A PARADIFFERENTIAL REDUCTION FOR THE GRAVITY-CAPILLARY WAVES SYSTEM AT LOW REGULARITY AND APPLICATIONS. Thibault de Poyferré & Quang-Huy Nguyen

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1 A PARADIFFERENTIAL REDUCTION FOR THE GRAVITY-CAPILLARY WAVES SYSTEM AT LOW REGULARITY AND APPLICATIONS Thibault de Poyferré & Quang-Huy Nguyen Abstract. We consider in this article the system of gravity-capillary waves in any dimension under the Zakharov/Craig-Sulem formulation. Using a paradifferential approach introduced by Alazard-Burq-Zuily we symmetrize this system into a quasilinear equation whose principal term is of order 3/. The main novelty compare to earlier studies is that this reduction is performed at the Sobolev regularity of quasilinear pdes: H s R d with s > 3/+d/ d is the horizontal dimension. From this reduction, we deduce a blow-up criterion and then an a priori estimate for the solution and the Lipschitz continuity of the flow map in terms of the Sobolev norm and the Strichartz norm. Contents 1. Introduction... Acknowledgment Elliptic estimates and the Dirichlet-Neumann operator Paralinearization and symmetrization of the system Blow-up criterion and a priori estimate Contraction estimates Appendix: Paradifferential Calculus and technical results.. 49 References Key words and phrases. gravity-capillary waves, paradifferential approach, blowup criterion, a priori estimate, contraction estimate.

2 THIBAULT DE POYFERRÉ & QUANG-HUY NGUYEN 1. Introduction We consider the system of gravity-capillary waves describing the motion of a fluid interface under the effect of gravity and surface tension. From the wellposedness result in Sobolev spaces Yosihara [54] see also Wu [50, 51] for pure gravity waves it is known that the system is quasilinear in nature. In a more recent work [1] Alazard-Burq-Zuily showed explicitly this quasilinearity by using a paradifferential approach see Appendix 6 to symmetrize the system into the following paradifferential equation 1.1 t +T Vt,x +it γt,x,ξ ut,x = ft,x where V is the horizontal component of the trace of the velocity field on the free surface, γ is a paradifferential symbol of order 3/, depending on the solution. This reduction has many consequences, among them are the local well-posedness and smoothing effect in [1], Strichartz estimates in [] for u L t Hs x Rd with s > +d/. As remarked in [1] s > +d/ is the minimal Sobolev index in term of Sobolev s embedding to ensure that the velocity filed is Lipschitz up to the boundary, without taking into account the dispersive property. From the works of Alazard-Burq-Zuily [3, 5], Hunter-Ifrim-Tataru [30] for pure gravity waves it seems natural to assume that the gradient of the velocity is Lipschitz so that the particles flow is well-defined. On the other hand, from the standard theory of quasilinear pdes, it is natural to ask if the reduction 1.1 holds at the Sobolev threshold s > 3/ + d/ and then, if a local-wellposedness theory holds at the same level of regularity? The two observations above motivate us to study the gravity-capillary system at the following regularity level: 1. u X := L t Hs x Lp t Wr, x with s > 3 + d, r >, which exhibits a gap of 1/ derivatives that may be filled up by Strichartz estimates. One of our main results will be a blow-up criterion at this scaling with p = 1 i.e. merely integrable in time, which states that the solution can be prolonged as long as the X-norm of u remained bounded at least in the case of infinite depth. To derive our criterion, the main difficulty compare to the reduction in [1] is that we have to keep all the estimates in the analysis to be tame, i.e., linear with respect to the highest norm-the Hölder norm W r,. First of all, let us recall the Zakharov/Craig-Sulem formulation of water waves.

3 A PARADIFFERENTIAL REDUCTION FOR THE GRAVITY-CAPILLARY WAVES The Zakharov/Craig-Sulem formulation. 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 = ηt,x} and above a fixed bottom Γ = Ω t \ Σ t. We make the following separation assumption H t on the domain at time t: Ω t is the intersection of the half space Ω 1,t = {x,y R d R : y = ηt,x} and an open connected set O containing a fixed strip around Σ t, i.e., there exists h > 0 such that 1.3 {x,y R d R : ηx h y ηt,x} O. Assume that the velocity field v admits a potential φ : Ω R, i.e, v = φ. Using the idea of Zakharov, we introduce the trace of φ on the free surface ψt,x = φt,x,ηt,x. Then φt,x,y is the unique variational solution of 1.4 φ = 0 in Ω t, φt,x,ηt,x = ψt,x. The Dirichlet-Neumann operator is then defined by Gηψ = φ 1+ x η n Σ = y φt,x,ηt,x x ηt,x x φt,x,ηt,x. The gravity-capillary water waves problem with surface tension consists in solving the following so-called Zakharov-Craig-Sulem system on η,ψ: t η = Gηψ, 1.5 t ψ +gη Hη+ 1 xψ 1 x η x ψ +Gηψ 1+ x η = 0. Here, Hη is the mean curvature of the free surface: η Hη = div. 1+ η It is important to introduce the vertical and horizontal components of the velocity on Σ, which can be expressed in terms of η and ψ: 1.6 B = v y Σ = xη x ψ +Gηψ 1+ x η, V = v x Σ = x ψ B x η.

4 4 THIBAULT DE POYFERRÉ & QUANG-HUY NGUYEN 1.. Main results. The Cauchy problem has been extensively studied, for example in Nalimov [40], Yosihara [54], Coutand- Shkoller [1], Craig [], Shatah-Zeng [41, 4, 43], Ming-Zhang [39], Lannes [36]: for sufficiently smooth solutions and Alazard-Burq-Zuily [1] for solutions at the energy threshold. See also Craig [], Wu [50, 51], Lannes [35] for the studies on gravity waves. Observe that the linearized system of 1.5 about the rest state η = 0,ψ = 0 when g = 0 reads { t η D x ψ = 0, t ψ η = 0 which becomes 1.7 t Φ+i D x 3 Φ = 0, with Φ = Dx 1 η +iψ. Therefore, it is natural to study 1.5 at the following algebraic scaling η,ψ H s+1 R d H s R d. From the formula 1.6 for the trace of velocity on the surface, we have that the Lipschitz threshold in [1] corresponds to s > + d/. On the other hand, the threshold s > 3/ +d/ suggested by the quasilinear nature 1.1 is also the minimal Sobolev index to ensure that the mean curvature Hη is bounded. The question we are concerned with is: Q If the Cauchy problem for 1.5 is solvable for initial data 1.8 η 0,ψ 0 H s+1 H s, s > 3 + d. Assume now that 1.9 η,ψ L [0,T];H s+1 H s L p [0,T];W r+1, W r, with 1.10 s > 3 + d, < r < s + 1 d, p 1. is a solution whose data is 1.8. We shall prove in Proposition 4.1 that the quasilinear reduction 1.1 of system 1.5 still holds with the right-hand-side term ft,x satisfying a tame estimate. To be concise in the following statements let us define the quantities that control the system: Sobolev norms : M σ,t = η,ψ L [0,T];H σ+1 H σ, M σ,0 = η 0,ψ 0 H σ+ 1 H σ, Blow-up norm : N σ,t = η,ψ L 1 [0,T];W σ+1, W σ,, Strichartz norm : Z σ,t = η,ψ L p [0,T];W σ+1, W σ,. Our first result concerns an a priori estimate for the Sobolev norm M s,t in terms of itself and the Strichartz norm Z r,t.

