Global well-posedness for the KP-II equation on the background of a non-localized solution

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1 Ann. I. H. Poincaré AN 8 (0) Global well-posedness for the KP-II equation on the background of a non-localized solution Luc Molinet a, Jean-Claude Saut b, Nikolay zvetkov c, a Laboratoire de Mathématiques et Physique héorique, UMR CNRS 6083, Faculté des Sciences et echniques, Université François Rabelais ours, Parc de Grandmont, 3700 ours Cedex, France b Université de Paris-Sud et CNRS, UMR de Mathématiques, Bât. 45, 9405 Orsay Cedex, France c Département de Mathématiques, Université de Cergy-Pontoise, avenue Adolphe Chauvin, 9530 Cergy-Pontoise Cedex, France Received 5 November 00; received in revised form 8 April 0; accepted 9 April 0 Available online 7 May 0 Abstract Motivated by transverse stability issues, we address the time evolution under the KP-II flow of perturbations of a solution which does not decay in all directions, for instance the KdV-line soliton. We study two different types of perturbations: perturbations that are square integrable in R and perturbations that are square integrable in R. In both cases we prove the global well-posedness of the Cauchy problem associated with such initial data. 0 Elsevier Masson SAS. All rights reserved.. Introduction.. Presentation of the problem he Kadomtsev Petviashvili (KP) equations (u t + u xxx + uu x ) x ± u yy = 0 () were introduced in [3] to study the transverse stability of the solitary wave solution of the Korteweg de Vries equation ( u t + u x + uu x + ) u xxx = 0, x R, t R. () 3 Here 0 is the Bond number which measures surface tension effects in the context of surface hydrodynamical waves. Actually the (formal) analysis in [3] consists in looking for a weakly transverse perturbation of the transport * Corresponding author. addresses: luc.molinet@lmpt.univ-tours.fr (L. Molinet), jean-claude.saut@math.u-psud.fr (J.-C. Saut), nikolay.tzvetkov@u-cergy.fr (N. zvetkov) /$ see front matter 0 Elsevier Masson SAS. All rights reserved. doi:0.06/j.anihpc

2 654 L. Molinet et al. / Ann. I. H. Poincaré AN 8 (0) equation u t + u x = 0. his perturbation amounts to adding to the equation a non-local term, namely x u yy. he same formal procedure is applied to the KdV equation () yielding the KP equation of the form ( u t + u x + uu x + ) u xxx + 3 x u yy = 0. (3) By change of frame and scaling, (3) reduces to () with the + sign (KP-II) when < 3 and the sign (KP-I) when > 3. Asfarasthetransverse stability of the KdV solitary wave ( -soliton ) ψ c (x ct,y), where ψ c (x, y) = 3c ( ) cx cosh, (4) is concerned, the natural initial condition associated to () should be u 0 = ψ c + v 0 where v 0 is either localized in x and y, or localized in x and y-periodic. In any case this rules out initial data in Sobolev spaces like H s (R ) or their anisotropic versions H s,s (R ), as was for instance the case considered in [4,8,9,9,0,,7] for the KP-II equation or in [7,4,,3] for the KP-I equation. Actually, it was proved in [8] that the Cauchy problem for the KP-I equation is globally well-posed for data which are localized perturbations of arbitrary size of a non-localized traveling wave solution such as the KdV N-soliton or the Zaitsev soliton (which is a localized in x and periodic in y solitary wave of the KP-I equation). he same result has been proven in [6] for localized perturbations of small size using Inverse Scattering techniques. No such result seems to be known for the KP-II equation for data of arbitrary size. he aim of the present paper is to fix this issue. Observe that the results in [8] concern initial data localized in y and periodic in x, for instance belonging to H s ( R) which excludes initial data of type ψ c + v 0 as above. On the other hand, the Inverse Scattering method has been used formally in [] and rigorously in [] to study the Cauchy problem for the KP-II equation with non-decaying data along a line, that is u(0,x,y)= u (x vy)+φ(x,y) with φ(x,y) 0asx + y and u (x) 0as x. ypically, u is the profile of a traveling wave solution with its peak localized on a moving line in the (x, y) plane. It is a particular case of the N-soliton of the KP-II equation discovered by Satsuma [] (see the derivation and the explicit form when N =, in the Appendix of [9]). As in all results obtained for KP equations by using the Inverse Scattering method, the initial perturbation of the non-decaying solution is supposed to be small enough in a weighted L space (see [, heorem 3]). As will be proven in the present paper, PDE techniques allow to consider arbitrary large perturbations of a (smaller) class of non-decaying solutions of the KP-II equation. We will therefore study here the initial value problem for the Kadomtsev Petviashvili (KP-II) equation (u t + u xxx + uu x ) x + u yy = 0. (5) We will either consider periodic in y solutions or we will suppose that u = u(t,x,y), (x, y) R, t R, with initial data u(0,x,y)= φ(x,y) + ψ(x,y), (6) where ψ is the profile of a non-localized (i.e. not decaying in all spatial directions) traveling wave of the KP-II equation and φ is localized, i.e. belonging to Sobolev spaces on R. We recall (see [3]) that, contrary to the KP-I equation, the KP-II equation does not possess any localized in both directions traveling wave solution. he background solution could be for instance the line soliton (-soliton) ψ c (x ct,y) moving with velocity c of the Korteweg de Vries (KdV) equation defined by (4). But it could also be the profile of the N-soliton solution of the KdV equation, N. he KdV N-soliton is of course considered as a two-dimensional (constant in y) object. Solving the Cauchy problem in both those functional settings can be viewed as a preliminary (and necessary!) step towards the study of the dynamics of the KP-II equation on the background of a non-fully localized solution, in particular towards a PDE proof of the nonlinear stability of the KdV soliton or N-soliton with respect to transversal perturbations governed by the KP-II flow. his has been established in [, Proposition 7] by Inverse Scattering methods and very recently by Mizumachi and zvetkov [5] who proved (by PDE techniques but using the Miura transform for the KP-II equation) the L stability of the KP-II line soliton with respect to transverse perturbations. We recall that it is established in [30] by Inverse Scattering methods and in [0,] by PDE techniques which extend to a large class of (non-integrable) problems, that the KdV -soliton is transversally nonlinearly unstable with respect to the KP-I equation.

