Quang-Huy Nguyen

Sharp Strichartz estimates for water waves systems

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Sharp Strichartz estimates for water waves systems

Introduction

Water waves systems govern the dynamic of an interface between a fluid domain and the vacuum. It is well-known that these systems are dispersive, i.e., waves at different frequency propagate at different speed. For approximate models of water waves in certain regimes such as Kadomtsev-Petviashvili equations, Korteweg-de Vries equations, Schrodinger equations, wave equations,...dispersive properties have been extensively studied. For the fully nonlinear system of water waves, dispersive properties are however less understood.

On the one hand, in global dynamic, dispersive properties have been considered in establishing the existence of global (or almost global) solutions for small, localized, smooth data by the works of Wu [START_REF] Wu | Almost global wellposedness of the 2-D full water waves problem[END_REF][START_REF] Wu | Global wellposedness of the 3-D full water waves problem[END_REF], Germain-Masmoudi-Shatah [START_REF] Germain | Global solutions for the gravity water waves equation in dimension 3[END_REF][START_REF] Germain | Global existence for capillary water waves[END_REF], Ionescu-Pusateri [START_REF] Alexandru | Global solutions for the gravity water waves system in 2d[END_REF][START_REF] Alexandru | Global regularity for 2d water waves with surface tension[END_REF], Alazard-Delort [START_REF] Alazard | Global solutions and asymptotic behavior for two dimensional gravity water waves[END_REF], Ifrim-Tataru [START_REF] Ifrim | Two dimensional water waves in holomorphic coordinates II: global solutions[END_REF][START_REF] Ifrim | The lifespan of small data solutions in two dimensional capillary water waves[END_REF]. On the other hand, in local dynamic, dispersive properties and more precisely Strichartz estimates have been exploited in proving the existence of local-in-time solutions with rough, generic data, initiated by the work of Alazard-Burq-Zuily [START_REF] Alazard | Strichartz estimate and the Cauchy problem for the gravity water waves equations[END_REF] and then followed by de Poyferré-Nguyen [START_REF] Thibault De | A paradifferential reduction for the gravity-capillary waves system at low regularity and applications[END_REF][START_REF] Thibault De | Strichartz estimates and local existence for the capillary water waves with non-Lipschitz initial velocity[END_REF]. Prior to these, a Strichartz estimate was proved for 2D gravity-capillary waves by Christianson-Hur-Staffilani [START_REF] Christianson | Strichartz estimates for the water-wave problem with surface tension[END_REF]. Unlike the case of semilinear Schrodinger (wave) equations, water waves systems are quasilinear in nature and thus how much regularity one can gain in Strichartz estimates depends also on the smoothness of solutions under consideration. In other words, in term of dispersive analysis (for generic solutions), the nonlinear systems are not obviously dictated by their linearizations. In fact, the Strichartz estimates proved in [START_REF] Alazard | Strichartz estimate and the Cauchy problem for the gravity water waves equations[END_REF], [START_REF] Thibault De | Strichartz estimates and local existence for the capillary water waves with non-Lipschitz initial velocity[END_REF] are non optimal compared to the linearized systems. We address in this paper the following problem: At which level of regularity, solutions to the fully nonlinear systems of water waves obey the same Strichartz estimates as its linearization? Remark first that due to the systematic use of symbolic calculus in the framework of semi-classical analysis in [START_REF] Alazard | Strichartz estimate and the Cauchy problem for the gravity water waves equations[END_REF], [START_REF] Thibault De | Strichartz estimates and local existence for the capillary water waves with non-Lipschitz initial velocity[END_REF] we were not able to reach sharp Strichartz estimates by simply adapting their method to the case of sufficiently smooth solutions.

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

In this paper, we choose to work on the The Zakharov-Craig-Sulem formulation of water waves, which is recalled now.

1.1. The The Zakharov-Craig-Sulem formulation of water waves. We consider an incompressible inviscid fluid with unit density moving in a time-dependent domain Ω = {(t, x, y)

∈ [0, T ] × R d × R : (x, y) ∈ Ω t }
where each Ω t is a domain located underneath a free surface

Σ t = {(x, y) × R d × R : y = η(t, x)}
and above a fixed bottom Γ = ∂Ω t \ Σ t . We make the following assumption on the domain: Assumption (H) Each Ω t is the intersection of the haft space

Ω 1,t = {(x, y) × R d × R : y = η(t, x)}
and an open connected set Ω 2 containing a fixed strip around Σ t , i.e., there exists h > 0 such that for all t ∈ [0, T ]

{(x, y) × R d × R : η(x) -h ≤ y ≤ η(t, x)} ⊂ Ω 2 .
Assume that the velocity field v admits a potential φ : Ω → R, i.e, v = ∇φ. Using the Zakharov formulation, we introduce the trace of φ on the free surface ψ(t, x) = φ(t, x, η(t, x)).

Then φ(t, x, y) is the unique variational solution of (1.1) ∆φ = 0 in Ω t , φ(t, x, η(t, x)) = ψ(t, x).

The Dirichlet-Neumann operator is then defined by 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)).

The gravity-capillary waves (see [START_REF] Lannes | Water waves: mathematical analysis and asymptotics[END_REF]) problem consists in solving the following system of (η, ψ):

(1.2)    ∂ t η = G(η)ψ, ∂ t ψ + gη -σH(η) + 1 2 |∇ x ψ| 2 - 1 2 (∇ x η • ∇ x ψ + G(η)ψ) 2 1 + |∇ x η| 2 = 0,
where σ is the surface tension coefficient and H(η) is the mean curvature of the free surface:

H(η) = div ∇η 1 + |∇η| 2 .
In the regime of large wavelengths, one can discard the effect of surface tension by taking σ = 0 in the system (1.3) to obtain the system of pure gravity water waves

(1.3)    ∂ t η = G(η)ψ, ∂ t ψ + gη + 1 2 |∇ x ψ| 2 - 1 2 (∇ x η • ∇ x ψ + G(η)ψ) 2 1 + |∇ x η| 2 = 0,
The physical dimensions are d = 1, 2. For terminologies, when d = 1 (respectively d = 2) we call (1.2), (1.3) the 2D (respectively 3D) waves systems. It is important to introduce the vertical and horizontal components of the trace of the velocity on Σ, 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.
We recall also that the Taylor coefficient defined by 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]).

1.2. Known results and main theorems.

1.2.1. Pure gravity water waves. For the system (1.3) of pure gravity water waves, the only existent Strichartz estimate, to our knowledge, is [START_REF] Alazard | Strichartz estimate and the Cauchy problem for the gravity water waves equations[END_REF] where the authors proved Strichartz estimates for rough solutions with a gain of (1.5) µ = 1 24

-when d = 1, µ = 1 12 -when d ≥ 2
The starting point of this result is the symmetrization of (1.2) into a quasilinear paradifferential equation of the following form (see Appendix A for the paradifferential calculus theory and Theorem 2.2 below for a precise reduction statement)

(1.6) (∂ t + T V • ∇ + iT γ ) u = f ∈ L ∞ t H s x , s > 1 + d 2 ,
where γ is a symbol of order 1 2 . Let us now look at the linearization of (1.3) (take g = 1 and infinite depth) around the rest state (0, 0):

∂ t η -|D x |ψ = 0, ∂ t ψ + η = 0 which is equivalent to, after imposing u := η + i|D x | 1 2 ψ, (1.7) ∂ t u + i|D x | 1
2 u = 0. For this Schrodinger-type dispersive equation we can prove classically the Strichartz estimates (1.8) u

L p W s-d 2 +µ opt ,∞ ≤ C(s, d) u| t=0 H s , µ opt = 1 8 , p = 4 if d = 1, µ opt = 1 4 -, p = 2 if d ≥ 2
, from which the estimates for the original unknowns η, ψ can be recovered. Our first result states that the fully nonlinear system (1.3) enjoys Strichartz estimates with the same gain as in (1.10), for solutions slightly smoother than the energy threshold in [START_REF] Alazard | On the Cauchy problem for gravity water waves[END_REF].

Notation 1.1. Denote H s = H s+ 1 2 (R d ) × H s+ 1 2 (R d ) × H s (R d ) × H s (R d ), W s = W r+ 1 2 ,∞ (R d ) × W r+ 1 2 ,∞ (R d ) × W r,∞ (R d ) × W r,∞ (R d ).
Theorem 1.2. Let d = 1, 2 and consider a solution (η, ψ) of (1.3) on the time interval I = [0, T ], T < +∞ such that Ω t satisfies H t for every t ∈ [0, T ] and

(η, ψ, B, V ) ∈ C([0, T ]; H s ). (see Theorem 1.2, [3]). Define s(d) = 5 3 + d 2 , µ opt (d) = 1 8 , p(d) = 4 if d = 1, s(d) = 2 + d 2 , µ opt (d) = 1 4 -, p(d) = 2 if d ≥ 2.
Then for any s > s(d) we have

(η, ψ, B, V ) ∈ L p(d) (I; W s-d 2 +µopt(d),∞ ).
1.2.2. Gravity-capillary waves. Let us now look at the linearization of (1.2) (with infinite depth) around the rest state (0, 0),

∂ t η -|D x |ψ = 0, ∂ t ψ -∆η = 0 or equivalently, with Φ = |D x | 1 2 η + iψ (1.9) ∂ t Φ + i|D x | 3 2 Φ = 0,
for which one can easily prove the following Strichartz estimates

(1.10) Φ L p W s-d 2 +µ opt ,∞ ≤ C(s, d) Φ| t=0 H s , µ opt = 3 8 , p = 4 if d = 1, µ opt = 3 4 -, p = 2 if d ≥ 2.
Turning to the nonlinear case, in high dimensions (d ≥ 2) the geometry can be non trivial and hence trapping can occur. As a consequence, natural dispersive estimates expected are the one constructed at small time scales which are tailored to the frequencies. The propagator e -it|Dx| 32 has the speed of propagation at order |ξ| 1 2 . Hence, for time |t| < |ξ| -1 2 , we expect no problem due to the global geometry. This leads to the so called semi-classical Strichartz estimate. This terminology appeared in [START_REF] Burq | Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds[END_REF] for a study of the Schrödinger equations on compact manifolds. To realize this heuristic argument, one multiplies both side of (1.9) by h Then the optimal dispersive estimates for (1.11) implies the semi-classical Strichartz estimates for (1.9) with a lost of 1 8 derivatives when d = 1 and 1 4 derivatives when d ≥ 2.

In [START_REF] Alazard | On the water waves equations with surface tension[END_REF] it was proved that if

(1.12) (η, ψ) ∈ C([0, T ]; H s+ 1 2 × H s ), s > 2 + d 2 then system (1.
2) can be symmetrized into a single equation analogous to its linearization (1.9):

(1.13) (∂ t + T V • ∇ + iT γ ) u = f ∈ L ∞ H s , γ ∈ Γ 3 2 ;
from which the local-wellposedness was obtain at this regularity level-(1.12). Using this reduction, Alazard-Burq-Zuily [START_REF] Alazard | Strichartz estimates for water waves[END_REF] established, for 2D waves, the semi-classical Strichartz estimate at the threshold (1.12) and the classical (optimal) Strichartz estimate when s > 5 + 1 2 . We remark that in [START_REF] Alazard | On the water waves equations with surface tension[END_REF], the semi-classical gain is achieved due to the fact that after a para change of variables, the highest order term T γ u in (1.13) is converted into the simple Fourier multiplier |D x | Theorem 1.3. Let d ≥ 2 and 0 < T < ∞. Consider a solution (η, ψ) of (1.2) on the time interval I = [0, T ] such that Ω t satisfies H(t) for every t ∈ [0, T ] and

(η, ψ) ∈ C([0, T ]; H s+ 1 2 (R d ) × H s (R d )).
If s > 5 2 + d 2 then for every ε > 0, there holds

(η, ψ) ∈ L 2 ([0, T ]; W s+1-ε-d 2 (R d ) × W s+ 1 2 -ε-d 2 ,∞ (R d )).
Remark 1.4. Our proof of Theorem 1.3 works equally for the 2D waves (d = 1), when (η,

ψ) ∈ C([0, T ]; H s+ 1 2 (R)×H s (R)) with s > 5 2 + 1 2 .
On the other hand, using the paracomposition reduction in [START_REF] Alazard | Strichartz estimates for water waves[END_REF] we can indeed improve the preceding regularity to s > 2 + 1 2 , which is the same as Theorem 1.1 in [START_REF] Alazard | Strichartz estimates for water waves[END_REF].

