Well-posedness for the fifth-order KdV equation in the energy space

We prove that the initial value problem (IVP) associated to the fifth order KdV equation {equation} \label{05KdV} \partial_tu-\alpha\partial^5_x u=c_1\partial_xu\partial_x^2u+c_2\partial_x(u\partial_x^2u)+c_3\partial_x(u^3), {equation} where $x \in \mathbb R$, $t \in \mathbb R$, $u=u(x,t)$ is a real-valued function and $\alpha, \ c_1, \ c_2, \ c_3$ are real constants with $\alpha \neq 0$, is locally well-posed in $H^s(\mathbb R)$ for $s \ge 2$. In the Hamiltonian case (\textit i.e. when $c_1=c_2$), the IVP associated to \eqref{05KdV} is then globally well-posed in the energy space $H^2(\mathbb R)$.


Introduction
Considered here is the initial value problem (IVP) associated to the fifth-order Korteweg-de Vries equation where x ∈ R, t ∈ R, u = u(x, t) is a real-valued function and α, c 1 , c 2 , c 3 are real constants with α = 0. Such equations and its generalizations (1.2) ∂ t u − α∂ 5 x u + β∂ 3 x u = c 0 u∂ x u + c 1 ∂ x u∂ 2 x u + c 2 ∂ x (u∂ 2 x u) + c 3 ∂ x (u 3 ) arise as long-wave approximations to the water-wave equation. They have been derived as second-order asymptotic expansions for unidirectional wave propagation in the so-called Boussinesq regime (see Craig, Guyenne and Kalisch [5], Olver [29] and the references therein), the first order expansions being of course the Kortewegde Vries (KdV) equation, (1.3) ∂ t u + β∂ 3 x u = c 0 u∂ x u. The equation in (1.1) was also proposed by Benney [2] as a model for interaction of short and long waves.
When c 1 = c 2 , the Hamiltonian as well as the quantity are conserved by the flow of (1.1). Indeed, it is easy to check that Thus the equation in (1.1) has the form ∂ t u = ∂ x grad H(u), so that d dt H(u) = grad H(u), ∂ t u L 2 = grad H(u), ∂ x grad H(u) L 2 = 0.
Moreover in the special case where c 2 = c 1 = −10α and c 3 = 10α, the equation in (1.1) is the equation following KdV in the KdV hierarchy discovered by Lax [30] and writes in the case α = 1 Therefore equation (1.6) is completely integrable and possesses an infinite number of conservation laws. We refer to the introductions in [12,33,34] for more details on this subject. Our purpose is to study the IVP (1.1) in classical L 2 -based Sobolev spaces H s (R). We shall say that the IVP is locally (resp. globally) well-posed in the function space X if it induces a dynamical system on X by generating a continuous local (resp. global) flow.
First, it is worth mentioning that without dispersion (i.e. when α = 0) and when c 1 = 0 or c 2 = 0, the IVP (1.1) is likely to be ill-posed in any H s (R) (see the comments in the introduction of [33]). This is in sharp contrast with the KdV equation. Indeed, when β = 0 in (1.3), we obtain the Burgers equation, which is still well-posed in H s (R) for s > 3/2 by using standard energy methods. However, the direct energy estimate for equation (1.1) (after fixing c 3 = 0 for simplicity) gives only Observe that the last term on the right-hand side of (1.7) has still higher-order derivatives and cannot be treated by using only integration by parts. To overcome this difficulty, Ponce [33] used a recursive argument based on the dispersive smoothing effects associated to the linear part of (1.1), combined to a parabolic regularization method, to establish that the IVP (1.1) is locally well-posed in H s (R) for s ≥ 4. Later, Kwon [25] improved Ponce's result by proving local well-posedness for (1.1) in H s (R) for s > 5/2. The main new idea was to modify the energy by adding a correctional lower-order cubic term to cancel the last term on the righthand side of (1.7). Note that he also used a refined Strichartz estimate derived by chopping the time interval in small pieces whose length depends on the spatial frequency. This estimate was first established by Koch and Tzvetkov [24] (see also Kenig and Koenig [18] for an improved version) in the Benjamin-Ono context.
On the other hand, it was proved 1 by the second author in [32], by using an argument due to Molinet, Saut and Tzvetkov for the Benjamin-Ono equation [28], that, in the case c 2 = 0, the flow map associated to (1.1) fails to be C 2 in H s (R), for any s ∈ R. This result was improved by Kwon [25] who showed that the flow map fails to be even uniformly continuous in H s (R) when s > 5 2 (and s > 0 in the completely integrable case). Those results are based on the fact that the dispersive smoothing effects associated to the linear part of (1.1) are not strong enough to control the high-low frequency interactions in the nonlinear term ∂ x (u∂ 2 x u). As a consequence, one cannot solve the IVP (1.1) by a Picard iterative method implemented on the integral equation associated to (1.1) for initial data in any Sobolev space H s (R) with s ∈ R.
However, the fixed point method may be employed to prove well-posedness for (1.1) in other function spaces. For example in [20,21], Kenig, Ponce and Vega proved that the more general class of IVPs (1.8) ∂ t u + ∂ 2j+1 x u + P (u, ∂ x u, . . . , ∂ 2j x u), x, t ∈ R, j ∈ N u(0) = u 0 , where P : R 2j+1 → R (or P : C 2j+1 → C) is a polynomial having no constant or linear terms, is well-posed in weighted Sobolev spaces of the type H k (R) ∩ H l (R; x 2 dx) with k, l ∈ Z + , k ≥ k 0 , l ≥ l 0 for some k 0 , l 0 ∈ Z + . We also refer to [32] for sharper results in the case of small initial data and when the nonlinearity in (1.8) is quadratic. Recently, Grünrock [12], respectively Kato [16], used variants of the Fourier restriction norm method to prove well-posedness in H s r (R) for 1 < r ≤ 4 3 and s > 1 4 + 3 2r ′ , respectively in H s,a (R) for s ≥ max{− 1 4 , −2a − 2} with − 3 2 < a ≤ − 1 4 and (s, a) = (− 1 4 , − 7 8 ). The spaces H s r (R) and H s,a (R) are respectively defined by the norms ϕ H s r = ξ s ϕ L r ′ with 1 r + 1 r ′ = 1 and ϕ H s,a = ξ s−a |ξ| a ϕ L 2 . Nevertheless, the L 2 -based Sobolev spaces H s (R) remain the natural 2 spaces to study well-posedness for the fifth order KdV equation. Our main result states that the IVP (1.1) is locally well-posed in H s (R) for s ≥ 2. 1 Strictly speaking the result was proved only in the case where c 3 = 0, but as observed in the introduction of [12], the cubic term ∂x(u 3 ) in (1.1) is well behaved and no cancellations occur, so that the proof remains true even when c 3 = 0. Remark 1.6. As a byproduct of the proof of Theorem 1.1, we obtain a priori estimates on smooth solutions of (1.1) in H s (R) for s ≥ 5 4 (see Proposition 6.2 below). In other word, the flow map data-solutions in H ∞ (R) satisfies for any s ≥ 5 4 and where T only depends on u 0 H s . However, we were not able to prove well-posedness at this level of regularity.