5 A PARADIFFERENTIAL REDUCTION FOR THE GRAVITY-CAPILLARY WAVES 5 Theorem 1.1. Let d 1, h > 0, p > 1. Then there exists a non-negative, non-decreasing function F such that: for all T 0,1] and all η,ψ solution to 1.5 on [0,T] with regularity 1.10 and initial data 1.8 and satisfies inf t [0,T] distηt,γ > h, there holds M s,t F M s,0 +TF M s,t +Z r,t. As a consequence, when s > +d/ one retrieves by Sobolev embeddings the a priori estimate in [1]. Our second result provides a blow-up criterion for solutions of 1.5. Theorem 1.. Let d 1, h > 0 and indices 3 + d < s 0 < s 1, < r < 1 d. Let T = T η 0,ψ 0,h be the maximal time of existence defined by 4.17 and η,ψ L [0,T ;H s+1 H s be the maximal solution of 1.5 with prescribed data η 0,ψ 0 satisfying distη 0,Γ > h. Then if T is finite, we have lim sup M s0,t +N r,t + 1 = +, T T ht where ht is the distance from the surface η to the bottom Γ over the time interval [0,T]. Remark that the Sobolev regularity in the above criterion is exactly the one given in question Q. In contrast, for pure gravity waves in which the surface tension is neglected, it was shown in [5] and [30] that boundedness of the curvature is irrelevant: that T 0 ηt W1/ dt < + is enough. In the survey paper [4] Craig-Wayne posed see Problem 3 the following questions on How do solutions break down?: Q1 For which α is it true that, if one knows a priori that sup [ T,T] η,ψ C α < + then C data η 0,ψ 0 implies that the solution is C over the time interval [ T,T]? Q It would be more satisfying to say that the solution fails to exist because the "curvature of the surface has diverged at some point", or a related geometrical and/or physical statement. Theorem 1. gives a partial answer to Q: our criteria on M s0,t and N r,t are directly imposed on the regularity of the solution yet they correspond to the following geometrical and physical statements: η = + is the minimal Sobolev-based criterion to say L [0,T ;H 1 that the curvature explodes,

6 6 THIBAULT DE POYFERRÉ & QUANG-HUY NGUYEN N r,t = + corresponds to the condition that the slightly above Lipschitz norm of the trace of velocity on the surface fail to be intergable in time, ht = 0 means that the bottom rises to the surface. Concerning question Q1 we have by virtue of Theorem 1. the following result on persistence of Sobolev regularity. Corollary 1.3. Let T 0, + and η, ψ be a distributional solution to system 1.5 on the time interval [0,T] such that inf [0,T] distηt,γ > 0. Then the following property holds: if one knows a priori that T 1.11 sup ηt,ψt [0,T] H + d + H 3 +d++ ηt,ψt 5 0 C dt < + + C+ then η0,ψ0 H R d implies that η,ψ L [0,T];H R d. As a consequence, by Sobolev s embedding one can replace 1.11 by a stronger condition involving only Sobolev regularity sup ηt,ψt 5 [0,T] H + d < + + H +d + and thus obtain an answer for the Sobolev version of Q1. Finally, we observe that the relation 1.10 exhibits a gap of 1/ derivative from H s to W r, in term of Sobolev embedding. To fill up this gap we need to take into account the dispersive property of water waves to prove a Strichartz estimate with a gain of 1/ derivative. As remarked in [6] this gain can be achieved for the 3D linearized system i.e. d = and corresponds to the semiclassical Strichartz estimate. By virtue of Theorems 1.1, 1. and Theorem 5.9 on the Lipschitz continuity of the solution map one would end up with an affirmative answer for Q. Therefore, the problem boils down to studying Strichartz estimates for 1.5. As a first effort in this direction, we prove in the companion paper [6] Strichartz estimates with an intermediate gain 0 < µ < 1/ which will yield a Cauchy theory see also [6] in which the initial velocity may fail to be Lipschitz up to the boundary but becomes Lipschitz at almost all later time; this is an analogue of the result in [5] for pure gravity waves. The article is organized as follows. Section is devoted for the elliptic estimates needed to study the Dirichlet-Neumann operator: bound estimates and paralinearizations. Next, in Section 3 we adapt the method in [1] to paralinrarize and then symmetrize system 1.5 at our level of regularity Having this reduction, we use the energy method to derive a blow-up criterion and then an a priori estimate in Section 4. Section 5 is devoted for contraction estimates, more precisely we establish the Lipschitz continuity of the flow map

7 A PARADIFFERENTIAL REDUCTION FOR THE GRAVITY-CAPILLARY WAVES 7 in spaces of 1-derivative less. Finally, we gather some basic features of the paradifferential calculus theory and technical results in Appendix 6, most of which comes from [3, 5]. Acknowledgment The authors would like to sincerely thank T.Alazard, N.Burq and C.Zuily for many fruitful discussions, suggestions when this work was preparing, as well as their helpful comments at the final stage of the work. Quang Huy Nguyen was partially supported by the labex LMH through the grant no ANR-11-LABX LMH in the "Programme des Investissements d Avenir".. Elliptic estimates and the Dirichlet-Neumann operator Notation.1. Throughout this article, for spatial regularity we shall denote for simplicity the Zygmund spaces C σ R d σ R by C σ ; while for temporal variable, C k k N are the usual spaces of functions having continuous derivatives up to order k..1. The elliptic problem. Let η W 1, R d and f H 1 R d. It was proved in [3] that there exists a unique variational solution φ to the boundary value problem.1 x,y φ = 0 in Ω, φ Σ = f, n φ Γ = 0. Define. and.3 { } Ω 1 := x,y : x R d,ηx h < y < ηx, Ω :={x,y O : y ηx h}, Ω :=Ω 1 Ω, { } Ω 1 := x,z : x R d,z I, I = 1,0, } Ω := {x,z R d, 1] : x,z +1+ηx h Ω, Ω := Ω 1 Ω.

8 8 THIBAULT DE POYFERRÉ & QUANG-HUY NGUYEN To study the regularity of φ, we follow [35], [3] straighten out the fluid domain using the map x,z ρx,z from Ω to Ω, defined as.4 { } ρx,z :=1+ze δz Dx ηx z e 1+zδ Dx ηx h if x,z Ω 1, ρx,z :=z +1+ηx h if x,z Ω, with δ > 0. It has been proven in [3] that if η W 1,, for δ = δ η W 1, R d small enough, the map x,z x,ρx,z is a Lipschitz diffeomorphism from Ω 1 to Ω 1. Introduce for µ R and J R the interpolation spaces.5 X µ I = C 0 zi;h µ R d L zi;h µ+1 R d, Y µ I = L 1 z I;Hµ R d +L z I;Hµ 1 R d. Remark that Y µ X µ 1 for any µ R. In these spaces, we have from [3] and some easy computations Lemma.. If s > 1 + d, there exists a positive function F such that for every η H s+1 R d there holds z ρ h X s η 1 F I W 1, R d s+ 1, R d z ρ η F X.6 s 3 I W 1, R d s+ 1, R d 3 z ρ η F X s 5 I W 1, R d s+ 1, R d x ρ X s η 1 F I W 1, R d s+ R 1. d Then if we take.7 vx,z = φx,ρx,z, x,z Ω, the pullback of φ by this diffeomorphism, it solves.8 z +α x +β x z γ z v = 0, where.9 α := zρ 1+ x ρ, β := zρ x ρ 1 1+ x ρ, γ := z ρ z ρ+α xρ+β x z ρ. We have the following control on those coefficients :

9 A PARADIFFERENTIAL REDUCTION FOR THE GRAVITY-CAPILLARY WAVES 9 Lemma.3. Assume s s 0 > 3/+d/. Then for η H s+1, there holds with X µ = X µ I:.10 α h X s 1 + β X s 1 + γ X s 3 F 1.11 s+ 1, α C 0 I;C r 1 + β C 0 I;C r 1 + γ C 0 I;C r 3 F 1.1 z α X s 3 + zβ X s 3 + zγ X s 5 F 1 s+ 1. η C r+ 1, Those are all consequences of the product rules and nonlinear estimates of Proposition 6.8. Now a consequence of Proposition 3.16 and the estimate 3.5 of [3] is that our solution v satisfies Proposition.4. Let d 1, s 0 > 1/+d/, 1/ σ s 0 1/ and η H s0+1/. If f H σ+1, then for any z 0 1,0, x,z v X σ [z 0,0], and.13 x,z v X σ [z 0,0] F 1 f H σ+1, for some non-decreasing positive function F depending only on s 0 and σ. It was deduced from the preceding Proposition the following Sobolev estimate for the Dirichlet-Neumann operator see Theorem 3.1, [3] Theorem.5. Let d 1, s 0 > 1 + d and 1 σ s Then there exists a non-decreasing function F: R + R + such that, for all η H s 0+ R 1 d and all f H σ R d, we have.14 Gηf H σ 1 R d F 1 R d f H σ R d. Since we authorize the control on our quantities to depend non-linearly on the H s 0 norms and only want linearity in the higher order H s norm, this means we can use Proposition.4 as a base case for a bootstrap to control the H s and C r norms. We want to prove the following proposition : Proposition.6. Let s s 0 > 3 + d,

