Bilinear Strichartz estimates for the ZK equation and applications

Transcription

1 Bilinear trichartz estimates for the ZK equation and applications Luc Molinet, Didier Pilod To cite this version: Luc Molinet, Didier Pilod. Bilinear trichartz estimates for the ZK equation and applications. Annales de l Institut Henri Poincaré (C) Non Linear Analysis, Elsevier, 015, 3 (), pp <hal > HAL Id: hal ubmitted on 8 ep 015 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 BILINEAR TRICHARTZ ETIMATE FOR THE ZAKHAROV-KUZNETOV EQUATION AND APPLICATION LUC MOLINET AND DIDIER PILOD LMPT, Université François Rabelais Tours, CNR UMR 7350, Fédération Denis Poisson, Parc Grandmont, 3700 Tours, France. Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, CEP: , Rio de Janeiro, RJ, Brazil. Abstract. This article is concerned with the Zakharov-Kuznetsov equation ZK0 (0.1) tu + x u + u xu = 0. We prove that the associated initial value problem is locally well-posed in H s (R ) for s > 1 and globally well-posed in H1 (R T) and in H s (R 3 ) for s > 1. Our main new ingredient is a bilinear trichartz estimate in the context of Bourgain s spaces which allows to control the high-low frequency interactions appearing in the nonlinearity of (0.1). In the R case, we also need to use a recent result by Carbery, Kenig and Ziesler on sharp trichartz estimates for homogeneous dispersive operators. Finally, to prove the global well-posedness result in R 3, we need to use the atomic spaces introduced by Koch and Tataru. 1. Introduction The Zakharov-Kuznetsov equation (ZK) ZK (1.1) t u + x u + u x u = 0, where u = u(x, y, t) is a real-valued function, t R, x R, y R, T or R and is the laplacian, was introduced by Zakharov and Kuznetsov in [8] to describe the propagation of ionic-acoustic waves in magnetized plasma. The derivation of ZK from the Euler-Poisson system with magnetic field was performed by Lannes, Linares and aut [10] (see also [13] for a formal derivation). Moreover, the following quantities are conserved by the flow of ZK, M (1.) M(u) = u(x, y, t) dxdy, and H (1.3) H(u) = 1 ( u(x, y, t) 1 3 u(x, y, t)3) dxdy. Therefore L and H 1 are two natural spaces to study the well-posedness for the ZK equation. In the D case, Faminskii proved in [3] that the Cauchy problem associated to (1.1) was well-posed in the energy space H 1 (R ). This result was recently improved 000 Mathematics ubject Classification. Primary ; econdary. Key words and phrases. Zakharov-Kuznetsov equation, Initial value problem, Bilinear trichartz estimates, Bourgain s spaces. 1

3 L. MOLINET AND D. PILOD by Linares and Pastor who proved well-posedness in H s (R ), for s > 3/4. Both results were proved by using a fixed point argument taking advantage of the dispersive smoothing effects associated to the linear part of ZK, following the ideas of Kenig, Ponce and Vega [7] for the KdV equation. The case of the cylinder R T was treated by Linares, Pastor and aut in [1]. They obtained well-posedness in H s (R T) for s > 3. Note that the best results in the 3D case were obtained last year by Ribaud and Vento [15] (see also Linares and aut [13] for former results). They proved local well-posedness in H s (R 3 ) for s > 1 and in B 1,1 (R3 ). However that it is still an open problem to obtain global solutions in R T and R 3. The objective of this article is to improve the local well-posedness results for the ZK equation in R and R T, and to prove new global well-posedness results. In this direction, we obtain the global well-posedness in H 1 (R T) and in H s (R 3 ) for s > 1. Next are our main results. theor Theorem 1.1. Assume that s > 1. For any u 0 H s (R ), there exists T = T ( u 0 H s) > 0 and a unique solution of (1.1) such that u(, 0) = u 0 and theor.1 (1.4) u C([0, T ] : H s (R )) X s, 1 + T. Moreover, for any T (0, T ), there exists a neighborhood U of u 0 in H s (R ), such that the flow map data-solution theor. (1.5) : v 0 U v C([0, T ] : H s (R )) X s, 1 + T is smooth. theort Theorem 1.. Assume that s 1. For any u 0 H s (R T), there exists T = T ( u 0 H s) > 0 and a unique solution of (1.1) such that u(, 0) = u 0 and theort.1 (1.6) u C([0, T ] : H s (R T)) X s, 1 + T. Moreover, for any T (0, T ), there exists a neighborhood Ũ of u 0 in H s (R T), such that the flow map data-solution theort. (1.7) : v 0 Ũ v C([0, T ] : H s (R T)) X s, 1 + T theortglobal theo3 is smooth. Remark 1.1. The spaces X s,b T are defined in ection As a consequence of Theorem 1., we deduce the following result by using the conserved quantities M and H defined in (1.) and (1.3). Theorem 1.3. The initial value problem associated to the Zakharov-Kuznetsov equation is globally well-posed in H 1 (R T). Remark 1.. Theorem 1.3 provides a good setting to apply the techniques of Rousset and Tzvetkov [16], [17] and prove the transverse instability of the KdV soliton for the ZK equation. Finally, we combine the conserved quantities M and H with a well-posedness result in the Besov space B 1,1 and interpolation arguments to prove : Theorem 1.4. The initial value problem associated to the Zakharov-Kuznetsov equation is globally well-posed in H s (R 3 ) for any s > 1.

4 BILINEAR TRICHARTZ ETIMATE FOR THE ZK EQUATION AND APPLICATION 3 Remark 1.3. Note that the global well-posedness for the ZK equation in the energy space H 1 (R 3 ) is still an open problem. The main new ingredient in the proofs of Theorems 1.1, 1. and 1.4 is a bilinear estimate in the context of Bourgain s spaces (see for instance the work of Molinet, aut and Tzvetkov for the the KPII equation [14] for similar estimates), which allows to control the interactions between high and low frequencies appearing in the nonlinearity of (1.1). In the R case, we also need to use a recent result by Carbery, Kenig and Ziesler on sharp trichartz estimates for homogeneous dispersive operators. This allows us to treat the case of high-high to high frequency interactions. With those estimates in hand, we are able to derive the crucial bilinear estimates (see Propositions 4.1 and 5.1 below) and conclude the proof of Theorems 1.1 and 1. by using a fixed point argument in Bourgain s spaces. To prove the global wellposedness in R 3 we follows ideas in [1] and need to get a suitable lower bound on the time before the norm of solution doubles. To get this bound we will have to work in the framework of the atomic spaces U and V introduced by Koch and Tataru in [9]. We saw very recently on the arxiv that Grünrock and Herr obtained a similar result [5] in the R case by using the same kind of techniques. Note however that they do not need to use the trichartz estimate derived by Carbery, Kenig and Ziesler. On the other hand, they use a linear transformation on the equation to obtain a symmetric symbol ξ 3 + η 3 in order to apply their arguments. ince we derive our bilinear estimate directly on the original equation, our method of proof also worked in the R T setting (see the results in Theorems 1. and 1.3). This paper is organized as follows: in the next section we introduce the notations and define the function spaces. In ection 3, we recall the linear trichartz estimates for ZK and derive our crucial bilinear estimate. Those estimates are used in ection 4 and 5 to prove the bilinear estimates in R and R T. Finally, ection 6 is devoted to the R 3 case. notation. Notation, function spaces and linear estimates.1. Notation. For any positive numbers a and b, the notation a b means that there exists a positive constant c such that a cb. We also write a b when a b and b a. If α R, then α +, respectively α, will denote a number slightly greater, respectively lesser, than α. If A and B are two positive numbers, we use the notation A B = min(a, B) and A B = max(a, B). Finally, mes or denotes the Lebesgue measure of a measurable set of R n, whereas #F or denotes the cardinal of a finite set F. We use the notation (x, y) = 3x + y for (x, y) R. For u = u(x, y, t) (R 3 ), F(u), or û, will denote its space-time Fourier transform, whereas F xy (u), or (u) xy, respectively F t (u) = (u) t, will denote its Fourier transform in space, respectively in time. For s R, we define the Bessel and Riesz potentials of order s, J s and D s, by J s u = F 1 xy ( (1 + (ξ, µ) ) s Fxy (u) ) and D s u = F 1 ( (ξ, µ) s F xy (u) ). Throughout the paper, we fix a smooth cutoff function η such that η C 0 (R), 0 η 1, η [ 5/4,5/4] = 1 and supp(η) [ 8/5, 8/5]. xy

