Well-posedness and scattering for a system of quadratic derivative nonlinear Schr\"odinger equations with low regularity initial data

In the present paper, we consider the Cauchy problem of a system of quadratic derivative nonlinear Schr\"odinger equations which was introduced by M. Colin and T. Colin (2004) as a model of laser-plasma interaction. The local existence of the solution of the system in the Sobolev space $H^{s}$ for $s>d/2+3$ is proved by M. Colin and T. Colin. We prove the well-posedness of the system with low regularity initial data. For some cases, we also prove the well-posedness and the scattering at the scaling critical regularity by using $U^{2}$ space and $V^{2}$ space which are applied to prove the well-posedness and the scattering for KP-II equation at the scaling critical regularity by Hadac, Herr and Koch (2009).


Introduction
We consider the Cauchy problem of the system of Schrödinger equations: where α, β, γ ∈ R\{0} and the unknown functions u, v, w are d-dimensional complex vector valued. The system (1.1) was introduced by Colin and Colin in [6] as a model of laser-plasma interaction. (1.1) is invariant under the following scaling transformation: First, we introduce some known results for related problems. The system (1.1) has quadratic nonlinear terms which contains a derivative. A derivative loss arising from the nonlinearity makes the problem difficult. In fact, Mizohata ([23]) proved that a necessary condition for the L 2 well-posedness of the problem: is the uniform bound sup x∈R n ,ω∈S n−1 ,R>0 Furthermore, Christ ([5]) proved that the flow map of the Cauchy problem: is not continuous on H s for any s ∈ R. While, there are positive results for the Cauchy problem: i∂ t u − ∆u = P (u, u, ∇u, ∇u), t ∈ R, x ∈ R d , u(0, x) = u 0 (x), x ∈ R d , P is a polynomial which has no constant and linear terms, (1.5) there are many positive results for the well-posedness in the weighted Sobolev space ( [1], [2], [3], [4], [21], [27]). Kenig, Ponce and Vega ( [21]) also obtained that (1.5) is locally well-posed in H s (without weight) for large enough s when P has no quadratic terms.
(ii) The statement in (i) remains valid if we replace the spaceḢ sc by H s for s ≥ s c .
We assume (α − γ)(β + γ) = 0 and s > s c . Then (1.1) is locally well-posed in H s . More precisely, for any r > 0 and for all initial data is Lipschitz continuous.
System (1.1) has the following conservation quantities (see Proposition 7.1): By using the conservation law for M and H, we obtain the following result.
(ii) We assume that α, β, γ ∈ R\{0} have the same sign and satisfy κ = 0 if While, we obtain the negative result as follows.  Grünrock ([12]) are not contained the critical case s = s c and global property of the solution. In this sense, Theorem 1.6 is the extension of the results by Grünrock ([12]).
The main tools of our results are U p space and V p space which are applied to prove the well-posedness and scattering for KP-II equation at the scaling critical regularity by Hadac,Herr and Koch ([13], [14]). After their work, U p space and V p space are used to prove the well-posedness of the 3D periodic quintic nonlinear Schrödinger equation at the scaling critical regularity by Herr, Tataru and Tzvetkov ( [17]) and to prove the well-posedness and the scattering of the quadratic Klein-Gordon system at the scaling critical regularity by Schottdorf ([26]).
Notation. We denote the spatial Fourier transform by · or F x , the Fourier transform in time by F t and the Fourier transform in all variables by · or F tx . For σ ∈ R, the free evolution e itσ∆ on L 2 is given as a Fourier multiplier We will use A B to denote an estimate of the form A ≤ CB for some constant C and write A ∼ B to mean A B and B A. We will use the convention that capital letters denote dyadic numbers, e.g. N = 2 n for n ∈ Z and for a dyadic summation we write N a N := n∈Z a 2 n and N ≥M a N := n∈Z,2 n ≥M a 2 n for brevity. Let χ ∈ C ∞ 0 ((−2, 2)) be an even, non-negative function such that χ(t) = 1 for |t| ≤ 1. We define ψ(t) := χ(t) − χ(2t) and ψ N (t) := ψ(N −1 t). Then, N ψ N (t) = 1 whenever t = 0. We define frequency and modulation projections The rest of this paper is planned as follows. In Section 2, we will give the definition and properties of the U p space and V p space. In Sections 3, 4 and 5, we will give the bilinear and trilinear estimates which will be used to prove the well-posedness.
In Section 6, we will give the proof of the well-posedness and the scattering (Theorems 1.1, 1.3, 1.6 and Corollary 1.2). In Section 7, we will give the a priori estimates and show Theorem 1.4. In Section 8, we will give the proof of C 2 -ill-posedness (Theorem 1.5). In Appendix A, we will give the proof of the bilinear estimates for the standard 1-dimensional Bourgain norm under the condition (α − γ)(β + γ) = 0 and αβγ(1/α − 1/β − 1/γ) = 0.