10 10 THIBAULT DE POYFERRÉ & QUANG-HUY NGUYEN f H s and η H s+1. Then for any z 0 1,0, x,z v X s 1 [z 0,0] and [ ] x,z v X s 1 [z 0,0] η F H 1, f H s 0 f H s 0 + f H s + s+ 1. for some non-decreasing positive function F depending only on s 0 and s. The proof will be a simple bootstrap procedure ons. CallingH s the proposition for s, Proposition.4 applied with σ = s 0 1 tells us that H s0 is true. We will show that if H s is true, then so is H s+ε with 0 < ε 1, ε < s 0 3 d. First we paralinearize equation.8 of v : Lemma.7. There is a function F such that for all I [ 1,0], v satisfies zv +T α x v +T β x z v = F := γ z v +T α α x v +T β β z v, [ ] F Y s 1+ε I η F H 1, f H s 0 f H s 0 + x,z v X s 1 I + η. H s+1 Proof. The above expression of F follows directly from equation.8 satisfied by v. Now, using 6.0 and Hölder inequality in z we have γ z v L I;H s 3 +ε γ L I;H s 0 1 zv L I;H s 1 + zv L I;H s 0 1 γ L I;H s 1, so that using.10 to control γ and Proposition.4 to control z v L I;H s 0 1 gives [ ] γ z v η F L I;H s 3 +ε H 1, f H s 0 x,z v X s 1 I + η. H s+1 Next by 6.15 we have T α α x v L 1 I;H s 1+ε xv L I;H s 0 3 [ 1+ α h ] L, I;H s so that again we can conclude using.10 and Proposition.4. The last remainder term can be controlled identically. We then decouple the equation into a forward and a backward parabolic equation : Lemma.8. There exist two symbols a 1,A 1 Γ 1 ε[ 1,0] satisfying M 1 1+εa 1 +M 1 1+εA 1 F 1, R a 1 +RA 1 c ξ for some constant c = c 1 > 0, such that z +T α x +T β x z = z T a 1 z T A 1+R,

11 A PARADIFFERENTIAL REDUCTION FOR THE GRAVITY-CAPILLARY WAVES 11 where R is of order 1 see Definition 6.3 having its norm bounded by F 1. In particular,.15 Rv Y s 1+ε I F 1 x,z v X s 1 I. Proof. Take.16 a 1 = 1 iβ ξ A 1 = 1 4α ξ β ξ, iβ ξ + 4α ξ β ξ so that a 1 +A 1 = iβ ξ, a 1 A 1 = α ξ. Then the control of the semi-norm of a 1 and A 1 is a consequence of the boundedness of the coefficients α, β from.10. From the expressions of α and β, and the fact that z ρ c 0 > 0, we get c > 0, 4α ξ β ξ c ξ, which gives the ellipticity. At last, R = T a 1T A 1 T α x T za 1. The first difference is of order 1+ε = 1 ε by Theorem 6.4 ii, and the second term z A 1 Γ 1 ε by.1. Consequently, the remainder R has order 1 and.15 follows. Here, we can replace v H s+ 1 by v since the paradifferential operator T p can be repalced by T p 1 ΨD x, for a low frequency H s 1 cutoff Ψ, at no cost. To conclude the proof of Proposition.6, we want to apply Theorem 6.10 two times. Take 0 > z 1 > z 0 > 1. Since H s is true, there holds [ ] x,z v X s 1 [z 0,0] η F H 1, f H s 0 f H s 0 + f H s + s+ 1. We will prove x,z v X s 1+ε [z 1,0] F 1, f H s 0 Since z 0 and z 1 are arbitrary, this will complete the proof. [ f H s0 + f H s+ε + s+1+ε ]. We now introduce a cutoff χ satisfying χ z<z0 = 0, χ z>z1 = 1, and set w = χz z T A 1v. From Lemma.8 we have z w T a 1w = F, with.17 F = χzf +Rv+χ z z T A 1v. We have the trivial estimate χ z z T A v Y s 1+ε [z 0,0] F 1 x,z v X s 1 [z 0,0].

12 1 THIBAULT DE POYFERRÉ & QUANG-HUY NGUYEN Together with the preceding lemmas, we obtain that F Y s 1+ε [z 0,0] F 1, f H s 0 [ f H s 0 + f H s + s+ 1 Since wz 0 = 0 and R a c ξ, Theorem 6.10 implies.18 [ ] w X s 1+ε [z 0,0] η F H 1, f H s 0 f H s 0 + f H s + s+ 1. Consequently, w Y s+ε [z 0,0] F 1, f H s 0 [ ] f H s 0 + f H s + s+ 1. Then because χ = 1 on [z 1,0], z v T A 1v = w for z [z 1,0]. At last, applying again Theorem 6.10 with v0 = f, after inversing z into z, we obtain ] v X s+ε [z 1,0] η F H 1, f H s 0 [ f H s0 + f H s+ε + η. H s+1+ε Using the relation z v = T A 1v+w and take into account the estimate.18 we can finally estimate x,z v as claimed. Next, we prove a Hölder estimate for x,z v. Proposition.9. Let s 0 > 3 + d, r < s 0 d + 1, 1 µ 5, and f H s 0 C r, η H s 0+ 1 C r+1. Then for any z 0 1,0, we have x,z v C 0 [z 0,0];C r µ η F H 1, f H s 0 f H s 0 µ+1 + f C r µ+1. for some non-decreasing positive function F depending only on s 0 and r. Proof. Similar to the proof above, we take 1 < z 0 < z 1 < 0, introduce a cutoff χ satisfying χ z<z0 = 0, χ z>z1 = 1 and set w = χz z T A 1v. We use the estimate 3.56 in [3]: for it holds that 0 ε 1, ε < s 0 1 d, 1 σ s 0 1 ε w X σ+ε [z 0,0] F 1 x,zv X σ [z 0,0]. Then, applying this inequality with ε = 1/, σ = s 0 µ gives w X s 0 µ+ 1 [z 0,0] F 1 x,zv X s 0 µ [z 0,0]. On the other hand, it follows from Proposition.4 that.19 x,z v X s 0 µ [z 0,0] F 1 f H s 0 µ+1 ; ].

13 A PARADIFFERENTIAL REDUCTION FOR THE GRAVITY-CAPILLARY WAVES 13 consequently,.0 w X s 0 µ+ 1 [z 0,0] F 1 f H s 0 µ+1. Now, on [z 1,0] z T A 1v = w so after inversing z to z one can apply Theorem 6.11with r 1 = r µ+1, r 0 < r 1, q = to get with J = [z 1,0] v CJ;C r µ+1 f F η H 1 C r µ+1 + w L J;C r µ+δ + v L J;C r 0. Using.0, Sobolev s embedding and the relation between r and s 0, one deduces w L J;C r µ+δ w L J;H s 0 µ+1 F 1 f H s 0 µ+1, where we have taken 0 < δ < 1/ d/ r. Finally, for the last term on the right-hand side, one chooses r 0 small enough so that the desired estimate can be deduced from.19 via Sobolev embeddings... Dirichlet-Neumann operator. We now apply the elliptic estimates in the previous paragraph to derive estimates for the Dirichlet-Neumann operator. Proposition.10. Let d 1, and s s 0 > 3 + d, 1 µ 5, µ+1 < r < s 0 d + 1. Then there exists a positive nondecreasing function F such that [.1 Gηf H s 1 F 1, f H s 0 f H f H s + s+ 1. Gηf C r µ F 1 [ f H s 0 µ+1 + f C r µ+1]. Proof. By definition the Dirichlet-Neumann operator is given by Gηf = 1+ xρ z v x ρ x v z=0. z ρ Thus the result is a consequence of Propositions.6,.9, of the estimations onρ of Lemma., and of the product and nonlinear estimates of Proposition 6.8. Here, we need to take some care for the second estimate. 1. If r µ 0, the rule 6.1 implies at z = 0 x ρ x v C r µ x ρ C r µ x v C r µ x ρ H s 0 µ+ 1 xv C r µ Then since s 0 µ+1/ s 0 1/ and x ρ H s 0 1 F η H 1 ],