3 L. Molinet et al. / Ann. I. H. Poincaré AN 8 (0) he advantage of the PDE approach of the present paper compared to an Inverse Scattering one is that it can be straightforwardly applied to non-integrable equations such as the higher order KP-II equations (see [5,6])... Statement of the results As was previously mentioned our main result is that the KP-II equation is globally well-posed for data of arbitrary size in the two aforementioned functional settings. heorem.. he Cauchy problem associated with the KP-II equation is globally well-posed in H s (R ) for any s 0. More precisely for every φ H s (R ) there is a global strong solution of (5) in C(R; H s (R )) with initial datum u(0,x,y)= φ(x,y). he solution is unique in C(R; H s (R )) if s>.fors [0, ] the uniqueness holds in the following sense. For every >0 there is a Banach space X continuously embedded in C([,]; H s (R )) and containing C([,]; H (R )) such that the solution u belongs to X and is unique in this class. Moreover the flow is Lipschitz continuous on bounded sets of H s (R ). Namely, for every bounded set B of H s (R ) and every >0 there exists a constant C such that for every φ,φ B the corresponding solutions u, u satisfy u u L ([,];H s (R )) C φ φ H s (R ). Finally the L -norm is conserved by the flow, i.e. u(t, ) L (R ) = φ L (R ), t R. We next state our result concerning localized perturbations. heorem.. Let ψ c (x ct,y) be a solution of the KP-II equation such that for every σ 0, ( x y ) σ ψ c : R R is bounded and belongs to L x L y (R ). Let s 0 be an integer. hen for every φ H s (R ) there exists a global strong solution u of (5) with initial data φ + ψ c. he uniqueness holds in the following sense. For every >0 there is a Banach space X continuously embedded in C([,]; H s (R )) and containing C([,]; H (R )) such that the solution u belongs to X and is unique in this class ψ c (x ct,y) + X. he spaces X involved in the statements of the above results are suitable Bourgain spaces defined below. Our proof is based on the approach by Bourgain to study the fully periodic case. We need however to deal with difficulties linked to several low frequencies cases (see for example Lemma 3.4 below) which do not occur in the purely periodic setting. Moreover in the proof of heorem. one needs to make a use of the Kato type smoothing effect for KP-II which was not present in previous works on the low regularity well-posedness of the KP-II equation. Remark.3. As was previously noticed, the result of heorem. holds (with the same proof) when we replace ψ c by the value at t = 0oftheN-soliton solution S N of the KdV equation which satisfies similar smoothness and decay properties as ψ c. his follows from the structure of the S N. For instance (see [5]) the -soliton of the KdV equation written in the form is u t 6uu x + u xxx = cosh(x 8t)+ cosh(4x 64t) S (x, t) = (3 cosh(x 8t)+ cosh(3x 36t)). On the other hand, a priori one cannot take instead of ψ c a function ψ which is non-decaying along a line {(x, y) x vy = x 0 }, as for instance the line soliton of the KP-II equation which writes ψ(x vy ct). However, as was pointed out to us by Herbert Koch the change of variables (X = x + vy v t,y = y vt) leaves invariant the KP-II equation and transforms the line soliton to an independent of y object. herefore our analysis applies equally well to the line soliton.

4 656 L. Molinet et al. / Ann. I. H. Poincaré AN 8 (0) Organization of the paper he second section is devoted to the proof of a bilinear estimate for localized functions in R Z. It is based on Bourgain s method in [4] who considers the case of functions in R Z. We prove heorem. in Section 3 as a consequence of two bilinear estimates in Bourgain X b,s spaces. Finally we prove heorem. in Section 4 by a fixed point argument in suitable Bourgain spaces, using in a crucial way the dispersive and smoothing properties of the KP-II linear group..4. Notations We will denote L p (resp H s ) the standard norm in the Lebesgue space L p (resp. the Sobolev space H s ), the domain being clear from the context. he Fourier transform on R x,y R t (resp. R x ) is denoted F or ˆ (resp. F x ). We will use the Japanese bracket notation = (+ ) /. he notation U(t)will stand for the (unitary in all H s (R ) or H s (R ) Sobolev spaces) KP-II group, that is U(t)= e t( 3 x + x y ).For(b, s) R R, the norm of the Bourgain space associated to U(t)is, for functions defined on R x,y R t (with the obvious modification that integration in η R is replaced by summation in q Z for functions defined on R x y R t ): u X b,b,s = U( t)u H b,b,s, where τ b u H b,b,s = ξ 3 ζ s τ b û(τ, ξ, η) dτ dξ dη, ζ = (ξ, η), that is u X b,b,s = R ξ,η R τ R ξ,η R τ σ(τ,ξ,η) b ζ s σ(τ,ξ,η) b û(τ, ξ, η) dτ dξ dη, ξ 3 where σ(τ,ξ,η)= τ ξ 3 + η /ξ. For any >0, the norm in the localized version X b,b,s u b,b X,s = inf { w X b,b,s,w(t)= u(t) on (,) }. For (b, s) R R the space Z b,s is the space associated to the norm u Z b,s = σ b ζ sû L ζ L τ. is defined as We define the restricted spaces Z b,s similarly to above. he notation is used for C where C is an irrelevant constant. For a real number s, s means for any r<sclose enough to s. If A,B R, we denote A B = min(a, B) and A B = max(a, B). A B means that there exists c>0 such that c A B c A. If S is a Lebesgue measurable set in R n, S, or mes S, denotes its Lebesgue measure. #S denotes the cardinal of a finite set S.. A bilinear estimate for localized functions in R Z In this section we will prove the following crucial bilinear estimate for functions having some localizations related to the KP-II dispersion relation in R Z. his is essentially a bilinear L 4 L Strichartz estimate associated to the linear KP-II group. We mainly follow the proof of Bourgain [4] that treats the case of functions in R Z. Proposition.. Let u and u be two real valued L functions defined on R R Z with the following support properties (τ,ξ,q) supp(u i ) ξ I i, ξ M i, τ ξ 3 + q /ξ K i, i =,. hen the following estimates hold, u u L (K K ) /( I I ) / (K K ) /4 (M M ) /4 u L u L (7) and if M M then

5 L. Molinet et al. / Ann. I. H. Poincaré AN 8 (0) u u L (K K ) / (K K ) /4 (M M ) /4 [ (M M ) /4 (K K ) (M M ) + ( I I ) / ] (K K ) /4 u L u L. (8).. hree basic lemmas Before starting the proof of the proposition let us recall the three following basic lemmas that we will use intensively. Lemma.. Consider a set Λ R ξ Z q. Let the projection of Λ on the ξ-axis be contained in a set I R. Assume in addition that for any fixed ξ 0 I the cardinal of the set Λ {(ξ 0,q), q Z} is bounded by a constant C. hen Λ C I. Lemma.3. Let I and J be two intervals on the real line and f : J R be a smooth function. hen and mes { x J : f(x) I } # { q J Z: f(q) I } I inf ξ J f (ξ) I inf ξ J f (ξ) +. Lemma.4. Let a 0, b, c be real numbers and I be an interval on the real line. hen and mes { x R: ax + bx + c I } I / a / # { q Z: aq + bq + c I } I / +. a /.. Proof of Proposition. According to [4, p. 30], we can suppose that ξ 0 on the support of u j (τ,ξ,q)(see also [4, p. 460] for a detailed discussion). We thus have to evaluate dτ dξ. q Z R R + q Z R R + χ {ξ ξ 0}u (τ,ξ,q )u (τ τ,ξ ξ,q q )dτ dξ By the Cauchy Schwarz inequality in (τ,ξ,q ) we thus get u u L sup A τ,ξ,q u L u L (9) (τ,ξ0,η) where A τ,ξ,q R Z is the set { A τ,ξ,q := (τ,ξ,q ): ξ I,ξ ξ I, 0 <ξ M, 0 <ξ ξ M, τ ξ 3 + q K, ξ A use of the triangle inequality yields A τ,ξ,q (K K ) B τ,ξ,q where τ τ (ξ ξ ) 3 + (q q ) ξ ξ K }. (0)