In the Hamiltonian case, the conserved quantities H and M defined in (1.4)-(1.5) provide a control on the H 2 -norm and allow to prove that the IVP (1.1) is globally well-posed in H 2 (R).  Remark 1. 10. In his study of stability of solitary waves for Hamiltonian fifth-order water wave models of the form (1.2) with quadratic nonlinearities 4 , Levandosky assumed well-posedness in H 2 (R) (c.f. Assumption 1.1 in [26]). Therefore, Corollary 1.7 provides an affirmative answer to this issue. We also refer to [1,27] for further results on stability/instability of such fifth-order water wave models.
We now discuss the main ingredients in the proof of Theorem 1.1. We follow the method introduced by Ionescu, Kenig and Tataru [15] in the context of the KP1 equation, which is based on the dyadic Bourgain's spaces F s α and their dual N s α , defined in Subsection 2.2. We refer to [4,23] for previous works using similar spaces to prove a priori bounds for the 1 D cubic NLS at low regularity and also to [9,10,31] for applications to other dispersive equations.
The F s α spaces enjoy a X s,b -type structure but with a localization in small time dependent intervals whose length is of order 2 −αk when the spatial frequency of the function is localized around 2 k . This prevents the time frequency modulation 5 |τ − w(ξ)| to be too small, which allows for suitable α, α = 2 in our case, to prove a bilinear estimate of the form (c.f Proposition 4.1 for a precise statment ) , as soon as s > 1. Of course 6 , we cannot conclude directly by using a contraction argument since the linear estimate estimate (1.13) u F s requires the introduction of the energy norm u B s (T ) , instead of the usual H s -norm of the initial data u 0 H s , in order to control the small time localization appearing in the F s α -structure. Therefore it remains to derive the frequency localized energy estimate x is small. The main new difficulty in our case is that after using suitable frequency localized commutator estimates, we are not able to handle directly the remaining lower-order terms (see Lemma 5.6 and Remark 5.7 below). This is somehow the price to pay for the choice of α = 2 which enabled to derive the bilinear estimate (1.12). Then, we modify the energy by adding a cubic lower-order term to u 2 B s (T ) in order to cancel those terms. This can be viewed as a localized version of Kwon's argument in [25].
We deduce the a priori bound (1.11) by combining (1.12)-(1.14) and using a scaling argument. To finish the proof of Theorem 1.1, we apply this method to the difference of two solutions. However, due to the lack of symmetry of the new equation, we only are able to prove the corresponding energy estimate for s ≥ 2. Finally, we conclude the proof by adapting the classical Bona-Smith argument [3].
Around the time when we completed this work, we learned that Guo, Kwak and Kwon [11] had also worked on the same problem and obtained the same results as ours (in Theorem 1.1 and Proposition 6.2). They also used the short-time X s,b method. However, instead of modifying the energy as we did, they put an additional weight in the X s,b structure of the spaces in order to derive the key energy estimates.
The rest of the paper is organized as follows: in Section 2, we introduce the notations, define the function spaces and prove some of their basic properties as well the main linear estimates. In Section 3, we derive the L 2 bilinear and trilinear estimates, which are used to prove the bilinear estimates in Section 4 and the energy estimates in Section 5. The proof of Theorem 1.1 is given in Section 6. We conclude the paper with an appendix explaining how to treat the cubic term ∂ x (u 3 ), which we omit in the previous sections to simplify the exposition.
2. Notation, function spaces and linear estimates 2.1. 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 denote a ∼ b when a b and b a. Moreover, if α ∈ R, α + , respectively α − , will denote a number slightly greater, respectively lesser, than α.
For a 1 , a 2 , a 3 ∈ R, it will be convenient to define the quantities a max ≥ a med ≥ a min to be the maximum, median and minimum of a 1 , a 2 and a 3 respectively. For a 1 , a 2 , a 3 , a 4 ∈ R, we define the quantities a max ≥ a sub ≥ a thd ≥ a min to be the maximum, sub-maximum, third-maximum and minimum of a 1 , a 2 , a 3 and a 4 respectively. Usually, we use k i and j i to denote integers and N i = 2 ki , L i = 2 ji to denote dyadic numbers.
For u = u(x, t) ∈ S(R 2 ), Fu = u will denote its space-time Fourier transform, whereas F x u = (u) ∧x , respectively F t u = (u) ∧t , will denote its Fourier transform in space, respectively in time. Moreover, we generally omit the index x or t when the function depends only on one variable. For s ∈ R, we define the Bessel and Riesz potentials of order −s, J s x and D s x , by The unitary group e t∂ 5 x associated to the linear dispersive equation where w(ξ) = ξ 5 . For k ∈ Z + , let us define if k ≥ 1 and I 0 = ξ ∈ R : |ξ| ≤ 2 . Throughout the paper, we fix an even smooth cutoff function η 0 : R → [0, 1] supported in [−8/5, 8/5] and such that η 0 is equal to 1 in [−5/4, 5/4]. For k ∈ Z ∩ [1, +∞), we define the functions η k and η ≤k respectively by Then, (η k ) k≥0 is dyadic partition of the unity satisfying supp η k ⊂ I k . Let ( η k ) k≥0 be another nonhomogeneous dyadic partition of the unity satisfying supp η k ⊂ I k and η k = 1 on supp η k .
Finally, for k ∈ Z ∩ [1, +∞), let us define the Fourier multiplier P k , P ≤0 and P ≤k by Then it is clear that P ≤0 + +∞ k=1 P k = 1. Often, when there is no risk of confusion, we also denote P 0 = P ≤0 .

2.2.
Function spaces. For 1 ≤ p ≤ ∞, L p (R) is the usual Lebesgue space with the norm · L p , and for s ∈ R, the Sobolev spaces H s (R) is defined via its usual norm φ H s = J s x φ L 2 . Let f = f (x, t) be a function defined for x ∈ R and t in the time interval [−T, T ], with T > 0 or in the whole line R. Then if X is one of the spaces defined above, we define the spaces L p T X x and L p t X x by the norms when 1 ≤ p < ∞, with the natural modifications for p = ∞.
We will work with the short time localized Bourgain spaces introduced in [15]. First, for k ∈ Z + , we introduce the l 1 -Besov type space X k of regularity 1/2 with respect to modulations, Let α ≥ 0 be fixed. For k ∈ Z + , we introduce the space F k,α possessing a X kstructure in short time intervals of length 2 −αk , Its dual version N k,α is defined by Now for s ∈ R + , we define the global F s α and N s α spaces from their frequency localized versions F k,α and N k,α , by using a nonhomogeneous Littlewood-Paley decomposition as follows We also define a localized (in time) version of those spaces. Let T be a positive time and Then, Finally for s ∈ R + and T > 0, we define the energy space B s (T ) by 2.3. First properties. Following [15], we state some important properties of the F s α (T ) spaces. First, we show that F s α (T ) ֒→ L ∞ ([−T, T ]; H s (R)).