14 14 THIBAULT DE POYFERRÉ & QUANG-HUY NGUYEN the right-hand side is bounded as claimed by virtue of Proposition.9.. If r µ < 0 one applies 6. with α := µ r < β := r 1 to get at z = 0 x ρ x v C r µ x ρ C r 1 x v C r µ x ρ H s 0 1 xv C r µ from which. follows. The first term in the expression of Gηf is treated in the same way by writing 1+ x ρ 1+ x ρ z v = 1 z v + 1 z ρ z ρ h h zv. Recall the expression of the trace of the velocity at the free surface B = η ψ +Gηψ 1+ η, V = ψ B η. As a consequence, we have the following estimates on V and B. Corollary.11. Let d 1, and s s 0 > 3 + d, < r < s 0 d + 1. Then there exists a positive nondecreasing function F such that B,V H s 0 1 H s 0 1 F 1, ψ H s 0 B,V H s 1 H s 1 F 1, ψ H s 0 B,V C r 1 C r 1 F 1, ψ H s 0, [ ] 1+ ψ H s + s+ 1, [ ] 1+ ψ C r + η C r+ 1. Proof. We only need to prove estimates for B, then those for V will follow immediately. This is done by decomposing B as.6 B = η 1+ η ψ η Gηψ =: K η ψ+l ηgηψ, with K and L smooth. The first estimate is a consequence of Theorem.5 and the fact that H s 0 1 is an algebra since s 0 > 3 + d. For the second and the third, we use estimates 6.4, 6.3,6.1, 6.1, and Proposition.10.

15 A PARADIFFERENTIAL REDUCTION FOR THE GRAVITY-CAPILLARY WAVES 15 We also prove, following [6], that the Dirichlet-Neumann operator can be paralinearized. We show that it is possible to obtain tame estimates on the remainder. Define.7 λ := λ 1 +λ 0 a symbol with an order one part.8 λ 1 := 1+ η ξ η ξ, and an order zero part.9 λ 0 := 1+ η [ λ 1 divα 1 η+i ξ λ 1 α 1], α 1 := Proposition.1. Let d 1, and s s 0 > 3 + d, r >. Then there exists a positive nondecreasing function F such that for η,ψ H s+1 H s C r+1 C r, there holds with.30 Gηψ = T λ ψ T B η T V η +fη,ψ, fη,ψ H s+ 1 F 1, ψ H s η λ1 +i η ξ. [ ][ ] 1+ ψ C r + η C r ψ H s + s+ 1. The rest of this section is devoted to the proof of this Proposition. Recall that in the preceding section we have straightened the domain using the diffeomorphism ρ to obtain from φ the potential velocity a new unknown v satisfying z +α x +β x z γ z v = 0. We then established in Proposition.6 that x,z v X s 1 [z 0,0] F 1, ψ H s 0 [ ] 1+ ψ H s + s+ 1. Now using the above equation on v, the estimates on its coefficients and their z-derivatives from Lemma.3 one gets z v X s [z 0,0] + 3 z v [ ] X s 3 [z 0,0] F 1, ψ H s 0 1+ ψ H s + s+ 1. On the other hand, by Proposition.9 we have x,z v C 0 [z 0,0];C r 1 F 1, ψ H s 0 [ ] 1+ ψ C r + η C r+ 1,

16 16 THIBAULT DE POYFERRÉ & QUANG-HUY NGUYEN and again with product rules, z v C 0 [z 0,0];C r + 3 z v [ ] C 0 [z 0,0];C r 3 F 1, ψ H s 0 1+ ψ C r + η C r+ 1. The result we want to prove is linked to the so-called good unknown of Alinhac cf [7, 8]: we introduce.31 b := zv z ρ, and u := v T bρ, so that b z=0 = B, u z=0 = ψ T B η. The interest of the good unknown is that we expect it to satisfy a better paradifferential equation than v itself. Indeed, we have the following lemma. Lemma.13. The good unknown u = v T b ρ satisfies the equation.3 z u+t α x u+t β x z u T γ z u = f, f Y s+ 1 [ 1,0] F 1, ψ H s 0 [ ][ ] 1+ ψ C r + η C r ψ H s + s+ 1. Proof. To simplify the proof, we will write f 1 f iff f 1 f Y s+ 1 [ 1,0] is bounded by the right-hand side of.3. In particular, f 1 f if f 1 f X s [ 1,0] 1 is bounded by the right-hand side of.3. Introduce E := z +α +β z γ z, P := z +T α +T β z T γ z. We have Ev = 0. By decomposing each term in Ev with the Bony decomposition, using the estimates 6.4, 6.1, the previous estimates on z v, zv, and the estimates on the coefficients.10 we obtain which gives since v = u+t b ρ that 0 = Ev Pv T zvγ, Pu+PT b ρ T zvγ 0. The proof then boils down to showing that.33 PT b ρ T zvγ 0. Now PT b ρ = zt b ρ+t α T b ρ+t β z T b ρ T γ z T b ρ. Using Leibniz rule for the z-derivatives and neglecting the terms 0 it holds that PT b ρ T b z ρ+t αt b ρ+t β T b z ρ. To suppress the terms in γ, we use T b Eρ = T b 0 = 0, which implies T b zρ+t b T α ρ+t b T β z ρ T b T zργ 0.

17 A PARADIFFERENTIAL REDUCTION FOR THE GRAVITY-CAPILLARY WAVES 17 Now again by the symbolic calculus, we get [T b,t α ] ρ+[t b,t β ] z ρ 0 and T zvγ = T b zργ T b T zργ, hence we obtain.33. The next step of the proof is again to decouple between a forward and a backward parabolic equations, using a refinement of Lemma.8. Lemma.14. For 0 ε < min r, 1, there exist two symbols a and A satisfying R a+ra c ξ for a constant c 1 > 0, such that z +T α x +T β x z T γ z = z T a z T A +R, where R is of order 1/ ε. In particular, for any z 0 1,0 we have [ ][ ] Ru Y s+ η 1 F [z 0,0] H 1, ψ H s 0 1+ η C r ψ H s + s+ 1. Proof. We look for symbols of the following form: a = a 1 +a 0 Γ 1 3 +ε + Γ 0 1 +ε, A = A1 +A 0 Γ 1 3 +ε + Γ 0 1 +ε. We already found a 1 = 1 A 1 = 1 which satisfy M 1 3 +ε A 1 z +M 1 3 A 1 z M 1 0 +M 1 0 iβ ξ 4α ξ β ξ, iβ ξ + A 1 z +ε A 1 z F 4α ξ β ξ, F 1 1. [ ] 1+ η C r+ 1, Then we take so that a 0 = A 0 = 1 A 1 a 1 i ξ a 1 x A 1 γa 1, 1 a 1 A 1 i ξ a 1 x A 1 γa 1 a 1 A i ξa 1 x A 1 +a 1 A 0 +a 0 A 1 = α ξ, a+a = iβ ξ +γ.

