Strichartz estimates for water waves

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(1)ISSN 0012-9593. ASENAH. quatrième série - tome 44. fascicule 5. septembre-octobre 2011. aNNALES SCIENnIFIQUES d� L ÉCOLE hORMALE SUPÉRIEUkE Thomas ALAZARD & Nicolas BURQ & Claude ZUILY. Strichartz estimates for water waves. SOCIÉTÉ MATHÉMATIQUE DE FRANCE.

(2) Ann. Scient. Éc. Norm. Sup.. 4 e série, t. 44, 2011, p. 855 à 903. STRICHARTZ ESTIMATES FOR WATER WAVES. ʙʏ Tʜ�� ALAZARD, Nɪ��ʟ�� BURQ �ɴ� Cʟ�� ZUILY. Aʙ��ʀ��. – In this paper we investigate the dispersive properties of the solutions of the two dimensional water-waves system with surface tension. First we prove Strichartz type estimates with loss of derivatives at the same low level of regularity we were able to construct the solutions in [2]. On the other hand, for smoother initial data, we prove that the solutions enjoy the optimal Strichartz estimates (i.e, without loss of regularity compared to the system linearized at (η = 0, ψ = 0)). R��. – Nous nous intéressons dans cet article aux propriétés dispersives du système des ondes de surface en dimension 2, avec tension de surface. Nous démontrons tout d’abord des estimées de Strichartz, avec pertes de dérivées, au niveau de régularité où nous avons construit des solutions dans [2]. Ensuite, pour des données initiales plus régulières, nous démontrons les estimées de Strichartz optimales (i.e. sans perte de régularité par rapport à celles du système linéarisé en (η = 0, ψ = 0)).. 1. Introduction In a time-dependent domain Ωt ⊂ R2 which is located between a free hypersurface Σt and a fixed known bottom Γ, consider a potential flow v = ∇x,y φ, with ∆x,y φ = 0 in Ωt ,. ∂n φ = 0 on Γ.. The surface-tension water-waves problem is given by two equations: a kinematic condition (which states that the free surface moves with the fluid), and a dynamic condition (that expresses a balance of forces across the free surface). The system reads  on Σt = {y = η(t, x)},  ∂t η = ∂y φ − ∇η · ∇φ (1.1)  ∂t φ + 1 |∇x,y φ|2 + gη = H(η) on Σt , 2. Support by the French Agence Nationale de la Recherche, project EDP Dispersives, référence ANR-07-BLAN0250, is acknowledged. 0012-9593/05/© 2011 Société Mathématique de France. Tous droits réservés ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(3) 856. T. ALAZARD, N. BURQ AND C. ZUILY. where ∇ = ∂x , g > 0 is the acceleration of gravity and Ç å ∇η H(η) = div � 1 + (∇η)2 is the mean curvature of the free surface. 1.1. Assumptions We work in a fluid domain such that there is uniformly a minimum depth of water, more precisely we assume that for each time t one has Ωt = Ω1,t ∩ Ω2. where Ω1,t is the half space located below the free surface Σt , Ω1,t = { (x, y) ∈ R × R : y < η(t, x) }. for some unknown function η and Ω2 contains a fixed strip around Σt , that means that there exists h > 0 such that, (1.2). {(x, y) ∈ R × R : η(t, x) − h ≤ y ≤ η(t, x)} ⊂ Ω2 ,. for all t ∈ [0, T ]. We shall also assume that the domain Ω2 (and hence the domain Ωt = Ω1,t ∩ Ω2 ) is connected. We emphasize that no regularity assumption is made on the bottom Γ = ∂Ωt \ Σt . We consider both cases of infinite depth and bounded depth bottoms (and all cases in-between). Finally, we could consider the cases where the free surface is a graph over a given smooth hypersurface and the bottom is time dependent. 1.2. Main results Following Zakharov we reduce the system to a system on the free surface. If ψ = ψ(t, x) ∈ R is defined by ψ(t, x) = φ(t, x, η(t, x)), then φ(t, x, y) is the unique variational solution of (1.3). ∆φ = 0 in Ωt ,. φ(t, x, η(t, x)) = ψ(t, x).. The Dirichlet-Neumann operator is then defined by  »  (G(η)ψ)(t, x) = 1 + |∇η|2 ∂n φ|y=η(t,x) = ∂y φ − ∇η · ∇φ. y=η(t,x). (we refer to Section 2 in [2] for a precise construction). Then (η, φ) is solution of the water-waves system (1.1) if and only if (η, ψ) solves the system  ∂ η − G(η)ψ = 0,   t � �2 (1.4) 1 1 ∂x η · ∂x ψ + G(η)ψ 2   ∂t ψ + gη − H(η) + |∂x ψ| − = 0. 2 2 1 + |∂x η|2 Concerning the Cauchy theory for the water waves with surface tension, there are many results starting from the pioneering work of K. Beyer and M. Günther [10]. See D. M. Ambrose and N. Masmoudi [6], B. Schweiser [24], T. Iguchi [19], D. Coutand and S. Shkoller 4 e SÉRIE – TOME 44 – 2011 – No 5.

(4) STRICHARTZ ESTIMATES FOR WATER WAVES. 857. [17], J. Shatah and C. Zeng [25], M. Ming and Z. Zhang [22], F. Rousset and N. Tzvetkov [23]. In [2], we established new local well posedness results for the system (1.4) under sharp (as long as no dispersive effects are taken into account) regularity assumptions on the initial data. We refer to the introduction of [2] for references and a short historical survey of the background of these problems. The purpose of this work is precisely, in the case d = 1, to investigate the dispersive properties of these solutions. Our results are twofold: first we prove Strichartz type estimates with loss of derivatives at the very same level of regularity we were able to construct the solutions in [2]. On the other hand, for smoother initial data, we prove that the solutions enjoy the optimal Strichartz estimates (i.e, without loss of regularity compared to the system linearized at (η = 0, ψ = 0)). Define the usual Besov space, � σ u ∈ B∞,2 (R) ⇐⇒ 22jσ �∆j (u)�2L∞ (R) < +∞, �. j∈N. where u = j ∆j (u) is the standard Littlewood-Paley decomposition of u. Notice that if σ∈ / N, we have (with continuous injection) σ B∞,2 (R) ⊂ W σ,∞ (R),. where W σ,∞ (R) is the usual Hölder C σ space (which, if σ ∈ / N, is characterized by the fact � � that 2jσ �∆j (u)�L∞ (R) j∈N ∈ �∞ (N), see for example [15, Proposition 2.3.1]). Our main results are the following. Tʜ��ʀ�� 1.1. – Let s > 5/2 and T > 0. Consider a solution (η, ψ) of (1.4) on the time interval I = [0, T ] such that Ωt satisfies (1.2) for t ∈ I. If � � 1 (η, ψ) ∈ C 0 I, H s+ 2 (R) × H s (R) , then. � � s+ 1 s− 1 (η, ψ) ∈ L4 I, B∞,24 (R) × B∞,24 (R) .. Tʜ��ʀ�� 1.2. – Let s > 11/2, T > 0. Consider a solution (η, ψ) of (1.4) on the time interval I = [0, T ] such that Ωt satisfies (1.2) for t ∈ I. If � � 1 (η, ψ) ∈ C 0 I, H s+ 2 (R) × H s (R) , then. � � s+ 3 s− 1 (η, ψ) ∈ L4 I, B∞,28 (R) × B∞,28 (R) .. R��ʀ� 1.3. – (i) Theorem 1.1 was obtained recently under the assumption s ≥ 15 by Christianson-Hur-Staffilani [16] . 1 (ii) Let s > 5/2 and (η0 , ψ0 ) ∈ H s+ 2 (R) × H s (R) satisfying dist(Σ0 , Γ) ≥ c > 0, we � � 1 proved in [2] that there exist T > 0 and a solution (η, ψ) ∈ C 0 [0, T ]; H s+ 2 (R) × H s (R) satisfying dist(Σt , Γ) ≥ c > 0. (iii) The gain of regularity exhibited in Theorem 1.2 is optimal as can be seen at the level of the linearized system around the trivial solution (η, ψ) = (0, 0) which reads (when g = 0), ∂t η − |Dx | ψ = 0,. ∂t ψ − ∆η = 0.. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(5) 858. T. ALAZARD, N. BURQ AND C. ZUILY 1. 3. Indeed u = |Dx | 2 η + iψ is a solution of the equation i∂t u − |Dx | 2 u = 0, for which one can prove the optimal estimate � � 3 � � �exp(−it |Dx | 2 )u0 � 4 Ä s− 18 ä ≤ C�u0 �H s (R) , L. I,B∞,2 (R). which gives the desired regularity on (η, ψ).. (iv) It is most likely that Theorem 1.1 remains valid when R is replaced by the one dimensional torus T. Indeed, our proof relies on a semi-classical parametrix (on time intervals tailored to the frequency) which exhibits finite speed of propagation and which can consequently be easily localized in space. (v) Notice that the dispersive estimates proved in this paper can be combined with our previous work to improve the regularity threshold obtained in [2] and give local well posednesss for initial data below the s = 2 + 12 threshold. This will be the matter of a forthcoming paper (including the 3-d water-waves system) [1]. (vi) Notice finally that dispersive properties of the operator linearized at (η = 0, ψ = 0) were used recently by Wu [30, 31] and Germain-Masmoudi-Shatah [18] to prove global existence results for gravity waves. 1.3. Strategy of the proofs Following the approach in Alazard-Métivier [3], after suitable paralinearizations, we have shown in [2] that the water waves system can be arranged into an explicit paradifferential symmetric equation of Schrödinger type, and we deduced the smoothing effect for the 2-d surface tension water waves. Here, we will also take benefit of this paralinearization reduction, and this reduced system will be our starting point. The guiding line for the rest of our proof is very classical: construction of a parametrix to prove dispersion (L1 − L∞ estimates, and then T T ∗ argument). There are two main difficulties in the analysis of this equation. First the coefficients of the operator are time dependent and consequently we cannot get rid of the lower order terms by simple conjugation arguments (see Burq-Planchon [14]). Second the coefficients enjoy poor regularity, and finally, whereas the principal part in the operator is of order 3/2, the subprincipal part in the operator is of order 1 which gives only a 1/2 difference compared to the usual 1 difference encountered for magnetic Schrödinger operators. As will be shown in our analysis, the presence of such subprincipal parts will produce non trivial oscillations which here have to be taken into account in the analysis. The first common step for both theorems is to perform several reductions for the paradifferential equation. The first one is to use Alinhac’s para-composition theory [4] (see also Burq-Planchon [14] where a similar idea was used) to reduce the matters to the study of a Schrödinger type operator with constant coefficients principal part. This is particular to space dimension 1 and reflects the fact that there is only one metric on R. The second reduction, inspired by works by Smith [26], Bahouri-Chemin [7], Tataru [28, 29] and Blair [11], consists in smoothing out the coefficients of the operator. Once this reduction has been achieved, we can construct the parametrix, for which the natural time is the semi-classical one: s = t|ξ|−1/2 . Here the differences between our 4 e SÉRIE – TOME 44 – 2011 – No 5.

