On special regularity properties of solutions of the benjamin-ono-zakharov-kuznetsov (bo-zk) equation

In this paper we study special properties of solutions of the initial value problem (IVP) associated to the Benjamin-Ono-Zakharov-Kuznetsov (BO-ZK) equation. We prove that if initial data has some prescribed regularity on the right hand side of the real line, then this regularity is propagated with infinite speed by the flow solution. In other words, the extra regularity on the data propagates in the solutions in the direction of the dispersion. The method of proof to obtain our result uses weighted energy estimates arguments combined with the smoothing properties of the solutions. Hence we need to have local well-posedness for the associated IVP via compactness method. In particular, we establish a local well-posedness in the usual \begin{document}$ L^{2}( \mathbb R^2) $\end{document} -based Sobolev spaces \begin{document}$ H^s( \mathbb R^2) $\end{document} for \begin{document}$ s>\frac{5}{4} $\end{document} which coincides with the best available result in the literature proved employing more complicated tools.


1.
Introduction. In this work we are interested in the study of some special properties for solutions of the Benjamin-Ono-Zakharov-Kuznetsov (BO-ZK) equation. We will consider the initial-value problem (IVP) for the Benjamin-Ono-Zakharov-Kuznetsov (BO-ZK) equation ∂ t u + αH∂ 2 x u + ∂ x ∂ 2 y u + u∂ x u = 0, u(x, y, 0) = φ(x, y), (1.1) where u = u(x, y, t) is a real-valued function, (x, y) ∈ R 2 , t > 0, α = 0, and H stands for the Hilbert transform defined as The equation (1.1) is a two-dimensional generalization of the Benjamin-Ono (BO) equation ∂ t u + H∂ 2 x u + u∂ x u = 0, u = u(x, t), x ∈ R, t > 0, (1.3) when the effects of long wave lateral dispersion are included. The Benjamin-Ono equation (1.3) was proposed as a model for unidirectional long internal gravity waves in deep stratified fluids (see [2] and [43]).
The BO-ZK equation can also be considered as a non-local version of the Zakharov-Kuznetsov (ZK) equation ∂ t u + α∂ 3 x u + ∂ x ∂ 2 y u + u∂ x u = 0, u = u(x, y, t), (x, y) ∈ R 2 , t > 0. (1.4) The ZK equation (1.4) was introduced by Zakharov and Kuznetsov as a higherdimensional extension of the Korteweg-de Vries model of surface wave propagation (see [45]). The BO-ZK equation was deduced in [21] and [31] and has applications to electromigration in thin nanoconductors on a dieletric substrate.
Moreover, for any v ≥ 0, > 0 and R > 0  (∂ x u) 2 (x, t) dxdt < c(R; T ; u 0 L 2 ). (1.10) The estimate (1.10) is known as Kato's smoothing effect. Roughly, the proof of (1.10) follows by noticing that a smooth solution to the KdV satisfies with l ∈ Z + ∪ {0}. Next, we select l = 0 and ψ ∈ C 3 (R) to be an appropriate nonnegative, nondecreasing cutoff function with ψ compactly supported and integrate in time to obtain (1.10). There exist a relation between Kato's smoothing effect and the propagation of regularity obtained in Theorem 1.1. We will see this also in the case we are considering. The mathematical study of the BO-ZK equation has given rise to some papers in recent years. Regarding existence and stability of solitary waves solutions for BO-ZK equation we refer [10] and [13] where the authors proved the orbital stability of ground state solutions in the energy space.
It is worth to notice that in [13] the authors established an anisotropic Gagliardo-Nirenberg type inequality whose optimal constant were later characterized by Esfahani and Pastor [11], in terms of the ground state solutions of BO-ZK equation.
As an application of their results, the authors in [11] proved the uniform bound of smooth solutions in the energy space.
Regarding unique continuation properties we refer [7] and [12]. More precisely, the authors in [12] established if a sufficiently smooth solution is supported in a rectangle (for all time), then it must vanish identically. Later, in [7] the authors improved this result by showing if a sufficiently smooth local solution has, in three different times (not for all time), a suitable algebraic decay at infinity, then it must be identically zero.
The authors in [7] also proved local well-posedness in the anisotropic Sobolev spaces H s1,s2 (R 2 ), s 2 > 2, s 1 ≥ s 2 . Esfahani and Pastor [9], following the ideas of [39] established the ill-posedness of (1.1) in the sense that it cannot be solved in the usual L 2 -based Sobolev space by using a fixed point argument. More precisely, the map data-solution cannot be C 2 -differentiable at the origin from H s1,s2 (R 2 ) to H s1,s2 (R 2 ) for any s 1 , s 2 ∈ R.