6 658 L. Molinet et al. / Ann. I. H. Poincaré AN 8 (0) { B τ,ξ,q := (ξ,q ): ξ I,ξ ξ I, 0 <ξ M, 0 <ξ ξ M, τ ξ 3 + q ξ + 3ξξ (ξ ξ ) + (ξq ξ q) } (K K ). ξξ (ξ ξ ) Let us now first prove (7) by bounding B τ,ξ,q in a direct way. he measure of the projection of B τ,ξ,q on the ξ - axis is bounded by ( I I ) and for a fixed ξ, using Lemma.4, the cardinal of its q -section is bounded by c(m M ) / (K K ) / +. herefore Lemma. and (9) (0) yield the bound (7). o prove (8) we divide B τ,ξ,q into two regions by setting Bτ,ξ,q {(ξ :=,q ) B τ,ξ,q : q q q } ξ ξ ξ () and Bτ,ξ,q {(ξ :=,q ) B τ,ξ,q : q q q ξ ξ ξ }. () In the next lemma we estimate the size of the first region. Lemma.5. For M M, the following estimate holds B τ,ξ,q (K K ) / (M M ) /. Proof. Recall that ξ and ξ ξ are non-negative real numbers and thus ξ ξ (ξ ξ ). Note also that (ξq ξ q) ξξ (ξ ξ ) = ξ ( (ξ ξ ) q q q ). (3) ξ ξ ξ ξ From the definition of Bτ,ξ,q we thus deduce that τ ξ 3 + q /ξ + 3ξξ (ξ ξ ) K K + M M. According to Lemma.4, the projection of B τ,ξ,q on the ξ -axis is thus bounded by (K K + M M ) / (M M ) / (K K ) / (M M ) / + (M M ) /. (4) (M M ) / We separate two cases: () M M K K. hen from the definition of Bτ,ξ,q and Lemma.3 we infer that for ξ fixed, the cardinal of the q -section of Bτ,ξ,q is bounded from above by M M M M. We thus get by Lemma. that in the considered case B τ,ξ,q (K K ) / (M M ) (M M ) / (K K ) / (M M ) /. () M M K K. In this case we subdivide once more: (.) q ξ q q ξ ξ (K K ) /. hen again by (4), Lemmas. and.3, we have that in the considered case (M M ) / B τ,ξ,q (M M ) / (K K ) / (M M ) (M M ) / (M M ) / (K K ) / (M M ) /. (.) q ξ q q ξ ξ (K K ) /. Since (M M ) / ( τ ξ 3 + q /ξ + 3ξξ (ξ ξ ) + ξ ( (ξ ξ ) q q q ) ) = q q ξ ξ ξ ξ q q ξ ξ ξ (5) it follows from Lemma.3 that the cardinal of the q -section at fixed ξ is bounded by (K K ) (M M ) / (K K ) / + (K K ) / (M M ) /.

7 L. Molinet et al. / Ann. I. H. Poincaré AN 8 (0) According to (4), the projection of this region on the ξ -axis is of measure less than a uniform constant and thus in the considered case (K K ) / (M M ) /. B τ,ξ,q his completes the proof of Lemma.5. and We now divide Bτ,ξ,q into three regions by setting { B, τ,ξ,q := (ξ,q ) Bτ,ξ,q : q q q ξ ξ ξ < (K K ) / } (M M ) /, { B, τ,ξ,q := (ξ,q ) Bτ,ξ,q : (K K ) / q (M M ) / q q ξ ξ ξ < (M M ) / } (M M ) / { ( B,3 τ,ξ,q := (ξ,q ) Bτ,ξ,q :max (K K ) / (M M ) /, (M M ) / ) (M M ) / Note that B, τ,ξ,q and B, τ,ξ,q may be empty. Lemma.6. he following estimates hold whenever M M and B, τ,ξ,q B, τ,ξ,q q q q } ξ ξ ξ. (K K ) (M M ) / (M M ) / (6) (K K ) / (M M ) /. (7) Proof. Assuming that B, τ,ξ,q is not empty, it follows from (3) and Lemma.4 that the measure of its projection on the ξ -axis can be estimated as (K K ) / (M M ) /. On the other hand, it follows from Lemma.3 that for a fixed ξ the cardinal of its q -section is bounded by (M M ) / (K K ) / + (M M ) / (K K ) /. (8) his proves (6) in view of Lemma.. Now coming back to (3) and using Lemma.4 we infer that the projection of B, τ,ξ,q on the ξ -axis is bounded by [ (K (M M ) / K ) + (M M ) ] / C since B, τ,ξ,q is empty as soon as K K M M. On the other hand, it follows from (5) and Lemma. that for a fixed ξ the cardinal of its q -section is also bounded by (8). his leads to (7) thanks to Lemma.. his completes the proof of Lemma.6. We finally estimate the size of B,3 τ,ξ,q. Lemma.7. For any 0 <ε it holds B,3 τ,ξ,q (M M ) /+ε/ Cε (M M ) / ε/ (K K ) + I I. (9)

8 660 L. Molinet et al. / Ann. I. H. Poincaré AN 8 (0) Proof. We subdivide B,3 τ,ξ,q by setting { B,3 τ,ξ,q (+ ) := (ξ,q ) B,3 τ,ξ,q : q q q } ξ ξ ξ K K and { B,3 τ,ξ,q (L) := (ξ,q ) B,3 τ,ξ,q : q q q } ξ ξ ξ L where L = l, l 0 l l, with ( ( M M l 0 ln max M M, K K M M )), l ln(k K ). In view of (5), for a fixed ξ,theq -section of B,3 τ,ξ,q (+ ) contains at most two elements and thus by Lemma., B,3 τ,ξ,q (+ ) I I. (0) Now, in B,3 τ,ξ,q (L) it holds τ ξ 3 + q /ξ + 3ξξ (ξ ξ ) K K + (M M )L and thus from Lemma.4 we infer that the measure of the projection of this region on the ξ -axis is bounded by (K K ) / (M M ) / + (M M ) / L. (M M ) / Interpolating with the crude bound M M we obtain the bound [ (K K ) / (M M ) / + (M M ) / ] ε (M M ) / L (M M ) ε. On the other hand, for a fixed ξ, by using again (5) and Lemma.3, we obtain that the cardinal of its q -section is bounded by K K + K K. L L We thus get by Lemma., [ B,3 τ,ξ,q (L) (K K ) / (M M ) / + (M M ) / ] ε (M M ) / L (M M ) ε (K K ) L (M M ) /+ε/ (K K ) (M M ) / ε/ L ε, where we used that in the considered case L (K K ) /. A summation over L yields the claimed bound. his (M M ) / completes the proof of Lemma.7. Now, using Lemmas.5.7, we get B τ,ξ,q (K K ) / (M M ) / (M M ) / (K K ) + C ε (M M ) / ε + I I, and thus according to (9) (0), u u L C ε (K K ) / (K K ) /4 [(M M ) /4 (K K ) /4 (M M ) /4 ε + ( I I ) / ] (K K ) /4 u L u L. his completes the proof of Proposition..