1.2.3. On the proof of the main results. In [START_REF] Alazard | Strichartz estimate and the Cauchy problem for the gravity water waves equations[END_REF][START_REF] Thibault De | Strichartz estimates and local existence for the capillary water waves with non-Lipschitz initial velocity[END_REF] the authors worked completely in the semi-classical formulism and proved dispersive estimate using the approximation WKB method. This allowed the authors to prove Strichartz estimates with nontrivial gains even for very rough backgrounds. However, we emphasize that with this method, we were not able to reach the classical or semi-classical level as in Theorem 1.2 and 1.3. The dispersive estimates for principally normal pseudodifferential operators in [START_REF] Koch | Dispersive estimates for principally normal pseudodifferential operators Comm[END_REF] require more regularity (C 2 ) of the symbols to control the Hamiltonian flow and apply the FBI transform technique. This indeed allows us to obtain sharp dispersive estimates when the characteristic set of the symbol has maximal nonvanishing curvatures. For the proof of our main results, we shall combine the para-differential reduction in the works of Alazard-Burq-Zuily with the phase transform method in the work [START_REF] Koch | Dispersive estimates for principally normal pseudodifferential operators Comm[END_REF] of Koch-Tataru. Notice that the later works effectively for operators of order 1, after renormalizing. For gravity-capillary waves (see (1.13)) the dispersive term has order 3 2 and thus the semi-classical time-scale brings it to the one of order 1 and hence leads to the semi-classical Strichartz estimate in Theorem 1.3. For the pure gravity waves (1.6), one observes that the dispersive term iT γ has order 1 2 which is lower than the transport term T V • ∇. Here, we follow [START_REF] Alazard | Strichartz estimate and the Cauchy problem for the gravity water waves equations[END_REF], suppressing this transport term by straightening the vector field ∂ t + T V • ∇ and then make another change of spatial variables to convert it to an operator of order 1. However, the new symbol then is not in the standard form p(x, ξ) to apply phase transforms and other technical issues appear. The proof of Theorem 1.2 thus requires much more care.

Preliminaries

2.1. Symmetrization of system (1.3). We first recall the symmetrization of system (1.3) to a single quasilinear equation, performed in [START_REF] Alazard | On the water waves equations with surface tension[END_REF]. This reduction requires the following symbols:

• Symbols of the Dirichlet-Neumann operator λ (1) 1) div(α (1) ∇η) + i∂ ξ λ (1) • ∇α (1) , α (1)

:= (1 + |∇η| 2 )|ξ| 2 -(∇η • ξ) 2 , λ (0) := 1 + |∇η| 2 2λ (
:= λ (1) + i∇η • ξ 1 + |∇η| 2 .
• Symbols of the mean-curvature operator:

ℓ (2) := (1 + |∇η| 2 ) -1 2 |ξ| 2 - (∇η • ξ) 2 1 + |∇η| 2 , ℓ (1) := - i 2 (∂ x • ∂ ξ )ℓ (2) ;
• Symbols using for symmetrization

q := 1 + (∇ x η) 2 -1 2 , p = 1 + (∇ x η) 2 -5 4 |ξ| 1 2 + p (-1 2 ) , where p (-1 2 ) := F (∇ x η, ξ)∂ α x η, with |α| = 2 and F ∈ C ∞ (R d × R d \ {0}; C) is homogeneous of order -1 2 in ξ.
• Symbols in the symmetrized equation:

γ := l (2) λ (1) = |ξ| 2 (1 + |∇η| 2 ) -(∇η • ξ) 2 1 + |∇η| 2 3 4 , ω := - i 2 (∂ ξ • ∂ x ) l (2) λ (1) , ω 1 := l (2) λ (1) ℜλ (0) 2 .
Then with the good-unknown U := ψ -T B η, it was proved in [START_REF] Alazard | On the water waves equations with surface tension[END_REF] that Theorem 2.1 ([1, Corollary 4.9]). Let s > 2 + d 2 and (η, ψ) ∈ C 0 ([0, T ]; H s+ 1 2 × H s ) be a solution to the gravity-capillary wave system (1.2) and satisfies condition (H t ) for every t ∈ [0, T ]. The complex-valued function u := T p η + iT q U then solves the following para-differential equation

(2.1) ∂ t u + T V • ∇u + iT γ+ω+ω 1 u = f,
where, there exists a nondecreasing function

F : R + × R + → R + , in dependent of (η, ψ), such that (2.2) f L ∞ ([0,T ];H s ) ≤ F (η, ψ) L ∞ ([0,T ];H s+ 1 2 ×H s )
.

2.2. Symmetrization of system (1.2). Define first the principle symbol of the Dirichlet-Neumann operator

λ = 1 + |∇η| 2 |ξ| 2 -ξ • ∇η 2 1 2 .
Next, set ζ = ∇η and introduce

U s := D x s V + T ζ D x s B, ζ s := D x s ζ.
Theorem 2.2 ([3, Proposition 4.10]). Let s > 1 + d 2 and (η, ψ) ∈ C 0 ([0, T ]; H s+ 1 2 × H s ) be a solution to the pure gravity water wave system (1.3) such that condition (H t ) is fulfilled for every t ∈ [0, T ] and the velocity trace

(B, V ) ∈ C 0 ([0, T ]; H s+ 1 2 × H s )
; and there exists c 0 > 0 such that a(t, x) ≥ c 0 . Then the complex-valued function

u := D x -s (U s -iT √ a/λ ζ s ) solves the following para-differential equation (2.3) ∂ t u + T V • ∇u + iT γ u = f, where γ = √ aλ and a -g L ∞ ([0,T ];H s-1 2 ) + f L ∞ ([0,T ];H s ) ≤ F (η, ψ)(t) H s+ 1 2 , (V, B)(t) H s . Remark 2.
3. The change of variables (η, ψ) → u in Theorem 2.1 and (η, ψ, B, V ) → u in Theorem 2.2 are essentially "invertible" in the sense that one can recover Sobolev estimates and Hölder estimates for (η, ψ, B, V ) from those for u by virtue of the symbolic calculus for para-differential operators contained in Theorem A.4.

Para and pseudo differential operators.

Since the para-differential setting is not suitable for proving dispersive estimates, we shall change it into the pseudo-differential one, whose standard definitions are recalled here. Definition 2.4. 1. For any m ∈ R, 0 ≤ δ 1 , δ 2 , ρ ≤ 1 we denote by S m ρ,δ 1 ,δ 2 the class of all symbols a(x, y, ξ)

: (R d ) 3 → C satisfying ∂ α x ∂ β y ∂ γ ξ a(x, y, ξ) ≤ C α,β,γ (1 + |ξ|) m+δ 1 |α|+δ 2 |β|-ρ|γ| .
The corresponding pseudo-differential operator is defined by

Op(a)u(x) = R d e i(x-y)ξ a(x, y, ξ)u(y)dydξ.
When a : (R d ) 2 → C we consider it as a symbol in S m ρ,δ 1 ,0 that does not depend on y and rename S m ρ,δ 1 ,0 ≡ S m ρ,δ 1 . 2. For any symbol a(x, ξ) ∈ S m ρ,δ the Weyl quantization Op w (a) ≡ a w (x, D x ) is defined by Op w (a)u(x) = Op(b)u(x) with b(x, y, ξ) := a( x+y 2 , ξ) ∈ S m ρ,δ,δ .

We shall later need to transform the operators Op(a) to Op w (a). This is done by means of the following result, which can be easily deduce from [START_REF] Taylor | Pseudodifferential operators and nonlinear PDE[END_REF], Proposition 0.3.A.

Proposition 2.5. For any symbol a ∈ S m ρ,δ with m ∈ R, 0 ≤ δ < ρ ≤ 1 there exists a symbol b ∈ S m ρ,δ such that Op w (a) = Op(b)
. Moreover, we have the following asymptotic expansion in the sense of symbolic calculus:

b(x, ξ) ∼ |α|≥0 (-i) |α| α!2 |α| ∂ α x • ∂ α ξ a(x, ξ).
Remark that for all α, ∂ α

x • ∂ α ξ a(x, ξ) ∈ S m-(ρ-δ)|α| ρ,δ
. Now, let a ∈ Γ m r , r > 0 be a para-differential symbol (see Definition A.2) and define

(2.4) ∀j ∈ Z, ∀δ > 0, S jδ (a)(x, ξ) = ψ(2 -jδ D x )a(x, ξ)
the spatial regularization of the symbol a, where ψ is the Littlewood-Paley function defined in (A.1). We first prove a Bernstein's type inequality for S jδ (a).

Lemma 2.6. If a ∈ Γ m ρ then for all α, β ∈ N d , |α| ≥ ρ, there exists a constant C α,β such that for all (x, ξ) ∈ R 2d |∂ α x ∂ β ξ S jδ (a)(x, ξ)| ≤ C α,β 2 j(|α|-ρ) ∂ β ξ a(•, ξ) W ρ,∞ (R d ) .
Proof.

If |α| = ρ the estimate is obvious by writing ∂ α x ∂ β ξ S jδ (a) as a convolution of ∂ α x ∂ β ξ a with a kernel. Considering now |α| > ρ. Recall the dyadic partition of unity (A.2): 1 = ∞ k=0 ∆ k where each ∆ k is spectrally supported in the annulus {2 k-1 ≤ |ξ| ≤ 2 k+1 }.
Using this partition, we can write

∂ α x ∂ β ξ S jδ (a)(x, ξ) = +∞ k=0 ∆ k ∂ α x ψ(2 -jδ D x )∂ β ξ a(x, ξ) := +∞ k=0 u k If 1 2 2 k ≥ 2 jδ then ∆ k ψ(2 -jδ D x ) = 0. Therefore ∂ α x ∂ β ξ S jδ (a)(x, ξ) = 2+[jδ] k=0 u k . Now, introducing ϕ 1 (ξ) ∈ C ∞ c (R d ), supported in { 1 3 ≤ ξ| ≤ 3} one has u k = 2 k|α| ϕ 1 (2 -k D x )ψ(2 -jδ D x )∆ k ∂ β ξ a(x, ξ). Consequently, u k L ∞ (R d ) ≤ 2 k|α| ∆ k D β ξ a(•, ξ) L ∞ (R d ) ≤ C2 k|α| 2 -kρ ∂ β ξ a(•, ξ) W ρ,∞ (R d ) . It follows that ∂ α x ∂ β ξ S jδ (a)(x, ξ) L ∞ (R d ) ≤ C 2+[jδ] k=0 2 k(|α|-ρ) D β ξ a(•, ξ) W ρ,∞ (R d ) .
Finally, since |α| -ρ > 0 we deduce that

∂ α x ∂ β ξ S jδ (a)(x, ξ) L ∞ (R d ) ≤ C2 jδ(|α|-ρ) ∂ β ξ a(•, ξ) W ρ,∞ (R d ) , which concludes the proof.
We show in the next Proposition that after localizing a distribution u in frequency, one can go from para-differential operators to pseudo-differential operators when acting on u.

