Well-Posedness for a System of Quadratic Derivative Nonlinear Schrödinger Equations with Radial Initial Data

In the present paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schrödinger equations. This system was introduced by Colin and Colin (Differ Integral Equ 17:297–330, 2004). The first and second authors obtained some well-posedness results in the Sobolev space Hs(Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{s}({\mathbb R}^d)$$\end{document}. We improve these results for conditional radial initial data by rewriting the system radial form.


Introduction
We consider the Cauchy problem of the system of nonlinear 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.(See, also [7], [8].)They also showed that the local existence of the solution of (1.1) in H s (R d ) for s > d 2 + 3. The system (1.1) is invariant under the following scaling transformation: and the scaling critical regularity is s c = d 2 − 1.We put We note that κ = 0 does not occur when θ ≥ 0 for α, β, γ ∈ R\{0}.
First, we introduce some known results for related problems.The system (1.1) has quadratic nonlinear terms which contain a derivative.A derivative loss arising from the nonlinearity makes the problem difficult.In fact, Mizohata ([21]) considered the Schrödinger equation and proved that the uniform bound is a necessary condition for the L 2 (R d ) well-posedness.Furthermore, Christ ([5]) proved that the flow map of the nonlinear Schrödinger equation is not continuous on H s (R d ) for any s ∈ R. From these results, it is difficult to obtain the well-posedness for quadratic derivative nonlinear Schrödinger equation in general.While for the system of quadratic derivative nonlinear equations, it is known that the well-posedness holds.In [15], the first author proved the well-posedness of (1.1) in H s (R d ), where s is given in Table 1 below.
θ > 0 WP for s ≥ 0 WP for s ≥ s c WP for s ≥ s c θ = 0 WP for s ≥ 1 WP for s ≥ 1 κ = 0 and θ < 0 WP for s ≥ 1 2 Table 1.Well-posedness (WP for short) for (1.1) proved in [15] Recently in [16], the first and second authors improved this result by using the generalization of the Loomis-Whitney inequality introduced in [2] and [3].They proved the well-posedness of (1.1) in H s (R d ) for s ≥ 1 2 if d = 2 and s > 1 2 if d = 3, under the condition κ = 0 and θ < 0. While in [15], the first author also proved that the flow map is not C 2 for s < 1 if θ = 0 and for s < 1  2 if θ < 0 and κ = 0. Therefore, the well-posedness obtained in [15] and [16] are optimal except the case d = 3 and s = 1 2 (which is scaling critical) as far as we use the iteration argument.In particular, the optimal regularity are far from the scaling critical regularity if d ≤ 3 and θ ≤ 0.
We point out that the results in [15] and [16] do not contain the scattering of the solution for d ≤ 3 under the condition θ = 0 (and also θ < 0).In [17], Ikeda, Katayama, and Sunagawa considered the system of quadratic nonlinear Schrödinger equations under the mass resonance condition m 1 + m 2 = m 3 (which corresponds to the condition θ = 0 for (1.1)), where u = (u 1 , u 2 , u 3 ) is and F j is defined by They obtained the small data global existence and the scattering of the solution to (1.5) in the weighted Sobolev space for d = 2 under the mass resonance condition and the null condition for the nonlinear terms (1.6).They also proved the same result for d ≥ 3 without the null condition.
In [18], Ikeda, Kishimoto, and Okamoto proved the small data global well-posedness and the scattering of the solution to (1.5) in H s (R d ) for d ≥ 3 and s ≥ s c under the mass resonance condition and the null condition for the nonlinear terms (1.6).