5 4 L. MOLINET AND D. PILOD For k N = Z [1, + ), we define φ(ξ) = η(ξ) η(ξ), φ k(ξ, µ) := φ( k (ξ, µ) ). and ψ k(ξ, µ, τ) = φ( k (τ (ξ 3 + ξµ ))). By convention, we also denote φ 1 (ξ, µ) = η( (ξ, µ) ), and ψ 1 (ξ, µ, τ) = η(τ (ξ 3 + ξµ )). Any summations over capitalized variables such as N, L, K or M are presumed to be dyadic with N, L, K or M 1, i.e., these variables range over numbers of the form { k : k N}. Then, we have that φ N (ξ, µ) = 1, supp (φ N ) { 5 8 N (ξ, µ) 8 5 N} =: I N, N, and N supp (φ 1 ) { (ξ, µ) 8 5 } =: I 1. Let us define the Littlewood-Paley multipliers by proj (.1) P N u = F 1 ( φn F xy (u) ), Q L u = F 1( ψ L F(u) ). xy Finally, we denote by e t x the free group associated with the linearized part of equation (1.1), which is to say, V (.) F xy ( e t x ϕ ) (ξ, µ) = e itw(ξ,µ) F xy (ϕ)(ξ, µ), where w(ξ, µ) = ξ 3 + ξµ. We also define the resonance function H by Resonance (.3) H(ξ 1, µ 1, ξ, µ ) = w(ξ 1 + ξ, µ 1 + µ ) w(ξ 1, µ 1 ) w(ξ, µ ). traightforward computations give that Resonance (.4) H(ξ 1, µ 1, ξ, µ ) = 3ξ 1 ξ (ξ 1 + ξ ) + ξ µ 1 + ξ 1 µ + (ξ 1 + ξ )µ 1 µ. We make the obvious modifications when working with u = u(x, y) for (x, y) R T and denote by q the Fourier variable corresponding to y... Function spaces. For 1 p, L p (R ) is the usual Lebesgue space with the norm L p, and for s R, the real-valued obolev space H s (R ) denotes the space of all real-valued functions with the usual norm u H s = J s u L. If u = u(x, y, t) is a function defined for (x, y) R and t in the time interval [0, T ], with T > 0, if B is one of the spaces defined above, 1 p and 1 q, we will define the mixed space-time spaces L p T B xy, L p t B xy, L q xyl p T by the norms and ( T ) u L p = u(,, t) p 1 T Bxy B dt p 0 u L q xy L p T (, u L p = t Bxy R ( ( T = u(x, y, t) p dt R 0 ) q p dx u(,, t) p B dt ) 1 p, if 1 p, q < with the obvious modifications in the case p = + or q = +. For s, b R, we introduce the Bourgain spaces X s,b related to the linear part of (1.1) as the completion of the chwartz space (R 3 ) under the norm Bourgain (.5) u X s,b = ( ) 1 q R 3 τ w(ξ, µ) b (ξ, µ) s û(ξ, µ, τ) dξdµdτ, ) 1,

6 BILINEAR TRICHARTZ ETIMATE FOR THE ZK EQUATION AND APPLICATION 5 where x := 1 + x. Moreover, we define a localized (in time) version of these spaces. Let T > 0 be a positive time. Then, if u : R [0, T ] C, we have that u X s,b = inf{ ũ X s,b : ũ : R R C, ũ R [0,T ] = u}. T We make the obvious modifications for functions defined on (x, y, t) R Z R. In particular, the integration over µ R in (.5) is replaced by a summation over q Z, which is to say 1 Bourgainper (.6) u X s,b = τ w(ξ, q) b (ξ, q) s û(ξ, q, τ) dξdτ, q Z R prop1.1 where w(ξ, q) = ξ 3 + ξq..3. Linear estimates in the X s,b spaces. In this subsection, we recall some well-known estimates for Bourgain s spaces (see [4] for instance). Lemma.1 (Homogeneous linear estimate). Let s R and b > 1. Then prop1.1. (.7) η(t)e t x f X s,b f H s. prop1. Lemma. (Non-homogeneous linear estimate). Let s R. Then for any 0 < δ < 1, prop1..1 (.8) prop1.3b η(t) t 0 e (t t ) x g(t )dt X s, 1 +δ g X s, 1 +δ. Lemma.3. For any T > 0, s R and for all 1 < b b < 1, it holds prop1.3b.1 (.9) u X s,b T T b b u X s,b. T 3. Linear and bilinear trichartz estimates 3.1. Linear strichartz estimates on R. First, we state a trichartz estimate for the unitary group {e t x } proved by Linares and Pastor (c.f. Proposition.3 in [11]). trichartz Proposition 3.1. Let 0 ɛ < 1 and 0 θ 1. Assume that (q, p) satisfy p = 1 θ and q = 6 θ(+ɛ). Then, we have that trichartz1 (3.1) D θɛ x e t x ϕ L q ϕ t Lp xy L trichartzcoro for all ϕ L (R ). Then, we obtain the following corollary in the context of Bourgain spaces. Corollary 3.. We have that trichartzcoro1 (3.) u L 4 xyt u X 0, 5, 6 + for all u X 0, Proof. Estimate (3.1) in the case ɛ = 0 and θ = 3 5 writes trichartzcoro (3.3) e t x ϕ L 5 xyt ϕ L for all ϕ L (R ). A classical argument (see for example [4]) yields u L 5 xyt u X 0, 1 +,

7 6 L. MOLINET AND D. PILOD CKZ CKZ1 (3.4) which implies estimate (3.) after interpolation with Plancherel s identity u L xyt = u X 0,0. In [], Carbery, Kenig and Ziesler proved an optimal L 4 -restriction theorem for homogeneous polynomial hypersurfaces in R 3. Theorem 3.3. Let Γ(ξ, µ) = (ξ, µ, Ω(ξ, µ)), where Ω(ξ, µ) is a polynomial, homogeneous of degree d. Then there exists a positive constant C (depending on φ) such that ( for all f L 4/3 (R 3 ) and where f(γ(ξ, ) µ)) K Ω (ξ, µ) dξdµ R CKZ (3.5) K Ω (ξ, µ) = det Hess Ω(ξ, µ). As a consequence, we have the following corollary. C f L 4/3, CKZcoro Corollary 3.4. Let K Ω (D) 1 8 and e itω(d) be the Fourier multipliers associated to K Ω (ξ, µ) 1 8 and e itω(ξ,µ), i.e. ( CKZcoro1 (3.6) F xy K Ω (D) 1 8 ϕ )(ξ, µ) = K Ω (ξ, µ) 1 8 Fxy (ϕ)(ξ, µ) where KΩ (ξ, µ) is defined in (3.5), and ( CKZcoro (3.7) F xy e itω(d) ϕ ) (ξ, µ) = e itω(ξ,µ) F xy (ϕ)(ξ, µ). Then, CKZcoro3 (3.8) for all ϕ L (R ). KΩ (D) 1 8 e itω(d) ϕ L 4 xyt ϕ L, Proof. By duality, it suffices to prove that CKZcoro4 (3.9) K Ω (D) 1 8 e itω(d) ϕ(x, y)f(x, y, t)dxdydt ϕ L xy f 4/3 L. R 3 xyt The Cauchy-chwarz inequality implies that it is enough to prove that CKZcoro5 (3.10) K Ω (D) 1 8 e itω(d) f(x, y, t)dt f 4/3 L L xy R in order to prove estimate (3.9). But straightforward computations give ( F x,y K Ω (D) ) 1 8 e itω(d) fdt (ξ, µ) = c K Ω (ξ, µ) 1 8 F x,y,t (f)(ξ, µ, Ω(ξ, µ)), R so that estimate (3.10) follows directly from Plancherel s identity and estimate (3.4). Now, we apply Corollary 3.4 in the case of the unitary group e t x. trichartzlin Proposition 3.5. Let K(D) 1 8 be the Fourier multiplier associated to K(ξ, µ) 1 8 where trichartzlin1 (3.11) K(ξ, µ) = 3ξ µ trichartzlin (3.1) Then, we have that K(D) 1 8 e t x ϕ L 4 xyt ϕ L xyt