Lemma 2.1. Let T > 0, s ∈ R + and α ∈ R + . Then it holds that Proof. Let f ∈ F s α (T ). We choose f ∈ F s α such that The Fourier inversion formula gives that On the other hand, the definition X k in (2.4) and the Cauchy-Schwarz inequality in τ implies that for all φ ∈ X k . Therefore, it is deduced from (2.5), (2.13) and (2.14) that for all k ∈ Z + . Then, estimate (2.10) follows gathering (2.11), (2.12), (2.15) and taking the supreme over t ∈ [−T, T ].
Then, we derive an important property involving the space X k (see [15] Proof. We fixl = [αl]. We begin proving estimate (2.16). Following [31], we use that (η k ) k≥0 is dyadic partition of the unity and the Cauchy-Schwarz inequality in τ ′ to get that Now, we get trivially that which concludes the proof of (2.16) recalling the definition of the space X k in (2.4). Next, we turn to the proof of estimate (2.17). The mean-value theorem yields which implies that where and Applying Young's theorem on convolution (L 2 To deal with II b we just proceed as in the proof of estimate (2.16) and obtain that In the case where j ≥ q + 5, we have that In the case where q ≥ j − 4, we get that Then, after summing in j, we deduce that in both cases Estimate (2.17) follows gathering (2.18)-(2.20), which concludes the proof of Lemma 2.2.
Corollary 2.3. Let k ∈ Z + , α ≥ 0,t ∈ R and γ ∈ S(R). Then it holds that Proof. Since γ ∈ S(R 2 ), we have that Therefore estimate (2.21) follows by using the definition of X k and applying estimates (2.16)-(2.17) to the right-hand side of (2.22).
Corollary 2.4. Let k ∈ Z + , α ≥ 0,t ∈ R and γ ∈ S(R). Then it holds that Proof. We have that (2.24) We treat the first term on the right-hand side of (2.24) by using Lemma 2.2 as in the proof of Corollary 2.3 and the second one by using Lemma 2.2 and duality. This implies estimate (2.23).
Remark 2.5. For s ∈ R + , the classical dyadic Bourgain space X s, 1 2 ,1 (introduced for instance in [35]) is defined by the norm Thus, if f ∈ X s, 1 2 ,1 , one deduce after applying estimate (2.21) to each P k f , taking the supreme in t and summing in k that f F s α f X s, 1 2 ,1 , for any α ≥ 0. In other words, we have that X s, 1 2 ,1 ֒→ F s α ֒→ L ∞ (R; H s (R)).
More generally for any k ∈ Z + and α ≥ 0, we define the set S k,α of k-acceptable time multiplication factors (c.f. [15]) as Corollary 2.6. Let k ∈ Z + , α ≥ 0 and m k ∈ S k,α . Then it holds that Proof. We prove estimate (2.25). The proof of estimate (2.26) would follow in a similar way. Arguing as in the proof of Corollary 2.3 it suffices to prove It follows from the definition of the Fourier transform that By using again basic properties of the Fourier transform and the Leibniz rule, we deduce that (2.29) Estimates (2.28)-(2.29) and the definition of S k,α imply estimate (2.27) which concludes the proof of Corollary 2.6.
The next Corollary of Lemma 2.2 will be useful in the proof of the bilinear and energy estimates (c.f. Sections 4 and 5).
Corollary 2.7. Let α ≥ 0,t ∈ R and l, k ∈ Z + be such that l + 5 ≥ k. Then it holds that Proof. Observe that Moreover, it follows from Corollary 2.6 that , since it allows to consider only regions where the modulation |τ − w(ξ)| is not too small, and therefore to avoid the regions giving troubles in the low-high frequency interactions (c.f. [32]).
2.4. Linear estimates. In this subsection, we derive the linear estimate associated to the spaces F s α (T ) (c.f. [15]). Proposition 2.9. Assume s ∈ R + , α > 0 and T ∈ (0, 1]. Then we have that Remark 2.10. Observe that, when working in the classical Bourgain space X 0, 1 2 ,1 (T ) defined in Remark 2.5, one would obtain an estimate of the form Here, we need to introduce the energy norm u B s (T ) instead of u(0) H s , since we are working on very short time intervals, whose length depends on the spatial frequency.
We first derive a homogeneous and a nonhomogeneous linear estimate in the spaces X k .
Lemma 2.11. [Homogeneous linear estimate] Let α ≥ 0 and k ∈ Z + . Then it holds that Proof. A direct computation shows that Thus, it follows from the definition of X k and Plancherel's identity that Moreover, it is clear since η 0 ∈ S(R) that for all f such that supp F(f ) ∈ I k × R.
On the one hand, we deduce from Lemma 2.2 that (2.39) On the other hand, it follows arguing as in the proof of Lemma 2.11 and using estimate (2.14) that (2.40) Finally, we conclude the proof of Proposition 2.12 gathering (2.37)-(2.40).
A proof of Proposition 2.9 is now in sight.
Then, it follows from (2.26) and the definition of θ that Moreover, for all k ∈ Z + , we also extend P k u on R 2 , by defining u k (t) as Next, we show that Thus, if t k > T , we get, We could argue similarly for t < T , which implies estimate (2.43). Now we fix t k ∈ [−T, T ]. Observe that and by the Duhamel principle, where m k ∈ S k,α . Thus, we deduce from estimates (2.25), (2.34) and (2.36) that which implies estimate (2.32) after taking the supreme in t k ∈ [−T, T ], summing over k ∈ Z + and using (2.26), (2.41)-(2.43).

Strichartz estimates.
We recall the Strichartz estimates associated to {e t∂ 5 x } proved by Kenig, Ponce and Vega in [19].
As a consequence, we obtain a Strichartz estimate in the context of the Bourgain spaces F s α (T ). Corollary 2.14. Assume 0 < T ≤ 1, α ≥ 0 and ǫ > 0. Then, it holds that Then we deduce using the Sobolev embedding W ǫ ′ ,r (R) ֒→ L ∞ (R), the square function theorem and Minkowski's inequality that where ǫ ′ and r(> 1/ǫ ′ ) will be chosen later. Therefore, according to the definition of F ǫ α in (2.7), it suffices to prove that for all k ≥ 0 in order to prove estimate (2.45). Indeed, it is enough then to choose r and ǫ ′ such that rǫ ′ > 1 and ǫ ′ + 3−α 2r < ǫ. Next, we prove estimate (2.47). For Then, we deduce applying Hölder's inequality in time that (2.48) Due to the Fourier inversion formula, we have that Thus, Minkowski's inequality, estimate (2.44), Plancherel's identity and the Cauchy-Schwarz inequality in q imply that Then, we observe that which together with (2.49) and the definition of F k,α in (2.5) implies that Finally, we deduce combining (2.48) and (2.50) that Next, we derive a bilinear Strichartz estimate for the group {e t∂ 5 x }, which is an extension of the one proved in [13] for the Airy equation (see also Lemma 3.4 in [14] for the dispersion generalized Benjamin-Ono equation). Let ζ ∈ C ∞ be an even function such that ζ | [−1,1] = 0, ζ | R\[−2,2] = 1 and 0 ≤ ζ ≤ 1. We define |x| 1 = ζ(x)|x|.