They also proved the local well-posedness in H s (R d ) for d = 1 and s ≥ 0, d = 2 and s > s c , and d = 3 and s ≥ s c under the same conditions.(The results in [15] for d ≤ 3 and θ = 0 say that if the nonlinear terms do not have null condition, then s = 1 is optimal regularity to obtain the well-posedness by using the iteration argument.) Recently in [23], Sakoda and Sunagawa considered (1.5) for d = 2 and j = where They obtained the small data global existence and the time decay estimate for the solution under some conditions for m 1 , • • • m N and the nonlinear terms (1.7), where the conditions contain (1.1) with θ = 0. While, it is known that the existence of the blow up solutions for the system of nonlinear Schrödinger equations.Ozawa and Sunagawa ( [22]) gave the examples of the derivative nonlinearity which causes the small data blow up for a system of Schrödinger equations.There are also some known results for a system of nonlinear Schrödinger equations with no derivative nonlinearity ( [12], [13], [14]).
The aim in the present paper is to improve the results in [15] and [16] for conditional radial initial data in R 2 and R 3 .The radial solution to (1.1) is only trivial solution since the nonlinear terms of (1.1) are not radial form.Therefore, we rewrite (1.1) into a radial form.Here, we focus on d = 2. Let S(R 2 ) denote the Schwartz then there exists a scalar potential Indeed, if we put for some a 1 , a 2 ∈ R, then W satisfies (1.9) by the first equality in (1.8).Furthermore, W also satisfies (1.10) by the second equality in (1.8).We note that the first equality in (1.8) is equivalent to Now, we consider the system of nonlinear Schrödingers equations: instead of (1.1), where d = 2 or 3, and which is well-defined since H s+1 (R d ) is a quotient space.The system (1.12) is obtained by substituting w = ∇W and w 0 = ∇W 0 in (1.1).
Theorem 1.2.Let d = 2 and θ = 0.Then, the flow map of (1.12) Remark 1.4.Theorem 1.2 says that the well-posedness in Theorem 1.1 for θ = 0 is optimal as far as we use the iteration argument.
Remark 1.5.It is interesting that the result for 2D radial initial data is better than that for 1D initial data.Actually, the optimal regularity for 1D initial data is s = 1 if θ = 0, and s = 1 2 if θ < 0 and κ = 0, which are larger than the optimal regularity for 2D radial initial data.The reason is the following.We use the angular decomposition and each angular localized term has a better property.
For radial functions, the angular localized bound leads to an estimate for the original functions.(See, (2.15) below.) We note that if ∇W 0 = w 0 holds and (u, v, [W ]) is a solution to (1.12) with The existence of a scalar potential W 0 ∈ H s+1 rad (R d ) will be proved for w 0 ∈ A s (R d ) with s > 1 2 (See, Proposition 3.3), where Therefore, we obtain the following.
Remark 1.6.For d = 3, Theorem 1.1 can be obtained by almost the same way as in [15].In Proposition 4.4 (i) of [15] , the author used the Strichartz estimate x with an admissible pair (q, r) = (3, 6d 3d−4 ) for d ≥ 4. But this trilinear estimate does not hold for d = 3.This is the reason why the well-posedness in H sc (R 3 ) could not be obtained in [15].For the radial function u 0 ∈ L 2 (R 3 ), it is known that the improved Strichartz estimate ( [24], Corollary 6.2) While, it holds that Therefore, for d = 3, we can obtain the same estimate in 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 ∈ N 0 := N ∪ {0} and for a dyadic summation we write N a N := n∈N 0 a 2 n and N ≥M a N := n∈N 0 ,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), ψ 1 (t) := χ(t), and ψ N (t) := ψ(N −1 t) for N ≥ 2.Then, N ψ N (t) = 1.We define frequency and modulation projections Furthermore, we define The rest of this paper is planned as follows.In Section 2, we will give the bilinear estimates which will be used to prove the well-posedness.In Section 3, we will give the proof of Theorems 1.1 and 1.3.In Section 4, we will give the proof of Theorem 1.2.

