The Zakharov–Kuznetsov equation in weighted Sobolev spaces

Accepted Manuscript The Zakharov–Kuznetsov equation in weighted Sobolev spaces

Eddye Bustamante, José Jiménez, Jorge Mejía

PII: DOI: Reference:

S0022-247X(15)00659-9 http://dx.doi.org/10.1016/j.jmaa.2015.07.024 YJMAA 19649

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Journal of Mathematical Analysis and Applications

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4 February 2015

Please cite this article in press as: E. Bustamante et al., The Zakharov–Kuznetsov equation in weighted Sobolev spaces, J. Math. Anal. Appl. (2015), http://dx.doi.org/10.1016/j.jmaa.2015.07.024

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THE ZAKHAROV-KUZNETSOV EQUATION IN WEIGHTED SOBOLEV SPACES ´ JIMENEZ ´ EDDYE BUSTAMANTE, JOSE AND JORGE MEJ´IA

Abstract. In this work we consider the initial value problem (IVP) associated to the two dimensional Zakharov-Kuznetsov equation * ut ` Bx3 u ` Bx By2 u ` uBx u “ 0, px, yq P R2 , t P R, upx, y, 0q “ u0 px, yq. We study the well-posedness of the IVP in the weighted Sobolev spaces H s pR2 q X L2 pp1 ` x2 ` y 2 qr dxdyq, with s, r P R.

1. Introduction In this article we consider the initial value problem (IVP) associated to the two dimensional Zakharov-Kuznetsov (ZK) equation, * ut ` Bx3 u ` Bx By2 u ` uBx u “ 0, px, yq P R2 , t P R, (1.1) upx, y, 0q “ u0 px, yq. This equation is a bidimensional generalization of the Korteweg-de Vries (KdV) equation and in three spatial dimensions was derived by Zakharov and Kuznetsov in [31] to describe unidirectional wave propagation in a magnetized plasma. A rigorous justificacion of the ZK equation from the Euler-Poisson system for uniformly magnetized plasma was done by Lannes, Linares and Saut in the chapter 10 of [17]. Lately, different aspects of the ZK equation and its generalizations have been extensively studied. With respect to the local and global well posedness (LWP and GWP) of the IVP (1.1) in the context of classical Sobolev spaces, Faminskii in [4], established GWP in H s pR2 q, for s ě 1, integer. For that, Faminskii followed the arguments developed by Kenig, Ponce and Vega for the Korteweg-de Vries equation in [16], which use the local smoothing effect, a maximal function estimate and a Strichartz type inequality, for the group associated to the linear part of the equation, to obtain LWP by the contraction mapping principle. Then the global result is a consequence of the conservation of energy. In [18], Linares and Pastor refined Faminskii’s method and obtained LWP for initial data in Sobolev spaces H s pR2 q, for s ą 3{4. Recently, symmetrizing the ZK equation and using the Fourier restriction norm method (Bourgain’s spaces, see [2]), Gr¨ unrock and Herr in [10] improved s the previous results, establishing LWP of the IVP (1.1) in H pR2 q for s ą 1{2. Independently, without symmetrization, the same result was obtained by Pilod and Molinet in [21]. For that they used new bilinear Strichartz estimates deduced directly from the original equation. In this manner their method also works for the case of periodic solutions in the space variable y. 2000 Mathematics Subject Classification. 35Q53, 37K05. Key words and phrases. Zakharov-Kuznetsov equation, local well-posedness, weighted Sobolev spaces. Supported by Universidad Nacional de Colombia-Medell´ın and Colciencias, Colombia, project 111865842951. 1

´ JIMENEZ ´ EDDYE BUSTAMANTE, JOSE AND JORGE MEJ´IA

2

LWP and GWP of the IVP (1.1) for the ZK equation and its generalizations also have been considered in the articles [1], [5], [19], [20], [25], [26] and references therein. In [14], Kato studied the IVP for the generalized KdV equation in several spaces, besides the classical Sobolev spaces. Among them, Kato considered weighted Sobolev spaces. In this work we will be concerned with the well-posedness of the IVP (1.1) in weighted Sobolev spaces. This type of spaces arises in a natural manner when we are interested in determining if the Schwartz space is preserved by the flow of the evolution equation in (1.1). Some relevant nonlinear evolution equations as the KdV equation, the non-linear Schr¨odinger equation and the Benjamin-Ono equation, have also been studied in the context of weighted Sobolev spaces (see [6], [7], [11], [12], [13], [22], [23] and [24] and references therein). We will study real valued solutions of the IVP (1.1) in the weighted Sobolev spaces Zs,r :“ H s pR2 q X L2 pp1 ` x2 ` y 2 qr dxdyq, with s, r P R. The relation between the indices s and r for the solutions of the IVP (1.1) can be found, after the following considerations, contained in the work of Kato: suppose we have a solution u P Cpr0, 8q; H s pR2 qq to the IVP (1.1) for some s ą 1. We want to estimate ppu, uq, where ppx, yq :“ p1 ` x2 ` y 2 qr and p¨, ¨q is the inner product in L2 pR2 q. Proceeding formally we multiply the ZK equation by up, integrate over px, yq P R2 and apply integration by parts to obtain: d 2 ppu, uq “ ´3ppx Bx u, Bx uq ´ ppx By u, By uq ´ 2ppy By u, Bx uq ` pppxxx ` pxyy qu, uq ` ppx u3 , 1q. dt 3 To see that ppu, uq is finite and bounded in t, we must bound the right hand side in the last equation in terms of ppu, uq and }u}2H s . The most significant terms to control in the right hand side in the equation are the three first ones. They may be controlled in the same way. Let us indicate how to bound the first term. Using the Interpolation Lemma 2.5 (see section 2), for θ P r0, 1s and u P Zs,r we have p1´θq }p1 ` x2 ` y 2 qp1´θqr{2 u}H θs ď C}p1 ` x2 ` y 2 qr{2 u}L2 }u}θH s . The term 3ppx Bx u, Bx uq can be controlled when θs “ 1 if |px | ď p1 ` x2 ` y 2 qp1´θqr .

(1.2)

Taking into account that |px | ď p1 ` x2 ` y 2 qr´1{2 , in order to have (1.2) it is enough to require that r ´ 1{2 “ p1 ´ θqr. This condition, together with θs “ 1, leads to r “ s{2. In this way the natural weighted Sobolev space to study the IVP (1.1) is Zs,s{2 . Our aim in this article is to prove that the IVP (1.1) is LWP in Zs,s{2 for s ą 3{4, s real. In order to do that we consider two cases: (i) 3{4 ă s ď 1 and (ii) s ą 1. (i) In the first case p3{4 ă s ď 1q we symmetrize the equation as it was done by Gr¨ unrock and Herr in [10]. In this manner we can establish the estimates for the group associated to the linear part of the symmetrization of the ZK equation, using directly the correspondent estimates for the group associated to the linear KdV equation. In particular, the method used by Faminskii in [4], in order to obtain an estimate for the maximal function associated to the group of the linear ZK equation, is simpler in the case of the linear symmetrized ZK equation. In fact, Faminskii’s method, in this case, combines in a transparent way the decay in t of the fundamental solution of the linear KdV equation with the procedure followed by Kenig, Ponce and Vega in [15], to obtain the maximal type estimate for the KdV equation.

3

On the other hand, we need a tool to treat fractional powers of p|x| ` |y|q. A key ingredient in this direction is a characterization of the generalized Sobolev space Lpb pRn q :“ p1 ´ Δq´b{2 Lp pRn q,

(1.3)

due to Stein (see [27] and [28]) (when p “ 2, L2b pRn q “ H b pRn q). This characterization is as follows. Theorem A. Let b P p0, 1q and 2n{pn ` 2bq ď p ă 8. Then f P Lpb pRn q if and only if (a) f P Lp pRn q, and ˆż ˙1{2 |f pxq ´ f pyq|2 b (b) D f pxq :“ dy P Lp pRn q, n`2b Rn |x ´ y| with }f }Lp :“ }p1 ´ Δqb{2 f }Lp » }f }Lp ` }Db f }Lp » }f }Lp ` }Db f }Lp , b

(1.4)

where Ds f is the homogeneous fractional derivative of order b of f , defined through the Fourier transform by pDb f q^ pξq “ |ξ|b fˆpξq,

(1.5)

(ξ P Rn is the dual Fourier variable of x P Rn ). From now on we will refer to Db f as the Stein derivative of f . As a consequence of Theorem A, Nahas and Ponce proved (see Proposition 1 in [23]) that for measurable functions f, g : Rn Ñ C: Db pf gqpxq ď}f }L8 pDb gqpxq ` |gpxq|Db f pxq, a.e. x P Rn , and }D pf gq}L2 ď}f D g}L2 ` }gD f }L2 . b

b

b

(1.6) (1.7)

