Well-posedness for the two dimensional generalized Zakharov–Kuznetsov equation in anisotropic weighted Sobolev spaces

Accepted Manuscript Well-posedness for the two dimensional generalized Zakharov-Kuznetsov equation in anisotropic weighted Sobolev spaces

G. Fonseca, M. Pachón

PII: DOI: Reference:

S0022-247X(16)30170-6 http://dx.doi.org/10.1016/j.jmaa.2016.05.028 YJMAA 20446

To appear in:

Journal of Mathematical Analysis and Applications

Received date:

14 October 2015

Please cite this article in press as: G. Fonseca, M. Pachón, Well-posedness for the two dimensional generalized Zakharov-Kuznetsov equation in anisotropic weighted Sobolev spaces, J. Math. Anal. Appl. (2016), http://dx.doi.org/10.1016/j.jmaa.2016.05.028

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

WELL-POSEDNESS FOR THE TWO DIMENSIONAL GENERALIZED ZAKHAROV-KUZNETSOV EQUATION IN ANISOTROPIC WEIGHTED SOBOLEV SPACES ´ G. FONSECA AND M. PACHON

Abstract. We consider the well-posedness of the initial value problem associated to the k-generalized Zakharov-Kuznetsov equation in fractional weighted Sobolev spaces H s (R2 ) ∩ L2 (( |x|2r1 + |y|2r2 ) dxdy), s, r1 , r2 ∈ R. Our method of proof is based on the contraction mapping principle and it mainly relies on the well-posedness results recently obtained for this equation in the Sobolev spaces H s (R2 ) and a new pointwise commutator type formula involving the group induced by the linear part of the equation and the fractional anisotropic weights to be considered.

1. Introduction Our aim is to study persistence properties of solutions of the two dimensional k-generalized Zakharov-Kuznetsov equation (gZK) in fractional weighted spaces. More precisely we consider the initial value problem (IVP):  ∂t u + ∂x Δu + uk ∂x u = 0, t ∈ R, (x, y) ∈ R2 , k ∈ Z+ , (1.1) u(x, y, 0) = u0 (x, y). Let us introduce the weighted Sobolev spaces of our interest (1.2)

Zs,(r1 ,r2 ) = H s (R2 ) ∩ L2 (( |x|2r1 + |y|2r2 ) dxdy), s, r1 , r2 ∈ R.

We want to show that for initial data in this function space the associated IVP is locally well-posed and with some additional assumptions it turns out to be globally well-posed. In general an IVP is said to be locally well-posed (LWP) in a function space X if for each u0 ∈ X there exist T > 0 and a unique solution u ∈ C([−T, T ] : X) ∩ · · · = YT of the equation, such that the map data → solution is locally continuous from X to YT . This notion of LWP includes the “persistence” property, i.e. the solution describes a continuous curve on X. In particular, this implies that the solution flow of the considered equation defines a dynamical system in X. Whenever T can be taken arbitrarily large we say that the IVP is globally well-posed (GWP). It is important to mention that this family of dispersive equations include the Zakharov-Kuznetsov (ZK) equation (k = 1) and the modified Zakharov-Kuznetsov (mZK) equation (k = 2) which are considered two dimensional versions of the famous Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations respectively. The ZK equation was introduced by Zakharov and Kusnetsov in [26] in the context of plasma physics in order to model the propagation of ion-acoustic 1991 Mathematics Subject Classification. Primary: 35Q53; Secondary: 35B65, 35Q60. Key words and phrases. Generalized Zakharov-Kuznetsov, Weighted Sobolev spaces. 1

´ G. FONSECA AND M. PACHON

2

waves in magnetized plasma, for a rigorous proof of this fact see [18]. On the other hand mKdV equation is used to describe the propagation of Alfv´en waves at a critical angle to an undisturbed magnetic field (see [13]) and mZK is related to the same type of phenomena in two dimensions (see [24]). In order to motivate our results we remark that for the gKdV IVP:  ∂t u + ∂x3 u + uk ∂x u = 0, t, x ∈ R, k ∈ Z+ , (1.3) u(x, 0) = u0 (x), Kato showed in [14] the persistence of solutions in the weighted Sobolev spaces Zs,m = H s (R) ∩ L2 ( |x|2m dx),

s ≥ 2m,

m = 1, 2, . . . .

The proof of this result is based on the commutative property of the operators (1.4)

Γ = x − 3t∂x2 ,

L = ∂t + ∂x3 ,

i.e.

[Γ; L] = 0.

Let us consider the linear IVP  ∂t v + ∂x3 v = 0, t, x ∈ R, (1.5) v(x, 0) = v0 (x), and let us denote by {U (t) : t ∈ R} the unitary group of operators describing its solution, that is 3

U (t)v0 (x) = (eitξ v0 )∨ (x).

(1.6) Then (1.4) is equivalent to (1.7)

x U (t)v0 (x) = U (t)(xv0 )(x) + 3tU (t)(∂x2 v0 )(x).

This equality clearly suggests that regularity and decay are strongly related and furthermore in order to obtain persistent properties for the flow in (1.5) in those weighted spaces Zs,r , at least twice of the decay rate r is expected to be required in regularity, that is, s ≥ 2r. Notice that Kato’s result strongly indicates us that this condition should hold even for the non-linear associated IVP. In fact, this was recently proved by Fonseca, Linares and Ponce in [9] where they extended (1.7) to fractional powers of |x| with the help of a point-wise version of the homogeneous derivative of order s introduced by Stein [25]. In that way, they improved Kato’s results for the gKdV in those fractional Sobolev weighted spaces. Their argument, via contraction principle, also required some detail on previous results on LWP and GWP results for the gKdV IVP on the classical Sobolev spaces H s (R) obtained by Kenig, Ponce and Vega, see [15], [16]. In view of the ideas just presented in the case of the gKdV equation, the first piece in our analysis is related to some of the existent theory on LWP and GWP for the gZK equation on classical Sobolev spaces H s (R2 ), see remark b) below. Let us define the regularity index sk by:  3/4 if 1 ≤ k ≤ 7, (1.8) sk = 1 − 2/k if k ≥ 8. We now state the results by Linares and Pastor [19], [20] and Farah, Linares, and Pastor [5] and notice that some detail of their proof will be included in Section 2.