Bilinear estimates
In this section, we prove the bilinear estimates.First, we define the radial condition for time-space function.
Next, we define the Fourier restriction norm, which was introduced by Bourgain in [4].
We put The following bilinear estimate plays a central role to show Theorem 1.1.
To prove Proposition 2.1, we first give the Strichartz estimate.
The Strichartz estimate implies the following.(See the proof of Lemma 2.3 in and (p, q) be an admissible pair of exponents for the Schrödinger equation.Then, we have ) Next, we give the bilinear Strichartz estimate.

Definition 5 ([1]
).We define the angular decomposition of R 2 in frequency.We define a partition of unity in R, .
For a dyadic number A ≥ 64, we also define a partition of unity on the unit circle, We observe that ω A j is supported in We now define the angular frequency localization operators R A j , where ξ = |ξ|(cos ϑ, sin ϑ).
Case 1: High modulation, L max N 2 max In this case, the radial condition is not needed.We assume L 1 N 2 max ∼ N 2 2 .By the Cauchy-Schwarz inequality and (2.5), we have where δ := 1 2 − c.Therefore, we obtain Thus, it suffices to show that Therefore, by choosing b ′ and c as max{ 3−s 6 , 3 8 } < c < b ′ < 1 2 for s > 0, we get (2.14).
Case 2: Low modulation, L max ≪ N 2 max By Lemma 2.6, we can assume N 1 ∼ N 2 ∼ N 3 thanks to L max ≪ N 2 max .We assume L max = L 3 for simplicity.The other cases can be treated similarly.
, by Lemma 2.8, we can write We note that #J(j 1 ) 1.By using the Hölder inequality and Corollary 2.3 with p = q = 4, we get Since u, v, and w are radial respect to x, we have (2.15) Therefore, we obtain This estimate gives the desired estimate (2.13) for s ≥ 1 2 by choosing b ′ and c as • The case θ < 0 We decompose R 3 × R 3 as follows: We can write For the former term, by using the Hölder inequality, Theorem 2.9, and (2.15), we get For the latter term, by using Proposition 2.10, (2.15), and The above two estimates give the desired estimate (2.13) for s > 0 by choosing b ′ and c as max{

Proof of the well-posedness
In this section, we prove Theorems 1.1 and 1.3.For a Banach space H and r > 0, we define B r (H) := {f ∈ H | f H ≤ r}.Furthermore, we define X s,b T as where X s,b α,rad,T and X s,b β,rad,T are the time localized spaces defined by Also, X s+1,b γ,rad,T is defined by the same way.Now, we restate Theorems 1.1 for d = 2 more precisely.Theorem 3.1.Let s ≥ 1  2 if θ = 0 and s > 0 if θ < 0. For any r > 0 and for all initial data (u 0 , v 0 , [W 0 ]) ∈ B r (H s (R 2 )), there exist T = T (r) > 0 and a solution (u, v, [W ]) ∈ X s,b T to the system (1.12) on [0, T ] for suitable b > 1 2 .Such solution is unique in B R (X s T ) for some R > 0.Moreover, the flow map To prove Theorem 3.1, we give the linear estimate.
(2) There exists C 2 > 0 such that for any F ∈ X s,−b ′ σ,T , we have (3) There exists C 3 > 0 such that for any u ∈ X s,b σ,T , we have .
For the proof of Proposition 3.2, see Lemma 2.1 and 3.1 in [10].
We define the map Φ(u, v, γ,[W 0 ] (u, v))]) as To prove the existence of the solution of (1.1), we prove that Φ is a contraction map on B R (X s T ) for some R > 0 and T > 0. For a vector valued function Similarly, .
Therefore if we choose R > 0 and T > 0 as T ).This implies the existence of the solution of the system (1.1) and the uniqueness in the ball B R (X s T ).The Lipschitz-continuity of the flow map is also proved by similar argument.
Next, to prove Theorem 1.3, we justify the existence of a scalar potential of w ∈ (H s (R 2 )) 2 .Let F 1 and F 2 denote the Fourier transform with respect to the first component and the second component, respectively.We note that F −1 x (and also To obtain Proposition 3.3, we use the next lemma.
Remark 3.2.In the proof of Proposition 3.3, we also used This implies given in the proof of Proposition 3.3 is radial.Indeed, this condition with ∇W (x) = w(x) yields (1.10).Remark 3.4.For s ≤ 1 2 , we do not know whether there exists a scalar potential of w ∈ (H s (R 2 )) 2 or not.But we point out that if s < 1  2 , then the 1D delta function appears in ∂ 2 w 1 − ∂ 1 w 2 for some w ∈ (H s (R 2 )) 2 .Then, the irrotational condition does not make sense for pointwise.
Next, we prove that A s (R 2 ) is a Banach space.

The lack of the twice differentiability of the flow map
The following proposition implies Theorem 1.2.Proof.Let N ≫ 1 and p := γ α−γ ( = 0).We note that p is well-defined since θ = 0 implies κ = 0 for α, β, γ ∈ R\{0}.For simplicity, we assume p > 0. Put We calculate that It implies that F is radial.Therefore, there exists G : R → R such that F (ξ) = G(|ξ|).We note that We also observe that Let ǫ > 0 be small.We define a new set E as and we decompose E into four sets:

Proposition 4 .
4 (i).Because of such reason, we omit more detail of the proof for d = 3, and only consider d = 2 in the following sections.

x 1 ∈
R by Lemma 3.4, we have lim h→0