Later, Cunha and Pastor [6] obtained local well-posedness for the BO-ZK equation (1.1) with α = 1, in the Sobolev spaces H s (R 2 ), s > 11 8 . Their proof is based on the refined Strichartz estimates introduced by Koch and Tzvetkov [30] in the context of the Benjamin-Ono equation.
Recently, Ribaud and Vento [44] showed that the initial value problem associated to the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov equation is locally well-posed in the spaces E s , s > 2 σ − 3 4 , endowed with the norm f E s = || |ξ| σ + η 2 s f || L 2 (R 2 ) . They also proved global well-posedness in the energy space E 1 2 as long as σ > 8 5 .
Observe that E s is nothing but the anisotropic Sobolev space H σs,2s (R 2 ). Their proof is based on the approach of the short time Bourgain spaces from Ionescu, Kenig and Tataru [17] combined with some new Strichartz estimates and modified energy.
In particular, we observe that when σ = 2 the equation (1.12) is the well-known ZK equation while for σ = 1 the equation (1.12) reduces to the BO-ZK equation. Thus, in the context of the BO-ZK, the authors in [44] obtained local well-posedness in the anisotropic Sobolev space H s,2s (R 2 ) for s > 5 4 . Our contribution in this direction is to improve the above results by pushing down the Sobolev regularity index of the classical H s (R 2 ) spaces. In fact, we obtain the following Theorem 1.3. Let s > 5 4 . The IVP (1.1) is locally well-posed in H s (R 2 ). More precisely, there exist T = T (||φ|| H s ) > 0 and a unique solution u to (1.1), such that (1.14) Moreover, the mapping φ → u ∈ C([0, T ]; H s (R 2 )) is continuous.
The main tool of the proof of this result is a refined Strichartz inequality (see Lemma 3.6 below). This type of estimate was first introduced by Koch and Tzvetkov [30] in the context of the Benjamin-Ono equation. It was extended by Kenig and Koenig [25] for BO equation and Kenig [24] for Kadomtsev-Petviashvilli (KP-I) equation. Roughly, one needs to control ||∂ x u|| L 1 T L ∞ xy . Before to enunciate our main result for BO-ZK equation, we need to define some weights independent of the variable y, which are a class of real functions χ ,b ∈ C ∞ (R) for > 0 and b ≥ 5 with χ ,b (x) ≥ 0 and which will be constructed in the following way. Let ρ ∈ C ∞ 0 (R) a function which is even, non-negative, with suppρ ⊆ (−1, 1) and ρ(x)dx = 1 and define We define η ,b by the identities 20) and observe that for any > 0 and b ≥ 5 , the function Our main result for BO-ZK equation reads as follows Then the solution u in [0, T ] of equation (1.1) with α = −1 provided by Theorem 1.3 satisfies for any ν > 0, for l = 0, 1, 2, ..., m with C = C T, φ H s (R 2 ) , ∂ m x φ L 2 ((x0,∞)×R) , , b, ν . In particular, for all times t ∈ (0, T ] and for all a ∈ R, ∂ m x u(t) ∈ L 2 ((a, ∞) × R). Moreover, for any ν > 0, > 0 and R > 0 ). The proof of Theorem 1.4 is based on an induction argument on m, combining some weighted energy estimates the properties (1.14) of the solutions to (1.1) and Gronwall's inequality. However, the presence of a non-local operator, the Hilbert transform (1.2), lead us to use some commutator estimates with the extension of the Calderón first commutator given by Bajvsank and Coifman in [1] (see Theorem 3.5 below) in order to obtain the local smoothing effect which is crucial to carry on our argument of induction. Besides, we notice that unlike KdV equation in which we gain one derivative in the local smoothing effect (see Theorem 6.1 in [22]), for the BO-ZK equation, we have the gain of the local smoothing effect of just 1/2 derivative, so the iterative argument has to be carried out in two steps, one for positive integers m and another one for m + 1/2.