It is unknown whether or not (1.7) still holds with Db instead of Db . Following a similar procedure to that done by Nahas and Ponce in [23], in order to obtain 2 a pointwise estimate for Db peit|x| qpxq (see Proposition 2 in [23]), we get to bound appro3 priately Db peitx1 qpx1 , x2 q for b P p0, 1{2s (see Lemma 2.6 in section 2). Using (1.4) (for p “ 2), (1.6), (1.7) and Lemma 2.6 we deduce an estimate for the weighted L2 -norm of the group associated to the linear part of the symmetrization of the ZK equation, }p|x| ` |y|qb V ptqf }L2 , in terms of t, }p|x| ` |y|qb f }L2 and }f }H 2b (see Corollary 2.7 in section 2). This estimate is similar to that, obtained by Fonseca, Linares and Ponce in [8] (see formulas 1.8 and 1.9 in Theorem 1) for the KdV equation. The linear estimates for the group of the linear part of the symmetrization of the ZK equation, together with the estimate for the weighted L2 -norm of the group, allow us to obtain LWP of the IVP (1.1) in a certain subspace of Zs,s{2 by the contraction mapping principle. (ii) In the second case (s ą 1) we use the LWP of the IVP (1.1) in H s pR2 q, obtained by Linares and Pastor in [18]. Then we perform a priori estimates on the ZK equation in order to prove that if the initial data belongs to Zs,s{2 then necessarily u P L8 pr0, T s; L2 pp1 ` x2 ` y 2 qs{2 dxdyqq. In this step of the proof we apply the interpolation inequality (Lemma 2.5 in section 2), mentioned before, which was proved in [9]. Finally, we conclude the proof of the LWP in Zs,s{2 in a similar manner as it was done in [3] for a fifth order KdV equation. Now we formulate in a precise manner the main result of this article. Theorem 1.1. Let s ą 3{4 and u0 P Zs,s{2 a real valued function. Then there exist T ą 0 and a unique u, in a certain subspace YT of Cpr0, T s; Zs,s{2 q, solution of the IVP (1.1). (The definition of the subspace YT will be clear in the proof of the theorem).

4

´ JIMENEZ ´ EDDYE BUSTAMANTE, JOSE AND JORGE MEJ´IA

Moreover, for any T 1 P p0, T q there exists a neighborhood V of u0 in Zs,s{2 such that the data-solution ˜ from V into YT 1 is Lipschitz. map u ˜0 ÞÑ u When 3{4 ă s ď 1, the size of T depends on }u0 }Zs,s{2 , and when s ą 1 the size of T depends only on }u0 }H s . Remark 1. We would like to point out that the global result of Faminskii in H 1 (see [4]), together with a regularity theorem, permits to affirm the existence of global solutions for the IVP (1.1) in H s pR2 q with s ą 1. On the other hand, taking into account the procedure followed in the proof of Theorem 1.1 when s ą 1, we can conclude that the IVP (1.1) is GWP in Zs,s{2 . This article is organized as follows: in section 2 we establish some linear estimates for the group associated to the linear part of the symmetrization of the ZK equation (subsection 2.1), we recall the Leibniz rule for fractional derivatives, deduced by Kenig, Ponce and Vega in [16] and an interpolation lemma proved in [9] and [23] (subsection 2.2), and we find (subsection 2.3) an appropriate 3 estimate for the Stein derivative of order b in R2 of the symbol eitx1 (Lemma 2.6), which has an important consequence (Corollary 2.7) that affirms that the weighted Sobolev space Zs,s{2 remains invariant by the group. In section 3, we use the results, obtained in section 2, in order to prove Theorem 1.1. 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 the notation for mixed space-time norms. For f : R2 ˆ r0, T s Ñ R (or C) we have ˜ż ˆż ż ˙p{q ¸1{p T q }f }Lpx Lq :“ |f px, tq| dtdy dx . Ty

R

R 0

When p “ 8 or q “ 8 we must do the obvious changes with essup. Besides, when in the space-time norm appears t instead of T , the time interval is r0, `8q. 2. Preliminary Results 2.1. Linear Estimates. In this section we consider the linear IVP vt ` Bx3 v ` By3 v “ 0, vpx, y, 0q “ v0 px, yq.

px, yq P R2 , t P R

* (2.1)

The solution of (2.1) is given by vpx, y, tq “ rV ptqv0 spx, yq,

px, yq P R2 ,

t P R,

where tV ptqutPR is the unitary group, defined by ż 1 3 3 eirtpξ `η q`xξ`yηs vp0 pξ, ηqdξdη. rV ptqv0 spx, yq “ 2π R2 For 0 ď ε ď 1{2, let us consider the oscillatory integrals ż 3 3 It px, yq :“ |ξ|ε eirtpξ `η q`xξ`yηs dξdη, 2 żR 3 3 |η|ε eirtpξ `η q`xξ`yηs dξdη. Jt px, yq :“ R2

and

(2.2)

(2.3)

(2.4) (2.5)

5

From lemma 2.2 in [15] it follows that ˇż ˇ ˇż ˇ ˇ ˇˇ ˇ C C C ε iptξ 3 `xξq ˇ ˇ iptη 3 `yηq ˇ |It px, yq| “ ˇ |ξ| e dξ ˇ ˇ e dη ˇˇ ď pε`1q{3 ¨ 1{3 “ p2`εq{3 . |t| |t| |t| R R

(2.6)

In a similar manner, we have |Jt px, yq| ď

C |t|p2`εq{3

.

(2.7)

Proceeding as in [18], from the estimates (2.6) and (2.7) we can obtain the following Strichartz-type estimates for the group. Lemma 2.1. (Strichartz type estimates). For ε P p0, 1{2s, }V p¨t qf }L2 L8 ď CT γ }Dx´ε{2 f }L2xy , xy

(2.8)

ď CT γ }Dy´ε{2 f }L2xy , }V p¨t qf }L2 L8 xy

(2.9)

T T

p1 ´ εq . (Let us recall that if pξ, ηq is the dual Fourier variable of px, yq P R2 , 6 ´ε{2 ´ε{2 pDx f q^ pξ, ηq :“ |ξ|´ε{2 fppξ, ηq and pDy f q^ pξ, ηq :“ |η|´ε{2 fppξ, ηq). where γ “

A standard procedure, which uses Plancherel’s theorem with respect to the variables t and y, and the variables t and x, respectively, permits to obtain the following local type estimates. Lemma 2.2. (Local type estimates). There exists a constant C such that }Bx V p¨t qv0 }L8 2 ď C}v0 }L2 , xy x Lty

(2.10)

}By V p¨t qv0 }L8 2 ď C}v0 }L2 . xy y Ltx

(2.11)

and,

In the next lemma we establish estimates of maximal type. Lemma 2.3. (Maximal type estimates). Let v0 P H s pR2 q, for some s ą 3{4. Then for all T ą 0 ď Cs p1 ` T q1{2 }Ds v0 }L2xy }V p¨t qv0 }L2x L8 yT

(2.12)

ď Cs p1 ` T q1{2 }Ds v0 }L2xy . }V p¨t qv0 }L2y L8 xT

(2.13)

and,

Proof. By the symmetry of the equation Bt v ` Bx3 v ` By3 v “ 0 in x and y, it is enough to establish estimate (2.12). Following Faminskii in [4], let μ P C 8 pRq a nondecreasing function such that μpξq “ 0 for ξ ď 0, μpξq “ 1 for ξ ě 1, and μpθq ` μp1 ´ θq “ 1 for θ P r0, 1s, and let us consider the sequence of functions tψk ukPNYt0u in C 8 pR2 q, defined by ψ0 pξ, ηq :“ μp2 ´ |ξ|qμp2 ´ |η|q, and for k ě 1, ψk pξ, ηq :“ μp2k`1 ´ |ξ|qμp2k`1 ´ |η|qμp|η| ´ 2k ` 1q ` μp2k`1 ´ |ξ|qμp|ξ| ´ 2k ` 1qμp2k ´ |η|q. It can be seen that for all pη, ξq P R2 8 ÿ k“0

ψk pξ, ηq “ 1.

(2.14)

´ JIMENEZ ´ EDDYE BUSTAMANTE, JOSE AND JORGE MEJ´IA

6

For k “ 0, 1, 2, . . . let us define

ż

Ik px, y, tq :“

eirtpξ

R2

3 `η 3 q`xξ`yηs

ψk pξ, ηqdξdη.

Let us estimate the oscillatory integrals Ik px, y, tq. For that we procceed as Faminskii in [4] (Lemma 2.2) and Kenig, Ponce and Vega in [15] (Proposition 2.6). Estimation of I0 px, y, tq. ˇż ˇ ˇż ˇ ˇż ˇ ˇ ˇˇ ˇ ˇ ˇ 3 `yηq 3 `xξq 3 `xξq iptη iptξ iptξ |I0 px, y, tq| “ ˇˇ e μp2 ´ |η|qdη ˇˇ ˇˇ e μp2 ´ |ξ|qdξ ˇˇ ď 4 ˇˇ e μp2 ´ |ξ|qdξ ˇˇ R

R

R

Let us define the phase function ϕ by ϕpξq :“ tξ 3 ` xξ. For t P r0, T s, |ξ| ď 2 and |x| ą 48T , |ϕ1 pξq| “ |3tξ 2 ` x| ě

|x| . 2

Integrating by parts, it can be shown that for t P r0, T s and |x| ą 48T , ˇ ˇˇż ˇż ˆ ˙1 j1 ˇˇ „ ˇ ˇ ˇ μp2 ´ |ξ|q 1 C ˇ iϕpξq iϕpξq ˇ e μp2 ´ |ξ|qdξ ˇˇ “ ˇ e dξ ˇ ď 2 . ˇ 1 pξq 1 pξq ˇ ˇ ϕ ϕ x R R For 0 ă T ď 1, if we define

# H0 pxq :“

then for all px, yq P R2 and t P r0, T s

if |x| ď 48,

16 4C x2

if |x| ą 48,

|I0 px, y, tq| ď H0 pxq, For T ą 1, let us define

# H0 pxq :“

16 4C x2

and

}H0 }L1x ď C.

if |x| ď 48T, if |x| ą 48T.