9 L. Molinet et al. / Ann. I. H. Poincaré AN 8 (0) Corollary.8. Let u and u be two real valued L functions defined on R R Z with the following support properties (τ,ξ,q) supp(u i ) ξ M i, τ ξ 3 + q /ξ K i, i =,. hen the following estimates hold, u u L ( ξ M) (K K ) / (M M M) / and if M M then (K K ) /4 (M M ) /4 u L u L () u u L ( ξ M) (K K ) / (K K ) /4 [ (M M ) /4 (K K ) /4 (M M ) + (M M M) / ] (K K ) /4 u L u L. () Proof. Rewriting the functions u i as u i = k Z u i,k with u i,k = u i χ {kmξ<(k+)m}, we easily obtain by support considerations and the Minkowsky inequality that u u L ( ξ M) = u,k u, k+ql k Z 0 q ( ξ M) u,k u, k+q L. 0q k Z he desired result follows by applying Proposition. with I = I =M for each k Z, and then Cauchy Schwarz in k. A rough interpolation between (7) and (8) on one side and between () and () on the other side gives the following bilinear estimates that we will use intensively in the next section. Corollary.9. here exists β 0 < / such that the following holds true. Let u and u be two real valued L functions defined on R R Z with the following support properties (τ,ξ,q) supp(u i ) ξ M i, τ ξ 3 + q /ξ K i, i =,. hen the following estimates hold, and u u L (K K ) β 0 (M M ) / u L u L (3) u u L ( ξ M) (K K ) β 0 [ (M M ) /4 + (M M M) /] u L u L. (4) Proof. Letusfirstprove(3).IfM M then we can easily conclude by only using (7). If M M then (8) gives u u L (K K ) / (K K ) /4 (M M ) /4 ( + (K K ) /4 (M M ) /4 + (M M ) /4 ) (K K ) /4 u L u L. We now distinguish cases according to which terms dominate in the sum + (K K ) /4 (M M ) /4 + (M M ) /4 (K K ) /4.

10 66 L. Molinet et al. / Ann. I. H. Poincaré AN 8 (0) If the first or the third term dominates then we get directly the needed bound. he only case we really need an interpolation between (7) and (8) is when the second term dominates. In this case we interpolate with weight θ on the bound (7) and weight θ on the second bound (8) and the conditions on θ to get the needed estimate are 3 θ>0, 4 θ< which of course can be achieved. Let us now turn to the proof of (4). Again, if M M then we can easily conclude by only using (). If M M then () gives u u L ( ξ M) (K K ) / (K K ) /4 [(M M ) ( /4 + (K K ) /4 ) (M M ) /4 + (M M M) / ] (K K ) /4 u L u L. Again, we distinguish cases according to which terms dominate in the sum (M M ) /4 + (M M ) /4 (K K ) /4 (M M ) /4 + (M M M) / (K K ) /4. If the first or the third term dominates then we get directly the needed bound. If the second term dominates than we interpolate with weight θ on the bound () and weight θ on the bound () and the conditions on θ to get the needed estimate are θ>0, θ <, the first assumption being imposed in order to ensure the K,K conditions and the second one for the M, M and M conditions. his completes the proof of Corollary Global well-posedness on R 3.. wo bilinear estimates in X b,b,s spaces he local well-posedness result is a direct consequence of the following bilinear estimates combined with a standard fixed point argument in X b,b,s spaces. Proposition 3.. here exist β</ and /4 <b < 3/8 such that for all u, v X /,b,s, the following bilinear estimates hold x (uv) X /,b,s u X /,b,s v X β,0,s + u X β,0,s v X /,b,s (5) and x (uv) Z /,s u X /,b,s v X β,0,s + u X β,0,s v X /,b,s, (6) provided s 0. o prove this bilinear estimate we first note that by symmetry it suffices to consider x Λ(u, v) where Λ(, ) is defined by ( ) F x Λ(u, v) := χ ξ ξ ξ (F x u)(ξ )(F x v)(ξ ξ )dξ. R he following resonance relation (see [4]) is crucial for our analysis: σ σ σ = 3ξξ (ξ ξ ) + (ξq ξ q) ξξ (ξ ξ ) 3 ξξ (ξ ξ ), (7) where σ := σ(τ,ξ,q):= τ ξ 3 + q /ξ, σ := σ(τ,ξ,q ),

11 L. Molinet et al. / Ann. I. H. Poincaré AN 8 (0) and σ := σ(τ τ,ξ ξ,q q ). We start by giving an estimate for interactions of high frequencies in x whose sum is not close to zero. Lemma 3.. here exists β</ such that x P Λ(P u, v) X /,b,s u X /,b,s v X β,0,s + u X β,0,s v X /,b,s, provided s 0 and /4 <b < 3/8. Proof. By duality we have to prove that I := τ,ξ,q f(τ,ξ,q )g(τ,ξ,q )h(τ, ξ, q) dτ dτ dξ dξ R 4 (q,q ) Z Γ τ,ξ,q ( ) f X /,b,s g X β,0,s + f X β,0,s g X /,b,s h (8) where Γ τ,ξ,q σ b τ,ξ,q := χ { ξ ξ ξ, ξ } σ / ξ 3 ξ ζ s ζ s ζ ζ s. For s 0, ζ s ζ s + ζ ζ s. (9) We therefore need to prove (8) only for s = 0. In addition, we obtain that thanks to (9) the estimate (8) holds with the following left-hand side ( ) f X /,b,s g X β,0,0 + f X /,b,0 g X β,0,s + f X β,0,s g X /,b,0 + f X β,0,0 g X /,b,s h. he above refinement allows us to get tame estimates which provides the propagation of regularity in the proof of heorem.. We will not further detail this (standard) aspect of the analysis in the sequel. We can of course assume that f, g and h are non-negative functions in R Z. For any L function w of (τ,ξ,q) and any (K, M) [, + [ R + we define the following localized versions of w: w K := wχ { σ K}, w M := wχ { ξ M} and w K,M := wχ { σ K, ξ M}. (30) We now consider the dyadic level D K,K,K M,M,M := { (τ,ξ,q,τ,ξ,q ): ξ M, ξ M, } ξ ξ M, σ K, σ K, σ K. Denoting by J K,K,K M,M,M the contribution of D K,K,K M,M,M to (8), then clearly I J K,K,K M,M,M (3) K,K,K,M,M M where K,K,K,M,M and M describe the dyadic level N. From the resonance estimate (7) it follows that D K,K,K M,M,M is empty whenever max(k, K,K ) MM M. We thus have only to consider the three following contributions: A. K MM M, B. K MM M and K MM M, C. K MM M and max(k, K ) MM M. Moreover, since M M, it is clear that either. M M M or. M M M.