It is worth notice that the proof of Theorem 1.4 can be extended to solutions of the IVP (1.27) (iii) The solutions of the defocusing BO-ZK equation We present now, some immediate consequences of Theorem 1.4. 29) then for l = 0, 1, ..., k any t ∈ (0, T ), ν > 0 and > 0 and for for l = 0, 1, ..., m any t ∈ [−T, 0) and α ∈ R As aforementioned, the authors in [26] extended the result in Theorem 1.1 for the generalized KdV for initial data with restricted H s ((x 0 , ∞))-norm, s real, instead of H l ((x 0 , ∞)), l integer. More precisely, Theorem 1.5. Let u 0 ∈ H 3/4 + (R). If for some s ∈ R, s > 3/4, and for some then the solution u = u(x, t) of the IVP (1.7) associated to the generalized KdV equation satisfies that for any v > 0 and > 0 It is an open problem to extend this result for nonlinear dispersive models in higher dimensions.
Roughly speaking, Theorem 1.6 tell us that polynomial decay of the initial data yields to more regular solutions. On the other hand, Cunha and Pastor [7] where s > 2, r ≥ 0, and s ≥ 2r. We would like to know whether solutions of the BO-ZK equation satisfy a property similar to the one described in Theorem 1.6. Recently, Linares, Miyazaki and Ponce in [33] considered the following IVP associated to the generalized KdV equation with low degree of non-linearity (1.36) They established that suitable solutions of the IVP (1.36) satisfy the propagation of regularity principle proven in Theorem 1.1 for solutions of the k-generalized KdV equation in (1.7) (see Theorem 1.6. in [33]). We would like to extend this result for the high dimensional models with non-linearity of fractional order such as generalized BO-ZK equation (1.37) The rest of this paper is organized in the following manner: Section 2 contains some notations that will be used in this work. Section 3 contains the proof of Theorem 1.3, our new local well-posedness result for BO-ZK equation. In Section 4 a local smoothing effect for BO-ZK equation is derived (see Proposition 3) which is an useful tool to carry on our argument of proof. Next, we perform the proof of Theorem 1.4. In Appendices we present several estimates that are crucial in our analysis.
2. Notation and the resolution space. Given any positives quantities C, D, by C D we mean that there exists a constant c > 0 such that C ≤ cD; and, by C ∼ D we mean C D and D C. For a real number r we shall denote r+ instead of r + , whenever is a positive number whose value is small enough. Given two operators A and B, we denote by [A, B] = AB − BA the commutator between A and B. By F{u} or u we will denote the Fourier transform of u with respect to the space variable, while F −1 {u} orǔ will denote its inverse Fourier transform. L p -norms will be written as · L p xy or · L p if no confusion is caused. For 1 ≤ p, q < ∞ and f : 3. Local well posedness in H s (R 2 ), s > 5 4 . In this section we will prove Theorem 1.3. We follow the argument in [24] to obtain the result. Our argument of proof uses energy estimates. The key point is to establish an estimate of the kind ||∂ x u|| L 1 Lemma 3.8 ). To this end, we prove a refined Strichartz estimates (see Lemma 3.6 ) for solutions of the linear problem.
3.1. Linear estimates. Consider the IVP associated to the BO-ZK equation, The solution of (3.1) is given by the unitary group {U (t)} ∞ t=−∞ such that where We recall the Strichartz-type estimates for solution (3.2).
Proof. Hölder's inequality in t followed by an application of Lemma 3.3 implies that for p > 2 and 2 q = θ(3+2δ)

Preliminary estimates. Consider the IVP
We begin this section establishing an energy estimate for solutions of the IVP (3.12). We will borrow the result of Cunha and Pastor [7] (see Theorem 1.2 above) where smooth solutions of (3.12) were obtained.
As usual we apply the operator J s to the equation in (3.12) and multiply it by J s u. Then we use integration by parts in x, combined with Kato-Ponce commutator estimates (see [23]) and Gronwall's Lemma to obtain the following: for 0 < t < T.
We present the following extension of the Calderón commutator Theorem [5] established by Bajvsank and Coifman in [1]. This estimate will be a crucial ingredient in the proof of Theorem 1.4. (3.14) Proof. See Lemma 3.1 in [8].

3.3.
Refined strichartz estimate. The next result is fundamental in our analysis.
) is a solution to the linear equation Then, Proof. In order to obtain (3.16), we will use a Littlewood-Paley decomposition of we obtain from Hölder's inequality in time, Corollary 2 and Bernstein's inequalities that To this end, we assume for simplicity that T = 1 and split the interval [0, 1] = j I j in subintervals From the Hölder's inequality in I j , and the fact that ξν(2 −k ξ) has inverse Fourier transform whose L 1 norm in x is bounded by cλ, we have (3.20), and Corollary 2 we have From (3.18) and (3.22) we obtain (3.16). This concludes the proof.

