A remark on the well-posedness of the degenerated Zakharov system

We extend the local well-posedness theory for the Cauchy problem associated to a degenerated Zakharov system. The new main ingredients are the derivation of Strichartz and maximal function norm estimates for the linear solution of a Schrodinger type equation with missing dispersion in one direction. The result here improves the one in [11].


Introduction
We consider the initial value problem associated to the degenerate Zakharov system ipB t E`B z Eq`∆ K E " nE, px, y, zq P R 3 , t ą 0, B 2 t n´∆ K n " ∆ K p|E| 2 q, Ep¨, 0q " E 0 p¨q, np¨, 0q " n 0 p¨q, B t np¨, 0q " n 1 p¨q, y , E is a complex-valued function, and n is a real-valued function. The system (1.1) describes the laser propagation when the paraxial approximation is used and the effect of the group velocity is negligible ( [13]).
We use the term degenerate in the sense that there is no dispersion in the z-direction for the system in (1.1) in contrast to the well known Zakharov system # iB t E`∆E " nE, px, y, zq P R 3 , t ą 0, B 2 t n´∆n " ∆p|E| 2 q, (1.2) which was introduced in [14] to describe the long wave Langmuir turbulence in a plasma. Regarding the IVP (1.1), Colin and Colin in [2] posed the question of the well-posedness. A positive answer was given by Linares, Ponce and Saut in [11], showing the local wellposedness of the IVP (1.1) in a suitable Sobolev space. The results proved in [11] extended previous ones for the Zakharov system (1.2), where transversal dispersion is taken into account (see [12], [5] and references therein). However, the system (1.1) is quite different from the classical Zakharov system (1.2) since the Cauchy problem for the periodic data exhibits strong instabilities of the Hadamard type implying ill-posedness (see [3]).
Our goal here is to extend the local well-posedness for the IVP (1.1) to a larger functional space than that in [11].
Before describing our main result and the new ingredients used in its proof we proceed as in [11] to study this problem.
First the IVP (1.1) is reduced into the IVP associated to a single equation, that is, # ipB t E`B z Eq`∆ K E " nE, px, y, zq P R 3 , t ą 0, Epx, y, z, 0q " E 0 px, y, zq, (1.3) where nptq " N 1 ptqn 0`N ptqn 1`ż t 0 N pt´t 1 q∆ K p|Ept 1 q| 2 qdt 1 , 4) and N 1 ptqf " cospp´∆ K q 1{2 tqf, (1.5) where p´∆ K q 1{2 f " ppξ 2 Then it is considered the integral equivalent formulation of the IVP (1.3), that is, Ept´t 1 qpN 1 pt 1 qn 0`N pt 1 qn 1 qEpt 1 qdt 1 where Eptq denotes the unitary group associated to the linear problem to (1.3) given by A smoothing effect for the unitary group Eptq similar to the one obtained for solutions of the linear Schrödinger equation was proved in [11] (see Proposition 2.1 below). This was the main tool used there to establish local well-posedness via contraction principle in the following functional space r H 2j`1 pR 3 q " tf P H 2j`1 pR 3 q, D 1{2 x B α f, D 1{2 y B α f P L 2 pR 3 q, |α| ď 2j`1, j P Nu, (1.8) where α P pZ`q 3 is a multiindex, D 1{2 x f " p|ξ 1 | 1{2f q _ and D 1{2 y f " p|ξ 2 | 1{2f q _ . Roughly the result in [11] guarantees the local well-posedness in r H 2j`1 pR 3 q, j ě 2, for data E 0 P r H 2j`1 pR 3 q, n 0 P H 2j pR 3 q and n 1 P H 2j´1 pR 3 q with B z n 1 P H 2j´1 pR 3 q, where H s pR 3 q is the usual Sobolev space.
To improve the previous result obtained in [11] we derive two new estimates for solutions of the linear problem. The first one is the following Strichartz estimate, where 2{q " 1´2{p, 2 ď p ă 8. We can observe that the lack of dispersion in the z-direction is reflected in the estimate above. The proof uses the explicit Fourier transform of e itx 2 and the usual method to prove Strichartz estimates for the linear Schrödinger equation.
The second new estimate for solutions of the linear problem is the following maximal function estimate }Eptqf } L 2 x L 8 yzT ď cpT, sq}f } H s pR 3 q , s ą 3{2. (1.10) The argument to prove (1.10) follows the ideas in [7], where they obtained a L 4 x -maximal function estimates for solutions of the linear problem associated to the modified Kadomtsev-Petviashvili (KPI) equation.
Remark 1.1. It is not clear whether the estimate (1.10) is sharp. In Proposition 2.5 below we show that this estimate is false in H s pR 3 q for s ă 1.
To state our result we shall slightly modify the space r H 2j`1 pR 3 q defined in (1.8). We define (1.11) With this notation, the main result here reads as: For initial data pE 0 , n 0 , n 1 q in r H 2 pR 3 qˆH 2 pR 3 qˆH 1 pR 3 q and B z n 1 P H 1 pR 3 q, there exist T ą 0 and a unique solution E of the integral equation (1.6) such that and Moreover, there exists a neighborhood V of pE 0 , n 0 , n 1 q P r H 2 pR 3 qˆH 2 pR 3 qˆH 1 pR 3 q such that the map F : pE 0 , n 0 , n 1 q Þ Ñ Eptq from V into the class defined by (1.12)-(1.15) is smooth.
One also has that n P Cpr0, T s : H 2 pR 3 qq.