8 BILINEAR TRICHARTZ ETIMATE FOR THE ZK EQUATION AND APPLICATION 7 trichartzlin3 (3.13) for all ϕ L (R ), and K(D) 1 8 u L 4 xyt u X 0, 1 + for all u X 0, 1 +. Proof. The symbol associated to e t x is given by w(ξ, µ) = ξ 3 + ξµ. After an easy computation, we get that det Hess w(ξ, µ) = 4(3ξ µ ). Estimate (3.1) follows then as a direct application of Corollary 3.4. Remark 3.1. It follows by applying estimate (3.1) with ɛ = 1/ and θ = /3+ that D 1 6 x e t x ϕ L 6 ϕ xyt L, for all ϕ L (R ), which implies in the context of Bourgain s spaces (after interpolating with the trivial estimate u L xyt = u X 0,0) that trichartzlinremark (3.14) D 1 8 x u L 4 xyt u X 0, 3, 8 + for all u X 0, Estimate (3.13) can be viewed as an improvement of estimate (3.14), since outside of the lines ξ = 1 3 µ, it allows to recover 1/4 of derivatives instead of 1/8 of derivatives in L 4. Remark 3.. it is interesting to observe that the resonance function H defined in (.4) cancels out on the planes (ξ 1 = µ1 3, ξ = µ 3 ) and (ξ 1 = µ1 3, ξ = µ 3 ). 3.. Bilinear trichartz estimates. In this subsection, we prove the following crucial bilinear estimates related to the ZK dispersion relation for functions defined on R 3 and R T R. BilintrichartzI Proposition 3.6. Let N 1, N, L 1, L be dyadic numbers in { k : k N } {1}. Assume that u 1 and u are two functions in L (R 3 ) or L (R T R). Then, BilintrichartzI0 (3.15) BilintrichartzI1 (3.16) basici (P N1 Q L1 u 1 )(P N Q L u ) L (L 1 L ) 1 (N1 N ) P N1 Q L1 u 1 L P N Q L u L Assume moreover that N 4N 1 or N 1 4N. Then, (P N1 Q L1 u 1 )(P N Q L u ) L (N 1 N ) 1 N 1 N (L 1 L ) 1 (L1 L ) 1 PN1 Q L1 u 1 L P N Q L u L. Remark 3.3. Estimate (3.16) will be very useful to control the high-low frequency interactions in the nonlinear term of (1.1). In the proof of Proposition 3.6 we will need some basic Lemmas stated in [14]. Lemma 3.7. Consider a set Λ R X, where X = R or T. Let the projection on the µ axis be contained in a set I R. Assume in addition that there exists C > 0 such that for any fixed µ 0 I X, Λ {(ξ, µ 0 ) : µ 0 X} C. Then, we get that Λ C I in the case where X = R and Λ C( I + 1) in the case where X = T.

9 8 L. MOLINET AND D. PILOD basicii The second one is a direct consequence of the mean value theorem. Lemma 3.8. Let I and J be two intervals on the real line and f : J R be a smooth function. Then, I basicii1 (3.17) mes {x J : f(x) I} inf ξ J f (ξ). basiciii In the case where f is a polynomial of degree 3, we also have the following result. Lemma 3.9. Let a 0, b, c be real numbers and I be an interval on the real line. Then, basiciii1 (3.18) mes {x J : ax + bx + c I} I 1. a 1 and basiciii (3.19) #{q Z : aq + bq + c I} I 1 a 1 BilintrichartzI3 (3.0) Proof of Proposition 3.6. We prove estimates (3.15) (3.16) in the case where (x, y, t) R 3. The case (x, y, t) R T R follows in a similar way. The Cauchy-chwarz inequality and Plancherel s identity yield (P N1 Q L1 u 1 )(P N Q L u ) L + 1. = (P N1 Q L1 u 1 ) (P N Q L u ) L sup A ξ µ,τ 1 PN1 Q L1 u 1 L P N Q L u L, (ξ,µ,τ) R 3 where { A ξ,µ,τ = (ξ 1, µ 1, τ 1 ) R 3 : (ξ 1, µ 1 ) I N1, (ξ ξ 1, µ µ 1 ) I N } τ 1 w(ξ 1, µ 1 ) I L1, τ τ 1 w(ξ ξ 1, µ µ 1 ) I L. it remains then to estimate the measure of the set A ξ,µ,τ uniformly in (ξ, µ, τ) R 3. To obtain (3.15), we use the trivial estimate A ξ µ,τ (L 1 L )(N 1 N ), for all (ξ, µ, τ) R 3. Now we turn to the proof of estimate (3.16). First, we get easily from the triangle inequality that BilintrichartzI4 (3.1) A ξ µ,τ (L 1 L ) B ξ µ,τ, where BilintrichartzI40 (3.) { B ξ,µ,τ = (ξ 1, µ 1 ) R : (ξ 1, µ 1 ) I N1, (ξ ξ 1, µ µ 1 ) I N } τ w(ξ, µ) H(ξ 1, ξ ξ 1, µ 1, µ µ 1 ) L 1 L and H(ξ 1, ξ, µ 1, µ ) is the resonance function defined in (.4). Next, we observe from the hypotheses on the daydic numbers N 1 and N that H (ξ 1, ξ ξ 1, µ 1, µ µ 1 ) = 3ξ 1 + µ 1 (3(ξ ξ 1 ) + (µ µ 1 ) ) (N 1 N ). ξ 1

10 BILINEAR TRICHARTZ ETIMATE FOR THE ZK EQUATION AND APPLICATION 9 Then, if we define B ξ,µ,τ (µ 1 ) = {ξ 1 R : (ξ 1, µ 1 ) B ξ,µ,τ }, we deduce applying estimate (3.17) that B ξ,µ,τ (µ 1 ) L 1 L (N 1 N ), for all µ 1 R. Thus, it follows from Lemma 3.7 that BilintrichartzI5 (3.3) B ξ,µ,τ N 1 N (N 1 N ) (L 1 L ). Finally, we conclude the proof of estimate (3.16) gathering estimates (3.0) (3.3). 4. Bilinear estimate in R R The main result of this section is stated below. BilinR Proposition 4.1. Let s > 1. Then, there exists δ > 0 such that BilinR.1 (4.1) x (uv) X s, 1 u +δ X s, 1 v +δ X s, 1, +δ technicalr technicalr.1 (4.) for all u, v : R 3 R such that u, v X s, 1 +δ. Before proving Proposition 4.1, we give a technical lemma. Lemma 4.. Assume that 0 < α < 1. Then, we have that (ξ 1 + ξ,µ 1 + µ ) (ξ 1, µ 1 ) (ξ, µ ) + f(α) max { (ξ 1, µ 1 ), (ξ, µ ) }, for all (ξ 1, µ 1 ), (ξ, µ ) R satisfying technicalr. (4.3) (1 α) 1 3 ξi µ i (1 α) 1 3 ξi, for i = 1,, and technicalr.3 (4.4) ξ 1 ξ < 0 and µ 1 µ < 0, and where f is a continuous function on [0, 1] satisfying lim α 0 f(α) = 0. We also recall te notation (ξ, µ) = 3ξ + µ. Proof. If we denote by u 1 = (ξ 1, µ 1 ), u = (ξ, µ ) and ( u 1, u ) e = 3ξ 1 ξ + µ 1 µ the scalar product associated to, then (4.) is equivalent to technicalr.4 (4.5) u 1 + u u 1 u + f(α) max { u 1, u }. Moreover, without loss of generality, we can always assume that technicalr.5 (4.6) ξ 1 > 0, µ 1 > 0, ξ < 0, µ < 0 and u 1 u. Thus, it suffices to prove that technicalr.6 (4.7) ( u 1 + u, u ) e f(α) u 1. By using (4.3) and (4.4), we have that technicalr.7 (4.8) ( u 1 + u, u ) e = 3(ξ 1 + ξ )ξ + (µ 1 + µ )µ 6(ξ 1 + ξ )ξ 3αξ 1 ξ + 3 ( (1 α) 1 1 ) ξ On the other hand, the assumptions ξ 1 > 0, ξ < 0, u 1 u and (4.3) imply that technicalr.8 (4.9) ξ 1 = ξ 1 (1 g(α)) ξ = (1 g(α))ξ

