Well-posedness of the periodic dispersion-generalized Benjamin–Ono equation in the weakly dispersive regime

We study the dispersion-generalized Benjamin–Ono equation in the periodic setting. This equation interpolates between the Benjamin–Ono equation ( α=1 ) and the inviscid Burgers’ equation ( α=0 ). We obtain local well-posedness in Hs(T) for s>32−α and α∈(0,1) by using the short-time Fourier restriction method.


Introduction
In this article we consider the dispersion-generalized Benjamin-Ono equation posed on R × T with α ∈ (0, 1).Here, the unknown u is a real-valued function, the initial datum u 0 is in a Sobolev space H s (T) and D α x denotes the Fourier multiplier defined by F (D α x u)(ξ) = |ξ| α F (u)(ξ).Famous examples of equations of the form (1.1) are the Korteweg-de Vries equation (α = 2) modelling unidirectional nonlinear dispersive waves as well as the Benjamin-Ono equation (α = 1) which models long internal waves in deep stratified fluids.For non-integer valued α, equation (1.1) can be seen as an interpolation between the two models and for α = 1/2 it is closely related to the Whitham equation for capillary waves, see below.In the case α = 0, equation (1.1) reduces to the viscous Burgers' equation.
It is well-known that the Korteweg-de Vries equation as well as the Benjamin-Ono equation are integrable PDEs having infinitely many conserved quantities along their flow, see [16,15] and the references therein.For general α we still have the conservation of the mean, the mass and the Hamiltonian.These quantities are given by u(t, x)dx, |u(t, x)| 2 dx, 1 2 D α/2 x u(t, x) If posed on the real line, equation (1.1) is invariant under the scaling u(t, x) → λ α u(λ 1+α t, λx) and hence the equation is scaling-critical in the homogeneous Sobolev space Ḣ1/2−α (R).In particular, the equation is L 2 -critical for α = 1/2 and it is energy-critical for α = 1/3.
The purpose of this article is to establish local well-posedness in H s (T), s > 3/2 − α, for the Cauchy problem (1.1) with periodic initial data and α ∈ (0, 1).By local well-posedness, we mean the existence and uniqueness of solutions as well as their continuous dependence on the initial datum.Before stating our main theorem, let us briefly discuss some recent results concerning the well-posedness theory of equation (1.1) for α ∈ [0, 2] posed on the torus and on the real line.
1.1.Known results and main theorem.A classical result due to Kato [13] yields the local wellposedness of the Cauchy problem (1.1) in H s (T) and H s (R) for any s > 3/2 and all α ∈ [0, 2].This result is sharp for α = 0, but the proof does note take advantage of the dispersive structure of (1.1) for α = 0.
Before we state some results in the range α < 2, let us recall that the behaviour of equation (1.1) changes significantly in comparison to the case α = 2. Indeed, Molinet, Saut and Tzvetkov [20] observed that the dispersion is too weak to deal with the nonlinearity by perturbative means.More precisely, they showed that the frequency-interaction u low ∂ x u high in the nonlinearity cannot be estimated appropriately in order to allow for a Picard iteration, see also [17,7].
In the well-posedness theory of the Benjamin-Ono equation (α = 1) a major breakthrough overcoming latter problem was obtained by Tao [24], who proved global well-posedness in H 1 (R).He applied a gauge transform which effectively cancels the worst behaving interaction in the nonlinearity.Following the same approach, Ionescu and Kenig [11] established global well-posedness in L 2 (R) and Molinet [19] proved global well-posedness in L 2 (T).Relying on the integrability of the equation, Gérard, Kappeler and Topalov [4] obtained global well-posedness in H s (T), s > −1/2, almost reaching the scaling critical regularity −1/2.Recently, Killip, Laurens and Vişan [15] obtained global well-posedness in H s (T) and in H s (R) for s > −1/2.
1.2.Strategy of the proof.Let us comment on the strategy of the proof.The general idea is to follow the approach by Ionescu, Kenig and Tataru [10] involving frequency-dependent time-localized function spaces.We begin by giving a heuristic argument indicating how these time-localizations improve the estimate of the aforementioned problematic term u low ∂ x u high .Let u K , respectively u N , denote the localization of a function u to frequencies of size K, respectively N .Fix K ≤ N , s ≥ 1/2 and let I be an interval of length N −1 .Then, we can bound the L 1 (I; H s (T))-norm of u K ∂ x u N using Hölder's and Bernstein's inequality by Note that the spatial regularity is the same on both sides thanks to the time-localization.
As in [10], we will introduce function spaces F s T , N s T and E s T .The first and third space can be thought of as C([0, T ]; H s ), while the second function space is close to L 1 ([0, T ]; H s ).The distinctive feature of the spaces F s T and N s T is that they are equipped with norms that involve frequency-dependent time-localizations.These time-localizations allow us to prove estimates of the form where F 1 is a function and d ∈ (0, 1] is a small but positive number, see Section 5 for the precise estimates.Above, the linear estimate is a consequence of Duhamel's principle, whereas the nonlinear estimate relies on the time-localization -similar to the heuristic argument above.The energy estimates follow with considerably more effort.Here, we will use quadrilinear estimates, commutator estimates and symmetrizations.
