Well-posedness and Ill-posedness for the cubic fractional Schr\"odinger equations

We study the low regularity well-posedness of the 1-dimensional cubic nonlinear fractional Schr\"odinger equations with L\'{e}vy indices $1<\alpha<2$. We consider both non-periodic and periodic cases, and prove that the Cauchy problems are locally well-posed in $H^s$ for $s \geq \frac {2-\alpha}4$. This is shown via a trilinear estimate in Bourgain's $X^{s,b}$ space. We also show that non-periodic equations are ill-posed in $H^s$ for $\frac {2 - 3\alpha}{4(\alpha + 1)}<s<\frac {2-\alpha}4$ in the sense that the flow map is not locally uniformly continuous.


Introduction
We consider the Cauchy problem for the one dimensional fractional Schrödinger equations with cubic nonlinearity in periodic and non periodic settings: i∂ t u + (−∆) α/2 u = γ|u| 2 u, u(0, ·) = φ ∈ H s ( Z), (1.1) where Z = R or T, α ∈ (1, 2) is the Lévy index, γ ∈ R \ {0} and s ∈ R. In this paper we are concerned with well-posedness of the Cauchy problem in low regularity Sobolev spaces. As the linear part generalizes the usual second-order Schrödinger equation, our interest is to investigate how the weaker dispersion affects dynamics and well-posedness. The fractional Schrödinger equations was introduced in the theory of the fractional quantum mechanics where the Feynmann path integrals approach is generalized to α-stable Lévy process [13]. Also it appears in the water wave models (for example, see [11] and references therein).
In what follows Z denotes R (non-periodic) or Z (periodic). Accordingly, the Sobolev space H s ( Z) is defined by where L 2 (Z) denotes L 2 (R) or ℓ 2 (Z) and F f is the Fourier transform or Fourier coefficient of f given by F f (ξ) = Z e −ixξ f dx for ξ ∈ Z.
We define the linear propagator U (t) by setting where F −1 denotes the inverse Fourier transform. Then, by Duhamel's formula the equation (1.1) is written as an integral equation Well-posedness. If s > 1/2, by the Sobolev embedding and the energy method one can easily show the local well-posedness in H s for 0 < α < 2 for both periodic and non periodic cases. The equation (1.1) also has the mass and energy conservation: Thus, for s ≥ α/2 and s > 1/2, the global well-posedness in H s follows from the conservation laws. (For instance see [4,5].) For the less regular initial data, i.e. s ≤ 1/2, particularly in the non periodic case, a plausible approach may be to use the Strichartz estimate for U (t). In fact, it is known that the estimate holds for 2/q + 1/r = 1/2, 2 ≤ q, r ≤ ∞ (see [8]). However, due to weak dispersion the estimate accompanies a derivative loss of order 2-α unless one imposes additional assumptions on φ ( [6,7]). This makes difficult for general data to use the usual iteration argument which relies on (1.3).
To get around the shortcoming of Strichartz estimates we use Bourgain's X s,b space, which has been widely used in the studies of dispersive equations for both non periodic and periodic setting. For the fractional Schrödinger equation, X s,b Z is defined by where ϕ(τ, ξ) is the Fourier transform of ϕ with respect to the time and space variables. Here · denotes 1 + | · |. For the standard iteration argument, the main step is to show the trilinear estimate in terms of X s,b spaces: We obtain this estimate by adapting the dyadic method in Tao [15] in which multilinear estimates in weighted L 2 spaces are systematically studied. The argument similarly applies to both non periodic and periodic cases. The following is our local well-posedness result.
Recently, for the periodic case, Demirbas, Erdogan and Tzirakis [9] showed that the equation (1.1) is locally well-posed for s > 2−α 4 and globally wellposed for s > 5α+1 12 . Our result gives local well-posedness at the missing endpoint s = 2−α 4 . The regularity threshold s = 2−α 4 is optimal in that below that number we do not expect to solve (1.1) via the contraction mapping principle. Firstly, the estimate (1.4) fails for s < 2−α 4 due to the resonant interaction of high-high-high to high (frequencies). Compared to the usual Schrödinger equation, the curvature of the characteristic curve is smaller ((frequency) α−2 ). So, the stronger such resonant interactions make the threshold regularity higher. See the counter-example in Section 4. In [10], the authors claimed that (1.1) is globally well posed if φ ∈ L 2 . But Theorem 1.2 below shows that their result is incorrect. Their proof is based on a trilinear estimate, namely (4.1) with s = 0 ([10, Theorem 3.2]), which is not true.
Ill-posedness. Now we consider ill-posedness in the non periodic setting. Following Christ, Colliander, and Tao [2], we approximate the fractional equations with the cubic NLS, at (N, N α ) in the Fourier space by Taylor expansion of the phase function. This allows to transfer an ill-posedness result of NLS to (1.1). A similar trick was also used in the fifth-order modified KdV equation [12]. The following is our second result.
Then the solution map of the initial value problem (1.1) fails to be locally uniformly continuous on C T H s (R) for any T > 0. More precisely, for 0 < δ ≪ ε ≪ 1 and T > 0 arbitrary, there are two solutions u 1 , u 2 to (1.1) with initial data φ 1 , φ 2 such that In view of the counter-example of the trilinear estimate (1.4) it seems natural to expect the similar ill-posedness result for the periodic equations. However, it is not so simple to set make up a counter example because the frequency supports are distributed in a wide region of length N 2−α 2 . Currently we are not able to prove ill-posedness 1 .
Organization of the paper. The paper is organized as follows. In section 2, we introduce notations and recall previously known estimates which we need in the subsequent section. In section 3, bilinear estimates in X s,b Z space are established. Finally, we prove Theorem 1.1 in section 4 and Theorem 1.2 in section 5.