Then, for all px, yq P R2 and t P r0, T s, |I0 px, y, tq| ď H0 pxq,

and

}H0 }L1x ď Cp1 ` T q.

(2.15)

In this manner we can conclude that, for T ą 0, there exists H0 P L1 pRq such that for all px, yq P R2 and t P r0, T s the assertion (2.15) holds. Estimation of Ik px, y, tq, k ě 1. Because of the form of ψk pξ, ηq, it is sufficient to bound the integral ż 3 3 Jpx, y, tq :“ eirtpξ `η q`xξ`yηs φpξqφpηqdξdη, R2

where φpνq :“ μp2k`1 ´ |ν|q. Let tρ1 , ρ2 u be a partition of unity of R subordinated to the open sets tξ : |ξ| ą 1u and tξ : |ξ| ă 2u, respectively. Then 2 ż ÿ 3 3 Jpx, y, tq “ eirtpξ `η q`xξ`yηs φpξqρj pξqφpηqdξdη ” J1 px, y, tq ` J2 px, y, tq. 2 j“1 R

For j “ 1, 2, let Φj pξq “ φpξqρj pξq. Estimation of J1 px, y, tq.

7

We consider two cases: i) First case: T ą 0 and k P N such that 48T 22k ď 2´k{2 . If |x| ď 2´k{2 it is obvious that |J1 px, y, tq| ď C22k for y P R and t P r0, T s.

(2.16)

If 2´k{2 ă |x| and ξ P supp Φ1 (1 ă |ξ| ă 2k`1 ), then for t P r0, T s |x| (2.17) ě 3tξ 2 . 2 Integrating twice by parts with respect to ξ, it can be seen that for t P r0, T s and y P R, ˆ ˙1 j1 ˇˇ ż ż ˇˇ„ Φ pξq C2k 1 ˇ ˇ 1 dξdη ď φpηq ˇ 1 . (2.18) |J1 px, y, tq| ď ˇ ˇ ϕ1 pξq x2 R R ˇ ϕ pξq |ϕ1 pξq| “ |3tξ 2 ` x| ě

$ & C22k if |x| ď 2´k{2 , Hk1 pxq :“ C2k % if |x| ą 2´k{2 . x2 Then from (2.16) and (2.18) we can conclude that

Let us define

|J1 px, y, tq| ď Hk1 pxq for px, yq P R2 , and t P r0, T s, where 2k ´k{2

}Hk1 }L1 ď C2 2

` C2

k

ż8 2´k{2

1 dx “ C23k{2 . x2

(2.19)

(2.20)

ii) Second case: T ą 0 and k P N such that 2´k{2 ă 48T 22k . If |x| ď 2´k{2 it is clear that (2.16) holds. If x ą 2´k{2 , ϕ1 pξq “ |ϕ1 pξq| “ 3tξ 2 ` x, and in consequence ϕ1 pξq ą 3tξ 2 and ϕ1 pξq ě x.

(2.21)

If x ă ´48T 22k and ξ P supp Φ1 , then for t P r0, T s |x| (2.22) ě 3tξ 2 . 2 From (2.21) and (2.22), proceeding as it was done in the first case, we have that if x ă ´48T 22k or x ą 2´k{2 , then |ϕ1 pξq| ą

C2k for y P R and t P r0, T s. (2.23) x2 Let us suppose that ´48T 22k ă x ă ´2´k{2 and t P r0, T s. If x ď ´48t22k and ξ P supp Φ1 , 1 48 ¨ 22k then then inequalities (2.22) and (2.23) hold. If ´48T 22k ď ´48t22k ă x, i.e. ď t |x| ˇż ˇ ˇż ˇ ˇ ˇˇ ˇ 3 iyη itη iϕpξq |J1 px, y, tq| “ ˇˇ e e φpηqdη ˇˇ ˇˇ e Φ1 pξqdξ ˇˇ R R ˇ ˇ ˇˇż ˇ ˇ ˇˇ ˇ ´1 ´1 itη 3 iϕpξq Φ1 pξqdξ ˇˇ “ C ˇrpF φq ˚ F pe qspyqˇ ˇ e ˇ ˇż ˆ R ˙ ˇ ˇż ˇ ˇ ˇˇ z 1 ´1 iϕpξq ˇ, ˇ ˇ ˇ e Φ pξqdξ (2.24) “ C ˇ pF φqpy ´ zq 1{3 Ai dz 1 ˇ ˇˇ t p3tq1{3 R R where Ai is the Airy function and |J1 px, y, tq| ď

}F ´1 φ}L1 ď Cpk ` 1q.

(2.25)

´ JIMENEZ ´ EDDYE BUSTAMANTE, JOSE AND JORGE MEJ´IA

8

Let us split R in the sets Ω1 :“ tξ : ξ 2 ą

|x| |x| u and Ω2 :“ tξ : ξ 2 ď u. If ξ P Ω1 , 48t 48t

|x|1{2 “ Ct1{2 |x|1{2 . t1{2 Hence, by the Vander Courput’s lemma (see [29], pages 309-311), we have that ˇ ˇż ˇ ˇ iϕpξq ˇ e Φ1 pξqdξ ˇˇ ď Cpt1{2 |x|1{2 q´1{2 “ Ct´1{4 |x|´1{4 . ˇ |ϕ2 pξq| “ 6t|ξ| ě Ct

(2.26)

Ω1

If ξ P Ω2 , |ϕ1 pξq| “ |3tξ 2 ` x| ě |x| ´ 3tξ 2 ě |x| ´

|x| |x| ě . 16 2

Then, integrating by parts with respect to ξ, we have: ˇ ˇż ˇż ˇ ˇ ˇ ˇ Φ1 pξq ˇˇ iϕpξq iϕpξq 1 ˇ ˇ ˇ e Φ1 pξqdξ ˇ “ ˇ e iϕ pξq 1 dξ ˇ ˇ iϕ pξq Ω2 Ω2 ˇ j1 ˇ „ ż ˇ ˇ ˇ Φ1 pξq iϕpξq ˇ Φ pξq C 1 iϕpξq ´ e dξ ˇˇ ď . “ ˇˇ 1 e ˇ 1 iϕ pξq ϕ pξq |x| ξPBΩ2 Ω2

(2.27) (2.28)

From (2.24) to (2.28), taking into account that the Airy function is bounded, we conclude that, |J1 px, y, tq| ď Cpk ` 1qt´1{3 pt´1{4 |x|´1{4 ` |x|´1 q ď Cpk ` 1qpt´7{12 |x|´1{4 ` t´1{3 |x|´1 q. 1 22k ďC , we have t |x| ¸ ˜ 27k{6 22k{3 ´1 ´1{4 |x| ` 1{3 |x| |J1 px, y, tq| ď Cpk ` 1q |x|7{12 |x|

Because of the fact that

ď Cpk ` 1qp27k{6 |x|´5{6 ` 22k{3 |x|´4{3 q.

(2.29)

Let us define $ ’ C22k if |x| ď 2´k{2 , ’ ’ ’ & C2k if x ď ´48T 22k or x ą 2´k{2 , 2 Hk1 pxq :“ x ˆ ˙ ’ ’ 2k ’ ’ ` pk ` 1qp27k{6 |x|´5{6 ` 22k{3 |x|´4{3 q if ´48T 22k ă x ă ´2´k{2 . % C x2 Then, taking into account inequalities (2.16), (2.23) and (2.29), it follows that |J1 px, y, tq| ď Hk1 pxq for px, yq P R2 and t P r0, T s,

(2.30)

}Hk1 }L1 ď Cp1 ` T 1{6 qpk ` 1qp23k{2 ` 25k{6 q ď Cp1 ` T 1{6 qpk ` 1q23k{2 .

(2.31)

and From (2.19) - (2.20) and (2.30)-(2.31) we have that for k ě 1, there exists Hk1 P L1 pRq such that for t P r0, T s and px, yq P R2 , |J1 px, y, tq| ď Hk1 pxq and estimate (2.31) for }Hk1 }L1 holds. Estimation of J2 px, y, tq. Let T ą 0, t P r0, T s and px, yq P R2 . If |x| ď 24T then |J2 px, y, tq| ď Areatpξ, ηq : |ξ| ă 2, |η ă 2k`1 |u ď C2k .