U p , V p spaces and their properties
In this section, we define the U p space and the V p space, and introduce the properties of these spaces which are proved by Hadac,Herr and Koch ([13], [14]).
We define the set of finite partitions Z as Definition 1. Let 1 ≤ p < ∞. For {t k } K k=0 ∈ Z and {φ k } K−1 k=0 ⊂ L 2 with K−1 k=0 ||φ k || p L 2 = 1 we call the function a : R → L 2 given by a "U p -atom". Furthermore, we define the atomic space Definition 2. Let 1 ≤ p < ∞. We define the space of the bounded p-variation Likewise, let V p −,rc denote the closed subspace of all right-continuous functions v ∈ V p with lim t→−∞ v(t) = 0, endowed with the same norm (2.1).
For Theorem 2.2 and its proof, see Theorem 2.8 Proposition 2.10 and Remark 2.11 in [13].
be a m-linear operator and I ⊂ R be an interval. Assume that for some 1 ≤ p, q < ∞ Then, there exists T : For Proposition 2.4 and its proof, see Proposition 2.19 in [13].
By Proposition 2.4 and 2.5, we have following: Corollary 2.6. Let σ ∈ R\{0} and (p, q) be an admissible pair of exponents for the Proposition 2.7. Let q > 1, E be a Banach space and T : U q σ → E be a bounded, linear operator with ||T u|| E ≤ C q ||u|| U q σ for all u ∈ U q σ . In addition, assume that for some 1 ≤ p < q there exists C p ∈ (0, C q ] such that the estimate ||T u|| E ≤ C p ||u|| U p σ holds true for all u ∈ U p σ . Then, T satisfies the estimate where implicit constant depends only on p and q.
For Proposition 2.7 and its proof, see Proposition 2.20 in [13].
and A N := {ξ ∈ R d |N/2 ≤ |ξ| ≤ 2N} for a dyadic number N. By the Plancherel's theorem and the duality argument, it is enough to prove the estimate for f ∈ L 2 τ ξ . We change the variables (τ 1 , τ 2 ) → (θ 1 , θ 2 ) as θ i = τ i + σ i |ξ i | 2 (i = 1, 2) and put Then, we have We consider only the estimate for K 1 . The estimates for other K j are obtained by the same way.