Proposition 2.7. For every j ∈ N * , define

R j u := T a ∆ j u -S j-3 (a)(x, D x )∆ j u.
Then the spectrum of R j u is contained in an annulus {c -1 1 2 j-1 ≤ |ξ| ≤ c 1 2 j+1 } and for every µ ∈ R we have

R j u H µ-m+r (R d ) ≤ CM m r (a) u H µ (R d )
where the constants c 1 , C > 0 are independent of a, u, j.

Proof. Recall first the definition (A.5) of T a u, where we have ̺ = 1 on the support of ϕ j for any j ≥ 1, so

R j u = T a ∆ j u -S j-3 (a)(x, D x )̺(D x )∆ j u.
In the following proof, we shall use the presentation of Métivier [START_REF] Métivier | Para-differential calculus and applications to the Cauchy problem for nonlinear systems[END_REF] on pseudodifferential and paradifferential operators. To be compatible with [START_REF] Métivier | Para-differential calculus and applications to the Cauchy problem for nonlinear systems[END_REF] we also abuse notations: by Γ m r we denote the class of symbols a satisfying the growth condition (A.3) for any ξ ∈ R d and by M m 0 the semi-norm (A.4) where the suppremum is taken over ξ ∈ R d .

1. By definition (A.5) it holds that T a v = Op(σ a ̺)v, where Op(σ a ̺) denotes the classical pseudodifferential operator with symbol

σ a (x, ξ)̺(ξ) = χ(D x , ξ)a(x, ξ)̺(ξ). Hence R j u = Op(a j )u with a j (x, ξ) = σ a (x, ξ)̺(ξ)ϕ j (ξ) -S j-3 (a)(x, ξ)̺(ξ)ϕ j (ξ).

Now, we write

a j = σ a ̺ϕ j -a̺ϕ j + a̺ϕ j -S j-3 (a)̺ϕ j = a 1 j + a 2 j .
Applying Proposition 5.8(ii) in [START_REF] Métivier | Para-differential calculus and applications to the Cauchy problem for nonlinear systems[END_REF] gives

a 1 j ∈ Γ m-r 0 and (remark that (ϕ j ) j is bounded in Γ 0 r ) M m-r 0 (a 1 j ) ≤ CM m r (a̺ϕ j ) ≤ CM m r (aρ).
On the other hand, if we denote b = a̺ϕ then a 2 j (x, ξ) = b(x, ξ) -ψ j-3 (D x , ξ)b(x, ξ). Taking into account the fact that supp ϕ j ⊂ B(0, C2 j ) we may estimate

|a 2 j (x, ξ)| ≤ k≥j-2 |∆ j b(x, ξ)| ≤ k≥j-2 2 -kr b(•, ξ) W r,∞ ≤ C2 -jr b(•, ξ) W r,∞ = C2 -jr |ϕ j (ξ)| a(•, ξ)̺(ξ) W r,∞ ≤ C(1 + |ξ|) m-r M m r (a̺), ∀ξ ∈ R d . By the same method for estimating |∂ α ξ a 2 j | we obtain that a 2 j ∈ Γ m-r 0 and hence a j ∈ Γ m-r 0 ; moreover M m-r 0 (a j ) ≤ CM m r (a̺). 2. Property (A.7) implies in particular that F x (σ a )(η, ξ) = 0 for |η| ≥ ε 2 (1 + |ξ|),
here we denote F x the Fourier transform with respect the the patial variable x. On the other hand, by definition of the smoothing operator

F x S j-3 (a)(x, ξ)̺(ξ)ϕ j (ξ) = ψ(2 -(j-3) η)F x a(η, ξ)̺(ξ)ϕ(2 -j ξ) which is vanishing if |η| ≥ 1 2 (1 + |ξ|). Indeed, if either |ξ| > 2 j+1 or |ξ| ≤ 2 j-1 then ϕ(2 -j ξ) = 0. Considering 2 j-1 < |ξ| ≤ 2 j+1 then |η| ≥ 1 2 (1 + |ξ|) > 2 j-2 and thus ψ(2 -(j-3) η) = 0. We have proved the existence of 0 < ε < 1 such that (2.5) F x a j (η, ξ) = 0 for |η| ≥ ε(1 + |ξ|).
3. By the spectral property (2.5) one can use the Bernstein's inequalities (see Corollary 4.1.7, [START_REF] Métivier | Para-differential calculus and applications to the Cauchy problem for nonlinear systems[END_REF]) to prove that a j is a pseudodifferential symbol in the class S m-r 1,1 . Then, applying Theorem 4.3.5 in [START_REF] Métivier | Para-differential calculus and applications to the Cauchy problem for nonlinear systems[END_REF] we conclude that

R j u H µ-m+r (R d ) = Op(a j )u H µ-m+r (R d ) ≤ CM m-r 0 (a j ) u H µ (R d ) .
Finally, the Fourier transform of R j u reads

F(R j u)(ξ) = R d F x (a j )(ξ -η, η)û(η)dη.
Using the spectral localization property (2.5) and the fact that F x (a j )(ξ -η, η) contains the factor ϕ j (η) we conclude that the spectrum of R j u is contained in an annulus of size 2 j as claimed.

2.4.

A result of Koch-Tataru. In this paragraph, we recall the dispersive estimates proved by Koch-Tataru [START_REF] Koch | Dispersive estimates for principally normal pseudodifferential operators Comm[END_REF] based on the technique of FBI transform on phase space. These estimates were established for the following class of operators.

Definition 2.8. For λ > 1, m ∈ R and k = 0, 1, ... consider classes of symbols p : T * R d → C, denoted by λ m S k λ , which satisfy (2.6) ∂ α x ∂ β ξ p(x, ξ) ≤ c α,β λ m-|β| , |α| ≤ k, ∂ α x ∂ β ξ p(x, ξ) ≤ c α,β λ m+ |α|-k 2 -|β| , |α| ≥ k.
The mentioned result reads Proposition 2.9 ([19, Proposition 4.7]). Let p(σ, x, ξ) ∈ λS 2 λ be a real symbol in (x, ξ), uniformly in σ ∈ [0, 1]. Assume that p satisfies the following curvature condition (A) for each

(σ, x, ξ) ∈ [0, 1] × R d × C λ , | det ∂ 2 ξ p| λ -d , where C λ = {c -1 λ ≤ |ξ| ≤ cλ}.
Denote by S(σ, σ 0 ) the flow maps of D σ +Op w (p). Then for any χ ∈ S 0 λ such that for all

x ∈ R d , χ(x, •) compactly supported in C ′ λ = {c ′-1 λ ≤ |ξ| ≤ c ′ λ}with 1 < c ′ < c, we have S(σ, σ 0 )(χ(x, D x )v 0 ) L ∞ λ d 2 |σ -σ 0 | -d 2 v 0 L 1 , ∀σ, σ 0 ∈ [0, 1].
Remark 2.10. In the statement of Proposition 4.7, [START_REF] Koch | Dispersive estimates for principally normal pseudodifferential operators Comm[END_REF], condition (A) is stated for (x, ξ) ∈ B λ := {|x| ≤ 1, |ξ| ≤ λ} and correspondingly, χ is supported in B; in addition, the usual quantization χ(x, D x ) is replaced by the Weyl quantization χ w . However, one can inspect easily its proof to see that if (A) is fulfilled globally in x then we have the above variant.

Remarks on the symbolic calculus for

λ m S k λ . Let a ∈ λ m S k m , k ≥ 0 and suppose that on the support of a(x, ξ), λ -1 |ξ| ∼ 1. It follows by definition of λ m S k λ that a ∈ S m 1, 1 2
. Observe however that when k ≥ 1, a behaves better than a general symbol in S m 1, 1

2

. In this paragraph we present some enhanced properties of λ m S k λ with k ≥ 1. First, we are concerned with the relation between the the Weyl quantization and the usual quantization. According to Proposition 2.5, for a ∈ S m 1,

1 2 there holds Op w (a) -Op(a) = Op(r), r ∈ S m-1 2 1, 1 2 .
In fact, we have

Op w (a) = Op( a), a(x, y, ξ) = a( x + y 2 , ξ) and a(x, y, ξ) = a(x + y -x 2 , ξ) = a(x, ξ) + 1 2 1 0 (∂ x a)(x + s y -x 2 , ξ)ds (y -x).
It follows that

(2.7) Op w (a) -Op(a) = Op(r), with (2.8) r(x, y, ξ) = - i 2 1 0 (∂ ξ ∂ x a)(x + s y -x 2 , ξ)ds.
We now show that in fact r is of order m -1 as in the case

a ∈ S m 1,0 . Lemma 2.11. Let a ∈ λ m S k λ , k ≥ 1 satisfying λ -1 |ξ| ∼ 1 on the support of a(x, ξ). Then we have the relation (2.7)-(2.8) with r ∈ S m-1 1, 1 2 , 1 2 .
Proof. For any α, β, ν ∈ N d we have

|∂ α x ∂ β y ∂ ν ξ r(x, y, ξ)| ≤ C α,β,ν λ m-|ν|-1 ½ λ -1 |ξ|∼1 (ξ), if |α| + |β| + 1 ≤ k, C α,β,ν λ m+ |α|+1+|β|-k 2 -|ν|-1 ½ λ -1 |ξ|∼1 (ξ), if |α| + |β| + 1 ≥ k. Since k ≥ 1, |α| + 1 + |β| -k 2 ≤ |α| + |β| 2 Consequently, it holds that |∂ α x ∂ β y ∂ ν ξ r(x, y, ξ)| ≤ C α,β,ν (1 + |ξ|) (m-1)+ |α|+|β| 2 -|ν| , ∀α, β, ν ∈ N d ; in other words, r ∈ S m-1 1, 1 2 , 1 2 .
For the composition rule we prove Lemma 2.12. Let p ∈ S n 1,0 and a ∈ λ m S k λ , k ≥ 1 satisfying λ -1 |ξ| ∼ 1 on the support of a(x, ξ). Then we have

Op(p) • Op(a) -Op(pa) = Op(r) with r ∈ S n+m-1 1, 1 2 .
Proof. According to Proposition 0.3.C [START_REF] Taylor | Pseudodifferential operators and nonlinear PDE[END_REF], one has Op(p)

• Op(a) = Op(b) with b ∼ |α|≥0 (-i) |α| α! ∂ α ξ p(x, ξ)∂ α x a(x, ξ)
in the sense of symbol asymptotic. The general term in the above expansion belongs to S n+m-

|α| 2 1, 1 2 , hence (b -pa) - |α|=1 (-i) |α| α! ∂ α ξ p(x, ξ)∂ α x a(x, ξ) ∈ S n+m-1 1, 1 2 .
It then suffices to show that

c α := ∂ α ξ p(x, ξ)∂ α x a(x, ξ) ∈ S n+m-1 1, 1 2 for |α| = 1 or again, ∂ α x a(x, ξ) ∈ S m 1, 1 2
for |α| = 1. The later follows along the same lines as in the proof of Lemma 2.11.

In the same spirit we have the following result on adjoint operators, taking into account Proposition 0.3.B [START_REF] Taylor | Pseudodifferential operators and nonlinear PDE[END_REF].

Lemma 2.13. Let a ∈ λ m S k λ , k ≥ 1 satisfying λ -1 |ξ| ∼ 1 on the support of a(x, ξ).

Then we have

Op * (a) -Op(ā) = Op(r) with r ∈ S m-1 1, 1 2 and ā is the complex conjugate of a. Notation 2.14. Throughout this article, we write A B if there exists a constant C > 0 such that A ≤ CB, where C may depend on the coefficients of the equations under consideration. If the constant C involved has some explicit dependency, say, on some quantity µ, we emphasize it by denoting A µ B.

Proof of Theorem 1.3

Throughout this section, the dimension d is greater than or equal 2 and s > 5 2 + d 2 is a Sobolev index.