Notations and Preliminaries
We will use the same notations as in [15]. Let us invoke that Z denotes R for the non-periodic case and Z for the periodic case. For any integer k ≥ 2, let Γ k (R × Z) denote the hyperplane where dζ j is the product of Lebesgue and the counting measure for the periodic case, and the Lebesgue measure on R 2 for the non-periodic case. Note that the integral is symmetric under permutations of ζ j .
Let us define a [k; R × Z]-multiplier to be any function m : Γ k (R × Z) → C. When m is a [k; R × Z]-multiplier, the norm m [k;R×Z] is defined to be the best constant so that the inequality holds for all test functions f j on R × Z. Here we recall some of the results about [k; R × Z]-multiplier from [15], which is to be used later. . Also, if m is a [k; R × Z]-multiplier, and g 1 , · · · , g k are functions from R × Z to R, then From this and Minkowski's inequality, we thus have the averaging estimate, for any finite measure µ on Γ k (R × Z), Lemma 2.3. Let k 1 , k 2 ≥ 1, and m 1 , m 2 be functions defined on (R × Z) k1 , (R × Z) k2 , respectively. Then As a special case, we have the T T * identity, for all functions m : And if J is a non-empty subset of {1, · · · , k}, we define the set supp J (m) ⊂ R J by supp J (m) := j∈J supp j (m). Lemma 2.4. Let J 1 , J 2 be disjoint non-empty subsets of {1, · · · , k} and A 1 , A 2 > 0. Suppose that (m a ) a∈I is a collection of [k; R × Z] multipliers such that In particular, if m a is non-negative and A 1 , We set, for j = 1, 2, 3, For the X s,b Z space estimates, we need to consider the [3; R × Z]-multiplier for a function m on R 3 which will be specified later. By averaging over unit time scale (Lemmas 2.1 and 2.2), one may restrict the multiplier to the region |λ j | ≥ 1.
And we define the function h : which plays an important role in what follows. Let N j , L j , H (j = 1, 2, 3) be dyadic numbers. By dyadic decomposition along the variables ξ j , λ j , as well as the function h(ξ 1 , ξ 2 , ξ 3 ), we have where X N1,N2,N3;H;L1,L2,L3 is the multiplier given by From the identities ξ 1 + ξ 2 + ξ 3 = 0 and λ 1 + λ 2 + λ 3 + h(ξ 1 , ξ 2 , ξ 3 ) = 0 on the support of the multiplier, we see that X N1,N2,N3;H;L1,L2,L3 vanishes unless Suppose for the moment that N 1 ≥ N 2 ≥ N 3 . Then we have N 1 ∼ N 2 1. As N 1 ranges over the dyadic numbers, the symbols in the summation in (2.1) are supported on essentially disjoint regions of ξ 1 and ξ 2 spaces. This is true for any permutation of {1, 2, 3}. Thus, by Lemma 2.4 we have Hence, one is led to consider in the low modulation case H ∼ L max and the high modulation case L max ∼ L med ≫ H. The following two lemmas give estimates for (2.2) in each case.
Let |E| denote the Lebesgue measure or counting measure of any measurable subset E of Z.