(6) STRICHARTZ ESTIMATES FOR WATER WAVES. 859. two theorems appear. Indeed, in the proof of Theorem 1.1, following the strategy in BurqGérard-Tzvetkov [13] (see also Staffilani-Tataru [27] and Koch-Tataru [20]), we construct the parametrix on small times |s| ≤ c) and the main difficulty is to handle sharp regularity threshold (for smooth enough initial data the proof would be much simpler). In the proof of Theorem 1.2 the difficulties are different: first we have to handle the oscillations generated by the subprincipal part and furthermore we have to prove very large time asymptotics (|s| ≤ c|ξ|1/2 ) in the high frequency regime |ξ| → +∞. Notice that, even for initial data with arbitrarily large smoothness, the analysis would be non trivial. Finally, once the parametrix is constructed, the dispersion estimate is obtained by using non classical stationary phase lemmas involving precise controls on the remainder terms. Acknowledgements. – The authors would like to thank the referee for a very careful reading of the manuscript, which led to improvements in the presentation. 2. Preliminaries In this section we recall some notations and results from [2] which will be used in the sequel. 2.1. Paradifferential calculus In this paragraph we review classical facts about Bony’s paradifferential calculus (see [12]). For ρ ∈ N, according to the usual definition, we denote by W ρ,∞ (R) the Sobolev spaces of L∞ functions whose derivatives of order ≤ ρ are in L∞ . For ρ ∈]0, +∞[\N, we denote by W ρ,∞ (R) the space of bounded functions whose derivatives of order [ρ] are uniformly Hölder continuous with exponent ρ − [ρ]. D��ɪɴɪ�ɪ�ɴ 2.1. – Given ρ ≥ 0 and m ∈ R, Γm ρ (R) denotes the space of functions a(x, ξ) on R × (R \ 0), which are C ∞ with respect to ξ and such that, for all α ∈ N and all ξ �= 0, the function x �→ ∂ξα a(x, ξ) belongs to W ρ,∞ (R) and there exists a constant Cα such that, (2.1). ∀ |ξ| ≥. 1 , 2. � α � �∂ξ a(·, ξ)� ρ,∞ ≤ Cα (1 + |ξ|)m−|α| . W (R). D��ɪɴɪ�ɪ�ɴ 2.2. – Σm ρ (R) denotes the space of symbols a(x, ξ) such that � a= a(m−j) (j ∈ N), 0≤j<ρ. where a. (m−j). ∈. Γm−j ρ−j (R). is homogeneous of degree m − j with respect to ξ.. Given a symbol a, we define the paradifferential operator Ta by � −1 � (2.2) Ta u(ξ) = (2π) χ(ξ − η, η)� a(ξ − η, η)ψ(η)� u(η) dη,. � where � a(θ, ξ) = e−ix·θ a(x, ξ) dx is the Fourier transform of a with respect to the first variable, χ, ψ are two fixed C ∞ functions such that ψ(η) = 0 for |η| ≤ 1,. ψ(η) = 1. for |η| ≥ 2,. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(7) 860. T. ALAZARD, N. BURQ AND C. ZUILY. χ(θ, η) is homogeneous of degree 0 and satisfies, for 0 < ε1 < ε2 small enough, χ(θ, η) = 1 if. |θ| ≤ ε1 |η| ,. χ(θ, η) = 0. if. |θ| ≥ ε2 |η| .. We shall use quantitative results from Métivier [21] about operator norms estimates in symbolic calculus. To do so we introduce the following semi-norms. D��ɪɴɪ�ɪ�ɴ 2.3. – For m ∈ R, ρ ≥ 0 and a ∈ Γm ρ (R), we set � � � � sup �(1 + |ξ|)|α|−m ∂ξα a(·, ξ)� (2.3) Mρm (a) = sup. W ρ,∞ (R). |α|≤ 12 +1+ρ |ξ|≥1/2. .. The main features of symbolic calculus for paradifferential operators are given by the following theorems. D��ɪɴɪ�ɪ�ɴ 2.4. – Let m ∈ R. An operator T is said to be of order ≤ m if, for all µ ∈ R, it is bounded from H µ (R) to H µ−m (R). Tʜ��ʀ�� 2.5. – Let m ∈ R. If a ∈ Γm 0 (R), then Ta is of order ≤ m. Moreover, for all µ ∈ R there exists a constant K such that �Ta �H µ →H µ−m ≤ KM0m (a).. (2.4). �. m Tʜ��ʀ�� 2.6 (Composition). – Let m ∈ R and ρ > 0. If a ∈ Γm ρ (R) and b ∈ Γρ (R) then Ta Tb − Ta#b is of order ≤ m + m� − ρ, where � 1 a#b= ∂ α a∂ α b. i|α| α! ξ x |α|<ρ. Moreover, for all µ ∈ R there exists a constant K such that (2.5). �. �Ta Tb − Ta#b �H µ →H µ−m−m� +ρ ≤ KMρm (a)Mρm (b).. If a = a(x) is a function of x only, the paradifferential operator Ta is called a paraproduct. Paraproducts can also be defined using the Littlewood-Paley decomposition of the frequency space. Indeed, let φ : R → R be a smooth even function with φ(t) = 1 for |t| ≤ 1 and φ(t) = 0 for |t| ≥ 2. For k ∈ N, we introduce the symbol �ξ � φk (ξ) = φ k , 2 and then the operators Sk and ∆k defined by ‘ � S k f (ξ) := φk (ξ)f (ξ),. ‘ � ∆ k f (ξ) := (φk (ξ) − φk−1 (ξ)) f (ξ).. For all f ∈ S � (R), the spectrum of ∆k f satisfies spec ∆k f ⊂ {ξ : 2k−1 ≤ |ξ| ≤ 2k+1 }. Hence ∆j ∆k = 0 if |j − k| ≥ 2. Moreover we have the Littlewood–Paley decomposition: � f = S0 f + ∆k f. k∈N∗. With this decomposition, paraproducts can be defined by � Ta f = Sk−3 (a)∆k f. k≥4. Notice that the difference between paraproducts defined in these two ways is a smoothing operator. Namely, if a ∈ W ρ,∞ (R) for some ρ > 0 then the difference is of order −ρ. 4 e SÉRIE – TOME 44 – 2011 – No 5.

(8) STRICHARTZ ESTIMATES FOR WATER WAVES. 861. Tʜ��ʀ�� 2.7. – Let α, β ∈ R be such that α + β > 0. If a ∈ H α (R) and b ∈ H β (R) 1 then ab − Ta b − Tb a ∈ H α+β− 2 (R) and �ab − Ta b − Tb a�. 1. H α+β− 2 (R). ≤ K�a�H α (R) �b�H β (R). for some positive constant K independent of a, b. We use the following result which is a consequence of (2.5) with m = m� = 0, ρ = 1. L�� 2.8. – Let a ∈ W 1,∞ (R). Then for all σ ∈ R there exists a constant C > 0 such that for all j ∈ N, �[∆j , Ta ]u�H σ+1 (R) ≤ C�a�W 1,∞ (R) �u�H σ (R) . 2.2. The Dirichlet-Neumann operator L�� 2.9. – Let s > 2 + 12 and 1 ≤ σ ≤ s. Then there exists an increasing function 1 C : R+ → R+ such that for all (η, ψ) ∈ H s+ 2 (R) × H s (R) �G(η)ψ�H σ−1 (R) ≤ C(�η�. 1. H s+ 2 (R). )�ψ�H σ (R) .. 1. Furthermore, if (η, ψ) ∈ L∞ (I; H s+ 2 (R) × H s (R)) is a solution of (1.4), then (2.6). ∂t (G(η)ψ) = G(η)(∂t ψ − B∂t η) − div(V ∂t η). where (2.7). B(t, x) :=. ∂x ψ∂x η + G(η)ψ , 1 + |∂x η|2. V (t, x) := ∂x ψ − B∂x η.. 2.3. Symmetrization We consider a solution (η, ψ) of (1.4) on the time interval I = [0, T ] with 0 < T < +∞, satisfying the assumption (1.2) for all t ∈ I and such that � � 1 (η, ψ) ∈ C 0 I, H s+ 2 (R) × H s (R) , for some s > 52 . Then we set (2.8). U = ψ − TB η,. where B has been defined in (2.7). It follows from the analysis in [2] that we have the following symmetrization of the equations. L�� 2.10 ([2, Corollary 4.9]). – Let c, c1 be defined by � �− 3 � �− 1 c = 1 + (∂x η)2 4 , c1 = 1 + (∂x η)2 2 . 1/2. There exists an elliptic symbol p ∈ Σs−1 such that the complex-valued unknown (2.9). Φ = Tp η + iTc1 U. satisfies a scalar equation of the form (2.10). 3. 3. ∂t Φ + TV ∂x Φ + i |Dx | 4 Tc |Dx | 4 Φ = F,. where V has been defined in (2.7) and F ∈ L∞ (I, H s (R)).. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(9) 862. T. ALAZARD, N. BURQ AND C. ZUILY. 3. Reductions 3.1. Change of variables We consider a solution (η, ψ) of (1.4) on the time interval I = [0, T ] with 0 < T < +∞, satisfying the assumption (1.2) for all t ∈ I and such that � � 1 (η, ψ) ∈ C 0 I, H s+ 2 (R) × H s (R) ,. for some s > 52 . Our aim in this section is to simplify the Equation (2.10) by a change of variable. To compute the effect of a change of variable we shall use Alinhac’s paracomposition operators and we refer to [4] for the general theory . Let κ be a C 1 diffeomorphism from R to R. We define the operator κ∗ by κ∗ u = u ◦ κ − T(∂x u)◦κ κ.. (3.1). One of the main properties of κ∗ is that there is a symbolic calculus theorem which allows to compute the equation satisfied by κ∗ u in terms of the equation satisfied by u (in analogy with the paradifferential calculus). Tʜ��ʀ�� 3.1. – Let m ∈ R, r > 1, ρ > 0 and set σ := inf{ρ, r − 1}. Consider a diffeomorphism χ such that ∂x χ ∈ W r−1,∞ (R) and set κ = χ−1 . Let a be a symbol in Σm ρ (R). Then there exists a∗ ∈ Σm (R) such that σ κ∗ Ta − Ta∗ κ∗. is of order ≤ m − σ. � Moreover one can give an explicit formula for a∗ . If a = am−k , then � 1 (3.2) a∗ (χ(x), η) = ∂αa (x, χ� (x)η)∂yα (eiΨx (y)·η )|y=x , |α| α! ξ m−k i α. where the sum is taken over all α ∈ N such that the summand is well defined, χ� (x) is the derivative of χ and Ψx (y) = χ(y) − χ(x) − χ� (x)(y − x).. (3.3). We are now ready to simplify (2.10). Define χ by � x � x» 2 (3.4) χ(t, x) = c(t, y)− 3 dy = 1 + (∂y η(t, y))2 dy, 0. 0. so that. ∂x χ(t, x) =. ». 2. 1 + (∂x η(t, x))2 = c(t, x)− 3 .. Then for each t ∈ [0, T ], x �→ χ(t, x) is a diffeomorphism from R to R. Introduce its inverse κ = χ−1 .. (3.5). 3.1.1. Notations: We shall set I = [0, T ] and we shall denote � � (3.6) A = C �(η, ψ)� ∞ s+ 1 s 2 L. (I,H. (R)×H (R). where C : R → R is an increasing function which may change from line to line. Moreover we shall denote by f ◦ κ the function +. +. (3.7). 4 e SÉRIE – TOME 44 – 2011 – No 5. (f ◦ κ)(t, x) = f (t, κ(t, x))..