gZK IN WEIGHTED SOBOLEV SPACES

3

Theorem 1. ([19], [20], [5]). For any u0 ∈ H s (R2 ), s > sk , there exist T = T (u0 H s ) > 0, a space XT ⊂ C([0, T ] : H s (R2 )) and a unique solution u ∈ XT of the IVP (1.1) defined in [0, T ]. Moreover for any T  ∈ (0, T ) there exists a 0 → u (t) from V into XT  is neighborhood V of u0 in H s (R2 ) such that the map u smooth. Remarks: (a) The estimate for the length of the time interval of existence with respect of the size of the initial data in H s (R2 ) can be explicitly obtained in the proof of Theorem 1. (b) The critical index for the gZK equation (1.1) turns out to be sc,k = 1 − k2 which can easily be computed by a scaling argument, therefore it coincides with the regularity index sk in (1.8) within the range k ≥ 8. Hence we have that for k ≥ 8 these results are optimal. Actually, in [20] it was proven that the gZK IVP is ill-posed for s = sc,k in the sense that the map data to solution is not uniformly continuous so other approach different from contraction arguments is required in order to lower the LWP regularity. However, for the ZK equation, k = 1, Gr¨ unrock and Herr in [11] and Molinet and Pilod in [21] were able to show LWP in H s (R2 ) for s > 1/2 in the context of Bourgain spaces X s,b , see [1]. Also, recently Ribaud and Vento in [23] showed local well-posedness in H s (R2 ) for s > 1/4 if k = 2, s > 5/12 if k = 3 and s > sc,k if k ≥ 4 by working in some Besov spaces. Notice that for 1 ≤ k ≤ 3 there is still a gap to be filled in the expected LWP theory. (c) Regarding GWP results, it is important to mention that for the ZK equation local solutions can be globally defined in H 1 with the help of the conserved energy and a Gagliardo-Nirenberg inequality. In the case of the gZK equation, global solutions in H 1 , and even in a larger space in the case of the mZK, k = 2, are obtained if in addition it is assumed that the initial data is small enough, see [19], [20] and [5]. Next, let us explicitly introduce the group associated to the linear ZK equation: (1.9)

W (t)v0 (x, y) = (eit(ξ

3

+ξη 2 )

v0 )∨ (x, y).

Following the strategy used to deal with the gKdV equation in weighted spaces in [9], our second task is directed to get an extension of formula (1.7) but with the linear group in (1.9) instead. More precisely we have our first result: Theorem 2. Let r1 , r2 ∈ (0, 1), s ≥ 2 max{r1 , r2 } and {W (t) : t ∈ R} be the unitary group of operators defined in (1.9). If (1.10)

u0 ∈ Zs,(r1 ,r2 ) = H s (R2 ) ∩ L2 (( |x|2r1 + |y|2r2 ) dxdy),

then for all t ∈ R and for almost every (x, y) ∈ R2 (1.11)

|x|r1 W (t)u0 (x, y) = W (t)(|x|r1 u0 )(x, y) + W (t){Φ1,t,r1 ( u0 )(ξ, η)}∨ (x, y)

with (1.12)

{Φ1,t,r1 ( u0 )(ξ, η)}∨ 2 ≤ c(1 + |t|)(u0 2 + Dxs u0 2 + Dys u0 2 )

and (1.13)

|y|r2 W (t)u0 (x, y) = W (t)(|y|r2 u0 )(x, y) + W (t){Φ2,t,r2 ( u0 )(ξ, η)}∨ (x, y)

´ G. FONSECA AND M. PACHON

4

with (1.14)

{Φ2,t,r2 ( u0 )(ξ, η)}∨ 2 ≤ c(1 + |t|)(u0 2 + Dxs u0 2 + Dys u0 2 ).

Moreover, if in addition to (1.10) we suppose that for β ∈ (0, min{r1 , r2 }) (1.15)

Dβ (|x|r1 u0 ), Dβ (|y|r2 u0 ) ∈ L2 (R2 )

and

u0 ∈ H β+s (R2 ),

then for all t ∈ R and for almost every (x, y) ∈ R2 Dβ (|x|r1 W (t)u0 )(x, y) (1.16) = W (t)(Dβ |x|r1 u0 )(x, y) + W (t)(D β ({Φ1,t,r1 ( u0 )(ξ, η)}∨ ))(x, y) and Dβ (|y|r2 W (t)u0 )(x, y) (1.17) = W (t)(D β |y|r2 u0 )(x, y) + W (t)(Dβ ({Φ2,t,r2 ( u0 )(ξ, η)}∨ ))(x, y) with (1.18) Dβ ({Φj,t,r2 ( u0 )(ξ, η)}∨ )2 ≤ c(1 + |t|)(u0 2 + Dxβ+s u0 2 + Dyβ+s u0 2 ), for j = 1, 2. Remarks: (a) As we mentioned above, this type of formula was recently established by Fonseca, Linares and Ponce [9] in the context of the Airy group and more generally it also holds for the group associated to the dispersion generalized Benjamin-Ono equation:  t, x ∈ R, 0 ≤ a < 1, ∂t u − Dx1+a ∂x u = 0, (1.19) u(x, 0) = u0 (x), where Ds denotes the homogeneous derivative of order s ∈ R, ∨  Ds = (−∂ 2 )s/2 so Ds f = cs |ξ|s f , with Ds = (H ∂x )s , x

and H denotes the Hilbert transform,  1 f (x − y) Hf (x) = lim dy = (−i sgn(ξ)f(ξ))∨ (x). π ↓0 y |y|≥

For the problem we are dealing with we adapt those one dimensional situations and carefully handle estimates in the two Fourier space variables. (b) The proof of Theorem 2 is based on a characterization of the generalized Sobolev space (1.20)

Lα,p (Rn ) = (1 − Δ)−α/2 Lp (Rn ),

α ∈ (0, 2), p ∈ (1, ∞),

due to E. M. Stein [25] (see Theorem 4 below). (c) The conclusions in (1.12) and (1.14) can be adapted for r1 , r2 ≥ 1 by means of an iterative process at every integer step. For example, provided that r1 = 1+λ with λ ∈ (0, 1) our estimate will involve the estimate of the L2 norm  |x|r1 x W (t)u0 L2xy for which we just have to compute, via Fourier transform, one ξ-derivative for 3 2 ˆ0 (ξ, η) obtaining that W (t)u0 (ξ, η) = eit(ξ +ξη ) u (1.21) 3 2 3 2 3 2  0 (ξ, η). ˆ0 (ξ, η)) = eit(ξ +ξη ) it(3ξ 2 + η 2 )ˆ u0 (ξ, η) + eit(ξ +ξη ) −ixu ∂ξ (eit(ξ +ξη ) u

gZK IN WEIGHTED SOBOLEV SPACES

5

From this point we can apply Theorem 2 on the right hand side terms with the remaining of the weight |x|λ , λ ∈ (0, 1), and use similar arguments to Lemma 1 in reference [7] to get rid of the L2 norms for mixed weight-derivatives arising terms. This is a bit technical and has no new ideas involved, therefore we omit the details. Now we state our second result concerning local well-posedness of the gZK equation in weighted spaces: Theorem 3. Let u ∈ C([0, T ] : H s (R2 )), s > sk denote the local solution of the IVP (1.1) provided by Theorem 1. Let us assume that (|x|r1 + |y|r2 )u0 ∈ L2 (R2 ) with s satisfying 0 < 2 max{r1 , r2 } ≤ s, then u ∈ C([0, T ] : Zs,(r1 ,r2 ) ).