Bilinear Estimates
In order to prove well-posedness for (1.1), we show the trilinear estimates (Proposition 4.1 below). For this purpose, we first prove a bilinear estimate for uv L 2 (R× Z) , which automatically gives the estimate for uv L 2 (R× Z) . Since the resonance func- To begin with, we establish estimate for (2.2). Here · Z denotes | · | for nonperiodic case and 1 + | · | for periodic case. So, Then we have the following.
min Z . By symmetry, the same estimates also hold for the case H ∼ L max ∼ L 3 .
Proof. Lemma 2.5 gives the high modulation case H ≪ L max ∼ L med . So we need only to show the estimates in the first four cases.
. This and (3.1) give the estimate for the second case.
We now consider the case by the Taylor expansion. This means that ξ 2 is contained in an interval of length med ) by the mean value theorem and the estimate for the third case follows from (3.1).
max and thus (3.1) and the mean value theorem shows that ξ 2 is contained in an interval of length O(N 1−α max L med ). Since ξ 2 is also contained in an interval of length ≪ N min , Proposition 3.1 follows from (3.1).
We now show some bilinear estimates for the periodic and non periodic cases.
For the periodic case the following is to be useful.
Proof of Lemma 3.3. We observe . This gives the desired estimate.  Here H-sum is bounded by an absolute constant. By summing in L min and then L 1 , we get If Z = R, then we separate N min sum as follows: Secondly, we deal with the case L 2 = L max and N max ∼ N min . Using Proposition 3.1, we have Since s ≥ 2−α 4 , we have We now handle the remaining three cases: L 1 = L max and N 2 ∼ N 3 ≫ N 1 ; L 2 = L max and N 3 ∼ N 1 ≫ N 2 ; L 3 = L max and N 1 ∼ N 2 ≫ N 3 .

1.
Case L 3 = L max and N 1 ∼ N 2 ≫ N 3 . In this case L 3 ∼ H ∼ N α−1 N min . By Proposition 3.1 and summation in L min , we have Since N α−1 N min ∼ L max 1 implies N min N 1−α , by breaking N min -sum into two parts, we have: For the second inequality we use Since s ≥ 2−α 4 , we get the desired result. 4. Proof of Theorem 1.1 For the proof Theorem 1.1, we need the trilinear estimate Failure of (4.1) for s < 2−α 4 . It is easy to see that the trilinear estimate fails when s < 2−α 4 . The counter-example is a resonant high-high-high to high interaction. For N ≫ 1, let Here, the number N 2−α 2 is chosen so that the parallelogram A N to be fit in a width 1 strip of τ = |ξ| α . Then, it follows that This and letting N → ∞ give the necessary condition s ≥ 2−α 4 for (4.1) . and 0 < ε ≪ 1. For any u 1 , u 2 , and u 3 ∈ X s, 1 2 +ε Z , we have Proof. By duality and Plancherel's theorem it suffices to show that 1.
Proof of Theorem 1.1. We define a nonlinear functional N by where ψ is a fixed smooth cut-off function such that ψ(t) = 1 if |t| < 1 and ψ(t) = 0 if |t| > 2, and 0 < T ≤ 1 is fixed. For s, b ∈ R we define the norm Then we recall the well-known properties of X s,b Z : Define a compete metric space B T,ρ by . From (4.2) and (4. .
If ε ′ is sufficiently small, from Proposition 4.1 we see Choosing ρ and T small enough so that ρ ≥ 2C φ H s and CT ε ′ −ε ρ 3 ≤ ρ/2 for some constant C, we see that the functional N is a map from B T,ρ to itself. Similarly one can show that N (u) is a contraction. Therefore there is a unique u ∈ X s, 1 2 +ε Z (J T ) satisfying (1.2).

Ill-posedness
In this section, we prove that the equation (1.1) in the non-periodic case is illposed for 2−3α 4(α+1) < s < 2−α 4 . For convenience we assume that γ = 1. Our strategy is to approximate the solution by the solutions of (5.1) which is ill-posed in H s , s < 0 (see [2] for the non-periodic case and [3,14] for the periodic one). For this purpose we recall ill-posedness result for the Schrödinger equation  .
We shall construct approximate solutions which is given by It is easy to see that Since v(s, y) is a solution of (5.1), we have y). We need to bound the error. First we show the following perturbation result relying on the local well-posedness.
In particular, we have Proof. Writing the equation for V in integral form, we have By taking X 2−α 4 , 1 2 + R (J) norm on both sides and applying (4.3), we get .
By continuity argument with sufficiently small ε, we obtain V ε.
Let w := u − V . Then w satisfies the equation which is written in integral form as Again taking X 2−α 4 , 1 2 + R (J) norms on both sides of the above equation and applying (4.3), we have If ε is sufficiently small, the continuity argument with respect to time gives the desired bound.
For the proof of this lemma, we make use of the following which is in [2].