(10) STRICHARTZ ESTIMATES FOR WATER WAVES. 863. 3.1.2. Estimates of χ and κ. – From (3.4), the equation ∂t η = G(η)ψ, the Lemma 2.9, the Hölder inequality and the fact that s > 2 + 12 we deduce, (3.8). �∂t χ�L∞ (I×R) ≤ A.. Now since ∂x χ(t, x) = 1 + f (∂x η), 1 2. we deduce from the assumption s > 2 + (3.9). �∂x χ(t, x) − 1�. f ∈ C ∞ (R), f (0) = 0,. and the Sobolev embedding that, 1. L∞ (I,H s− 2 (R)). + �∂x χ�L∞ (I×R) ≤ A.. Let us consider the function κ. Since ∂t κ = − ∂∂xt χχ ◦ κ we have, using (3.8),. (3.10). �∂t κ�L∞ (I×R) ≤ A.. On the other hand we have ∂x κ = 1 + f (∂x η) where f ∈ C ∞ (R), f (0) = 0. It follows that, (3.11). �∂x κ − 1�. ≤ A.. 1. L∞ (I,H s− 2 (R)). Let p = [s] if s ∈ / N, p = s − 1 if s ∈ N; then p ≥ s − 1 and it follows from (3.11) that,. (3.12). �∂x κ�L∞ (I,W p−1,∞ (R)) ≤ A.. To go further we shall need the following elementary lemma. L�� 3.2. – Let p ∈ N∗ and κ : R → R be a diffeomorphism such that ∂x κ ∈ W p−1,∞ (R). Set χ = κ−1 . Then for all F ∈ H µ (R) with 0 ≤ µ ≤ p we have F ◦ κ ∈ H µ (R) and � � �F ◦ κ�H µ (R) ≤ �χ� �L∞ (R) C �∂x κ�W p−1,∞ (R) �F �H µ (R) where C is an increasing function from R+ to R+ .. We deduce from Lemma 3.2 and (3.12) that for 0 ≤ µ ≤ s − 1 and F ∈ L∞ (I, H µ (R)) we have, (3.13). �F ◦ κ�L∞ (I,H µ (R)) ≤ A�F �L∞ (I,H µ (R)) .. Coming back to the regularity of χ we deduce from (3.4) that, ∂x2 χ =. (∂x η)(∂x2 η). .. 1. (1 + (∂x η)2 ) 2. It follows from (3.13) that, (3.14). �(∂x2 χ) ◦ κ�. 3. L∞ (I,H s− 2 (R)). ≤ A.. On the other hand we have, ∂x ∂t χ =. (∂x η)∂x (G(η)ψ) 1. (1 + (∂x η)2 ) 2. .. So using Lemma 2.9 and (3.13) we obtain, (3.15). �(∂x ∂t χ) ◦ κ�L∞ (I,H s−2 (R)) ≤ A.. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(11) 864. T. ALAZARD, N. BURQ AND C. ZUILY. Now we would like to estimate ∂t2 χ. Since ∂t η = G(η)ψ we have, � x � x [∂x η∂x (G(η)ψ)]2 [∂x (G(η)ψ)]2 2 (3.16) ∂t χ(t, x) = − dy + dy 3 2 1 (1 + (∂x η)2 ) 2 0 0 (1 + (∂x η) ) 2 � x ∂x η∂x ∂t (G(η)ψ) + dy. 1 (1 + (∂x η)2 ) 2 0 Since s > 2 + 12 , the Hölder inequality and Lemma 2.9 show that the first two terms are pointwise bounded by A. By the Hölder inequality the last term can be pointwise bounded by � � �∂x η�L∞ (I,L2 (R)) �∂x ∂t G(η)ψ �L∞ (I,L2 (R)) . Using (2.6) and the equation satisfied by (η, ψ) we find, if s > 3 + 12 , � � �∂x ∂t G(η)ψ �L∞ (I,L2 (R)) ≤ A.. Therefore if s > 3 +. 1 2. we obtain,. �∂t2 χ�L∞ (I×R) ≤ A.. (3.17). Finally let us estimate the term ∂x ∂t2 χ. Using again (2.6) and the equation satisfied by (η, ψ) we find, if s > 4, that, �∂x ∂t2 χ�. (3.18). 7. L∞ (I,H s− 2 (R)). ≤ A.. 3.1.3. Reduction of the equation. – With V defined in (2.7) and Φ defined in (2.9) let us set (see (3.7)), (3.19). W = V ◦ κ(∂x χ ◦ κ) + ∂t χ ◦ κ,. (3.20). Φ∗ = κ∗ Φ = Φ ◦ κ − T(∂x Φ)◦κ κ.. Then we have the following result. Pʀ��ɪ�ɪ�ɴ 3.3. – Let s > 2 + 12 and I = [0, T ]. There exists a real valued function g such that ∂x g ∈ Σ0s− 3 and the function u = Teig Φ∗ satisfies the equation 2. 3. (3.21). (∂t + TW ∂x + i |Dx | 2 )u = F,. with F ∈ L∞ (I, H s (R)) and W is defined by (3.19).. Proof. – We apply the operator κ∗ to the Equation (2.10). We first show that (3.22). κ∗ (∂t + TV ∂x )Φ = (∂t + TW ∂x )Φ∗ + R(Φ) �R(Φ)�L∞ (I,H s (R)) ≤ C(�(η, ψ)�. 1. L∞ (I,H s+ 2 (R)×H s (R). )�Φ�L∞ (I,H s (R)) .. To estimate the remainders we shall use the estimates (3.8) – (3.15) on κ and χ obtained above. We begin by showing that (3.23). κ∗ (∂t Φ) = (∂t − T(∂t χ)◦κ )Φ∗ + R1 (Φ). where R1 satisfies the estimate in (3.22). We have κ∗ (∂t Φ) = (∂t Φ) ◦ κ − T(∂x ∂t Φ)◦κ κ. = ∂t (Φ ◦ κ) − (∂t κ)(∂x Φ ◦ κ) − T(∂x ∂t Φ)◦κ κ,. 4 e SÉRIE – TOME 44 – 2011 – No 5.

(12) 865. STRICHARTZ ESTIMATES FOR WATER WAVES. therefore, κ∗ (∂t Φ) = ∂t (κ∗ Φ) + B1 + B2 , (3.24). B1 = T(∂x2 Φ◦κ)∂t κ κ B2 = T(∂x Φ)◦κ ∂t κ − (∂t κ)(∂x Φ ◦ κ).. Let us consider the term B1 in (3.24) and let us set a = ∂t κ(∂x2 Φ ◦ κ). We have, � � � � −j D)(∂x κ) = Ta κ = Sj−3 (a)∆j (κ) = 2−j Sj−3 (a)φ(2 gj j≥4. j≥4. where φ� ∈ C ∞ (R), supp φ� ⊂ ‹j (∂x κ) = ∆ ‹j (f (∂x η)) so, ∆. { 12. j≥4. ≤ |ξ| ≤ 2}. Since ∂x κ = 1 + f (∂x η) with f (0) = 0, we have 1. �gj �L2 (R) ≤ 2−j �a�L∞ (R) 2−j(s− 2 ) cj C(�η�. 1. H s+ 2 (R). ),. (cj ) ∈ l2 .. On the other hand using (3.8), (3.9) we can write, �a�L∞ (I×R) ≤ �Φ�L∞ (I,H s (R)) �∂t χ(∂x χ)−1 �L∞ (I×R) ≤ �Φ�L∞ (I,H s (R)) C(�(η, ψ)�. 1. L∞ (I,H s+ 2 (R)×H s (R). ).. It follows that, (3.25). �B1 �L∞ (I,H s (R)) ≤ C(�(η, ψ)�. 1. L∞ (I,H s+ 2 (R)×H s (R). )�Φ�L∞ (I,H s (R)) .. Let us consider the term B2 . We have, ∂t κ = ab where a = ∂t χ ∈ Γ01 , b = ∂x κ ∈ Γ01 . It follows from Theorem 2.6 that a # b = ab and Tab − Ta Tb is of order −1. Let us set (3.26). Using (2.5) we obtain,. B21 = �(Tab − Ta Tb )(∂x Φ ◦ κ)�L∞ (I,H s (R)) .. B21 ≤ �∂t χ�L∞ (I,W 1,∞ (R)) �∂x κ�L∞ (I,W 1,∞ (R)) �(∂x Φ ◦ κ)�L∞ (I,H s−1 (R)) .. Since s − (3.27). 3 2. > 1, using (3.13) with µ = s − 1 we obtain, � � B21 ≤ C �η� ∞ �Φ�L∞ (I,H s (R)) . s+ 1 2 L. (I,H. (R)). Therefore using (3.24), (3.25), (3.27) and Theorem 2.7 we obtain, (3.28). κ∗ (∂t Φ) = ∂t κ∗ Φ − T∂t χ T∂x κ ∂x Φ ◦ κ + R2 (Φ),. where R2 satisfies (3.22). Now let us set. 1. a = ∂x κ ∈ L∞ (I, H s− 2 (R)), b = ∂x Φ ◦ κ ∈ L∞ (I, H s−1 (R)).. It follows from Theorem 2.7 that (3.29). �ab − Ta b − Tb a�L∞ (I,H 2s−2 (R)) ≤ �a�. 1. L∞ (I,H s− 2 (R)). �b�L∞ (I,H s−1 (R)) .. Therefore we obtain (3.30). κ∗ (∂t Φ) = ∂t (κ∗ Φ) − T∂t χ ∂x κ∂x (Φ ◦ κ) + T∂t χ T∂x Φ◦κ ∂x κ + R3 ,. where R3 satisfies (3.22). Using (3.1) we obtain. κ∗ (∂t Φ) = (∂t − T∂t χ ∂x )(κ∗ Φ) − T∂t χ ∂x (T∂x Φ◦κ κ) + T∂t χ T∂x Φ◦κ ∂x κ + R3 , ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(13) 866. T. ALAZARD, N. BURQ AND C. ZUILY. where R3 satisfies (3.22). It follows that κ∗ (∂t Φ) = (∂t − T∂t χ ∂x )(κ∗ Φ) − T(∂x2 Φ◦κ)∂x κ κ + R3 . Now the term T(∂x2 Φ◦κ)∂x κ κ can be estimated exactly by the same method as the term B1 , therefore we obtain κ∗ (∂t Φ) = (∂t − T∂t χ ∂x )(κ∗ Φ) + R4 ,. where R4 satisfies (3.22). This is precisely (3.23). Now we claim that. κ∗ (TV ∂x Φ) = T(V ∂x χ)◦κ ∂x κ∗ Φ + R5 (Φ),. (3.31). where R5 satisfies (3.22). But this is precisely a consequence of Theorem 3.1. Indeed we have 3 for (almost all) fixed t, a(x, ξ) = iV (t, x)ξ ∈ Σ1s−1 , and the diffeomorphism κ is in W s− 2 (R), so σ = s − 32 and the remainder term is of order less than 1 − (s − 32 ) = 52 − s < 0. Then (3.22) follows from (3.23) and (3.31). Let us consider now the principal part. Applying again Theorem 3.1 we find that, 3. 3. 3. κ∗ (|Dx | 4 Tc |Dx | 4 Φ) = |Dx | 2 κ∗ Φ + Ta κ∗ Φ, where a is of order 12 . Finally, it remains to reduce to the case where a = 0. Indeed, let g be a real-valued symbol such that ∂x g ∈ Γ0s−3/2 (R) and {|ξ|3/2 , g} = −a,. then if we set. u = Teig Φ∗ ,. (3.32). we obtain by symbolic calculus that u satisfies 3. (∂t + TW ∂x + i |Dx | 2 + iTa + Tb )u = F, with F ∈ L∞ (I, H s (R)) and b = i{|ξ|3/2 , g}. This completes the proof of Proposition 3.3. 3.1.4. Regularity of W. – The following result gives some informations on the function W defined in (3.19). L�� 3.4. – Let I = [0, T ], E = L∞ (I × R), F = L∞ (I, H s−2 (R)). 1. If s > 2 + 12 , we have W ∈ E, ∂x W ∈ F , and �W �E + �∂x W �F ≤ C(�(η, ψ)�. 1. L∞ (I,H s+ 2 (R)×H s (R)). ).. 2. If s > 4, we have ∂t W, ∂x2 W, ∂t ∂x W ∈ E and �∂t W �E + �∂x2 W �E + �∂t ∂x W �E ≤ C(�(η, ψ)�. 1. L∞ (I,H s+ 2 (R)×H s (R)). 4 e SÉRIE – TOME 44 – 2011 – No 5. )..