11 10 L. MOLINET AND D. PILOD with ( g(α) = 1 α ) ( (1 α) 1 1 ) Thus, it follows gathering (4.8) and (4.9) that 0. α 0 ( u 1 + u, u ) e 6g(α)ξ 3αξ 1 ξ + 3 ( (1 α) 1 1 ) ξ, which implies (4.7) by choosing f(α) = 1g(α) + 6α + 6 ( (1 α) 1 1 ) α 0 0. Proof of Proposition 4.1. By duality, it suffices to prove that BilinR. (4.10) I u L x,y,t v L x,y,t w L x,y,t, where BilinR.0 (4.11) I = Γ ξ1,µ1,τ1 ξ,µ,τ R 6 ŵ(ξ, µ, τ)û(ξ 1, µ 1, τ 1 ) v(ξ, µ, τ )dν, û, v and ŵ are nonnegative functions, and we used the following notations Γ ξ1,µ1,τ1 ξ,µ,τ = ξ (ξ, µ) s σ 1 +δ (ξ 1, µ 1 ) s σ 1 1 δ (ξ, µ ) s σ 1 δ, dν = dξdξ 1 dµdµ 1 dτdτ 1, ξ = ξ ξ 1, µ = µ µ 1, τ = τ τ 1, σ = τ w(ξ, µ) and σ i = τ i w(ξ i, µ i ), i = 1,. By using dyadic decompositions on the spatial frequencies of u, v and w, we rewrite I as BilinR.3 (4.1) I = I N,N1,N, where I N,N1,N = Γ ξ1,µ1,τ1 ξ,µ,τ R 6 N 1,N,N P N w(ξ, µ, τ) P N1 u(ξ 1, µ 1, τ 1 ) P N v(ξ, µ, τ )dν. ince (ξ, µ) = (ξ 1, µ 1 ) + (ξ, µ ), we can split the sum into the following cases: (1) Low Low Low interactions: N 1, N, N. In this case, we denote I LL L = I N,N1,N. N 4,N 1 4,N 4 () Low High High interactions: 4 N, N 1 N /4 ( N / N N ). In this case, we denote I LH H = I N,N1,N. 4 N,N 1 N /4,N / N N (3) High Low High interactions: 4 N 1, N N 1 /4 ( N 1 / N N 1 ). In this case, we denote I HL H = I N,N1,N. 4 N 1,N N 1/4,N 1/ N N 1

12 BILINEAR TRICHARTZ ETIMATE FOR THE ZK EQUATION AND APPLICATION 11 (4) High High Low interactions: 4 N 1, N N 1 /4 ( N 1 / N N 1 ) or 4 N, N N /4 ( N / N 1 N ). In this case, we denote I HH L = I N,N1,N. 4 N 1,N N 1/4,N / N 1 N (5) High High High interactions: N 4, N 1 4, N / N 1 N, N 1 / N N 1 and N / N N. In this case, we denote I HH H = I N,N1,N. N / N 1 N,N 1/ N N 1,N / N N Then, we have BilinR.4 (4.13) I = I LL L + I LH H + I HL H + I HH L + I HH H. BilinR.40 (4.14) 1. Estimate for I LL L. We observe from Plancherel s identity, Hölder s inequality and estimate (3.) that which yields I N,N1,N ( P N1 u ) L ( P N v ) L σ 1 1 +δ 4 4 P σ 1 +δ N w L P N1 u L P N v L P N w L, BilinR.400 (4.15) I LL L u L v L w L.. Estimate for I LH H. In this case, we also use dyadic decompositions on the modulations variables σ, σ 1 and σ, so that BilinR.5 (4.16) I N,N1,N = I L,L1,L N,N 1,N, L,L 1,L where I L,L1,L N,N 1,N = Γ ξ1,µ1,τ1 ξ,µ,τ R 6 P N Q L w(ξ, µ, τ) P N1 Q L1 u(ξ 1, µ 1, τ 1 ) P N Q L v(ξ, µ, τ )dν. Hence, by using the Cauchy-chwarz inequality in (ξ, µ, τ), we can bound I L,L1,L N,N 1,N by N N1 s L 1 +δ L 1 δ 1 L 1 δ (P N1 Q L1 u)(p N Q L v) L P N Q L w L. Now, estimate (3.16) provides the following bound for I LH H, L 1 +δ L δ 1 L δ L,L 1,L N N,N 1 N /4 N (s 1 ) 1 P N1 Q L1 u L P N Q L v L P N Q L w L. Therefore, we deduce after summing over L, L 1, L, N 1 and applying the Cauchy- chwarz inequality in N N that I LH H u L P N v L P N w L N N ( u L P N v ) 1 ( L P N w ) 1 BilinR.7 (4.17) L N N u L v L w L. 3. Estimate for I HL H. Arguing similarly, we get that BilinR.8 (4.18) I HL H u L v L w L.

13 1 L. MOLINET AND D. PILOD 4. Estimate for I HH L. We use the same decomposition as in (4.16). By using the Cauchy-chwarz inequality, we can bound I L,L1,L N,N 1,N by BilinR.9 (4.19) L 1 +δ L 1 δ 1 L 1 δ N s+1 N1 sn ( s P N1 Q L1 u)(p N Q L w) L P N Q L v L, BilinR.10 (4.0) where f(ξ, µ, τ) = f( ξ, µ, τ). Moreover, observe interpolating (3.15) and (3.16) that ( P N1 Q L1 u)(p N Q L w) L (N 1 N) 1 (1+θ) (N 1 N) 1 θ (L 1 L) 1 (1 θ) (L 1 L) 1 PN1 Q L1 u L P N Q L w L, for all 0 θ 1. Without loss of generality, we can assume that L = L L 1 (the case L 1 = L L 1 is actually easier). Hence, we deduce from (4.19) and (4.0) that BilinR.11 (4.1) I L,L1,L N,N 1,N L δ 1 1 L δ L δ θ 1 N +θ N (s θ) 1 P N1 Q L1 u L P N Q L w L P N Q L v L BilinR.1 (4.) Now, we choose 0 < θ < 1 and δ > 0 satisfying 0 < θ < s 1 and 0 < δ < θ 4. It follows after summing (4.1) over L, L 1, L and performing the Cauchy-chwarz inequality in N and N 1 that I HH L N (s 1 θ) ( 1 P N1 u L N 1 N u L w L v L. P N w L ) 1 v L 5. Estimate for I HH H. Let 0 < α < 1 be a small positive number such that f(α) = , where f is defined in Lemma 4.. In order to simplify the notations, we will denote (ξ, µ, τ) = (ξ 0, µ 0, τ 0 ). We split the integration domain in the following subsets: D 1 = { (ξ 1, µ 1, τ 1, µ, ξ, τ) R 6 : (1 α) 1 3 ξi µ i (1 α) 1 3 ξi, i = 1, }, D = { (ξ 1, µ 1, τ 1, µ, ξ, τ) R 6 : (1 α) 1 3 ξi µ i (1 α) 1 3 ξi, i = 0, 1 }, D 3 = { (ξ 1, µ 1, τ 1, µ, ξ, τ) R 6 : (1 α) 1 3 ξi µ i (1 α) 1 3 ξi, i = 0, }, 3 D 4 = R 6 \ D j. j=1 Then, if we denote by I j HH H the restriction of I HH H to the domain D j, we have that 4 BilinR.13 (4.3) I HH H = I j HH H Estimate for IHH H 1. We consider the following subcases. (i) Case { ξ 1 ξ > 0 and µ 1 µ > 0 }. We define D 1,1 = { (ξ 1, µ 1, τ 1, µ, ξ, τ) D 1 : ξ 1 ξ > 0 and µ 1 µ > 0 } j=1 and denote by I 1,1 HH H the restriction of I1 HH H to the domain D 1,1. We observe from (.4) and the frequency localization that BilinR.i.1 (4.4) max{ σ, σ 1, σ } H(ξ 1, µ 1, ξ, µ ) N1 3