(1.22) 

For any T ∈ (0, T ) there exists a neighborhood V of u0 in H s (R2 ) ∩ L2 ((|x|2r1 + 0 → u (t) from V into the class defined by XT in |y| )dxdy) such that the map u Theorem 1 and (1.22) with T  instead of T is smooth. 2r2

Remarks: (a) We observe that Theorem 3 guarantees that the persistent property in the weighted space Zs,(r1 ,r2 ) holds in the same time interval [0, T ] given by Theorem 1, where T depends only on u0 H s . (b) It is expected that the condition s ≥ 2 max{r1 , r2 } in Theorem 3 is optimal as it was shown in [12] in the case of the gKdV equation. More precisely, (1.22) would hold if and only if s ≥ 2 max{r1 , r2 }. (c) These results imply that the generalized Zakharov-Kuznetsov equations are two dimensional extensions of the generalized Korteweg-de Vries equations that inherit the persistence property in the Schwartz space. (d) Notice that for k = 1 the LWP results in [11] and [21] hold in a much larger space involving Bourgain spaces X s,b , s > 1/2, and similarly for 2 ≤ k ≤ 7 LWP results in [23] involve Besov spaces but so far it is not clear for us how to handle our weights in those spaces. (e) It is interesting to mention an important difference in the way we obtain persistent properties in these weighted spaces for dispersive type equations. Theorem 3 is established via the contraction principle as it was made for semi-linear Schr¨ odinger, gKdV, regularized Benjamin-Ono and the fifth order KdV equations, see [22], [9], [10] and [2] respectively. However, this technique could not be used for the dispersive family DGBO in (1.19) with the quadratic non-linearity u∂x u nor for the famous Benjamin-Ono equation. We point out that even that we continue having at hand Theorem 1, the dispersion on these equations is too weak to overcome the nonlinear effects in the contraction argument. Nevertheless, optimal persistency results in weighted Sobolev spaces were indeed attained via energy estimates for the associated IVPs, see [8] and [7]. See also [4] regarding a 2D ZK-BO equation. (f) Recently, it was proved in [3] a similar result for isotropic weights for the ZK equation, k = 1 in (1.1). Their argument of proof depends upon the range of values for the regularity of the initial data in H s (R2 ). For those values of s such that 34 < s ≤ 1, it relies on a symmetrization argument performed to the ZK equation as in [11], the proof of a LWP theory for the resultant symmetrized evolution equation on these weighted Sobolev spaces via a contraction argument and the help of another characterization of the generalized Sobolev spaces Lα,p (Rn ) in (1.20) obtained by an alternative Stein’s derivative to the one in Theorem 4 below which has also been used in previous works related for the BO and GDBO equations

´ G. FONSECA AND M. PACHON

6

found in [7] and [8] respectively. Here, the existence time depends on the size of the initial data on Zs,r . In the case s > 1, they use the LWP theory in [19], continuos dependence for the flow and energy type arguments in order to prove persistence for the weights. In this case, the time of existence solely depends on the size of the initial data on H s (R2 ) as in our results. We consider that our proof for this case is simpler and that Lemma 1 below which basically establishes a commutator property between weights and groups associated to the linear evolution of dispersive equations can be applied in many other multidimensional models. Last, we could recover the contraction argument to get solutions in a more general setting with anisotropic weights and a wider class of power non-linearities. (g) As a consequence of Theorem 1, its remark (c) and our proof of Theorem 3, local results globally extend when s ≥ 1 with arbitrary size of the initial data for k = 1 and for small enough initial data for k ≥ 3 (see [5]). For mZK, s > 53/63 and appropriate smallness of the initial data guarantee such extension of the local solutions (see [20]). The paper is organized as follows. In section 2 we introduce Stein’s derivatives and some detail on known results on LWP for the gZK equation. The proof of Theorem 2 will be given in Section 3. In Section 4 we will present the proof of Theorem 3. Notations.  · p denotes the norm in the Lebesgue space Lp (Rn ). Let α be a complex number, the homogeneous derivatives Dxα , Dyβ for functions in αˆ α α   R2 are defined via Fourier transform by D x f (ξ, η) = cα |ξ| f (ξ, η) and Dy f (ξ, η) = cα |η|α fˆ(ξ, η) respectively. We consider the Lebesgue space-time LrT Lpx Lqy spaces with 1 ≤ p, q, r < ∞ endowed with the norm ⎛



f LrT Lpx Lqy = ⎝

T 0



∞



−∞

∞ −∞

|f (x, y, t)|q dy

pr

pq dx

⎞ r1 dt⎠ ,

with the usual modifications when p = ∞ or q = ∞ or r = ∞. In general c denotes a universal constant which may change, increase, from line to line. 2. Preliminary results Let us start with a characterization of the Sobolev space (2.1)

Lα,p (Rn ) = (1 − Δ)−α/2 Lp (Rn ),

α ∈ (0, 2), p ∈ (1, ∞),

due to E. M. Stein [25]. For α ∈ (0, 2) define 1 Dα f (x) = lim →0 dα

(2.2) n/2

 |y|≥

f (x + y) − f (x) dy, |y|n+α

−α

2 Γ(−α/2)/Γ((n + 2)/2). where dα = π As it was remarked in [25] for appropriate f , for example f ∈ S(Rn ), one has (2.3)

α   D α f (ξ) = |ξ| f (ξ).

The following result concerning the Lα,p (Rn ) = (1 − Δ)α/2 Lp (Rn ) spaces was established in [25]:

gZK IN WEIGHTED SOBOLEV SPACES

7

Theorem 4. ([25]) Let α ∈ (0, 2) and p ∈ (1, ∞). Then f ∈ Lα,p (Rn ) if and only if ⎧ p n ⎪ ⎨ (a) f ∈ L (R ), (2.4) ⎪ ⎩ (Dα f (x) defined in (2.2)), (b) Dα f ∈ Lp (Rn ), with (2.5)

f α,p = (1 − Δ)α/2 f p f p + Dα f p f p +  Dα f p .