Lemma 2.15. For s ∈ R, we define the bilinear operator I s by Then, it holds that for any u 1 , u 2 ∈ L 2 (R).
For k ∈ Z + and j ∈ Z + , let us define D k,j by We state a useful lemma (see also Lemma 2.3 in [7]).
Lemma 3.2. Assume that k 1 , k 2 , k 3 ∈ Z + , j 1 , j 2 , j 3 ∈ Z + and f i : (a) Then it follows that If moreover k min ≥ 1, then (3.8) In all the others cases, we have that (c) In the case |k min − k max | ≤ 10, k min ≥ 10, then we have that Proof. First, we begin with the proof of item (a). We observe that . Therefore, we can always assume that j 1 = j min . Moreover, let us define f ♯ i (ξ, θ) = f i (ξ, θ + w(ξ)), for i = 1, 2, 3. In view of the assumptions on f i , the functions f ♯ i are supported in the sets D ♯ ki,ji = (ξ, θ) : ξ ∈ I ki and |θ| ≤ 2 ji .
We also note that f i L 2 = f ♯ i L 2 . Then, it follows changing variables that where . Thus, it follows by applying the Cauchy-Schwarz and Young inequalities in the θ variables that (3.13) Estimate (3.6) is deduced from (3.13) by applying the same arguments in the ξ variables.
Next we turn to the proof of item (b). According to (3.11), we can assume that j 3 = j max . Moreover, it is enough to consider the two cases k min = k 2 and k min = k 3 (since by symmetry the case k min = k 1 is equivalent to the case k min = k 2 ).
We prove estimate (3.9) in the case j 3 = j max and k min = k 2 . It suffices to prove that if g i : R → R + are L 2 functions supported in I ki for i = 1, 2 and g : Indeed, if estimate (3.15) holds, let us define , for θ 1 and θ 2 fixed. Hence, we would deduce applying (3.15) and the Cauchy-Schwarz inequality to (3.12) that which is estimate (3.9) in this case. To prove estimate (3.15), we apply twice the Cauchy-Schwarz inequality to get that We observe that since 2 k1 ∼ 2 kmax by the frequency localization. Then, the change of variables (3.15), which concludes the proof of estimate (3.9) in this case.
To prove estimate (3.8) in the case (k min , j max ) = (k 3 , j 3 ) and k 3 ≥ 1, we observe arguing as above that it suffices to prove that where J(g 1 , g 2 , g) is defined in (3.14). First, we change variables ξ ′ 1 = ξ 1 and The Cauchy-Schwarz inequality implies that We compute that since |ξ ′ 1 | ∼ 2 kmax and |ξ ′ 2 | ∼ 2 kmin due to the frequency localization. Therefore estimate (3.18) is deduced by performing the change of variables On the other hand, by writing, and arguing as in (3.16), we get estimate (3.7) in the case (k min , j max ) = (k 3 , j 3 ). Estimate (3.10) is stated in Lemma 2.3 (c) of [7] and its proof follows closely the one for the dispersion generalized BO in [9]. However, for sake of completeness we will derive it here. According to (3.11), we may assume that j max = j 3 . Furthermore, we have following (3.12) that where and R is a positive number which will be chosen later. First we prove that which would imply after interpolating with estimate (3.6). To prove (3.21), we argue as for (3.9), so that it suffices to prove where J(g 1 , g 2 , g) is defined as in (3.15). By symmetry, we can always assume that |ξ 1 | ≤ |ξ 2 |. We apply twice the Cauchy-Schwarz inequality and perform the change due to the frequency localization and the restriction ξ ′ 1 · ξ ′ 2 ≤ 0 (which is a consequence of the assumptions ξ 1 · ξ 2 < 0 and |ξ 1 | ≤ |ξ 2 |). Therefore, the change of (3.23). To deal with I 2 , we get as in (3.13) that Then, we obtain by letting (ξ ′ 1 , ξ ′ 2 ) = (ξ 1 − ξ 2 , ξ 2 ) and applying twice the Cauchy-Schwarz inequality that (3.25) Next, we observe that in the region R 3 , since R will be chosen large enough. Thus, the Cauchy-Schwarz inequality implies that where the definition of |·| 1 is given just before Lemma 2.15. By Plancherel's identity, the L 2 -norm of the integral on the right-hand side of (3.26) is equal to This implies after changing variables τ i = θ i +w(ξ i ) for i = 1, 2 and using Minkowski's inequality that where the bilinear operator I 2 is defined in Lemma 2.15. Therefore, we deduce from estimate (2.51) and the Cauchy-Schwarz inequality that (3.27) Finally, we conclude estimate (3.10) gathering estimates (3.20), (3.22), (3.25), (3.27) and choosing R = 2 −3kmax/2 2 j med /2 . This finishes the proof of Lemma 3.2.
As a consequence of Lemma 3.2, we have the following L 2 bilinear estimates.
(a) Then it follows that If moreover k min ≥ 1, then In all the others cases, we have that Proof. Corolloray 3.3 follows directly from Lemma 3.2 by using a duality argument.

34)
and In all the others cases, we have that (3.36) Proof. Estimate (3.33) can be proved exactly as estimate (3.6). To prove part (b), we follow closely the arguments of Guo for the mBO equation [10]. Let us define Observe that . Therefore, we can always assume that In view of the assumptions on f i , the functions f ♯ i are supported in the sets D ♯ ki,ji = (ξ, θ) : ξ ∈ I ki and |θ| ≤ 2 ji . We also note that f i L 2 = f ♯ i L 2 . Then, it follows changing variables that (3.39) Since k thd ≤ k max − 5 by hypothesis, we always have that k max ∼ k sub . Thus, we only need to treat the following cases: k 4 ∼ k max , k 4 = k thd and k 4 = k min . Case k 4 ∼ k max . By symmetry, we can assume that k 1 ≤ k 2 ≤ k 3 ≤ k 4 in this case. For g i : R → R + , L 2 functions supported in I ki for i = 1, 2, 3 and g : Then, arguing as in (3.41), it suffices to show that in order to prove (3.36) in this case. To prove estimate (3.41), we change variables and apply twice the Cauchy-Schwarz inequality in the ξ ′ 1 and ξ ′ 2 to deduce that by using the frequency localization. Thus estimate (3.41) is deduced by performing the change of variables (µ 2 , µ 3 ) = ( Ω, ξ ′ 3 ) in the inner integral on the right-hand side of (3.42) and by applying the Cauchy-Schwarz inequality in the variable ξ ′ 1 . Case k 4 = k min . In this case, we can assume without loss of generality that k 4 ≤ k 1 ≤ k 2 ≤ k 3 . It suffices to show that estimate (3.41) remains valid in this case. First, we change variables Thus the Cauchy-Schwarz inequality in ξ ′ 1 implies that (3.43) Moreover, we have that due to the frequency localization, so that we deduce through the change of variable Therefore, we deduce inserting (3.44) in (3.43) and applying twice the Cauchy-Schwarz inequality that which is exactly (3.41).