3.4.
A priori estimates. In this section we derive an a priori estimate for the norm ∂ x u L 1 T L ∞ xy based on the refined Strichartz estimate deduced in Lemma 3.6. Notice that we also need to control u L 1 xy and prove the following Lemma 3.7. Let u be a solution to the Cauchy problem (3.12) with φ ∈ H ∞ (R 2 ), provided by Theorem 1.2 and defined for some T > 0. Then, for any s > 5 4 , for a fixed, universal constant C.

Thus, an application of Lemma 3.4 leads to
Using Lemma A.2 item (A.2) in the y variable, we have We argue as we did in II 1 to conclude that since > 0 is chosen small enoug so that 1 + ≤ 5 4 + 0 = s.
We observe that Young's inequality implies that where γ ∼ 1 4 − and > 0 are chosen such that 1 2 − γ + 2 ≤ 5 4 + 0 = s. From Placherel's theorem and Lemma 3.4, we have We proceed similarly, using that for γ ∼ 1 4 − and > 0 such that 3 2 − γ + 2 ≤ 5 4 + 0 = s and using Lemma 3.4, to see that We note that where > 0 is chosen such that 1 + ≤ 5 4 + 0 = s. From Lemma 3.4, and (A.9) of the Lemma A.8, we have We observe that, We apply Hölder's inequality in t, Remark A.8, (A.11), (A.12), Lemma 3.4, and the estimate (3.24) to obtain that Combining the estimates for I, II, III and IV, we obtain The estimate for u L 1 T L ∞ xy , can be obtained arguing along the same lines as above to get From the (3.25) and (3.26) we obtain (3.23) and hence (3.27). This finishes the proof.
Next, note that u(x,y,t) is a solution of (3.12), with initial data φ, if and only if u λ (x,y,t) = λ 2 u(λ 2 x,λy,λ 4 t) is a solution of (3.12) with initial data φ λ (x,y,t) = we see that all the exponents of λ are different of zero, so we can first choose λ = λ( φ H s ) such that φ λ H s ≤ 0 , and apply the above conclusion to u λ to obtain (3.27). Finally, an application of Lemma 3.4 concludes the proof of (3.27).

3.5.
Local well posedness result. We follow a similar argument to that employed in [16] (see [7] for more details). We will use essentially Lemmas 3.4 and 3.7 to obtain our result. Let φ ∈ H s , s > 5 4 and > 0, we consider the Cauchy problem From Theorem 1.2 and its proof, it follows, the problem (3.29) has a unique solution u such that in H s (R 2 ) uniformly in , when → 0. Moreover, we have This implies that u ∈ L ∞ ([0, T ] : H s (R 2 )). Let u and u be solutions of IVP (3.29). Setting ω(t) = (u − u )(t) for > > 0, we observe that ω satisfies (3.32) We multiply (3.32) by ω and integrate by parts to obtain From (3.30) and Gronwall's inequality we have as , → 0. Thus, we conclude that {u } is a Cauchy sequence in L ∞ ([0, T ] : L 2 (R 2 )) and ∂ x u 2 converges to ∂ x u 2 in the distributional sense. Then, u satisfies (3.12) in the distributional sense. Therefore, we have obtained . The uniqueness of u follows from the same use of Gronwall's inequality as in (3.34). In fact, Suppose that u 1 and u 2 are two solutions of IVP (3.12). We set w = u 1 − u 2 and observe that w satisfies with w(·, 0) = φ 1 − φ 2 . We multiply (3.37) by w, integrate by parts to obtain that Using Hölder's inequality we obtain that which implies from Gronwall's inequality that This concludes the proof of uniqueness.
It remains to show that u ∈ C([0, T ]; H s ). In fact, let ϕ ∈ H s , ϕ H s = 1 we have Thus, continuity at zero is a consequence of weak continuity. If τ ∈ (0, T ] we obtain continuity from continuity at zero and uniqueness. Since the equation in founded previously satisfies some special properties. We start our task by proving several needed estimates in our analysis. Before stating our result we define the class of solutions to the IVP (4.1) to which it applies. We shall consider solutions provided by Theorem 1.3, satisfying the properties (1.14) namely, for s > 5 4 , the solution u provided by Theorem 1.3 satisfies, in addition to belong to C([0, T ]; H s (R 2 )) We also deduce a Kato's smoothing effect (see Proposition 3) that will be an important ingredient of our analysis.

