Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation

We show that the initial value problem associated to the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov equation$$ u\_t-D\_x^\alpha u\_{x} + u\_{xyy} = uu\_x,\quad (t,x,y)\in\R^3,\quad 1\le \alpha\le 2,$$is locally well-posed in the spaces $E^s$, $s\textgreater{}\frac 2\alpha-\frac 34$, endowed with the norm$\|f\|\_{E^s} = \|\langle |\xi|^\alpha+\mu^2\rangle^s\hat{f}\|\_{L^2(\R^2)}.$As a consequence, we get the global well-posedness in the energy space $E^{1/2}$ as soon as $\alpha\textgreater{}\frac 85$. The proof is based on the approach of the short time Bourgain spaces developed by Ionescu, Kenig and Tataru \cite{IKT} combined with new Strichartz estimates and a modified energy.


Introduction
In this paper we study a class of two-dimensional nonlinear dispersive equations which extend the well-known Korteweg-de Vries (KdV) and Benjamin-Ono (BO) equations.There are several ways to generalize such 1D models in order to include the effect of long wave lateral dispersion.For instance one can consider the Kadomstev-Petviashvili (KP) and Zakharov-Kuznetsov (ZK) equations.Here we are interested with the effect of the dispersion in the propagation direction applied to the initial value problem for the ZK equation.More precisely we consider the generalized g-BOZK equation (1.1) u t − D α x u x + u xyy = uu x , (t, x, y) ∈ R 3 where D α x is the Fourier multiplier by |ξ| α , 1 ≤ α ≤ 2. When α = 2, (1.1) is the well-known ZK equation introduced by Zakharov and Kuznetsov in [21] to describe the propagation of ionic-acoustic waves in magnetized plasma.We refer to [14] for a rigorous derivation of ZK.For α = 1, equation (1.1) is the so-called Benjamin-Ono-Zakharov-Kuznetsov (BOZK) equation introduced in [11] and [15] and has applications to thin nanoconductors on a dielectric substrate.
We notice that (1.1) enjoys the two following conservation laws: where and Therefore, it is natural to study the well-posedness of g-BOZK in the functional spaces E 0 and E 1/2 , and more generally in E s defined for any s ∈ R by the norm Observe that E s is nothing but the anisotropic Sobolev space H αs,2s (R 2 ).In particular when α = 2, then E s = H 2s (R 2 ).Let us recall some well-known facts concerning the associated 1D model The Cauchy problem for (1.3), and especially the cases α = 1, 2 (respectively the BO and KdV equation), has been extensively studied these last decades, and is now well-understood.The standard fixed point argument in suitable functional spaces allows to solve the KdV equation at very low regularity level (see [13] for instance).This is in sharp contrast with what occurs in the case α < 2, since it was shown by Molinet-Saut-Tzvetkov [17] that the solution flow map for (1.3) cannot be C 2 in any Sobolev spaces (due to bad low-high interactions).Therefore the problem cannot be solved using such arguments.In view of this result, three approaches were developed to lower the regularity requirement.The first one consists in introducing a nonlinear gauge transform of the solution that solves an equation with better interactions (see [20]- [8]).This method was proved to be very efficient but as pointed out in [3], it is not clear how to find such a transform adapted to our 2D problem (1.1).The second one was introduced very recently by Molinet and the second author [18] and consists in an improvement of the classical energy method by taking into account the dispersive effect of the equation.This method is more flexible with respect to perturbations of the equation but requires that the dispersive part of the equation does not exhibit too strong resonances.Unfortunately, the cancelation zone of the resonance function Ω associated to g-BOZK (see (2.2) for the definition) seems too large to apply this technique to equation (1.1).Finally the third method introduced to solve (1.3) consists in improving dispersive estimates by localizing it in space frequency depending time intervals.In the context of the Bourgain spaces, this approach was successfully applied by Guo in [6] to solve (1.3) (see also [9] for an application to the KP-I equation) and seems to be the best way to deal with the g-BOZK equation.Now we come back to the 2D problem (1.1).The initial value problem for the ZK equation (α = 2) has given rise to many papers these last years.In particular, Faminskii proved in [4] that it is globally well-posed in the energy space H 1 (R 2 ).The best result concerning the local well-posedness was recently independently obtained by Grünrock and Herr in [5] and by Molinet and Pilod in [16] where they show the LWP of (1.1) in H s (R 2 ), s > 1/2.Similarly to the KdV equation, all these results were proved using the fixed point procedure.Concerning the case α = 1, using classical energy methods and parabolic regularization that does not take into account the dispersive effect of the equation, Cunha and Pastor [3] have proved the well-posedness of (1.1) in H s (R 2 ) for s > 2 as well as in the anisotropic Sobolev spaces H s1,s2 (R 2 ), s 2 > 2, s 1 ≥ s 2 .Also, it was proved in [7] that the solution mapping fails to be C 2 smooth in any H s1,s2 (R 2 ), s 1 , s 2 ∈ R. Moreover this result even extends to the case 1 ≤ α < 4  3 .In the intermediate cases 1 < α < 2, there is no positive results concerning the well-posedness for (1.1).Our main theorem is the following.Theorem 1.1.Assume that 1 ≤ α ≤ 2 and s > s α := 2 α − 3 4 .Then for every u 0 ∈ E s , there exists a positive time T = T ( u 0 E s ) and a unique solution u to (1.1) in the class C([−T, T ]; E s ) ∩ F s (T ) ∩ B s (T ).
