On Fractional Schrodinger Equations in sobolev spaces

Let $\sigma\in(0,1)$ with $\sigma\neq\frac{1}{2}$. We investigate the fractional nonlinear Schr\"odinger equation in $\mathbb R^d$: $$i\partial_tu+(-\Delta)^\sigma u+\mu|u|^{p-1}u=0,\, u(0)=u_0\in H^s,$$ where $(-\Delta)^\sigma$ is the Fourier multiplier of symbol $|\xi|^{2\sigma}$, and $\mu=\pm 1$. This model has been introduced by Laskin in quantum physics \cite{laskin}. We establish local well-posedness and ill-posedness in Sobolev spaces for power-type nonlinearities.

1 is the infinitesimal generator of some Levy processes [Ber96]. A rather extensive study of the potential theoretic aspects of this operator can be found in [Lan72]. The previous equation is a fundamental equation of fractional quantum mechnics, a generalization of the standard quantum mechanics extending the Feynman path integral to Levy processes [Las02].
The purpose of the present paper is to develop a general well-posedness and illposedness theory in Sobolev spaces. The one-dimensional case has been treated in [CHKL14] for cubic nonlinearities, i.e. p " 3, and σ P p 1 2 , 1q. Here, we consider a higher-dimensional version and other types of nonlinear terms. We also include all σ P p0, 1q except σ " 1 2 ; furthermore, contrary to [CHKL14] where the use of Bourgain spaces was crucial (since the main goal of their paper was to derive well-posedness theory on the flat torus), we rely only on standard Strichartz estimates and functional inequalities in R d . In the case of Hartree-type nonlinearities, the local well-posedness and blow-up have been investigated in [CHHO13].
In the present paper, we will not consider global aspects with large data. For that, we refer the reader to [GSWZ13] for a study of the energy-critical equation in the radial case, following the seminal work of Kenig and Merle [KM08,KM06]. As a consequence, we do not consider blow-up phenomena, an aspect we will treat in a forthcoming work.
We introduce two important exponents for our purposes: Here, s c is the scaling-critical regularity exponent in the following sense: for λ ą 0, the transformation upt, xq Þ Ñ 1 λ 2σ{pp´1q u´t λ 2σ , x λ¯, u 0 pxq Þ Ñ 1 λ 2σ{pp´1q u 0´x λk eeps the equation invariant and one can expect local-wellposedness for s ě s c , since the scaling leaves the 9 H sc norm invariant. On the other hand, s g is the critical regularity in the "pseudo"-Galilean invariance (see the proof of ill-posedness below). Under the flow of the equation (NLS σ ), the following quantities are conserved: An important feature of the equation under study is a loss of derivatives for the Strichartz estimates as proved in [COX11]. Unless additional assumptions are met such as radiality as in [GW14], one has a loss of dp1´σq derivatives in the dispersion (see (2.1)). This happens to be an issue in several arguments.
Main results. The goal of this paper is to show that pNLS σ q is locally well-posed in H s for s ě maxps c , s g , 0q, and it is ill-posed in H s for s P ps c , 0q. We start with well-posedness results. Then, pNLS σ q is locally well-posed in H s .
Theorem 1.2 (Local well-posedness in critical cases). Suppose that # p ą 5 when d " 1, Then, pNLS σ q is locally well-posed in H sc .
The proof of Theorem 1.2 is based on a new method, improving on estimates in [CKS`08]. This improvement, based on controlling the nonlinearity in a suitable space, is necessary due to the loss of derivatives in the Strichartz estimates.
As a by-product, we also prove small data scattering.
Finally, our last theorem is the ill-posedness result. Note that our result is not optimal, since one should expect ill-posedness in H s up to s g " 1´σ 2 , which is nonnegative. We hope to come back to this issue in a forthcoming work. Theorem 1.5 (Ill-posedness). Let d " 1, 2 or 3 and σ P p d 4 , 1q. If p is not an odd integer, we further assume that p ě k`1, where k is an integer larger than d 2 . Then, pNLS σ q is ill-posed in H s for s P ps c , 0q.
An interesting feature of the previous ill-posedness result is the fact that, contrary to the standard NLS equation (σ " 1) there is no exact Galilean invariance. However, one can introduce a new "pseudo-Galilean invariance" which is enough to our purposes. More precisely, for v P R d , we define the transformation G v upt, xq " e´i v¨x e it|v| 2σ upt, x´2tσ|v| 2pσ´1q vq.