4.1.
Preliminares. We present some useful properties of the weights functions originally constructed in [20] and [26] to prove our main result.

Remark 3. From Proposition 2 and the fact that
provided by definition (1.20) we notice that the function η ,b satisfies We introduce now a local smoothing properties of the BO-ZK equation.
Proposition 3. Let s > 5 4 and φ ∈ H s (R 2 ). Then the corresponding solution u of (4.1) satisfies, for any R > 0 and T > 0, x J s x u(x, y, t)) 2 +(J s x ∂ y u(x, y, t)) 2 dxdydt ≤ C(R, T, φ H s ). (4.5) Proof. We consider ψ = ψ(x) ∈ C ∞ (R) an increasing function such that ψ ∈ C ∞ 0 (R). Formally, we apply J s x to equation (4.1) multiply by J s x u ψ, and integrate the result in the space variable to obtain, We obtain from integration by parts that Using the fact that Hilbert transform is skew symmetric, from integration by parts we have

This implies that
Using the commutator estimate (3.14) of the Theorem 3.5, we obtain Next, we use the notation η 2 = ψ to rewrite B as follows We notice that B 1 in (4.10) is positive and represents the smoothing effect.
In order to estimate the remaining terms in (4.10) we use the fact that H is an isometry in L 2 and the commutator estimate (3.14) of the Theorem 3.5, to obtain Integration by parts and Kato-Ponce's commutator estimate [23] implies that Combining the estimates (4.7) − (4.12), we obtain that Using the a priori estimate (3.13) the fact that the solution u provided by the Theorem 1.3 satisfies (4.2), i.e., ∂ x u L 1 T L ∞ xy < ∞ and integrating in time we conclude that . This finishes the proof.

4.2.
Proof of Theorem 1.4. By translation if necessary, we may assume that x 0 = 0. We apply ∂ l x to equation (4.1), multiply by ∂ l x u χ ,b (x + νt) and integrate in R 2 . If we have enough regularity to apply integration by parts we obtain that We recall that g ≥ 0 is a function with T 0 g(t)dt ≤ C. Notice that, eventually, we will mix several cases together to perform our analysis and then we will carry on denoting by g a generic nonnegative integrable function on [0, T ].
Since given T > 0, > 0, b > 5 , ν > 0, there exist c > 0 and R > 0 such that Thus, we obtain that We observe that the term A 2 is positive and provides a smoothing effect.
To estimate the terms A 3 and A 4 we consider the two cases l = 0 and l = 1 separately. Regarding the first case l = 0 we use integration by parts to rewrite the term A 3 in (4.15) as follows Using the fact that Hilbert transform is skew symmetric, from integration by parts we have Thus, we have Using the commutator estimate (3.14), we obtain Now, we use that η 2 = χ to rewrite A 32 as follows First of all, we observe that A 321 (t) in (4.21) is positive and represents the smoothing effect.
To estimate A 322 we use the fact that H is an isometry in L 2 to write From the commutator estimate (3.14) in Theorem 3.5, and integration in time interval [0, T ] we see that To estimate the term A 4 in (4.15) we use Sobolev embedding, the conservation of L 2 norm of the solutions (1.5) and (4.16), to obtain that Next, we consider the case l = 1. In order to control the contribution of A 3 in (4.15) we perform integration by parts and write

An argument similar to the one used in (4.19) leads to
Using the commutator estimate (3.14) we obtain Similar to the argument in (4.21) Firstly, we observe that A 321 (t) in (4.27) is positive and represents the smoothing effect.
To estimate A 322 we use the fact that H is an isometry in L 2 to obtain . (4.28) The commutator estimate (3.14), after integration in time interval [0, T ], Sobolev embedding and the conservation of L 2 norm of the solutions (1.5) leads to .

(4.29)
To estimate the term A 4 in (4.15) we use integration by parts to observe that and This gives the result for all indices l = 0, 1.