Remark 1.3.
Observe that in comparison with the result in [11] we could considerably weaken the regularity required to prove local well-posedness for the IVP (1.1). Notice also that it would be possible to lower the regularity a little further because the maximal function works well in H s pR 3 q, s ą 3{2.
Remark 1.4. The Strichartz estimates were essential in our analysis. It may be possible to use them in the Bourgain spaces framework to obtain better results (see for instance [1], [5], and references therein for the Zakharov system). Regarding global well-posedness, we do not know any conserved quantity that might be useful to extend globally the local results.
The plan of the paper is the following. In Section 2 we prove the new linear estimates commented above and recall some known ones established in [11]. Some useful lemmas will also be presented in this section. In Section 3 we establish estimates involving the nonlinear term that allow us to simplify the exposition of the proof of the main result. Finally our main result will be proved in Section 4.
Before leaving this section we introduce the notation used throughout the paper. We use standard notation in Partial Differential Equations. In addition we will use c to denote various constants that may change from line to line.
Let x " px 1 , x 2 , x 3 q and ξ " pξ 1 , ξ 2 , ξ 3 q. For f " f px, tq P SpR 4 q, p f will denote its Fourier transform in space, whereas p f px i x l q , respectively p f px i q , will denote its Fourier transform in the x i x l and x i variables, i, l " 1, 2, 3. For s P R, we define the Bessel and Riesz potentials of order´s, J s x and D s x , by y J s x f " p1`|ξ| 2 q s{2 p f and z D s x f " |ξ| s p f .
(1.16) We also use the notation J s x i x l and J s x i to denote the operators We introduce the next notation to set together all the terms involving the Strichartz norms in our analysis. (1.18)

Linear estimates
Consider the linear problem:

The solution of the linear IVP (2.19) is given by the unitary group
(2.23) These estimates hold exchanging x and y. Here D 1{2 Proof. We refer to [11] for a proof of this proposition. Now we give the precise statement of the inequality (1.10) and its proof.
The same estimate holds exchanging x and y.
The proof of Proposition 2.2 is a direct consequence of the next lemma, as we shall see later.
Lemma 2.3. For every T ą 0 and k ě 0, there exist a constant cpT q ą 0 and a positive function H k,T p¨q such that for |t| ď T and x " px 1 , x 2 , x 3 q and ξ " pξ 1 , ξ 2 , ξ 3 q in R 3 where ψ i pξ i q " ψp2 k`1´| ξ i |q, and ψ denotes a C 8 pRq function such that ψ " 1 for x ě 1 and ψ " 0 for x ď 0.
To prove this lemma we will employ the argument introduced in [4] Proof. Denote by Jpx, y, z, tq the integral on the left-hand side in (2.26). We can rewrite Jpt, x, y, zq as: Next we consider the following three cases: ‚ For |x 1 | ă 1 we use the support of ψ j , j " 1, 2, 3, and get |J| ď c 2 3k .
Proof of Proposition 2.2. Using the same notation as in Lemma 2.3, i.e., ψ j " ψp2 k`1| ξ j |q, j " 1, 2, 3, we define the sequence t r ψ k u as follows: and for k ě 1,ψ for |t| ď T and g P C 8 0 pR 4 q. Since from this point on the argument to complete the proof of the proposition is well understood (see for instance [9]) we will omit it. Thus the result follows. Now, following ideas from Kenig and Ziesler for the KPI equation (see [7]), we show that (2.24) does not hold for s ă 1.
Now we establish Strichartz estimates to the linear problem (2.19). Before that we state and prove an essential lemma: Lemma 2.6. If t ‰ 0, 1 p`1 p 1 " 1 and p 1 P r1, 2s, then the group Eptq defined in (2.20) is a continuous linear operator from L p 1 xy L 2 z pR 3 q to L p xy L 2 z pR 3 q and Proof. From Plancherel's theorem we have that Using Fourier's transform properties we obtain pEptqf qpx, y, zq " pe´i tpξ 2 and˚x 1 x 2 is the convolution in the first two variables, i.e., By Plancherel's theorem and Minkowski's inequality we have }Eptqf px, y,¨q} L 2 z " }gpx, y,¨q} L 2 z " } ż ż e ippx´x 1 q 2`p x´x 2 q 2 qq{4|t| Therefore, from the last inequality we obtain Proof. To prove this proposition we use standard by now arguments. First one shows that the three inequalities are equivalent. The main ingredient is the Stein-Thomas argument. Thus it is enough to establish for instance the estimate (2.34). To obtain (2.34) we use Lemma 2.6 and the Hardy-Littlewood-Sobolev theorem.
(2.40) Remark 2.9. From this lemma one can easily deduce that ÿ

43)
and These estimates hold exchanging x and y.
In our argument we shall use some of the calculus inequalities involving fractional derivatives proved in [8]. More precisely, we recall the following estimate which is a particular case of those established in ( [8], Theorem A.8).