For differences of solutions v = u 1 − u 2 and w = u 1 + u 2 , we analogously obtain the estimates as well as After bootstrapping the previously given sets of estimates, we get an a priori estimate in H s (T) for smooth solutions of (1.1) as well as estimates for the difference of two smooth solutions in H −1/2 (T) and in H s (T).Equipped with these results, we obtain Theorem 1.1 following the classical Bona-Smith argument [1].
Next, we want to comment on the choice of the time-localization.To prove Theorem 1.1, we will choose a time-localization that restricts a function u N to a time interval of length N −1−ǫ , ǫ > 0. Note that the localization does not depend on α and that it fits to the heuristic argument given above.In comparison to our approach, Guo [5] and Schippa [23], when studying equation (1.1) in the range α ∈ (1, 2), used a time-localization that restricts a function u N to an interval of size N α−2−ǫ , ǫ > 0. This localization appears in the heuristic argument if we use a bilinear estimate instead of Hölder's inequality.Choosing latter time-localization, we can repeat our proof with minor modifications and obtain an improved a priori estimate, but no improvement for the difference estimates, see Section 5.1.
At last, we argue that Theorem 1.1 holds for a slightly larger class of dispersive PDEs than (1.1).As in [21], let us consider Here, L iω is the Fourier multiplier defined by F (L iω u) = iωF (u) and ω is an odd, real-valued function in We need two further assumptions on ω to guarantee that Theorem 1.1 also holds for (1.2):Firstly, we require that for some κ > 0 we have the following control of the first two derivatives of ω: Secondly, denoting by |ξ Under these conditions on ω, Theorem 1.1 also holds for (1.2).A non-trivial example is for which equation (1.2) becomes the Whitham equation for capillary waves, see Chapter 12 in [25].
1.3.Structure of the paper.In Section 2 we fix the notation and define the frequency-dependent time-localized function spaces F s T , N s T and E s T .Moreover, we recall some properties to indicate their connection.In particular, we state the linear estimate required for the bootstrap argument.In Section 3 we prove estimates exploiting the structure of the involved function spaces leading to the nonlinear estimates.We also prove a quadrilinear estimate that is relevant for the energy estimates, which are dealt with in Section 4. In the last section we briefly recall the arguments due to Bona and Smith and complete the proof of Theorem 1.1.In the following, without mentioning, we assume t ∈ R, τ ∈ R, x ∈ T = R/2π and ξ ∈ Z to denote variables in time, Fourier-time, space and frequency.Given two variables ξ 1 and ξ 2 , we occasionally abbreviate their sum by ξ 12 = ξ 1 + ξ 2 .Moreover, for variables ξ 1 , . . ., ξ n , n ≥ 2, we denote by ξ

Preliminaries
The Fourier transform on the torus, respectively on the real line, is given by Here, the indices x and t indicate the transformed variable.We abbreviate F t F x by F t,x .We fix an even smooth function Based on this, we define Littlewood-Paley projectors by Similarly, we fix η 0 ∈ C ∞ (R) satisfying 1 [−5/4,5/4] ≤ η 0 ≤ 1 [−8/5,8/5] and define for any k ≥ 1. Mostly, we will use latter functions to restrict the modulation variable τ − ω(ξ) to dyadic ranges, where the dispersion relation ω is given by 2.2.Function spaces.In this section we will define the function spaces F s T , N s T and E s T .Essentially, these function spaces appeared first in [10] in the context of the analysis of the KP-I equation posed on R 2 .For an adaption and thorough analysis of these spaces on T, we refer to [6].
The definition of the spaces F s T and N s T is based on the space X k , k ∈ N, given by Note that X k is closely related to the Bourgain space X 0,1/2 and shares many properties with it.The following proposition recalls three of these properties.The first two properties follow from direct calculations, whereas the third property can be proved similarly to Lemma 3.5 in [6].
Proposition 2.1.We have the following statements: (1) Let k, L ∈ N and write η L for η ≤L .For all f k ∈ X k we have (2) Let k, l ∈ N, t 0 ∈ R and let γ ∈ S(R) be a Schwartz function.For all f k ∈ X k we have T ∈ (0, 1] and write η L for η ≤L .Let I be an interval with 0 ∈ I such that |I| ∼ T holds.For all u supported in R × T we have Let I ≥ 0 determine the strength of the frequency-dependent time-localization.Then, we define Observe that I = 0 corresponds to a frequency-independent time-localization.It will become evident that choosing I slightly larger than one is sufficient for our proof.