12 664 L. Molinet et al. / Ann. I. H. Poincaré AN 8 (0) We will now estimate all these contributions. A.. K MM M and M M M. In this case we can write K l MM with l Z +. Moreover one has clearly K M 3. From (4) we thus have ) I fm K g M,K,h M,K K,K,K,M,MM MK /( + K b /M 3b (M /4 (K K ) β0 + M / )M 3b K,K,MM,l Z (/ b)l M / b M b + [ ( (/ b)l (K K ) β0 M /4 M + K,K,MM,l Z + M f M,K g M,K h M, l MM ) b ] f M,K g M,K h M, l MM. Here and in the sequel we use a slight abuse of notation by denoting still by I the contribution of the region of dyadic values under consideration to I (see (3)). Summing over K, K, l and over M taking values M, we get for β (β 0, /), I M f M X β,0,0 g M X β,0,0 h and Cauchy Schwarz in M yields ( ) / ( ) / I f M X β,0,0 g M X β,0,0 h f X β,0,0 g X β,0,0 h. M M A.. K MM M and M M M. In this case we can write K l M M with l Z +. We distinguish between the cases K<M 3 and K M 3. (i) K<M 3. hen according to (3), it holds I K,K M M,l0 K,K M M,l0 (K K ) β 0 M / M l/ M / f M,K g M,K h M, M l M M l/ (K K ) β 0 f M,K g M,K h M, l M M. Summing over K, K and applying Cauchy Schwarz in (M,M ) it leads to I ( ) / ( ) / ( ) / l/ f M X β,0,0 g M X β,0,0 h M, l M M, l0 M M M,M provided β (β 0, /) which leads to the needed bound. (ii) K M 3. hen using (4) we have I K,K M M,l0 K,K M M,l0 (K K ) β M / 0 M 3b (/ b)l M / b M b f M,K g M,K h M, l MM (/ b )l (K K ) β 0 ( M M ) b f M,K g M,K h M, lmm. Summing as in A., exchanging the role of M and M and using that b > 0 yields the needed bound. B.. K MM M, K MM M and M M M. hen we can write K l MM with l N. We separate again the cases K<M 3 and K M 3 and use that K K.

13 L. Molinet et al. / Ann. I. H. Poincaré AN 8 (0) (i) K<M 3. hen we get using Corollary.9 I MK /( ) + K b /M 3b fm K, ǧ M,K h M,K K,K,K,MM,M K,K MM,l0 K,K MM,l0 (KK ) β 0 M/ M( l/ M / M ) ( l/ M / M )K / f M, l MM g M,K h M,K ( ) l/ K (/ β 0) M ( l/ M / ) M fm, M lmm K β 0 g M,K h M,K. Summing as in A. yields for β (β 0, /), I f X /,0,0 g X β,0,0 h. Observe that in this case and in some places of the sequel we use that the localization assumptions in the bilinear estimates established in the previous section are invariant under the transformation (τ,ξ,q) ( τ, ξ, q). (ii) K M 3. hen we have using Corollary.9 I K,K MM,l0 K β 0 M / M 3b (/ b )l M / b M b K / ( l/ M / ) M fm, l MM K β 0 g M,K h M,K ( ) (/ b)l K (/ β 0) M b K,K MM,l0 M ( l/ M / ) M fm, l MM K β 0 g M,K h M,K. his implies the result as above. B.. K MM M, K MM M and M M M. hen one can write K l M M K M 3 M 3. Using Corollary.9, we obtain I K,K M M,l0 K (/ β 0) (M/ + M /4 )M l/ M / M ( l/ M / ) M fm, l M M K β 0 g M,K h M,K. Next, we can write (M / + M /4 )M l/ M / l/ M ( M b M b ( M b l/ M b + M / b M b /4 ) b K + Mb /4 M b /4 M 3 ) b K M 3 with l N and clearly and we can conclude as in the previously considered cases thanks to our assumptions on b. Note that this is a case where we need to introduce the additional factor in the Fourier transform restriction norm. C.. K MM M,max(K, K ) MM M and M M M. hen we can write K l MM with l Z +. Since M M this contribution can be treated exactly as the contribution of the case B. by exchanging the roles of K and K. C.. K MM M,max(K, K ) MM M and M M M. hen we can write K l M M with l N. In the considered case K M 3 and thus using Corollary.9, we get

14 666 L. Molinet et al. / Ann. I. H. Poincaré AN 8 (0) I K,K M M,l N K, K M M,l N (KK ) β 0 M / M ( l/ M / M )K f ( / M,K l/ M / ) M gm, l M M h M,K l/ K (/ β 0) K β 0 f M,K ( l/ M / M ) gm, l M M h M,K. Summing as in A. case (i) yields the result. his completes the proof of Lemma 3.. Let us now treat interactions of frequencies in x whose sum is closed to zero. Lemma 3.3. here exists β</ such that x P Λ(u, v) X /,b,s u X /,b,s v X β,0,s + v X /,b,s u X β,0,s, provided s 0 and /4 <b < 3/8. Proof. By duality it is equivalent to prove (8) with Γ τ,ξ,q σ b τ,ξ,q := χ { ξ ξ ξ, ξ } σ / ξ 3 ξ ζ s ζ s ζ ζ s. Again we can restrict our attention to the case s = 0. We proceed in a similar way as in the proof of Lemma 3.. he only difference is that here M describes the dyadic levels N and M,M describe the dyadic levels Z.We distinguish between the region. ξ ξ and the region. ξ ξ.. ξ ξ.inthisregion ξ ξ. We treat only the case σ σ since, as ξ ξ, the case σ σ is similar. We subdivide this region in the two subregions σ σ and σ < σ. In the first one, according to (7) we infer that I K,K,K M,M (K K ) / (K K ) /4 M / f K,M g K,M h K,M K / b (K K ) 3/8 K b / M / f K,M g K,M h K,M. K,K,K M,M Summing in M, K, K, K, using that b < / and applying Cauchy Schwarz in M implies the desired result. Now in the subregion σ < σ we know from the resonance relation (7) that σ ξ ξ ξ and thus according to () I K,K,K M, M > + K,K,K M, 0<M (K K ) / (K K ) /4 M /4 (MM )/8 K 3/8 b M f K,M g K,M h K,M (K K ) / (K K ) /4 M K / b f K,M g K,M h K,M (K K ) 3/8 K b 3/8 M 7/8 f K,M g K,M h K,M K,K,K M,M which can be summed in the same way as above for b < 3/8.. ξ ξ.inthisregion ξ ξ and M, M and M describe only the dyadic levels N. Since M M M, it follows from (7) that