Note that when σ " 1, G v is simply a Galilean transformation, and that NLS is invariant under this transformation, that is, if uptq solves NLS, so does G v uptq. However, when σ ‰ 1, pNLS σ q is not exactly symmetric with respect to pseudo-Galilean transformations. This opens the construction of solitons for pNLS σ q which happen to be different from the ones constructed in the standard case σ " 1. Indeed, if we search for exact solutions of the type upt, xq " e itp|v| 2σ´ω2σ q e´i v¨x Q ω px´2tσ|v| 2pσ´1q vq, (1.1) then the profile Q ω solves the pseudo-differential equation i.e., P v is a Fourier multiplier y P v f pξq " p v pξqf pξq, wiht symbol p v pξq " |ξ´v| 2σ´| v| 2σ`2 σ|v| 2σ´2 v¨ξ. (1.4) We plan to come back to this issue in future works.
We define the Strichartz norm by }u} S s q,r pIq :" }|∇|´d p1´σqp 1 2´1 r q u} L q tPI W s,r x , where I " r0, T q. Let ψ : R d Ñ r0, 1s be a compactly supported smooth function such that ř N P2 Z ψ N " 1, where ψ N pξq " ψp ξ N q. For dyadic N P 2 Z , let P N be a Littlewood-Paley projection, that is, z P N f pξq " ψp ξ N qf pξq. Then, we define a slightly stronger Strichartz norm by }u}Ss q,r pIq :"´ÿ Proposition 2.1 (Strichartz estimates [COX11]). For an admissible pair pq, rq, we have Sketch of Proof. By the standard stationary phase estimate, one can show that }e itp´∆q σ P 1 } L 1 ÑL 8 À |t|´d 2 , and by scaling, Then, it follows from the argument of Keel-Tao [KT98] that for any I Ă R, Squaring the above inequalities and summing them over all dyadic numbers in 2 Z , we prove Strichartz estimates.
The loss of derivatives is due to the Knapp phenomenon (see [GW14]). However, in the radial case, one can overcome this loss as proved in [GW14], restricting then the admissible powers of the fractional laplacian. Indeed, in [GW14], this is proved that one has optimal Strichartz estimates if σ P pd{p2d´1q, 1q. In particular, the number d{p2d´1q is larger than 1{2 and there is a gap between the Strichartz estimates for the wave operator σ " 1{2 and the one occuring for higher powers. This issue suggests that a new phenomenon might occur for this range of powers.

Local Well-posedness
We establish local well-posedness of the fractional NLS by the standard contraction mapping argument based on Strichartz estimates. Due to loss of regularity in Strichartz estimates, our proof relies on the L 8 x bounds (see Lemma 3.2 and 3.3).
3.1. Subcritical cases. First, we consider the case that d " 1 and 2 ď p ă 5. In this case, the equation is scaling-subcritical in H s for s ą s g , since s g ą s c . We remark that in the proof, we control the L 4 tPI L 8 x norm simply by Strichartz estimates (see (3.1) and (3.2)).
Proof of Theorem 1.1 when d " 1 and 2 ď p ă 5. We define x , where I " r0, T q. Then, applying the 1d Strichartz estimates we get where s ą 0 and 1 q " 1 p 1`1 p 2 , and Hölder inequality, we obtain For the fractional chain rule (3.3), we refer [CW91], for example. We remark that one can choose p 1 " 8 in (3.3). Indeed, this can be proved by a little modification of the last step in the proof of Proposition 3.1 in [CW91]. Thus, we have Similarly, by Strichartz estimates, Then, applying the fractional Leibniz rule and the fractional chain rule in [CW91], we get X s q}u´v} X s . Choosing sufficiently small T ą 0, we conclude that Φ u 0 is a contraction on a ball B :" tu : }u} X s ď 2}u 0 } H s u equipped with the norm }¨} X s .
Next, we will prove Theorem 1.1 when d " 1 and p ě 5, or d ě 2 and p ě 3. In this case, we do not have a good control on the L 8 x norm from Strichartz estimates. Instead, we make use of Sobolev embedding.