(14) 867. STRICHARTZ ESTIMATES FOR WATER WAVES. Proof. – Let us recall that we have set, � (3.33) A = C �(η, ψ)�. 1. L∞ (I,H s+ 2 (R)×H s (R). �. where C : R+ → R+ is an increasing function which may change from place to place. Since s > 2 + 12 using (2.7) we obtain, (3.34). �V �E ≤ �∂x ψ�L∞ (I,H s−2 (R)) + �B�L∞ (I,H s−2 (R)) �∂x η�L∞ (I,H s−2 (R)) ≤ A.. Then the estimate �W �E ≤ A follows fom (3.8) and (3.9). Now we have ∂x W = ∂x V ◦ κ + V ◦ κ(∂x2 χ ◦ κ)∂x κ + (∂t ∂x χ ◦ κ)∂x κ.. (3.35). Using (3.13) we see that, (3.36). �V ◦ κ�F + �∂x V ◦ κ�F ≤ A�V �L∞ (I,H s−1 (R)) ≤ A.. Now using (3.11), (3.14) and the fact that H s−2 (R) is an algebra we deduce, �V ◦ κ(∂x2 χ ◦ κ)∂x κ�F ≤ A.. (3.37). Then the estimate �∂x W �F ≤ A follows from (3.15) and (3.11). Let us now prove 2. We have (3.38). ∂t W = ∂t V ◦ κ(∂x χ ◦ κ) + ∂x V ◦ κ(∂x χ ◦ κ)∂t κ + V ◦ κ(∂t ∂x χ ◦ κ) + V ◦ κ(∂x2 χ ◦ κ)∂t κ − ∂t2 χ ◦ κ − ∂t ∂x χ ◦ κ(∂x κ) =:. 6 �. Bi .. i=1. It follows from (3.13), (3.9), (3.10), (3.14), (3.15), and the Sobolev embedding that (3.39). |B2 | + |B3 | + |B4 | + |B6 | ≤ A.. Now we have ∂t V = ∂x ∂t ψ − (∂t B)∂x η − B∂x ∂t η. So using the equations satisfied by (η, ψ), the Sobolev embedding and Lemma 2.9 we obtain (3.40). �∂t V �. 5. L∞ (I,H s− 2 (R)). ≤ A.. It follows that (3.41). |B1 | ≤ A.. The term B5 is estimated by A using (3.17). Therefore using (3.39) and (3.41) we deduce that �∂t W �E ≤ A. The claim on ∂x2 W follows from the first part of the lemma and the Sobolev embedding since s > 3 + 12 . It remains to consider the quantity ∂t ∂x W. We go back to (3.38). The term 5 ∂t V ◦ κ(∂x χ ◦ κ) is bounded by A in L∞ (I, H s− 2 (R)). The third term V ◦ κ(∂t ∂x χ ◦ κ) is bounded by A in L∞ (I, H s−2 (R)). The term ∂t ∂x χ ◦ κ(∂x κ) is bounded by A in L∞ (I, H s−2 (R)). Therefore the ∂x derivative of these three terms are bounded by A 7 in L∞ (I, H s− 2 (R)). By (3.8) we have, �∂x V ◦ κ(∂x χ ◦ κ)∂t κ�L∞ (I×R) ≤ A�∂x V ◦ κ(∂x χ ◦ κ)�L∞ (I×R). ≤ A�∂x V ◦ κ(∂x χ ◦ κ)�L∞ (I,H s−2 ) ≤ A.. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(15) 868. T. ALAZARD, N. BURQ AND C. ZUILY. We can apply the same argument for the term V ◦ κ(∂x2 χ ◦ κ)∂t κ. Finally we bound the term 7 2 V ◦ κ(∂x ∂tt χ ◦ κ) in the space L∞ (I, H s− 2 (R)) by using (3.13) and (3.18).This completes the proof of our lemma. 3.2. Symbol smoothing In this section we follow an idea of Smith [26] (see also Bahouri-Chemin [7]), and we are going to smooth out the coefficients of the function W with respect to x. As already mentioned, here is the main place where the idea of allowing loss in remainder terms enters. We define for 0 < δ ≤ 1, � δ TW = S[δ(j−3)] (W )∆j , j≥4. δ The key difference between TW and TW is made clear below. δ L�� 3.5. – The operator TW − TW is of order −δ(s − 32 ).. Proof. – Since for almost all fixed t we have, ∂x W (t, ·) ∈ H s−2 (R) we have, j � � � �Sj (W ) − S[δj] (W )� ∞ ≤ �∆n (W )�L∞ (R) L (R) n=[δj] j �. ≤K. 3. n=[δj]. 3. 2−n(s− 2 ) ≤ K2−δj(s− 2 ) .. In the sequel we shall set    . (3.42).   . h = 2−j , j ∈ N,. Whδ = S[δ(j−3)] (W ), 3. a(ξ) = χ0 (ξ)|ξ| 2 ,. where χ0 ∈ C0∞ (R), supp χ0 ⊂ { 14 ≤ |ξ| ≤ 4}, χ0 = 1 in { 12 ≤ |ξ| ≤ 2}. L�� 3.6. – Let s > 2 + 12 and u ∈ L∞ (I, H s (R)) be a solution of (3.21). There exist 1 δ < 12 , � > 0 and fh ∈ L∞ (I, H s+�− 2 (R)) such that � � 1 �fh � ∞ ≤ C �(η, ψ)� , s+�− 1 s+ 2 (R)) L (I,H L∞ (I,H 2 (R)×H s (R)) (3.43) 1 −1 supp(f� ≤ |ξ| ≤ 2h−1 } h) ⊂ { h 2 and the functions uh = ∆j u satisfy (3.44). 1 (∂t + (Whδ ∂x + ∂x Whδ ) + ia(Dx ))uh = fh . 2. Furthermore we have (3.45). �. j∈N. 4 e SÉRIE – TOME 44 – 2011 – No 5. �∆j u�2L∞ (I;H s (R)) < +∞..

(16) 869. STRICHARTZ ESTIMATES FOR WATER WAVES 3. Proof. – First of all we remark that we have |Dx | 2 uh = a(Dx )uh (and consequently 3 h− 2 a(hDx )uh = a(Dx )uh ). Now, applying the operator ∆j to (3.21), we obtain (3.46). 3. (∂t + TW ∂x + i |Dx | 2 )uh = ∆j F − [∆j , TW ]∂x u := gh1 .. Let us prove (3.45). We deduce from the usual energy estimates that, � � �uh �L∞ (I;H s (R)) ≤ C �uh |t=0 �H s (R) + �gh1 �L1 (I;H s (R)) � � ≤ C � �uh |t=0 �H s (R) + �gh1 �L2 (I;H s (R)) .. Since by Lemma 3.4 we have ∂x W ∈ L∞ (I, H s−2 (R)), using Lemma 2.8 we can write, � � ‹j u�2 2 �gh1 �2L2 (I;H s (R)) ≤ 2 �∆j F �2L2 (I;H s (R)) + �[∆j , TW ]∂x ∆ L (I;H s (R)) � � ‹j u�2 2 ≤ C �∆j F �2L2 (I;H s (R)) + �∆ L (I;H s (R)) ‹j = � where ∆ |k−j|≤3 ∆k . It follows that � � � �gh1 �2L2 (I;H s (R)) ≤ C �F �2L2 (I;H s (R)) + �u�2L2 (I;H s (R)) j∈N. from which (3.45) follows easily.. δ Now we replace TW by TW in (3.46) to obtain � � 3� δ (3.47) ∂t + S[δ(k−3)] (W )∆k ∂x + i |Dx | 2 uh = gh1 + (TW − TW )∂x uh := gh1 + gh2 |k−j|≤1. where, according to Lemma 3.5, gh2 satisfies (3.43) with � = δ(s − 32 ) − chosen close enough to 12 . Now, we have � S[δ(j−3)] (W )∂x uh = S[δ(j−3)] (W )∆k ∂x uh .. 1 2. > 0 if δ <. 1 2. |k−j|≤1. Consequently, we obtain 3. (3.48) (∂t + S[δ(j−3)] (W )∂x + i |Dx | 2 )uh � = gh1 + gh2 + (S[δ(j−3)] (W ) − S[δ(k−3)] (W ))∆k ∂x uh = gh1 + gh2 + gh3 , |k−j|≤1. and using that for |k − j| ≤ 1, � � �S[δ(k−3)] (W ) − S[δ(j−3)] (W )� ∞ ≤ C2−jδ(s− 32 ) , L we obtain that gh3 satisfies (3.43). Finally, we obtain. 3 1 (∂t + (Whδ ∂x + ∂x Whδ ) + i |Dx | 2 )uh = gh1 + gh2 + gh3 + gh4 , 2. where gh4 = 12 S[δ(j−3)] (∂x W )uh satisfies (3.43) (for any 0 < � ≤ 12 ). L�� 3.7. – Let s >. 11 2. and set δ=. 1 s−. 3 2. 1 ∈]0, [. 4. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE. is.