3.1. Littlewood-Paley reduction. We shall prove Strichasrtz estimate for solution u to (2.1), which is a quasilinear para-differential equation with timedependent coefficients. Remark that since

(η, ψ) ∈ C 0 ([0, T ]; H s+ 1 2 × H s ), we have V ∈ C 0 ([0, T ]; H s-1 ), γ(•, ξ) ∈ C 0 ([0, T ]; H s-1 2
). The first step in our proof consists in localizing (2.1) at frequency 2 j using the Littlewood-Paley decomposition (cf. Definition A.1 1.). For every j ≥ 0, the dyadic piece ∆ j u solves

(3.1) (∂ t + T V • ∇ + iT γ+ω ) ∆ j u = F j , with (3.2) F 1 j := ∆ j f -i∆ j (T ω 1 u) + i [T γ , ∆ j ] + i [T ω , ∆ j ] u + [T V , ∆ j ] • ∇u. Remark that for each j ≥ 1, ∆ j u is spectrally supported in {2 j-1 ≤ |ξ| ≤ 2 j+1 }.
In views of Proposition 2.7 and the fact that γ ∈ Γ

3 2 3 2 , ω ∈ Γ 1 2 1 2 and V • ξ ∈ Γ 1
1 , equation (3.1) can be rewritten as

(3.3) (∂ t + S j-3 (V ) • ∇ + iS j-3 (γ)(x, D x )) ∆ j u = F 2 j , with (3.4) F 2 j := F 1 j + R j , R j u is spectrally supported in an annulus {c -1 1 2 j-1 ≤ |ξ| ≤ c 1 2 j+1 } and satisfies (3.5) R j H s u H s .
Next, we use (2.4) to smooth out the symbols by δ = 1 2 derivatives. Now, let 1 2

< c 1 < c 2 < c 3 , C k := {(2c k ) -1 ≤ |ξ| ≤ 2c k }, k = 1, 2, 3 and ϕ ∈ C ∞ , supp ϕ ⊂ C 3 , ϕ ≡ 1 on C 2 .
Then, equation (3.3) is equivalent to

(3.6) L j ∆ j u := ∂ t + S (j-3) 1 2 (V ) • ∇ ϕ(2 -j D x ) +iS (j-3) 1 2 (γ)(x, D x ) ϕ(2 -j D x ) + iS (j-3) 1 2 (ω)(x, D x ) ϕ(2 -j D x ) ∆ j u = F j , with (3.7) F j = F 2 j + F 3 j := F 2 j + i S (j-3) 1 2 γ(x, D x ) -S j-3 γ(x, D x ) ∆ j u + i S (j-3) 1 2 ω(x, D x ) -S j-3 ω(x, D x ) ∆ j u + S (j-3) 1 2 (V ) -S j-3 (V ) • ∇∆ j u.
3.2. Semi-classical time scale. Observe that the highest order operator on the left-hand side of (3.3) has order 3 2 , which does not match the result in [START_REF] Koch | Dispersive estimates for principally normal pseudodifferential operators Comm[END_REF] that we want to apply. Therefore, we reduce it to an operator of order 1 by multiplying both side by h 1 2 , where h := 2 -j , then making a change of temporal variables t = h 1 2 σ. For this purpose, let us reset the symbols in this new time scale:

Γ h (σ, x, ξ) = h 1 2 S (j-3) 1 2 (γ)(h 1 2 σ, x, ξ) ϕ(hξ), (3.8) Ω h (σ, x, ξ) = h 1 2 S (j-3) 1 2 (ω)(h 1 2 σ, x, ξ) ϕ(hξ), (3.9) V h (σ, x, ξ) = h 1 2 S (j-3) 1 2 (V )(h 1 2 σ, x) • ξ ϕ(hξ). (3.10) Next, set w h (σ, x) = ∆ j u(h 1 2 σ, x), G h (σ, x) = -ih 1 2 F j (h 1 2 σ, x). Equation (3.6) is then equivalent to (3.11) (D t + Γ h (σ, x, D x ) + Ω h (σ, x, D x ) + V h (σ, x, D x )) w h (σ, x) = G h (σ, x).
In what follows, we shall prove the classical Strichartz estimate for (3.11), from which the semi-classical Strichartz estimate for (3.6) follows.

We now replace the pseudo-differential operators in (3.11) by the corresponding Weyl operators using Proposition 2.5. Noticing that ω = -i 2 γ, we set

(3.12) Op w (Γ h ) = Op(Γ h ) + Op(Ω h ) + R 1 h , Op w (V h ) = Op(V h ) + R 2 h .
Inserting these identities into (3.11) leads to

(3.13) L h w h (σ, x) := (D t + Γ w h (σ, x, D x ) + V w h (σ, x, D x )) w h (σ, x) = G h (σ, x) + R 1 h (σ, x)w h (σ, x) + R 2 h (σ, x)w h (σ, x).
3.3. Classical Strichartz estimate for (3.11) (⇔ (3.13)). In this step, we shall show that Theorem 2.9 can be applied to the real symbol

p h := Γ h + V h . Set λ = h -1 = 2 j . First,
= C d (1+ ∇η L ∞ (I×R d ) )
we have (3.14) sup

(t,x,ξ)∈I×R d ×C det ∂ 2 ξ γ(t, x, ξ) ≥ c 0 .
2. For any 0 < δ ≤ 1 there exists j 0 ∈ N large enough such that

(3.15) sup (t,x,ξ)∈I×R d ×C det ∂ 2 ξ S jδ (γ)(t, x, ξ) ≥ c 0 .
Proof. 1. For the proof of part 1, we refer to Corollary 4.7, [START_REF] Alazard | Strichartz estimate and the Cauchy problem for the gravity water waves equations[END_REF]. Part 2. is a consequence of part 1. because S jδ (γ) is a small perturbation of γ when j is large enough (see for instance Proposition 4.5, [START_REF] Alazard | Strichartz estimate and the Cauchy problem for the gravity water waves equations[END_REF]).

Lemma 3.2. 1. We have Γ h ∈ λS 2 λ , V h ∈ λ 3 4 S 2
λ and hence p h ∈ λS 2 λ . 2. There exists h 0 > 0 small enough such that for 0 < h ≤ h 0 , the symbol p h satisfies condition (A) in Theorem 2.9 with C λ = λC 2 .

Proof. 1. Remark that since γ and V are respectively W 2,∞ and W 3 2 ,∞ in x, assertion 1. then follows easily from Lemma 2.6 and the fact that on the support of ϕ(hξ) we have |ξ| ∼ λ. 2. Let ξ ∈ λC 2 . We have ϕ( hξ) = 1, hence ∂ 2 ξ V h vanishes and since γ is homogeneous of order 3 2 in ξ,

∂ 2 ξ Γ h (σ, x, ξ) = ∂ 2 ξ h 1 2 S (j-3) 1 2 (γ)(h 1 2 σ, x, ξ) = h ∂ 2 ξ S (j-3) 1 2 (γ) (h 1 2 σ, x, hξ).
Therefore, condition (A) is verified by virtue of (3.15).

Calling S h (σ, σ 0 ) the flow map of the evolution operator L h = D t + Op w (p h ) (see (3.13)), we have

Lemma 3.3. If v 0 h is spectrally supported in λC 1 = (2c 1 ) -1 h -1 ≤ |ξ| ≤ 2c 1 h -1 then (i) S h (σ, σ 0 )v 0 h L ∞ (R d ) h -d 2 |σ -σ 0 | -d 2 v 0 h L 1 (R d ) ,
for all σ, σ 0 ∈ [0, 1] and 0 < h ≤ h 0 .

(ii) with q > 2 and

2 q = d 2 -d r , S h (•, 0)v 0 h L q ([0,1],L r ) h -1 q v 0 h L 2 .
Proof. By Lemma 3.2, (i) is a direct consequence of Proposition 2.9 if one chooses

χ(ξ) ∈ C ∞ , supp χ ⊂ {(2c 1,2 ) -1 λ ≤ |ξ| ≤ 2c 1,2 λ}, c 1 < c 1,2 < c 2 , χ ≡ 1 in C 1 .
For (ii) we remark that since Op w (Γ h ) and Op w (V h ) are self-adjoint, S h (σ, σ 0 ) is isometric in L 2 . This combining with the dispersive estimate (i) and a standard T T * argument (see the abstract semi-classical Strichartz estimate in Theorem 10.7, [START_REF] Zworski | Semiclassical analysis[END_REF]) yields (ii). Lemma 3.4. For any µ ∈ R, the operators S h (σ, τ ) are bounded on H µ (R d ) uniformly in t, s ∈ I .

Proof. This result bases on a standard energy estimate. However, the proof requires more care since we are not working on standard operators of classes S m 1,0 . Without loss of generality we assume τ = 0 and let f (t, x) be a solution of

(∂ t + i Op w (p h )) f (t, x) = 0, f (0, x) = f 0 (x).
We first apply Lemma 2.11 to obtain

Op w (p h ) = Op(p h ) + Op(r h ), r h ∈ S 0 1, 1 2 , 1 2 .
Then f solves the problem

(∂ t + i Op(p h ) + i Op(r h )) f (t, z) = 0, f (0, x) = f 0 (x).
Let µ ∈ R and set f µ := D z µ f then

d dt f µ 2 L 2 = -i (Op(p h ) -Op * (p h )) f µ , f µ + 2ℜ F, f µ where F := i [Op(p h ), D z µ ] f -i D z µ Op(r h )f.
According to Lemma 2.12 one has [Op(p h ),

D z µ ] ∈ S µ 1, 1 2
. This combining with the

fact that r h ∈ S 0 1, 1 2 , 1 2 gives F L 2 f H µ . On the other hand, since p h is real, Lemma 2.13 implies Op(p h ) -Op * (p h ) ∈ S 0 1, 1 2 .

Consequently, we have

Op(p 0 h ) -Op * (p 0 h ) f µ L 2 f µ L 2 .
Finally, we conclude by Gronwall's inequality that

f (t) H µ f 0 H µ , ∀t ∈ I.
Proposition 3.5. If w h is a solution to (3.11) with data w h (0) = w 0 h and supp w h , supp w 0 h ⊂ λC 1 then we have for all ε > 0

w h L 2+ε ([0,1],W s-d 2 + 1 2 -ε,∞ ) ε w 0 h H s + G h L 1 ([0,1],H s ) + h 1 2 w h L 1 ([0,1],H s ) .
Proof. To simplify notations, let us write S h (σ, τ ) = S(σ, τ ). If w h is a solution to (3.11) and it is also a solution to (3.13). By Duhamel's formula, there holds

w h (σ, 0) = S h (σ, 0)w 0 h + σ 0 S(σ, τ )[G h (τ )]dτ + σ 0 S(σ, τ )[(R 1 h w h + R 2 h w h )(τ )]dτ.
Let us call (I) and (II), respectively, the first and the second integral on the righthand side. Choosing c 1 large enough such that supp

G h ⊂ λC 1 , Lemma 3.3 (ii) gives (I) L q ([0,1],L r ) h -1 q G h L 1 ([0,1];L 2 )
For (II) we set

b α h = (-i) |α| α!2 |α| ∂ α x ∂ α ξ Γ h (σ, x, ξ), |α| ≥ 2; c α h = (-i) |α| α!2 |α| ∂ α x ∂ α ξ V h (σ, x, ξ), |α| ≥ 1.
For each |α| ≥ 2, since γ is W 2,∞ in x we have by applying Lemma 2.6

∂ µ x ∂ ν ξ (∂ α x ∂ α ξ Γ h ) (1 + |ξ|) 1+ |µ|+|α|-2 2 -(|ν|+|α|) (1 + |ξ|) -|α| 2 + |µ| 2 -|ν| , hence b α h ∈ S - |α| 2 1, 1 2 . Similarly, it holds that c α h ∈ S - |α| 2 1, 1 2 for V ∈ W 1,∞ . Taking q > 2 and 2 q = d 2 -d r , we claim that uniformly in τ ∈ [0, 1] (3.16) S(σ, τ )[(R 1 h w h )(τ )] L q σ L r x h -1 q +1 w h (τ ) L 2 x .
Indeed, by the asymptotic expansion in Proposition 2.5 it holds that

R 1 h = N -1 |α|=2 Op(b α h ) + Op(r N h ), r N h ∈ S 1-N 2 1, 1 2
, ∀N ≥ 3.