Time global estimates for d ≥ 2
In this and next section, implicit constants in ≪ actually depend on σ 1 , σ 2 , σ 3 .
Proof. We define f j,N j ,T := 1 [0,T ) P N j u j (j = 1, 2, 3). For sufficiently large constant C, we put M : . We divide the integrals on the left-hand side of (4.3) into eight piece of the form . By the Plancherel's theorem, we have where c is a constant. Therefore, Lemma 4.1 (i) implies that By the Hölder's inequality and the Sobolev embeddingḢ Furthermore, by the L 2 orthogonality and (2.2) with p = 2, we have While by (2.6) and (2.3), we have Therefore, we obtain Next, we consider the case Q σ 3 3 = Q σ 3 ≥M . By the Cauchy-Schwarz inequality, we have Furthermore, by (2.2) with p = 2, we have While by (3.5), (2.3) and the Cauchy-Schwarz inequality for the dyadic sum, we (4.7) Therefore, we obtain 8) where Proof. We define f j,N j ,T := 1 [0,T ) P N j u j (j = 1, 2, 3). For sufficiently large constant C, we put M := C −1 N 2 max and decompose Id = Q σ j <M + Q σ j ≥M (j = 1, 2, 3). We divide the integrals on the left-hand side of (4.8) into eight piece of the form 1, 2, 3). By the same argument of the proof of Proposition 4.2, we consider only the case that Q First, we consider the case Q σ 1 1 = Q σ 1 ≥M . By the Cauchy-Schwarz inequality, we have Furthermore by (2.2) with p = 2, we have when N 3 ≪ N 2 . Therefore, we obtain ||f || V 2 σ for any σ ∈ R and any T ∈ (0, ∞]. Next, we consider the case Q σ 3 3 = Q σ 3 ≥M . We define P N 3 = P N 3 /2 + P N 3 + P 2N 3 . By the Cauchy-Schwarz inequality, we have Therefore, we obtain σ for any σ ∈ R and any T ∈ (0, ∞]. For the case Q σ 2 2 = Q σ 2 ≥M is proved in exactly same way as the case Q σ 1 For any σ 1 , σ 2 , σ 3 ∈ R\{0} and any dyadic numbers 14) where N max := max 1≤j≤3 N j .
In the proof of proposition 4.2, for L.H.S of (4.5), we use the Sobolev embeddinġ For the other part, by the same way of the proof of proposition 4.2, we obtain (5.1).
Next, we assume d = 1. In the proof of proposition 4.2, for L.H.S of (4.5), we use the Hölder's inequality as follows: x .
We note that (8,4) is the admissible pair of the Strichartz estimate for d = 1.
Furthermore for the first inequality in (4.7), we use (3.7) instead of (3.5). For the other part, by the same way of the proof of proposition 4.2, we obtain (5.1) with T = 1.
In the proof of proposition 4.3, for L.H.S of (4.10) and (4.12), we use the Hölder's inequality and (2.2) with p = 2/(1 − 2δ) instead of p = 2. Then we have For the other part, by the same way of the proof of proposition 4.3, we obtain (5.2).
Next, we assume d = 1. In the proof of proposition 4.3, for L.H.S of (4.11), we use (3.7) instead of (3.5) and for the third inequality in (4.13), we use (3.9) and V 2 −,rc ֒→ U 8 instead of (3.8) and V 2 −,rc ֒→ U 4 . For the other part, by the same way of the proof of proposition 4.3, we obtain (5.2) with T = 1.
Then for any 0 < T < ∞, and any dyadic numbers Then for any 0 < T < ∞, and any dyadic numbers for some δ > 0.

Proof of the well-posedness and the scattering
In this section, we prove Theorems 1.1, 1.3, 1.6 and Corollary 1.2. To begin with, we define the function spaces which spaces will be used to construct the solution.
(i) We defineŻ s σ := {u ∈ C(R;Ḣ s (R d )) ∩ U 2 σ | ||u||Ż s σ < ∞} with the norm Remark 6.1. Let E be a Banach space of continuous functions f : R → H, for some Hilbert space H. We also consider the corresponding restriction space to the interval also a Banach space (see Remark 2.23 in [13]).
We define the map Φ(u, v, w) = (Φ T,γ,w 0 (u, v)) as To prove the existence of the solution of (1.1), we prove that Φ is a contraction map . To obtain (6.4), we use the argument of the proof of Theorem 3.2 in [13]. We define where implicit constants in ≪ actually depend on σ 1 , σ 2 , σ 3 .
We assume that α, β, γ ∈ R\{0} and s ∈ R satisfy the condition in Theorem 1.3. Then there exists δ > 0 such that for any 0 < T < ∞, we have

A priori estimates
In this section, we prove Theorem 1.4. We define We have the conservation law for M by calculating and for H by calculating The following a priori estimates imply Theorem 1.4.
By the same argument as above, we obtain for some constant C > 0 and d ≤ 4. Therefore if (7.6) holds for some ǫ with 0 < ǫ ≪ 1, we have By choosing ǫ sufficiently small, we have H 0 < 2ǫ 2 for d = 3 (and also d = 4).

C 2 -ill-posedness
In this section, we prove Theorem 1.5. We rewrite Theorem 1.5 as follows: Proof. We prove only for d = 1. For d ≥ 2, it is enough to replace D 1 , D 2 and D by in the following argument. We use the argument of the proof of Theorem 1 in [24]. For the sets D 1 , D 2 ⊂ R, we define the functions f , g ∈ H s (R) as First, we consider the case (α − γ)(β + γ) = 0. We assume α − γ = 0. (For the case β + γ = 0 is proved by similar argument. ) We put M := −(β + γ)/2γ, then we have For N ≫ 1, we define the sets D 1 , D 2 and D ⊂ R as Then, we have for any ξ ∈ D 1 satisfying ξ − ξ 1 ∈ D 2 and 0 ≤ t ≪ 1. This implies Therefore we obtain (8.1) because s − 1/2 > s − 1 for any s ∈ R.

acknowledgements
The author would like to express his appreciation to Kotaro Tsugawa for many discussions and very valuable comments.