(17) 870. T. ALAZARD, N. BURQ AND C. ZUILY. Then there exists fh ∈ L∞ (I, H s (R)) such that � � �fh �L∞ (I,H s (R)) ≤ C �(η, ψ)� ∞ , s+ 1 s 2 L (I,H (R)×H (R)) (3.49) 1 −1 supp(f� ≤ |ξ| ≤ 2h−1 } h) ⊂ { h 2 and the functions uh = ∆j u satisfy (3.50). 1 (∂t + (Whδ ∂x + ∂x Whδ ) + ia(Dx ))uh = fh . 2. Proof. – The proof is identical to that of Lemma 3.6, the only difference being that now we take δ such that δ(s − 32 ) = 1. 4. Semi-classical parametrix The purpose of this section is to prove the main step toward Theorem 1.1. Tʜ��ʀ�� 4.1. – Under the assumptions of Theorem 1.1, let u be defined by (3.32). Then there exists C = C(�(η, ψ)� ∞ ) such that s+ 1 s 2 L. (I,H. (R)×H (R)). �u�. s− 1. L4 (I,B∞,24 (R)). ≤ C.. Following [13] we shall reduce the analysis to establishing semi-classical estimates. Recall that 2−j = h and Whδ = S[δ(j−3)] (W ) = φ(hδ Dx )W , φ ∈ C0∞ (R). Tʜ��ʀ�� 4.2. – Let χ ∈ C0∞ (R) with supp χ ⊂ {ξ : 12 ≤ |ξ| ≤ 2} and t0 ∈ R. For any initial data u0,h = χ(hDx )u0 , where u0 ∈ L1 (R), let Uh := S(t, t0 , h)u0,h be the solution of (4.1). 1 ∂t Uh + (Whδ ∂x + ∂x Whδ )Uh + ia(Dx )Uh = 0, 2. Uh |t=t0 = u0,h .. 1. Then for any 0 < h ≤ 1 and any |t − t0 | ≤ h 2 , (4.2). �S(t, t0 , h)u0,h �L∞ (R) ≤. C �u0,h �L1 (R) . h1/4 |t − t0 |1/2. To prove this result, we shall follow a very classical trend and construct a parametrix. Notice that our assumptions being time-translation invariant we can assume t0 = 0. The parametrix will take the following form, �� i ‹h (t, x) = 1 ‹ x, z, ξ, h)u0,h (z)dzdξ, (4.3) U e h (Φ(t,x,ξ,h)−zξ) B(t, 2πh where Φ will satisfy the eikonal equation and (4.4). ‹ x, z, ξ, h) = B(t, x, ξ, h)ζ(x − z − th− 12 a� (ξ)), B(t,. where B will satisfy the transport equations and ζ ∈ C0∞ (R), ζ(s) = 1 if |s| ≤ 1, ζ(s) = 0 if |s| ≥ 2. 4 e SÉRIE – TOME 44 – 2011 – No 5.

(18) 871. STRICHARTZ ESTIMATES FOR WATER WAVES. In addition to χ (introduced in Theorem 4.2), we shall use two more cut-off functions χj ∈ C0∞ (R), j = 1, 2, such that  1   supp χ1 ⊂ {ξ : ≤ |ξ| ≤ 3}, χ1 = 1 on supp χ, 3 (4.5)   supp χ0 ⊂ {ξ : 1 ≤ |ξ| ≤ 4}, χ0 = 1 on supp χ1 . 4 4.1. The eikonal and transport equations. We introduce some space of symbols in which we shall solve our equations. D��ɪɴɪ�ɪ�ɴ 4.3. – For small h0 to be fixed, we introduce the sets ¶ © 1 Ω = (t, x, ξ, h) ∈ R4 : h ∈ (0, h0 ), |t| < h 2 , 1 < |ξ| < 3 , � � O = (σ, x, ξ, h) ∈ R4 : h ∈ (0, h0 ), |σ| < 1, 1 < |ξ| < 3 .. m ( O)) the set of all functions f on Ω If m ∈ R and � ∈ R+ , we denote by S�m (Ω) (resp.S�, ∞ which are C with respect to (t, x, ξ) (resp.(σ, x, ξ)) and satisfy the estimate. |∂xα f (t, x, ξ, h)| (resp. |∂xα f (σ, x, ξ, h)|) ≤ Cα hm−�α ,. (4.6). for all (t, x, ξ, h) ∈ Ω (resp.(σ, x, ξ, h) ∈ O).. �. �. R��ʀ� 4.4. – (i) If f ∈ S�m , g ∈ S�m then f g ∈ S�m+m ; if f ∈ S�m , (m ≥ 0) and 0 F ∈ C ∞ (C) then F (f ) ∈ S�m ; if f ∈ S�m , (m ≤ 0) and F ∈ Cb∞ (C) then F (f ) ∈ S�−m . µ µ−� m m � Let f ∈ S� , then ∂x f ∈ S� . Moreover S� ⊂ S�� if � ≥ � . (ii) Let W be such that ∂x W ∈ H s−2 (R) with s > 2 + 12 and set Wh� = γ(h� Dx )W where 0 γ ∈ S (R). Then ∂x Wh� ∈ S�, . Let δ ∈ (0, 12 ). We fix (4.7). µ0 =. 1 2. Å. ã 1 −δ . 2. Finally we set, 1 2. 3. L 0 = ∂t + (Whδ ∂x + ∂x Whδ ) + iχ0 (hDx ) |Dx | 2 ,. (4.8). 3. a(ξ) = χ0 (ξ) |ξ| 2 .. The main result of this section is the following. Pʀ��ɪ�ɪ�ɴ 4.5. – There exist a phase Φ of the form 1. 1. Φ(t, x, ξ, h) = xξ − h− 2 ta(ξ) + h 2 Ψ(t, x, ξ, h). ‹ defined in (4.4), with ∂x Ψ ∈ Sδ0 (Ω) and an amplitude B ∈ Sδ0 (Ω) such that, with B Ä i ä i ‹ = e h Φ Rh (4.9) L 0 e h ΦB and for all N ∈ N we have, � �� i � � � (4.10) � e h (Φ(t,x,ξ,h)−zξ) Rh (t, x, z, ξ, h)u0,h (z) dz dξ �. H 1 (Rx ). 1. for all t in [0, h 2 ].. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE. ≤ CN hN �u0,h �L1 (R) ,.

(19) 872. T. ALAZARD, N. BURQ AND C. ZUILY. Proof. – We set, 1. 1. t = h 2 σ, (4.11). ϕ(σ, x, ξ, h) = Φ(σh 2 , x, ξ, h),. 1 1 �b(σ, x, ξ, h) = B(σh ‹ 2 , x, ξ, h), Vh (σ, x) = Whδ (σh 2 , x), � 1 1� L = h∂σ + h 2 Vh (h∂x ) + h∂x Vh + ia(hDx ). 2 3. Multiplying (4.9) by h 2 we see that it is equivalent to, Ä i ä i (4.12) L e h ϕ�b = e h ϕ r(σ, x, z, ξ, h), and (4.10) becomes, � �� i � � � (4.13) � e h (ϕ(σ,x,ξ,h)−zξ) r(σ, x, z, ξ, h)u0,h (z) dz dξ �. H 1 (Rx ). ≤ CN hN �u0,h �L1 (R) .. In the proof of (4.13), z, ξ, h will be considered as parameters. We shall take ϕ of the form (4.14). 1. ϕ(σ, x, ξ, h) = xξ − σa(ξ) + h 2 ψ(σ, x, ξ, h),. where ψ is the solution of the problem ® ∂σ ψ + a� (ξ)∂x ψ = −ξVh , (4.15) ψ|σ=0 = 0.. Differentiating (4.15) with respect to x and ξ, using an induction on k and the fact that ∂x Vh ∈ Sδ0 ( O), we see easily that, +. |∂ξk ∂xα ψ(σ, x, ξ, h)| ≤ Ckα |σ|h−δ(α+k−1) ,. (4.16). for every (σ, x, ξ, h) ∈ O, where a+ = sup(a, 0). It follows in particular that ∂x ψ ∈ Sδ0 ( O), ∂σ ψ ∈ Sδ0 ( O) . Now, since (see (4.11)) �b = b ζ we have, 3. � � i i 1 1 h2 e− h ϕ h∂σ + (Vh ∂x + ∂x Vh ) (e h ϕ�b) = i[h 2 ξVh − a(ξ) + h 2 ∂σ ψ + hVh ∂x ψ]�b 2 (4.17) 1 1 1 1 + h[∂σ b + h 2 Vh ∂x b + h 2 (∂x Vh )b]ζ + h[−a� (ξ) + h 2 Vh ]b ζ � . 2 On the other hand recall that we have for all M ∈ N∗ (see the appendix), Ä i ä i (4.18) e− h ϕ a(hDx ) e h ϕ�b = A + r1 + r2 , where (4.19). A=. M −1 � k=0. with (4.20). ρ(x, y) =. © hk k ¶ k �b(y)  ∂ (∂ a) (ρ(x, y))  y ξ ik k! y=x �. 0. 1. ∂ϕ (σ, λx + (1 − λ)y, ξ, h) dλ, ∂x. and the remainder r1 , r2 are given by, �� 1 i (4.21) r1 = c hM −1 e h (x−y)η κ0 (η)(1 − λ)M −1 ∂yM [a(M ) (λη + ρ(x, y))�b(y)]dλdydη 0. 4 e SÉRIE – TOME 44 – 2011 – No 5.