If we choose c 1 large enough then each Op(b α h w h )(τ ) is spectrally supported in λC 1 (and so is w h ) so that Lemma 3.3 (ii) can be applied to derive

(3.17) S(σ, τ )[Op(b α h )w h (τ )] L q σ L r x ≤ h -1 q Op(b α h )w h L 2 x h -1 q + |α| 2 w h (τ ) L 2 x .
For Op(r N h ) one uses the Sobolev embedding

H d 2 ֒→ L r , ∀r ∈ [2, +∞) to estimate S(σ, τ )[Op(r N h )w h (τ )] L r x S(σ, τ )[Op(r N h )w h (τ )] H d 2 x
.

On the other hand, we know from Lemma 3.4 that S(σ, τ ) is bounded from

H µ to H µ uniformly in σ, τ ∈ [0, 1] for all µ ∈ R. Hence (3.18) S(σ, τ )[Op(r N h )w h (τ )] L r x h -1+ N 2 -d 2 w h (τ ) L 2 x .
Choosing N = N (d) large enough, we conclude the claim (3.16) from (3.17) and (3.18).

In the same way, we obtain the following estimate for R 2 h (which is also uniformly in τ ∈ [0, 1])

S(σ, τ )[(R 2 h w h )(τ )] L q σ L r x h -1 q + 1 2 w h (τ ) L 2 x .

Putting otgether the estimates above leads to

w h L q L r ≤ h -1 q w 0 h L 2 + G h L 1 ([0,1];L 2 ) + h 1 2 w h L 1 L 2 .
Taking q = 2 + ε then h

-1 q ≤ h -1
2 . We multiply both sides by h -s and use the frequency localization of w h , w 0 h to get

w h L 2+ε W s-1 2 ,r ε w 0 h H s + G h L 1 H s + h 1 2 w h L 1 H s .
On the other hand, on J = [0, 1 2 h

1 2 ] one can apply Corollary 3.6 and Hölder's inequality to have (3.20)

u j L 2 (J;W s-d 2 + 1 2 -ε,∞ ) h 1 4 u j L 2 (J;W s-d 2 + 3 4 -ε 2 ,∞ ) ε h 1 4 u 0 j H s + h 1 2 F j L 2 (J;H s ) + h 1 2 u j L 2 (J,H s ) ε u j L ∞ (I,H s-1 4 ) + F j L 2 (I;H s-1 2 ) + u j L 2 (I,H s-1 2 )
.

Likewise, we have (3.20

) with J = [(N h -1 2 )h 1 2
, T ] and thus (3.19) concludes that

u j L 2 (I;W s-d 2 + 1 2 -ε,∞ ) ε F j L 2 (I;H s-1 2 ) + u j L ∞ (I,H s ) .
In the finally step, we shall glue the estimates for ∆ j u over different frequency regimes to obtain an estimate for u, from which the corresponding estimates for (η, ψ) follow.

3.5. Concluding the proof of Theorem 1.3. If u is a solution to (2.1) with data u 0 then by (3.6), the dyadic piece ∆ j u is a solution to L j ∆ j u = F j with F j given by (3.7). Applying Corollary 3.7 one gets

(3.21) ∆ j u L 2 (I;W s-d 2 + 1 2 -ε,∞ ) ε F j L 2 (I;H s-1 2 ) + ∆ j u L ∞ (I;H s ) .
Recall that F j = F 1 j + R j + F 3 j where F k j are given in (3.2), (3.7) and from (3.5) that R j H s ≤ u H s .

Using the symbolic calculus Theorem A.4 one obtains without any difficulty that

F 1 j L 2 H s-1 2 u L 2 (I;H s ) .
For F 3 j we consider for example

A j := S (j-3) 1 2 γ(x, D x )∆ j u -S j-3 γ(x, D x )∆ j u = S (j-3) 1 2 γ(x, D x )∆ j u -γ(x, D x )∆ j u + (γ(x, D x )∆ j u -S j-3 γ(x, D x )∆ j u) = A 1 j + A 2 j . More generally, let a ∈ Γ m ρ , homogeneous of degree m in ξ. Using the spherical harmonic decomposition we can assume a(x, ξ) = b(x)c(ξ) with b ∈ W ρ,∞ and c is homogeneous of order m. Then with S δj (a)(x, ξ) = S δj (b)(x)c(ξ) one has (3.22) (S δj (a)(x, D x ) -a(x, D x ))v L 2 ≤ S δj (b) -b L ∞ c(D x )v L 2 2 -δjρ v H m . Since γ ∈ Γ 3 2
2 is homogeneous of degree 3 2 in ξ, the H s-1 2 -norm of A 1 j now can be bounded by

2 j(s-1 2 )-1 2 j2+ 3 2 j ∆ j u L 2 u H s while A 2 j H s-1 2 2 j(s-1 2 )-2j+ 3 2 j ∆ j u L 2 u H s-1 .
Similarly, we have

F 3 j L 2 H s-1 2 u L 2 (I;H s ) + f L 2 (I;H s ) .
The estimate (3.21) then implies

∆ j u L 2 (I;W s-d 2 + 1 2 -ε,∞ ) ε u L ∞ (I;H s ) + f L 2 (I;H s-1 2 )
.

Gluing these estimates together one derives

(3.23) u L 2 (I;W s-d 2 + 1 2 -2ε,∞ ) ≤ j 2 -jε ∆ j u L 2 (I;W s-d 2 + 1 2 -ε,∞ ) ε u L ∞ (I;H s ) + f L 2 (I;H s-1 2 )
.

Recall that u = T p η+iT q (ψ-T B η). From (3.23) one can use the symbolic calculus for para-differential operators in Theorem A.4 to recover the corresponding estimates for (η, ψ) (cf. [START_REF] Alazard | On the water waves equations with surface tension[END_REF], [START_REF] Alazard | Strichartz estimates for water waves[END_REF]):

η L 2 (I;W s-d 2 +1-2ε,∞ ) + ψ L 2 (I;W s-d 2 + 1 2 -2ε,∞ ) F ε (η, ψ) L ∞ ([0,T ];H s+ 1 2 ×H s )
.

The proof of Theorem 1.3 is complete.

Proof of Theorem 1.2

We consider three parameters δ ∈ (0, 1), r 0 ∈ [0, 1], r 1 ∈ [0, 1 2 ], which shall be determined later and assume that

(4.1) V ∈ L ∞ (I; W 1+r 0 ,∞ (R d )), γ(•, ξ) ∈ L ∞ (I; W 1 2 +r 1 ,∞ (R d )).
4.1. Littlewood-Paley reduction. For every j ≥ 0, the dyadic piece ∆ j u solves the equation

(4.2) (∂ t + T V • ∇ + iT γ ) ∆ j u = F j ,
where (4.3)

F 1 j := ∆ j f + i [T γ , ∆ j ] + [T V , ∆ j ] • ∇u. In view of Proposition 2.7 one has (4.4) (∂ t + S j-3 (V ) • ∇ + iS j-3 (γ)(x, D x )) ∆ j u = F 2 j , with (4.5) F 2 j := F 1 j + R j , R j u is spectrally supported in an annulus {c -1 1 2 j-1 ≤ |ξ| ≤ c 1 2 j+1 } and (4.6) R j H s u H s . Now, let 1 2 < c 1 < ... < c 5 , C k := {(2c k ) -1 ≤ |ξ| ≤ 2c k }, k = 1, 5 and ϕ ∈ C ∞ , supp ϕ ⊂ C 5 , ϕ ≡ 1 on C 4 .
Equation (4.4) is then equivalent to (4.7)

∂ t + S (j-3)δ (V ) • ∇ + iS (j-3)δ (γ)(x, D x ) ϕ(2 -j D x ) ∆ j u = F j , with (4.8) F j = F 2 j + F 3 j := F 2 j + i S (j-3)δ γ(x, D x ) -S j-3 γ(x, D x ) ∆ j u + S (j-3)δ (V ) -S j-3 (V ) • ∇∆ j u.
Let us define the operator corresponding to the homogeneous problem of (4.7) (4.9)

L j := ∂ t + S (j-3)δ (V ) • ∇ + iS (j-3)δ (γ)(x, D x ) ϕ(2 -j D x ).
To prove Strichartz estimate for ∆ j u as a solution to (4.7), we shall first establish a "pseudo dispersive estimate" for L j . Set

h := 2 -j , h := h 1 2 .
4.2. Straightening the vector field. Following [START_REF] Alazard | Strichartz estimate and the Cauchy problem for the gravity water waves equations[END_REF] we straighten the vector field ∂ t + S (j-3)δ • ∇ by considering the system (4.10)

Ẋk (s) = S (j-3)δ (V k )(s, X(s)), 1 ≤ k ≤ d, X = (X 1 , . . . , X d ) X k (0) = x k . Since V ∈ L ∞ ([0, T ]; L ∞ x )
, system (4.10) has unique solution on I = [0, T ], which shall be denoted for simplicity by X(s, x; h), or even X(s, x). Estimates for the flow s → X(s, •) is given in the next Proposition.

Proposition 4.1. For fixed (s, h) the map x → X(s; x, h) belongs to C ∞ (R d , R d ).
Moreover, for all (s, h) ∈ I × (0, 1] we have

(∂ x X)(s, •; h) -Id L ∞ (R d ) ≤ F V L ∞ ([0,T ];W 1,∞ |s|, (4.11) (∂ α x X)(s, •; h) L ∞ (R d ) ≤ α F V L ∞ ([0,T ];W 1+r 0 ,∞ h -δ(|α|-(1+r 0 )) |s|, |α| ≥ 2.
(4.12)

Proof. Here we follow the poof Proposition 4.10, [START_REF] Alazard | Strichartz estimate and the Cauchy problem for the gravity water waves equations[END_REF]. The improvement is due to the following estimate (by applying Lemma 2.6)

(4.13) ∂ β x S jδ (V )(s) L ∞ (R d ) ≤ C β h -δ(|β|-1-r 0 ) V (s) W 1+r 0 ,∞ , ∀|β| ≥ 2.
(i) To prove (4.11) we differentiate the system with respect to x l to obtain

           ∂X k ∂x l (s) = d q=1 S jδ ∂V k ∂x q (s, X(s)) ∂X q ∂x l (s) ∂X k ∂x l (0) = δ kl from which we deduce (4.14) ∂X k ∂x l (s) = δ kl + s 0 d q=1 S jδ ∂V k ∂x q (σ, X(σ)) ∂X q ∂x l (σ) dσ. Setting |∇X| = d k,l=1 | ∂X k ∂x l | we obtain from (4.14) |∇X(s)| ≤ C d + s 0 |∇V (σ, X(σ))| |∇X(σ)| dσ.
The Gronwall inequality implies that

(4.15) |∇X(s)| ≤ F( V W 1,∞ ) ∀s ∈ I.
Coming back to (4.14) and using (4.15) lead to

∂X ∂x (s) -Id ≤ F( V W 1,∞ ) s 0 ∇V (σ, •) L ∞ (R d ) dσ ≤ F 1 ( V W 1,∞ )|s|.
(ii) We shall prove (4.12) for |α| = 2 first and then prove by induction on |α| that the estimate

(∂ α x X)(s; •, h) L ∞ (R d ) ≤ F α ( V W 1+r 0 ,∞ )h -δ(|α|-1-r 0 )
for 2 ≤ |α| ≤ k implies (4.12) for |α| = k + 1. Differentiating |α| times (|α| ≥ 2) the system (4.10) and using the Faa-di-Bruno formula we obtain

(4.16) d ds ∂ α x X (s) = S jδ (∇V )(s, X(s))∂ α x X + (1)
where the term (1) is a finite linear combination of terms of the form

A β (s, x) = ∂ β x S jδ (V ) (s, X(s)) q i=1 ∂ L i x X(s) K i where 2 ≤ |β| ≤ |α|, 1 ≤ q ≤ |α|, q i=1 |K i |L i = α, q i=1 K i = β.
1. When |α| = 2, we have

A β (s, x) = ∂ β x S jδ (V ) (s, X(s)) q i=1 ∂ L i x X(s) K i with |L i | = 1 and |β| = |α| = 2. It then follows from (i) that | q i=1 ∂ L i x X(s) K i | ≤ F( V W 1,∞ ).
On the other hand, we have by (4.13)

∂ β x S jδ (V )(s) L ∞ (R d ) ≤ Ch -δ(|α|-1-r 0 ) V (s) W 1+r 0 ,∞ .
Consequently, it holds that

(1) L ∞ (R d ) ≤ h -δ(|α|-1-r 0 ) F( V W 1+r 0 ,∞ ),
from which we obtain (4.12) for |α| = 2 by using (4.16) and Gronwall's inequality.

