Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation

In this paper, we investigate the Cauchy problem for the fourth order nonlinear Schrodinger equation \begin{document}$i \partial_{t}u+\partial_{x}^{4}u=u^{2},\ \ (t,x)∈[0,T]× \mathbb{R}.$ \end{document} Zheng (Adv. Differential Equations, 16(2011), 467-486.) has proved that the problem is locally well-posed in \begin{document}$H^{s}(\mathbb{R})$\end{document} with \begin{document}$-\frac{7}{4} In this paper, we aim at extending Zheng's work to a lower regularity index. We prove that the equation is locally well-posed in \begin{document}$H^{s}(\mathbb{R})$\end{document} when \begin{document}$s≥q -2$\end{document} and ill-posed when \begin{document}$s in the sense that the solution map is discontinuous for \begin{document}$s . The key ingredient used in this paper is Besov-type space introduced by Bejenaru and Tao (Journal of Functional Analysis, 233(2006), 228-259.).

1. Introduction. In this paper, we consider the Cauchy problem of the following quadratic fourth order nonlinear Schrödinger equation where u = u(t, x) is a complex-valued unknown function, and ϕ(x) is prescribed complex-valued function. Several authors have investigated the Cauchy problem for where λ ∈ R, µ is a nonzero real number, and n ∈ N + , see, for instance [4,10,12,13,14,15,16,18].

YUANYUAN REN, YONGSHENG LI AND WEI YAN
When λ = 0, µ = 0 and F (u) = |u| 2σ u with σ > 0, (1.3) be reduced to the biharmonic nonlinear Schrödinger equation i∂ t u + µ∆ 2 u = |u| 2σ u, (t, x) ∈ R × R n , (1.4) which was studied by Ivanov and Kosevich [9] and Turitsyn [20] in the context of stability of solutions in magnetic materials where the effective quasi-particle mass becomes infinite. Pecher and von Wahl in [17] proved the existence of classical global solutions of (1.4) for the space dimensions n ≤ 14. Guo and Wang [4] considered the existence and scattering theory for this equation. They proved the local wellposedness for any initial data and the global well-posedness for small initial data in H s (R n ) for some s ≥ 0. Hao et al. [5,6] discuss the Cauchy problem in a high-regularity setting. Miao et al. [10] studied the focusing energy-critical case for n ≥ 5 with µ = 1 and σ = 1 n−4 and proved that the solution is global and scatters with initial data inḢ 2 (R n ). Segata [18] proves scattering in the case the space dimension is one. By using the [k; Z]-multiplier norm method [19], Zheng [21] studied the Cauchy problem for the fourth order Schrödinger equation (1.3) in the case µ = 1 and λ ∈ {−1, 0, 1}. More precisely, Zheng proved the local wellposedness in H s (R) with s > − 7 4 when F (u) = u 2 and F (u) = u 2 and the local well-posedness in H s (R) with s > − 3 4 when F (u) = |u| 2 . Recently, Miao and Zheng in [11] study the global well-posedness and scattering theory in the critical Sobolev space for the space dimensions n ≥ 8.
In this paper, we mainly study the Cauchy problem of the fourth order nonlinear Schrödinger equation (1.1). Inspired by [1,8], we get a lower regularity index of the solution by introducing a new Besov-type space and establishing some important bilinear estimates, which extend Zheng's result in [21]. More precisely, we prove that the equation is locally well-posed in H s (R) when s ≥ −2 and ill-posed when s < −2 in the sense that the solution map is discontinuous for s < −2.
We give some notations before presenting the main results. For any k ∈ N + , let η k ∈ C ∞ c (R) be a smooth cut-off function such that η k (t) ≡ 1 for |t| ≤ k and η k (t) ≡ 0 for |t| ≥ 2k, and let a(t) := 1 2 sgn(t)η 5 (t). Note that, the identity is valid for t ∈ (0, 2) and t ∈ (−3, 3), then we have for all 0 ≤ t ≤ 1 and any g : R → R with suppg ⊂ [−2, 2] (see [1,2,3]). From now on, for any 1 ≤ p, q ≤ ∞, L p L q always denotes the mixed Lebesgue space of time-space variable (τ, ξ): . For any domain Ω ⊂ R 2 , we also use the restricted norm, where 1 Ω denotes the characteristic function of Ω. We call a function f : R × R → C reasonable if f ∈ L ∞ (R 2 ) and supp f is compact. For any given s, b ∈ R, we definê X s,b as the completion of the reasonable functions with the following norm WELL-POSEDNESS FOR THE FOURTH ORDER NLS EQUATION   489 where · := (1 + | · | 2 ) 1/2 . Denote bỹ For any j ≥ 0 and d ≥ 0, we define Thus the sets A j ∩ B d for j, d ≥ 0 partition the frequency space. By the definition of the spaceX s,b , we have (1.6) For convenience, we define the following notations and similarly define A ≥j , B <d , B ≥d , etc. Also, the first (or the second) subscript of a function means the restriction to {A j } (or {B d }), for example and other cases can be similarly defined. For s, b ∈ R, we defineX −2, 1 2 ,1 as follows: Throughout this paper, C is a positive constant which may vary from line to line. We use the notation A B if there exists a constant C > 0 independent of any variable appearing in the estimate, such that A ≤ CB, while A ∼ B means A B and B A. In addition, A B equals to B A. Let a ∨ b = max {a, b} and a ∧ b = min {a, b}.
For a Banach space (H, · H ), we use B H (r) := {f ∈ H : · H ≤ r} to denote the usual open ball of radius r around the origin.