Nonlinear Estimates
In this section we will establish estimates for the nonlinear terms involving in our analysis We begin by rewriting the integral equivalent form of the IVP (1.3) as

47) and
Lptq " In the next lemma we treat the nonlinearity L in the Sobolev norm }¨} H 2 .
Lemma 3.1. Let α, β 1 , β 2 be multi-indices, then ÿ |α|ď2 }B α L} L 2 xyz ďc T 1{2 }E} L 2 Proof. Using the definition of L in (3.48) and the inequality (2.40) we have that ÿ (3.50) Next it will be enough to consider one of the terms inside the sum on the right hand side of (3.50). By Leibniz' rule we have (3.51) Now we will consider just one of the terms on the right hand of the last inequality, the other one can be similarly treated. To simplify the exposition we choose the terms }B 3 x EĒ} L 2 xyzT and }B 2 x EB xĒ } L 2 xyzT to show the next estimates since they have the same structure of the reminder terms in the sum in (3.51).
The Hölder inequality implies On the other hand, using the Hölder inequality and the Sobolev lemma in the z-direction we obtain (3.53) Using the information in inequalities (3.52) and (3.53) in (3.51) and then in (3.50) the estimate (3.49) follows.
Lemma 3.2. Let α, β 1 , β 2 be multiindexes, then (3.54) Proof. Using Lemma 2.8 we have that Thus applying the argument used in Lemma 3.1 the result follows. (3.56) Proof. To obtain the estimate (3.55) we first use properties of Sobolev spaces to obtain ÿ |α|ď2 }B α pEF q} L 2 Then Lemma 3.1 yields the result. (3.59) These estimates holds exchanging x and y.
Proof. Let β i P pZ`q 3 , i " 1, 2, be multi-indices. The Leibniz rule and Proposition 2.1 yield (3.60) Using the Holder inequality Lemma 2.10 and Remark 2.9 we deduce that ÿ (3.61) On the other hand, the use of the fractional Leibniz rule (2.11), the Holder inequality and the Sobolev embedding in several stages yield the next chain of inequalities ÿ Thus combining (3.61), (3.62) and (3.60) the result follows.
The estimate holds exchanging x and y.
Proof. We follow the argument in the previous lemma. More precisely, Let β i P pZ`q 3 , i " 1, 2, be multi-indices. The Leibniz rule and Proposition 2.1 yield Using the Holder inequality we deduce that ÿ (3.64) On the other hand, the use of the fractional Leibniz rule (2.11), the Holder inequality and the Sobolev embedding yield ÿ As we mention in the introduction we will use the contraction mapping principle. We first define the metric space X a,T " tE P Cpr0, T s : r H 2 pR 3 qq :~E~ď au, wherẽ E~:"}E} L 8 and the integral operator on X a,T , where F and L were defined in (3.47) and (3.48), respectively. We will show that for appropriate a and T the operator Ψp¨q defines a contraction on X a,T . We begin by estimating the H 2 pR 3 q-norm of ΨpEq. Let E P X a,T . By Fubini's Theorem, Minkowski's inequality and group properties we have (4.68) Therefore }ΨpEqptq} L 8 T H 2 ď}E 0 } H 2`cT~E~`}n 0 } H 2`}n 1 } H 1 pR 3 q`T }B z n 1 } H 1 pR 3 qc T 3{2~E~3 . (4.69) Next, we estimate the norms By symmetry is enough to estimate the first two norms. Thus, using the definition of Ψ in (4.66), Proposition 2.2 and the inequalities (4.67) and (4.68) it follows that }ΨpEq} L 2 x L 8 yzT ď}E 0 } H 2`cT~E~`}n 0 } H 2`}n 1 } H 1 pR 3 q`T }B z n 1 } H 1 pR 3 qc T 3{2~E~3 . ď }E 0 } H 2`cT~E~`}n 0 } H 2`}n 1 } H 1 pR 3 q`T }B z n 1 } H 1 pR 3 q˘`c T 3{2~E~3 .

4.72)
It remains to estimate the norms ř cpT q T 1{2~E~`} n 0 } H 2`}n 1 } H 1`T }B z n 1 } H 1˘. (4.73) Hence, a suitable choice of a " ap}E 0 }H 3 , T q and T ( T sufficiently small depending on }n 0 } r H 3 , }n 1 } r H 2 and }B z n 1 } r H 2 ), we see that Ψ maps X a,T into X a,T . Since the reminder of the proof follows a similar argument we will omit it.