Moreover, for any 0 < T ′ < T , there exists a neighbourhood U of u 0 in E s such that the flow map data-solution Remark 1.1.We refer to Section 2.2 for the definition of the functional spaces F s (T ) and B s (T ).
We discuss now some of the ingredients in the proof of Theorem 1.1.We will adapt the approach introduced by Ionescu, Kenig and Tataru [9] to our model (see also [6]- [12] for applications to other equations).It consists in an energy method combined with linear and nonlinear estimates in the short-time Bourgain's spaces F s (T ) and their dual N s (T ).The F s (T ) spaces enjoys a X s,1/2 -type structure but with a localization in small time intervals whose length is of order H 1− 2 α when the space frequency (ξ, µ) satisfies |ξ| α + µ 2 ∼ H.When deriving bilinear estimates in these spaces, one of the main obstruction is the strong resonance induced by the dispersive part of the equation.To overcome this difficulty, we will derive some improved Strichartz estimates for free solutions localized outside the critical region {2µ 2 = α(α + 1)|ξ| α }.Finally, we need energy estimates in order to apply the classical Bona-Smith argument (see [1]) and conclude the proof of Theorem 1.1.To derive such energy estimates, we are led to deal with terms of the form where P H localizes in the frequencies {|ξ| α + µ 2 ∼ H}.Unfortunately, in the twodimensional setting, we cannot put the x-derivative on the lower frequency term via commutators and integrations by parts without loosing a y-derivative.Therefore, we need to add a cubic lower-order term to the energy in order to cancel those bad interactions.
Assuming that s α < 1 2 , we may use the conservation laws (1.2) combined with the embedding E 1/2 ֒→ L 3 (R 2 ) to get an a priori bound of the E 1/2 -norm of the solution and then iterate Theorem 1.1 to obtain the following global well-posedness result.
Finally, as in the one dimensional case, we show that as soon as α < 2, the solution map S s T given by Theorem 1.1 is not of class C 2 for all s ∈ R.This implies in particular that the Cauchy problem for (1.1) cannot be solved by direct contraction principle.Theorem 1.2.Fix s ∈ R and 1 ≤ α < 2. Then there does not exist a T > 0 such that (1.1) admits a unique local solution defined on the interval [−T, T ] and such that the flow-map data-solution The rest of the paper is organized as follows: in Section 2, we introduce the notations, define the function spaces and state some associated properties.In Section 3, we derive Strichartz estimates for free solutions of (1.1).In Section 4 we show some L 2 -bilinear estimates which are used to prove the main short time bilinear estimates in Section 5 as well as the energy estimates in Section 6. Theorem 1.1 is proved in Section 7. We conclude the paper with an appendix where we show the ill-posedness result of Theorem 1.2.

Notations and functions spaces
2.1.Notations.For any positive numbers a and b, the notation a b means that there exists a positive constant c such that a ≤ cb.By a ∼ b we mean that a b and b a.Moreover, if γ ∈ R, γ+, respectively γ−, will denote a number slightly greater, respectively lesser, than γ.