14 BILINEAR TRICHARTZ ETIMATE FOR THE ZK EQUATION AND APPLICATION 13 in the region D 1,1. Therefore, it follows arguing exactly as in (4.14) that BilinR.i. (4.5) I 1,1 HH H u L v L w L. (ii) Case { ξ 1 ξ > 0 and µ 1 µ < 0 } or { ξ 1 ξ < 0 and µ 1 µ > 0 }. We define D 1, = { (ξ 1, µ 1, τ 1, µ, ξ, τ) D 1 : ξ 1 ξ > 0, µ 1 µ < 0 or ξ 1 ξ < 0, µ 1 µ > 0 } and denote by I 1, HH H the restriction of I1 HH H to the domain D 1,. Moreover, we use dyadic decompositions on the variables σ, σ 1 and σ as in (4.16). Plancherel s identity and the Cauchy-chwarz inequality yield BilinR.i.3 (4.6) I L,L1,L N,N 1,N N 1 s L 1 +δ L 1 δ 1 L 1 δ (P N1 Q L1 u)(p N Q L v) L w L. BilinR.i.4 (4.7) Next, we argue as in (3.0) to estimate (P N1 Q L1 u)(p N Q L v) L. Moreover, we observe that H (ξ 1, ξ ξ 1, µ 1, µ µ 1 ) = µ1 ξ 1 µ ξ N µ 1 in the region D 1,. Thus, we deduce from Lemma 3.7, estimates (3.17) and (3.0) and (3.1) that (P N1 Q L1 u)(p N Q L v) L N 1 (L1 L ) 1 (L1 L ) 1 PN1 Q L1 u L P N Q L v L. Therefore, we deduce combining estimates (4.6) and (4.7) and summing over L, L 1, L and N N 1 N that BilinR.i.5 (4.8) I 1, HH H u L v L w L. BilinR.i.6 (4.9) (iii) Case { ξ 1 ξ < 0 and µ 1 µ < 0 }. We define D 1,3 = { (ξ 1, µ 1, τ 1, µ, ξ, τ) D 1 : ξ 1 ξ < 0 and µ 1 µ < 0 } and denote by I 1,3 HH H the restriction of I1 HH H to the domain D 1,3. Moreover, we observe due to the frequency localization that there exists some 0 < γ 1 such that (ξ, µ ) (ξ 1, µ 1 ) { γ max (ξ1, µ 1 ), (ξ, µ ) } in D 1,3. Indeed, if estimate (4.9) does not hold for all 0 < γ , then estimate (4.) with f(α) = would imply that (ξ, µ) max { (ξ 1, µ 1 ), (ξ, µ ) } which would be a contradiction since we are in the High High High interactions case. Thus, we deduce from (4.9) that H (ξ 1, ξ ξ 1, µ 1, µ µ 1 ) = (ξ, µ ) (ξ 1, µ 1 ) N. ξ 1 We can then reapply the arguments in the proof of Proposition 3.6 to show that estimate (4.7) still holds true in this case. Therefore, we conclude arguing as above that BilinR.i.7 (4.30) I 1,3 HH H u L v L w L.

15 14 L. MOLINET AND D. PILOD Finally, estimates (4.5), (4.8) and (4.30) imply that BilinR.i.8 (4.31) IHH H 1 u L v L w L. 5.. Estimate for IHH H and I3 HH H. Arguing as for I1 HH H, we get that BilinR.ii.1 (4.3) IHH H + IHH H 3 u L v L w L. We explain for example how to deal with IHH H. It suffices to rewrite I N,N 1,N as I N,N1,N = P N w(ξ, µ, τ) P N1 u( ξ 1, µ 1, τ 1 ) P N v(ξ, µ, τ )d ν, where 1, µ 1, τ 1 Γ ξ ξ,µ,τ D d ν = dξdξ dµdµ dτdτ, ξ1 = ξ ξ, µ 1 = µ µ, τ 1 = τ τ, and Γ ξ 1, µ 1, τ 1 ξ,µ,τ is defined as in (4.11). Moreover, we observe that satisfies H = H(ξ, ξ ξ, µ, µ µ) = w(ξ, µ ) w(ξ, µ) w(ξ ξ, µ µ) H ξ = 3ξ + µ (3 ξ 1 + µ 1) and H µ = ξµ ξ1 µ 1. Therefore, we divide in the subregions { ξ ξ 1 > 0, µ µ 1 > 0}, { ξ ξ 1 < 0, µ µ 1 > 0}, { ξ ξ1 > 0, µ µ 1 < 0} and { ξ ξ 1 < 0, µ µ 1 < 0} and use the same arguments as above Estimate for I 4 HH H. Observe that in the region D 4, we have BilinR.iv.1 (4.33) µ i 3ξi > α (ξ i, µ i ) and µ j 3ξj > α (ξ j, µ j ), for at least a combination (i, j) in {0, 1, }. Without loss of generality 1, we can assume that i = 1 and j = in (4.33). Then, we deduce from Plancherel s identity and Hölder s inequality that I 4 HH H N (s 1 ) N N 1 1 K(D) 1 8 ( P N1 u ) L ( P 4 K(D) 1 N v ) L σ 1 1 +δ 8 4 w σ 1 +δ L, where the operator K(D) 1 8 is defined in Proposition 3.5. Therefore, estimate (3.13) implies that BilinR.iv. (4.34) IHH H 4 u L v L w L. Finally, we conclude the proof of estimate (4.1) gathering estimates (4.13), (4.15), (4.17), (4.18), (4.), (4.3), (4.31), (4.3) and (4.34). At this point, we observe that the proof of Theorem 1.1 follows from Proposition 4.1 and the linear estimates (.7), (.8) and (.9) by using a fixed point argument in a closed ball of X s, 1 +δ T (see for example [14] for more details). 1 in the other cases, we cannot use estimate (3.13) directly, but need to interpolate it with estimate (3.) as previously.

16 BILINEAR TRICHARTZ ETIMATE FOR THE ZK EQUATION AND APPLICATION 15 BilinRT 5. Bilinear estimate in R T The main result of this section is stated below. Proposition 5.1. Let s 1. Then, there exists δ > 0 such that BilinRT.1 (5.1) x (uv) X s, 1 u +δ X s, 1 v +δ X s, 1, +δ for all u, v : R T R R such that u, v X s, 1 +δ. Proof. By duality, it suffices to prove that BilinRT. (5.) J u L x,y,t v L x,y,t w L x,y,t, where BilinRT.3 (5.3) J = q,q 1 Z Γ ξ1,q1,τ1 ξ,q,τ R 4 ŵ(ξ, q, τ)û(ξ 1, q 1, τ 1 ) v(ξ, q, τ )dν, û, v and ŵ are nonnegative functions, and we used the following notations Γ ξ1,q1,τ1 ξ,q,τ = ξ (ξ, q) s σ 1 +δ (ξ 1, q 1 ) s σ 1 1 δ (ξ, q ) s σ 1 δ, dν = dξdξ 1 dτdτ 1, ξ = ξ ξ 1, q = q q 1, τ = τ τ 1, σ = τ w(ξ, q) and σ i = τ i w(ξ i, q i ), i = 1,. By using dyadic decompositions on the spatial frequencies of u, v and w, we rewrite J as BilinRT.4 (5.4) J = J N,N1,N, where J N,N1,N = q,q 1 Z Γ ξ1,q1,τ1 ξ,q,τ R 4 Now, we use the decomposition N 1,N,N P N w(ξ, q, τ) P N1 u(ξ 1, q 1, τ 1 ) P N v(ξ, q, τ )dν. BilinRT.5 (5.5) J = J LL L + J LH H + J HL H + J HH L + J HH H, where J LL L, J LH H, J HL H, J HH L, respectively J HH H, denote the Low Low Low, Low High High, High Low High, High High Low, respectively High High High contributions for J as defined in the proof of Proposition Estimate for J LH H + J HL H + J HH L. ince Proposition 3.6 also holds in the R T case, we deduce arguing as in (4.17), (4.18) and (4.) that BilinRT.6 (5.6) J LH H + J HL H + J HH L u L v L w L.. Estimate for J HH H. We recall that N N 1 N in this case. We divide the integration domain in several regions..1 Estimate for J HH H in the region ξ 100. We denote by JHH H 1 the restriction of J HH H to the region ξ 100 and use dyadic decompositions on the variables σ, σ 1, σ and ξ, so that BilinRT.7 (5.7) J N,N1,N = J L,L1,L N,N, 1,N,k k 0 L,L 1,L