For T ∈ (0, 1] we define the corresponding restriction spaces to the interval [−T, T ] by When working with functions in F k T or N k T , it will be sufficient to consider extensions of them by functions in F k or N k , which are supported in [−2T, 2T ].More precisely, Lemma 3.6 in [6] implies: Proposition 2.2.Let k ∈ N and T ∈ (0, 1].For every u ∈ F k T and every v ∈ N k T there are extensions ũ ∈ F k and ṽ ∈ N k having temporal support in [−2T, 2T ] and satisfying the estimates We equip the inhomogeneous Sobolev space of real-valued functions on T for s ∈ R with the norm We also write H ∞ (T) := s∈R H s (T) and denote by H ∞ c (T) the set of H ∞ (T)-functions u with mean û(0) = c.For T ∈ (0, 1] and s ∈ R, define the function space F s T by and the energy space E s T by In the subsequent proposition we recall the continuity of the norms E s T and N s T with respect to T and state a linear estimate which is needed for the bootstrap argument in Section 5.For the proofs, we refer to Lemma 3.3, Proposition 8.1 and Proposition 4.1 in [6].
Proposition 2.3.Let T ∈ (0, 1] and s ∈ R. (1) For all u ∈ F s T we have (2.4) are non-decreasing continuous functions for T ∈ (0, 1].Moreover, we have In particular, we conclude from the proposition that every smooth solution u of (1.1) with mean zero is contained in F s T and in E s T , whereas the nonlinearity ∂ x (u 2 ) is an element of N s T .

Multilinear estimates
In this section we derive the estimate of the nonlinearity for the bootstrap argument as well as a quadrilinear estimate that is used repetitively to obtain the energy estimates in the next section.As we will see, we need to exploit the frequency-dependent time-localized structure of the function spaces F s T and N s T for both estimates.
Before we start with the estimate of the nonlinearity, let us provide two auxiliary lemmata.The first one deals with convolution estimates for functions localized both in modulation and frequency.
Then, we have 3.1.Estimate of the nonlinearity.We continue with proving the nonlinear estimate for the bootstrap argument.More precisely, we need to bound the N r T -norm of ∂ x (u 2 ) as well as the vw), where u, u 1 and u 2 are solutions of (1.1) and v = u 1 − u 2 , w = u 1 + u 2 .The proof makes use of the structure of the spaces N • T and F • T , the strength of the time-localization I as well as of the estimate (3.2).Notably, as (3.2) is independent of α, the obtained bound holds independently of the precise choice of α ∈ (0, 1) and for arbitrary, but sufficiently regular functions.
Proof.According to Proposition 2.2, there exist extensions f k1 and g k2 of P k1 f and P k2 g satisfying Here, the implicit constants are independent of k 1 , k 2 , f and g.Since f k1 g k2 is an extension of P k1 f P k2 g, the definition of N b T yields In the following, we will restrict the right-hand side of (3.8) to the index sets and then estimate each contribution separately.Here, we understand Estimating the contribution for With we can write Observe that (3.5) implies that the X k3 -norm above vanishes for all but at most 2 2+6I many values of m.By the definition of X k3 , we get Above, we used the notation introduced in (3.1) and write D ≤⌊k3I⌋,k3 := ∪ l≤⌊k3I⌋ D l,k3 .Moreover, in the last inequality, we used which yields the estimate Now, the functions -as well as their convolution -are localized in modulation and frequency.Assume l * 3 ≤ l 2 and k * 3 = k 2 .We continue the estimation above by an application of (3.2) leading to Fix d ∈ [0, 1/2).We estimate the sum over l 1 by combining Hölder's inequality with estimate (2.3), which is applicable due to f k1 = 1 [−2T,2T ] f k1 .Thus, we obtain Using (2.1), the sum over l 2 can be estimated by Now, we apply (2.2) multiple times, loosing the additional time-localizations γ.Then, we take the supremum in t k3 for each factor (loosing the dependence on m and k 3 ), evaluate the sum over l 3 and apply the bounds in (3.7).It follows Estimating the contribution for k 2 ∼ k 3 ≫ k 1 .We may repeat all steps of the first case.Note that the application of (3.2) will yield the factor 2 k1/2 instead of 2 k2/2 .For b < 1/2, we obtain Similarly, the estimate follows for b ≥ 1/2.

Estimating the contribution for
and the right-hand side does not vanish for at most 2 (k1−k3)I many values of m.Proceeding as in the first case, we obtain the estimate Then, together with (3.2) and l * 3 ≤ l 3 (1/2 − ǫ) + l 1 ǫ, ǫ > 0, it follows Then, choosing ǫ, d > 0 such that 1 < I(1 + ǫ) and I(ǫ Estimating the contributions for In these cases, we can repeat the proof of the first case by choosing V = 6I, respectively V = 100I. 3.2.Quadrilinear estimate.In this section we will use (3.3) in order to derive a quadrilinear estimate.
Then, we have the following estimate: Proof.We localize each ûi = j χ kj ûi , i ∈ [4], apply the triangle inequality and obtain It remains to prove Let us denote by u i the function determined by ûi = |û i |.Due to the definition of F ki T and Proposition 2.2, there exist for each i ∈ [4] hold.More precisely, we interpret 1 [0,T ] P ki u i as an element of the space F ki [0,T ] (that is F ki restricted to [0, T ]), apply Proposition 2.2 for that space and use that any extension of 1 [−T,T ] P ki u i is already an for each j ∈ [4] and conclude Now, we split the sum over m into three parts.For this, define as well as . Clearly, we have S k1,k2,k3,k4 | M3 = 0. Thus, it suffices to consider M 1 and M 2 .