2. Assuming now that (4.12) holds with |α| = k ≥ 2, we shall prove it for |α| = k +1. Indeed, from (4.11) and the induction hypothesis it holds for any 1 1) |s|. Therefore, with 1 ≤ |L i | < |α| we can write in view of the estimate (4.13)

≤ |ν| ≤ k that (∂ ν x X)(s, •; h) L ∞ (R d ) ≤ α F V L ∞ ([0,T ];W 1+r 0 ,∞ h -δ(|ν|-
A β (s, •) L ∞ (R d ) ≤ ∂ β x S jδ (V ) (s, •) L ∞ (R d ) q i=1 ∂ L i x X(s, •) |K i | L ∞ (R d ) ≤ Ch -δ(|β|-1-r 0 ) V (s, •) W 1+r 0 ,∞ h -δ q i=1 |K i |(|L i |-1) F( V W 1+r 0 ,∞ ) ≤ h -δ(|α|-1-r 0 ) F( V W 1+r 0 ,∞ ) V (s, •) W 1+r 0 ,∞ .
As before, we conclude by (4.16) and Gronwall's inequality.

In view of (4.11) the mapping x → X(s, x; h) is a C ∞ -diffeomorphism when 0 ≤ s ≤ T 0 small enough. This restriction of T is harmless for one can iterate the final estimate over time intervals of length T 0 which depends only on V L ∞ ([0,T ];W 1,∞ ). Now, in (4.9) we first make the change of spatial variables (4.17) v h (t, y) = u j (t, X(t, y; h)) so that (4.18)

∂ t + S (j-3)δ (V ) • ∇ u j (t, X(t, y; h) = ∂ t v h (t, y). Denoting (4.19) q h (x, ξ) := S (j-3)δ (γ)(x, ξ) ϕ(hξ),
let us compute this dispersive term after the above change of variables. To this end, one sets (4.20)

H(y, y ′ ) = 1 0 ∂X ∂x (λy + (1 -λ)y ′ ) dλ, M (y, y ′ ) = t H(y, y ′ ) -1 M 0 (y) = t ∂X ∂x (y) -1 , J(y, y ′ ) = det ∂X ∂x (y ′ ) | det M (y, y ′ )|.
Then,

(Op(q h )u j ) • X(y) = (2π) -d e i(X(y)-x ′ )•η q h (X(y), η)u j (x ′ )dx ′ dη.
Now, we make two changes of variables x ′ = X(y ′ ), then η = M (y, y ′ )ξ to obtain

(Op(q h )u j ) • X(y) = (2π) -d e i(y-y ′ )•ζ q h X(y), M (y, y ′ )ζ J(y, y ′ )v h (y ′ ) dy ′ dζ.
Observe that the pseudo-differential operator above is still of order 1 2 . To change its order to 1, we need to make in addition another change of spatial variables

(4.21) y = h 1 2 z, y ′ = hz ′ , w h (z ′ ) = v h ( hz ′ ), ξ = hζ so that (4.22) (Op(q h )u j ) • X(y) = (2π) -d e i(z-z ′ )•ξ q h X( hz), M hz, hz ′ h -1 ξ × × J hz, hz ′ w h (z ′ ) dz ′ dξ.
Summing up, with (4.23)

p h (z, z ′ , ξ) := q h X( hz), M hz, hz ′ h -1 ξ J hz, hz ′ , w h (t, z) = u j (t, X(t, hz))
it holds that (Op(q h )u j ) • X( hz) = Op(p h )w h (z), which combines with (4.18) and (4.9) yields (4.24)

(L j u j ) (t, X(t, hz)) = (∂ t + i Op(p h )) w h (t, z), w h (t, z) = u j (t, X(t, hz)).
We have transformed the operator L j of order 1 2 into the right-hand side of (4.24), which has order 1.

Approximation of the symbol p h .

Observe that p h in (4.23) depends on (z, z ′ , ξ) which is not in the standard form to use the phase space transform in [START_REF] Koch | Dispersive estimates for principally normal pseudodifferential operators Comm[END_REF]. Therefore, one need to approximate p h by some symbol depend only on (z, ξ). A general result can be found in Proposition 0.3A [START_REF] Taylor | Pseudodifferential operators and nonlinear PDE[END_REF]. However, here we need the difference between p h and its approximation to be more regular, so we have to inspect more carefully the smoothness of p h . To do this, one writes as for (2.7)-(2.8)

p h (z, z ′ , ξ) = p h (z, z, ξ) + 1 0 ∂ z ′ p h (z, z + s(z ′ -z), ξ)ds(z ′ -z) := p 0 h (z, ξ) + r 0 h (z, z ′ , ξ)(z ′ -z), where (4.25) p 0 h (z, ξ) = p h (z, z, ξ) = q h X( hz), M 0 ( hz) h -1 ξ . On the other hand, Op(r 0 h .(z ′ -z))w(z) = -i Op(r h )w(z) with r h (z, z ′ , ξ) = 1 0 ∂ ξ ∂ z ′ p h (z, z + s(z ′ -z), ξ)ds.
To simplify notations, we denote [z,

z ′ ] s = z + s(z ′ -z) so that (4.26) r h (z, z ′ , ξ) = 1 0 ∂ ξ ∂ z ′ q h X( hz), M hz, h[z, z ′ ] s h -1 ξ J hz, h[z, z ′ ] s ds.
In conclusion, Op(p h ) = Op(p 0 h ) -i Op(r h ). 4.3.1. The symbol p 0 h . First, Lemma 4.1 implies directly the estimates for M and J.

Lemma 4.2. It holds for (α, α ′ ) ∈ (N d ) 2 that |∂ α z ∂ α ′ z ′ M (z, z ′ )| + |∂ α z ∂ α ′ z ′ J(z, z ′ )| α,α ′ 1, if |α| + |α ′ | = 0, h -2δ(|α|+|α ′ |-r 0 ) , if |α| + |α ′ | ≥ 1.
On the other hand, by Bernstein's inequalities and the fact that on the support of ϕ(hξ), |ξ| ∼ h -2 , we can estimate the derivatives of q h given by (4.19) as follows.

Lemma 4.3. We have for (α,

β) ∈ (N d ) 2 ∂ α x ∂ β ξ q h (x, ξ) α,β h -1+2|β| , if |α| = 0, h -1-2δ(|α|-( 1 2 +r 1 ))+2|β| , if |α| ≥ 1.
We now prove study the regularity of the symbol p 0 h . Proposition 4.4. Choosing r 0 , r 1 satisfying

(4.27) 2δ(1 -r 0 ) ≤ 1, 2δ(2 -r 0 ) ≤ 2, 2δ( 1 2 -r 1 ) ≤ 1, 2δ( 3 2 -r 1 ) ≤ 2 then the symbol p 0 h verifies (i) for all (α, β) ∈ N d , |α| ≤ 2 (4.28) ∂ α z ∂ β ξ p 0 h (z, ξ) α,β h -1+|β| ½ h|ξ|∼1 (ξ), (ii) for all (α, β) ∈ N d , |α| ≥ 3 (4.29) ∂ α z ∂ β ξ p 0 h (z, ξ) α,β h -1-(2δ-1)(|α|-2)+|β| ½ h|ξ|∼1 (ξ).
Proof. To simplify notations, we shall denote in this proof q ≡ q h . (i) Observe that (4.28) is trivial when α = 0. The argument below is independent of the dimension, however let us further simplify the notations by writing as if d = 1. For |α| = 1, we compute

∂ α z p 0 (z, ξ) = q x X( hz), M 0 ( hz) h -1 ξ hX ′ ( hz) (4.30) + q ξ X( hz), M 0 ( hz) h -1 ξ ( h -1 ξ) hM ′ 0 ( hz). (4.31)
For |α| = 2, we have

∂ α z p 0 h (z, ξ) = q xx (• • • ) h 2 (X ′ ) 2 + 2q xξ (• • • )X ′ M ′ 0 hξ + q ξξ (• • • )(M ′ 0 ) 2 ξ 2 , (4.32) + q x (• • • ) h 2 X ′′ + q ξ (• • • )( hξ)M ′′ 0 . (4.33)
Remark that on the support of p 0 (z, ξ), h|ξ| ∼ 1,using Proposition 4.1, Lemmas 4.2 and 4.3 one deduces easily that

∂ α z p 0 (z, ξ) α h -1-2δ( 1 2 -r 1 )+1 + h -1+2-1-2δ(1-r 0 ) , |α| = 1, ∂ α z p 0 (z, ξ) α h -1-2δ( 3 2 -r 1 )+2 + h -1-2δ( 1 2 -r 1 )+2-2δ(1-r 0 ) + h -1+4-4δ(1-r 0 )-2 + h -1-2δ( 1 2 -r 1 )+2-2δ(1-r 0 ) + h -1+2-2δ(2-r 0 ) , |α| = 2.
Under conditions (4.27), it follows that