17 16 L. MOLINET AND D. PILOD where J L,L1,L q,q 1 Z N,N 1,N,k Γ ξ1,q1,τ1 ξ,q,τ E k is given by the expression P N Q L w(ξ, q, τ) P N1 Q L1 u(ξ 1, q 1, τ 1 ) P N Q L v(ξ, q, τ )dν, with E k = {(ξ, ξ 1, τ, τ 1 ) R 4 : (k+1) 100 ξ k 100}. Thus, by using the Cauchy-chwarz inequality, we get that BilinRT.8 (5.8) J L,L1,L N,N 1,N,k k N1 s L 1 +δ L 1 δ 1 L 1 δ (P N1 Q L1 u)(p N Q L v) L w L. BilinRT.9 (5.9) Next, we argue as in (3.0) to estimate (P N1 Q L1 u)(p N Q L v) L. Moreover, we observe that H (ξ, ξ 1, q, q 1 ) = 6 ξ k. ξ 1 Thus, it follows from Lemma 3.7, estimates (3.18), (3.0) and (3.1) that (P N1 Q L1 u)(p N Q L v) L k 4 N 1 1 (L 1 L ) 1 (L1 L ) 1 4 PN1 Q L1 u L P N Q L v L. Therefore, we deduce combining (5.8) and (5.9) and summing over L, L 1, L, N N 1 N and k N that BilinRT.10 (5.10) JHH H 1 u L v L w L.. Estimate for J HH H in the region ξ 100, and ξ 1 ξ 100. We denote by JHH H the restriction of J HH H to this region and use dyadic decompositions on the variables σ, σ 1, σ, so that BilinRT.11 (5.11) J N,N1,N = J L,L1,L N,N 1,N, L,L 1,L BilinRT.110 where J L,L1,L N,N 1,N (5.1) q,q 1 Z is given by the expression Γ ξ1,q1,τ1 ξ,q,τ R 4 P N Q L w(ξ, q, τ) P N1 Q L1 u(ξ 1, q 1, τ 1 ) P N Q L v(ξ, q, τ )dν. Thus, the Caucy-chwarz inequality implies that BilinRT.1 (5.13) J L,L1,L N,N 1,N L 1 +δ L 1 δ 1 L 1 δ (P N1 Q L1 u)(p N Q L v) L w L, where we used the bound ξ N N 1 N and s 1. This time, we observe that H (ξ, ξ 1, q, q 1 ) = ξ 1. q 1 in order to estimate (P N1 Q L1 u)(p N Q L v) L. Then, since ξ 1 ξ 1, it follows from Lemma 3.7, estimates (3.19), (3.0) and (3.1) that BilinRT.13 (5.14) (P N1 Q L1 u)(p N Q L v) L (L 1 L ) 1 ( 1 + (L1 L ) 1 4 ) PN1 Q L1 u L P N Q L v L. Therefore, we deduce combining (5.13) and (5.14) and summing over L, L 1, L and N N 1 N (here we use the Cauchy-chwarz inequality in N 1 ) that BilinRT.14 (5.15) JHH H u L v L w L.

18 BILINEAR TRICHARTZ ETIMATE FOR THE ZK EQUATION AND APPLICATION 17 BilinRT.15 (5.16).3 Estimate for J HH H in the region ξ i 100 for i = 1,, 3. We denote by J 3 HH H the restriction of J HH H to this region. Once again, we use dyadic decompositions on the variables σ, σ 1 and σ as in (5.11). In order to simplify the notations, we will denote (ξ, q, τ) = (ξ 0, q 0, τ 0 ). Next, for 0 < δ 1, we split the integration domain in the following subregions F 3.1 = { (ξ, ξ 1, τ, τ 1, q, q 1 ) R 4 Z : ξ i 100, i {0, 1, } and (i, j) {0, 1, } with (ξi, q i ) (ξ j, q j ) NL 6δ }, F 3. = { (ξ, ξ 1, τ, τ 1, q, q 1 ) R 4 Z : ξ i 100, i {0, 1, } and denote by J 3,1 respectively F 3.. and (ξi, q i ) (ξ j, q j ) NL 6δ, (i, j) {0, 1, } }. HH H.3.1 Estimate for J 3,1 HH H, respectively J 3, HH H, the restriction of J HH H to F 3.1,. Without loss of generality, we can assume that (ξ, q) (ξ 1, q 1 ) NL 6δ. By using the Cauchy-chwarz inequality and the fact that ξ N N 1 N and s 1, we obtain that BilinRT.16 (5.17) J L,L1,L N,N 1,N L 1 +δ L 1 δ 1 L 1 δ ( P N1 Q L1 u)(p N Q L w) L P N Q L v L, where f(ξ, q, τ) = f( ξ, q, τ). Moreover, we observe arguing exactly as in the proof of Proposition 3.6 and by using (5.16) that ( P N1 Q L1 u)(p N Q L w) L BilinRT.17 (5.18) (N 1 N ) 1 (L N 1 1 L) 1 (L1 L) 1 L 3δ PN1 Q L1 u L P N Q L w L. Therefore, we deduce combining (5.17) and (5.18) and summing over L, L 1, L and N N 1 N (by using the Cauchy-chwarz inequality in N) that BilinRT.18 (5.19) J 3,1 HH H u L v L w L..3. Estimate for J 3, HH H. In the region F 3,, it holds that BilinRT.19 (5.0) (ξi, q i ) (ξ j, q j ) NL 6δ, (i, j) {0, 1, }. BilinRT.0 (5.1) Then, we deduce from the definition of H in (.3), the definition (ξ, q) = 3ξ + q and the assumptions (5.0) that H(ξ, ξ 1, q, q 1 ) = (ξ ξ 1 ξ ) (ξ i0, q i0 ) 6ξξ 1 ξ + Θ(ξ, ξ 1, q, q 1 ) = 6ξξ 1 ξ + Θ(ξ, ξ 1, q, q 1 ), for i 0 {1,, 3} such that ξ i0 = max{ ξ j : j = 1,, 3} and Θ(ξ, ξ 1, q, q 1 ) satisfies BilinRT.1 (5.) Θ(ξ, ξ1, q, q 1 ) ξ i (ξi, q i ) (ξ j, q j ) ξmed NL 6δ. i i 0 It follows combining (5.1) and (5.) that ( BilinRT. (5.3) H(ξ, ξ 1, q, q 1 ) ξ med 6 ξ max ξ min NL 6δ). Then, we subdivide the region F 1. in the following subregions F 3..1 = { (ξ, ξ 1, τ, τ 1, q, q 1 ) F 1. : ξ max ξ min NL 6δ},

19 18 L. MOLINET AND D. PILOD and denote by J 3,,1 respectively F 3... F 3.. = { (ξ, ξ 1, τ, τ 1, q, q 1 ) F 1. : ξ max ξ min NL 6δ}, HH H 3,, 3,, respectively JHH H, the restriction of JHH H to F 3..1,.3..1 Estimate for J 3,,1 HH H. Due to (5.3), we have that BilinRT.3 (5.4) max{ σ, σ 1, σ } ξ min ξ max, in F Without loss of generality, we assume that max{ σ, σ 1, σ } = σ. Then, by using the Cauchy-chwarz inequality, we deduce that BilinRT.4 (5.5) J L,L1,L N,N 1,N N 1 1 L δ L 1 δ 1 L 1 δ (P N1 Q L1 u)(p N Q L v) L w L. where we used that 1 ξ N s ( ξmax ξ min ) 1 +3δ 1 for s 1 and 0 < δ 1. BilinRT.5 (5.6) Moreover, we use that 1 H ξ 1 (ξ, ξ 1, q, q 1 ) = 6 ξ 1, Lemma 3.7, estimates (3.18), (3.0) and (3.1) lead to (P N1 Q L1 u)(p N Q L v) L N 1 1 (L 1 L ) 1 (L1 L ) 1 4 PN1 Q L1 u L P N Q L v L. We deduce combining (5.5) and (5.6) and summing over L, L 1, L and using the Cauchy-chwarz inequality in N 1 N that BilinRT.6 (5.7) J 3,,1 HH H u L v L w L..3.. Estimate for J 3,, HH H. This time, we perform also dyadic decompositions in the ξ 1, ξ and ξ variables. We ( denote by R K the Littlewood-Paley projectors, i.e. R K is defined by R K u = Fx 1 φ(k 1 ξ)f x (u) ), for any dyadic number K 1. Then, we have that BilinRT.7 (5.8) J L,L1,L N,N 1,N = J L,L1,L N,N 1,N (K 1, K, K 3 ), 100 K 1,K,K 3 N where J L,L1,L N,N 1,N (K 1, K, K 3 ) is defined by the expression J L,L1,L N,N 1,N (K 1, K, K 3 ) = ( PN Q L R K w ) (ξ, q, τ) q,q 1 Z Γ ξ1,q1,τ1 ξ,q,τ R 4 ( P N1 Q L1 R K1 u ) (ξ1, q 1, τ 1 ) ( P N Q L R K v ) (ξ, q, τ )dν. By using the Cauchy-chwarz inequality, we can bound J L,L1,L N,N 1,N (K 1, K, K 3 ) by BilinRT.8 (5.9) KK 1 min K 1 maxn 1 s L 1 +8δ L 1 δ 1 L 1 δ (P N1 Q L1 u)(p N Q L v) L w L. since K min K max NL 6δ in the region F 3,,. Moreover, noticing that H (ξ, ξ 1, q, q 1 ) = 6 ξ K, q 1 In the other cases we need to interpolate (5.6) with (3.15) as previously.