(2.32)

9

If |x| ą 24T , t P r0, T s and ξ P supp Φ2 we have |x| ą 3tξ 2 . 2 Integrating twice by parts with respect to ξ, and using (2.33), it follows that ˆ ˙ jˇ ż ż ˇ„ ˇ 1 Φ2 pξq 1 ˇˇ 2k ˇ dξdη ď C . |J2 px, y, tq| ď φpηq ˇ 1 ˇ ϕ pξq ϕ1 pξq x2 |ϕ1 pξq| “ |3tξ 2 ` x| ě |x| ´ 3tξ 2 ě |x| ´ 12T ą

For T ą 1, let us define

$ & C22k Hk2 pxq :“ C2k % x2 and for 0 ă T ď 1, let us define $ & C22k Hk2 pxq :“ C2k % x2 From (2.32) and (2.34) we have that

(2.33)

(2.34)

if |x| ď 24T, if |x| ą 24T, if |x| ď 24, if |x| ą 24,

|J2 px, y, tq| ď Hk2 pxq for px, yq P R2 and t P r0, T s,

(2.35)

}Hk2 }L1 ď Cp1 ` T q2k .

(2.36)

and From estimates (2.30) and (2.35) for J1 and J2 , respectively, taking into account (2.31) and (2.36), we conclude that there exists Hk P L1 pRq such that for px, yq P R2 and t P r0, T s, |Jpx, y, tq| ď Hk pxq, @x P R

(2.37)

}Hk }L1 pRq ď Cp1 ` T qpk ` 1q23k{2 .

(2.38)

and Because of the form of ψk pξ, ηq, the assertions (2.37) and (2.38) also are true for Ik px, y, tq instead of Jpx, y, tq. We apply now the results obtained for the integrals Ik px, y, tq to estimate the group V . For k “ 0, 1, 2, . . . , let ż 3 3 rVk ptqv0 spx, yq :“ eirtpξ `η q`xξ`yηs ψk pξ, ηqp v0 pξ, ηqdξdη. R2

Then rV ptqv0 spx, yq “

8 ÿ

rVk ptqv0k spx, yq,

k“0

where vp0k pξ, ηq :“ vp0 pξ, ηqχsupp Ψk pξ, ηq. (Here χsupp Ψk is the characteristic function of the set supp ψk in R2 ). Therefore ď }V p¨qv0 }L2x L8 Ty

8 ÿ k“0

}Vk p¨qv0k }L2x L8 . Ty

(2.39)

Using duality, an argument due to Tomas [30], and taking into account estimates (2.15) and (2.38) it can be proved that }Vk p¨qv0k }L2x L8 ď Cp1 ` T q1{2 pk ` 1q1{2 23k{4 }v0k }L2xy , Ty

k “ 0, 1, 2, 3, . . .

´ JIMENEZ ´ EDDYE BUSTAMANTE, JOSE AND JORGE MEJ´IA

10

3 Then, for s ą , 4 ďCp1 ` T q }V p¨qv0 }L2x L8 yT Let us observe that

ˆż

2 }v0k }L2xy “2 sk

2

sk R2

|p v0k pξ, ηq| dξdη

1{2

˙1{2

8 ÿ

pk ` 1q1{2 2p3{4´sqk 2sk }v0k }L2xy .

(2.40)

k“0

ˆż “

R2

2sk

2

2

2

|χsupp Ψk pξ, ηq| |p v0 pξ, ηq| dξdη

˙1{2 .

1 For pξ, ηq P supp ψk , 2k ´ 1 ă |ξ| ă 2k`1 or 2k ´ 1 ă |η| ă 2k`1 . In particular, 2k ă |ξ| ă 2 ¨ 2k 2 1 k k 2k 2 2 or 2 ă |η| ă 2 ¨ 2 . In this manner, for pξ, ηq P supp ψk , 2 ă 4pξ ` η q and in consequence it 2 follows that ˆż ˙1{2 sk s 2 2 s 2 2 }v0k }L2xy ď 4 pξ ` η q |p v0 pξ, ηq| dξdη ď C2s }Ds v0 }L2xy . (2.41) R2

From (2.40) and (2.41) we conclude that ˜ ¸ 8 ÿ s 1{2 p3{4´sqk }V p¨qv0 }L2x L8 p1 ` T q1{2 }Ds v0 }L2xy ď Cs p1 ` T q1{2 }Ds v0 }L2xy , ďC2 pk ` 1q 2 Ty k“0



and Lemma 2.3 is proved.

2.2. Leibniz rule and interpolation lemma. In this subsection we recall the Leibniz rule for fractional derivatives, obtained in [16], and an interpolation inequality, which was deduced in [23] and [9]. Lemma 2.4. (Leibniz rule). Let us consider 0 ă α ă 1 and 1 ă p ă 8. Thus }Dα pf gq ´ f Dα g ´ gDα f }Lp pRq ď C}g}L8 pRq }Dα f }Lp pRq , where Dα denotes Dxα or Dyα . With respect to the weight xry :“ p1 ` r2 q1{2 , for N P N, we will consider a truncated weight wN of xry, such that wN P C 8 pRq, " p1 ` r2 q1{2 if |r| ď N, wN prq :“ (2.42) 2N if |r| ě 3N, wN is non-decreasing in |r| and, for j P N and r P R, pjq

|wN prq| ď

cj , j´1 wN prq

where the constant cj is independent from N . Lemma 2.5. (Interpolation lemma). Let a, b ą 0 and assume that J a f :“ p1 ´ Δqa{2 f P L2 pRn q and x|x|yb f P L2 pRn q, ˘ `řn 2 1{2 . where x “ px1 , ¨ ¨ ¨ , xn q and |x| “ i“1 xi Then, for any θ P p0, 1q, . }x|x|yθb J p1´θqa f }L2 ď C}x|x|yb f }θL2 }J a f }1´θ L2

(2.43)

11

Moreover, the inequality (2.43) is still valid with wN p|x|q instead of x|x|y with a constant C independent of N . 2.3. Stein derivative. In this subsection, we obtain in Lemma 2.6 an appropriate bound for 3 Db peitξ qpξ, ηq. Then, using properties (1.6) and (1.7) of the Stein derivative and Lemma 2.6, we succeed, in Corollary 2.7, to bound in an adequate manner the weighted L2 -norm }p|x| ` |y|qb V ptq}L2xy , for the group ot the symmetrized ZK equation. Lemma 2.6. Let b P p0, 1{2s. For any t ą 0, ´ ¯ 3 Db peitx1 qpx1 , x2 q ď Cb tb{3 ` tpb`1q{3 ` tb{3 |x1 |b ` pt1{3`2b{3 ` t2b{3 q|x1 |2b . Proof. Let x :“ px1 , x2 q and y :“ py1 , y2 q. After the change of variables w :“ t1{3 px ´ yq we have that D pe b

itx31

˜ż qpx1 , x2 q “

3

R2

˜ż

“t

3

|eitx1 ´ eity1 |2 dy |x ´ y|2`2b 2 2{3 w

|eip´3x1 t

b{3

¸1{2

1 `3x1 t

1{3 w 2 ´w 3 q 1 1

´ 1|2

|w|2`2b

R2

¸1{2 dw

” tb{3 I.

(2.44)

Let us observe that |ip´3x21 t2{3 w1 ` 3x1 t1{3 w12 ´ w13 q| ď |w1 |p3x21 t2{3 ` 3|x1 |t1{3 |w1 | ` w12 q. In consequence, for w1 such that 3x21 t2{3 ą 3|x1 |t1{3 |w1 |, i.e. for w1 such that |x1 |t1{3 ą |w1 |, it follows that | ´ 3x21 t2{3 w1 ` 3x1 t1{3 w12 ´ w13 | ď |w1 |p6x21 t2{3 ` |w1 |2 q ď |w1 |p6x21 t2{3 ` x21 t2{3 q ď 7x21 t2{3 |w1 |. In order to estimate I we split the R2 -plane in three regions Ei , i “ 1, 2, 3. First, we define E2 :“ tw “ pw1 , w2 q : |w1 | ă t1{3 |x1 |, |w1 | ă pt1{3 x21 q´1 u, and we estimate ˜ż E2

2 2{3 w

|eip´3x1 t

1 `3x1 t

1{3 w 2 ´w 3 q 1 1

|w|2`2b

Two cases will be consider to estimate this integral. Case 2.1. t1{3 |x1 | ď t´1{3 x´2 1 .

´ 1|2

¸1{2 dw

.

(2.45)

´ JIMENEZ ´ EDDYE BUSTAMANTE, JOSE AND JORGE MEJ´IA

12

In this case, taking into account (2.45), we have ˜ż ¸1{2 2 2{3 1{3 2 3 |eip´3x1 t w1 `3x1 t w1 ´w1 q ´ 1|2 dw |w|2`2b E2 ˜ż 1{3 ¸1{2 t |x1 | ż w12 2 2{3 ď Cx1 t 2 2 1`b dw2 dw1 0 R pw1 ` w2 q ˜ż 1{3 « ff ¸1{2 ż8 t |x1 | ż w1 w12 w12 2 2{3 ď Cx1 t dw2 ` dw2 dw1 2`2b w12`2b 0 w1 w 2 0 ˜ż 1{3 ¸1{2 t |x1 | 1´2b 1´2b 2 2{3 ď Cx1 t pw1 ` w1 qdw1 ď Cx21 t2{3 pt1{3 |x1 |q1´b ď Ct1{3´b{9 , 0

(2.46) where in the last inequality the condition |x1 |3 ă t´2{3 was used. Case 2.2. t1{3 |x1 | ą t´1{3 x´2 1 . A simple calculation shows that ˜ż ´1{3 ´2 ¸1{2 ˜ż ¸1{2 2 2{3 1{3 2 3 t x1 |eip´3x1 t w1 `3x1 t w1 ´w1 q ´ 1|2 dw ď Cx21 t2{3 w11´2b dw1 |w|2`2b E2 0 ď Ct1{3`b{3 |x1 |2b .