YUANYUAN REN, YONGSHENG LI AND WEI YAN
For simplicity, we rewrite (1.9) as follows: where L is the linear operator L(ϕ)(t) := η 1 (t)S(t)ϕ (1.11) and N 2 is the bilinear operator Now we state our main results as follows.  It is easy to know that (1.1) is invariant under the scaling transformation a direct computation yields for λ > 1. When λ is sufficiently large, ϕ (λ) H −2 (R) becomes small. Thus, it suffices to prove Theorem 1.1 when T = 1 and ϕ H −2 (R) is sufficiently small.
The rest of the paper is arranged as follows. In Section 2, we construct a new weighted space. In Section 3, we establish some important bilinear estimates and complete the proof of Proposition 2.4. In Section 4, we give the proof of Theorem 1.1. In Section 5, we prove Theorem 1.2.
2. Construction of the work space W . We want to study the optical regularity index of the solution to the fourth order nonlinear Schrödinger equation, as the classical Bourgain space X s,b is not so perfect to our situation, then we will construct a weaker space. In this section, motivated by [1], we modify the working space based on the special property of the nonlinearity u 2 . We will construct a new space W to prove the main theorems.
We need to introduce the spaces Y and Z. Y is defined by the following norm and Z is the sum space ofX −2, 1 2 ,1 and Y , whose norm is defined as It is easy to deduce that Z has an equivalent norm: On the other hand, for a Banach space Z and a linear operator T , if there is a where It's easy to check that ω ≥ 1 and ω(τ, ξ) τ − ξ 4 10 . Firstly, for the relationship betweenX −2, 1 2 ,1 andX −2,b , we have the following lemma.
,1 , then square-summing in j yields the claim.
When b > 1 2 , by using Hölder's inequality, we have then square-summing in j yields the claim.
Furthermore, motivated by [1], we have the following basic estimates.

Proposition 2.2.
For any reasonable function f , we have Proof. (2.15)-(2.18) can be proved similarly to (32)-(35) of [1]. By using the definition of the space Z and (2.15), we obtain that (2.19) is valid. This completes the proof of the proposition.
By using Proposition 2.2, we easy get the following lemma.
Proof. For the proof, we refer the readers to Lemma 2 of [1]. Now, we give the following proposition which plays an important role in the proof of main theorems.

Proposition 2.4.
For any reasonable functions f and g satisfy the following properties. ( where the last inequality, we have used the fact that the estimate ω(τ, ξ) τ −ξ 4 10 . Thus (2.21) is proved.
Finally, to prove (2.22), by (2.14) and (i), it suffices to show for any reasonable functions f and g.
For (2.25), in order for the inner summand to be non-zero, we have the following two cases: Noting that under (2.24) that the weight ω satisfies Combining Schur's lemma with the above inequality, to obtain the bilinear estimate (2.23), it suffices to show that the following lemmas are valid.

