The Cauchy problem for a family of two-dimensional fractional Benjamin-Ono equations

In this work we prove that the initial value problem (IVP) associated to the fractional two-dimensional Benjamin-Ono equation $$\left. \begin{array}{rl} u_t+D_x^{\alpha} u_x +\mathcal Hu_{yy} +uu_x&=0,\qquad\qquad (x,y)\in\mathbb R^2,\; t\in\mathbb R, u(x,y,0)&=u_0(x,y), \end{array} \right\}\,,$$ where $0<\alpha\leq1$, $D_x^{\alpha}$ denotes the operator defined through the Fourier transform by \begin{align} (D_x^{\alpha}f)\widehat{\;}(\xi,\eta):=|\xi|^{\alpha}\widehat{f}(\xi,\eta)\,, \end{align} and $\mathcal H$ denotes the Hilbert transform with respect to the variable $x$, is locally well posed in the Sobolev space $H^s(\mathbb R^2)$ with $s>\dfrac32+\dfrac14(1-\alpha)$.


Introduction
In this article we consider a class of initial value problems (IVP) associated to the family of fractional twodimensional Benjamin-Ono (BO) equations u t + D α x u x + Hu yy + uu x = 0, (x, y) ∈ R 2 , t ≥ 0, u(x, y, t) ∈ R, u(x, y, 0) = u 0 (x, y), (1.1) where 0 < α ≤ 1, D α x denotes the operator defined through the Fourier transform by (D α x f ) (ξ, η) := |ξ| α f (ξ, η) , (1.2) and H denotes the Hilbert transform with respect to the variable x, which is defined through the Fourier transform by x ∈ R, t ≥ 0, (1.4) and was deduced by Pelinovsky and Shrira in [20] in connection with the propagation of long-wave weakly nonlinear two-dimensional perturbations in parallel boundary-layer type shear flows. Very recently, Esfahany and Pastor in [5] studied for this equation existence, regularity and decay properties of solitary waves. Different two-dimensional generalizations of the BO equation can be found, among others, in the references [3], [14], [1], and [2].
Using the abstract theory developed by Kato in [6] and [7], it can be established the local well-posedness (LWP) of the IVP (1.1) in H s (R 2 ) with s > 2 (see Lemma 6.1 below for α ∈ (0, 1], and [22] for α = 1). However, this approach does not employ sufficiently the dispersive terms in the equation in order to have LWP in Sobolev spaces with less regularity.
Ponce in [21] and Koch and Tzvetkov in [15] used the dispersive character of the linear part of the unidimensional BO equation (1.4) to prove the LWP of the IVP associated to this equation in Sobolev spaces H s (R), with s = 3/2 and s > 5/4, respectively. In this manner, they improved the classical result (s > 3/2) obtained from a nonlinear commutator argument (see [8] and [23]).
In particular, in [15] the improvement is a consequence of a nonlinear estimate, which, by means of a standard Littlewood-Paley decomposition, follows from a Strichartz type inequality for a linearized version of the BO equation.
In [4] Cunha and Pastor applied the technique, introduced by Koch and Tzvetkov in [15], in order to study the Benjamin-Ono-Zakharov-Kuznetsov equation in low regularity Sobolev spaces (H s (R 2 ) with s > 11/8).
In [10] Kenig and Koenig, studying the unidimensional BO equation and based on previous ideas of Koch and Tzvetkov, obtained a refined version of the Strichartz estimate, by dividing the time interval into small subintervals, whose length depends on the spatial frequency of the function. This new Strichartz estimated allowed them to establish LWP of the Cauchy problem in H s (R), with s > 9/8. Kenig in [9], in the context of the Kadomtsev-Petviashvili equation (KP-I), adapted the argument of Koch and Tzvetkov in [15], to obtain a refined Strichartz estimate (see Lemma 1.7 in [9]), similar to that in [10], in order to prove the LWP of the IVP of the KP-I equation in for s > 3/2.
As it is observed by Linares, Pilod and Saut in [17], the study of the well-posedness for nonlinear dispersive equations is based on the comparison between nonlinearity and dispersion. It is usual to fix the dispersion and vary the nonlinearity. The approach we use in this work, inspired in [17], is to fix the quadratic nonlinearity and vary the dispersion.