Let us start with the sum over M 1 .In this case, we can drop the temporal indicator function due to (3.10).Moreover, since the support of γ is of unit size, we have . Now, we can apply (3.3) and (2.1) to the sums over l i and arrive at Lastly, we use (2.2) and take the supremum in t ki for each factor separately leading to a loss of the dependency on m.Then, by (3.9), we arrive at the desired bound: It remains to consider the summation over M 2 in S k1,k2,k3,k4 .Note that m ∈ M 2 implies that either Now, we localize in the modulation variable and attach the function 1 [0,T ] to the factor with the highest modulation variable, which we assume to be l 1 .More precisely, we set Then, we proceed as before and obtain To treat the norm involving the sharp time-cutoff 1 [0,T ] , we fix d ∈ (0, 1/2).Using Hölder's inequality as well as (2.3), we obtain As before, we apply (2.2) to each factor, take the supremum in each t ki , i ∈ [4], and use (3.9) leading to Since T ∈ (0, 1] and d < 1 < I hold, the claim follows.

Energy estimates
In this section we prove the following three energy estimates: There exist c, d > 0 such that: (1) For all smooth solutions u of (1.1) with mean zero we have (2) For all smooth solutions u 1 and u 2 of (1.1) (3) For all smooth solutions u 1 and u 2 of (1.1) All three estimates will be proved with the help of the quadrilinear estimate from Lemma 3.4, commutator estimates and cancellations due to symmetry.Additionally, we need to control the resonance function, which will appear in the calculations.
By applying (4.5), we can bound the denominator above by To bound the numerator, we consider two cases: First, we assume k 2 ≫ k b .Then, the integers ξ a , ξ a3 , ξ ab and ξ ab3 all have the same sign and have modulus of size 2 ka .This allows to apply the double mean value theorem leading to the bound 2 ka(α−1) 2 k3 2 k b .
4.1.Proof of the first energy estimate.As the title suggests, this section is devoted to the proof of (4.1).We apply P k1 to equation (1.1), multiply it by P k1 u and integrate over T. Since χ k1 is even and u is real, so is P k1 u.Moreover, recall that ∂ x D α x has the purely imaginary symbol iω.We conclude Now, we integrate over [0, t k1 ], use the fundamental theorem of calculus, take the supremum over t k1 in [0, T ], multiply by 2 k12r and take the sum over k 1 in N leading to Above, the last sum is restricted to the regime Clearly, this estimate is not correct at first.However, we will justify (4.7) at the end of this section showing that it is sufficient to treat only this case.We want to shift the derivative to the low frequency.For ξ 123 = 0 we write and observe the trivial bound |σ j (ξ 1 , ξ 2 , ξ 3 )| 2 k3 for each j ∈ [3].The above implies Hence, for a parameter n ∈ N, we obtain Let us continue with bounding the right-hand side of (4.9) in the next two lemmata.First, we estimate the low-frequency contribution.
Proof.Using the bound |σ j | 2 k3 as well as Hölder's and Jensen's inequalities, we obtain The claim follows by an application of the estimate (2.4) and summation in k 1 and k 3 .Now, we have to estimate the second term on the right-hand side of (4.9).Recalling the definition (4.4) and the fact that u is a smooth solution of (1.1), we have In view of (4.5), this equation will turn out to be very helpful in the next lemma.
Lemma 4.4.Let r ≥ s > 3/2 − α.Then, there exist c, d > 0 such that we have Proof.We apply (4.10) to each summand I(k 1 , k 3 ) and obtain where the terms above are given by Here, B stands for boundary term and the subscript i of I corresponds to the variable ξ i being split into the sum ξ a + ξ b .Without specifically mentioning it, we can assume ξ i = 0 for all i ∈ [3].Indeed, any summand in I(k 1 , k 3 ) with ξ i = 0 for some i ∈ [3] vanishes.This follows immediately from the fact that all F s T -functions have mean zero.In particular, (4.5) guarantees that the resonance function Ω does not vanish in any of the expressions above.