Lemma 3.1 (L p´1 tPI L 8 x bound). Suppose that d " 1 and p ě 5, or d ě 2 and p ě 3. Let s ą s c . Then, we have where pq 0 , r 0 q " ppp´1q`,´2 dpp´1q dpp´1q´4¯´q is an admissible pair. Here, we denote by cà number larger than c but arbitrarily close to c, and similarly for c´.
Proof. We observe that Thus, by Sobolev inequality, We also employ a standard persistence of regularity argument.
, is a complete metric space.
Proof. We recall: Theorem 3.3 (Theorem 1.2.5 in [Caz03]). Consider two Banach spaces X ãÑ Y and 1 ă p, q ď 8. Let pf n q ně0 be a bouned sequence in L q pI, Y q and let f : I Ñ Y be such that f n ptq á f ptq in Y as n Ñ 8, for a.e. t P I. If pf n q ně0 is bounded in L p pI; Xq and if X is reflexive, then f P L p pI; Xq and }f } L p pI;Xq ď lim inf nÑ8 }f n } L p pI;Xq .
Suppose that pf n q 8 n"1 be a Cauchy sequence in B. Then, f n converges to f in L q tPI W s 2 ,r x . Moreover, it follows from Theorem 1.2.5 in [Caz03] that and thus f P B. Therefore, we conclude that B is complete.
Proof of Theorem 1.1 when d " 1 and p ě 5, or d ě 2 and p ě 3. Define the map Φ u 0 as above, and let X α :" L 8 tPI H α x X S α q 0 ,r 0 pIq, where pq 0 , r 0 q is an admissible pair in Lemma 3.2. Then, by Strichartz estimates, the fractional chain rule and (3.4), we get and similarly (3.5) Thus, for sufficiently small T ą 0, Φ u 0 is contractive on a ball B :" tu : }u} X s ď 2}u 0 } H s u equipped with the norm }¨} X 0 , which is complete by Lemma 3.2.
Remark 3.4. The standard persistence of regularity argument allows us to avoid derivatives in (3.5). Indeed, for u P B, }x∇y s u} L p´1 tPI L 8 x is not necessarily bounded. 3.2. Scaling-critical cases. In the scaling-critical case, we use the following lemma, which plays the same role as (3.4). We note that the norms in the lemma are defined via the Littlewood-Paley projection in order to overcome the failure of the Sobolev embedding W s,p ãÑ L q , 1 q " 1 p´s d , when q " 8. (3.6) Proof. We will prove the lemma only when d ě 3. By interpolation }f } L p θ ď }f } θ L p 0 }f } 1´θ L p 1 , 1 p θ " θ p 0`1´θ p 1 , 0 ă θ ă 1, it suffices to show the lemma for rational pp´1q " m n ą 2 with gcdpm, nq " 1. First, we estimate Observe from Bernstein's inequality that H sc x . As a consequence, we have where θ " 1 p´2 . Hence, applying (3.7) for i " 1,¨¨¨, n and (3.9) for i " n`1,¨¨¨, m, we bound Aptq by For an arbitrarily small ǫ ą 0, we let Then, since d N ďd N andd N i ď p N 1 N i q ǫd N 1 and similarly for primes, Aptq is bounded by Summing in N m , N m´1 , ..., N n`1 and using m´n " pp´2qn, and then summing in N n , N n´1 , ..., N 1 , we obtain that which is, by Hölder inequality and Young's inequality, bounded by Finally, by the estimate for Aptq, we prove that Proof of theorem 1.2. For simplicity, we assume that d ě 3. Indeed, with little modifications, we can prove the theorem when d " 1, 2. We define Φ u 0 puq as in the proof of Theorem 1.1. Then, by Strichartz estimates, the fractional chain rule and (3.6), we have Now we let δ " δpc, }u 0 } H sc q ą 0 be a sufficiently small number to be chosen later, and then we pick T " T pu 0 , δq ą 0 such that Then, for u P B, we have pIq ď δ`cp2δq 2 p2c}u 0 } H sc q p´2 ď 2δ, Choosing sufficiently small δ ą 0, we prove that Φ u 0 maps B to itself. Similarly, one can show Therefore, it follows that Φ u 0 is a contraction mapping in B.