The Fourier variables of (t, x, y) are denoted (τ, ξ, µ).Let U (t) = e t∂x(D α x −∂yy ) be the linear group associated with the free part of (1.1) and set Let h the partial derivatives of ω with respect to ξ : We define the set of dyadic numbers D = {2 ℓ , ℓ ∈ N}.If β ≥ 0 and H = 2 ℓ ∈ D, we will denote by ⌊H β ⌋ the dyadic number such that ⌊H β ⌋ ≤ H β < 2⌊H β ⌋.In other words we set ⌊H β ⌋ = 2 [βk] where [•] is the integer part.Let χ ∈ C ∞ 0 satisfies 0 ≤ χ ≤ 1, χ = 1 on [−4/3, 4/3] and χ(ξ) = 0 for |ξ| > 5/3.Let ϕ(ξ) = χ(ξ) − χ(2ξ) and for any N ∈ D \ {1}, define ϕ N (ξ) = ϕ(ξ/N ) and ϕ 1 = χ.For H, N ∈ D, we consider the Fourier multipliers P x N and P H defined as For N, H ∈ D \ {1}, let us define and ∆ H = {(ξ, µ) : h(ξ, µ) ∈ I H } .We also define P H = H1 H P H1 , P ≫H = Id−P H and P ∼H = Id−P H −P ≫H .We will use similarly the notation ϕ ≤ , ϕ ≥ ... Let η : R 4 → C be a bounded measurable function.We define the pseudoproduct operator Π η on S(R 2 ) 2 by This bilinear operator enjoys the symmetry property for any real-valued functions f, g, h ∈ S(R 2 ).This operator behaves like a product in the sense that it satisfies for any f, g ∈ S(R 2 ) where ∂ holds for Estimate (2.5) follows from (2.3), Plancherel's theorem and the fact that Its dual version N H is defined by the norm (2.7) Now if s ≥ 0, we define the global F s and N s spaces from their frequency localized version F H and N H by using a nonhomogeneous Littlewood-Paley decomposition as follows We define next a time localized version of those spaces.For T > 0 and Y = F s or Y = N s , the space Y (T ) is defined by its norm For s ≥ 0 and T > 0 we define the Banach spaces for the initial data E s by and their intersections are denoted by E ∞ = s≥0 E s .Finally, the associated energy spaces B s (T ) are endowed with norm 2.3.Properties of the function spaces.In this section, we state without proof some important results related to the short time function spaces introduced in the previous section.They all have been proved in different contexts in [9]- [12]- [6].
The F s (T ) and N s (T ) spaces enjoy the following linear properties.
Lemma 2.1.Let T > 0 and s ≥ 0. Then it holds that for all f ∈ F s (T ).
Proposition 2.1.Assume T ∈ (0, 1] and s ≥ 0. Then we have that for all u ∈ B s (T ) and f ∈ N s (T ) satisfying We will also need the following technical results.
Lemma 2.2.Let H, H 1 ∈ D be given.Then it holds that and Then it holds that and for all f such that F (f ) ∈ X H .

Strichartz estimates
For 1 ≤ α ≤ 2 we set B = α(α + 1)/2, and for δ > 0 small enough, let us define We also consider a function and The main result of this section is the following.
Remark 3.1.We notice that in the case α = 2 and θ = 1/2+, estimate (3.2) was already used in [16] and is a direct consequence of a more general theorem related to homogeneous polynomial hypersurfaces proved by Carbery, Kenig and Ziesler [2].However, this result does not apply as soon as α < 2 since the symbol ω defined in (2.1) is no more homogeneous.
To prove Proposition 3.1, we will need the following result.