Let us suppose f, g : Rn → R such that g ∈ L∞ (Rn ) ∩ C 2 (Rn ) and f and f g are Lα,p (Rn ) functions and let us consider Stein’s derivatives on each j−th direction in Rn , Stein’s partial derivatives, then we have, with dα as in (2.2) with n = 1, that 1 →0 dα



Dj,α (f g)(x) = lim

1 →0 dα

|y|≥



= lim (2.6)

f (x + y ej ) g(x + y ej ) − f (x) g(x) dy |y|1+α

g(x) |y|≥

1 →0 dα



+ lim

|y|≥

f (x + y ej ) − f (x) dy |y|1+α

(g(x + y ej ) − g(x))f (x + y ej ) dy |y|1+α

= g(x) Dj,α f (x) + Λj,α ((g(· + y ej ) − g(·))f (· + y ej )) (x). In particular, if g(x) = ei t ϕ(x) , then Λj,α ((g(· + y ej ) − g(·))f (· + y ej ))(x) 1 →0 dα



= lim (2.7)

|y|≥

= ei t ϕ(x) lim

→0

i t ϕ(x)

=e

(g(x + y ej ) − g(x))f (x + y ej ) dy |y|1+α

1 dα

 |y|≥

ei t(ϕ(x+y ej )−ϕ(x)) − 1 f (x + y ej ) dy |y|1+α

Φj,ϕ,α (f )(x).

Thus, we obtain the identity (2.8)

Dj,α (ei t ϕ(·) f )(x) = ei t ϕ(x) Dj,α f (x) + ei t ϕ(x) Φj,ϕ,α (f )(x).

Now we restrict to R2 and choose as the phase function the one from the group associated to the linear ZK equation in (1.9) (2.9)

ϕ(x1 , x2 ) = x31 + x1 x22 .

For j = 1, 2, we shall obtain a bound for      ei t(ϕ(x+y ej )−ϕ(x)) − 1   (2.10) Φj,α,ϕ (f )p =  lim f (x + y ej ) dy  . 1+α  →0 |y|≥  |y| p

Indeed this is achieved in our first result

´ G. FONSECA AND M. PACHON

8

Lemma 1. Let α ∈ (0, 1), and p ∈ (1, ∞). If f ∈ Lα,p (R2 ) and f ∈ Lp ((1 + x21 + x22 )αp dx1 dx2 ), then for all t ∈ R and for almost every (x1 , x2 ) ∈ R2 3

(2.11)

2

3

2

Dj,α (eit(x1 +x1 x2 ) f )(x1 , x2 ) =eit(x1 +x1 x2 ) Dj,α f (x1 , x2 ) 3

2

+ eit(x1 +x1 x2 ) Φj,t,α (f )(x1 , x2 ),

with 1 →0 dα



(2.12) Φ1,t,α (f )(x1 , x2 ) = lim and

1 →0 dα

(2.13) Φ2,t,α (f )(x1 , x2 ) = lim

|y|≥

eit(ϕ(x1 +y,x2 )−ϕ(x1 ,x2 )) − 1 f (x1 +y, x2 ) dy |y|1+α

|y|≥

eit(ϕ(x1 ,x2 +y)−ϕ(x1 ,x2 )) − 1 f (x1 , x2 +y) dy |y|1+α



satisfying (2.14)

Φj,t,α (f )p ≤ cα (1 + |t|)(f p +  (1 + x21 + x22 )α f p ),

for j = 1, 2. Proof. Since our interest resides in the parameter α ∈ (0, 1), then we are allowed to pass the absolute value inside the integral sign in (2.10). At different parts of our work we will make use of the elementary estimates  (a) ∀ θ ∈ R |eiθ − 1| ≤ 2, (2.15) (b) ∀ θ ∈ R |eiθ − 1| ≤ |θ|. Let us consider first Stein’s derivative with respect to x1 , i.e. j = 1 in (2.11). In order to obtain (2.14) we divide the integral in (2.12) into three cases depending on whether y is close or far away from the origin. 1 . Case I: |y| ≥ 100 From (2.15) (a) and Minkowski’s integral inequality it follows that     eit(ϕ(x1 +y,x2 )−ϕ(x1 ,x2 )) − 1   f (x + y, x ) dy   1 2 1+α  |y|≥ 1  |y| 100 p     2f (x1 + y, x2 )    dy ≤   1 |y|1+α |y|≥ 100 p  (2.16) f (x1 + y, x2 )p dy ≤c 1 |y|1+α |y|≥ 100  1 ≤ c f p dy 1+α 1 |y| |y|≥ 100 ≤ cα f p . Case II: |y| ≤

1 100

and (x1 , x2 ) ∈ B10 (0) = {(x1 , x2 ) ∈ R2 /x21 + x22 < 100}.

gZK IN WEIGHTED SOBOLEV SPACES

9

Inequality (2.15) (b) yields

(2.17)

   it(ϕ(x1 +y,x2 )−ϕ(x1 ,x2 ))  − 1 ≤ |t(ϕ(x1 + y, x2 ) − ϕ(x1 , x2 ))| e   1   ∂x1 ϕ(x1 + sy, x2 ) ds . = |t||y|  0

And for x1 , x2 in the ball B10 (0) we obtain

|∂x1 ϕ(x1 + sy, x2 )| = 3(x1 + sy)2 + x22 ≤ 6x21 + 6s2 y 2 + x22

(2.18)

≤ c, and therefore    it(ϕ(x1 +y,x2 )−ϕ(x1 ,x2 ))  − 1 ≤ c |t| |y|. e Hence Minkowski’s inequality yields

(2.19)

    eit(ϕ(x1 +y, x2 )−ϕ(x1 , x2 )) − 1   f (x + y, x ) dy   1 2 1+α  p  |y|≤ 1 |y| 100 L (B10 (0))     |t||y||f (x1 + y, x2 )|    ≤c dy   p 1 |y|1+α |y|≤ 100 L (B10 (0))  1 f (x1 + y, x2 )Lp (B10 (0)) dy ≤ c|t| α 1 |y| |y|≤ 100  1 ≤ c|t| f p dy 1 |y|α |y|≤ 100 ≤ cα |t| f p .