18 18 THIBAULT DE POYFERRÉ & QUANG-HUY NGUYEN We can easily verify that M 0 1 +ε A 0 z +M 1 1 A 0 z M 0 0 +M 0 0 The remainder will be A 0 z +ε A 0 z F F 1 1. [ ] 1+ η C r+ 1, R = T a T A T α +T a +T A +T β T γ z = T a T A T α. Using the symbolic calculus we obtain that R is of order 1 ε, hence by virtue of Proposition.6 we conclude [ Ru Y s+ 1 Ru L H s F F 1, f H s 0 1 [ 1+ η C r η C r+ 1 ] u L H s 1 ][ 1+ f H s + s+ 1 ]. Proof of Proposition.1. For the sake of conciseness we denote in this proof by Ξ the right-hand side of.30. Again we introduce w := χz z T A u with χ satisfying χ z<z0 = 0 and χ z>z1 = 1, for 1 < z 0 < z 1 < 0. Then z w T a w = χzru+χ z z T A u, with Ru Y s+ 1 [z 0,0] Ξ. We turn to estimate ω := χ z z T A u in Y s+1, it is non-zero only on z 0,z 1 and satisfies We have trivially z ω T a ω = χ zru+χ zω := f 0. z T A u Y s [z 0,0] F 1 x,z u X s 1 [ 1,0]. From the study of v and the expression u = v T b ρ, it holds that x,z u X s 1 [z 0,0] Ξ. Consequently, ω Y s Ξ and f 0 Y s Ξ. Applying Theorem 6.10 with the boundary condition ωz 1 = 0 gives ω X s [z 1,0] Ξ. Since X s Y s+1, we have proved that z w T a w = f with f Y s+ 1 [z 1,0] Ξ. Then using Theorem 6.10 once again gives.34 z u T A u X s+ 1 [z 1,0] Ξ. To finish the proof of Proposition.1, we recall that by definition Gηf = 1+ ρ z v z=0 ρ v z=0. z ρ

19 A PARADIFFERENTIAL REDUCTION FOR THE GRAVITY-CAPILLARY WAVES 19 We will say f 1 f if f 1 f X s+ 1 Ξ. By paralinearizing we have [z 1,0] 1+ ρ z v ρ v T z ρ 1+ ρ z v+t b ρ ρ T 1+ ρ z ρ T b ρ v T v ρ. zρ Then replacing v with u+t b ρ we have after some computations 1+ ρ z v ρ v T z ρ 1+ ρ z u T ρ u+t b ρ v ρ. zρ Lemma.34 gives T 1+ ρ z u T ρ u T Λ u, zρ with Λ z=0 = λ as announced. Now at z = 0, ρ z=0 = η, u z=0 = ψ T B η, v b ρ z=0 = V, so Gηψ T λ ψ T B η T V η as claimed. The proof of Proposition.1 is complete. Now, we want a paralinearization result for Gηf in term of the principle symbol λ 1 only, with a remainder of order 0. Proposition.15. Let d 1 and s > 3/+d/. Let 1/ µ s 1 then there exists a non-decreasing function F such that for any f H µ there hold Gηf = T λ 1f +Fη,f, Fη,f H µ F s+ 1 f H µ. Remark.16. The same result was proved in [1] when s > +d/. Proof. Step 1. Again, with v a solution to.8, Proposition.6 gives with I = [ 1,0].35 x,z v X µ 1 I F s+ 1 f H µ. According to Lemma.7 we have the paralinearization of.8.36 zv+t α x v+t β x z v = F 0 := γ z v+t α α x v+t β β z v. We claim that.37 F 0 Y µ F s+ 1 x,zv H µ 1. Indeed, for the first term one estimates using the product rule 6.14 with s 0 = µ 1/, s 1 = s 3/, s = µ 1/ to get γ z v γ L zv F η H µ 1 L H s 3 L H µ 1 H s+ 1 x,z v X µ 1. For the second term, the rule 6.15 yields T α α x v H µ 1 α v F η L H s 1 L H µ 3 H s+ 1 x,z v X µ 1. zρ

20 0 THIBAULT DE POYFERRÉ & QUANG-HUY NGUYEN The last term is estimated identically, we thus obtain.37. Step. Next, according to Lemma.8 z +T α x +T β x z = z T a 1 z T A 1+R, with R is of order 1 and thus Rv Y µ F s+ 1 x,zv X µ 1. In view of.35 there holds z T a 1[ z T A 1v] = F 1, F 1 Y µ F s+ 1 f H µ. By virtue of Theorem 6.10 we can obtain as before.38 z T A 1v X µ [z 1,0] F s+ 1 f H µ. Step 4. Writing f 1 f iff the X µ [z 1,0]-norm of f 1 f is bounded by the right-hand side of.38, we have notice that A 1 Γ 1 1 with semi-norms bounded by F s ρ z v ρ v T z ρ 1+ ρ z v T ρ v zρ T 1+ ρ T A 1v T ρ v T 1+ ρ A 1v T ρ v zρ zρ which concludes the proof since at z = 0, 1+ ρ zρ A 1 i ρ ξ = λ 1. To conclude this section, let us recall the following result on the shape derivative of the Dirichlet-Neumann operator. Theorem.17 [36, Theorem 3.1]. Let ψ H 3 and s > 1/+d/, d 1. Then the map is differentiable and G ψ : H s+1 H 1 d η Gηψ f = lim {Gη +εfψ Gηf} = GηBf divvf ε ε 0 1 where B and V are functions of η,ψ as in Paralinearization and symmetrization of the system 3.1. Paralinearization of the system. We want to replace all the nonlinear terms in the Zakharov-Craig-Sulem system 1.5 with paradifferential expressions. We have already paralinearized the Dirichlet-Neumann map, so we need to transform the nonlinear terms appearing in the second equation.

21 A PARADIFFERENTIAL REDUCTION FOR THE GRAVITY-CAPILLARY WAVES 1 Throughout this paragraph, we fix d 1, p [1,+ ], I = [0,T] and η,ψ be a solution to system 1.5 such that s s 0 > 3 + d, < r < s 0 d + 1, 3.1 ψ L I;H s L p I;W r,, η L I;H s+1 L p I;W r+1,, infdistηt,γ h > 0. t I Lemma 3.1. There exists a nondecreasing function F such that Hη = T l η +f, where l = l +l 1 with 3. l = 1+ η 1 ξ η ξ 1+ η, l 1 = i x ξ l, and f H s+r satisfying f H s+r F 1 η C r+ 1 η. H s+1 Proof. Applying Theorem 6.9 with u = η, µ = s 1 and ρ = r 1 we obtain η = T 1 η η p η +f 1, p = I 1+ η 1+ η 1 1+ η 3 and f 1 satisfies Since s 0 > 3 + d, this yields f 1 H s+r 1 F η L η C r 1 η H s f 1 H s+r 1 F Hence, 1 η C r+ 1 η. H s+1 Hη = divt p η +f 1 = T pξ ξ+idivpξ η +divf 1. This gives the conclusion with l = pξ ξ, l 1 = idivpξ, f = divf 1. We next tackle the other nonlinear terms. Recall the notations η ψ +Gηψ B = 1+ η, V = ψ B η. Lemma 3.. There exists a nondecreasing function F such that 1 ψ 1 η ψ Gηψ 1+ η = T V ψ T V T B η T B Gηψ +f,

22 THIBAULT DE POYFERRÉ & QUANG-HUY NGUYEN with f H s+r and [ ][ ] f H s+r F 1, ψ H s 0 1+ ψ C r + η C r ψ H s + s+ 1. Proof. Consider We compute a F = ab+c 1+ a Fa,b,c = 1 ab+c 1+ a, a,b,c Rd R d R. b ab+c 1+ a a, b F = ab+c 1+ a a, cf = ab+c 1+ a. Now we take a = η, b = ψ, and c = Gηψ. Using Proposition.10 and the hypothesis s 0 > 3 + d, we have a,b,c L F 1, ψ H s 0, [ ] a,b,c H s 1 F 1, ψ H s 0 1+ ψ H s + s+ 1, [ ] a,b,c C r 1 F 1, ψ H s 0 1+ ψ C r + η C r+ 1. Then combining this with Theorem 6.9 gives with 1 η ψ Gηψ 1+ η = T VB η +T B η ψ +T B Gηψ +f 1, f 1 H s+r F [ ][ ] 1, ψ H s 0 1+ ψ C r + η C r ψ H s + s+ 1. By the same theorem, there holds 1 ψ = T ψ ψ +f, f H s+r F ψ H s 0 ψ C r ψ H s. At last, we deduce from 6.5 and the estimates on B,V from Corollary.11 that T BV T V T B η H s+r F 1, ψ H s 0 η H s 1, from which we can conclude the proposition. To replace the original unknown with the new good unknown, we will need an estimate on T tbη. This is contained in the following lemma. Lemma 3.3. We have [ ] T tbη H s F 1, ψ H s 0 1+ ψ C r + η C r+ 1 s+ 1.

