 ≤ s 1 and s 0 < s 1 + s 2 -d 2 then there exists a constant C > 0 such that

The first term is estimated by the same method as (A. [START_REF] Wu | Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent[END_REF]) with the use of Theorem 2.9 (i) in [START_REF] Alazard | On the Cauchy problem for gravity water waves[END_REF]. The remainder wχ k S k (a, u) is estimated by using (A.14) and (A.15). (ii) and (iii) are direct consequences of (i) and Proposition A.12.

Remark A.14. We remark that with the methods in the proofs above, the commutator estimate in Lemma 7.20, [START_REF] Alazard | Cauchy theory for the gravity water waves system with nonlocalized initial data[END_REF] still holds for uniformly local Sobolev spaces with the weight w ∈ W po ( ), ≥ 0.


Gizem Okyay
email: gizem.okyay@ecp.fr

Franck Enguehard

EFFECT OF FRACTAL PARAMETERS OF SOOT AGGREGATES ON THEIR ABSORPTION AND SCATTERING PROPERTIES SIMULATED BY DISCRETE DIPOLE APPROXIMATION

The objective of this study is to determine the impact of the fractal properties of methane soot issued from incomplete combustions on their absorption and scattering properties. The fractal properties are selected according to the bibliography. Sample numerical aggregates are generated accordingly. The absorption and scattering cross-sections, as well as the asymmetry factor are determined by our in-house developed DDA (discrete dipole approximation) code. Simulation results indicate a higher sensitivity of the radiative properties to the fractal dimension than to the prefactor. This sensitivity analysis aims to guide the choice of relevant characterization experiments of soot samples about the necessary precision on measurements and data collection, and also to decide on the extent of soot morphological properties to be considered in heat transfer simulations.

Introduction

The interaction of soot with radiation is of interest for the control of heat transfer within combustion chambers and the improvement of their design. The overall size of soot aggregates formed by incomplete combustions may reach several hundreds of nanometers. In high temperature applications, radiation is the main heat transfer phenomenon, and a strong interaction is expected between an incident radiation and soot aggregates reaching the size of the incident wavelength. Under such conditions, the complex morphology of these aggregates cannot be ignored in the radiation-aggregate interaction because the incident wave is able to probe the spatial arrangement of the material within the aggregate. In this work, the electromagnetic theory "DDA" is used to account for the complex shape of the aggregates in the computation of their radiative properties i.e. the features of their interaction with the electromagnetic radiation.

Soot aggregates

Inside a combustion chamber, the morphology, the composition and the number of monomers of a flame soot depend strongly on the operating conditions like the temperature, type of fuel, stoichiometry, type of flame. This research focuses on soot aggregates from the combustion of methane with air and oxygen. These aggregates are of complex shape formed by hundreds of sphere like carbonaceous monomers.

A complete 3D description of the aggregate is needed as input for the electromagnetic modelling. Unfortunately, the current microscopy techniques do not allow access to this information, which is one of the fundamental input data for DDA simulation.

Studies found in the literature propose the application of the fractal theory for the 3D geometry description of soot aggregates [START_REF] Megaridis | Morphological Description of Flame-Generated Materials[END_REF][START_REF] Mackowski | A simplified model to predict the effects of aggregation on the absorption properties of soot particles[END_REF]. It is therefore possible to correlate the geometrical parameters N (number of monomers in the aggregate), r (monomer radius), Rg (radius of gyration of the aggregate) by use of the dimensionless parameters kf (prefactor) and Df (fractal dimension) specific to the material and the conditions of its elaboration: A partial list of r, N, Df and kf values found in the literature for our study on methane soot is presented in Table 1. Df values are all around 1.7 while the range of kf values is much larger (from 1.2 to 6.5) depending on the experimental conditions. In order to quantify the impacts of these fractal parameters on the radiative properties, we generate, for each couple (Df , kf) examined, several samples of soot aggregates and calculate their interaction cross-sections with the DDA theory.

N =k f (R g / r ) D f (1)
The reference case selected for methane soot has the parameters Df = 1.7 and kf = 2.0. The fractal properties are slightly modified around the reference values to observe the variations in the absorption and scattering crosssections. The number of monomers per aggregate N ranges from a few tens to several hundreds and the monomer radius r from 5nm to 10nm according to the literature data. To represent correctly the methane soot and to ensure a high aggregate-radiation interaction, higher limits are selected as N = 500 and r = 10nm. Numerical soot aggregates are generated using the DLCCA algorithm of Mackowski [START_REF] Mackowski | A simplified model to predict the effects of aggregation on the absorption properties of soot particles[END_REF], as represented in Figure 1.

DDA simulations

The DDA theory, originally proposed by Purcell and Pennypacker [START_REF] Purcell | Scattering and absorption of light by nonspherical dielectric grains[END_REF], is now a well-known modeling tool and the solution procedure is well established [START_REF] Enguehard | Mie Theory and the Discrete Dipole Approximation Calculating Radiative Properties of Particulate Media with Application to Nanostructured Materials[END_REF]. The method consists of material discretization into a sufficient number of dipoles where each dipole is polarized under the effect of the incident light. Each dipole contributes to the material-light interaction depending on its composition and its position in the discretized geometry. The limiting condition proposed by Draine [START_REF] Draine | The discrete dipole approximation for scattering calculations[END_REF] ensures a sufficient discretization under an incident wave :

m kd< 1 ( 2
)
where m is the complex refractive index of the material, k is the wavenumber and δ is the characteristic size of the volume elements of the discretization.