In [17] Linares, Pilod and Saut investigated how a weakly dispersive perturbation of the inviscid Burgers equation enlarges the space of resolution of the local Cauchy problem. Specifically they studied the family of fractional KdV equations u t − D α x u x + u∂ x u = 0, x ∈ R, t ≥ 0, u(x, t) ∈ R, (1.6) Throughout the paper the letter C will denote diverse constants, which may change from line to line, and whose dependence on certain parameters is clearly established in all cases.
Finally, let us explain some used notation. We will denote the Fourier transform and its inverse by the symbols ∧ and ∨ , respectively. For variable expressions A and B the notation A B and the notation A B mean that there exists a universal positive constant C such that A ≤ CB and A ≥ CB, respectively, and the notation A ∼ B means that there exist universal positive constants c and C such that cA ≤ B ≤ CA.

2.
Picard iterative scheme on the Duhamel's formula does not work In this section, inspired in the work [19], we prove that the IVP (1.1) can not be solved by a Picard iterative scheme based on the Duhamel's formula. In other words, if is the integral equation corresponding to the IVP (1.1) with initial datum φ, where the following assertion holds.

Linear Estimates
The main objective of this section is to establish a Strichartz type estimate for the spatial derivatives of the solutions of the non homogeneous linear equation from the classical Strichartz type estimate of the group {U α (t)}, defined by (2.2), associated to the linear part of the fractional two-dimensional BO equation. The Strichartz estimate for the spatial derivatives of the solutions of (3.1) will allow us to control the norm ∇u L 1 T L ∞ xy for sufficiently smooth solutions u of the fractional two-dimensional BO equation.
We have the following decay estimate in time for the group.
Proof. Without loss of generality we can suppose that t > 0. Let us observe that Now we will analyze the decay properties of the oscillatory integral Rη e −it sgn(ξ)η 2 e iyη dη e i(xξ−t|ξ| α ξ) dξ .
Taking into account that we have that and, in consequence Making in the last integral the change of variable ξ ′ := t 1 1+α ξ we obtain Let us define for λ ∈ R, By applying Lemma 2.7 in [12] with φ(ξ) := −|ξ| α ξ we may conclude that there exists a positive constant C such that for all λ ∈ R J(λ) ≤ C .
Definition 1. Given (q, p) ∈ R 2 we will say that (q, p) is an admissible pair for the fractional two-dimensional BO equation if 2 ≤ q, p ≤ ∞, q > 2 and 1 q The following Strichartz's type estimate is a consequence of the decay estimate in time (3.2).
Let us observe that (2, ∞) is not an admissible pair. We need to lose a little bit of regularity in both x and y directions in order to control the norm L 2 T L ∞ xy .
Proof. Let us recall the Sobolev embedding (see [24], page 336) if 1 < p < ∞ and δ > 0 is such that δ > 2/p, where W δ,p x,y (R 2 ) denotes the Sobolev space given by the closure of the Schwartz functions under the norm . Here J δ is the operator with symbol (1 + ξ 2 + η 2 ) δ/2 . Thus, if we take δ > 0 and p such that δ > 2/p and then we consider the admissible pair (q, p), we have that Using Hölder's inequality in t withq andq ′ , we obtain
Taking into account that w 0 is a solution of the integral equation we have that From (3.17), (3.18) and (3.10) we conclude that This way, Let us observe that Hence ∂ x w Λ L ∞ xy CΛ w Λ L ∞ xy . In consequence, using Cauchy-Schwarz inequality, we obtain Next, using Duhamel's formula in each I j , we obtain, for t ∈ I j , Thus combining (3.20) and (3.21) and taking into account (3.10), we deduce that Let us define k δ := 1/2+k δ ∈ (1/2, 1). We observe that in the support of (w Λ (a j )) ∧ and (

Energy Estimates
Using energy estimates on the fractional two-dimensional BO equation and the classical Kato-Ponce commutator inequality, in this section we obtain the following a priori estimate for sufficiently smooth solutions of the fractional BO equation.