Case l = 1 in (1.24). After apply D 1/2 x ∂ x to (4.1), multiply the resulting by D 1/2 x ∂ x uχ and integrate in the space variable we get the identity 1 2 We use that η 2 = χ to write We notice that the first term on the RHS of (4.35) is bounded, after integration in time, by the former case l = 1 in (1.22) estimate (4.33). To estimate the term A 12 on the RHS of (4.35) we use the fact that We employ Lemma A.5, to obtain which is bounded, after integration in time, by the former case of order l = 1 and the estimate (4.17) since φ ,b is compactly supported. Since dist(supp(η ,b ), supp(ψ )) ≥ 2 , we employ Lemma A.7, to obtain Thus, we obtain Regarding the term A 3 , using integration by parts we have A similar argument to that one used in (4.19) − (4.76) yields Now, we use the fact that η 2 = χ and a familiar argument analogous to that one used in (4.21) to write We notice that A 321 (t) in (4.42) is positive and represents the smoothing effect. Now, an argument similar to that one employed in (4.22) − (4.23) provide us The two term on RHS of (4.43) are similar, and the first one is simply a multiply of the term A 1 above, so it is treated exactly as we did before in (4.39).
It only remains to handle the term A 3 in (4.34). Since χ = χ /5, ≡ 1 on supp χ, we write Firstly, we rewrite A 41 as follows Combining (A.2), Lemma A.5 and Lemma A.6 in the variable x and then Hölder's inequality in the variable y, we have and similarly We recall that by construction so an application of Lemma A.7 in the variable x and then Hölder's inequality in the variable y, yields where the first identity is obtained taking ( , b) instead of ( /5, ).
Analogously to (4.53) we write Now, we apply the commutator estimates (3.14) and Lemma A.5 in the variable x and then Hölder's inequality in the variable y, to obtain and where we have chosen ( , b) instead of ( /5, ). Since the supports of χ ,b and ψ are separated, a familiar argument similar to that one performed in (4.48) yields where the first identity is obtained taking ( , b) instead of ( /5, ). Next, we write We notice that the L 2 norm of the first term on the RHS of (4.53) is the very quantity to be estimated. The control of the L 2 of the second one can be performed by employing a familiar argument, similar to that one used in the analysis of the terms A 41 , A 42 and A 43 above. Hölder's inequality, Proposition 2 and Theorem A.3 in the variable x provide us (4.54) Next, employ the L 2 norm in the variable y to obtain (4.55) The second term on the RHS of (4.55) can be controlled, by a familiar argument similar to that one performed in the analysis of A 1 in (4.39) while the third one is bounded by Sobolev's embedding. Lastly, we apply integration by parts to obtain The last integral term is the quantity to be estimated, and the other term will be controlled after integration in time by (4.2). We apply Sobolev's embedding to obtain where the last term is bounded, after integration in time, by using a familiar argument analogous to that one used in the analysis of the term A 1 in (4.39) above. We insert the above information contained in (4.39) − (4.56) in (4.34), apply Gronwall's inequality and (4.2) to obtain This gives the desired estimate (1.25) with l = 1. Next, we consider the case l = 2 Case l = 2 in (1.21).So, (4.15) can be written as follows We write (4.59) We notice that I 2 L 2 xy is bounded, after integration in time, by the former case (4.57). The term I 1 L 2 xy is controlled by using commutator estimate (3.14) while I 3 L 2 xy is bounded, after integration in time, by employing a similar argument to that one used in the analysis (4.46) − (4.48) above, combined with the former case (4.57). Since To control the contribution of A 3 in (4.58) we perform integration by parts and write Using the same argument performed to obtain (4.19) yields From the commutator estimate (3.14) we obtain Now, we use that η 2 = χ to rewrite A 32 and perform the same argument given in (4.21) to get First, we observe that A 321 (t) in (4.77) is positive and represents the smoothing effect.
To estimate A 322 we use the fact that H is an isometry in L 2 , Proposition 2, (4.4) and the estimate (4.60) to obtain Using the commutator estimate (3.14), after integration in time and the estimate (4.60) one finds that Finally, we consider A 4 in (4.58). Using integration by parts, we see that The first term on the RHS of (4.67) is the very quantity to be estimated while the last term can be handled as we did in the former case A 1 estimate (4.60). Inserting the above information (4.60) − (4.67) in (4.58) applying Gronwall's inequality and (4.2) we obtain that This gives the result for the case l = 2.