Case k 4 = k thd . Estimate (3.34) follows arguing exactly as in the case k 4 = k min . On the other hand, estimate (3.35) can also be proved applying the arguments of the cases k 4 ∼ k max or k 4 = k min , depending on wether j med = j 1 , j 2 or j 3 and using the symmetry relation (3.38).
As a consequence of Lemma 3.4, we have the following L 2 trilinear estimates.
(a) Then it follows that (b) Let us suppose that k thd ≤ k max − 5. If we are in the case where (k i , j i ) = (k thd , j max ) for some i ∈ {1, 2, 3, 4}, then it holds that In all the others cases, we have that Proof. Corolloray 3.5 follows directly from Lemma 3.4 by using a duality argument.

Short time bilinear estimates
The main results of this section are the following bilinear estimates in the F s α (T ) spaces. Note that to overcome the high-low frequency interaction problem (c.f. [32]), we need to work with α = 2 (see Lemma 4.3 below). Therefore, we will fix α = 2 in the rest of the paper and denote respectively F s 2 (T ), N s 2 (T ), F s 2 , N s 2 , F k,2 and N k,2 by F s (T ), N s (T ), F s , N s , F k and N k . The main results of this section are the bilinear estimates at the H s and L 2 level. and for all u, v ∈ F s (T ).
We split the proof of Propositions 4.1 and 4.2 in several technical lemmas. Then, for all u k1 ∈ F k1 and v k2 ∈ F k2 .
Remark 4.4. In the case k 1 = 0, the function u 0 ∈ F 0 is localized in spatial low frequencies corresponding to the projection P ≤0 , since we choose to use a nonhomogeneous dyadic partition of the unity to define the function spaces F s and N s (see Section 2).
Proof of Lemma 4.3. We only prove estimate (4.5), since the proof of estimate (4.6) is similar (and even easier). First, observe from the definition of N k in (2.6) that Now, we set for j i > 2k. Thus, we deduce from (4.7) and the definition of X k that where D k,j is defined in (3.5). Here, we use that since τ − w(ξ) + i2 2k −1 ≤ 2 −2k the sum from j = 0 to 2k − 1 appearing implicitely on the right-hand side of (4.7) is controlled by the term corresponding to j = 2k on right-hand side of (4.8). Therefore, according to Corollary 2.7 and estimate (4.8) it suffices to prove that (4.9) 2 3k with j 1 , j 2 ≥ 2k, in order to prove estimate (4.5).
But, we deduce from estimates (3.29) and (3.31) that 2 3k which implies estimate (4.9) after summing over j. This finishes the proof of Lemma 4.3.
Proof. Once again we only prove estimate (4.10). Arguing as in the proof of Lemma 4.3, it is enough to prove that where f ki,ji is localized in D ki,ji with j i ≥ 2k for i = 1, 2.
We deduce by applying estimate (3.28) to the left-hand side of (4.12) that 2 3k According to Lemma 3.1 and the frequency localization, we have that (4.14) 2 jmax ∼ max{2 j med , 2 5k }.
Finally, we observe that (4.13) and (4.14) imply estimate (4.12). This is clear in the cases where j max = j 1 or j 2 by using that 2 jmax 2 5k and summing over j ≥ 2k. In the case where j max = j, we have from (4.14) that either 2 j ∼ 2 5k or 2 j ∼ 2 j med . When 2 j ∼ 2 5k , estimate (4.12) follows directly from (4.13) since we do not need to sum over j, whereas when 2 j ∼ 2 j med , we can use one of the cases 2 jmax = 2 j1 or 2 jmax = 2 j2 to conclude.
We observe from the definition of N k in (2.6) that , for j i > 2k 2 . Thus, we deduce from (4.18) and the definition of X k that with j 1 , j 2 ≥ 2k 2 , in order to prove estimate (4.17).
In the cases j max = j 1 or j max = j 2 , say for example j max = j 1 , we deduce from estimate (3.31) that 2 3k2 which implies estimate (4.19) by summing over j.
Proof. Once again we only prove estimate (4.21). Arguing as in the proof of Lemma 4.3, it is enough to prove that where f ki,ji is localized in D ki,ji with j i ≥ 0 for i = 1, 2, which is a direct consequence of estimate (3.28).
Finally, we give the proof of Proposition 4.1. Note that the proof of Proposition 4.2 would be similar.
Proof of Proposition 4.1. We only prove estimate (4.2), since the proof of estimate (4.1) would be similar. We choose two extensionsũ andṽ of u and v satisfying Therefore ∂ xũ ∂ 2 xṽ is an extension of ∂ x u∂ 2 x v on R 2 and we have from the definition of N s (T ) and Minkowski inequality that where we took the convention P 0 = P ≤0 . Moreover, we denote Note that for a given k ∈ Z + , some of these regions may be empty and others may overlap, but due to the frequency localization, we always have that (4.25) To handle the sum S 1 , we use estimate (4.6) and the Cauchy-Schwarz inequality to obtain that where we assumed without loss of generality that max(k, k 2 ) = k. Similarly, we deduce from remark 4.5 that Estimate (4.11) leads to Next, we deal with the sum S 4 . Without loss of generality, assume that max(k 1 , k 2 ) = k 2 . It follows from estimate (4.17) and the Cauchy-Scwarz inequality in k 2 that Therefore, we conclude the proof of estimate (4.2) gathering (4.24)-(4.30).

Energy estimates
As indicated in the introduction we assume for sake of simplicity that c 3 = 0. We also recall that, due to the short time bilinear estimates derived in the last section, we need to work with α = 2 in the definition of the spaces F s α , F s α (T ) and F k,α and therefore we will omit the index α = 2 to simplify the notations.

Energy estimates for a smooth solution.
Due to the linear estimate (2.32), we need to control the norm · B s (T ) of a solution u to (1.1) as a function of u 0 H s and u F s (T ) . However, we are not able to estimate u B s (T ) directly. We need to modify the energy by a cubic term to cancel some bad terms appearing after a commutator estimate (see Remark 5.7 below).