20 BILINEAR TRICHARTZ ETIMATE FOR THE ZK EQUATION AND APPLICATION 19 Lemma 3.7, estimates (3.19), (3.0) and (3.1) yield BilinRT.9 (5.30) (P N1 Q L1 u)(p N Q L v) L (K 1 K ) 1 (L1 L ) 1 (1 + K 1 4 (L1 L ) 1 4 ) PN1 Q L1 u L P N Q L v L. Now, we observe that BilinRT.30 (5.31) K(K 1 K ) 1 K 1 min K 1 max K 1 min. BilinRT.31 (5.3) Assume without loss of generality that K min = K. Therefore, it follows combining (5.8) (5.31), summing over L, L 1, L and K min and applying Cauchy-chwarz in K 1 K and in N 1 N that J 3,, HH H N N 1 N 100 K 1 K N ( P N1 R K1 u L P N R K v L P N w L ) 1 ( P N1 R K1 u L P N R K v L N 1 N K 1 N 1 K N ( ) 1 ( ) 1 P N1 u L P N v L w L N 1 N u L v L w L. Thus, we deduce combining (5.10), (5.15), (5.19), (5.7) and (5.3) that BilinRT.3 (5.33) J HH H u L v L w L. ) 1 w L.3 Estimate for J LL L. We get arguing exactly as in the cases.1 and. that BilinRT.33 (5.34) J LL L u L v L w L. Finally, we conclude the proof of estimate (5.1) gathering (5.5), (5.6), (5.33) and (5.34). We observe that the proof of Theorem 1. follows from Proposition 5.1 and the linear estimates (.7), (.8) and (.9) by using a fixed point argument in a closed ball of X s, 1 +δ T (see for example [14] for more details). 6. Global existence in H s (R 3 ) for s > 1 In this section we prove the global well-posedness in H s (R 3 ) for s > 1. To this aim we combine the conservation laws (1.) and (1.3), a well-posedness result in the Besov space B 1,1 (R3 ) and follow ideas in [1] (see [18] for the same kind of arguments). One crucial tool will also be the atomic spaces U and V introduced by Koch-Tataru in [9]. Recall that the Besov space B 1,1 (R3 ) is the space of all functions g (R 3 ) such that defb (6.1) g B 1,1 := N N P N g L <, where the Fourier projector P N is the R 3 -version of the one defined in (.1). Before stating the local existence theorem let us give the definition of a doubling time that will appear in the statement of this theorem. Let be given a Cauchy

21 0 L. MOLINET AND D. PILOD prop problem locally well-posed in some Banach space B with a minimum time of existence depending on the B-norm of the initial data and let C 0 1 be given. For any u 0 B we call doubling time, the infinite or finite positive real number { } T C0 (u 0 ) = sup t > 0 : u(θ) B C 0 u 0 B on [0, t]. Theorem 6.1. The Cauchy problem associated to (1.1) is locally well-posed in H s (R 3 ) for s > 1. Moreover, there exists C 0 1 and C > 0 such that for any u 0 H s (R 3 ), the doubling time T C0 satisfies Z3 (6.) T C0 (u 0 ) C u 0 B 1,1 Remark 6.1. The local well-posedness of ZK in H s (R 3 ) for s > 1 was already proven in [15]. The only new result here is the estimate from below of the doubling time. With Theorem 6.1 in hand we will now prove the Theorem 1.4. The proof of Theorem 6.1 is postponed at the end of this section.. Proof of Theorem 1.4. Let us fix s > 1. For any g H s (R 3 ) and any k 1 it holds k 1 g B 1,1 = j P j g L + j(1 s) js P j g L j=0 Therefore, taking k = ln(1+ g H s ) (s 1) ln j=k k g H 1 + k(1 s) g H s. we get ( z1 (6.3) g B 1,1 C s 1 + g H 1 ln(1 + g H s) 1/) for some C s > 0. Now, let u 0 H s (R 3 ) and u be the solution of ZK emanating from u 0. Combining Theorem 6.1 and (6.3) we get T C0 (u 0 ) C 1 + u 0 H 1 ln(1 + u 0 H s). If T C0 (u 0 ) = + then we are done. Otherwise we set u 1 := u(t C0 (u 0 )). In the same way as above we have T C0 (u 1 ) C 1 + u 1 H 1 ln(1 + u 1 H s). From the definition of the doubling time, it holds u 1 H s = C 0 u 0 H s and from the conservation of the quantities M(u) and H(u) and classical obolev inequalities we infer that u 1 H 1 C E(u 1 ) = C E(u 0 ), for some positive constant C independent of u 1. Therefore, setting E 0 := E(u 0 ), we obtain C T C0 (u 1 ) 1 + C E 0 ln(1 + C 0 u 0 H s).

22 BILINEAR TRICHARTZ ETIMATE FOR THE ZK EQUATION AND APPLICATION 1 Repeating this argument n-times (assuming that all doubling times T C0 (u k ), k = 1,,.., n 1, are finite, since otherwise we are done), we get C z (6.4) T C0 (u n ) 1 + C E 0 ln(1 + (C 0 ) n u 0 H s) 1 n. ince 1/n = + this ensures that for any given T > 0 there exists n 1 such n 1 that T C0 (u k ) > T and thus the solution is global in time. k=0 Remark 6.. Actually, it is not too hard to check that the lower bound (6.4) leads to a double exponential upper bound on the solution u, i.e. there exists constants K 1, K and K 3 only depending on u 0 H s such that for all t 0, ( ) u(t) H s K 1 exp K exp(k 3 t) Proof of Theorem Resolution spaces. We start by recalling the definition of the function spaces U and V (see [9] and [6]). Definition 6.. Let Z be the set of finite partitions = t 0 < t 1 < < t K = +. For {t k } K k=0 Z and {φ k} K 1 k=0 L (R 3 ) with K 1 k=0 φ k L = 1 and φ 0 = 0 we call the function a : R L (R 3 ) given by K a = 1 [tk 1,t k )φ k 1 k=1 a U -atom and we define the atomic space { U := u = λ j a j : a j U -atom and λ j R with j=1 j=1 j=1 j=1 } λ j < with norm { } u U := inf λ j : u = λ j a j with λ j R and a j U -atom The function space V is defined as the normed space of all functions v : R L (R 3 ) such that lim t v(t) exists and for which the norm v V := sup ( K ) 1/ v(t k ) v(t k 1 ) L {t k } K k=0 Z k=1 is finite, where we use the convention that v( ) = lim t v(t) and v(+ ) = 0. The spaces U and V are Banach spaces. They will serve as substitutes of the 1/,1 Besov type spaces B (L (R 3 1/, )) and B (L (R 3 )) that where first used in [19] in the context of Bourgain s method. Denoting by j the Fourier multiplier by 3 φ( j τ) for j 1 and η(τ) for j = 0, these last spaces are respectively endowed with the norms u B1/,1 (L (R 3 )) := j/ j u L (R 4 ) j 0 3 ee ection for the definition of φ and η.

23 L. MOLINET AND D. PILOD and u B1/, (L (R 3 )) := sup j/ j u L (R 4 ). j 0 The crucial point for us will be that, from the definition of the function space V, for a smooth function ψ Cc (R) and any 0 < T < 1, it holds zzz (6.5) ψ( /T )f L (R 4 )) T 1/ f L t L (R 3 ) T 1/ f V, f Cc (R 4 ), whereas we only have ψ( /T )f L (R 4 )) T 1/ ln T f B1/, (L (R 3 )), f C c (R 4 ). This last inequality would lead to a lower bound T (u 0 ) u 0 B 1,1 1 ln( u 0 B 1,1 of the doubling time that will not be sufficient to get the global existence result. This is the reason why we will work with the couple of spaces U and V and not 1/,1 with the more usual couple of spaces B (L (R 3 1/, )) and B (L (R 3 )). Then denoting by (t) := e t x the linear group associated with ZK, we define the spaces U = ( )U with norm u U = ( )u U and V = ( )V with norm u V = ( )u V. The properties of these spaces we need in the sequel are summarized in the following propositions (see [6]). prop3 Proposition 6.3. Let ψ C c (R) then and ψ(t) t 0 ) ψ( )u 0 U u 0 L, u 0 L (R 3 ) (t t )f(t, ) dt U sup v V =1 fv, f Cc (R 4 ). R 4 prop4 Proposition 6.4. Let T 0 : L L L 1 loc (R3 ; R) be a n-linear operator. Assume that for some p, q, n T 0 (( )φ 1,, ( )φ n ) L p t (R;Lq (R 3 )) φ i L. Then there exists T : U U Lp t (R; L q (R 3 )) satisfying n T (u 1,, u n ) L p t (R;Lq (R 3 )) u i U such that T (u 1,, u n )(t)(x, y, z) = T 0 (u 1 (t),, u n (t))(x, y, z) almost everywhere. We are now ready to define our resolution spaces : we denote by Y 1,1 the space of all functions u (R 4 ) such that i=1 i=1 u Y 1,1 := N N P N u U <