Case III: |y| ≤ 1/100 and x21 + x22 ≥ 100. We sub-divide this region into two sub-regions:

(2.20)

(a) |y| ≤

We first assume |y| ≤

1 , 1 + x21 + x22

1 1+x21 +x22

(b) |y| ≥

1 . 1 + x21 + x22

and observe that |∂x1 ϕ(x1 + sy, x2 )| ≤ c(1 + x21 + x22 )

in this region. With the help of the change of variable y˜ = (1 + x21 + x22 )y, inequality

´ G. FONSECA AND M. PACHON

10

(2.15) (b), the argument in (2.17) and Minkowski’s inequality we obtain     it(ϕ(x1 +y, x2 )−ϕ(x1 , x2 ))   e − 1   f (x + y, x ) dy 1 2   1+α 1 |y|  p  |y|≤ 1+x2 +x2 1 2 L (B10 (0)c )      |t||y|(1 + x21 + x22 )|f (x1 + y, x2 )|  ≤ c dy    |y|1+α  |y|≤ 1+x21+x2  p 1 2 L (B10 (0)c )   y ˜ 2 2 α  |t|(1 + x1 + x2 ) |f (x1 + 1+x2 +x2 , x2 )|    1 2 ≤ c d˜ y  |˜y|≤1  p |˜ y |α L (B10 (0)c ) α     |t| 1 + (x1 + 1+xy2˜+x2 )2 + x22 |f (x1 + 1+xy2˜+x2 , x2 )|    1 2 1 2 ≤ cα  d˜ y   α |˜ y|  p  |˜y|≤1 L (B10 (0)c )     2α    |t| 1+xy2˜+x2 |f (x1 + 1+xy2˜+x2 , x2 )|    1 2 1 2 + cα  d˜ y .  |˜y|≤1  |˜ y |2α   p c L (B10 (0) )

Now we perform a second change of variables in the Lp norm u = x1 +

(2.21)

y˜ , 1 + x21 + x22

We notice that for the Jacobian determinant holds |

v = x2 . ∂(u,v) ∂(x1 ,x2 )

the following lower bound

2˜ y x1 2|˜ y ||x1 | 49 ∂(u, v) | = |1 − , |≥1− ≥ ∂(x1 , x2 ) (1 + x21 + x22 )2 (1 + x21 + x22 )2 50

since x21 + x22 ≥ 100. Then we can conclude after Minkowski’s integral inequality that       |y|≤ 1+x21+x2 1 2  ≤ cα |t|

   eit(ϕ(x1 +y, x2 )−ϕ(x1 , x2 )) − 1 f (x1 + y, x2 ) dy   |y|1+α 

   (1 + x21 + x22 )α |f (x1 , x2 )|    d˜   y |˜ y |α |˜ y |≤1 p     |f (x1 , x2 )|    d˜ + cα |t|   y |˜ y |α |˜ y |≤1

Lp (B10 (0)c )

p

≤ cα |t| (f p + (1 + x21 + x22 )α f p ). Next suppose that |y| ≥ 1+x12 +x2 . Changing variable, y˜ = (1 + x21 + x22 )y, using 1 2 (2.15) part (a), a second change of variables as in (2.21) and Minkowski’s inequality

gZK IN WEIGHTED SOBOLEV SPACES

11

we get     it(ϕ(x1 +y, x2 )−ϕ(x1 , x2 ))   e −1   f (x + y, x ) dy 1 2   1+α 1 |y|  p  1+x21+x2 ≤|y|≤ 100 1 2 L (B10 (0)c )      |f (x1 + y, x2 )|  ≤ c dy    1 |y|1+α  1+x21+x2 ≤|y|≤ 100  p 1 2 L (B10 (0)c )    (1 + x21 + x22 )α |f (x1 + 1+xy2˜+x2 , x2 )|    1 2 ≤ c d˜ y 1+α  1≤|˜y|≤ 1+x21 +x22  p |˜ y | 100 L (B10 (0)c )     α  1 + (x1 + 1+xy2˜+x2 )2 + x22 f (x1 + 1+xy2˜+x2 , x2 )χA    1 2 1 2 ≤ cα  d˜ y   1+α |˜ y|  p  |˜y|≥1 L (B10 (0)c )     2α   y˜  f (x1 + 1+xy2˜+x2 , x2 )χA  1+x21 +x22   1 2 + d˜ y  |˜y|≥1  |˜ y |1+α   p c L (B10 (0) )

≤ cα (f p + (1 + x21 + x22 )α f p ),  y | − 1}. where A = {˜ y / |(x1 , x2 )| ≥ 100|˜ On the other hand, regarding Stein’s derivative with respect to x2 , i.e. j = 2 in (2.11), we notice that the useful inequality, 2x1 x2 ≤ x21 + x22 , allows us to perform the same computations and obtain exactly the same bound (2.14).  Next we focus in the local H s theory in Theorem 1 for the gZK IVP in 2D from the works by Linares, Pastor and Farah, see [19], [20], [5] and references therein. Although for every k = 1, 2, 3, ... the proof is accomplished by the contraction principle, the XT space is different according to the nonlinearity degree k (different space-time norms are involved in the choice of XT ). Our persistence result in weighted spaces for gZK strongly depends on this theorem so for the sake of clearness we emphasize some aspects of its proof for all included nonlinearities with k = 1, 2, 3, .... We start by noticing that the method of proof is performed via the Picard iteration applied to the integral version of the gZK IVP given by:  (2.22)

Ψ(u(t)) = W (t)u0 −

t 0

W (t − t )(uk ∂x u)(t )dt , t ∈ [0, T ],

where the solution space XT ⊂ C([0, T ] : H s (R2 )) is determined by the norms • For k = 1, s > 3/4.

(2.23)

s s s + D ∂x u ∞ 2 2 μT1,1 (u) =uL∞ Lx LyT + Dy ∂x uL∞ x T H x LyT

+ ∂x uL2T L∞ + uL2x L∞ . yT xy

´ G. FONSECA AND M. PACHON

12

• For k = 2, s > 3/4.

(2.24)

s s s + D ∂x u ∞ 2 2 μT1,2 (u) =uL∞ Lx LyT + Dy ∂x uL∞ x T H x LyT

+ uL3T L∞ + ∂x u xy

9

LT4 L∞ xy

+ uL2x L∞ . yT

• For 3 ≤ k ≤ 7, s > 3/4.

(2.25)

s s s + D ∂x u ∞ 2 2 μT1,k (u) =uL∞ Lx LyT + Dy ∂x uL∞ x T H x LyT

+ uLpk L∞ + ∂x u xy where pk =

12(k−1) 7−12γ

12

LT5 L∞ xy

T

+ uL4x L∞ , yT

and γ ∈ (0, 1/12).

• For k ≥ 8, s > sk = 1 − 2/k.