(2.47)

From (2.46) and (2.47) we have that ˜ż ¸1{2 2 2{3 1{3 2 3 |eip´3x1 t w1 `3x1 t w1 ´w1 q ´ 1|2 dw ď Cpt1{3´b{9 ` t1{3`b{3 |x1 |2b q. |w|2`2b E2

(2.48)

For the region E1 :“ tw “ pw1 , w2 q : |w1 | ą pt1{3 x21 q´1 u, one has

˜ż

2 2{3 w

|eip´3x1 t

ďC

1{3 w 2 ´w 3 q 1 1

´ 1|2

|w|2`2b

E1

«ż

1 `3x1 t

pt1{3 x21 q´1 0

ż8 pt1{3 x21 q´1

¸1{2 dw

ˆż ď2 E1

1 w12`2b

dw1 dw2 `

ż8 pt1{3 x21 q´1

1 dw |w|2`2b

ż w1 pt1{3 x21 q´1

” ı1{2 ď Cb pt1{3 x21 q´1 pt1{3 x21 q1`2b ` pt1{3 x21 q2b ď Cb tb{3 |x1 |2b .

˙1{2

1 w12`2b

ff1{2 dw2 dw1

From (2.48) and (2.49), if mintt1{3 |x1 |, pt1{3 x21 q´1 u “ pt1{3 x21 q´1 , we obtain that ˜ż ¸1{2 2 2{3 1{3 2 3 |eip´3x1 t w1 `3x1 t w1 ´w1 q ´ 1|2 dw ď Cb rt1{3´b{9 ` pt1{3`b{3 ` tb{3 q|x1 |2b s. 2`2b |w| 2 R

(2.49)

(2.50)

Now we consider the case mintt1{3 |x1 |, pt1{3 x21 q´1 u “ t1{3 |x1 |, i.e. |x1 |3 t2{3 ă 1, and for that purpose we define E3 :“ tw “ pw1 , w2 q : t1{3 |x1 | ă |w1 | ă pt1{3 x21 q´1 u.

13

In order to estimate

˜ż

2 2{3 w

|eip´3x1 t

1 `3x1 t

1{3 w 2 ´w 3 q 1 1

´ 1|2

|w|2`2b

E3

¸1{2 dw

,

we need to consider three cases. Case 3.1. 1 ă t1{3 |x1 |. For this case we note that ¸1{2 ˜ż 2 2{3 1{3 2 3 |eip´3x1 t w1 `3x1 t w1 ´w1 q ´ 1|2 dw |w|2`2b E3 ¸1{2 ˜ż ż 1{3 2 ´1 ˙1{2 ˆż 8 pt x1 q 1 1 ďC ďC dw1 dw2 2`2b pw12 ` w22 q1`b 0 E3 |w| t1{3 |x1 | ¸ ˜ż 1{3 2 ´1 ¸ ff1{2 «ż 1{3 2 ´1 ˜ż 1{3 2 ´1 ż8 pt x1 q pt x1 q pt x1 q 1 1 ďC dw1 dw2 ` dw1 dw2 w12`2b w22`2b pt1{3 x21 q´1 0 t1{3 |x1 | t1{3 |x1 | ”CpI31 ` I32 q1{2 .

(2.51)

It is easy to check that ˙1{2 ˆ 1 Cpt1{3 x21 q´1 1{2 1{3 2 ´1 2 rpt x q s “ “ Ct1{3 t´p3`bq{3 |x1 |´p3`bq ď Ct1{3 , pI31 q ď C 1 pt1{3 |x1 |q2`2b pt1{3 |x1 |q1`b (2.52) where in the last inequality we use the condition t´1{3 |x1 |´1 ă 1. Besides, pI32 q1{2 ď

˜ż

8

¸1{2

1

pt1{3 x21 q´1

w22`2b

pt1{3 x21 q´1 dw2

ď Ctb{3 |x1 |2b .

Hence, from (2.51) to (2.53) we conclude that ¸1{2 ˜ż 2 2{3 1{3 2 3 |eip´3x1 t w1 `3x1 t w1 ´w1 q ´ 1|2 dw ď Cpt1{3 ` tb{3 |x1 |2b q. |w|2`2b E3

(2.53)

(2.54)

Case 3.2. t1{3 |x1 | ă 1 ă pt1{3 x21 q´1 . Let us observe that for |w1 | ă 1, |w1 p´3x21 t2{3 ` 3x1 t1{3 w1 ´ w12 q| ď |w1 |p3 ` 3|w1 | ` |w1 |2 q ď C|w1 |, and then ˜ż

2 2{3 w

|eip´3x1 t

1 `3x1 t

ďC

´ 1|2

|w|2`2b

E3

˜ż

1{3 w 2 ´w 3 q 1 1

8ż1 0

t1{3 |x

1|

|w1 |2 dw1 dw2 ` |w|2`2b

(2.55)

¸1{2 dw

ż 8 ż pt1{3 x2 q´1 1

0

1

1 dw1 dw2 |w|2`2b

¸1{2 ” CpII31 ` II32 q1{2 . (2.56)

´ JIMENEZ ´ EDDYE BUSTAMANTE, JOSE AND JORGE MEJ´IA

14

For pII31 q1{2 we have pII31 q

1{2

˜ż

ďC `

t1{3 |x1 | ż 1 t1{3 |x1 |

0

ż8ż1 1

t1{3 |x

1|

w12 dw1 dw2 ` w12`2b ¸1{2

w12 dw1 dw2 w22`2b

ż1

ż1

t1{3 |x1 | t1{3 |x1 |

w12 dw1 dw2 |w|2`2b

” CpII311 ` II312 ` II313 q1{2 .

(2.57)

” ı1{2 “ t1{3 |x1 | lnpt1{3 |x1 |q´1 ď C.

(2.58)

Let us estimate pII311 q1{2 . For b P p0, 1{2s, it follows that ˜ż 1{3 ż pII311 q1{2 ď

t

|x1 |

0

1

t1{3 |x1 |

1 dw1 dw2 w1

¸1{2

Now we estimate pII312 q1{2 . Taking into account that b P p0, 1{2s we conclude that ¸1{2 ˜ż ż w1 ż1 ż1 1 2 2 w w 1 1 dw2 dw1 ` dw2 dw1 pII312 q1{2 ď 2`2b 2`2b t1{3 |x1 | t1{3 |x1 | w1 t1{3 |x1 | w1 w2 ˆ ˙1{2 1 ď 1` “ C. 1 ` 2b

(2.59)

And for pII313 q1{2 it is clear that pII313 q1{2 ď C

˜ż

8

1

1 w22`2b

¸1{2 dw2

ď C.

(2.60)

From (2.57) to (2.60) we have that pII31 q1{2 ď C.

(2.61)

The estimation of pII32 q1{2 is as follows: ˜ż ż 1{3 2 ´1 ż pt1{3 x2 q´1 ż pt1{3 x2 q´1 1 pt x1 q 1 1 1 1 1{2 pII32 q ď dw dw ` dw1 dw2 1 2 2`2b 2`2b |w| w1 0 1 1 1 ¸1{2 ż pt1{3 x2 q´1 ż8 1 1 ` dw1 dw2 w22`2b pt1{3 x21 q´1 1 ďpC ` Cb ` t2b{3 |x1 |4b q1{2 ď Cb ` tb{3 |x1 |2b . From (2.61) and (2.62) we can affirm that, for b P p0, 1{2s, ˜ż ¸1{2 2 2{3 1{3 2 3 |eip´3x1 t w1 `3x1 t w1 ´w1 q ´ 1|2 dw ď Cb ` tb{3 |x1 |2b . |w|2`2b E3 Case 3.3. pt1{3 x21 q´1 ă 1.

(2.62)

(2.63)

15

In this final case we obtain, using (2.55), ˜ż ¸1{2 ˜ż 1{3 2 2{3 1{3 2 3 t |x1 | ż pt1{3 x21 q´1 |eip´3x1 t w1 `3x1 t w1 ´w1 q ´ 1|2 w12 dw ďC dw1 dw2 |w|2`2b w12`2b 0 E3 t1{3 |x1 | ż pt1{3 x2 q´1 ż pt1{3 x2 q´1 1 1 w12 ` dw1 dw2 w12`2b ` w22`2b t1{3 |x1 | t1{3 |x1 | ¸1{2 ż pt1{3 x2 q´1 ż8 1 w12 ` dw1 dw2 w22`2b pt1{3 x21 q´1 t1{3 |x1 | ”CpIII31 ` III32 ` III33 q1{2 . It is easily seen that ˜ż 1{3 t

pIII31 q1{2 ď

¸1{2

|x1 | ż 1 t1{3 |x1 |

0

w1´2b dw1 dw2

and that pIII32 q

1{2

˜ż ď

pt1{3 x21 q´1 t1{3 |x1 |

˜ż ď

t1{3 |x1 | ż 1 t1{3 |x1 |

0

ż pt1{3 x2 q´1 1

t1{3 |x1 |

(2.64) ¸1{2

w1´1 dw1 dw2

ď C, (2.65)

¸1{2 w1´2b dw1 dw2

.