24 BILINEAR TRICHARTZ ETIMATE FOR THE ZK EQUATION AND APPLICATION 3 and by Y s, the space of all functions u (R 4 ) such that ( ) 1/ u Y s, := N s P N u U <. N Here, the Fourier projector P N is the R 3 -version of the one defined in (.1), i.e. P 1 localized in frequencies 3ξ + µ + η 1 while for N, P N localized in frequencies 3ξ + µ + η N Local existence estimate. Note that Proposition 6.3 ensures that 65 (6.6) ψ( )( )u 0 Y 1,1 u 0 B 1,1, u 0 B 1,1 (R3 ), and (6.7) ψ( )( )u 0 Y s, u 0 H s, u 0 H s (R 3 ). Moreover, Proposition 6.4 lead to the following estimates in U : lem5 Lemma 6.5. Let ψ Cc (R). For any u U it holds ψu L 4 u U. For any couple u, v U and any couple (N 1, N ) of dyadic number such that N 1 4N it holds ψp N1 up N v L N N 1 P N1 u U P N v U. Proof. The first estimate is a direct combination of the trichartz estimate for the ZK equation in R 3 (see [13] 4 ) strichartzr3 (6.8) ψ( )g L4 (R 4 ) g L (R 3 ) with Proposition 6.4. To prove the second estimate we notice that since H (ξ 1, ξ ξ 1, µ 1, µ µ 1, η 1, (η η 1 )) ξ 1 = 3ξ 1 + µ 1 + η 1 (3(ξ ξ 1 ) + (µ µ 1 ) + (η η 1 ) ) N 1. where H is the resonance function in dimension 3, the R 3 -version of the bilinear estimate (3.16) reads (P N1 Q L1 u 1 )(P N Q L u ) L N N 1 (L 1 L ) 1 (L1 L ) 1 PN1 Q L1 u 1 L P N Q L u L. ince for ψ C c this ensures that (R), g L (R 3 ) and any dyadic number L 1 it holds Q L ψ( )g L L 4 g L (ψp N1 ( )g)(ψp N ( )f) L N N 1 P N1 g L P N f L. The desired estimate follows by applying Proposition 6.4. We are now in position to prove the needed estimates on the retarded Duhamel operator. 4 Estimate (6.8) would correspond to the case ɛ = 0 and θ = 1/ of Proposition 3.1 in [13], but the case ɛ = 0 is not included in the hypotheses. Note however that this case follows by arguing exactly as in [13].

25 4 L. MOLINET AND D. PILOD prop6 Proposition 6.6. Let 0 < T < 1. For all u, v Y 1,1 with compact support in time in ] T, T [ it holds t po1 (6.9) ψ(t/t ) (t t ) x (uv)(t ) dt Y T 1/ u 1,1 Y 1,1 v Y 1,1. 0 For all u, v Y s,, s > 1, with compact support in time in ] T, T [ it holds po (6.10) t ψ(t/t ) (t t ) x (uv)(t ) dt Y T 1/( ) u s, Y 1,1 v Y s, + u Y s, v Y 1,1. 0 Proof. We separate the contribution of N 1 N P N1 up N v and the one of N 1 N P N1 up N v. We use Proposition 6.3, Lemma 6.5 and (6.5). For the first one we assume without loss of generality that N 1 4N to get t ψ(t) (t t ) x P N (P N1 up N v)(t ) dt U N N 1 4N N 0 sup w V =1 T 1/ ( N 1 x (P N1 up N v) L ψ( T )P N 1 w L N 1 4N ( N ) P N1 u N U P N v U P N1 w V 1 sup w V =1 N 1 4N N 1 T 1/ u Y 1,1 v Y 1,1. Whereas the contribution of the second one is easily estimated by t N ψ(t) (t t ) x P N (P N1 up N v)(t ) dt U N N 1 N N 0 ( sup w V =1 T 1/ N 1 N N 1 N N 1 N N P N1 u U P N v U ψ( ) T )P Nw L l N1 (N 1 P N1 u U )(N 1 P N1 v U ) l 0 T 1/ u Y 1,1 v Y 1,1. Finally the proof of (6.10) follows the same lines and thus will be omitted. Note that the definition of the function space U ensures that for any 0 < T < 1 and any smooth function ψ Cc (R) it holds ψ( /T )u U u U, u U. Therefore, combining (6.6) and Proposition 6.6, we deduce that for any 0 < T < 1, the functional G T (w)(t, ) := ψ(t)(t)u 0 1 maps Y 1,1 into itself and satisfies t G T (w) Y 1,1 u 0 B 1,1 0 (t t ) x (ψ( /T )w) (t, ) dt + T 1/ w Y 1,1. This ensures that there exists C 1 such that, for T u 0 B 1,1 contractive in the ball of Y 1,1 centered at the origin of radius C u 0 B 1,1 ), G T is strictly. By the

26 BILINEAR TRICHARTZ ETIMATE FOR THE ZK EQUATION AND APPLICATION 5 Banach fixed point theorem, it follows that G T has got a fixed point u satisfying u Y 1,1 C u 0 B 1,1. ince Y 1,1 L t B 1,1, this proves the local existence and uniqueness in the time restriction space Y 1,1 T of the solution u C([ T, T ]; B 1,1 ) of ZK emanating from u 0 B 1,1 (R3 ) with a doubling time satisfying (6.) for some constant C 0 1. The result for u 0 H s (R 3 ), s > 1, follows by noticing that (6.10) implies that G T maps as well Y s, into itself with G T (w) Y s, u 0 H s + T 1/ w Y 1,1 w Y s,. This completes the proof of Theorem 6.1. Acknowledgements. The authors were partially supported by the Brazilian- French program in mathematics. This work was initiated during a visit of the first author at the Mathematical Institute of the Federal University of Rio de Janeiro (IM-UFRJ) in Brazil. He would like to thank the IM-UFRJ for the kind hospitality References Bou [1] J. Bourgain, Exponential sums and nonlinear chrödinger equations, Geom. Funct. Anal.,3 (1993), CKZ [] A. Carbery, C. E. Kenig and. Ziesler, Restriction for homogeneous polynomial surfaces in R 3, to appear in Trans. Amer. Math. oc., (011) arxiv: Fa [3] A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation, Differential Equations, 31 (1995), no. 6, Gi [4] J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables déspace (d après Bourgain), Asterisque, 37 (1996), GH [5] A. Grünrock and. Herr, The Fourier restriction norm method for the Zakharov-Kuznetsov equation, preprint (013), arxiv: HHK [6] M. Hadac,. Herr and H. Koch, Well-posedness and sacttering for the KP-II equation in a critical space, Ann. I. H. Poincaré-AN, 6 (009), ERRATUM Ann. I. H. Poincaré- AN, 7 (010), KPV [7] C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation, Comm. Pure Appl. Math., 46 (1993), ZK [8] E. A. Kuznetsov and V. E. Zakharov, On three dimensional solitons, ov. Phys. JETP., 39 (1974), KT [9] H. Koch and D. Tataru, Dispersive estimates for principally normal pseudodifferential operators, Comm. Pure Appl. Math., 58 (005), LL [10] D. Lannes, F. Linares and J.-C. aut, The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation, preprint (01), arxiv: LP [11] F. Linares and A. Pastor, Well-posedness for the two-dimensional modified Zakharov- Kuznetsov equation, IAM J. Math. Anal. 41 (009), no. 4, LP [1] F. Linares, A. Pastor and J.-C. aut, Well-posedness for the ZK equation in a cylinder and on the background of a KdV soliton, Communications PDE, 35 (010), L [13] F. Linares and J.-C. aut, The Cauchy problem for the 3D Zakharov-Kuznetsov equation, Discrete Contin. Dyn. yst., 4 (009), no., MT [14] L. Molinet, J.-C. aut and N. Tzvetkov, Global well-posedness for the KPII equation on the background of non localized solution, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (011), no. 5, RV [15] F. Ribaud and. Vento, Well-posedness results for the 3D Zakharov-Kuznetsov equation, IAM J. Math. Analysis 44 (01), RT1 [16] F. Rousset and N Tzvetkov, Transverse nonlinear instability for two-dimensional dispersive models, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (009), RT [17] F. Rousset and N Tzvetkov, Transverse nonlinear instability of solitary waves for some Hamltonian PDE s, J. Math. Pures Appl. 90 (008), Plan [18] F. Planchon, On the cubic NL on 3D compact domains, preprint (013), arxiv: tataru [19] D. Tataru, On global existence and scattering for the wave maps equation, Amer. J. Math. 13 (1) (001),