(2.26)

s s s + D ∂x u ∞ 2 2 μT1,k (u) =uL∞ Lx LyT + Dy ∂x uL∞ x T H x LyT 2 + ∂x uL∞ + u x LyT

3k +

LT2

L∞ xy

+ ∂x u

3k

LTk+2 L∞ xy

+ u

k

Lx2 L∞ yT

.

The norms involved in these spaces reflect important properties of the associated group to the gZK equation, i.e. linear estimates like the smoothing effect, Strichartz estimates, maximal function estimates, ... . The standard argument is then carried out in the closed ball BaT = {u ∈ XT ; μT1,k (u) ≤ a = 2cu0 H s (R2 ) }, of the metric space XT = {u ∈ C([0, T ] : H s (R2 )); μT1,k (u) < ∞}. By applying to the integral equation (2.22) each norm in the definition of μT1,k (u), linear estimates yield (2.27)

μT1,k (u) ≤ cu0 H s + cT γ (μT1,k (u))k+1 ,

where γ is a positive constant. From this point, the local existence time is chosen so that (2.28)

cak T γ ≤ 1/2, −k

which implies that the time size T ∼ u0 H sγ and that the local solution satisfies T μ1,k (u) ≤ 2cu0 H s . 3. Proof of Theorem 2 We consider the unitary group of operators {W (t) : t ∈ R} in L2 (R2 ) defined as (3.1)

W (t)u0 (x, y) = (eit(ξ

Thus, for α = r1 ∈ (0, 1), (2.3) yields

3

+ξη 2 )

u 0 (ξ, η))∨ (x, y).

gZK IN WEIGHTED SOBOLEV SPACES

|x|r1 W (t)u0 (x, y) = |x|r1 (eit(ξ

3

+ξη 2 )

u 0 (ξ, η))∨ (x, y) = (D1,r1 (eit(ξ

13

3

+ξη 2 )

u 0 (ξ, η)))∨ (x, y).

and from Lemma 1 then (3.2) 3 2 3 2 3 2 D1,r1 (eit(ξ +ξη ) u 0 )(ξ, η) = eit(ξ +ξη ) D1,r1 u 0 (ξ, η) + eit(ξ +ξη ) Φ1,t,r1 ( u0 )(ξ, η), with

u0 )p ≤ cr1 (1 + |t|)( u0 p +  (1 + ξ 2 + η 2 )r1 u 0 p ). Φ1,t,r1 ( Hence, if we take p = 2 and via Fourier transform we obtain from here the identity (3.3) |x|r1 W (t)u0 (x, y) = W (t)(|x|r1 u0 )(x, y) + W (t)({Φ1,t,r1 ( u0 )(ξ, η)}∨ )(x, y). with Φ1,t,r1 as in (2.12) and u0 )(ξ, η)}∨ 2 = Φ1,t,r1 ( u0 )2 {Φ1,t,r1 ( ≤ cr1 (1 + |t|)( u0 2 +  (1 + ξ 2 + η 2 )r1 u 0 2

(3.4)

≤ cr1 (1 + |t|)(u0 2 +  Dxs u0 2 +  Dys u0 2 ), where we have used that s ≥ 2 max{r1 , r2 }. On the other hand, if β ∈ (0, r1 ), then (3.5)

Dxβ (|x|r1 W (t)u0 )(x, y) =W (t)(Dxβ |x|r1 u0 )(x, y) u0 )(ξ, η)}∨ )(x, y). + W (t)(Dxβ {Φ1,t,r1 (

In order to prove (1.18) we need to show that  it(ϕ(ξ+τ,η)−ϕ(ξ,η)) −1 e u 0 (ξ + τ, η) dτ )∨ 2 Dxβ ( |τ |1+r1 (3.6) ≤ cα,β (1 + |t|)(u0 2 + Dxβ+s u0 2 + Dyβ+s u0 2 ). Thus, we write Dxβ (



 = 

(3.7)

eit(ϕ(ξ+τ,η)−ϕ(ξ,η)) − 1 u 0 (ξ + τ, η) dτ )∨ 2 |τ |1+r1 |ξ|β (eit(ϕ(ξ+τ,η)−ϕ(ξ,η)) − 1) u 0 (ξ + τ, η) dτ 2 |τ |1+r1

|ξ|β |eit(ϕ(ξ+τ,η)−ϕ(ξ,η)) − 1| | u0 (ξ + τ, η)| dτ 2 |τ |1+r1  |ξ + τ |β |eit(ϕ(ξ+τ,η)−ϕ(ξ,η)) − 1| ≤ cβ  | u0 (ξ + τ, η)| dτ 2 |τ |1+r1  |τ |β |eit(ϕ(ξ+τ,η)−ϕ(ξ,η)) − 1| + cβ  | u0 (ξ + τ, η)| dτ 2 |τ |1+r1 = I 1 + I2 . ≤

Following the argument used in the proof of Lemma 1 to get (2.14) it holds that 0 2 +  (1 + ξ 2 + η 2 )r1 |ξ|β u 0 2 ) I1 ≤ cr1 ,β (1 + |t|)(|ξ|β u (3.8)

≤ cr1 ,β (1 + |t|)(Dxβ u0 2 + Dyβ u0 2 + Dxβ+2r1 u0 2 + Dyβ+2r1 u0 2 ) ≤ cr1 ,β (1 + |t|)(u0 2 +  Dxβ+s u0 2 +  Dyβ+s u0 2 ).

´ G. FONSECA AND M. PACHON

14

To bound I2 we observe that this estimate is similar to that one used in the proof of Lemma 1 with r1 − β ≥ 0 instead of α. Hence, u0 2 +  (1 + ξ 2 + η 2 )(r1 −β) u 0 2 ) I2 ≤ cr1 ,β (1 + |t|)( (3.9)

≤ cr1 ,β (1 + |t|)(u0 2 + Dx2(r1 −β) u0 2 + Dy2(r1 −β) u0 2 ) ≤ cr1 ,β (1 + |t|)(u0 2 +  Dxβ+s u0 2 +  Dyβ+s u0 2 ).

For the weight in the y direction the analysis follows similar arguments and hence we get that for r2 ∈ (0, 1) u0 )(ξ, η)}∨ )(x, y). (3.10) |y|r2 W (t)u0 (x, y) = W (t)(|y|r2 u0 )(x, y) + W (t)({Φ2,t,r2 ( with Φj,t,r2 as in (2.13) and (3.11)

u0 )(ξ, η)}∨ 2 ≤ cr2 (1 + |t|)(u0 2 +  Dxs u0 2 +  Dys u0 2 ), {Φ2,t,r2 (

and similarly (1.18) for j = 2 is obtained. This completes the proof of Theorem 2.