Proof.First, recall that the semi-convergent integral I t may be understood as and ρ ± δ = ρ δ 1 R± .We are going to bound |I ± t |, uniformly in x, y and M .Let ε ∈ (0, 1) be a small number to be chosen later and define Then I ± t may be decomposed as We estimate I ± t,1 and rewrite it as (3.7) where the phase function ψ 1 is defined by ψ 1 (ξ) = xξ + tξ(|ξ| α + µ 2 ), and where Then we easily check that

and
(3.10) Using (3.8)-(3.9)-(3.10),an integration by parts yields Coming back to (3.7) we infer It remains to estimate we get Performing the change of variables u → |ξ| −α/2 u, a dilatation argument leads to where the new phase function ψ 2 is defined by We argue similarly to estimate Hence we have, (3.15) , which is acceptable as soon as |t| < N −(α+1) .Therefore we assume now that |t| ≥ N −(α+1) .Observe that since (3.13) and (3.15) also holds for Differentiating the phase function we get Let γ ∈ (0, 1) be a small parameter that we will choose later, and define We decompose J ± as From the definition of C γ , we have The Van der Corput lemma applies and provides (3.24) To estimate J ± 1 , we will take advantage of the first derivative of ψ 2 given by (3.20).Let ξ ∈ C γ .Then we easily see that Since we can always choose ε, γ > 0 small enough so that (1 Therefore J ± 1 is estimated thanks to (3.23)-(3.25)and integration by parts as follows as desired.Estimates for K are similar, since (3.23) is replaced with We obtain the bound On the other hand, we get from (3.1) that ).Thus, thanks to Young inequality and estimate (3.29), we infer , for any t ∈ R * .Therefore, by interpolation with the straightforward equality U (t)φ L 2 xy = φ L 2 we deduce that for any θ ∈ [0, 1), Remark that we exclude the case θ = 1 because the operator P A c δ is not continuous on L 1 (R 2 ).The previous estimate combined with the triangle inequality and Hardy-Littlewood-Sobolev theorem lead to (3.30) for all f ∈ S(R 3 ), where 1 q + 1 q ′ = 1 and 2 q = 1 − ε 2 .Estimate (3.1) is then obtained from (3.30) by the classical Stein-Thomas argument.
Corollary 3.1.Assume δ ∈ (0, 1), H, N ∈ D and f ∈ X H . Then for all s > −α/8, it holds that Proof.We apply Proposition 3.1 with θ = 1 2−ε and obtain for any φ ∈ L 2 (R 2 ).Setting f ♯ (θ, ξ, µ) = f (θ + ω(ξ, µ), ξ, µ) it follows then from Minkowski and Cauchy-Schwarz in θ that Interpolating this with the trivial bound We conclude this section by stating a global Strichartz estimate that will not be used in the proof of Theorem 1.1, but that may be of independent interest for future considerations.
Remark 3.2.It follows by applying estimate (3.32) with θ = 1/2 that Therefore, arguing as in the proof of Corollary 3.1 we infer that for all f ∈ X H such that supp Consequently, (3.31) can be viewed as an improvement of estimate (3.35) since outside the curves µ 2 = B|ξ| α , it allows to recover α 8 derivatives instead of 1 12 (α− 1 2 ) derivatives in L 4 .Assume that H i , N i , L i ∈ D are dyadic numbers and f i :

L 2 bilinear estimates
(2) Let us suppose that H min ≪ H max and Otherwise we have (3) If H min ∼ H max and f i are supported in D Hi,Ni,Li for i = 1, 2, 3, then Before proving Proposition 4.1 we give a technical lemma.
Proof.Without loss of generality, we may assume (4.9) Thus, it suffices to prove that (4.10) Thanks to (4.8) and (4.9), we have that This implies that On the other hand, using (4.11) again we infer Estimate (4.10) follows then by choosing Proof of Proposition 4.1.First we show part (1).We observe that (4.12) Therefore we can always assume that L 1 = L min .Moreover, let us define In view of the assumptions on f i , the functions f ♯ i are supported in the sets We also note that . Thus applying the Cauchy-Schwarz and Young inequalities in the θ variable we get 3) is deduced from (4.14) by applying the same arguments in the ξ, µ variables.