Proof. First of all, let us observe that the operator D α x ∂ x + H∂ yy is skew-adjoint in L 2 (R 2 ). In fact, if we denote by (·, ·) the inner product in L 2 we have x v x + Hv yy ) . If we take in the last equality u = v a real function, then we obtain Applying the operator J s ≡ (1 − ∆) s/2 to the equation in (1.1), multiplying by J s u(t), integrating in R 2 xy and, taking into account (4.2),we obtain, for t ∈ [0, T ], that Kato-Ponce's commutator inequality (see [8]) affirms that From Kato-Ponce's commutator inequality it follows that In consequence, from (4.3) to (4.6) we can conclude that Now, we integrate (4.7) in [0, t] to obtain Then, (4.1) follows from the last inequality.

5.
Estimates for the norm u L 1 We will derive an a priori estimate for the norm   (1 − α), there exist k s ∈ (1/2, 1) and C s > 0 such that Proof. Let us fix a constant δ 0 such that 0 < δ 0 < s − s α . We first estimate ∇u L 1 T L ∞ xy . Taking F := −u∂ x u in equations (3.14) and (3.15), from Lemma 3.4 we get where δ is such that 0 < δ < δ 0 and will be determined during the proof.
Let us bound each one of the terms in the right hand side of (5.3). By choosing δ > 0 such that 0 < δ < δ 0 /2, it is clear that s α + 2δ < s α + δ 0 < s and in consequence On the other hand Let us estimate I 1 : Now we estimate I 2 : The fractional Leibniz rule, proved in [13], says that for σ ∈ (0, 1), it holds that Taking in (5.6) σ := s α − 1 + 2δ ∈ (0, 1) (δ > 0 small enough), and p 1 = p 2 := 2, q 1 = q 2 := ∞ we have that In a similar way, we obtain In order to estimate the norm u L 1 T L ∞ xy we observe that u is a solution of the integral equation Then, using Hölder's inequality in time and (3.10) we have that where k δ := 1 2 +k δ ∈ (1/2, 1). Proceeding as it was done to estimate (5.11) From (5.9) and (5.11) it follows (5.2).
6. Proof of Theorem 1.1 Using the abstract theory, developed by Kato in [6] and [7], to prove LWP of quasi-linear evolution equations, it can be established the following result of LWP of the IVP (1.1) for initial data in H s (R 2 ), with s > 2.
is continuous.
Next we will describe the abstract theorem of Kato and how the IVP (1.1), considered by us, satisfies the hypotheses of this abstract theorem when the initial data belong to H s (R 2 ) with s > 2. In other words, we will show how to prove Lemma 6.1 using Kato's theory.
Let X, Y be reflexive and separable Banach spaces such that Y ⊂ X with continuous and dense immersion. Suppose that there is a surjective isometry S : Y −→ X. Let A be a function defined on Y and with values in the set of linear operators in X. We say that the system (X, Y, A, S) is admissible for the equation if, for R > 0, there are positive constants β(R), λ(R), µ(R), ν(R), ρ(R), called the parameters of the system, such that, if v Y < R, and w Y < R, then: Theorem 6.2. (see [6] and [7]) If the system (X, Y, A, S) is admissible for the evolution equation 6.1, then, is the unique noncontinuable solution of (6.2) and 0 < T < T max , then there exists a neighborhood V of u 0 in Y such that the mapũ 0 →ũ is continuous from V to C([0, T ]; Y ).
To IVP (1.1) we associate the abstract IVP (6.2) by defining A(v)u := D α x u x + Hu yy + vu x . Let s > 2 and S := (1 − ∆) s/2 ≡ J s . Then the system (L 2 (R 2 ), H s (R 2 ), A, S) is admissible for the quasilinear equation in IVP (6.2). Properties (i) and (ii) can be easily verified by using standard methods of the theory of linear semigroups. With respect to condition (iii), we can use the Kato-Ponce's commutator inequality (4.4). In fact, taking into account that s > 2, we have that H s−1 (R 2 ) ֒→ L ∞ (R 2 ), and in consequence which implies (iii) with ν(R) = CR and ρ(R) = C, for some constant C. In this manner Lemma 6.1 has been proved. Lemma 6.1 will allow us to apply the Bona-Smith method for proving the existence part of Theorem 1.1.  Proof. Let A be the set {T ′ ∈ (0, T * ) : Since u ∈ C([0, T * ); H ∞ (R 2 )), the set A is not empty. Let T 0 := sup A. We will prove that T 0 > T . We argue by contradiction by assuming that 0 < T 0 ≤ T < 1. By continuity we have that u 2 , 1) be the exponent in (5.2).