15 I K,K,K M, M K,K,K M, M L. Molinet et al. / Ann. I. H. Poincaré AN 8 (0) (K K ) / (K K ) /4 M / K / b f K,M g K,M h K,M K 3/8 K 3/8 K b / M / f K,M g K,M h K,M. Summing in M, K, K, K, using that b < / and applying Cauchy Schwarz in M we obtain the desired result. his completes the proof of Lemma 3.3. It remains to treat interactions between very low frequencies and high frequencies. For this purpose we define the operators Λ i, i = 0,,, by where ( F t,x,y Λi (u, v) ) (τ,ξ,q) := χ {(τ,τ,ξ,ξ,q,q ) A i }(F t,x,y u)(τ,ξ,q )(F t,x,y v)(τ τ,ξ ξ,q q )dξ dτ, R q Z with A 0 := { (τ, τ,ξ,ξ,q,q ) B: σ ξ ξ ξ ξ }, A := { (τ, τ,ξ,ξ,q,q ) B: σ ξ ξ ξ ξ > max ( σ, σ )}, A := { (τ, τ,ξ,ξ,q,q ) B: σ ξ ξ ξ ξ > σ }, B := { (τ, τ,ξ,ξ,q,q ) R 4 Z : ξ ξ ξ }. Lemma 3.4. here exists β</ such that x Λ 0 (P u, P v) X /,b,s u X /,b,s v X β,0,s + u X β,0,s v X /,b,s, (3) x Λ (P u, P v) X /,b,s u X /,b,s v X β,0,s + u X β,0,s v X /,b,s (33) and x Λ (P u, P v) X /,b,s u X /,b,s v X β,0,s + u X β,0,s v X /,b,s, (34) provided /4 <b < 3/8. Proof. By duality it is equivalent to prove (8) with Γ τ,ξ,q σ b τ,ξ,q := χ Ai χ { ξ, ξ ξ } σ / ξ 3 ξ ζ s ζ s ζ ζ s. We thus proceed similarly to the previous propositions. he only difference is that here M describes the dyadic level N and M, M describe the dyadic level N. Again, we only consider the case s = 0. We first prove (3). Note that in A 0 we have ξ, ξ ξ and σ ξ ξ ξ. We can thus write K l M M with l N. We distinguish between the two cases K<M3 and K M 3. (i) K<M 3. In this case we can write I K,K M, M, l N K,K M, M, l N (K K ) / (K K ) /4 M /4 M / M l/ M / f M,K g M,K h M, M l M M l/ K 3/8 K 3/8 f M,K g M,K h M, l M M.

16 668 L. Molinet et al. / Ann. I. H. Poincaré AN 8 (0) Summing in K, K and then applying Cauchy Schwarz in (M,M ) implies I ( ) / ( ) / ( l/ f M X β,0,0 g M X β,0,0 h M, l M M l N M M M,M f X β,0,0 g X β,0,0 h, provided β (3/8, /). (ii) K M 3. hen we have for β (3/8, /), I (K K ) / (K K ) /4 M /4 K,K M, M, l N M / M 3b (/ b)l M / b M b f M,K g M,K h M, l M M (/ b)l (K K ) 3/8 β M b M b f X β,0,0 g X β,0,0 h. K,K M, M, l N A direct summing in l,k,k,m and M yields the desired result. Let us now prove (33). In A,wehave ξ, ξ ξ and σ ξ ξ ξ > σ. We can thus write K l M M with l N. (i) K<M 3. In this case we can write by using (7) I (K K ) / (K K ) /4 K,K M, M, l N M / M l/ M / M K f ( / M,K l/ M / ) M gm, l M M h M,K l/ K /8 K 3/8 ( f M,K l/ M / ) M gm, l M M h M,K. K,K M M,l N Summing in K, K and then applying Cauchy Schwarz in (M,M ), it results for β (3/8, /), I ( ) / ( ) / ( ) / l/ f M X β,0,0 h M l M M g M, l M M l N M M M,M f X β,0,0 g X /,0,0 h. (ii) K M 3. hen using that K K we have K/M 3 K /M 3 and thus we can proceed exactly as in the case K<M 3, the obtained bound being f X β,0,0 g X /,b,0 h. Let us finally prove (34). In this case we have K K + K, K M 3, K M M. hus using (8) and (7) we can write I K,K,K M, M K / K /4 M / M K / f M,K g M,K h ) /

17 L. Molinet et al. / Ann. I. H. Poincaré AN 8 (0) (we simply neglect the h localizations). We now naturally split the sum into two contributions, the contribution of M / M and M / M. In the first case we can write for δ (0,b /4) and ε>0 such that b δ /4 ε>0, I K,K,K M, M, M M K,K M, M, M M M, M, M M M, l N K /4 M / M K / K /+b ε M / M K /+δ K b δ /4 ε (M M )δ f M X /,b,0 g M X β,0,0 h δl f l M X /,b,0 g M X β,0,0 h f X /,b,0 g X β,0,0 h, K /+b ε f M,K K / ε g M,K h K /+b ε f M,K K / ε g M,K h where β ]/ ε, /[ and in the last inequality we use the Cauchy Schwarz inequality in the M summation (at fixed l). In the second case the argument is even easier since we do not use the lower bound on K. Namely, we can write for β (3/8, /), K / K /4 K,K,K M, M, M M M, l N his competes the proof of Lemma 3.4. M / M K / f M,K g M,K h l/ f l M X β,0,0 g M X β,0,0 h f X β,0,0 g X β,0,0 h. Now (5) follows by combining Lemmas Proof of (6). Since X /+,0,s is continuously embedded in Z s and since in the proof of Lemmas , except in the case A in Lemma 3. and in the proof of (3) in Lemma 3.4, we can keep a factor K 0+ with h K, it remains to treat the corresponding regions. Actually it is obvious to see that we can even restrict ourselves to the intersection of these regions with the region σ. By duality we have to prove J := χ ξ ξ ξ Θ τ,ξ,q τ,ξ,q f(τ,ξ,q )g(τ,ξ,q )h(ξ, q) dτ dτ dξ dξ R 4 (q,q ) Z f L (R ) g L (R ) h L (R ) where Θ τ,ξ,q τ,ξ,q := σ ξ ζ s ζ s ζ ζ s. Again we can suppose that s = 0 and moreover we will not make use of the factors involving b. Recall that in the region A of Lemma 3. we have M,M,M and we can write K l MM M with l N. We separate the cases M M and M M. A.. K MM M and M M M. hen we can write K l MM with l Z +. From (7) and (4) we thus have