For b P p0, 1{2q, pIII32 q1{2 ď Cb ppt1{3 x21 q´1 q1´b ď Cb , and, for b “ 1{2, ˜ż pIII32 q1{2 ď

pt1{3 x21 q´1

t1{3 |x1 |

¸1{2 lnpt1{3 |x1 |q´1 dw2

¯1{2 ´ ď pt1{3 x21 q´1 pt1{3 |x1 |q´1 pt1{3 |x1 |q lnpt1{3 |x1 |q´1

1{2 ďCt´1{3 x´2 ď C|x1 |1{2 ď C|x1 |b . 1 |x1 |

Therefore, for b P p0, 1{2s, pIII32 q1{2 ď Cb ` C|x1 |b .

(2.66)

´ ¯1{2 pIII33 q1{2 ď C pt1{3 x21 q´3 pt1{3 x21 q1`2b “ Cppt1{3 x21 q´1 q1´b ď C.

(2.67)

Finally, we estimate pIII33 q1{2 .

From (2.64), (2.65), (2.66) and (2.67) we have that, for b P p0, 1{2s, ˜ż ¸1{2 2 2{3 1{3 2 3 |eip´3x1 t w1 `3x1 t w1 ´w1 q ´ 1|2 dw ď Cb ` C|x1 |b . 2`2b |w| E3 Consequently, from (2.54), (2.63) and (2.68), in any case, for b P p0, 1{2s, ¸1{2 ˜ż 2 2{3 1{3 2 3 |eip´3x1 t w1 `3x1 t w1 ´w1 q ´ 1|2 dw ď Cb p1 ` t1{3 ` |x1 |b ` tb{3 |x1 |2b q. |w|2`2b E3

(2.68)

(2.69)

´ JIMENEZ ´ EDDYE BUSTAMANTE, JOSE AND JORGE MEJ´IA

16

Summarizing estimates (2.44), (2.50) and (2.69) imply that, for b P p0, 1{2s, ¯ ´ 3 Db peitx1 qpx1 , x2 q ďCb tb{3 1 ` t1{3´b{9 ` t1{3 ` |x1 |b ` pt1{3`b{3 ` tb{3 |x1 |2b q ´ ¯ ďCb tb{3 ` tpb`1q{3 ` tb{3 |x1 |b ` pt1{3`2b{3 ` t2b{3 q|x1 |2b . Lemma 2.6 is proved.  Corollary 2.7. Let tV ptqutPR be the group defined by (2.3). For b P p0, 1{2s, there exists Cb ą 0 such that for t ě 0 and f P H 2b pR2 q X L2 pp|x| ` |y|q2b dxdyq ” Z2b,b , }p|x| ` |y|qb V ptqf }L2xy ď Cb rp1 ` tb{3 ` tpb`1q{3 q}f }L2xy ` ptb{3 ` t1{3`2b{3 ` t2b{3 q}D2b f }L2xy ` }p|x| ` |y|qb f }L2xy s. (2.70) Proof. Taking into account the definition of Db (see (1.5)), Plancherel’s theorem, the properties (1.4), (1.7) and (1.6) of the Stein derivative of Db , and Lemma 2.6 and using the notation _ for the inverse Fourier transform, we have: }p|x|`|y|qb V ptqf }L2 3 3 fˆq_ }L2xy “ }p| ´ x| ` | ´ y|qb peitξ `itη fˆq^ p´x, ´yq}L2xy ´ ¯ 3 3 3 3 3 3 ďC}rDb peitξ `itη fˆqs_ px, yq}L2xy ď C }eitξ `itη fˆ}L2 ` }Db peitξ `itη fˆq}L2 ξη ξη ´ ¯ 3 3 3 3 ďC }f }L2xy ` }fˆDb peitξ `itη q}L2 ` }Db pfˆqpeitξ `itη q}L2 ξη ξη ´ ¯ 3 3 ďC }f }L2xy ` }fˆpDb peitξ q ` Db peitη qq}L2 ` }Db pfˆq}L2 ξη ξη ´ ďC }f }L2xy ` Cb }fˆptb{3 ` tpb`1q{3 ` tb{3 p|ξ|b ` |η|b q ¯ `pt1{3`2b{3 ` t2b{3 qp|ξ|2b ` |η|2b qq}L2 ` Cp}fˆ}L2 ` }Db fˆ}L2 q ξη ξη ξη ” ı ďCb p1 ` tb{3 ` tpb`1q{3 q}f }L2xy ` ptb{3 ` t1{3`2b{3 ` t2b{3 q}D2b f }L2xy ` }p|x| ` |y|qb f }L2xy .

“}p|x| ` |y|qb peitξ

3 `itη 3

 3. Proof of the main theorem Proof. Case 3{4 ă s ď 1. Following Gr¨ unrock and Herr in [10] we perform a linear change of variables in order to symmetrize the equation. Let (3.1) x1 :“ μx ` λy, y 1 :“ μx ´ λy, t1 :“ t and vpx1 , y 1 , t1 q :“ upx, y, tq, ? where μ “ 4´1{3 and λ “ 3μ. Then u satisfies the Z-K equation iff v satisfies the equation Bt1 v ` pBx31 v ` By31 vq ` μpvBx1 v ` vBy1 vq “ 0. On the other hand, if v0 px1 , y 1 q :“ u0 px, yq, it easily can be seen that v0 P Zs,s{2 iff u0 P Zs,s{2 .

(3.2)

17

In this manner we may consider the IVP Bt v ` pBx3 v ` By3 vq ` μpvBx v ` vBy vq “ 0, vpx, y, 0q “ v0 px, yq P Zs,s{2 ,

* (3.3)

instead of IVP (1.1), and the integral operator żt Ψpvqptq :“ V ptqv0 ´ μ V pt ´ t1 qpvBx v ` vBy vqpt1 qdt1 ,

(3.4)

0

where tV ptqutPR is the unitary group associated to the linear part of the equation in (3.3), defined in (2.3). Proceeding as in [18], let us define, for T ą 0, the metric space XT :“ tv P Cpr0, T s; H s q : |}v|} ă 8u,

(3.5)

where s s ` }Dx vx } 8 2 ` }Dys vx }L8 ` }vx }L2 L8 ` }v}L2x L8 ` }Dxs vy }L8 |}v|} :“}v}L8 2 2 Lx L T Hxy x L xy y L yT yT

yT

T

` }Dys vy }L8 ` }vy }L2 L8 ` }v}L2y L8 ` }v}L8 ” 2 2 s y L xy xT T L pp|x|`|y|q dxdyq xT

(When s “ 1 in (3.6) we change For a ą 0, let

XTa

T

Dxs

and

Dys

xT

10 ÿ

ni pvq.

(3.6)

i“1

by Bx and By , respectively).

be the closed ball in XT defined by XTa :“ tv P XT : |}v|} ď au.

(3.7)

We will prove that there exist T ą 0 and a ą 0 such that the operator Ψ : XTa Ñ XTa is a contraction. First of all let us prove that for v0 P Zs,s{2 , V p¨qv0 P XT . Indeed s “ }v0 }H s ă 8. n1 pV p¨t qpv0 qq ” }V p¨qv0 }L8 T Hxy

(3.8)

Using local type estimate (2.10), we have “ }Bx V p¨qDxs v0 }L8 ď C}Dxs v0 }L2xy ď C}v0 }H s ă 8; (3.9) n2 pV p¨t qpv0 qq ” }Dxs Bx V p¨qv0 }L8 2 2 x L x L yT

yT

and ď C}Dys v0 }L2xy ď C}v0 }H s ă 8. n3 pV p¨t qpv0 qq ” }Dys Bx V p¨qv0 }L8 2 x L yT

(3.10)

From the Strichartz-type estimate (2.8) it follows that, for ε P p0, 1{2s, “}V p¨qBx v0 }L2 L8 ď CT γ }Dxp1´ε{2q v0 }L2xy , n4 pV p¨t qpv0 qq ” }Bx V p¨qv0 }L2 L8 xy xy T

T

1´ε . 6 Since s ą 3{4, taking ε P p0, 1{2q such that s ą 1 ´ ε{2, from the last inequality we obtain with γ “

n4 pV p¨t qpv0 qq ď CT γ }v0 }H s ă 8.