Then, using (5.2), we have We use the estimate (4.1) with s = 3 to obtain This way, H 3 , which implies, taking into account (6.4), that T 0 < T * .
On the other hand, by using the energy estimate (4.1), we have that u 0 2 H s , and by continuity, for some T ′ ∈ (T 0 , T * ), we have that This contradicts the definition of T 0 . Then we conclude that T < T 0 and thus Thus, since T < 1, and in consequence which completes the proof of Lemma 6.3.
The existence of the solutions u n is guaranteed by Lemma 6.1.
First of all, let us see that {u n } is a Cauchy sequence in C([0, T ]; L 2 (R 2 )). For m ≥ n ≥ 1 we define v n,m := u n − u m . Let us prove that v n,m L ∞ It is clear that v n,m satisfies the differential equation We multiply this differential equation by v n,m (t) and integrate in R 2 xy to obtain 1 2 where K is the constant in (6.9). This way, v n,m L ∞ T L 2 xy ≤ e K/2 u 0,n − u 0,m L 2 . Estimation (6.10) follows if we prove that u 0,n − u 0,m L 2 = o(n −s ). In fact, Since u 0 ∈ H s (R 2 ), by the Dominated Convergence Theorem, it is clear that For 0 ≤ σ < s, making interpolation and taking into account inequalities (6.8) and (6.10) it follows that Since lim n→∞ n s v n,m L ∞ T L 2 xy = 0, we have that lim n→∞ n s(1−σ/s) v n,m ). (6.11) In particular, from (6.11) it follows that {u n } is a Cauchy sequence in C([0, T ]; H σ (R)) for 0 ≤ σ < s. Let us prove now that {u n } is a Cauchy sequence in Hence, using Cauchy-Schwarz's inequality in the variable t and (3.10), it follows that for δ > 0 v n,m L 1 Taking into account that H 1+1/4(1−α)+δ (R 2 ) is an algebra, from the last inequality we conclude that v n,m L 1 . Since, for 0 < δ < s − 1 − 1/4(1 − α), {u n } is a Cauchy sequence in L ∞ ([0, T ]; H 1+1/4(1−α)+δ (R 2 )), from the last inequality we conclude that {u n } is a Cauchy sequence in L 1 ([0, T ]; L ∞ xy ). On the other hand, from Lemma 3.4, it follows that ∇v n,m L 1 If δ > 0 is such that s α + 2δ < s, then {u n } is a Cauchy sequence in L ∞ ([0, T ]; H sα+2δ (R 2 )). Therefore, from the last inequality it follows that {∇u n } is a Cauchy sequence in L 1 ([0, T ]; L ∞ xy ). Hence, we have that {u n } is a Cauchy sequence in L 1 ([0, T ]; W 1,∞ xy (R 2 )). It remains to show that u ∈ C([0, T ]; H s (R 2 )) and that u is a solution of the IVP (1.1).