Case l = 2 in (1.24). We assume that (1.21) holds for l = 2 and that (1.22) holds for any > 0 and b ≥ 5 with x 0 = 0 and l = 2 we shall prove (1.25). We apply D 1/2 x ∂ 2 x to (4.1), multiply the result by D 1/2 x ∂ 2 x uχ and integrate in the space variable to get the identity 1 2 Firstly, using that η 2 = χ we write The first term on the RHS of (4.70) is bounded, after integration in time, by the former case l = 2 in (1.22) estimate (4.68). Regarding the term A 12 on the RHS of (4.70) we use the fact that We employ Lemma A.5, to obtain which is bounded, after an application of Leibniz rule from calculus and integration in time, by the former cases of order l = 1, 2 and the smoothing effect (4.60). Notice that we have used the fact that φ ,b is compactly supported, so the familiar argument in (4.60) can be employed. Next, we recall that by construction dist(supp(η ,b ), supp(ψ )) ≥ 2 , so applying Lemma A.7, we have Concerning the term A 3 , we use integration by parts to write We perform a similar argument to that one used in (4.19) to obtain From the commutator estimate (3.14) we obtain Now, we use the fact that η 2 = χ and the same argument given in (4.21) to write Firstly, we observe that A 321 (t) in (4.77) is positive and represents the smoothing effect. Now, an argument similar to that one in (4.65) yields Firstly, we remark that the two terms on the RHS of (4.78) are similar, thus we restrict ourselves the first one. In fact, the first one is nothing but a multiply of the term A 1 above, so it is treated exactly as we did before in (4.70). Finally, a familiar argument yields (4.79) Once again, the first term on the RHS of (4.79) is treated exactly as we did before for A 1 in (4.70). Finally, we consider the A 4 term in (4.69). Since χ = χ /5, ≡ 1 on supp χ, we rewrite A 4 as follows We employ Proposition 2 in the first two terms on the RHS of (4.80) to obtain Hölder's inequaity yields In order to estimate A 47 (t) on the RHS of (4.82) we apply integration by parts to obtain Firstly, where the last integral is the very quantity to be estimated, and the other term will be controlled after integration in time by (4.2). Sobolev's embedding yields where the last term can be controlled using a familiar argument analogous to that one performed in the analysis of the term A 1 above. Combining (A.2), Lemma A.5 and Lemma A.6 in the variable x and then Hölder's inequality in the variable y, one gets and Regarding the last quadratic term we apply Lemma A.7 in the variable x and then Hölder's inequality in the variable y, to obtain Applying Lemma A.5 in the variable x and then Hölder's inequality in the variable y, we have where we have chosen ( , b) instead of ( /5, ). Similarly, we obtain We recall that by construction so a familiar application of Lemma A.7 provide us where the first identity is obtained taking ( , b) instead of ( /5, ). In order to finish the estimates in (4.84) − (4.87) only remains to bound D xy . The second and third term above are analogous, so we will restrict ourselves to the analysis of the last one. We perform the analysis of the first term by observing that We notice that the L 2 norm of the first term on the RHS of (4.89) is the very quantity to be estimated while the control of the L 2 of the second one can be performed by using a familiar argument, similar to that one employed in (4.71) above. An application of Hölder's inequality, Proposition 2 and Theorem A.3 in the variable x yields (4.90) Next, an argument similar to that one employed in (4.55) provide us (4.91) The second and third term on the RHS of (4.91) can be controlled, after integration in time, as we did before for A 1 in (4.70) and (4.39), respectively, taking ( , b) instead of ( /24, b + 7 /24). We insert the above information (4.70) − (4.90) in (4.69), apply Gronwall's inequality and (4.2) to obtain This gives the desired estimate (1.25) with l = 2.
We follows an induction argument by assuming that (1.22) holds for l ≤ m ∈ Z + , m ≥ 3. More precisely, we assume An analogous argument to that one used in (4.70) yields The term A 11 is bounded, after integration in time, by induction hypothesis (4.93).
In order to control term A 12 we perform a similar argument to that one used in the analysis (4.71) − (4.73) above, applying the smoothing effect as (4.60) combined with (4.93) for l = 1, 2..., m.
Concerning the term A 3 in (4.94) a familiar argument provides A similar argument to that one used in (4.75) yields From the commutator estimate (3.14) we obtain Using the fact that η 2 = χ and the same argument employed in (4.21) we obtain Firstly, we observe that A 321 (t) in (4.99) is positive and represents the smoothing effect. Next, an argument similar to that one performed in (4.78) − (4.79) yields The two terms on the RHS of (4.100) are similar and can be treated in the same manner as we did before for A 1 in (4.70).