Next we turn to the proof of part (2).From (4.12), we may assume Indeed, if estimate (4.16) holds, let us define for θ 1 and θ 2 fixed.Hence, we would deduce applying (4.16) and the Cauchy-Schwarz inequality to (4.13) that which implies (4.4) and (4.5).To prove estimate (4.16), we apply twice the Cauchy-Schwarz inequality to get that , and noting that the Jacobi determinant satisfies , which lead to (4.16) after integrating in µ 2 .Now we show part (3) and assume that the functions f ♯ i are supported in the sets In order to simplify the notations, we will denote We split the integration domain in the following subsets: Then, if we denote by I j the restriction of I given by (4.13) to the domain D j , we have that Estimate for I 1 .From (4.12) we may assume L max = L 3 .Since H min ∼ H max , it follows that N min ∼ N max and ∼ N α+1 max in the region D 1 .We infer that I 1 is non zero only for L 3 N α+1 max and it suffices to show that (4.20) Arguing as in (4. , where J 2 is the restriction of the integral J defined by (4.15) to the domain D 2 .This leads to Estimate for I 3 .First we notice that in D 3 , we have Let 0 < δ ≪ 1 be a small positive number such that f (δ) = 1 1000 where f is defined in Lemma 4.1.We split again the integration domain D 3 in the following subsets: Then, if we denote by I j 3 the restriction of I 3 to the domain D j 3 , we have that Estimate for I 1 3 .We consider the following subcases.(1) Case {ξ 1 ξ 2 > 0 and µ 1 µ 2 > 0}.We define and denote by I 1,1 3 the restriction of I 1 3 to the domain D 1,1 3 .We observe from (4.19) that 3 .Therefore, it follows arguing exactly as in (4.20) that (4.21) and denote by I 1,2 3 the restriction of . Thus, arguing as in the proof of (4.16), we get that the restriction of J to D 1,2 3 satisfies min .(3) Case {ξ 1 ξ 2 < 0 and µ 1 µ 2 < 0}.We define and denote by I 1,3 3 the restriction of I 1 3 to the domain D 1,3 3 .We observe due to the frequency localization that there exists some 0 < γ ≪ 1 such that in D 1,3 3 .Indeed, if estimate (4.23) does not hold for all 0 < γ ≤ 1 1000 , then estimate (4.7) with f (δ) = 1 1000 would imply that which would be a contradiction since H min ∼ H max .Thus we deduce from (4.23) that . We can then reapply the arguments in the proof of (4.16) to show that (4.24) 3 and I 3 3 .The estimates for these terms follow the same lines as for I 1  3 .
Estimate for I 4 3 .Without loss of generality, we can assume that ζ 1 , ζ 2 ∈ R 2 \ A δ .Then we may take advantage of the improved Strichartz estimates derived in Section 3. We deduce from Plancherel's identity and Hölder's inequality that We conclude from Corollary 3.1 that which is acceptable since min .As a consequence of Proposition 4.1, we have the following L 2 bilinear estimates.Corollary 4.1.Assume that H i , N i , L i ∈ D are dyadic numbers and f i : R 3 → R + are L 2 functions for i = 1, 2, 3.
(1) If f i are supported in D Hi,∞,Li for i = 1, 2, 3, then Proof.Corollary 4.1 follows directly from Proposition 4.1 by using a duality argument.

Short time bilinear estimates
Proposition 5.1.
(1) If s > 1/4, T ∈ (0, 1] and u, v ∈ F s (T ), then ( We split the proof of Proposition 5.1 into several technical lemmas. Then, for all u H1 ∈ F H1 and v H2 ∈ F H2 . Proof.First observe from the definition of N H in (2.6) that (5.4) where for L > ⌊H β ⌋ and we define similarly g H2,L for L ≥ ⌊H β ⌋.Thus we deduce from (5.4) and the definition of X H that (5.5) where D H,∞,L is defined in (4.2).Here we use that since the sum for L < ⌊H β ⌋ appearing implicitly on the RHS of (5.4) is controlled by the term corresponding to L = ⌊H β ⌋ on the RHS of (5.5).Therefore, according to Corollary 2.1 and estimate (5.5) it suffices to prove that for all u H1 ∈ F H1 and v H2 ∈ F H2 .
Proof.Arguing as in the proof of Lemma 5.1, it is enough to prove that where f H1,N1,L1 and g H2,N2,L2 are localized in D Hi,Ni,Li , with L, L We observe from the definition of N H in (2.7) that (5.10) , where for L > ⌊H β 1 ⌋ and we define similarly g m H2,L for L ≥ ⌊H β 1 ⌋.Thus we deduce from (5.4) and the definition of X H that (5.11) Therefore, according to Lemma 2.2 and estimate (5.11) it suffices to prove that (5.12) 1 ⌋ in order to prove (5.9).As in the proof of Lemma 5.1, estimate (5.12) follows from (4.26)-(4.27)and the fact that L max ∼ max(L med , |Ω|).
Proof.Arguing as in the proof of Lemma 5.1, it is enough to prove that (5.14 where f H1,L1 and g H2,L2 are localized in D Hi,∞,Li , with L 1 , L 2 ∈ D, which is a direct consequence of estimate (4.25).
Proof of Proposition 5.1.We only prove part (1) since the proof of estimate (5.2) follows the same lines.We choose two extensions u and v of u and v satisfying (5.15) We have from the definition of N s (T ) and Minkowski inequality that .