23 A PARADIFFERENTIAL REDUCTION FOR THE GRAVITY-CAPILLARY WAVES 3 Proof. First, using the equations 1.5, the product and nonlinear estimates, and the estimates on Gηψ of Proposition.10, we have, t η H s t ψ H s 0 3 F 1, ψ H s 0 t η C r 1 + t ψ C r 3 F 1, ψ H s 0 [ 1+ ψ C r + η C r+ 1 Then using Theorem.17 for the shape derivative of the Dirichlet-Neumann, we have t [Gηψ] = Gη t ψ B t η divv t η. Then using the preceding estimates and the estimate on B from Corollary.11,, t ψ B t η H s 0 3 F 1, ψ H s 0 t ψ B t η C r 3 F 1, ψ H s 0 Thus the last estimates of Proposition.10 give Gη t ψ B t η C r 5 F 1, ψ H s 0 There also holds divv t η C r 5 V tη C r 3 F 1, ψ H s 0 so that t Gηψ C r 5 F 1, ψ H s 0 At last, as in.6, B = K η ψ +L ηgηψ. [ 1+ ψ C r + η C r+ 1 ]. ]. [ ] 1+ ψ C r + η C r+ 1. [ ] 1+ ψ C r + η C r+ 1, [ ] 1+ ψ C r + η C r+ 1. Differentiating this expression and using the preceding estimates on the time derivatives, we have [ ] t B C r η 5 F H 1, ψ H s 0 1+ ψ C r + η C r+ 1, from which the lemma follows immediately by 6.16 and the fact that r 5 > 1. We now have all the ingredients needed to paralinearize the equations. Recall that λ has been defined in.7, and l in 3.. Proposition 3.4. There exists a nondecreasing function F such that with U := ψ T B η there holds { t η +T V η T λ U =f 1, 3.4 t U +T V U T l η =f,

24 4 THIBAULT DE POYFERRÉ & QUANG-HUY NGUYEN with f 1,f satisfying [ ] f 1,f H s+ η 1 F H s H 1, ψ H s 0 1+ ψ C r + η C r+ 1 [ ] 1+ ψ H s + s+ 1. Proof. The first equation is just Proposition.1. For the second one, we use the equation satisfied by ψ and Lemmas to see that with R H s F and since t ψ +T l η +T V ψ T V T B η T B Gηψ = R [ ][ ] 1, ψ H s 0 1+ ψ C r + η C r ψ H s + s+ 1, t U = t ψ T B t η T tbη, we can use Lemma 3.3, the fact that t η = Gηψ, and with to conclude. T V ψ T V T B η = T V U +R R [ ] H s F 1, ψ H s 0 1+ ψ H s + s Symmetrization of the system. As in [1] we shall deal with a class of symbols having special structure that we recall here for the reader s convenience. Definition 3.5. Given m R, Σ m denotes the class of symbols a of the form a = a m +a m 1 with such that a m x,ξ = F ηx,ξ, a m 1 x,ξ = F α ηx,ξ xηx α α = 1. T a maps real-valued functions to real-valued functions;. F is a C real-valued function of ζ,ξ R d R d \{0}, homogeneous of order m in ξ, and there exists a function K = Kζ > 0 such that Fζ,ξ Kζ ξ m, ζ,ξ R d R d \{0}; 3. the F α s are complex-valued functions of ζ,ξ R d R d \{0}, homogeneous of order m 1 in ξ.

25 A PARADIFFERENTIAL REDUCTION FOR THE GRAVITY-CAPILLARY WAVES 5 In the sequel, we often need an estimate for u from T a u. For this purpose, we prove Proposition 3.6. Let m, µ R, and s 0 > 3 + d. Then there exists a function F such that for all η H s 0 1, for all a Σ m, we have u H µ+m F s 0 1 u C µ+m F T a u H µ + u L, s 0 1 Ta u C µ + u C 0. Remark The same result was proved in Proposition4.6 of [1] where the constant in the right hand side is F ηt H s 1. Here, for less regular η we prove a worse estimate. However, it turns out that 3.5 is sufficient to obtain a priori bounds.. In 3.5 resp. 3.6 one can freely replace u L reps. u C 0 by any lower order Sobolev resp. Hölder norm. Proof. We give the proof for 3.5, the one of 3.6 follows identically. We write a = a m +a m 1. Introduce b = 1 and a { m 0 < ε < min 1,s 0 3 d }. Applying Theorem 6.4 ii with ρ = ε gives T b T a m = I + r where r is of order ε and 3.7 ru H µ+ε F η C ε u H µ F η C 1+ε u H µ F s 0 1 u H µ. Then, setting R = r T b T a m 1 we have I Ru = T b T a u. Let us consider the symbol a m 1. For any α N d with α = and fixed ξ, since s 0 > 3 + d 3, Sobolev embedding and estimates 6.19, 6.3 give F α η,ξ α x η C 1+ε F α η,ξ x α η F α η,ξ H 1+ε+d H s 0 3 α x η H s 0 5 F η L η F η H s 0 1 H s 0 1. Consequently, one deduces a m 1 m 1 Γ 1+ε T a m 1u H µ m+ε F and thus by Proposition 6.6, u H µ. s 0 1 Because b Γ m 0 with semi-norm bounded by F s 0 1 we get 3.8 T b T a m 1u H µ+ε F s 0 1 u H µ.

26 6 THIBAULT DE POYFERRÉ & QUANG-HUY NGUYEN Combining 3.7 with 3.8 yields Ru H µ+ε F ηt H s 0 1 u H µ. The rest of the proof is identical to that of Proposition 4.6 in [1]. For the sake of conciseness, we give the following definition. Definition 3.8. Let m R and consider two families of operators of order m, {At : t [0,T]}, {Bt : t [0,T]}. Let s 0 > 3 + d and < r 1 d. We write A B if A B is of order m 3 and the following condition is fulfilled: for all µ R, there exists a nondecreasing function F such that for a.e. t [0,T], At Bt F ηt H µ H µ m+3 H 1 1+ ηt C r+ 1. Remark 3.9. Let a = a m + a m 1 Σ m. We make the following remarks. i Because the principal symbol a m t contains only the first derivative η C r 1 R d H s 0 1 R d with r 1 > 3, s 0 1 > 1 + d, applying the nonlinear estimate 6.3 we obtain On the other hand, M m 3 a m t F M m 0 a m t F ηt H s 0 1 ηt C r+ 1. s 0 1. ii The subprincipal symbol a m 1 t depends on α η, α = which belongs to C r 3 R d C R 1 d. Hence, a m 1 Γ m 1 1/ and by 6.1 and 6.3 we have uniformly for ξ = 1, F α ηt,x,ξ α xηt,x C 1 [F α ηt,,ξ F α 0,ξ] α xηt, C 1 + F α 0,ξ α xηt, C 1 F ηt L ηt C 1 η C r+ 1 + F α0,ξ ηt C r+ 1 F ηt L ηt H s 0 η C r+ 1 + F α0,ξ ηt C r+ 1. The same estimates hold when one takes derivatives in ξ, consequently M m 1 1 a m 1 t F s 0 ηt C r+ 1.

27 A PARADIFFERENTIAL REDUCTION FOR THE GRAVITY-CAPILLARY WAVES 7 On the other hand, due to the fact that s 0 > 3 + d M0 m 1 a m 1 t F s 0 1 η C F we have 1 From i and ii we observe that when one applies the symbolic calculus Theorem 6.4, the operator-norm estimates are always linear in the highest norm of η, namely η C r+ 1. Using this remark, one can verify easily that Proposition 4.3 in [1] is still valid and hence so is Proposition 4.8, [1]: Proposition Let q Σ 0, p Σ 1, γ Σ 3 defined by where q = 1+ x η 1, p = 1+ x η 5 4 λ 1 +p 1/, γ = l λ 1 + l Rλ 0 λ 1 i ξ x l λ 1, p 1/ = 1 γ 3/ { q 0 l 1 γ 1/ p 1/ +i ξ γ 3/ x p 1/}. Then, it holds that T p T λ T γ T q, T q T l T γ T p, T γ T γ. Using this Proposition, we now perform the symmetrization of the system 3.4. Remark that in [], for s > + 1, this is achieved by using a technical result in Lemma 4.4, []: for any m,µ R there exists a function C such that for all a Σ m and t [0,T], T at u H µ m C ηt H s 1 u H µ which says that the operator norm of T at depends only on ηt H s 1 instead of ηt H s when one applies Theorem 6.4 i. In our situation, we shall use Proposition 6.6 to handle symbols with negative regularity. Proposition Introduce two new unknowns Φ 1 = T p η, Φ = T q U. Then Φ 1, Φ C 0 [0,T],H s R and satisfy { t Φ 1 +T V Φ 1 T γ Φ = F 1, 3.9 t Φ +T V Φ +T γ Φ = F,.