Remark 3.6. piq In the proofs, the L 8 x norm bounds are crucial for the following reason. In Proposition 2.1, there is a loss of regularity except the trivial ones, x . Hence, when we estimate the L 8 tPI H s x norm of the integral term in Φ u 0 puq, we are forced to use the trivial one (3.10) Indeed, otherwise, we have a higher regularity norm on the right hand side. Then, we cannot close the contraction mapping argument. Moreover, if u 0 P H s , there is no good bound for }e itp´∆q σ u 0 } L q tPI W s,r x except the trivial one pq, rq " p8, 2q. Thus, we are forced to bound the right hand side of (3.10) by x . Therefore, we should have a good control on }u} L p´1 tPI L 8 x . piiq When p ă 3, the L p´1 tPI L 8 x norm is scaling-supercritical. Thus, based on our method, the assumptions on p in Theorem 1.1 and 1.2 are optimal except p " 3 in the critical case.

Small Data Scattering
Proof of Theorem 1.3. For simplicity, we consider the case d ě 3 only. It follows from the estimates in the proof of Theorem 1.2 that if }u 0 } H s is small enough, then pRq`} uptq}Ssc 8,2 pRq À }u 0 } H sc ă 8.
By Strichartz estimates, the fractional chain rule and (3.6), we prove that as T 1 , T 2 Ñ˘8. Thus, the limits u˘" lim tÑ˘8 e´i tp´∆q σ uptq exist in H sc . Repeating the above estimates, we show that

Ill-posedness
We will prove Theorem 1.5 following the strategy in [CCT03a]. Throughout this section, we assume that d " 1, 2 or 3 and d 4 ă σ ă 1. If p is not an odd integer, we further assume that p ě k`1, where k is the smallest integer greater than d 2 . First, we construct an almost non-dispersive solution by small dispersion analysis.
Lemma 5.1 (Small dispersion analysis). Given a Schwartz function φ 0 , let φ pνq pt, xq be the solution to the fractional NLS iB t u`ν 2σ p´∆q σ u`µ|u| p´1 u " 0, up0q " φ 0 , (5.1) and φ p0q pt, xq be the solution to the ODE with no dispersion Then there exist C, c ą 0 such that if 0 ă ν ď c is sufficiently small, then for all |t| ď c| log ν| c .
Proof. The proof closely follows the proof of Lemma 2.1 in [CCT03a].
Obviously, φ pνq pt, νxq is a solution to pNLS σ q. Moreover, φ pνq pt, νxq is bounded and almost flat in the following sense.
Remark 5.4. When p " 3, in [CHKL14] the authors could use the counterexample in [CCT03b]. This counterexample is constructed by pseudo-conformal symmetry and Galilean transformation. A good thing is that this solution is very small in high Sobolev norms, too. Somehow, this smallness allows [CHKL14] to show that the error in pseudo-Galilean transformation is also small. However, when p ą 3, the counterexample in [CCT03b] does not work. Later, Christ, Colliander and Tao [CCT03a] constructed a different counterexample which works for more general p. Unfortunately, this counterexample is not small in high Sobolev norms. It is very large instead. In particular, for our purposes, it is hard to control the error from pseudo-Galilean transformation. But, our new counterexample still has small high Sobolev norm after translating it to its frequency center; this is the term e iv¨x in equation (5.9). Using this smallness, we can prove that pseudo-Galilean transformation is almost invariant. We also remark that the condition σ ą d 4 is to guarantee smallness of the error (see (5.10)).
Hence, by a trivial estimate, we get Ipsq`IIpsqds.
Collecting all, for |t| ď c| log ν| c . Then, by the standard nonlinear iteration argument, we prove the lemma.
Since we have solutions almost symmetric with respect to the pseudo-Galilean transformations, we can make use of the following decoherence lemma to construct counterexamples for local well-posedness.
Proof. The proof closely follows the proof of Lemma 3.1 in [CCT03a].
Proof of Theorem 1.5. The proof is very similar to that of Theorem 1 in [CCT03a] except that in the last step, we need to use Lemma 5.3 due to lack of exact symmetry. We give a proof for the readers' convenience. Let ǫ ą 0 be a given but arbitrarily small number. Let ν " λ α , where α ą 0 is a small number to be chosen later. Then, we pick v P R d such that λ´2 σ p´1 |v| s pλ{νq d{2 " ǫ ô |v| " ν 1 s p dp1´αq 2`2 ασ p´1 q ǫ 1{s .