18 670 L. Molinet et al. / Ann. I. H. Poincaré AN 8 (0) J K,K MM,l N K,K MM,l N M (K K ) β M/ 0 l MM f M,K g M,K h M χ { σ l MM } l/ (K K ) β 0 ( ) M / f M,K g M,K h M. M Summing over M M, K, K and l and then applying Cauchy Schwarz in M we get the desired result. A.. K MM M and M M M.Wehave J K,K M M,l N M l M M (h M χ { σ l M M },f K,M g K,M ) L. (35) If K K, we can easily conclude by using (4). We shall therefore suppose that K K. For fixed (ξ, q), ( ) χ { σ K} χ { σ K} τ K χ { τ K}. Using that the function χ { τ K} is pair, the L scalar product in (35) can be estimated by ( ( ))) K h M χ { σ l M M },f K,M (g K,M τ χ { σ lmm } We have that g K,M τ ( K χ { σ l M M }) is of the form g M,K+K with ( ) g M / K,K+K g K,M K. (36) Indeed the linear operator K,K : v K v( )χ { K } χ { K} is a continuous endomorphism of L (R) and of L (R) with K, K v L (R) sup x R K v(y)χ { y K } χ { x y K} dy K K v L (R) and R K, K v L (R) K v L (R) χ { K} L (R) v L (R). herefore, by Riesz interpolation theorem K,K is a continuous endomorphism of L (R) with ( ) / K K, K v L (R) v K L (R). Applying (3) with (36) at hand we get J K,K M M,l N K,K M M,l N K β 0 Kβ 0 L. M / M l M M f M,K g M,K+K h M χ { σ l M M } L t,x,y ( ) / l/ K β 0 Kβ 0 K l M M f M,K g M,K h M L x,y l/ K β 0 Kβ 0 ( Kδ M M ) δ fm,k g M,K h M L. x,y By choosing δ (0, / β 0 ), we can sum over K,K, M, M and l which yields the needed bound.

19 L. Molinet et al. / Ann. I. H. Poincaré AN 8 (0) It remains to treat the part of the region A 0 where ξ, ξ. We can write J K M (h M χ { σ K},f K,M g K,M ) L. K,K,K M, M We now consider separately the contributions of the regions M M and M M > toj. Let us denote by J the contribution of M M. hen we can write by using (3) for some β 0 <β</ J K,K,K M, M, M M M M / K / (K K ) β 0 h M f K,M g K,M M M / h f M X β,0,0 g M X β,0,0 M, M, M M l N, M l/ h f l M X β,0,0 g M X β,0,0 f X β,0,0 g X β,0,0 h, where in the last inequality we used the Cauchy Schwarz inequality for fixed l and then summing in a straightforward way in l. We next estimate the contribution of M M >. Denote by J this contribution to J. Since σ ξ ξ ξ,we can write σ l M M with l N. We only consider the case K K, the case K K being simpler. Using (7) and proceeding as in A. above, we obtain J (K K ) / (K K ) /4 M /4 K,K M, M, l N, M M > M/ M l M M f M,K g M,K+K h M χ { σ l M M }, where again g M, l M M +K satisfies (36). hus J (K K ) / (K K ) /4 M /4 K,K, l M M K M,M,l N, M M > ( M/ M K ) / ( l M M ) / fm,k g M,K h M. l M M l M M We therefore obtain that for a suitable δ>0 and β</, ( J M M ) δ f X β,0,0 g M X β,0,0 h M M,M,M M > f X β,0,0 g X β,0,0 h, where in the last inequality we write M M = q with q N, we apply the Cauchy Schwarz in M and then we sum in q. his completes the proof of Proposition Global well-posedness Let s 0 and /4 <b < 3/8. Noticing that in all the estimates in Sections we can keep a factor K 0+ or K 0+ with f M,K or g M,K and that for any function v X /,s supported in time in ],[, v X β,0,s (/ β) v X /,0,s, (37)

20 67 L. Molinet et al. / Ann. I. H. Poincaré AN 8 (0) we infer that the following restricted bilinear estimates hold: x (uv) /,b X,s ν u /,b X,s v /,b X,s and x (uv) Z /,s ν u X /,b,s (38) v /,b X,s, (39) with ν>0. Estimates (38), (39) in conjugation with the linear estimate (see [7]) t ψ(t) U ( t t ) x F ( t ) dt X F X /,b,s Z /,s, /,b,s Z /,s 0 with ψ C0 (R), lead to the well-posedness result by a standard fixed point argument in X/,b,s small enough, on the Duhamel formulation of (5): u(t) = U(t)ϕ t 0 Z /,s, >0 U ( t t ) x ( u ( t )) dt. (40) Also standard considerations prove that the time of existence of the solution only depends on ϕ L (see (9)). he uniqueness statement for s>follows from the Gronwall lemma. Next, from (40) and the fact that H s (R ) is an algebra for s>weinfer that for s>, x u belongs to C([0, ]; H s (R )) provided ϕ H s (R ) where ϕ H s = ξ s (ξ, η) ˆϕ(ξ,η) dξ. η Z R ξ aking the L scalar product of (5) with x u it is then easy to check that the L (R )-norm of such solutions is a constant of the motion. he density of H s (R ) in L (R ) combining with the continuity with respect to initial data in L (R ) ensures that the L -norm is a constant of the motion for our solutions (see [6] for details on this point). his proves that the solutions exist for all time. his completes the proof of heorem.. 4. Proof of heorem. We write the solution u of (5), (6) as u(t,x,y) = ψ c (x ct,y) + v(t,x ct,y) where v(t,.,.) is an L (R ) function. his v satisfies the equation ( vt cv x + v xxx + vv x + x (ψ c v) ) x v yy = 0, v(0,x,y)= φ(x,y). (4) Our strategy is to perform a fixed point argument in some Bourgain s type spaces on the Duhamel formulation of (4). In the context of (4) some straightforward modifications taking into account the term cv x of the unitary group U(t) and the Bourgain spaces should be done. We will use in a crucial way the following linear estimates of Strichartz or smoothing type (see for instance [3]) injected in the framework of the Bourgain spaces. Lemma 4.. One has v L 4 txy v X b,0,0 (4) and v x L x L ty v X b,0,0, (43) provided b>/.