4. Proof of Theorem 3 We consider the most interesting case s = 2 max{r1 , r2 }, with sk < s < 1 as in the LWP theory in H s . Case 1: k = 1. From the previous assumption on s, and with s > s1 = 3/4. Let u ∈ C([0, T ] : H s (R2 )) be the unique solution of the ZK IVP satisfying the integral equation  (4.1)

u(t) = Ψ(u(t)) = W (t)u0 −

t 0

W (t − t )(u ∂x u)(t )dt , t ∈ [0, T ],

with T = T (u0 H s ) < 1 in (2.28) chosen to satisfy cμT1,1 (u)T γ ≤ 1/2,

(4.2)

where γ = 1/2 for k = 1, T ∼ u0 −2 H s and a = 2cu0 H s is the radius of the ball in the contraction argument in H s so that μT1,1 (u) ≤ a = 2cu0 H s .

(4.3)

Now we suppose additionally that u0 ∈ Zs,(r1 ,r2 ) = H s (R2 ) ∩ L2 (( |x|2r1 + |y|2r2 ) dxdy), and introduce the new norm (4.4)

2 2 . +  |y|r2 uL∞ μT2,1 (u) = μT1,1 (u) +  |x|r1 uL∞ T Lxy T Lxy

Let us estimate μT2,1 (u) in the integral equation (4.1) where we note that it remains to estimate only the last two norms in (4.4) 2 2 and ii)  |y|r2 uL∞ i)  |x|r1 uL∞ T Lxy T Lxy

For i) we have from Theorem 2 and (4.3) that

gZK IN WEIGHTED SOBOLEV SPACES

(4.5)  |x|r1 u(t)L2xy ≤  |x|r1 u0 2 + c(1 + T )u0 H 2r1 +  |x|r1 

r1

≤  |x| u0 2 + c(1 + T )u0 H s + 

T

+ c(1 + T ) 0



T

0

 |x|

15

t 0

r1

W (t − t )(u ∂x u)(t )dt L2xy u∂x uL2xy dt

u ∂x uH s dt 1

≤  |x|r1 u0 2 + c(1 + T )u0 H s + c(1 + T )T 2 (μT1,1 (u))2 + I1,x , where



(4.6)

I1,x =

T 0

 |x|r1 u∂x uL2xy dt .

We note that from Theorem 2 we obtain a similar estimate for ii) and therefore we conclude that (4.7) 1 μT2,1 (u) ≤ c(1+T )(u0 H s + |x|r1 u0 L2x y + |y|r2 u0 L2x y )+c(1+T )T 2 (μT1,1 (u))2 +I1,x +I1,y , where

 I1,y =

(4.8)

T 0

 |y|r2 u∂x uL2xy dt .

Now we have for I1,x : 2 ∂x uL1 L∞ I1,x ≤  |x|r1 uL∞ T Lxy T xy 1

≤ μT2,1 (u) T 2 ∂x uL2T L∞ xy

(4.9)

1

≤ T 2 μT1,1 (u)μT2,1 (u). We observe that a similar estimate to (4.9) holds for I1,y , so in summary we get

(4.10)

μT2,1 (u) ≤c(1 + T )(u0 H s +  |x|r1 u0 L2x y +  |y|r2 u0 L2x y ) 1

1

+ c(1 + T )T 2 (μT1,1 (u))2 + cT 2 μT1,1 (u)μT2,1 (u).

From the time size in (4.2), we can pass the last term to the left side and obtain (4.11) μT2,1 (u) ≤ 2c(1 + T )(u0 H s +  |x|r1 u0 L2x y +  |y|r2 u0 L2x y ) + 2c(1 + T )u0 H s . This basically completes the proof of this theorem in the k = 1 case. Case 2: k ≥ 2. For these nonlinearities the argument follows exactly the same ideas as in the former case and we provide some details at the points where the estimates depend on the norms involved in the associated local theory in H s . We again define a new norm as in (4.4) (4.12)

2 2 , μT2,k (u) = μT1,k (u) +  |x|r1 uL∞ +  |y|r2 uL∞ T Lxy T Lxy

´ G. FONSECA AND M. PACHON

16

with μT1,k the norm associated to the solution space XT in Theorem 1 and given in section 2. Now we consider the second term in (4.12) of the solution u represented in Duhamel’s formula and observe that it is enough consider the norms 2 2 . i)  |x|r1 uL∞ and ii)  |y|r2 uL∞ T Lxy T Lxy

In order to estimate the first norm in i),we apply Theorem 2 and the estimates in the local well-posedness theory in H s (R2 ) in (2.27): (4.13)  |x|r1 u(t)L2xy ≤  |x|r1 u0 2 + c(1 + T )u0 H 2r1 +  |x|r1 ≤  |x|r1 u0 2 + c(1 + T )u0 H s + 

T

+ c(1 + T ) 0



T



t 0

W (t − t )(uk ∂x u)(t )dt L2xy

 |x|r1 uk ∂x uL2xy dt

0

uk ∂x uH s dt

≤  |x|r1 u0 2 + c(1 + T )u0 H s + c(1 + T )T γ (μT1,k (u))k+1 + Ik,x , where

 Ik,x =

(4.14)

T 0

 |x|r1 uk ∂x uL2xy dt .

In a similar fashion, Theorem 2 can be applied again to estimate the norm in ii), and we now obtain (4.15)  |y|r2 u(t)L2xy ≤  |y|r2 u0 2 + c(1 + T )u0 H s + c(1 + T )T γ (μT1,k (u))k+1 + Ik,y , where



(4.16)

Ik,y =

T 0

 |y|r2 uk ∂x uL2xy dt .

Let us estimate Ik,x for the different values of k: • For k = 2. 2 u∂x uL1 L∞ I2,x ≤  |x|r1 uL∞ T Lxy T xy

≤ μT2,2 (u) u (4.17)

9

LT5 L∞ xy

∂x u

9

LT4 L∞ xy

2

≤ T 9 μT2,2 (u)uL3T L∞ ∂x u xy

9

LT4 L∞ xy

2

≤ T 9 (μT1,2 (u))2 μT2,2 (u). • For 3 ≤ k ≤ 7. k−1 2 u ∂x uL1T L∞ Ik,x ≤  |x|r1 uL∞ T Lxy xy

≤ μT2,k (u)uk−1  (4.18)

12

LT7 L∞ xy

≤ T γ μT2,k (u)uk−1 12(k−1)

∂x u

LT7−12γ L∞ xy

≤ T γ (μT1,k (u))k μT2,k (u).