(3.11)

The maximal type estimate (2.12) implies s ă 8. ď Cs p1 ` T q1{2 }Ds v0 }L2xy ď Cs p1 ` T q1{2 }v0 }Hxy n5 pV p¨t qpv0 qq ” }V p¨qv0 }L2x L8 yT

(3.12)

´ JIMENEZ ´ EDDYE BUSTAMANTE, JOSE AND JORGE MEJ´IA

18

On the other hand, using local type estimate (2.11), Strichartz type estimate (2.9) and maximal type estimate (2.13), we obtain ď C}Dxs v0 }L2xy ď C}v0 }H s ă 8; n6 pV p¨t qpv0 qq ”}Dxs By V p¨qv0 }L8 2 y L

(3.13)

xT

n7 pV p¨t qpv0 qq

”}Dys By V

n8 pV p¨t qpv0 qq ”}By V

p¨qv0 }L8 ď C}Dys v0 }L2xy ď 2 y LxT p¨qv0 }L2 L8 ď CT γ }v0 }H s ă 8; T xy

C}v0 }H s ă 8;

(3.14) (3.15)

ď Cs p1 ` T q1{2 }Ds v0 }L2xy ď Cs p1 ` T q1{2 }v0 }H s ă 8. n9 pV p¨t qpv0 qq ”}V p¨qv0 }L2y L8 xT

(3.16)

Finally, from Corollary 2.7 in section 2.3, we have “ sup }p|x| ` |y|qs{2 V ptqv0 }L2xy n10 pV p¨t qpv0 qq ” }V p¨qv0 }L8 2 s T L pp|x|`|y|q dxdyq tPr0,T s

”

ďCs p1 ` T s{6 ` T ps`2q{6 q}v0 }L2xy ` pT s{6 ` T 1{3`s{3 ` T s{3 q}Ds v0 }L2xy ` }p|x| ` |y|qs{2 v0 }L2xy ă8.

ı

(3.17)

Estimates (3.8) to (3.17) imply that V p¨qv0 P XT . Let v P XTa . We proceed to estimate |}Ψpvq|}. For that it is necessary to bound all the norms ni that appear in the definition of |} ¨ |}. Estimation of n1 pΨpvqq. For t P r0, T s, using (3.8), we have }Ψpvqptq}L2xy ď}V ptqv0 }L2xy ` C

żT 0

1

żT

1

}pvvx qpt q}L2xy dt ` C

ď}v0 }H s ` CT 1{2 p}vvx }L2 L2xy ` }vvy }L2 L2xy q T

ď}v0 }H s ` CT

1{2

0

}pvvy qpt1 q}L2xy dt1

T

p}v}L8 2 }vx }L2 L8 ` }v}L8 L2 }vy }L2 L8 q xy xy T Lxy T xy T

T

ď}v0 }H s ` CT 1{2 |}v|}2 ; and }Dxs Ψpvqptq}L2xy ď }Dxs v0 }L2xy ` C

żT 0

(3.18)

}Dxs pvvx qpt1 q}L2xy dt1 ` C

żT 0

}Dxs pvvy qpt1 q}L2xy dt1 .

(3.19)

We only estimate the first integral in (3.19), being the estimation of the second one similar. From Cauchy-Schwarz inequality and Leibniz rule for fractional derivatives (Lemma 2.4 in section 2.2) it follows that ˆż Tż ˙1{2 żT }Dxs pvvx qpt1 q}L2xy dt1 ď T 1{2 }Dxs pvvx qpt1 qp¨, yq}2L2x dydt1 0

ďCT

1{2

ďCT

1{2

0 R

ˆż Tż 0 R

ˆż T 0

}vx pt

}vx pt

1

1

qp¨, yq}2L8 }Dxs vpt1 qp¨, yq}2L2x dydt1 x

q}2L8 }Dxs vpt1 q}2L2xy dt1 xy

`

żT 0

}vpt

1

`

ż Tż 0 R

qDxs vx pt1 qp¨, yq}2L2x dydt1

qDxs vx pt1 q}2L2xy dt1

´ ¯1{2 2 2 s 2 ďCT 1{2 }v}2L8 Hxy s }vx } 2 8 ` }v}L2 L8 }Dx vx } 8 2 LT Lxy Lx LyT x yT T ´ ¯ s 8 s ďCT 1{2 }v}L8 }v } ` }v} }D v } . 2 2 2 8 8 x L Lxy Lx LyT x x Lx L T Hxy T

}vpt

1

yT

˙1{2

˙1{2

(3.20)

19

From (3.19), (3.20) and the similar estimation for the second integral in (3.19) we can conclude that ” s p}vx } 2 8 ` }vy } 2 8 q }Dxs Ψpvqptq}L2xy ď}Dxs v0 }L2xy ` CT 1{2 }v}L8 LT Lxy LT Lxy T Hxy ı s s 8 `}v}L2x L8 }D v } ` }v} }D v } 2 2 2 8 8 Ly LxT x x Lx L x y Ly L yT yT

ď}Dxs v0 }L2xy

` CT

1{2

xT

2

|}v|} .

(3.21)

Similarly, it can be established that, for all t P r0, T s, }Dys Ψpvqptq}L2xy ď }Dys v0 }L2xy ` CT 1{2 |}v|}2 .

(3.22)

Estimates (3.18), (3.21) and (3.22) imply that 1{2 s ď C}v0 }H s ` CT |}v|}2 . n1 pΨpvqq ” }Ψpvq}L8 T Hxy

Estimation of

9 ř n“2

(3.23)

ni pΨpvqq.

Using estimates from (3.9) to (3.16) and proceeding in a similar manner as it was done in the estimation of n1 pΨpvqq it can be proved that: 9 ÿ

ni pΨpvq ď Cs p1 ` T q1{2 }v0 }H s ` Cs p1 ` T q1{2 T 1{2 |}v|}2 .

(3.24)

i“2

Estimation of n10 pΨpvqq. Applying Corollary 2.7 in section 2.3 we have, for t P r0, T s, that n10 pΨpvqq ”}Ψpvqptq}L2 pp|x|`|y|qs dxdyq

żt

ď}V ptqv0 }L2 pp|x|`|y|qs dxdyq ` C} V pt ´ t1 qppvvx qpt1 q ` pvvy qpt1 qqdt1 }L2 pp|x|`|y|qs dxdyq 0 ” s{6 ps`2q{6 ďCs p1 ` t ` t q}v0 }L2xy ` pts{6 ` t1{3`s{3 ` ts{3 q}Ds v0 }L2xy żt ” ı s{2 `}p|x| ` |y|q v0 }L2xy ` C Cs p1 ` pt ´ t1 qs{6 ` pt ´ t1 qps`2q{6 qp}pvvx qpt1 q}L2xy 0 1 s{6

1

` pt ´ t1 q1{3`s{3 ` pt ´ t1 qs{3 qp}Ds pvvx qpt1 q}L2xy ı `}Ds pvvy qpt1 q}L2xy q ` }p|x| ` |y|qs{2 ppvvx qpt1 q ` pvvy qpt1 qq}L2xy dt1 ” ı 1{3`s{3 s{2 s ďCs p1 ` T q}v0 }H ` }p|x| ` |y|q v0 }L2xy żT 1{3`s{3 q p}pvvx qpt1 q}L2xy ` }pvvy qpt1 q}L2xy qdt1 ` Cs p1 ` T ` }pvvy qpt q}L2xy q ` ppt ´ t q

0

` Cs p1 ` T 1{3`s{3 q ` Cs T

1{2

żT

p}Ds pvvx qpt1 q}L2xy ` }Ds pvvy qpt1 q}L2xy qdt1

0 s{2

p}p|x| ` |y|q

pvvx q}L2 L2xy ` }p|x| ` |y|qs{2 pvvy q}L2 L2xy q. T

T

´ JIMENEZ ´ EDDYE BUSTAMANTE, JOSE AND JORGE MEJ´IA

20

Taking into account that 1{3 ă 4s{9, it follows, for t P r0, T s, that ” ı n10 pΨpvqq ďCs p1 ` T 7s{9 q}v0 }H s ` }p|x| ` |y|qs{2 v0 }L2xy ` Cs p1 ` T 7s{9 qT 1{2 |}v|}2 ` Cs T 1{2 p}p|x| ` |y|qs{2 v}L8 2 }vx }L2 L8 xy T Lxy T

` }p|x| ` |y|q v}L8 2 }vy }L2 L8 q T Lxy T xy ” ı 7s{9 ďCs p1 ` T q}v0 }H s ` }p|x| ` |y|qs{2 v0 }L2xy ` Cs p1 ` T 7s{9 qT 1{2 |}v|}2 . s{2

From estimates (3.23) to (3.25), taking into account that 7s{9 ą 1{2, we obtain ” ı |}Ψpvq|} ď Cs p1 ` T 7s{9 q}v0 }H s ` }p|x| ` |y|qs{2 v0 }L2xy ` Cs p1 ` T 7s{9 qT 1{2 |}v|}2 . If we choose

(3.25)

(3.26)