Only remains to estimate A 4 to finish this part of the proof. We will assume l an even integer. The case where l is odd follows by an argument similar to the case l = 1. A familiar argument similar to that one used in (4.81) yields An application of Hölder's inequaity provide us A similar argument to that one used in (4.83) implies The terms A 471 and A 472 can now be controlled, exaclty as we did before in the analysis of (4.83). More precisely, we obtain and where the last integral term above can be controlled using an argument similar to that one performed in the analysis of the term A 1 in (4.95). A familiar argument similar to that one performed in (4.84) − (4.85) yields Similarly, we have Once again, we employ the same procedure as in the previous analysis in (4.86) − (4.87) to obtain and (4.107) Analogously to the analysis in (4.88) we obtain To conclude the estimates in (4.104) − (4.107), we observe that Similarly to the analysis in (4.89), we notice that the L 2 norm of the first term on the RHS of (4.109) is the very quantity to be estimated. The second term can be controlled by using a familiar argument, similar to that one employed in (4.71) above, combining local theory, smoothing effect, estimate (4.93) and the former case for l = 1, 2, ..., m.
A familiar argument similar to that one used in (4.90) − (4.91) provide us with O 1 (m), O 2 (m) denoting the odd integers and even integers in {1, 2, ..., m} respectively. We employ Proposition 2 to obtain (4.111) The term on the RHS of (4.111) can be controlled, after integration in time, by applying the induction hypothesis (4.93) for l = 1, 2, ..., m combined with the same argument performed in the analysis of A 1 in (4.70) with ( , b) instead of ( /24, b + 7 /24).

Similarly, one gets
Finally, employing an argument similar to that one used in (4.109), we write in order to control the term on the RHS of (4.112), by combining local theory, smoothing effect and the induction hypothesis (4.93) for l = 1, 2, ..., m. Part b): Next, we assume that (1.22) holds with x 0 = 0 for l = m + 1, our induction hypothesis, i.e., for l = 1, 2, ..., m (4.114) and remember that in part a) of our argument of proof we have obtained (1.25) with l = m. A familiar argument yields For the first term we write (4.116) We notice that, I 2 L 2 xy is bounded, after integration in time, by the induction hypothesis (1.25) with l = m. The term I 1 L 2 xy is controlled by using commutator estimate (3.14) while I 3 L 2 xy is bounded, after integration in time, by employing a similar argument to that one used in the analysis (4.71) − (4.73) above, combined with induction hypothesis (1.25) for l = m.
In order to control the contribution of A 3 in (4.115) we write x H∂ x u∂ m+2 x uχdxdy+ ∂ m+1 x H∂ x u∂ m+1 x uχ dxdy = A 31 +A 32 (4.117) A familiar argument yields Once again, we observe that A 321 (t) in (4.119) is positive and represents the smoothing effect. To take care of the remaining terms on the RHS of (4.119) we employ a familiar argument similar to that one used in the analysis (4.65) − (4.66), combined with the former case A 1 above, treated in (4.116). Finally, one just needs to handle the contribution of the nonlinear term A 4 in (4.115). From integration by parts and Leibniz rule, we have We notice that A 40 is bounded, after integration in time, by the former case A 1 above, treated in (4.116). Next, we have We use (4.2) to bound the first term on the right hand side of (4.121) when we latter apply Gronwall's Lemma. The last integral term is the quantity to be estimated. We now consider the term A 42 in (4.120). Firstly, we denote χ = χ /5, . Using the identity χ ,b = χ /5, χ ,b , Hölder's inequality, Sobolev embedding, Young's inequality, we obtain that We begin observing that the last term on the right hand side of (4.122) is the very quantity to be estimated. The remaining terms are bounded, after integration in time interval, by the former cases l = 2, 3 combined with (4.93).
To estimate m−1 l=3 A 4l in (4.120) we employ an argument analogous to that in (4.122) to get |A 4( l+1) (t)| ∂ l x u χ /5, where the last term integral on the right hand side of (4.123) is the quantity to be estimated. To deal with the remaining terms on the RHS of (4.123) we observe they are all of order ≤ m so bounded, after integration in time interval, by the former cases l = 2, 3, ..., m combined with (4.93). This completes the induction argument.
Recently, D. Li [32] extended fractional Leibniz rule for the nonlocal operator D s , s > 0 and related ones, including various end-point situations.
Lemma A.4. For each s ∈ R and σ > d 2 there exists a constant C = C s,σ > 0 such that for all ϕ ∈ S(R d ) and f ∈ H s−1 (R d ) where l = |s − 1| + 1 + σ.
Next, we present a result that were first introduced in [26] for the operator J s , and more recently established in [37] for the non-local operator D s , which is a crucial ingredient in our analysis.
Proof. The argument of proof is analogous to the one given by Kenig in [24] (see for instance Lemma 4.6 in [34]).