Let us define ψ(ξ) := ξη ′ (ξ), where η is defined in (2.3) and ′ denote the derivative, i.e. η ′ (ξ) = d dξ η(ξ). Then, for k ≥ 1, we define ψ k (ξ) = ψ(2 −k ξ). We also denote by Q k the Littlewood-Paley multiplier of symbol ψ k , i.e. Q k u = F x ψ k F x u . From the definition of η k in (2.3), we observe that Finally, we define the new energy by for any k ≥ 1, and where α and β are two real numbers which will be fixed later. This modified energy may be seen as a localized version of the one introduced by Kwon in [25]. The next lemma states that when u L ∞ T H s x is small, then E s T (u) and u 2 B s (T ) are comparable.
Lemma 5.1. Let s > 1 2 . Then, there exists 0 < δ 0 such that for all k ≥ 1. It follows that for any t ∈ [−T, T ] and k ≥ 1. Thus, if we choose u L ∞ T H s ≤ δ 0 with δ 0 small enough, we obtain that which implies the first inequality in (5.4) after taking the supreme over t ∈ [−T, T ] and summing in k ≥ 1. The second inequality in (5.4) follows similarly. is a solution to (1.1) with c 3 = 0, we have that As a Corollary to Lemma 5.1 and Proposition 5.2, we deduce an a priori estimate in · B s (T ) for smooth solutions to (1.1). Corollary 5.3. Assume s ≥ 1 and T ∈ (0, 1]. Then, there exists 0 < δ 0 ≤ 1 such that for all solutions u to (1.1) with c 3 = 0 and satisfying u ∈ C([−T, T ]; H ∞ (R)) and u We split the proof of Proposition 5.2 in several lemmas.
Lemma 5.4. Assume that T ∈ (0, 1], k 1 , k 2 , k 3 ∈ Z + and that u j ∈ F kj for j = 1, 2, 3. (a) In the case k min ≤ k max − 5, it holds that If moreover k min ≥ 1, we also have that (b) In the case |k min − k max | ≤ 10, it holds that The following technical result will be needed in the proof of Lemma 5.4.
Lemma 5.5. Assume k ∈ Z + and I ⊂ R is an interval. Then (5.10) sup Proof. Fix j ∈ Z + . We can also assume that j ≥ 5. By writing we have that On the one hand, Plancherel's identity implies that On the other hand, we have that |F t (1 I )(τ )| 1 |τ | , since I is an interval of R. Thus, we deduce by applying the Cauchy-Schwarz inequality in τ ′ that since |τ − τ ′ | ∼ 2 j in this case. We deduce estimate (5.10) gathering (5.11)-(5.13) and taking the supreme in j.
Proof of Lemma 5.4. Assume without loss of generality that k 1 ≤ k 2 ≤ k 3 . Moreover, due to the frequency localization, we must have |k 2 − k 3 | ≤ 4. We first prove estimate (5.7). Let β : R → [0, 1] be a smooth function supported in [−1, 1] with the property that Then, it follows that Now we observe that the sum on the right-hand side of (5.9) 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 (3.7), (3.9) that 2 ji/2 f m ki,ji L 2 .
Finally, we only give a sketch of the proof of estimate (5.9) since it follows the same lines as the proof of estimate (5.7). Note that under the assumption |k min − k med | ≤ 4, we have that 2 k1 ∼ 2 k2 ∼ 2 k3 . Moreover, we can assume that k 1 ≥ 10, since the proof is trivial otherwise by using (3.6). We introduce the same decomposition as in (5.14) and split the summation domain in A and B. The estimates for the sum over the regions A and B follow by using (3.10) instead of (3.7) and (3.9) and the fact that 2 jmax 2 5 2 k3 (c.f. Lemma 3.1).
Remark 5.7. Lemma 5.4 does not permit to control the terms without loosing a 2 k factor, which would not be good to derive the energy estimates.
For that reason, we need to modify the energy by a cubic term (c.f. (5.3)) in order to cancel those two terms.
Proof of Lemma 5.6. We first prove estimate (5.18). After integrating by part, we rewrite the term on the left-hand side of (5.18) as where [A, B] = AB−BA denotes the commutator of A and B. Now, straightforward computations using (5.1) lead to due to the Taylor-Lagrange theorem and the frequency localization on ξ and ξ 1 . Therefore estimate (5.18) follows arguing exactly as in the proof of Lemma 5.4. To prove of estimate (5.19), we first observe integrating by parts that First, we apply estimates (5.7) and (5.8) to obtain that On the other hand, we observe that An easy computation gives due to the mean value theorem and the frequency localization on ξ and ξ 1 . We finish the proof of estimate (5.19) arguing exactly as in the proof of Lemma 5.6.
Lemma 5.8. Assume that T ∈ (0, 1], k 1 , k 2 , k 3 , k 4 ∈ Z + and that u j ∈ F kj for j = 1, 2, 3, 4. If k thd ≤ k max − 5, then it holds that If instead, k min ≪ k thd ∼ k sub ∼ k max , then it holds that Proof Then, for any k ∈ Z + ∩[1, +∞) and t ∈ [−T, T ], we differentiate E k (u) with respect to t and deduce using (1.1) that Now, we fix t k ∈ [−T, T ]. Without loss of generality, we can assume that 0 < t k ≤ T . Therefore, we obtain integrating (5.23) between 0 and t k that Next we estimate the right-hand side of (5.25).
Estimates for the cubic terms. We deduce after some integrations by parts that Similarly it holds that We choose α = − 2c2 5 and β = c2−4c1
Estimates for the fourth order terms. We estimate the fourth order term corresponding to N 2 k (u). After a few integration by parts in (5.24), we get that for each k ≥ 1, whith uP k u∂ x u P k udxdt .
We use the Strichartz estimate (2.45) with α = 2, estimate (2.10) and Hölder's inequality to deduce that To deal with X 2 (k), we perform dyadic decompositions over u and ∂ x u. Then By using Hölder's inequality and the Cauchy-Schwarz inequality, we can bound the sum over D 1 ∪ D 2 by Thus, it follows from estimates (2.10) and (2.46) that A similar bound holds over D 3 . In the region D 4 , we have that Hence, we deduce after taking the supreme of t k over [0, T ], summing over k ∈ Z + ∩ [1, +∞) and using estimate (2.46) that Similarly, we get that To deal with X 4 (k), we use the following decomposition where Observe that, according to estimates (2.10) and (2.45) Now, by using estimate (5.20), we get that Over the region E 3 , we deduce from estimates (2.10) and (2.46) that Finally, estimate (2.10) gives Thus, we deduce from (5.37)-(5.41) that (5.42) Therefore, we conclude gathering (5.32)-(5.36) and (5.42) that (5.43) By using the same arguments, we could obtain a similar bound for N 1 k (u). We finish the proof of Proposition 5.2 recalling the definition of the energy in (5.3) and gathering estimates (5.25), (5.31) and (5.43).

5.2.