(20) STRICHARTZ ESTIMATES FOR WATER WAVES. 873. and (4.22). r2 =. M −1 �. ck,M hM +k. ��. 0. k=0. 1. z M κ̂0 (z)(1 − λ)M −1 ∂yM +k [a(k) (ρ)�b]|y=x−λhz dλdz,. where cM , ck,M ∈ C, κ0 ∈ C0∞ (R), κ0 = 1 in a neighborhood of the origin. Now since �b(σ, x, z, ξ, h) = b(σ, x, ξ, h)ζ(x − z − σa� (ξ)),. writing for simplicity b(y) = b(σ, y, ξ, h) and ζ = ζ(x − z − σa� (ξ)) we have,  M −1 �    A = ( Ak )ζ + r3 ,     k=0    � � hk (4.23) Ak = k ∂yk (∂ξk a) (ρ(x, y)) b(y) |y=x ,  i k!    M −1 � k  �  � �   cjk hk ∂yk−j (∂ξk a)(ρ(x, y))b(y) |y=x ζ (j) .   r3 = k=1 j=1. 1. The term A0 in (4.23) is equal to a(ξ + h 2 ∂x ψ)b. Then � 2 � � 1 1 � 1 2 ∂ ψ)3 1 1 (h x (4.24) A0 = a(j) (ξ)(h 2 ∂x ψ)j + (1 − λ)2 ∂ξ3 a(ξ + λh 2 ∂x ψ) dλ b. j! 2 0 j=0 The term A1 in (4.23) can be written as ï ò 1 1 h � 1 1 A1 = a (ξ + h 2 ∂x ψ)∂x b + h 2 (∂x2 ψ)a�� (ξ + h 2 ∂x ψ)b . i 2. Therefore h A1 = i. (4.25). ñ® �. 1 2. a (ξ) + h ∂x ψ. �. 1. ´ ��. 1 2. a (ξ + λh ∂x ψ) dλ ∂x b. 0. ò 1 1 1 + h 2 (∂x2 ψ)a�� (ξ + h 2 ∂x ψ)b . 2. Since ∂x ψ ∈ Sδ0 , hδ ∂x2 ψ ∈ Sδ0 , we deduce from (4.24) and (4.25) that ï ò 1 h �� h � 2 2 (4.26) A0 + A1 = a(ξ) + h a (ξ)∂x ψ + a (ξ)(∂x ψ) b + a� (ξ)∂x b 2 i + hhµ0 (c1 b + c2 hδ ∂x b) for some cj ∈ Sδ0 , where µ0 has been defined in (4.7). Now, consider the term Ak with k ≥ 2. We have � � k � � hk � k Ak = k ∂yk1 (∂ξk a)(ρ(x, y)) y=x ∂xk−k1 b. i k! k1 k1 =0. 1 2. Since h ∂x ψ ∈ Sδ0 , we obtain,. � � ck,k1 := hk1 δ ∂yk1 (∂ξk a)(ρ(x, y)) y=x ∈ Sδ0 .. It follows that the generic term in Ak can be written as. hhk−1 h−k1 δ ck,k1 h−δ(k−k1 ) (hδ ∂x )k−k1 b.. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(21) 874. T. ALAZARD, N. BURQ AND C. ZUILY. We have, since k ≥ 2,. 1 1 (4.27) k − 1 − k1 δ − δ(k − k1 ) ≥ k(1 − δ) − 1 ≥ 2(1 − δ) − 1 ≥ 2( − δ) ≥ − δ ≥ µ0 2 2 so that k � Ak = hhµ0 h c� (hδ ∂x )� b, c� ∈ Sδ0 . �=0. We deduce from (4.26) that � 2 � M −1 M −1 � � 1 � 1 h (j) j 2 (4.28) Ak = a (ξ)(h ∂x ψ) b + a� (ξ)∂x b + hhµ0 d� (hδ ∂x )� b j! i j=0 k=0. with d� ∈. �=0. Sδ0 .. Then it follows from (4.12), (4.17), (4.18) and (4.28) that Ä ä 1 1 r = i −a(ξ) + h 2 ξVh + h 2 ∂σ ψ + hVh ∂x ψ b ζ. 1 1 1 + h[∂σ b + h 2 Vh ∂x b + h 2 (∂x Vh )b]ζ 2 ï ò 1 h � + i a(ξ) + h 2 a (ξ)∂x ψ + a�� (ξ)(∂x ψ)2 b ζ + ha� (ξ)∂x b ζ 2. + hhµ0. M −1 �. d� (hδ ∂x )� b ζ +. 3 �. rj .. j=1. �=0. Gathering the terms in powers of h, noting that the coefficient of h0 vanishes and using the 1 eikonal equation to see that the coefficient in h 2 vanishes, we are left with � � 4 M −1 � � (4.29) r = h ∂σ b + a� (ξ)∂x b + if b + hµ0 h e� (hδ ∂x )� b ζ + i rj , j=1. �=0. where f = Vh ∂x ψ + a (ξ)(∂x ψ) is real valued, e� ∈ ��. 2. (4.30). r4 =. Sδ0. and. 1 1 h[−a� (ξ) + h 2 Vh ]b ζ � . i. It follows from (4.16) that |∂xα ∂ξk f (σ, x, ξ, h)| ≤ Ckα σh−k h−δ(α+k) ,. (4.31). for every (σ, x, ξ, h) ∈ O. In particular f ∈ Sδ0 ( O). Now we shall seek b under the form. (4.32). b=. J−1 �. hjµ0 bj ,. j=0. where the (4.33). b�j s. are the solutions of the following problems   ∂b0 + a� (ξ) ∂b0 + if b = 0, 0 ∂σ ∂x  b0 |σ=0 = χ1 (ξ),. 4 e SÉRIE – TOME 44 – 2011 – No 5.

(22) STRICHARTZ ESTIMATES FOR WATER WAVES. 875. where χ1 ∈ C0∞ (R) has been introduced in (4.5) and  M −1 �    ∂bj + a� (ξ) ∂bj + if bj = − e� (hδ ∂x )� bj−1 , ∂σ ∂x (4.34) �=0   b | = 0. j σ=0. It is easy to see that for all j we have,. (4.35). bj (σ, x, ξ, h) = χ1 (ξ)cj (σ, x, ξ, h).. For the estimates we shall use the following elementary lemma. L�� 4.6. – If u is a solution of the problem ∂σ u + a� (ξ)∂x u + if u = g,. u|σ=0 = z ∈ C,. where f be real-valued, then it satisfies the estimate � σ |u(σ, x, ξ, h)| ≤ |z| + |g(σ � , x + (σ � − σ)a� (ξ), ξ, h)|dσ � 0. for every (σ, x, ξ, h) ∈ O.. Proof. – Indeed, the solution is given by �σ � −i f (σ ,x+(σ � −σ)a� (ξ),ξ,h) dσ � 0 u(σ, x, ξ, h) = e ® ´ � σ � σ� i f (t,x+(t−σ)a� (ξ),ξ,h) dt � � � � × z+ e 0 g(σ , x + (σ − σ)a (ξ), ξ, h) dσ . 0. Using this lemma we deduce the following.. L�� 4.7. – The problems (4.33), (4.34) have unique solutions bj = χ1 (ξ)cj where the cj satisfy the estimates (4.36). |∂xα ∂ξk cj (σ, x, ξ, h)| ≤ Cαkj h−(α+k)δ. for all (σ, x, ξ, h) ∈ O, all α, k ∈ N, and all j = 0, . . . , M. � jµ0 In particular c = M belongs to Sδ0 ( O). j=0 cj h. Proof. – Let us look to the case j = 0. Then c0 satisfies the same equation and c0 |σ=0 = 1. We show first (4.36) for k = 0 and all α. By Lemma 4.6 we have |c0 | ≤ C. So assume that (4.36) is true (for k = 0) up to the order α − 1 and let us differentiate the Equation (4.33) α time with respect to x. It follows that U = ∂xα c0 satisfies the equation α. (4.37). � ∂U ∂U + a� (ξ) + if U = −i Clα (∂xl f )∂xα−l c0 . ∂σ ∂x l=1. Using (4.31), Lemma 4.6 and the induction we deduce that α � |U | ≤ C h−lδ h−(α−l)δ ≤ Ch−αδ . l=1. This proves (4.36) for k = 0 and all α. Then using an induction on k we differentiate the Equation (4.37) k times with respect to ξ and we use again (4.31), Lemma 4.6 and the induction to prove (4.36) for all k and α. The proof of (4.36) for j ≥ 1 is similar. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(23) 876. T. ALAZARD, N. BURQ AND C. ZUILY. It follows from (4.29), (4.33), (4.34) that r=. 5 �. rj .. j=1. where for any J ∈ N. r5 = hJµ0 bJ−1 ζ.. (4.38). 4.1.1. End of the proof of Proposition 4.5. – For rj , j = 1, 2 defined in (4.21), (4.22) we have,. We are left with the proof of (4.13).. �x − z − σa� (ξ)�|rj (σ, x, z, ξ, h)| ≤ ChM (1−δ)−1 |χ1 (ξ)|.. (4.39). Let us prove (4.39) for r1 . Recall that �� 1 i M −1 r1 = c h e h (x−y)η κ0 (η)(1 − λ)M −1 ∂yM [a(M ) (λη + ρ(x, y))�b(y)]dλdydη. where κ0 ∈. 0 ∞ C0 (R), κ0. = 1 in a neighborhood of the origin and. �b(y) = χ1 (ξ)c(σ, y, ξ, h)ζ(y − z − σa� (ξ)). with c ∈ Sδ0 and δ < 1. We estimate separately r1 , (x − y)r1 and (y − z − σa� (ξ))r1 . Since c ∈ Sδ0 we can write, � � |r1 | ≤ ChM −1−δM |κ0 (η)|dη |ζ1 (u)|du. where ζ1 ∈ C0∞ (R), ζ1 = 1 on the support of ζ. The term (y − z − σa� (ξ))r1 is estimated similarly, since (y − z − σa� (ξ)) is bounded on the support of ζ1 (y − z − σa� (ξ)). Finally to i i estimate (x − y)r1 we integrate by parts using the fact that hi ∂η e h (x−y)η = (x − y)e h (x−y)η . Recall that �� 1 M −1 � r2 = ck,M hM +k z M κ̂0 (z)(1 − λ)M −1 ∂yM +k [a(k) (ρ)�b]|y=x−λhz dλdz. k=0. 0. As above one can estimate separately r2 , (x − z − λhz − σa� (ξ)) + λhz)r2 and λhzr2 to obtain the desired bound. An estimate of the type of (4.39) for r5 follows immediately from (4.38). Let us prove (4.13) for rj , j = 1, 2, 5. Let us first bound the L2 norm of the left hand side. Using (4.39) we can see that for any N ∈ N one can find M and J so large that � �rj (σ, ·, z, ξ, h)�L2 (Rx ) dξ ≤ CN hN .. It follows that �� i � h (ϕ(σ,x,ξ,h)−zξ) r (σ, x, z, ξ, h)u (4.40) � e (z) dz dξ j 0,h � �. L2 (Rx ). ≤ CN hN. �. |u0,h (z)|dz.. The estimate of the L2 norm of the x derivative is analogous. The terms corresponding to r3 and r4 defined in (4.23) and (4.30) will be treated in the same manner and will use the fact that on the support of a derivative of the function ζ one 1 1 1 has |x − z − σa� (ξ)| ≥ 1. Since by (4.16) we have h 2 |∂ξ ψ| ≤ Ch 2 |σ| ≤ Ch 2 we deduce from (4.14) that |∂ξ (ϕ(σ, x, ξ, h) − zξ)| ≥ 12 if h is small enough. Therefore we can obtain an 4 e SÉRIE – TOME 44 – 2011 – No 5.