We first show that u : [0, T ] → H s (R 2 ) is weakly continuous (u ∈ C w ([0, T ]; H s (R 2 ))), i.e. that for each φ ∈ H s , the function t → (u(t), φ) s is continuous from [0, T ] in R (here (·, ·) s denotes the inner product in H s ). In order to show this, it is enough to prove that if f n : [0, T ] → R is defined by f n (t) := (u n (t), φ) s , then the sequence {f n } is uniformly Cauchy in [0, T ]. For φ ∈ H s , we take ψ ∈ C ∞ 0 (R 2 ) such that φ − ψ H s < η (where η > 0 will be specified later). Hence, for t ∈ [0, T ], For σ ∈ (1, s), we proved that {u n } es a Cauchy sequence in C([0, T ]; H σ (R 2 )). Taking into account the immersion H σ (R 2 ) ֒→ L ∞ (R 2 ) we conclude that u n (t) → u(t) in L ∞ xy uniformly for t ∈ [0, T ]. Given ǫ > 0, define η such that Cη < ǫ 2 and define N ∈ N such that for each m ≥ n ≥ N and t ∈ [0, T ], Lemma 6.4. (see [4]) There is a constant C > 0 such that for any w ∈ L 2 (R 2 ) and any v such that ∇v ∈ L ∞ (R 2 ) Lemma 6.5. (see [15]) Let u ∈ C([0, T ]; H s (R 2 )) ∩ L 1 ([0, T ]; W 1,∞ xy (R 2 )) be a solution of the fractional twodimensional BO equation in [0, T ] with s > s α . Let δ and k be such that 1 < δ < k. Assume that the dyadic sequence of positive numbers (w λ ) λ satisfies δw λ ≤ w 2λ ≤ kw λ , for each dyadic integer λ and that λ w 2 λ u λ (0) 2 L 2 < ∞. Then for each τ and t in [0, T ] it follows that where Proof. It is easy to see that u λ satisfies the differential equation We multiply this equation by u λ (t), integrate in R 2 xy and use integration by parts to conclude that Let us integrate this equation with respect to time in [τ, t] to obtain Hence, Taking into account that ∆ λ∆λ = ∆ λ , it could be seen that Therefore, Now, from Lemma 6.4, it follows that On the other hand, since for µ dyadic, µ ≤ λ 16 , we have that ∆ λ (u µ ∂ x (I −∆ λ )u) = 0 and the operator (I −∆ λ ) is bounded from L ∞ to L ∞ , we see that (6.17) From (6.15), (6.16), and (6.17), it follows that t τ λ From (6.14) and (6.18) we conclude that Since δw λ ≤ w 2λ for each λ, it can be proved by induction, that for all j ≥ 0, and since w λ ≤ kw λ/2 , it can also be seen that Let us see that for any (d λ ) λ , such that In fact, from (6.20) and (6.21) it follows that Defining a j := k −j for j = −3, −2, −1 and a j := 1 δ j for j ≥ 0, we have Taking in (6.22) d λ := w λ u λ (σ) L 2 , we conclude that From (6.19) and (6.23) it follows that Hence, (6.13) follows from (6.24) using Gronwall's inequality.
Proof. Let us consider the space l 1 (N) of summable sequences. We define a n i := λ 2s ν n 2 i 2 L 2 and a i := λ 2s ν 2 i 2 L 2 , (6.27) where λ := 2 i . From the hypothesis it follows that (a n i ) i∈N → (a i ) i∈N in l 1 (N), as n → ∞. In fact, Since ν n → ν in H s (R 2 ), J s (ν n − ν) L 2 → 0 as n → ∞. Therefore, from (6.28) and (6.29), it follows that (a n i ) i∈N → (a i ) i∈N in l 1 (N) as n → ∞.
Let us construct a sequence {µ i } such that 0 < µ i < µ i+1 ≤ 2µ i , such that lim i→∞ µ i = ∞ and sup n ∞ i=1 µ i a n i < ∞. (6.30) (Observe that sup n ∞ i=1 a n i < ∞ because (a n i ) i∈N → (a i ) i∈N in l 1 (N)).
Let us see that, for each k ∈ N, there exists N k such that sup n ∞ i=N k a n i < 2 −k . (6.31) Since (a n i ) i∈N → (a i ) i∈N in l 1 (N) as n → ∞, there exists N such that, for each n ≥ N , On the other hand, there exists N * * k such that, for each n ∈ {1, . . . , N − 1}, i=N * * k a n i < 2 −k 2 . We define N k := max{N * k , N * * k }. Hence, for each n ≥ N , This way, inequality (6.31) follows.
We can assume that the sequence {N k } k∈N is strictly increasing. For fixed i ∈ N, there exists a unique k ∈ N such that N k−1 ≤ i < N k . Let us define µ i := 2 k/2 . It is clear that 0 < µ i ≤ µ i+1 ≤ 2µ i . Using (6.31) we obtain that ∞ i=1 µ i a n i = Proof of the continuous dependence on the initial data. This proof follows the same ideas of Koch and Tzvetkov in [15].