Energy estimates for the differences of two solutions. In this subsection, we assume that s ≥ 2. Let u 1 and u 2 be two solutions to the equation in (1.1) with c 3 = 0 in the class (1.9) satisfying u 1 (·, 0) = ϕ 1 and u 2 (·, 0) = ϕ 2 . Then by setting v = u 1 − u 2 , we see that v must satisfy , with v(·, 0) = ϕ := ϕ 1 − ϕ 2 . As in subsection 5.1, we introduce the energy E s T (v) associated to (5.44). For k ≥ 1, and where α and β are two real numbers which will be fixed later. As in Lemma 5.1, x is small enough. Lemma 5.9. Let s > 1 2 . Then, there exists 0 < δ 1 such that for all v ∈ B s (T ) as soon as u 1 L ∞ T H s x ≤ δ 1 . Proposition 5.10. Assume T ∈ (0, 1] and s ≥ 2. Then, if v is a solution to (5.44), we have that

48)
and . As a Corollary to Lemma 5.9 and Proposition 5.10, we deduce an a priori estimate in · B s (T ) for the solutions v to the difference equation (5.44).
Corollary 5.11. Assume T ∈ (0, 1]. Then, there exists 0 < δ 1 ≤ 1 such that for all solutions v to (5.44) with u 1 Proof of Proposition 5.10. We argue as in the proof of Proposition 5.2. First, we choose extensions v, u 1 and u 2 of v, u 1 and u 2 over R 2 satisfying Then, for any k ∈ Z + ∩[1, +∞) and t ∈ [−T, T ], we differentiate E k (v) with respect to t and deduce using (5.44) that Now, we fix t k ∈ [−T, T ]. Without loss of generality, we can assume that 0 < t k ≤ T . Therefore, we obtain integrating (5.53) between 0 and t k that (5.55) Next we estimate the right-hand side of (5.55).
Estimates for the cubic terms. We deduce after some integrations by parts that and We choose α = − 2c2 5 and β = c2−4c1

5
. Then it follows, after performing a dyadic decomposition on v, that for each k ≥ 1, with Clearly, Lemma 5.6 and the Cauchy-Schwarz inequality imply that Similarly, we get applying estimate (5.7) if k 1 = 0, and estimate (5.8) if k 1 > 0, that Now, estimate (5.9) leads to (5.59) Arguing exactly as in (5.30), we get that This implies after taking the suprem of t k over [0, T ] and summing in k ∈ Z + ∩ [1, +∞) that at the L 2 -level. Note that to obtain (5.62), we need to modify the first term on the right-hand side of (5.60) by putting all the derivative on P k ′ u 1 F k ′ . To bound T 5 (k) and T 6 (k), we split the domain of summation over the {D j } 4 j=1 defined in (5.34). For example, we explain how to deal with T 6 (k). We have that By using estimates (5.7) when k 2 = 0, (5.8) when k 2 ≥ 1 and the Cauchy-Schwarz inequality in k 2 , we deduce that We treat the summation over D 2 similarly. Estimate (5.7) when k 1 = 0, estimate (5.8) when k 1 ≥ 1 and the Cauchy-Schwarz inequality in k 1 imply that (5.65) Estimate (5.9) gives that Finally, it follows from estimate (5.8) that at the L 2 -level. Therefore, we deduce gathering (5.57)-(5.61) and (5.68) that at the L 2 level.
Estimates for the fourth order terms. We estimate the fourth order term corresponding to N 2 k (v). After a few integration by parts in (5.54), we get that for each k ≥ 1, whith We use the Strichartz estimate (2.45) with α = 2, estimate (2.10) and Hölder's inequality to deduce that for any s ≥ 0.
To handle X 2 (k), we perform the following decomposition where By applying Hölder's inequality, we can bound X 2,1 (k) by (5.75) which implies after suing the Sobolev embedding, the Cauchy-Schwarz inequality and estimate (2.46) for any s ≥ 0. On the other by putting the L ∞ T L 2 x norm on P k2 v and the L 2 T L ∞ x norm on P k3 u 2 in (5.75), we get that at the L 2 level. By using similar arguments, we get that for any s ≥ 0. Finally, we use estimate (5.20) to bound X 2,4 (k) by which implies after summing over k ∈ Z + ∩ [1, +∞) for all s ≥ 0. Therefore, we conclude gathering estimates (5.52) and (5.74)-(5.79) that for any s ≥ 0 and at the L 2 level. By using the same arguments as for X 2 (k), we have that at the L 2 level.
To deal with X 4 (k) at the L 2 level, we observe after integrating by parts that Arguing exactly as for X 2 (k) in (5.35), we deduce that at the L 2 level. To estimate X 4,3 (k) at the H s -level, we use the same decomposition as for X 4 (k) in (5.37). It follows that which implies together with (5.84) and (5.85) for all s ≥ 0.
Finally, we treat the term X 5 (k). After integrating by parts, we obtain that (5.88) By using the same arguments as above, we deduce that Arguing exactly as for for all s ≥ 0 and at the L 2 level. To handle X 5,3 (k), we perform the same decomposition as for X 4 (k) in (5.37). It follows that for any s ≥ 0 and for any s ≥ 0, and from (5.90) and (5.92) that which together with (5.55) and (5.70) implies estimate (5.52). This concludes the proof of Proposition 5.10.
6. Proof of Theorem 1.1 We recall that, for sake of simplicity, we are proving Theorem 1.1 in the case c 3 = 0. The starting point is a well-posedness result for smooth solutions which follows from Theorem 3.1 in [33]. The following technical lemma will be needed in the proof of Proposition 6.2.
Lemma 6.3. Assume s ∈ R + , T > 0 and u ∈ C([−T, T ]; H ∞ (R)). We define for any 0 ≤ T ′ ≤ T . Then : T ′ → Λ s T ′ (u) is nondecreasing and continuous on [0, T ). Moreover In order to deal with the other components of Λ s T ′ (u) in (6.2), it suffices to prove that given f ∈ C([−T, T ]; H ∞ (R)), (6.5) : is continuous and nondecreasing, and (6.6) lim It is clear from the definition of N s that x , for any f ∈ L 2 t H s x . Then, we deduce by applying estimate (6.7) to f ( which gives (6.6). Now, we turn to the proof of (6.5). The fact that : is a nondecreasing function follows directly from the definition of N s (T ′ ). To prove the continuity of : T ′ ∈ (0, T ) → f N s (T ′ ) at some fixed time T ′ 0 ∈ (0, T ), we introduce the scaling operator D r (f )(x, t) := f (x/r 1 5 , t/r), for r close enough to 1. Hence, we have from (6.8) and the triangle inequality that since f ∈ C([−T, T ]; H ∞ (R)). Then, it remains to show that , to conclude the proof of (6.5). We observe that (6.10) would follow from the inequalities First, we begin with the proof of (6.11). Let ǫ be an arbitrarily small positive number. For r close to 1, we choose an extension f r of D r (f ) outside of [−rT ′ 0 , rT ′ 0 ] satisfying where M is a positive constant independent of r. We also observe that D 1/r ( f r ) is an extension of f outside of [−T ′ 0 , T ′ 0 ], so that (6.14) f N s (T ′ 0 ) ≤ D 1/r ( f r ) N s . Moreover, we will prove that (6.15) where ψ is a continuous function defined in a neighborhood of 1 and satisfying lim r→1 ψ(r) = 1. Then estimate (6.11) would be deduced gathering estimates (6.13), (6.14) and (6.15).