We begin by estimating B. Here, we necessarily have k 2 ∼ k 1 and, similarly to the proof of Lemma 4.3, conclude In order to bound the terms I 1 , I 2 , I 12 and I 3 , we localize the variables ξ a and ξ b to dyadic frequency ranges, i.e. we insert the factor 1 = ka,k b χ ka (ξ a )χ k b (ξ b ).Using the bound σj Ω (ξ ab , ξ 2 , ξ 3 )ξ ab 2 k1(1−α) as well as Lemma 3.4, I 1 can be estimated by Similarly, I 2 can be handled with the help of σj Ω (ξ 1 , ξ ab , ξ 3 )ξ ab 2 k1(1−α) and Lemma 3.4.We obtain Next, we bound I 12 .Firstly, consider the case k 1 ∼ k a ≫ k b .A direct estimate of the modulus of the symbol σj Ω yields the bound 2 k1(1−α) , which is insufficient in this case.Instead, we benefit from cancellations in the symbol since the rest of the integrand -that is the factor û(ξ 1 )û(ξ 2 )û(ξ 3 )û(ξ 4 ) -is symmetric in the high-frequency variables ξ a and ξ 2 .Indeed, this symmetry yields Thus, we can rewrite the new symbol as follows: Using |σ j | 2 k3 , (4.5) as well as the equation ξ ab + ξ 2b = ξ b − ξ 3 , we obtain The estimate in (4.6) combined with |σ j | 2 k3 once again yields To bound s 3 , we observe that Above, each summand on the right-hand side has modulus bounded by 2 −k1 2 k3 2 k b .Using (4.5), we obtain Consequently, the symbol s 1 + s 2 + s 3 has modulus bounded by 2 −αk1 2 max{k3,k b } and after localizing ξ a and ξ b to dyadic frequency ranges, an application of Lemma 3.4 yields where the second estimate follows by the same arguments.Finally, let us bound I 3 .Making use of Lemma 3.4 and the bound σj Ω (ξ 1 , ξ 2 , ξ ab )ξ ab 2 −k1α 2 k3 , it follows where k * 3 (resp.k * 4 ) denotes the third (resp.fourth) largest number of k 1 , k 2 , k a and k b .Now, we sum over the bounds of B, I 1 , I 2 , I 12 and I 3 in k 1 and k 3 and additionally invoke (2.4) for the bound of B. This concludes the proof for every c ∈ (0, α).
Using the estimates proved in Lemmata 4.3 and 4.4, we obtain a bound for the modulus of (4.7), which is given by To complete the proof of estimate (4.1), it remains to deduce appropriate bounds for the term above for any other possible restrictions on k 1 , k 2 and k 3 .The bound for the case k 1 ∼ k 3 ≫ k 2 can be derived similarly to that for k 1 ∼ k 2 ≫ k 3 .To treat the case k 2 ∼ k 3 ≫ k 1 , we can omit the application of (4.8) at the beginning of our calculations since the derivative is already on the low-frequency term.After that, we proceed as before.It remains to analyze the case k 1 ∼ k 2 ∼ k 3 .Again, we can omit the application of (4.8) and argue as before with the minor difference that Corollary 4.2 cannot be used in this case.Hence, we must prove the bound for I 12 in Lemma 4.4 differently.However, we can simply use the direct bound on the symbol, which is of size Thus, the proof of (4.1) is complete.

4.2.
Proof of the second energy estimate.In this section we will prove estimate (4.2).Recall that u 1 and u 2 are smooth solutions of (1.1) and that we write v = u 1 − u 2 , w = u 1 + u 2 .Observe that v satisfies Hence, by similar arguments as used in Section 4.1, we obtain (4.12) Moreover, we have the following analogue of (4.10): Let us begin by estimating the low-frequency interaction.
Next, we bound those summands, in which one factor v is localized to a high frequency and the other factor v is localized to a low frequency.Lemma 4.6.Let s > 3/2 − α.Then, there exist c, d > 0 such that we have Proof.We apply (4.13) to each summand II(k 1 , k 3 ) and obtain where the terms on the right-hand side are defined by The term BB can be estimated similarly to the boundary term B in Lemma 4.4.Indeed, we must have k 2 k 1 , which -together with (4.5) -leads to Let us continue with bounding the summands II 1 , II 2 and II 3 .Analogously to the proof of (4.1), we insert the factor 1 = ka,k b χ ka (ξ a )χ k b (ξ b ) into each of the terms II 1 , II 2 and II 3 in order to localize the variables ξ a and ξ b .Using the bound Ω −1 (ξ ab , ξ 2 , ξ 3 )(−iξ ab ) 2 2 k3(2−α) 2 −k2 and applying Lemma 3.4, we obtain Similarly, from Ω −1 (ξ 1 , ξ ab , ξ 3 )(−iξ 1 )(−iξ 2 ) 2 k1(1−α) and Lemma 3.4, we conclude 2 k1(2−α) 2 −k2 and Lemma 3.4 imply The claim follows after summation of the obtained bounds in k 1 and k 3 .
Proof.We can repeat the proof of the previous lemma since we still have one high-and one low-frequency factor v.
It remains to bound those terms, where both high-frequency factors are given by v. Here, the first step is to shift the derivative to the low-frequency factor as in the beginning of Section 4.1.
Lemma 4.8.Let s > 3/2 − α.Then, there exist c, d > 0 such that we have Proof.Recall that we have We can rewrite the spatial integral of I(k 1 , k 3 ) as follows: Then, defining Now, as in Lemma 4.6, we apply equation (4.13) to II 1 (k 1 , k 3 ) and II 2 (k 1 , k 3 ).It follows where the terms on the right-hand side for j = 1 are given by The corresponding terms for j = 2 follow by obvious modifications.It remains to bound the quantities II 1 , II 2 and II 3 .