∂ α z p 0 (z, ξ) α h -1 ½ h|ξ|∼1 (ξ), |α| ≤ 2.
To obtain (i) it remains to estimate ∂ β ξ (∂ α z p h 0 ) for |α| ≤ 2 and β ∈ N d . From the explicit expressions (4.30), (4.32) of ∂ α z p 0 h above, we see that there are two possibilities when differentiating in ξ. A possibility is that a derivative falls down to q. This makes appear the factor M 0 ( hz) h -1 while we gain h 2 when differentiating q in ξ (by Lemma 4.3), we thus gain h.Another possibility is that a derivative falls down to ξ ν , ν = 1, 2, which results in νξ ν-1 . Since ξ ∼ h -1 on the support of p 0 h one deduces that ξ ν-1 ∼ ξ ν h, which means that we still gain h. Therefore, in both cases we gain h when differentiating once in ξ and thus (4.28) follows. (ii) As just explained above, it suffices to prove (4.29) with β = 0. From the formula (4.32), the proof of (4.29) reduces to showing for |α| ≥ 0

|∂ α z A j (z, ξ, h)| α,β h -1-(2δ-1)|α| ½ h|ξ|∼1 (ξ), j = 1, 5 with                    A 1 = q xξ X( hz), M 0 ( hz) h -1 ξ h -1 , A 2 = q xx X( hz), M 0 ( hz) h -1 ξ h 2 , A 3 = q ξξ X( hz), M 0 ( hz) h -1 ξ h -4 , A 4 = q x X( hz), M 0 ( hz) h -1 ξ h, A 5 = q ξ X( hz), M 0 ( hz) h -1 ξ h -2 and |∂ α z B j (z, h)| α,β h -(2δ-1)|α| , j = 1, 4 with              B 1 = X ′ ( hz), B 2 = hM ′ 0 ( hz), B 3 = hX ′′ ( hz), B 4 = h 2 M ′′ ( hz).
1. B j . It follows from Lemma (4.11) that

|∂ α z B 1 | = |∂ α z X ′ ( hz)| = h |α| |(∂ α+1 x X)( hz)| h |α|-2δ|α| h -1-(2δ-1)|α| .
On the other hand, (4.12) and the condition 2δ(1 -r 0 ) ≤ 1 imply

|∂ α z B 3 | = h|∂ α z X ′′ ( hz)| = h 1+|α| |(∂ α+2 x X)( hz)| h 1+|α|-2δ(|α|+1-r 0 ) h -(2δ-1)|α| .
Remark that M ′ 0 ( hz) is as smooth as X ′′ ( hz), the preceding estimate also holds for B 2 . For B 4 , we apply (4.12) and use the condition 2δ(2 -r 0 ) ≤ 2 to estimate

|∂ α z B 4 | = h 2 |∂ α z M ′′ 0 ( hz)| = h 2+|α| |(∂ α+2 x M 0 )( hz)| h |α|+2-2δ(|α|+2-r 0 ) h -(2δ-1)|α| .
2. A 1 . For for α = 0, Lemma 4.3 gives

|A 1 | h -1-2δ( 1 2 -r 1 )+2-1 h -1
since 2δ( 1 2 -r 1 ) ≤ 1. Considering now |α| ≥ 1, we write using the Faa-di-Bruno formula that ∂ α z A 1 is a linear combination of terms of the form

C 1 = h |α|-1 ∂ a+1 x ∂ b+1 ξ q (• • • ) r j=1 (∂ L j x X)( hz) P j (∂ L j x M 0 )( hz) h -1 ξ Q j where 1 ≤ |a| + |b| ≤ |α|, |L j | ≥ 1, ∀j = 1, r and r j=1 P j = a, r j=1 Q j = b, r j=1 (|P j | + |Q j |)L j = α.
According to Lemma 4.3,

| ∂ a+1 x ∂ b+1 ξ q (• • • )| h -1-2δ(|a|+ 1 2 -r 1 )+2(|b|+1) .
On the other hand, since |L j | ≥ 1 Lemmas 4.3, 4.2 allow us to estimate the product appearing in C 1 as follows

| r j=1 | h K , K = r j=1 -2δ(|L j | -1)|P j | -2δ(|L j | -r 0 )|P j | -2 r j=1 |Q j | = -2δ|α| + 2δ|a| + 2δr 0 |b| -2|b|. Therefore, |C 1 | h L with L = |α| -1 -1 -2δ(|a| + 1 2 -r 1 ) + 2(|b| + 1) -2δ|α| + 2δ|a| + 2δr 0 |b| -2|b| ≥ -1 -(2δ -1)|α| + 1 -2δ( 1 2 -r 1 ) ≥ -1 -(2δ -1)|α|,
where we have used again the condition that 2δ( 1 2 -r 1 ) ≤ 1. The proof for A 1 is complete. 3. A 2 , A 3 , A 4 , A 5 . The estimate for these terms can be derived along the same lines as for A 3 , where one need to make use of the condition 2δ( 32 -r 1 ) ≤ 2 for A 2 and the condition 2δ( 12 -r 1 ) ≤ 1 for A 4 .

From now on, we always assume condition (4.27) for r 0 and r 1 .

4.3.2.

The symbol r h . The next lemma provides the order of r h and shows that it decays in ξ faster than in (z, z ′ ) which shall be important in our "pseudo-dispersive estimates" in paragraph 4.4.

Lemma 4.5. For any (α, α ′ , ξ) ∈ (N d ) 3 ∂ α z ∂ α ′ z ′ ∂ β ξ r h (z, z ′ , ξ) α,α ′ ,β h 1-2δ(1-r 0 )-(2δ-1)(|α|+|α ′ |)+|β| ½ { h|ξ|∼1} (ξ).
Consequently, we have r h ∈ S -1+2δ(1-r 0 )

1,(2δ-1),(2δ-1) .

Proof. Recall the definition (4.26) of r h , on the support of this symbol we have |ξ| ∼ h -1 . In this proof, all the estimates are uniform in s ∈ [0, 1]. It follows from Lemma 4.2 that

∀(α, α ′ ) ∈ (N d ) 2 , ∂ α z ∂ α ′ z ′ J hz, h[z, z ′ ] s α,α ′ h -(2δ-1)(|α|+|α ′ |) .
Next, setting q(x, ξ) = S (j-3)δ (γ)(x, ξ) ϕ(ξ) we see that

q X( hz), M hz, h[z, z ′ ] s h -1 ξ = h -1 q X( hz), M hz, h[z, z ′ ] s hξ .
The proof of this lemma then boils down to showing for any (α, α

′ , β) ∈ (N d ) 3 , (4.34) ∂ β ξ ∂ α z ∂ α ′ z ′ ∂ ξ ∂ z ′ q(X(z), M (z, [z, z ′ ] s )ξ) α,α ′ ,β h -2δ(|α|+|α ′ |)-2δ(1-r 0 ) . We compute Ξ := ∂ ξ ∂ z ′ q(X(z), M (z, [z, z ′ ] s )ξ) = s q ξξ (• • • )M z ′ ξM + s q ξ (• • • )M z ′ ,
which is bounded by h -2δ(1-r 0 ) in view of Lemma 4.2 and the fact that |ξ| ∼ 1 on the support of q. For the same reason we see that taking ξ-derivatives of Ξ is harmless (notice that M is bounded), so we only need to prove (4.34) for |β| = 0. Indeed, one has again by Lemma 4.2

(4.35) ∂ α z ∂ α ′ z ′ M (•) + ∂ α z ∂ α ′ z ′ M z ′ (•) h -2δ(|α|+|α ′ |)-2δ(1-r 0 )
On the other hand, using the Faa-di-Bruno formula (as in the proof of Proposition 4.4) we can prove that

∂ α z ∂ α ′ z ′ (∂ γ ξ q) X(z), M (z, [z, z ′ ] s )ξ h -2δ(|α|+|α ′ |) ,
from which we conclude the proof.

In view of equation (4.24) we have proved that (4.36) (L j u j ) (t, X(t, hz) = ∂ t + i Op(p 0 h ) w h (t, z) -i Op(r h )w h (t, z) via the relation w h (t, z) = u j (t, X(t, hz)).

4.4.

A "pseudo dispersive estimate" for L j . In this step, we shall show that Theorem 2.9 can be applied to the evolution operator

L h := D t + Op w (p 0 h ). Henceforth, we set δ = 3 4 , λ = h -1 . Proposition
) χ(z, D z )f L ∞ h -d 2 |t -t 0 | -d 2 f L 1 ,
for all t, t 0 ∈ [0, 1] and 0 < h ≤ h 0 . If in addition, χ(z, D z ) : L 2 → L 2 then for any r ∈ [2, ∞] there holds by interpolation

(4.40) S h (t, t 0 ) χ(z, D z )f L r h -d( 1 2 -1 r ) |t -t 0 | -d( 1 2 -1 r ) f L r ′ .
where r ′ is the conjugate exponent of r, i.e., 1 r + 1 r ′ = 1.

Proof. We have seen that p 0 h ∈ λS 2 λ . On the other hand, ϕ ≡ 1 in C 4 , Proposition 4.7 then gives sup

(t,x,ξ,h)∈I×R d ×C 4 ×(0,h 0 ] det ∂ 2 ξ S δj (γ)(t, x, ξ) ϕ(ξ) 1.
Remark that (4.11) implies |M 0 (y)| ≥ c 0 for all y ∈ R d (by choosing T small enough as explained after Proposition 4.1). Consequently, sup

(t,x,ξ,h)∈I×R d ×(λC 3 )×(0,h 0 ] det ∂ 2 ξ p 0 h λ -d
if c 3 < c 4 is chosen appropriately. In other word, condition (A) in Theorem 2.9 is fulfilled with c = c 3 and thus the Proposition follows.

Let ϕ 1 be a smooth function verifying

supp ϕ 1 ⊂ {(2c 2 ) -1 ≤ |ξ| ≤ 2c 2 }, ϕ 1 ≡ 1 in {(2c 1 ) -1 ≤ |ξ| ≤ 2c 1 }.
Lemma 4.9. For f (t, z) = g(t, X(t, hz)) we have S h (t, t 0 ) g • X(t 0 , hz)

(4.41) (ϕ 1 (hD x )g) (t, X(t, hz)) = ϕ * h (z, D z )f (t, z) -i Op(r 2 h )f (t, z), with ϕ * h (z, ξ) = ϕ 1 M 0 ( hz) hξ , (4.42) r 2 h (z, z ′ , ξ) = 1 0 ∂ ξ ∂ z ′ ϕ 1 M ( hz, h[z, z ′ ] s ) hξ J hz, h[z, z ′ ] s ds. (4.43) Moreover, for every (α, α ′ , ξ) ∈ (N d ) 3 there hold ∂ α z ∂ β ξ ϕ * h (z, ξ) α,β h -(2δ-1)+|β| 1 { h|ξ|∼1} , (4.44) ∂ α z ∂ α ′ z ′ ∂ β ξ r 2 h (z, z ′ , ξ) α,α ′ ,β
L r h -d( 2 r ′ -1 2 ) |t -t 0 | -d( 1 2 -1 r ) g L r ′
+ S(t, t 0 ) Op(r 2 h ) g • X(t 0 , hz) L r , for all t, t 0 ∈ [0, 1] and 0 < h ≤ h 0 .

Proof. We first apply the identity (4.41) at t = t 0 and notice that ϕ 1 (hξ) = 1 if ξ ∈ 1 h C 1 to have g • X(t 0 , hz) = ϕ * h (z, D z ) g • X(t 0 , hz) (z) -i Op(r 2 h ) g • X(t 0 , hz) (z). The estimate (4.44) implies that ϕ * h ∈ S 0 λ ∩ S 0 + S(t, t 0 ) Op(r 2 h ) g • X(t 0 , hz) L r . Finally, since X is Lipschitz we have g • X(t 0 , hz)

L r ′ h -d r ′ g L r ′ ,
from which we conclude the proof.

To control the right-hand side of the estimate in the preceding Corollary, we use the following Lemma, whose proof is identical to that of Lemma 3.4. Lemma 4.11. For any µ ∈ R, the operators S h (t, s), S j (t, s) are bounded on H µ (R d ) uniformly in t, s ∈ I .

Henceforth, we choose ε 0 > 0 arbitrarily small and Proposition 4.12. If L j u j (t, x) = 0 and u j (t), t ∈ [0, T ] are spectrally supported in 1 h C 1 then for any t 0 ∈ [0, T ] and r ∈ [2, ∞] we have u j (t) L r h -d( Finally, putting together the above estimates we conclude from (4.54) that u L q (I;W s 0 -d 2 +µ,∞ ) ≤ j ∆ j u L q (I;W s 0 -d 2 +µ,∞ ) u 0 H s 0 + u L 1 (I;H s 0 ) + f L 1 (I;H s 0 ) u 0 H s 0 + f L 1 (I;H s 0 ) , where in the last inequality, we have used the energy estimate u L ∞ (I;H s 0 ) u 0 H s 0 + f L 1 (I;H s 0 ) . The proof is complete.

According to Remark 2.3, after having the estimate for u-solution to (2.3) one can use the symbolic calculus to obtain the desired estimates for the original solution (η, ψ, B, V ) as stated in Theorem 1.2.