The proof of estimate (6.12) follows in a similar way (it is actually easier).
Proof of Proposition 6.2. Fix s ≥ 5 4 . First, it is worth noticing that we can always assume that the initial data u 0 have small H s -norm by using a scaling argument.
Indeed, if u is a solution to the IVP (1.1) on the time interval [0, T ], then u λ (x, t) = λ 2 u(λx, λ 5 t) is also a solution to the equation in (1.1) with initial data u λ (·, 0) = λ 2 u 0 (λ·) on the time interval [0, λ −5 T ]. For ǫ > 0, let us denote by B s (ǫ) the ball of H s (R) centered at the origin with radius ǫ. Since we can always force u λ (·, 0) to belong to B s (ǫ) by choosing λ ∼ ǫ H s . Therefore, it is enough to prove that if u 0 ∈ B s (ǫ), then Proposition 6.2 holds with T = 1. This would imply that Proposition 6.2 holds for arbitrarily large initial data in H s (R) with a time T ∼ λ 5 ∼ u 0 − 10 3 H s . Now, fix u 0 ∈ H ∞ (R) ∩ B s (ǫ) and let u ∈ C([−T, T ]; H ∞ ) the solution to (1.1) given by Theorem 6.1 where 0 < T ≤ 1. We obtain gathering the linear estimate for i = 1, 2, and where c is the implicit constant appearing in the first inequality of (6.34) below.
Remark 6.6. Observe that the convergence of {u λ } in C([−1, 1]; H 1 (R)) would be enough to obtain that the limit u satisfies the equation in (1.1) in the weak sense.
6.4. Continuity of the flow map data-solution. Observe that for s ≥ 4, the result was already proved in Theorem 3.1 in [33]. Then it is enough to prove it for 2 ≤ s < 4. Let u 0 ∈ H s (R). Once again we can assume by using a scaling argument that u 0 ∈ B s (ǫ) with 0 < ǫ ≤ǭ < ǫ s and where ǫ s was determined in the previous subsection. Then, the solution u emanating from u 0 is defined on the time interval [−1, 1] and satisfies u ∈ C([−1, 1]; H s (R)).
Let θ > 0 be given. It suffices to prove that for any initial data v 0 ∈ B s (ǫ) with u 0 − v 0 H s ≤ δ, where δ = δ(θ) > 0 will be fixed later, the solution v ∈ C([−1, 1]; H s (R)) emanating from v 0 satisfies For any λ > 0, we normalize the initial data u 0 and v 0 by defining u 0,λ = ρ λ * u 0 and v 0,λ = ρ λ * v 0 as in the previous subsection and consider the associated smooth solutions u λ , v λ ∈ C([−1, 1]; H ∞ (R)). Then it follows from the triangle inequality that x . On the one hand, according to (6.41), we can choose λ 0 small enough so that . On the other hand, we get from (6.35) that Therefore, by using the continuity of the flow map for smooth initial data (c.f. Theorem 3.1 in [33]), we can choose δ > 0 small enough such that (6.45) u λ0 − v λ0 L ∞ 1 H s x ≤ θ/3. Estimate (6.42) is concluded gathering (6.43)-(6.45). 7. Appendix: how to deal with the cubic term ∂ x (u 3 ).
In this appendix, we explain what are the main modifications needed to deal with cubic term ∂ x (u 3 ) (i.e. in the case where c 3 = 0). As above, we fix α = 2 in the definition of the spaces F s α (T ), N s α (T ), F s α , N s α , F k,α , N k,α and write those spaces without the index α = 2, since there is no risk of confusion. 7.1. Short time trilinear estimate. In this subsection, we prove the trilinear estimate for the nonlinear term ∂ x (u 3 ).
Proposition 7.1. Let s ≥ 0 and T ∈ (0, 1] be given. Then, it holds that for all u, v, w ∈ F s (T ).
We split the proof of Proposition 7.1 in several technical lemmas depending of the frequency interactions.
Proof. We argue exactly as in the proof of Lemma 7.3 and observe that estimate (7.5) leads to estimate (7.3) even without using that j max ≥ 5k − 20, which is not always satisfied in this case. Instead, it is sufficient to use that j, j i ≥ 2k for all i = 1, 2, 3.
Proof. Following the proof of Lemma 7.5, we need to prove that estimate (7.8) still holds in this case. This is a direct consequence of estimates (3.47) and (3.48).
Proof. It follows arguing as in Lemma 4.10.
Finally, we give the proof of Proposition 7.1.
Proof of Proposition 7.1. Fix s ≥ 0. We choose two extensionsũ,ṽ andw of u, v and w satisfying ũ F s ≤ 2 u F s (T ) , ṽ F s ≤ 2 v F s (T ) and w F s ≤ 2 w F s (T ) .

7.2.
Modifications to the energy estimates. We only explain how to deal with the a priori estimates, since the modifications would be similar to derive estimates for the differences of two solutions. The main point is to derive an analog to Proposition 5.2 in the case where c 3 = 0. . Proof. The proof of Proposition 7.8 follows the same strategy as the one of Proposition 5.2. The unique difference is that we need to add the terms M 1 k (u), αM 1 k (u) and βM 2 k (u) to the right-hand side of (5.23), where K k (u) = 2c 3 R P k uP k ∂ x u 3 dx, Therefore, it suffices to bound K k (u) + αM 1 k (u) + βM 2 k (u) dt by the the terms appearing on the right-hand side of (7.12). We first treat the fourth-order term corresponding to K k (u). We perform the same dyadic decomposition as in the proof of Proposition 7.1. Thus, P k uP k ∂ x P k1 uP k2 uP k3 u dx . (7.14) By using respectively estimate (5.20) for the sums over G 1 and G 5 and estimate (5.21) for the sums over G 2 and G 4 , the corresponding terms on the right-hand side of (7.14) can be bounded by . In the regions G 3 and G 6 , we use estimates (2.10) and (2.46) to bound the corresponding terms by (7.16) u F 0 (T ) u F 3 4 + (T ) u 2 B s (T ) . Observe that (7.15) and (7.16) are controlled by the second term on the right-hand side of (7.12).
Next, we deal with the fifth order term corresponding to M 2 k (u) and observe that the one corresponding to M 1 k (u) could be treated similarly. It follows from estimate (2.10) that which leads to the bound in (7.12) after summing over k ∈ Z + ∩ [1, +∞) and taking the supreme over t k ∈ [0, T ]. Finally, to deal with the second term on the right-hand side of (7.13), we introduce a dyadic decomposition ∂ x (u 3 ) = k1,k2,k3 ∂ x P k1 uP k2 uP k3 u , and use estimates (2.10) and (2.46) to obtain the right estimate. This finishes the proof of Proposition 7.8.