(24) STRICHARTZ ESTIMATES FOR WATER WAVES. 877. estimate analogous to (4.40) by integrating N times by parts in the integral appearing in the left hand side of (4.40) using the vector field L=. h ∂ξ . i(∂ξ (ϕ(σ, x, ξ, h) − zξ)). The proof of Proposition 4.5 is complete. 4.2. Refined Van der Corput estimate. Let us recall that we have set (see (4.3)) � ‹ ‹ x, z, h)u0h (z) dz (4.41) Uh (t, x) = K(t, where (4.42). ‹ x, z, h) = 1 K(t, 2πh 1. �. ‹ x, z, ξ, h) dξ. e h (Φ(t,x,ξ,h)−zξ) B(t, i. In the variable σ = th− 2 we have ‹ x, z, h) = K(σ, x, z, h) K(t, where. � i 1 K(σ, x, z, h) = e h (ϕ(σ,x,ξ,h)−zξ) �b(σ, x, z, ξ, h) dξ, 2πh where ϕ and b have been determined in (4.14), (4.33) and (4.34). Pʀ��ɪ�ɪ�ɴ 4.8. – There exists C > 0 such that Å ã1 C h 2 (4.43) |K(σ, x, z, h)| ≤ , h σ for all (σ, x, z, h) in ]0, 1/2] × R × R×]0, h0 [. Proof. – Since b ∈ Sδ0 is bounded with compact support in ξ, the estimate (4.43) is trivial for |σ| ≤ Ch. Let us assume that |σ| ≥ Ch. We have by (4.11), Å ã 1 1 1 3 L = h∂σ + h 2 Whδ (h 2 σ, x)(h∂x ) + h 2 (∂x Whδ ) + ia(hDx )) . 2 By a scaling argument we can assume without loss of generality that σ = σ0 = 1/2. Indeed, otherwise, setting σ x h τ= , x �= , � h= , σ0 σ0 σ0 we see that in the new variables, the operator reads where. 3 �=� �hδ � � δ ) + i|� L h∂τ + � h1/2 W h∂x + � h 2 (∂� W hD� |3/2 x h x. and consequently we have. 1/2 � δ (τ, x W �) = σ0 Whδ (σ0 τ, σ0 x �) h. � δ ∈ L∞ (H s−1 ), W h. with bounds uniform with respect to σ0 .. �δ ∈ S 0 ∂� W δ x h. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(25) 878. T. ALAZARD, N. BURQ AND C. ZUILY. Assume now that the dispersion estimate has been proved for the kernel of the operator � and σ = 1/2. Since we have L σ x x Sh (σ)u0 (x) = (S��h ( )� u0 )( ), u �0 ( ) = u0 (x), σ0 σ0 σ0 we can write σ (4.44) �Sh (σ0 )u0 �L∞ (R) = �S��h ( )� u0 �L∞ (R) σ0 C C|σ0 |1/2 C ≤ �� u0 �L1 (R) ≤ �u0 �L1 (R) ≤ 1/2 1/2 � |h| |σ0 | |hσ0 |1/2 |h| which is the dispersion estimate for the kernel of the operator L and σ = σ0 . Let us set (4.45). 1. θ(x, y, ξ, h) = ϕ(σ, x, ξ, h) − zξ = (x − z)ξ + a(ξ)σ + h 2 ψ(σ, x, ξ, h).. Then 1. ∂ξ2 θ(x, y, ξ, h) = ∂ξ2 a(ξ)σ + h 2 ∂ξ2 ψ(σ, x, ξ, h). 3. Now by (4.5) and (4.8), on the support of χ1 we have a(ξ) = |ξ| 2 . Therefore ∂ξ2 a(ξ) = 3 − 12 ≥ c0 > 0. On the other hand from (4.16) we have |∂ξ2 ψ| ≤ Cσh−δ which implies 4 |ξ| 1 1 h 2 |∂ξ2 ψ| ≤ Cσh 2 −δ ≤ Cσhµ0 . It follows that on the support of χ1 one can find a constant c1 > 0 such that 1 (4.46) 0 < c1 σ ≤ ∂ξ2 θ(x, y, ξ, h) ≤ σ, c1 if h0 is small enough. In the sequel we shall omit to note the variables (x, z, h) which are fixed. However, we shall take care of the fact that all the constants are independent of (x, z, h) ∈ R × R×]0, h0 [. Let us denote by [α, β] ⊂ [ 13 , 3] the support of χ1 .We deduce from (4.46) that the function ξ → ∂ξ θ(ξ) is increasing on [α, β]. Therefore one can find ρ ∈ [α, β] such that ∂ξ θ(ξ) ≤ 0. for ξ ∈ [α, ρ],. ∂ξ θ(ξ) ≥ 0. for ξ ∈ [ρ, β].. Noting b(σ, x, ξ, h) = b(ξ) and assuming that ]ρ, β[ is non empty, we shall estimate � β i 1 K+ (σ, x, ξ, h) = e h θ(ξ) b(ξ)ζ(x − z − a� (ξ))dξ, 2πh ρ. the estimate corresponding to the interval [α, ρ] being similar. We write for small h,  1  K+ = (I1 + I2 ),    2πh    � ρ+( σh ) 12   i I1 = e h θ(ξ) b(ξ)ζ(x − z − a� (ξ))dξ, (4.47)  ρ   � β    i  I =  e h θ(ξ) b(ξ)ζ(x − z − a� (ξ))dξ.  2 1 h 2 ρ+( σ ). We have obviously,. (4.48) 4 e SÉRIE – TOME 44 – 2011 – No 5. h 1 |I1 | ≤ C( ) 2 . σ.

(26) STRICHARTZ ESTIMATES FOR WATER WAVES. 879. In the integral I2 using (4.46) and the Taylor formula we see that, (4.49). ∂ξ θ(ξ) ≥ c1 σ. � h � 12 1 = C1 (hσ) 2 , σ. h 1 ∀ξ ∈ [ρ + ( ) 2 , β]. σ. Let us estimate the integral I2 . We can state the following lemma which is a refined version of the well known Van der Corput Lemma. L�� 4.9. – For all k ∈ N there exists C > 0 such that � � � � h �k β � h � 12 h 1 � � k i θ(ξ) (4.50) e ∂ q(ξ)dξ �I2 − (−1)k �≤C ξ 1 k h 2 i (∂ξ θ(ξ)) σ ρ+( σ ) where q(ξ) = b(ξ)ζ(x − z − a� (ξ).. Proof. – Let us denote by Jk the integral term in (4.50). The lemma is true for k = 0. i i Assume it is true up to the order k. Using the fact that i∂ξhθ(ξ) ∂ξ e h θ(ξ) = e h θ(ξ) and integrating by parts in Jk we obtain, � � � � h �k+1 β h 1 i θ(ξ) ∂ Jk = (−1)k+1 e ∂ξk q(ξ)dξ ξ 1 k+1 h 2 i (∂ θ(ξ)) ξ ρ+( σ ) � β � � h h 1 k+1 ∂ k+1 q(ξ)dξ + (−1)k+1 e i θ(ξ) 1 h 2 i (∂ξ θ(ξ))k+1 ξ ρ+( σ ) � � h �k+1 h θ(ξ) � 1 + (−1)k+1 [e i ∂ξk q(ξ)]β h 1 = Jk1 + Jk2 + Jk3 . k+1 ρ+( σ ) 2 i (∂ξ θ(ξ)) First of all we have Jk2 = Jk+1 . Now using (4.36) and (4.49) we can write, |Jk3 | ≤ Chk+1. h−k(δ+4) (hσ). k+1 2. ≤C. � h � 12 k( 1 −δ−4) � h � 12 h 2 ≤C , σ σ. since σ ≥ 1. Now using again (4.36) we obtain, � β |Jk1 | ≤ Chk+1−k(δ+4). 1. h 2 ρ+( σ ). � � �� 1 �∂ξ � dξ. � (∂ξ θ(ξ))k+1 �. Since by (4.46) the function ∂ξ θ is increasing we have � � �� 1 1 �∂ξ � = −∂ξ . � (∂ξ θ(ξ))k+1 � (∂ξ θ(ξ))k+1 Therefore we can write,. � |Jk1 | ≤ Chk+1−k(δ+4) [−. We deduce exactly as for Jk3 that,. It follows that |Jk − Jk2 | ≤ C. � h � 12 σ. |Jk1 | ≤ C. � 1 ]β h 1 . k+1 ρ+( σ ) 2 (∂ξ θ(ξ)). � h � 12 . σ. , which proves our induction.. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(27) 880. T. ALAZARD, N. BURQ AND C. ZUILY. Now using Lemma 4.9, (4.36) and (4.49) we can write, � h � 12 k( 1 −δ)− 1 � h � 12 2kµ0 − 1 1 −kδ 2 ≤ C 2, |Jk | ≤ Chk ≤C h 2 h k h σ σ (hσ) 2 � �1 so taking k such that 2kµ0 ≥ 12 and using (4.7) we deduce that |Jk | ≤ C σh 2 . It follows from � �1 � �1 Lemma 4.9 that |I2 | ≤ C σh 2 and from (4.48), (4.47) that |K+ | ≤ Ch σh 2 which completes the proof of Proposition 4.8. 4.3. End of the proof of Theorem 4.2 1. Let us set J = [0, h 2 ]. It follows from (4.3) and Proposition 4.5 that (4.51). ‹h + 1 (Whδ ∂x + ∂x Whδ )U ‹h + ia(Dx )U ‹h = Fh , ∂t U 2. ‹h |t=0 = U ‹h (0, x), U. with sup �Fh (s, ·)�H 1 (R) ≤ CN hN �u0,h �L1 (R) .. (4.52). s∈J. We claim that, (4.53). ‹h (0, ·) = u0,h + v0,h , U. �v0,h �H 1 (R) ≤ CN hN �u0,h �L1 (R) .. Indeed using (4.3), (4.4), (4.14), (4.33) and (4.34) we see that, �� i (4.54) v0,h (x) = (2πh)−1 e h (x−z)ξ (ζ(x − z) − 1)χ1 (ξ)u0,h (z)dzdξ.. Since on the support of 1 − ζ(x − z) we have |x − z| ≥ 1 we can integrate by parts as much as we want to obtain that for all N ≥ 1, �� ï ò i 1 − ζ(x − z) v0,h (x) = cN hN −1 e h (x−z)ξ (∂ξN χ1 )(ξ)u0,h (z)dzdξ. (x − z)N Using the Hölder inequality we deduce that, � � �� 1 − ζ(x − z) ��2 � 2 N −1 � � |u0,h (z)|dz �u0,h �L1 (R) |v0,h (x)| ≤ CN h � (x − z)N � from which we deduce that,. �v0,h �L2 (R) ≤ CN hN −1 �u0,h �L1 (R) .. Differentiating (4.54) with respect to x and using the same trick we obtain the estimate in (4.53). Now by (4.51), the Duhamel formula and the definition in Theorem 4.2 we can write, ‹h (t, x), where D1 = U � t D2 = −S(t, 0, h)v0,h (x), D3 = − S(t, s, h)[Fh (s, x)]ds. S(t, 0, h)u0,h = D1 + D2 + D3 (4.55). 0. First of all the estimate (4.56). �D1 (t)�L∞ (R) ≤. follows from Proposition 4.8 and (4.41). 4 e SÉRIE – TOME 44 – 2011 – No 5. C �u0,h �L1 (R) h1/4 |t|1/2.