12

LT5 L∞ xy

∂x u

12

LT5 L∞ xy

gZK IN WEIGHTED SOBOLEV SPACES

17

• For k ≥ 8. k−1 2 u ∂x uL1T L∞ Ik,x ≤  |x|r1 uL∞ T Lxy xy

≤ μT2,k (u)uk−1 

3k 2(k−1)

LT

(4.19)

≤ μT2,k (u)uk−1 3k

LT2 L∞ xy

∂x u

≤ T γ μT2,k (u)uk−1 3k + LT2

L∞ xy

L∞ xy

∂x u

3k

LTk+2 L∞ xy

3k

LTk+2 L∞ xy

∂x u

3k

LTk+2 L∞ xy

≤ T γ (μT1,k (u))k μT2,k (u). We remark that for the integrals Ik,y in (4.15) the computations are almost the same to conclude the same bounds as we did for the integral Ik,x , therefore we omit the details. Hence the following estimate for the norm in (4.12) holds

(4.20)

μT2,k (u) ≤c(1 + T )(u0 H s +  |x|r1 u0 L2x y +  |y|r2 u0 L2x y ) + c(1 + T )T γ (μT1,k (u))k+1 + cT γ (μT1,k (u))k μT2,k (u).

From the time size in the contraction argument in Theorem 1, we can again pass the last term to the left side and obtain (4.21) μT2,k (u) ≤ 2c(1 + T )(u0 H s +  |x|r1 u0 L2xy +  |y|r2 u0 L2xy ) + 2c(1 + T )u0 H s . Which completes the proof of the theorem. ACKNOWLEDGMENTS: G. Fonseca was partially supported by Universidad Nacional de Colombia-Sede Bogot´ a. M. Pach´on was partially supported by Universidad Central, Bogot´ a, Colombia. The authors are grateful with Felipe Linares for fruitful conversations regarding this work. The authors also thank the anonymous referees for their helpful remarks. References [1] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I, II, Geom. Funct. Anal. 3 (1993) 107–156, 209–262. [2] E. Bustamante, J. Jim´ enez and J. Mej´ıa The Cauchy problem for a fifth order KdV equation in weighted Sobolev spaces, Electron. J. Differential Equations 141 (2015), 24 pp. [3] E. Bustamante, J. Jim´ enez and J. Mej´ıa, The Zakharov-Kuznetsov equation in weighted Sobolev spaces, J. Math. Anal. Appl. 433 (2016) 149–175. [4] A. Cunha, A. Pastor, The IVP for the Benjamin-Ono-Zakharov-Kuznetsov equation in weighted Sobolev spaces, J. Math. Anal. Appl. 417 (2014) 660–693. [5] L. Farah, F. Linares and A. Pastor, A note on 2D generalized Zakharov-Kuznetsov equation: Local, global and scattering results, J. Differential equations 253 (2012) 2558–2571. [6] A.V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation, Differ. Equ. 31 (1995) 1002–1012. [7] G. Fonseca, G. Ponce, The IVP for the Benjamin-Ono equation in weighted Sobolev spaces, J. Func. Anal. 260 (2011) 436–459. [8] G. Fonseca, F. Linares and G. Ponce, The IVP for the dispersion generalized Benjamin-Ono equation in weighted Sobolev spaces, Annales IHP. Analyse non Lin´ eaire 30 (2013) 763–790. [9] G. Fonseca, F. Linares and G. Ponce, On persistence properties in fractional weighted spaces, Proc. Amer. Math. Soc. 143 (2015) 5353–5367. [10] G. Fonseca, G. Rodriguez and W. Sandoval, Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces, Comm. Pure Appl. Anal. 14 (2015) 1327–1341.

18

´ G. FONSECA AND M. PACHON

[11] A. Gr¨ unrock, S. Herr, The Fourier restriction norm method for the Zakharov-Kuznetsov equation, Discrete Contin. Dyn. Syst. 34 (2014) 2061–2068. [12] P. Isaza, F. Linares and G. Ponce, On decay properties of solutions of the k-generalized KdV equation, Comm. Math. Phys. 324, (2013) 129–146. [13] T. Kakutani, H. Ono, Weak nonlinear hydromagnetic waves in a cold collision-free plasma, J. Phys. Soc. Japan 26 (1969) 1305–1318. [14] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Advances in Mathematics Supplementary Studies, Studies in Applied Math. 8 (1983) 93–128. [15] C.E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), 33–69. [16] C.E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993) 527–620. [17] C.E. Kenig, G. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106 (2001) 617–633. [18] D. Lannes, F. Linares and J.C. Saut, The Cauchy problem for the Euler-Poisson equation and derivation of the Zakharov-Kuznetsov equation, Chapter 10 in Studies in Phase Space Analysis with Applications to PDEs, Progress in Nonlinear Differential Equations and Their Applications 84 Birkh¨ auser (2013), 181–213. [19] F. Linares, A. Pastor, Well-posedness for the 2D modified Zakharov-Kuznetsov equation, SIAM J. Math. Anal. 41 (2009) 1323–1339. [20] F. Linares, A. Pastor, Local and global well-posedness for the 2D generalized ZakharovKuznetsov equation, J. Funct. Anal. 260 (2011) 1060–1085. [21] L. Molinet, D. Pilod, Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications, Annales IHP. Analyse non Lin´ eaire 32 (2015) 347–371. [22] J. Nahas, G. Ponce, On the persistent properties of solutions to semi-linear Schr¨ odinger equation, Comm. P.D.E. 34 (2009) 1208–1227. [23] F. Ribaud, S. Vento, A note on the Cauchy problem for the 2D generalized ZakharovKuznetsov equations, C. R. Math. Acad. Sci. Paris, 350 (2012) 499–503. [24] R. Sipcic, D.J. Benney, Lump interactions and collapse in the modified Zakharov-Kuznetsov equation, Studies in Appl. Math. 105 (2000) 385–403. [25] E.M. Stein, The Characterization of Functions Arising as Potentials, Bull. Amer. Math. Soc. 67 (1961) 102–104. [26] V.E. Zakharov, E.A. Kuznetsov, On three dimensional solitons, Sov. Phys. JETP 39 (1974) 285–286. ´ ticas, Universidad Nacional de Colombia, Bo(G. Fonseca) Departamento de Matema ´ , Colombia gota E-mail address: [email protected] ´ ticas, Universidad Central, Bogota ´ , Colombia (M. Pach´ on) Departamento de Matema E-mail address: [email protected]