28 8 THIBAULT DE POYFERRÉ & QUANG-HUY NGUYEN and there exists a nondecreasing function F independent of η, ψ such that for each t [0,T], there holds 3.10 F 1,F H s H η s F H 1, ψ H s 0 1+ η + ψ C r+1 C 1+ η r + ψ H s+1 H s. Proof. It follows directly from the parlinearizedsystem 3.4 and Proposition 3.10 that Φ 1, Φ satisfy { t Φ 1 +T V Φ 1 T γ Φ = T p f 1 +T tpη +[T V,T p ]η, 3.11 t Φ +T V Φ +T γ Φ = T q f +T tqu +[T V,T q ]U. For simplicity in notation, we denote the right-hand side of 3.10 by RHS. First, Remark 3.9 and the symbolic calculus from Theorem 6.4 ii applied with ρ = 1 gives It remains to estimate [T V,T p ]η H s + [T V,T q ]U H s RHS. T tp H s+ 1 H s, T tq H s H s. Recall that we have from the estimates on the Dirichlet-Neumann in Proposition.10 t η H s 0 F 1, ψ H s 0, and 3.1 t η C r F We thus get by Theorem 6.4 i that H T tp η 1/ F s+ 1 H s H 1, ψ H s 0 Thus, we are left with the estimate of is of the form ] 1, ψ H s 0 [1+ η + ψ C r+1 C r. [1+ η C r+1 + ψ C r ]. H T tp 1/ s+ H 1 s. Recall that p 1/ p 1/ = F α η,ξ xη, α α = where the F α s are smooth functions of ξ and homogeneous of order 1. Hence, t p 1/ = [ t F α η,ξ] xη α + F α η,ξ t xη. α α = i Since s 0 > 3 + d, we have for all α = α = α x η L α x η H s

29 A PARADIFFERENTIAL REDUCTION FOR THE GRAVITY-CAPILLARY WAVES 9 This estimate together with 3.1 implies that t F α η,ξ xη α Γ 1 0 and ] M 1 0 [ t F α η,ξ] x η α η F H 1, ψ H s 0 [1+ η + ψ C r+1 C r. Theorem 6.4 i then yields T[ tf α η,ξ] α x η η H s M 1 0 [ t F α η,ξ] α xη s 1 RHS. ii Let G be an arbitrary smooth function of η. For any α =, we apply 6. with 1 < s 0 1 d to get G η t xη α C 1 G η C s 0 1 t d xη C 1 G η G0 + G0 H s0 1 t xη α C 1. Clearly, G η G0 H s 0 1 F η H 1. On the other hand, by virtue of Proposition.9, Consequently, t xη α C 1 Gηψ C 1 Gηψ C r 1 ] F 1, ψ H s 0 [1+ η + ψ C r+1 C r. G η t α x η C 1 F 1, ψ H s 0 [1+ η C r+1 + ψ C r ]. Therefore, according to Definition 6.5, F α η,ξ t α xη Γ 1 1 M 1 1 F α η,ξ t xη α F 1, ψ H s 0 We then obtain by virtue of Proposition 6.6 T Fα η,ξ t α xηη H s RHS. with semi-norm [1+ η C r+1 + ψ C r ]. For t q, the proof is the same as for the principal part of t p, and we only need to remark that ] 3.13 U H s F 1, ψ H s 0 [ η + ψ H s+1 H s. This concludes the proof of the Proposition.

30 30 THIBAULT DE POYFERRÉ & QUANG-HUY NGUYEN 4. Blow-up criterion and a priori estimate First of all, it follows straightforwardly from Proposition 3.11, that one can reduce the water waves system to a single equation of a complex-valued unknown: Proposition 4.1. Assume that 3.1 holds and let Φ 1,Φ be as in Proposition 3.11 then Φ := Φ 1 +iφ = T p η +it q U satisfies Ft H s F t +T V +it γ Φ = F, ] 1, ψ H s 0 [1+ η + ψ C r+1 C ][1+ η r + ψ H s+1 H s. To obtain estimates in Sobolev spaces, we shall commute equation 4.1 with an elliptic operator of order s and then perform an L -energy estimate. Since γ 3/ is of order 3/ > 1 we need to choose function of γ 3/ as in [1]: 4.3 := γ 3/ s/3, and take ϕ = T Φ. Since we want to obtain energy estimates in terms of the original variables η and ψ, we have to link them with this new variable ϕ. Lemma 4.. For s s 0 > 3 + d, there is a function F such that there holds ϕ L F ] [ η 1 + ψ H s+1 H s s+ 1 + ψ H s F 1, ψ H s 0 [1+ ϕ L ]. Proof. Recall that p Σ s, q Σ 0, and Σ s since γ 3 Σ 3. Thus we have [ ] ϕ L F 1 Φ H s F 1 U H s + s+ 1 ] F [ η 1 + ψ H s+1 H s, where we have used 3.13 to estimate U. To prove 4.5 we apply Proposition 3.6 two times to get ] 4.6 s+ 1 F s 0 [ T 1 T p η L + η, H 1 [ T 4.7 ψ H s F s 0 1 T q ψ L + ψ L ].

31 A PARADIFFERENTIAL REDUCTION FOR THE GRAVITY-CAPILLARY WAVES 31 Clearly, T T p η L ϕ L. On the other hand, 4.8 T T q ψ L T T q U L + T T q T B η L and 4.9 T T q T B η L F 1, ψ H s 0 F s+ 1 1, ψ H s 0 [ T T p η L +1 ], using 4.6. Putting together these estimates proves the proposition. For the blow-up criterion and energy estimate below, we recall the following quantities controlling the system 1.5. Notation 4.3. The Sobolev norm, blow-up norm and Strichartz norm for η,ψ are denoted by M s,t, N r,t, Z r,t respectively: M s,t = η,ψ L [0,T];H s+1 H s, M s,0 = η,ψ t=0 H s+ 1 H s, 4.10 Nr,T = η,ψ 4.11 Zr,T = η,ψ L 1 [0,T];W r+1, W r,, 4.1 L p [0,T];W r+1, W r,. Proposition 4.4. Let d 1, h > 0, and indices 3 + d < s 0 s, < r < 1 d. Then there exists a non-negative, non-decreasing function F h such that for all T 0 and η,ψ L [0,T];H s+1 H s L 1 [0,T];W r+1, W r, solution of the Zacharov-Craig-Sulem system 1.5 satisfying condition 1.3, there holds ϕ L [0,T];L F hm s,0 +F M s0,t[t +N r,t ]. Remark 4.5. In general, F h depends also on d, s, r, s 0. Proof. Using Grönwall lemma and the fact that ϕ0 L FM s,0, we see that the Proposition will be a consequence of the following estimate for ϕ : 4.13 d ] dt ϕ L F 1, ψ H s 0 [1+ η + ψ C r+1 C r [1+ ϕ L ] ϕ L. To prove this estimate, we see from 4.1 that ϕ solves the equation 4.14 t +T V +it γ ϕ = T F +G

32 3 THIBAULT DE POYFERRÉ & QUANG-HUY NGUYEN where G = T t Φ+[T V,T ]Φ+i[T γ,t ]Φ. First, remark that since ξ x γ 3/ = ξ γ 3/ x we can apply Theorem 6.4 ii with m = s, m = 3, ρ = 3 to find keep in mind Remark 3.9 [ ] [T,T γ ] H s L η F H 1, ψ H s 0 1+ ψ C r + η C r+ 1. The same theorem applied with m = 1, m = s, ρ = 1 also gives [T V,T ] H s L F 1, ψ H s 0 [ 1+ ψ C r + η C r+ 1 Finally, one can write t = L η, t η,ξ for some smooth function L homogeneous of order s in ξ, so that [ ] T t H s L η F H 1, ψ H s 0 1+ ψ C r + η C r+ 1. The estimates above imply [ ] G L F 1, ψ H s 0 1+ ψ C r + η C r+ 1 Φ H s. On the other hand, Proposition 3.6 applied to u = Φ, a = Σ s yields Φ H s F 1, ψ H s 0 [ ϕ L +1]. Therefore, G L F [ ] 1, ψ H s 0 1+ ψ C r + η C r+ 1 [1+ ϕ L ]. On the other hand, we see from 4. that [ T F L F 1, ψ H s 0 1+ ψ C r + η C r+ 1 so that thanks to Lemma 4. we have T F L F 1, ψ H s 0 Now, using Theorem 6.4 iii we see easily that 4.15 [ 1+ ψ C r + η C r+ 1 T V +T V L L F 1, ψ H s 0 and 4.16 T γ +T γ L L F 1, ψ H s 0 ] ]. [ 1+ ψ H s + s+ 1 ], ] [1+ ϕ L ]. [ ] 1+ ψ C r + η C r+ 1 [ ] 1+ ψ C r + η C r+ 1. Then using equation 4.14 we conclude the proof of 4.13 and thus of the Proposition.