” ı a :“ 2Cs p1 ` T 7s{9 q}v0 }H s ` }p|x| ` |y|qs{2 v0 }L2xy ,

and T ą 0 such that

Cs p1 ` T 7s{9 qT 1{2 a ă 1{2,

it can be seen that Ψ maps XTa into itself. Moreover, for T small enough, Ψ : XTa Ñ XTa is a contraction. In consequence, there exists a unique v P XTa such that Ψpvq “ v. In other words, for t P r0, T s, żt vptq “ V ptqv0 ´ μ V pt ´ t1 qpvBx v ` vBy vqpt1 qdt1 , 0

i.e., the IVP (3.3) has a unique solution in XTa . Using standard arguments, it is possible to show that for any T 1 P p0, T q there exists a neighborhood W of v0 in Zs,s{2 such that the map v˜0 Ñ v˜ from W into the metric space XT 1 , with T 1 instead of T , is Lipschitz. Then the assertion of Theorem 1.1 follows if we take YT :“ tu P Cpr0, T s; H s q : |}v|} ă 8u, where the relations between u and v, and between u0 and v0 are given by the equations (3.1) and (3.2), respectively. Case s ą 1. By Theorem 1.6 in [18] there exist T “ T p}u0 }H s q and a unique u in the class defined by the conditions u P Cpr0, T s; H s pR2 qq, }Dxs ux }L8 2 x LyT

`

}Dys ux }L8 2 x LyT

(3.27) ` }ux }L2 L8 ` }u}L2x L8 ă 8, xy yT T

(3.28)

which is solution of the IVP (1.1). Moreover, for any T 1 P p0, T q there exists a neighborhood V of u0 in H s such that the data-solution map u ˜0 ÞÑ u ˜ from V into the class defined by (3.27) and (3.28) with T 1 instead of T is Lipschitz. Let tu0m umPN be a sequence in C08 pR2 q such that u0m Ñ u0 in H s pR2 q and let um P Cpr0, T s; H 8 pRqq be the solution of the equation in (1.1) corresponding to the initial data u0m . By Theorem 1.6 in [18], um Ñ u in Cpr0, T s; H s pR2 qq. For N P N, let wN be the function defined in section 2.2. Let p ” pN be the function defined in R2 by

a ppx, yq :“ pwN p x2 ` y 2 qqs .

21

We multiply the equation Bt um ` Bx Δum ` um Bx um “ 0 by um p, and for a fixed t P r0, T s we integrate in R2 with respect to x and y, and use integration by parts to obtain d 2 pum ptq, um ptqpq “ ´ 3pBx um ptq, Bx um ptqpx q ` pum ptq, um ptqpxxx q ` pu3m ptq, px q dt 3 ´ pBy um ptq, By um ptqpx q ´ 2pBx um ptq, By um ptqpy q ` pum ptq, um ptqpxyy q, where p¨, ¨q denotes the inner product in L2 pR2 q. Integrating last equation with respect to the time variable in the interval r0, ts, we have żt żt pum ptq, um ptqpq “pu0m , u0m pq ´ 3 pBx um pt1 q, Bx um pt1 qpx qdt1 ´ pBy um pt1 q, By um pt1 qpx qdt1 0 0 żt żt ´ 2 pBx um pt1 q, By um pt1 qpy qdt1 ` pum pt1 q, um pt1 qppxxx ` pxyy qqdt1 0 0 ż 2 t 3 1 ` pu pt q, px qdt1 . (3.29) 3 0 m Since um Ñ u in Cpr0, T s; H s pR2 qq, ps ą 1q, and the weights p, px , py , pxxx ` pxyy are bounded functions, it follows from (3.29), after passing to the limit when m Ñ 8, that puptq, uptqpq

żt

1

1

1

żt

“pu0 , u0 pq ´ 3 pBx upt q, Bx upt qpx qdt ´ pBy upt1 q, By upt1 qpx qdt1 0 0 ż żt żt 2 t 3 1 ´ 2 pBx upt1 q, By upt1 qpy qdt1 ` pupt1 q, upt1 qppxxx ` pxyy qqdt1 ` pu pt q, px qdt1 3 0 0 0 ”I ` II ` III ` IV ` V ` V I. (3.30) Let us estimate the terms in the right-hand side of (3.30). First of all I ď }u0 }2L2 pp1`x2 `y2 qs{2 dxdyq .

(3.31)

a s´1 With respect to the term II, since |px | ď CwN p x2 ` y 2 q, we have |II| ďC ďC

żt 0 żt 0

1

1

pBx upt q, Bx upt

żt a a ps´1q{2 1 2 2 x ` y qqdt “ C }wN p x2 ` y 2 qBx upt1 q}2L2 pR2 q dt1

s´1 qwN p

a x2 ` y 2 qJupt1 q|}2L2 pR2 q dt1 ,

0

ps´1q{2 }wN p

where J :“ p1 ´ Δq1{2 .

a Using estimate (2.43) in Lemma 2.5 with wN p x2 ` y 2 q we obtain II ďC ďC

żt 0

żt 0

a s´1 s 1 }wN p x2 ` y 2 q s 2 J s s upt1 q}2L2 dt1 a s{2 2ps´1q{s 2{s }wN p x2 ` y 2 qupt1 q}L2 }J s upt1 q}L2 dt1 .

(3.32)

´ JIMENEZ ´ EDDYE BUSTAMANTE, JOSE AND JORGE MEJ´IA

22

Since u P Cpr0, T s; H s q, then from (3.32) it follows that, for t P r0, T s, żt żt a a s{2 s{2 1 2ps´1q{s 1 2 2 |II| ďC }wN p x ` y qupt q}L2 dt ď C p1 ` }wN p x2 ` y 2 qupt1 q}2L2 qdt1 0 0 żt żt a s ďCt ` C pupt1 q, upt1 qwN p x2 ` y 2 qqdt1 “ Ct ` C pupt1 q, upt1 qpqdt1 . (3.33) 0

0

a s´1 In a similar manner, taking into account that |py | ď CwN p x2 ` y 2 q, it can be seen that żt (3.34) |III|, |IV | ďCt ` C pupt1 q, upt1 qpqdt1 . 0

a s´3 With respect to the term V, since |pxxx ` pxyy | ď CwN p x2 ` y 2 q, we have żt żt a s´3 1 1 1 2 2 |V | ď C pupt q, upt qwN p x ` y qqdt ď C pupt1 q, upt1 qpqdt1 .

(3.35)

In order to estimate |V I|, since s ą 1, we have żt a s´1 p x2 ` y 2 qqdt1 |V I| ďC }upt1 q}L8 pupt1 q, upt1 qwN 0 żt żt a 1 1 1 s 1 2 2 ďC }upt q}H s pR2 q pupt q, upt qwN p x ` y qqdt ď C pupt1 q, upt1 qpqdt1 .

(3.36)

0

0

0

0

From equality (3.30) and estimates (3.31) to (3.36) it follows that, for t P r0, T s, żt 2 puptq, uptqpN q ď }u0 }L2 pp1`x2 `y2 qs{2 dxdyq ` Ct ` C pupt1 q, upt1 qpN qdt1 . 0

Gronwall’s inequality enables us to conclude that, for t P r0, T s, żt 1 2 puptq, uptqpN q ď }u0 }L2 pp1`x2 `y2 qs{2 dxdyq ` Ct ` C p}u0 }2L2 pp1`x2 `y2 qs{2 dxdyq ` Ct1 qeCpt´t q dt1 . 0

(3.37)

Passing to the limit in (3.37) when N Ñ 8 we obtain, for t P r0, T s, }uptq}2L2 pp1`x2 `y2 qs{2 dxdyq ď}u0 }2L2 pp1`x2 `y2 qs{2 dxdyq ` Ct żt 1 ` C p}u0 }2L2 pp1`x2 `y2 qs{2 dxdyq ` Ct1 qeCpt´t q dt1 ,

(3.38)

0

which implies that u P L8 pr0, T s; L2 pp1 ` x2 ` y 2 qs{2 dxdyqq. Proceeding as it was done in [3], it can be seen that u P Cpr0, T s; L2 pp1 ` x2 ` y 2 qs{2 dxdyqq and that if u ˜m P Cpr0, T s; Zs,s{2 q is the solution of the ZK equation, corresponding to the initial data u ˜m0 Ñ u0 , where u ˜m0 in Zs,s{2 when m Ñ 8, then u ˜m Ñ u0 in Cpr0, T s; Zs,s{2 q. This fact, together with the continuous dependence proved in [18], allow us to conclude that the assertion of theorem is true for the subspace YT of Cpr0, T s; Zs,s{2 q given by YT “ tu P Cpr0, T s; Zs,s{2 q : inequality (3.28) holdsu. 

23

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´ JIMENEZ ´ EDDYE BUSTAMANTE, JOSE AND JORGE MEJ´IA

[29] Stein, E. M., Oscillatory integrals in Fourier Analysis, Beijing Lectures in Harmonic Analysis, Princeton University Press (1986), 307-355. [30] Tomas, P., A restriction theorem for the Fourier transform, Bull. AMS 81 (1975) 477-478. [31] Zakharov, V.E., Kuznetsov, E.A., On three-dimensional solitons, Soviet Phys. JETP 39 (1974), 285-286. ´ Jime ´nez U., Jorge Mej´ıa L. Eddye Bustamante M., Jose ´ ticas, Universidad Nacional de Colombia Departamento de Matema A. A. 3840 Medell´ın, Colombia E-mail address: [email protected], [email protected], [email protected]