Estimating II 1 .Note that k 1 ∼ k 2 holds.The boundary term can be estimated similar to the boundary term B in Lemma 4.4 by Now, we estimate II 1 3 .Using Ω(ξ ab , ξ 2 , ξ 3 ) −1 (−iξ 3 ) 2 2 −k1α 2 k3 as well as Lemma 3.4, we obtain Similarly, the bound Ω −1 (ξ ab , ξ 2 , ξ 3 )(−iξ ab )(−iξ 3 ) 2 k1(1−α) and Lemma 3.4 lead to Note that we did not cover the case k 2 ∼ k a ≫ k b in the previous estimate.In that case, we have where Here, a direct estimate of the modulus of m only yields the bound 2 k1(1−α) 2 −k3 , which -after an application of Lemma 3.4 -would lead to a factor 2 −k1α 2 k3/2 2 k b /2 .As α < 1, this is insufficient.Thus, we need to use the symmetry of II 1 1 in the variables ξ 2 and ξ a .The spatial integral of (4.14) can be written as For all ξ a , ξ b , ξ 2 , ξ 3 satisfying ξ ab23 = 0 direct calculations yield Note that m 6 = −m(ξ 2 , ξ b , ξ a , ξ 3 ) holds.Hence, with (4.16), we obtain The next step consists of proving For m 1 the desired bound follows from (4.5), whereas for m 5 it is a consequence of Corollary 4.2.For m 2 , m 3 and m 4 the claim follows from estimates of the form Thus, applying Lemma 3.4 to the right-hand side of (4.17), we conclude The term II 1 2 (k 1 , k 3 ) can be dealt with by the same arguments.
Estimating II 2 .To handle the term II 2 , we can repeat the calculations made for II 1 replacing χ 2 k1 and χ k2 by χ k1 and omitting the sum over k 2 .These changes only have an impact on the implicit constants.
Estimating II 3 .Now, we consider Applying (4.13), we obtain , where the terms on the right-hand side are given by Here, II 3 12 contains those summands, which are excluded in the summation of II 3 1 and II 3 2 .When estimating II 3  12 , we will emphasize the advantage of this definition.Also, note that the variable ξ 2 is already of size 2 k1 due to the localization in ν(ξ 1 , ξ 2 , ξ 3 ).Nonetheless, we localize each ξ 2 to dyadic frequency ranges to improve the notation.Similar to the estimation for the boundary term B in Lemma 4.4, we bound BB 3 by Using Lemma 3.4 and the bounds ν as well as It remains to bound II 3  12 .A direct estimate leads to the insufficient factor 2 −k1α 2 k3/2 2 k b /2 forcing us to use some cancellation in the symbol.We have To bound A 1 , we use |ν| 2 k3 , (4.5) as well as ξ ab + ξ 2b = ξ b − ξ 3 , which holds due to ξ ab23 = 0.It follows We observe Trivially, each term on the right-hand side has modulus bounded by 2 −k1 2 k b 2 k3 .Hence, it follows Concerning A 3 , we can use Lemma 4.2 and immediately arrive at Thus, we can bound the modulus of the symbol appearing in II 3 12 by 2 −k2α 2 k b .Lemma 3.4 yields This concludes the proof after summation of the obtained bounds in k 1 and k 3 .

4.3.
Proof of the third energy estimate.We end Section 4 with the proof of estimate (4.3).Recall that u 1 , u 2 are smooth solutions of (1.1) and that we defined v = u 1 − u 2 and w = u 1 + u 2 .Rewriting (4.11), we obtain that v satisfies the equation Thus, as in the beginning of Section 4.1, it follows The advantage of splitting the term above on the right-hand side into X and Y is that X can be estimated by proceeding as in Section 4.1, whereas the bound for Y follows by modifying arguments used in Section 4.3.Let us begin with estimating X.As in Section 4.1, we frequency-localize the second and third factor in the integrand and restrict ourselves to the case k 1 ∼ k 2 ≫ k 3 .Then, using instead of (4.10) and noting that w = v + 2u 2 holds, we deduce Here, we slightly abused the notation introduced in (4.9) -the additional term v indicates that we replace the factors uuu in the spatial integral of (4.9) by vvv.Repeating the proof of Lemma 4.3, it follows Similar to Lemma 4.4, we obtain where we again changed the notation.Here, each term in the first line has its integrand uuu replaced by vvv, whereas each term in the second line has its integrand uuuu replaced by vvvv.The terms in the third line have their integrand uuuu replaced by vu 2 vv, vvu 2 v or vvvu 2 .An inspection of the proof of Lemma 4.4 shows that the bounds of B, I 1 , I 2 and I 3 do not depend on the integrand, whereas the estimate for I 12 requires an integrand that is symmetric in its high-frequency variables.Hence, repeating the proof of Lemma 4.4, we get In order to end the estimation of X, it remains to consider I 12 (k 1 , k 3 ; u 2 ) given by Lemma 4.9.Let s > 3/2 − α.Then, there exists d > 0 such that we have Proof.In the case k a ≫ k b we have k a ∼ k 2 and the integrand v(ξ a )û 2 (ξ b )v(ξ 2 )v(ξ 3 ) is symmetric in the high-frequency variables ξ a and ξ 2 .As in Lemma 4.4, it follows In the remaining case k b ≫ k a we have k b ∼ k 2 and the integrand is no longer symmetric in the highfrequency variables ξ b and ξ 2 .Hence, the argument given in Lemma 4.4 does not apply.Instead, we use the direct bound of the symbol as well as the fact that u 2 can be estimated in a high-regularity norm.An application of Lemma 3.4 yields Recall that we assumed k 1 ∼ k 2 ≫ k 3 .Thus, summation in k 1 and k 3 concludes the proof.