3 2 1 2 1 2 3 2

 3113 with h = 2 -j , j ∈ N and make a change of temporal variables t = h σ, u(σ, x) = Φ(h σ, x) to derive the semi-classical equation(1.11) h∂ σ u + |hD x | u = 0.

  the characteristic set of γ has maximal (d) nonvanishing curvatures: Proposition 3.1. Let C be a fixed annulus in R d . 1. There exists an absolute constant C d > 0 such that with c 0

5 . 4 . 10 .

 5410 h 2-2δ(1-r 0 )-(2δ-1)(|α|+|α ′ |)+|β| 1 { h|ξ|∼1} . (4.45) Proof. The formulas (4.41) and (4.42), (4.43) are derived along the same lines as in paragraph 4.2, 4.3 where we performed the change of variables x = X(t, hz) to derive (4.36). 1. Proof of (4.44). Observe first that ∂ β ξ ϕ * h (z, ξ) = (∂ γ ϕ 1 ) M 0 ( hz) hξ M 0 ( hz) γ h |β| where |γ| = |β|. Next, Lemma 4.2 implies for every α ∈ N d ,∂ α z (∂ γ ϕ 1 ) M 0 ( hz) hξ + ∂ α z M 0 ( hz) γ h -(2δ-1)|α| , and thus (4.44) follows. 2. For (4.45) one proceeds exactly as in the proof of Lemma 4.Corollary If g is spectrally supported in the annulus 1 h C 1 then for any r ∈ [2, ∞] we have (4.46)

1 , 1 2 ( 1 r 1 2 - 1 r

 12111 λ = h -1 ), so the estimate (4.40)applied to χ = ϕ * h and f (z) = g • X(t 0 , hz) gives for r ∈ [2, ∞] S h (t, t 0 ) g • X(t 0 , hz) ) |t -t 0 | -d( ) g • X(t 0 , hz) L r ′

  2δ(2 -r 0 ) = 2 -3 2 ε 0 when d = 1, 2δ(1 -r 0 ) = 0, 2δ(2 -r 0 ) = 3 2 when d = 2and (4.27) is fulfilled.The next proposition proves what we called "pseudo-dispersive estimates" above.

H d 2 +εr 2 r ′ - 1 2 1 r 2 u

 22112 -1+2δ(1-r 0 ) ds. Putting together the estimates above, one obtainsw h (t) L r h -d( ) |t -t 0 | -d( 1 2 -) u j (t 0 ) L r ′ + t t 0 w h (s) H d 2 +εr -1+2δ(1-r 0 ) ds. When d = 1, r = ∞, d 2 + ε r -1 + 2δ(1 -r 0 ) = 0 and since X(t, •) ∈ W 1,∞ (R d ) we have for all s ∈ [t 0 , t] w h (s) j (s) L 2 . When d = 2, r ∈ [2, ∞), d 2 + ε r -1 + 2δ(1 -r 0 ) = 0 and thus by Lemma 4.

  Next, we recall the following proposition, which shows that the the characteristic set of p 0 h has d nonvanishing curvatures.Proposition 4.7 ([4,Proposition 4.16]). Let C be a fixed annulus in R d . For any 0 < δ < 1 there exist m 0 > 0, h 0 > 0 such that Calling S j , S h are, respectively, the propagator of L j , L h , we are now in position to apply Theorem 2.9 to derive dispersive estimates for S h .

	4.4 then shows that p 0 h belongs to λS 2 λ . Using Lemma 2.11 to replace h ) in (4.36) by Op w (p h ) we have Op(p 0 (4.37) Op w (p 0 h ) = Op(p 0 h ) + Op(r ′ h ) with r ′ h ∈ S 0 1, 1 2 , 1 2 . On the other hand, since 2δ(1 -r 0 ) ≤ 1, Lemma 4.5 combining with (4.36), (4.37) leads to Proposition 4.6. There holds for some symbol r 1 h ∈ S 0 1, 1 2 , 1 2 that (4.38) 1 i (t,x,ξ,h)∈I×R d ×(C×(0,h 0 ] det ∂ 2 ξ S δj (γ)(t, x, ξ) ≥ m 0 . Proposition 4.8. For any symbol χ ∈ S 0 λ satisfying for all z ∈ R d , supp χ(z, •) ⊂ λC 2 we have (L sup (4.39) S h (t, t 0

j u j ) (t, X(t, hz) = D t + Op w (p 0 h ) w h (t, z) + Op(r 1 h )w h (t, z).

  ) u j (t 0 ) L r ′ + h -d 4 u j (t 0 ) L 2 where r = ∞ when d = 1 and r ∈ [2, ∞) when d = 2.Proof. First, equation (4.38) implies that w h (t, z) := u j (t, X(t, hz)) satisfiesw h (t) = S h (t, t 0 )w h (t 0 ) -Op h (r 1 h ) w h (s)ds =: (1) + (2).Applying Corollary 4.10 to g(x) = u j (t 0 , x) and then using the Sobolev embeddingsH d 2 +ε ֒→ L ∞ , H d 2 ֒→ L r , ∀r ∈ [2, ∞) together with Lemma 4.11 one gets (1) L r h -d( 2 r ′ -1 2 ) |t -t 0 | -d( 1 2 -1 r ) u j (t 0 ) L 1 + Op(r 2 h )w h (t 0 )

		2 r ′ -1 2 ) |t -t 0 | -d( 1 2 -1
					t
					t 0	S(t, s) H	d 2 +εr
	where
	ε The estimate (4.45) gives r 2 h ∈ S	-2+2δ(1-r 0 ) 1, 1 2 2 , 1	, hence
	Op(r 2 h )w h (t 0 )	H	d 2 +εr	w h (t 0 )	H	d 2 +εr -2+2δ(1-r 0 ) .
	Similarly, since r 1 h ∈ S	-1+2δ(1-r 0 ) 1, 1 2 2 , 1	one deduces with the aid of Lemma 4.11
					t
		(2) L r		t 0	w h (s)

r r = 0, if r ∈ [2, ∞), ε r = 3 2 ε 0 , if r = ∞.

  [START_REF] Thibault De | Strichartz estimates and local existence for the capillary water waves with non-Lipschitz initial velocity[END_REF] it holds for all s ∈ [t 0 , t] thatw h (s) (s) L 2 ds h -d 2 u j (s) L 2 .On the other hand, by Lemma (4.11), u j (s) L 2 u j (t 0 ) L 2 for all s ∈ [t 0 , t]. Finally, noticing that u j (t) L r h Remark 4.13. Strictly speaking, the preceding estimate is not a standard dispersive estimate since it does not show decay in time on the right hand side. The appearance of the non-decaying term h -d 4 u j (t 0 ) L 2 is however harmless for the purpose of proving Strichartz estimate in the next paragraph.

	H	d 2 +εr -1+2δ(1-r 0 ) ds h -d 2 u j 1 r w h L r	w h L r
	we conclude the proof.

. Unfortunately, such a reduction can not work for the

3D case and hence, semi-classical Strichartz estimate in this case is much more difficult to establish, especially at the regularity level (1.12). In the present paper, we want to investigate the semi-classical Strichartz estimate for (1.2) when d ≥ 2, assuming that the solution is slightly smoother than (1.12) (1/2 derivatives). Our second result reads as follows.

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 sincerely thank Prof. Nicolas Burq and Prof. Claude Zuily for many helpful discussions and encouragements. I thank Prof. Daniel Tataru for suggesting me his joint work [19] with Herbert Koch, with whom I shared an interesting discussion.

Consequently, the Sobolev embedding W s-1 2 ,r ֒→ W s-d 2 + 1 2 -ε,∞ concludes the proof.

3.4. Semi-classical Strichartz estimate for (3.6). From the preceding Proposition, one deduces the corresponding Strichartz estimate for u j ≡ ∆ j u as a solution of (3.6) via the change of temporal variables w h (σ, x) = u j (h 1 2 σ, x) as follows.

Corollary 3.6. If u j is solution to (3.6), i.e., L j u j = F j with data u 0 j and u j , u 0 j , F j are spectrally supported in 2 j C 1 then u j satisfies with

] and ε > 0

The next step consists in gluing the estimates on small time scales above to derive an estimate on the whole interval of time [0, 1] to the price of loosing 1 4 derivatives.

Corollary 3.7. If u j is solution to L j u j = F j with data u 0 j and u j , u 0 j , F j are spectrally supported in C 1 h then u j satisfies with I = [0, T ] and ε > 0

, u j,k = χ j,k (t)u j .

Then each u j,k is a solution to

2 ) = 0, from which we deduce by virtue of Corollary 3.6

we get

Squaring both sides of the above inequality and then summing in 0

or, equivalently after multiplying by h

+ u j L 2 (I;H s ) .

Strichartz estimates.

Proposition 4.14. Suppose that L j u j (t, x) = 0 and u j (t),

consequently, for any s 0 ∈ R and ε > 0

Proof. For the two estimates (4.48), (4.49), using the T T * argument, one need to show that

for every f spectrally supported in 1 h C 1 . In view of the "pseudo dispersive estimate" in Proposition 4.12,

. By the Hardy-Littlewood-Sobolev inequality, (1) L q t is bounded by the right-hand side of (4.51). On the other hand, (2) can be estimated using Sobolev embedding as

which concludes the proof of (4.48).

(ii) d = 2, q > 2, 2 q + 2 r = 1. Again, the Hardy-Littlewood-Sobolev inequality yields

For (2) one uses the embedding

The estimate (4.49) then follows. Now, for any ε > 0, let q = 2 + ε then

.

Multiplying both sides of (4.49) by h -s 0 yields (4.52) u j L q (I;W

Recall from (4.1) and (4.47) that we have required

In the above estimates, the dependent constants depend on a finite number of seminorms of the symbols V and γ.

Proof. If u is a solution to (2.3) with data u 0 then by (4.7), the dyadic piece ∆ j u is a solution to L j ∆ j u = F j with F j given by (4.8) spectrally supported in 1 h C 1 if c 1 large enough. Under the regularity of V, γ given in 1. and 2., condition (4.53) is fulfilled. Using Duhamel's formula and applying the Strichartz estimates in Proposition 4.14 we deduce that (4.54) ∆ j u

We are left with the estimate for F j = F 1 j +R j +F 3 j where F k j are given by (4.3), (4.8).

it follows from (4.5) that R j H s 0 ∆ j u H s 0 . Using the symbolic calculus Theorem A.4 one obtains without any difficulty that

For F 3 j we use (3.22) to obtain that: if

Then we define χ(θ, η)

Given a temperate distribution u and an integer k in N we also introduce S k u and

Then we have the formal decomposition

(Hölder spaces) For

) the space of functions whose derivatives up to order k are bounded and uniformly Hölder continuous with exponent σ.

Let us review notations and results about Bony's paradifferential calculus (cf. [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF], [START_REF] Métivier | Para-differential calculus and applications to the Cauchy problem for nonlinear systems[END_REF]).

Definition A.2. 1. (Symbols) Given ρ ∈ [0, ∞) and m ∈ R, Γ m ρ (R d ) denotes the space of locally bounded functions a(x, ξ) on R d ×(R d \0), which are C ∞ with respect to ξ for ξ = 0 and such that, for all α ∈ N d and all ξ = 0, the function x → ∂ α ξ a(x, ξ) belongs to W ρ,∞ (R d ) and there exists a constant C α such that,

Let a ∈ Γ m ρ (R d ), we define the semi-norm

.

2. (Paradifferential operators) Given a symbol a, we define the paradifferential operator T a by

where a(θ, ξ) = e -ix•θ a(x, ξ) dx is the Fourier transform of a with respect to the first variable; χ and ρ are two fixed C ∞ functions such that:

We remark that the cut-off function χ in the preceding definition has the following properties for some 0

Symbolic calculus for paradifferential operators is summarized in the following theorem. Moreover, for all µ there exists a constant K such that