(28) 881. STRICHARTZ ESTIMATES FOR WATER WAVES. Let us estimate D2 . We have by Sobolev inequality, �D2 (t)�L∞ (R) ≤ C1 �D2 (t)�H 1 (R) ≤ C2 �v0,h �H 1 (R) , therefore by (4.53), �D2 (t)�L∞ (R) ≤ CN hN �u0,h �L1 (R) .. (4.57). Let us look now to the term D3 . We have, � � �D3 (t)�L∞ (R) ≤ C �S(t, s, h)Fh (s, ·)�H 1 (R) ds ≤ C � �Fh (s, ·)�H 1 (R) ds, J. J. from which we deduce,. �D3 (t)�L∞ (R) ≤ CN hN �u0,h �L1 (R) .. (4.58). Then Theorem 4.2 follows from (4.55), (4.56), (4.57), and (4.58). 4.4. The T T ∗ argument Having proved the dispersion estimate, the Strichartz estimates for the solution of (4.1) follow very classically. Pʀ��ɪ�ɪ�ɴ 4.10. – There exist > 0, C > 0 such that for any 0 < h < 1 and any initial data u0,h = χ(hDx )u0 , we have (4.59). �S(t, 0, h)u0,h �. 1. L4 ((0,h 2 ),L∞ (R)). ≤ C�u0,h �. 1. H 8 (R). .. Proof. – Indeed, applying the usual T T ∗ argument, it suffices to prove that the operator �. 1. h2. S(t, 0, h)S(s, 0, h)∗ f (s)ds. 0. 4. 1. 1. maps continuously L 3 ((0, h 2 ), L1 (R)) to L4 ((0, h 2 ), L∞ (R)). But a direct calculation shows that since 12 (Whδ ∂x + ∂x Whδ ) is self adjoint, one has S(s, 0, h)∗ = S(0, s, h), and consequently, Proposition 4.10 follows from the classical Hardy-Littlewood-Sobolev inequality and the dispersion estimate (4.2). C�ʀ�ʟʟ�ʀʏ 4.11. – Let u be a solution of the problem 1 ∂t u + (Whδ ∂x + ∂x Whδ )u + ia(Dx )u = f, 2. u|t=0 = 0. with supp fˆ ⊂ { 12 h−1 ≤ |ξ| ≤ 2h−1 }. Then we have, �u�. 1. L4 ((0,h 2 ),L∞ (R)). ≤ Kh−1/8 �f �. 1. L1 ((0,h 2 ),L2 (R)). ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE. ..

(29) 882. T. ALAZARD, N. BURQ AND C. ZUILY. Proof. – Indeed we have, u(t, ·) = 1. �. t. 0. S(t, 0, h)S ∗ (s, 0, h)f (s, ·) ds.. Let us set J = [0, h 2 ]. It follows from Proposition 4.10 that, �u�L4 (J,L∞ (R)) ≤ C ≤C. �. 1. h2. �. �. �S(s, 0, h)∗ f (s, ·)�. 1. H 8 (R). 0. ds. 1. h2. �f (s, ·)�. 0. 1. H 8 (R). ds ≤ C �� h−1/8 �f �L1 (J,L2 (R)) ,. since fˆ is supported in { 12 h−1 ≤ |ξ| ≤ 2h−1 }. 4.5. Gluing the estimates It remains to glue the estimates which up to now have been proved on small time intervals 1 of size h 2 . Recall that from Lemma 3.6 we have 3 1 1 ∂t uh + (Whδ ∂x + ∂x Whδ )uh + i |Dx | 2 uh = fh ∈ L∞ ((0, T ); H s+ε− 2 ). 2 1. Let ϕ ∈ C0∞ (0, 2), equal to 1 on ( 12 , 32 ). For −1 ≤ k ≤ T h− 2 , define uh,k = ϕ which satisfies. � t − kh 12 � 1. h2. uh ,. 3 1 (4.60) ∂t uh,k + (Whδ ∂x u + ∂x Whδ )uh,k + i |Dx | 2 uh,k 2 � t − kh 12 � � t − kh 12 � 1 =ϕ fh + h− 2 ϕ� uh , 1 1 h2 h2 As a consequence, using Corollary 4.11 we obtain,. (4.61). �uh,k �. 1. uh,k |. 1. t=kh 2. = 0.. 1. L4 ((kh 2 ,(k+2)h 2 ), L∞ (R)). 1 � � � t − kh 12 � � � 2 1� � − 12 � t − kh ≤ h− 8 �ϕ f + h ϕ u � 1 h h 1 1 1 1 L ((kh 2 ,(k+2)h 2 ), L2 (R)) 2 2 h h � � 1 1 1 ≤ Ch 2 − 8 �fh �L∞ (0,T ), L2 (R)) + h− 2 �uh �L∞ ((0,T ), L2 (R)) 1. ≤ Chs− 8 (hε + �uh �L∞ ((0,T ),H s (R)) ). 1. where in the last inequality we used that by (3.43) we have fh ∈ L∞ ((0, T ), H s+ε− 2 (R)). Eventually, noticing that, 1. �uh �4L4 ((0,T ), L∞ (R)) we obtain (4.62). ≤. T� h− 2. k=−1. 1. �uh,k �4 4. 1. 1. L ((kh 2 ,(k+2)h 2 ), L∞ (R)). 1. ,. �uh �4L4 ((0,T ), L∞ (R)) ≤ Ch− 2 h4(s− 8 ) (hε + �uh �L∞ ((0,T ),H s (R)) )4 .. 4 e SÉRIE – TOME 44 – 2011 – No 5.

(30) 883. STRICHARTZ ESTIMATES FOR WATER WAVES. For h = 2−j , let us set cj = 2−jε + �uh �L∞ ((0,T ),H s (R)) .. Notice that according to (3.45), we have � (4.63) c2j < +∞. j∈N. We deduce �u�. s− 1 L4 ((0,T ), B∞,24. (R)). � � 2j(s− 1 ) � 12 4 �∆ u�2 ∞ =� 2 �L4 (0,T ) j L (R) j∈N. =� (4.64). �. j∈N. ��. ≤. j∈N. ��. =. j∈N. 1. 1. 22j(s− 4 ) �∆j u�2L∞ (R) �L2 2 (0,T ) 1. �22j(s− 4 ) �∆j u�2L∞ (R) �L2 (0,T ) 1. 22j(s− 4 ) �∆j u�2L4 ((0,T ),L∞ (R)). � 12. � 12. which, by (4.62) and (4.63) completes the proof of Theorem 4.1.. 5. Classical time parametrix In this section we take s > 11 2 and we prove the usual Strichartz estimates. The main step is, as before, the dispersion estimate. To do so, we seek a parametrix. The main difference with respect to the previous section is that (in the semi-classical framework), we are looking for a large ( O(h−1/2 )) time parametrix. As a consequence, the lower order term TW ∂x induces oscillations. This is reflected in the fact that the new eikonal equation will be quasi-linear. We begin by an analogue of Theorem 4.1. Tʜ��ʀ�� 5.1. – Under the assumptions of Theorem 1.2, let u be defined by (3.32). Then there exists C = C(�(η, ψ)� ∞ ) such that s+ 1 s 2 L. (I,H. (R)×H (R)). �u�. s− 1. L4 (I,B∞,28 (R)). ≤ C.. Tʜ��ʀ�� 5.2. – Let χ ∈ C0∞ (R) with supp χ ⊂ {ξ : 12 ≤ |ξ| ≤ 2} and t0 ∈ R. For any initial data u0,h = χ(hDx )u0 where u0 ∈ L1 (R) let us denote by S(t, t0 , h)u0,h := Uh the solution of (5.1). 1 ∂t Uh + (Whδ ∂x + ∂x Whδ )Uh + ia(Dx )Uh = 0, 2. Uh |t=t0 = u0,h .. Then there exists τ0 > 0 such that for any 0 < h ≤ 1 and any |t − t0 | ≤ τ0 , (5.2). �S(t, t0 , h)u0,h �L∞ (R) ≤. h1/4 |t. C �u0,h �L1 (R) . − t0 |1/2. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(31) 884. T. ALAZARD, N. BURQ AND C. ZUILY. In the remaining of this section, we shall prove Theorem 5.2. We need first to refine the constructions in Section 4 to handle large times. An important point in the construction of the phase function is that handling large times leads us to non linear geometric optics. Our parametrix will be of the form (4.3), (4.4) that is, �� i 1 ‹ ‹ x, z, ξ, h)u0,h (z)dzdξ, (5.3) Uh (t, x) = e h (Φ(t,x,ξ,h)−zξ) B(t, 2πh where Φ will satisfy the eikonal equation and. ‹ x, z, ξ, h) = B(t, x, ξ, h)ζ(x − z − th− 12 a� (ξ)), B(t,. (5.4). where B will satisfy the transport equations and ζ ∈ C0∞ (R), ζ(s) = 1 if |s| ≤ 1, ζ(s) = 0 if |s| ≥ 2. 5.1. Notations In this section we fix s>. 11 1 and δ = 2 s−. 3 2. <. 1 . 4. As before we shall set 2−j = h, where j ∈ N and we shall work with the semiclassical 1 time σ = th− 2 . In addition to the function χ introduced in Theorem 5.2, we shall use two more cut-off functions χj ∈ C0∞ (R), j = 1, 2, such that,  1   supp χ1 ⊂ {ξ : ≤ |ξ| ≤ 3}, χ1 = 1 on the support of χ, 3 (5.5)   supp χ0 ⊂ {ξ : 1 ≤ |ξ| ≤ 4}, χ0 = 1 on the support of χ1 . 4 Recall that we have (i) (ii). W ∈ L∞ ([0, T ], W 2,∞ (R)), ∂t W ∈ L∞ ([0, T ], W 1,∞ (R)) (Lemma 3.4), � � � � Whδ = S[δ(j−3)] (W ) satisfies �∂xα Whδ � ∞ ≤ Cα �∂xα W �L∞ , t,x Lt,x. (iii). 3 2. a(ξ) = χ0 (ξ) |ξ| ,. D��ɪɴɪ�ɪ�ɴ 5.3. – For small h0 , τ0 to be fixed, we introduce the sets � � Ω = (t, x, ξ, h) ∈ R4 : h ∈ (0, h0 ), |t| < τ0 , 1 < |ξ| < 3 ¶ © 1 O = (σ, x, ξ, h) ∈ R4 : h ∈ (0, h0 ), |σ| < τ0 h− 2 , 1 < |ξ| < 3 . If m ∈ R and � ∈ R+ , we denote by S�m (Ω) (resp. S�m ( O)) the set of all functions f on Ω (resp. O) which are C ∞ with respect to (t, x, ξ) (resp. (σ, x, ξ)) and satisfy the estimate (5.6). |∂xα f (t, x, ξ, h)| (resp. |∂xα f (σ, x, ξ, h)|) ≤ Cα hm−�α ,. for all (t, x, ξ, h) ∈ Ω (resp. (σ, x, ξ, h) ∈ O). 4 e SÉRIE – TOME 44 – 2011 – No 5.