33 A PARADIFFERENTIAL REDUCTION FOR THE GRAVITY-CAPILLARY WAVES 33 Now, taking s > + d and η 0,ψ 0 H s+1 H s such that distη 0,Γ > h > 0, we know from Theorem 1.1, [1] that there exists a time T 0, such that the Cauchy problem for system 1.5 with initial condition η 0,ψ 0 has a unique solution η,ψ C [0,T];H s+1 H s The maximal time of existence T > 0 then can be defined as 4.17 T = T η 0,ψ 0,h := sup { T > 0 : the Cauchy problem for 1.5 with data η 0,ψ 0 has a solution η,ψ C[0,T ];H s+1 H s } satisfying inf distηt,γ > 0. [0,T ] By the uniqueness statement of Proposition 6.4, [1] it is because of this Proposition that we require the separation condition in the definition 4.17 the solution η,ψ is defined for all t < T and η,ψ C [0,T ;H s+1 H s, which will be called the maximal solution. Theorem 4.6. Let d 1, h > 0 and indices 3 + d < s 0 < s 1, < r < 1 d. Let T = T η 0,ψ 0,h be the maximal time of existence defined by 4.17 and 4.18 η,ψ L [0,T ;H s+1 H s be the maximal solution of 1.5 with prescribed data η 0,ψ 0 satisfying distη 0,Γ > h. Then if T is finite, it holds that lim sup M s0,t +N r,t + 1 = +, T T ht where ht is the distance from the surface η to the bottom Γ over the time interval [0,T]. Proof. Suppose that T < + and K := limsup M s0,t +N r,t + 1 < +. T T ht

34 34 THIBAULT DE POYFERRÉ & QUANG-HUY NGUYEN Let T [0,T arbitrary then ht > 1/K > 0. It follows from Proposition 4.4 and the estimate 4.5 that 4.19 M s,t F 1/K T,M s,0,m s0,t,n r,t L for some function F 1/K increasing in each argument and L = LK,M s,0,t. On the other hand, from the a priori estimate in Proposition 5., [1] we deduce that the existence time for local solutions can be chosen uniformly for data lying in a bounded subset of H s+1 H s and satisfy uniformly the separation condition H 0. In particular, call T 1 be the time of existence for data in the ball B0,L of H s+1 H s whose surface is away from the bottom a distance at least 1/K. Choosing ηt T 1 as such a datum we can prolong the solution up to the time T + T 1 which contradicts the maximality of T, the theorem is proved. The preceding result means that one can continue a solution satisfying the separation condition H t as long as the Sobolev norm M s0 for any index s 0 > 3 +d stays bounded, and the time integral of the Hölder norms at regularityr = +ε, N r, is finite. Now we give the proof of Corollary 1.3 which is stated again for reader s convenience. Corollary 4.7. Let T 0, + and η, ψ be a distributional solution to system 1.5 on the time interval [0,T] such that inf [0,T] distηt,γ > 0. Then the following property holds: if one knows a priori that T 4.0 sup ηt,ψt [0,T] H + d + H 3 +d++ ηt,ψt 5 0 C dt < + + C+ then η0,ψ0 H R d implies that η,ψ L [0,T];H R d. Proof. From condition 4.0 one can find s 0, r verifying such that 3 + d < s 0, < r < 1 d 4.1 M s0,t +N r,t <. Take s > 1 arbitrary, it suffices to prove that if η0,ψ0 Hs+1 H s then η,ψ L [0,T];H s+1 H s. Since s > + d, according to the Cauchy theory in [1] one has a maximal solution η,ψ L [0,T s ;H s+1 H s.

35 A PARADIFFERENTIAL REDUCTION FOR THE GRAVITY-CAPILLARY WAVES 35 By uniqueness, we only need to show that T s T. Suppose that T s < T < + we get by applying Theorem 4.6 lim sup M s0,t +N r,t + 1 T T s ht = +. By our assumption, ht ht s > 0, hence which contradicts 4.1. lim sup M s0,t +N r,t = + T T s Remark 4.8. In fact, the proof of Corollary 4.7 above shows a stronger regularity result: under condition 4.0, for anys > + d + we have η0,ψ0 H s+1 H s implies η,ψ L [0,T];H s+1 H s. The value of this result over the Cauchy theory in [1] is that here the existence time T is given and is independent of the Sobolev index s. We next derive from Proposition 4.4 an a priori estimate for the Sobolev norm M s,t by means of itself and the Strichartz norm Z r,t. Theorem 4.9. Let d 1, h > 0, p > 1, and indices s > 3 + d, < r < s + 1 d. Then there exists a non-negative, non-decreasing function F h such that for all T [0,1 and η,ψ L [0,T];H s+1 H s L p [0,T];W r+1, W r, solution of the Zacharov-Craig-Sulem system 1.5 satisfyinginf t [0,T] distηt,γ > h we have 4. M s,t F h M s,0 +T δ F M s,t +Z r,t, where δ = min{1 1 p, 1 }. Proof. First, by Hölder inequality we have N r,t T 1 1 pz r,t and thus Proposition 4.4 applied with s 0 = s implies that 4.3 T T p η L [0,T];L + T T q U L [0,T];L M F s,0 +T 1 1 pf M s,t +Z r,t.

36 36 THIBAULT DE POYFERRÉ & QUANG-HUY NGUYEN We denote by Ξ the right-hand side of the preceding inequality, where F may change from line to line. Using the estimate for the Dirichlet-Neumann operator in Proposition.10 we get 4.4 ηt η0 H s 1 Consequently, t 0 t ηm H s 1dm = t 0 Gηψm H s 1dm TF M s,t. 4.5 ηt H s 1 η0 H s 1 + ηt η0 H s 1 η0 H s 1 + ηt η0 1 H s 1 ηt η0 1 H s M s,0 +T 1 F Ms T. The estimates 4.6, 4.3 and 4.5 then give 4.6 η L H s+1 Ξ. We turn to estimate ψ L Hs, for which we use the second equation in 1.5 to get ψt ψ0 H s 3 TF M s,t. By interpolation as in 4.4, there holds 4.7 ψt H s 1 ψ0 H s 1 + TF M s,t. Then, in views of 4.7 and 4.5 it remains to estimate T T q ψ L [0,T],L. To do this, one writes by definition of U T T q ψ L [0,T],L T T q U L [0,T],L + T T q T B η L [0,T],L. The second term on the right-hand side is bounded by 4.3. For the second term, one uses 4.6 to have T T q T B η L [0,T],L Ξ T B η L H s Ξ B L C 1 η L H s+1. Thus, to complete the proof we are left with B, for which we use L C 1 again the decomposition.6 for B: B = K η ψ +L ηgηψ. Then by 6. and the estimate for Gηψ in Theorem.5, there hold B C 1 K η C 1 ψ + L η C 1 C 1 Gηψ C 1 F s+ 1 ψ H s + Gηψ H s F s+ 1 ψ H s 1. The estimates 4.6 and 4.7 then conclude the proof.