Finally, we obtain Let us continue by estimating Y .We will proceed similar to Section 4.2.However, as the regularity is different, we need to modify some steps, which is why we will provide a complete proof.As before, we have Proof.Writing v = k2 v k2 and noticing that k 2 n holds, the claim follows from the estimate and (2.4).
In order to estimate the remaining frequency interactions, we need another analogue of (4.10), which is given by Proof.Applying (4.22) to each summand III(k 1 , k 3 ), we obtain where the terms on the right-hand side are given by Similar to the boundary term B in Lemma 4.4, the term BBB can be estimated by An application of Lemma 3.4 in combination with the bounds After summation in k 1 and k 3 , the proof is completed.Lemma 4.12.Let s > 3/2 − α.Then, there exist c, d > 0 such that we have Proof.Since we still have one high-and one low-frequency factor v (see (4.21)), we can repeat the proof of the previous lemma. .
Together with we conclude Next, we apply (4.22) to each III j (k 1 , k 3 ), j ∈ [2], and obtain , where the terms on the right-hand side for j = 1 are given by As before, the terms for j = 2 follow from obvious modifications.We proceed by bounding the terms III 1 , III 2 and III 3 separately.
Estimating III 1 .Note that k 1 ∼ k 2 holds.As in Lemma 4.4, the boundary term admits the estimate Let us continue with III 1 3 .An application of Lemma 3.4 together with To treat the case k a ∼ k 1 ≫ k b , we proceed as in Lemma 4.8.The modulus of , where m is defined in (4.15).Using the symmetry of r in its first and third variable, we write [m(ξ a , ξ b , ξ 2 , ξ 3 ) + m(ξ 2 , ξ b , ξ a , ξ 3 )] r(ξ a , ξ b , ξ 2 , ξ 3 ).
Moreover, recall that holds.Combining the last observations with Lemma 3.4, we arrive at In the case k b ∼ k 1 ≫ k a , r is not symmetric in its second and third variable.Hence, we only obtain Similar to Lemma 4.8, we can write [m(ξ a , ξ b , ξ 2 , ξ 3 ) + m(ξ a , ξ 2 , ξ b , ξ 3 )] = j∈ [5] m ′ j for some appropriately chosen summands m ′ j with modulus bounded by 2 max{k3,ka} 2 −k2α .This allows us to estimate the first sum on the right-hand side of (4.23) as before.For the second sum, observe that

Proof of the main theorem
In this section we will use the results of the previous sections in order to prove Theorem 1.1.The proof is divided into four steps: First, we recall a result guaranteeing the existence and uniqueness of smooth solutions of (1.1) as well as the continuity of the data-to-solution-map. Second, we collect the estimates obtained in the previous sections.This leads to an a priori estimate in H s (T) for smooth solutions of (1.1) with mean zero as well as to two estimates for the difference of smooth solutions with mean zero -one estimate in H −1/2 (T) and another in H s (T).In the third step, we approximate an initial datum u 0 ∈ H s 0 (T) by smooth initial data (u k,0 ) k∈N and show that the sequence of smooth solutions (S ∞ T (u k,0 )) k∈N converges in C([0, T ]; H s (T)).Denoting the limit by S s T (u 0 ), this yields a continuous extension of S ∞ T to H s 0 (T).In the last step, we upgrade this to an extension of S ∞ T to H s (T) by using the conservation of the mean along the flow of equation (1.1).
Step 1.The following statement is a direct consequence of Theorem 6 and 7 proved in [13]: Proposition 5.1.Let R > 0 and σ > 3/2.Then, there exists a positive time In particular, we obtain the existence, uniqueness and continuity of the map Step 2: Fix α ∈ (0, 1), T ∈ (0, 1] and r ≥ s.Let u 0 , u 1,0 and u 2,0 be H ∞ 0 (T)-functions and denote the corresponding solutions of ( (5.2) Moreover, we obtain two sets of estimates for differences of solutions.More precisely, we have According to Proposition 2.3, all quantities above are finite, which allows to bootstrap each set of estimates.Doing so, we obtain a common time T = T ( u 0 H s ) > 0, for which we have an a priori estimate in H r 0 (T) given by u L ∞ T H r and a difference estimate in H s 0 (T) given by (5.7) Now, we specify u 1,0 = u 0 and u 2,0 = P ≤n u 0 for n > 0. In that case, we can improve (5.7).From the inequality (5.6), we obtain Combining these estimates with (5.7), we arrive at .