Our research is focused on the radiation in combustion chambers at 1500-2000K. The DDA simulations are performed for an incident wavelength of λ = 1μm ; this wavelength is in the range of high temperature radiation and is of the same order of magnitude as the characteristic size of the soot aggregates, so that unconventional interaction effects can be expected. The non-polarized thermal radiation is modeled by averaging the simulations with orthogonal polarizability directions for the same incident wave direction [START_REF] Bohren | Absorption and scattering of light by small particles[END_REF].

For our monomers of radius r = 10nm, the spatial discretization of one dipole per monomer satisfies the condition given in Eq. ( 2) under λ = 1μm and with a typical methane soot refractive index of m = 2.0+0.5i [START_REF] Van-Hulle | Caractérisation des Aérosols Emis par la Combustion des Hydrocarbures : Application à la Mesure de l'Indice de Réfraction des Suies[END_REF].

Soot being a highly absorbing material, errors may arise in DDA simulations due to the badly explained surface interactions (surface dipole polarizabilities) and the shading effect on inner dipoles at high refractive indices, even when the discretization condition is satisfied [START_REF] Collinge | Discrete-dipole approximation with polarizabilities that account for both finite wavelength and target geometry[END_REF][START_REF] Rahmani | On the importance of local-field corrections for polarizable particles on a finite lattice : Application to the discrete dipole approximation[END_REF]. In a previous work, we have compared the Mie cross-sections of a soot monomer with the DDA solutions using different discretization finesses, and decided to consider one dipole per monomer in order to avoid such discretization errors [START_REF] Okyay | Modélisation des propriétés radiatives des suies issues de combustions incomplètes au moyen de l'approximation dipolaire discrète[END_REF]. Furthermore, the final aim is to obtain a qualitative comparison of the sensitivities to the fractal parameters.

Absorption and scattering from soot aggregates

It is to be reminded that the scope is to find the impacts of the morphology and fractal properties of numerically generated aggregates on their radiative properties, all other variables (material composition, r and N) being fixed. In the first part, a comparison with the Mie solution shows the relevance of considering complex aggregate shapes. In the second part, the evolution of the radiative properties is presented as a function of Df and kf.

Effect of complex aggregate geometry

Our present in-house heat transfer simulations consider Mie solution properties for the radiative properties of soot. The DDA results for the reference aggregate presented in Figure 1 are compared to the Mie solution for a sphere having an equivalent material volume.

Cross-section DDA (m 2 )

Mie (m The results presented in Table 2 show the relevance of DDA simulations for the determination of the radiative properties of soot. Despite the well-known overestimation tendency of absorption by DDA [START_REF] Fuller | Electromagnetic Scattering by Compounded Spherical Particles[END_REF], the DDA solution for absorption is more than 10% below the Mie solution. Also the asymmetry factor predicted by the DDA indicates a strong forward scattering unlike the Mie solution, which proves the sensitivity of the DDA to the morphology of the scatterer.

Effect of fractal properties

Aggregate samples are simulated first by varying Df around its reference value with all other parameters fixed. The trends of the DDA results as a function of Df are represented in Figures 3 and4. As can be seen on these figures, several numerical aggregates are generated for the same fractal parameter in order to account for the random nature of the aggregation process.

The same procedure is applied to study the sensitivity of the radiative properties to the prefactor kf. The trends extracted for kf are globally similar to the ones presented in Figures 3 and4. However, the sensitivities on Df and kf are slightly different, as summarized in Table 3. Results indicate that the interaction of soot with radiation is more affected by the value of the Df parameter. In particular, varying Df from 1.6 to 1.8 modifies the absorption by only 1.5% but the scattering by 46%. But as the absorption phenomenon dominates the scattering by soot, the sensitivity of soot radiative properties to the fractal parameters can be considered relatively small in the range of study if no particular interest is shown for scattering in the radiative heat transfer simulations. 3 Variation of the radiative properties of methane soot aggregates due to modification of their fractal parameters.

Conclusion and perspectives

In this study, the effects of the fractal properties on the radiative properties of methane soot are investigated. Methane soot is represented by numerically generated aggregates and the radiation-soot interaction is simulated using the DDA theory. Results indicate a small dependence for absorption but an important one for scattering versus the Df and kf parameters. If absorption is considered as the dominating phenomenon, the uncertainties on Df and kf around the reference values can be ignored.

The choice of our future characterization techniques and the necessary precision in data extraction depends an extended version of the presented sensitivity study. Also, numerical soot aggregates will be gathered in groups of identical radiative properties merely dependent on the fractal parameters ; these groups are to be inserted as input for heat transfer simulations.

Figure 1

 1 Figure 1 Numerical aggregate with the reference parameters.

Figure 3

 3 Figure 3 Absorption (left), scattering (right) cross-sections of methane soot as a function of Df (kf, N and r are fixed).

Figure 4

 4 Figure 4 Asymmetry factor for methane soot as a function of Df (kf, N and r are fixed).

Table 1

 1 Bibliographic data on methane soot fractal properties

	Once the Df, kf and N parameters are decided, different aggregation algorithms like DLA (Diffusion Limited Aggregation) or DLCCA (Diffusion Limited Cluster Cluster Aggregation) are able to generate aggregates in a specific range of Df, kf and N.
	Reference	r	N	Df	kf
	Zhang et al. (1998)	10	5-200	1.62	-
	Koylu et al. (1995)	-	-	1.65	2.71
	Cai et al. (1995)	-	-	1.74	1.23
	Cai et al. (1995)	-	-	1.74	2.45
	Sorensen et al. (1992)	-	-	1.73	-
	Lee et al. (2000)	5 -10	10-600	1.75	6.46

Table 2

 2 DDA compared to Mie solution for the reference case

	2 )	|ΔC|/CDDA

Acknowledgement

The authors kindly acknowledge the support of this research within the industrial chair OXYTEC of Ecole Centrale Paris, jointly conducted and funded by the laboratory EM2C, Air Liquide, CNRS and ANR.