Let us denote
Due to the frequency localization, we have j=1 H∈D H 2s (H1,H2)∈Aj To handle the sum S 1 , we use estimate (5.9) to obtain that (5.17) Similarly we deduce by symmetry that (5.19) Next it follows from estimate (5.7) and Cauchy-Schwarz inequality that (5.20) Finally it is clear from estimate (5.13) that (5.21) Therefore we conclude the proof of (5.1) gathering (5.16)-(5.21).

Energy estimates
The aim of this section is to derive energy estimates for the solutions of (1.1) and the solutions of the equation satisfied by the difference of two solutions of (1.1).In order to simplify the notations, we will instead derive energy estimates on the solutions v of the more general equation (6.1) ).Here we assume c 1 , c 2 ∈ R * and that all the functions u, v, u 1 , u 2 are real-valued.
Let us define our new energy by for any H ∈ D \ {1} and where η is a bounded function uniformly in H that will be fixed later.Finally we set (6.4) Note that for the integral in (6.3) to be non zero, the first occurrence of the function v must be localized in ∆ ∼H .
are solutions of (6.1)-(6.2),we have that . and . Moreover in the case where u = v it holds that The following result will be of constant use in the proof of Proposition 6.1.Lemma 6.2.Assume that T ∈ (0, 1], H 1 , H 2 , H 3 ∈ D and that u i ∈ F Hi for i = 1, 2, 3. (1) In the case H min ≪ H max it holds that (6.11) ( Observe that in the right-hand side of (6.11), we have H Proof.From (2.3) we may always assume H 1 ≤ H 2 ≤ H 3 .We first prove estimate (6.11).Let γ : R → [0, 1] be a smooth function supported in [−1, 1] with the property that m∈Z Then it follows that (6.13) Now we observe that the sum on the right-hand side of (6.13) is taken over the two disjoint sets To deal with the sum over A, we set , for each m ∈ A and i ∈ {1, 2, 3}.Therefore, we deduce by using Plancherel's identity and estimates (4.4)-(4.5)that m∈A This implies together with Corollary 2.1 that (6.15) Now observe that #B 1.We set Then, we deduce using again (4.4)-(4.5)as well as Lemma 2.3 that We deduce estimate (6.11) gathering (6.13)-(6.16).Finally, the proof of (6.12) follows the same lines by using (4.6) instead of (4.4)- (4.5).We also need to interpolate (4.6) with (4.3) to get for ε ∈ (0, 1).With this estimate in hand, we are able to control the contribution of the sum in the region B.
Proof of Proposition 6.1.Let v, u, u 1 , u 2 ∈ C([−T, T ], E ∞ ) be solutions to (6.1)-(6.2).We choose some extensions v, u, u 1 , u 2 of v, u, u 1 , u 2 respectively on R 3 satisfying v F s v F s (T ) , u F s u F s (T ) and u i F s u i F s (T ) for i = 1, 2. We fix s > s α and set σ ∈ {0, s}.Then, for any H ∈ D \ {1}, we differentiate E H (v) with respect to t and deduce using (6.1)-(6.2) as well as (2.4) that (6.17) with and Without loss of generality, we can assume that 0 < t H < T .Therefore we obtain integrating (6.17) between 0 and t H that (6.18) Using Hölder and Bernstein inequalities, the first term in the right-hand side of (6.18) is easily estimated by (6.19) Next we estimate the second term in the right-hand side of (6.18).
Estimates for the cubic terms.By localization considerations, we obtain Note that in the case where u = v, we have Clearly we get by estimate (6.12) that Similarly, we get applying estimate (6.11) that From this and Cauchy-Schwarz inequality we infer (6.21) In the case u = v we estimate I Therefore, it remains to estimate Using a Taylor expansion of ψ H we may decompose I 1 H (v) as where η i , i = 1, 2, 3 are bounded uniformly in H and defined by To estimate the contribution of I 11 H (v), we integrate by parts and use (6.11) to obtain (6.23) Estimates for I 12 H (v) and I 13 H (v) are easily obtained thanks to (6.11): Combining estimates (6.23)-( 6.24) we infer (6.25) Note that due to the lack of derivative on the lowest frequencies term P ≪H u, Lemma 6.2 does not permit to control the term I 14 H (v) without loosing a H factor.This is why we modify the energy by adding the cubic term in (6.3).Let us rewrite L H (v) as After a few integrations by parts, we obtain thanks to (2.4) that Choosing η = − 1 c1 η 3 , a cancellation occurs and we get In the case u = v, it suffices to set η = − 1 2c1 η to obtain 2I )(H It follows that (6.26) Finally to deal with L 2 H (v), we integrate by parts and use that We deduce Noticing that η is bounded on ∆ ≪H × ∆ ∼H we easily get from Lemma 6.2 that (6.27) Gathering (6.20)-(6.27)we conclude (6.28) Estimates for the fourth order terms.We get using (2.5) and Hölder inequality that Noticing that which combined with (6.18)-(6.19)and (6.28) concludes the proof of Proposition 6.1.