21 L. Molinet et al. / Ann. I. H. Poincaré AN 8 (0) Proof. We only briefly recall the proof of (43), for (4) we refer to [8] for instance. By usual considerations (see [7]) it suffices to prove that ( ) x U(t)ϕ L x L ϕ L ty. (44) xy We first notice that ( ) x U(t)ϕ (x, y) = c (iξ)e i(xξ+yη) e i(ξ3 +cξ η /ξ)t ˆϕ(ξ,η)dξ dη R and that, for any fixed η, ξ ξ 3 +cξ η /ξ is an increasing bijection from R + into R and from R into R. herefore splitting the ξ integral into two pieces corresponding to positive and negative values of ξ and after performing the change of variables ξ ξ 3 + cξ η /ξ a short computation leads to x U(t)ϕ L = c ty his yields (44). R ˆϕ(ξ,η) ξ 3ξ + c + η dξ dη. /ξ Lemma 4.. For any s 0, 0 <, b>/ one has x (ψ c v) ( X 0,0,s x ψ c W s, + α x,y ψ L c x L y α s ) v X b,0,s. (45) Proof. We write x (ψ c v) as x ψ c v + v x ψ c and we treat each term separately. Write x ψ c v X 0,0,s Further, we have ψ c v x X 0,0,s = x ψ c v L H s xy x ψ c W s, v L H s x ψ c W s, v X b,0,s. ( α s x,y α ψ c L x L y his leads to (45) thanks to Lemma 4.. )( x,y β v x β s L x L ty Proposition 4.3. here exists ε 0 > 0 such that the following bilinear estimate holds x (uv) X /+ε,b,s u X /+ε,b,s v X /+ε,b,s (46) provided /4 <b < 3/8, s 0 and 0 <ε<ε 0. Proof. We have that Proposition., Corollaries.8.9 and their proofs are still valid for functions of R 3. Since as noticed in Section 3. we can always keep a factor K except in the cases σ dominant and ξ, we deduce that there exists ε 0 > 0 such that Ftxy (χ { σ <max( σ, σ )}χ { ξ <} ξ û ˆv) X /+ε,b,s u X /+ε,b,s v X /+ε,b,s provided /4 <b < 3/8, s 0 and 0 <ε<ε 0. It thus remains to treat this region σ max( σ, σ ), ξ. We thus have to prove that J := τ,ξ,q Γτ,ξ,η f(τ,ξ,η )g(τ,ξ,η )h(τ, ξ, η) dτ dτ dξ dξ dηdη f g h where R 6 ).

22 674 L. Molinet et al. / Ann. I. H. Poincaré AN 8 (0) Γ τ,ξ,η τ,ξ,η := χ { ξ ξ ξ, ξ, σ max( σ, σ )} σ b ξ ζ s ζ ξ 3 s ζ ζ s σ / ε σ /+ε σ ξ b σ 3 /+ε σ ξ ξ b 3 with σ := σ(τ,ξ,η):= τ ξ 3 cξ + η /ξ, σ := σ(τ,ξ,η ) and σ := σ(τ τ,ξ ξ,η η ). Notice that the crucial non-resonance relation (7) for σ, σ and σ still holds with the slight modification of the definition of σ. We can of course assume that f, g and h are non-negative functions in R 3. We separate the domain of integration into two regions.. 00 σ ξ 3. By Plancherel and then Hölder inequality, using the Strichartz inequality (4), we infer that for ε, J R 3 R 3 3b ξ 3(/ ε b ) h(τ, ξ, η) R 3 f(τ,ξ,η )g(τ τ,ξ ξ,η η ) σ /+ε σ /+ε dτ dξ dη dτ dξ dη ( ) ( ) F f 4 τ,ξ,η σ /+ε F g τ,ξ,η σ /+ε h L f L g L h L. L σ ξ 3. Applying Cauchy Schwarz in (τ,ξ,η ) and setting (τ,ξ,η ) := (τ τ,ξ ξ,η η ) we get [ J I(τ,ξ,η) f(τ,ξ,η )g(τ,ξ,η ) ] / dτ dξ dη h(τ,ξ,η)dτ dξ dη, (48) where, using the elementary inequality, dθ θ a +ε θ b +ε a + b +ε R it holds, by the resonance relation, [ ( ξ I(τ,ξ,η) σ / ε R 3 R ξ ( σ / ε max( σ, σ ) σ R { ξξ (ξ ξ ) σ } We perform the change of variables (ξ,η ) (ν, μ) with { ν = 3ξξ (ξ ξ ), μ = σ + σ. Noticing that ν [ 3 σ, 3 σ ], ν / dνdμ dξ dη = c ξ 3/ ( 3 4 ξ 3 ν) / σ + ν μ / and using the elementary inequality dθ θ a +ε θ b / a + b /, R we thus infer that ) ] dτ / σ +ε σ +ε dξ dη ) dξ dη / σ + σ +ε. (47)

23 L. Molinet et al. / Ann. I. H. Poincaré AN 8 (0) ξ /4 I σ / ε [ 3 σ + 3 σ [ 3 σ ξ /4 σ / ε 3 σ ν / dνdμ μ +ε ( 3 4 ξ 3 ν) / σ + ν μ / ν / dν ( 3 4 ξ 3 ν) / σ + ν / ] /. By the definition of ν it holds ν 3 8 ξ 3,i.e. 3 4 ξ 3 ν ξ 3 and thus ξ / I σ / ε [ 3 σ 3 σ ] / ] / ν / dν σ + ν / ξ / σ ε ξ /+6ε, provided 0 <ε /. his concludes the proof of (46) by applying Cauchy Schwarz in (τ,ξ,η) in (48). his completes the proof of Proposition 4.3. For and ε, we have the following estimate (see [7,8]) t U ( t t ) F ( t ) dt F /+ε,b /+ε,b X,s X,s. (49) 0 Moreover, using [8, heorem 3.], we obtain that for some ν>0, F X /+ε,b,s ν F /+ε,b X,s. Combining Proposition 4.3, Lemma 4. and the bounds (49), (50), we infer that the map t G : v U(t)φ 0 U ( t t ) x ( v / + ψ c v ) dt is a strictly contractive map in the ball of radius R := φ H s of X /+ε,b,s provided = (R)>0 is small enough. herefore there exists a unique local solution. Moreover, arguing as in Section 3.3 it is easy to check that the following differential identity holds for our solutions: d v = x (ψ c )v dt R R and thus v(t) L exp ( t ψ x L ) φ L. his leads to the global well-posedness result. Acknowledgements he authors acknowledge support from the project ANR-07-BLAN-050 of the Agence Nationale de la Recherche. We are grateful to Herbert Koch for his remarks on an earlier version of the paper. References [] M.J. Ablowitz, J. Villarroel, he Cauchy problem for the Kadomtsev Petviashvili II equation with non-decaying data along a line, Stud. Appl. Math. 09 (00) 5 6. [] M.J. Ablowitz, J. Villarroel, he Cauchy problem for the Kadomtsev Petviashvili II equation with data that do not decay along a line, Nonlinearity 7 (004) [3] A. de Bouard, J.-C. Saut, Solitary waves of the generalized KP equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 () (997) 36. (50)

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