For n sufficiently large (depending on the implicit constant in the inequality above, the H s (T)-norm of u 0 and on 3/2 − α − s), we conclude Recalling that we chose u 1 = S ∞ T (u 0 ) and u 2 = S ∞ T (P ≤n u 0 ), we arrive at the estimate S ∞ T (u 0 ) − S ∞ T (P ≤n u 0 ) L ∞ H s P >n u 0 H s (5.8) holding for any u 0 ∈ H ∞ 0 (T), every s > 3/2 − α and all sufficiently large n.
Step 3: In the third step, we show that S ∞ T extends continuously to B R (0) ⊂ H s 0 (T).Fix u 0 ∈ H s 0 (T) and let (u k,0 ) k∈N be a sequence of smooth functions with mean zero converging to u 0 in H s (T).For n, k, l ∈ N, we have T (u l,0 ) − S ∞ T (P ≤n u l,0 ) L ∞ H s + S ∞ T (P ≤n u k,0 ) − S ∞ T (P ≤n u l,0 ) L ∞ H s . (5.9) We will prove that the right-hand side converges to zero as k and l tend to infinity.Fix ǫ > 0. There exists k 0 such that for all k > k 0 and all n ∈ N we have Moreover, we find n 0 such that for all n > n 0 the inequality P >n u 0 H s < ǫ holds.Hence, choosing n > n 0 , we get P >n u k,0 H s < 2ǫ.Applying (5.8), we conclude that for all k > k 0 and all n > n 0 we have S ∞ T (u k,0 ) − S ∞ T (P ≤n u k,0 ) L ∞ H s 2ǫ.
(5.10) Due to P ≤n u k,0 − P ≤n u 0 L ∞ H r ≤ P ≤n (u k,0 − u 0 ) L ∞ H s if r ≤ s, 2 n(r−s) P ≤n (u k,0 − u 0 ) L ∞ H s if r > s, convergence of (u k,0 ) k∈N to u 0 in H s (T) implies convergence of (P ≤n u k,0 ) k∈N to P ≤n u k,0 in H r (T) for any r ∈ R. In particular, this yields convergence of (P ≤n u k,0 ) k∈N to P ≤n u 0 in H ∞ (T).Thus, the continuity of the map S ∞ T implies S ∞ T (P ≤n u k,0 ) − S ∞ T (P ≤n u 0 ) L ∞ H s ǫ for sufficiently large k leading to S ∞ T (P ≤n u k,0 ) − S ∞ T (P ≤n u l,0 ) L ∞ H s 2ǫ. (5.11) From (5.9), (5.10) and (5.11), we deduce S ∞ T (u k,0 ) − S ∞ T (u l,0 ) L ∞ H s 6ǫ for all sufficiently large k and l.Thus, (S ∞ T (u k,0 )) k∈N is a Cauchy sequence in C([0, T ]; H s (T)), which implies the existence of a unique limit denoted by S s T (u 0 ) ∈ C([0, T ]; H s (T)).Hence, the map S s T extends continuously to H s 0 (T).
Step 4: We recall that the mean is conserved along the flow of (1.1).In particular, if u 0 ∈ H ∞ (T) and c ∈ R, then the solution with initial datum u 0 + c can be written as T (u 0 + c)(t, x) = S ∞ T (u 0 )(t, x + 2ct) + c.Fix u 0 ∈ H s (T) and let (u k,0 ) k∈N be a sequence converging to u 0 in H s (T).Using the previous equation and the continuous extension of S ∞ T to H s 0 (T) from Step 3, it is easy to conclude that (S ∞ T (u k,0 )) k∈N is a Cauchy sequence in C([0, T ]; H s (T)).Then, denoting the unique limit of (S ∞ T (u k,0 )) k∈N by S s T (u 0 ), we conclude that the map S ∞ T extends continuously to H s (T).5.1.Improved a priori estimate.As indicated in the introduction, the a priori estimate (5.5) can be improved for α ∈ (1/2, 1) by some minor modifications of the proofs given in the previous sections.We omit the exact details and just point out which estimates need to be replaced.
From now on, let the parameter I in the definitions of F s T , N s T and E s T be slightly larger than 2 − α.We will show a linear, a nonlinear and an energy estimate as in (5.2).The linear estimate (2.5) holds for all r ∈ R and any I > 0, see Proposition 2.3.Next, we claim that the nonlinear estimate holds for all r > 0. Indeed, if we repeat the proof of Lemma 3.3, the case k 1 ∼ k 2 ≫ k 3 can be improved using I > 2 − α.For all remaining cases, we replace each application of (3.2) by an application of In the range α ∈ (1/2, 1), we can repeat the proofs given in Section 4.1 with the estimate above.As a consequence, we obtain the energy estimate for all r > max{2(1 − α), 5/4 − α}.

2. 1 .
Notation.For positive real numbers a, b > 0 we write a b if a ≤ Cb holds for some constant C > 0.Moreover, we write a ≫ b if a b as well as a ∼ b if a b and b a are satisfied.