YUANYUAN REN, YONGSHENG LI AND WEI YAN
In fact, if we assume that Lemma 2.5 is valid, then for each j, by (2.26) and the Cauchy-Schwarz inequality, we have j1=j−10 j2≤j+11 Then square-summing over j and by the triangle inequality, we obtain (2.23) in the High-low interaction case. Similarly, if we assume Lemma 2.6 is valid, then for each j 1 , we have Then square-summing over j 1 and by the triangle inequality and the Cauchy-Schwarz inequality, we obtain (2.23) in the High-high interaction case. From the discussion above, in order to complete the proof of (2.22), it suffices to prove Lemmas 2.5 and 2.6. Thus, we will give the proofs of Lemmas 2.5 and 2.6 in the next section.
3. The proofs of Lemmas 2.5 and 2.6. In this section, motivated by [1,8], we first show two bilinear estimates.
Proof. Without loss of generality, we may assume f and g to be non-negative. Let suppf d1 ⊂ B d1 and suppg d2 ⊂ B d2 . By the definition of the spaceX −2, 1 2 ,1 , we have using the triangle inequality, it suffices to prove that, for any d 1 , d 2 ≥ 0, We may assume d 1 ≥ d 2 by symmetry. Applying Schwarz's inequality, we have where m 1 (τ, ξ) is the measure of the set Thus, it remains to be proved ). Then elementary algebra shows that (ξ 1 −ξ 2 ) 2 +3ξ 2 belongs to at most two intervals of size O(2 d1/2 ). This in turn implies that ξ 1 − ξ 2 belongs to at most four intervals of size O(2 d1/4 ), which means that the variation of ξ 1 is estimated by 2 d1/4 for fixed (τ, ξ).
If we also fix ξ 1 , then the estimates imply that the variation of τ 1 is bounded by 2 d2 . Thus we obtain (3.29). This completes the proof of Proposition 3.1.
We also have the following estimate with respect to time variable.
Now, we give the proof of Lemma 2.5.
Proof. Note that, for this case, we have a priori assumptions |j 1 − j| ≤ 10 and j 2 ≤ j 1 + 11. We firstly show that the simple case j 2 = 0. By Lemma 2.1, Young's inequality and Proposition 2.2, we have then (2.27) holds when j 2 = 0. From now on, we may assume j 2 > 0. Then the assumption |j 1 − j| ≤ 10 and the following relation 20) . Therefore, we split the L.H.S. of (2.27) into four parts, =:I + II + III + IV.
We will estimate I. By using (2.13), we have Applying Young's inequality and Proposition 2.2, we have Putting these estimates together, we get Next we consider II. By using Lemma 2.1 and the definition ofX s,b , we obtain On the one hand, if we measure g j2 inX −2, 1 2 ,1 , then from Proposition 3.1 and Lemma 2.3, we have On the other hand, if we measure g j2 in Y , then from Young's inequality, Proposition 2.2 we have Therefore, we have Inserting this last inequality into (3.30) yields Next we consider III. On the one hand, according to Propositions 3.2 and 2.2, we get On the other hand, let b ∈ ( 1 2 , 2 3 ), applying Lemma 2.1, Hölder's inequality and Young's inequality we deduce that Combining these two estimates, we have Finally, we estimate IV . By using Proposition 3.2, Lemma 2.3 and Proposition 2.2, we obtain From the above estimates, we obtain we get the desired inequality. This completes the proof of Lemma 2.5.
Next, we give the proof of Lemma 2.6.
Proof. Note that, in this case, we have a priori assumption |j 1 −j 2 | ≤ 1. Meanwhile, we may also assume j 1 ≥ 10. Now, from Hölder's inequality, Young's inequality and Proposition 2.2 we deduce that (3.31) According to Proposition 3.1, we have On the other hand, by using Young's inequality and Proposition 2.2, we have ,1 , and similarly, we obtain Putting all these estimates together, we obtain Inserting (3.31) and the above inequality into (2.13), we get (3.32) We now return to (2.28). Let us first restrict A <j1−10 to the region A <j1−10 ∩ B ≥4j1−10 . In this case, from (2.26) and (3.32), we deduce thus, (2.28) holds in this case. Next, we consider the domain A <j1−10 ∩ B <4j1−10 . We discuss in two different cases: For the first case, by the inequality |j 1 − j 2 | ≤ 1 it suffices to assume that suppf j1 ⊂ A j1 ∩ B <4j1−100 , the other case can be proved in the similar way.
For the second case, we may measure f j1 and g j2 in L 2 L 2 instead of in Z by Proposition 2.2. Applying (2.26), we get where , For I 1 , by using Young's inequality and Proposition 2.2, we have that For I 2 , by the definition of Y , we have For any (τ, ξ) ∈ Ω 2 , it is easy to know that the relation τ − ξ 4 ∼ τ ≥ ξ 4 holds. By using Lemma 3.3, Fubini's theorem and Hölder's inequality, we have Thus we have (2.28). This completes the proof of Lemma 2.6.
From what has been discussed above, we complete the proof of (2.22) and thus the proof of Proposition 2.4.