The proof of Theorem 1.1 closely follows the proof of existence and uniqueness given in [12].We start with a well-posedness result for smooth initial data u 0 in E ∞ = H ∞ (R 2 ).This result can be easily obtained with a parabolic regularization of (1.1) by adding an extra term −ε∆u and going to the limit as ε → 0. We refer the reader to [10] for more details.
Theorem 7.1.Assume that u 0 ∈ E ∞ .Then there exist a positive time T and a unique solution u ∈ C([−T, T ]; E ∞ ) of (1.1) with initial data u(0, .)= u 0 (.).Moreover T = T ( u 0 E 3 ) is a nonincreasing function of u 0 E 3 and the flow-map is continuous.

7.1.
A priori estimates for E ∞ solutions.Theorem 7.2.Assume that s > s α .For any M > 0 there exists T = T (M ) > 0 such that, for all initial data u 0 ∈ E ∞ satisfying u 0 E s ≤ M , the smooth solution u given by Theorem 7.1 is defined on [−T, T ] and moreover To obtain Theorem 7.2 we will need the following result proved in [12]. .
We prove now that for T < T ′ small enough, we also have (7.14)u i,λ F s ( T ) ε .
Since u i,λ F s (T ) ≤ C, we can always find H ∈ D such that (7.15) P >H u i,λ F s ( T ) ≤ P >H u i,λ F s (T ) ≤ ε , i = 1, 2.
Moreover since u N s ( T ) ≤ C u L 2 T E s , we infer from (2.9), Hölder inequality and the Sobolev embedding E s ֒→ H by choosing T small enough.Gathering estimates (7.13), (7.15) and (7.20), we thus obtain that the smallness condition (7.8) holds, which shows that u 1 = u 2 on [− T , T ] (since (7.9) holds).Using the same argument a finite number of time we obtain that u 1 = u 2 on [−T ′ , T ′ ] and so on [−T, T ] by dilatation.7.3.Existence.Let s α < s < 3 and u 0 ∈ E s .By scaling considerations we can always assume that u 0 ∈ B s (ε).Following [12] we are going to use the Bona-Smith argument to obtain the existence of a solution u with u 0 as initial data.
1 , L 2 ≥ ⌊H β ⌋ and N, N 1 , N 2 H 1/α .Observe that the sums over N, N 1 , N 2 are easily controlled by log(H 1/α ) H 0+ .Using that 1 − 1 α ≥ 0 and 1 α − β 2 − 1 2 = 0, this is a consequence of estimate (4.28) in the case L = L min or L med ∼ L max .Otherwise, we have L max ∼ |Ω| H 1+ 1 α so that the sum over L is bounded by H 0+ and (5.8) still holds.Lemma 5.3 (high × high → low).Assume that H, H 1 , H 2 ∈ D satisfy H ≪ H 1 ∼ H 2 .Then, 2 max P ∼H P x N1 u FH P ∼H P x N2 v FH P H P x N3 v FH H sα+ P ∼H u FH P ∼H v FH P H v FH , ∼H u FH P H1 v FH P H v FH H1≪HH sα+ P 2, for all β ≥ s > s α .Using(7.5)with β = s, a continuity argument and that lim By estimate (2.9) together with the short time estimate (5.1) it follows then that for u 0 E s ≤ ε, for all β ≥ s as soon as u 0 E s ≤ ε.Using this above estimate with β = 3 we can apply Theorem 7.1 a finite number of time and thus extend the solution u of (1.1) on the time interval [−1, 1].7.2.L 2 -Lipschitz bounds and uniqueness.Let us consider two solutions u 1 and u 2 defined on [−T, T ], with initial data ϕ 1 and ϕ 2 and assume moreover that t→0Λ s t (u) = 0, we have Λ s T (u) ε as soon as u 0 E s ≤ ε.