4.
The proof of Theorem 1.1. In this section, we will give the proof of the first main theorem. First, we take S s and N s to be the closure of the Schwartz functions under the norms Then we have the following proposition.
Proposition 4.1. For any s ∈ R, Banach spaces S s (R 2 ) and N s (R 2 ) have the following properties.
(i) The Schwartz functions on R 2 are dense in S s (R 2 ) and in N s (R 2 ).
(v) Let F ∈ N s (R 2 ), and η(t) is a smooth bump function, then (vi) Let F ∈ N s (R 2 ), and η(t) is a smooth bump function, then (viii) Let u ∈ S s (R 2 ), and η(t) is a smooth bump function, then Proof. By Proposition 2.4, we can easily give the proof of (i)-(vii) where completely similar to the proof of Proposition 2 in [1]. So we omit it here. Now, we only prove (viii). By the definition of the space S s , it suffices to show It is not difficult to show that ω(τ, ξ) ω(τ 1 , ξ)ω(τ 2 , ξ), where τ = τ 1 + τ 2 . On the one hand, by using Young's inequality, we have Combining (4.34) with (4.35), we deduce from the definition of spaces Z and Y that On the other hand, by the definition of the spaceX −2, 1 2 ,1 and (4.34), we obtain So, the definition of the space Z and the above inequality imply that Combining (4.36)-(4.37) gives (4.33). Thus, (viii) is proved.
We are now in a position to give the proof of Theorem 1.1.
Proof. We use the above proposition to study the local well-posedness to the Cauchy problem (1.1)-(1.2) by a contraction argument. To be more specific, local solution of (1.1)-(1.2) is constructed as a fixed point u of contraction mapping u → Γu, where Γu(t) := L(ϕ) + N 2 (u, u), on a suitable complete metric space of functions. Applying (iv)-(viii) of Proposition 4.1, we deduce from (1.11) and (1.12) that , (4.39) for some constant C > 0.
Thus, for ϕ ∈ B H −2 (R) (r), we choose r = 1 8C 2 , by a simple computation, it is easy to know that Γ is a strict contraction on B S −2 ( 1 4C ), and thus has a unique fixed point u ∈ B S −2 ( 1 4C ). In fact, for u, v ∈ B S −2 ( 1 4C ), by using (4.38)-(4.39), we have It is easy to see by the property (iii) that the space S −2 (R 2 ) is continuously embedded into C(R; H s (R)). Thus, we see that (1.10) is well-posed in the spaces C(R; H s (R)).
Similarly, for any s ≥ −2, by using (4.38)-(4.39), we have , by Theorem 4 in [1], we see that (1.10) has persistence of regularity for the spaces H s (R) and S s (R 2 ) for any s ≥ −2. Finally, combining with (ii) and (iii) of Proposition 4.1, we complete the proof of Theorem 1.1.

5.
The proof of Theorem 1.2. In this section, we prove Theorem 1.2.