Well-posedness and scattering for fourth order nonlinear Schr\"odinger type equations at the scaling critical regularity

In the present paper, we consider the Cauchy problem of fourth order nonlinear Schr\"odinger type equations with a derivative nonlinearity. In one dimensional case, we prove that the fourth order nonlinear Schr\"odinger equation with the derivative quartic nonlinearity $\partial _x (\overline{u}^4)$ is the small data global in time well-posed and scattering to a free solution. Furthermore, we show that the same result holds for the $d \ge 2$ and derivative polynomial type nonlinearity, for example $|\nabla | (u^m)$ with $(m-1)d \ge 4$.


Introduction
We consider the Cauchy problem of the fourth order nonlinear Schrödinger type equations:    (i∂ t + ∆ 2 )u = ∂P m (u, u), (t, x) ∈ (0, ∞) × R d where m ∈ N, m ≥ 2, P m is a polynomial which is written by ∂ is a first order derivative with respect to the spatial variable, for example a linear combination of ∂ ∂x 1 , . . . , ∂ ∂x d or |∇| = F −1 [|ξ|F ] and the unknown function u is C-valued. The fourth order Schrödinger equation with P m (u, u) = |u| m−1 u appears in the study of deep water wave dynamics [2], solitary waves [14], [15], vortex filaments [3], and so on. The equation (1.1) is invariant under the following scaling transformation: and the scaling critical regularity is s c = d/2 − 3/(m − 1). The aim of this paper is to prove the well-posedness and the scattering for the solution of (1.1) in the scaling critical Sobolev space.
Hayashi and Naumkin ( [8]) considered (1.1) for d = 1 with the power type nonlineality ∂ x (|u| ρ−1 u) (ρ > 4) and proved the global existence of the solution and the scattering in the weighted Sobolev space. Moreover, they ( [9]) also proved that the large time asymptotics is determined by the self similar solution in the case ρ = 4. Therefore, derivative quartic nonlinearity in the one spatial dimension is the critical in the sense of the asymptotic behavior of the solution.
We firstly focus on the quartic nonlinearity ∂ x (u 4 ) in one space dimension. Since this nonlinearity has some good structure, the global solution scatters to a free solution in the scaling critical Sobolev space. Our argument does not apply to (1.1) with P (u, u) = |u| 3 u because we rely on the Fourier restriction norm method. Now, we give the first results in this paper. For a Banach space H and r > 0, we define of (1.1) on (0, T ). Such solution is unique inŻ −1/2 r ([0, T )) which is a closed subset ofŻ −1/2 ([0, T )) (see Definition 2.11 and (4.2)). Moreover, the flow map is Lipschitz continuous.
Moreover, we obtain the large data local in time well-posedness in the scaling critical Sobolev space. To state the result, we put Furthermore, the same statement remains valid if we replace H −1/2 byḢ −1/2 as Remark 1.5. For s > −1/2, the local in time well-posedness in H s follows from the usual Fourier restriction norm method, which covers for all initial data in H s .
It however is not of very much interest. On the other hand, since we focus on the scaling critical cases, which is the negative regularity, we have to impose that thė H −1/2 part of initial data is small. But, Theorem 1.4 is a large data result because the L 2 part is not restricted.
The main tools of the proof are the U p space and V p 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 ( [4], [5]).
We also consider the one dimensional cubic case and the high dimensional cases.
The second result in this paper is as follows. The smoothing effect of the linear part recovers derivative in higher dimensional case. Therefore, we do not use the U p and V p type spaces. More precisely, to establish Theorem 1.6, we only use the Strichartz estimates and get the solution Accordingly, the scattering follows from a standard argument.
Since the condition (m − 1)d ≥ 4 is equivalent to s c + 1/(m − 1) ≥ 0, the solution space L pm ([0, T ); W qm,sc+1/(m−1) ) has nonnegative regularity even if the data belongs to H sc with −1/(m − 1) ≤ s c < 0. Our proof of Thorem 1.6 (ii) cannot applied for for s ≥ 0 by using the iteration argument since the fractional Leibnitz rule (see [1]) and the Hölder inequality imply We give a remark on our problem, which shows that the standard iteration argument does not work. (ii) Let m ≥ 2, s < s c and ∂ = |∇| or ∂ ∂x k for some 1 ≤ k ≤ d. Then the flow map of (1.1) from H s to C(R; H s ) is not smooth.
More precisely, we prove that the flow map is not C 3 if d = 1, m = 3, s < 0 and Since the resonance appears in the case d = 1, m = 3 and P 3 (u, u) = |u| 2 u, there exists an irregular flow map even for the subcritical Sobolev regularity.
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 . The free evolution S(t) := e it∆ 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 Furthermore, we define Q S ≥M := N ≥M Q S N and Q S <M := Id − Q S ≥M . 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 Section 3, we will give the multilinear estimates which are main estimates to prove Theorems 1.1 and 1.4. In Section 4, we will give the proof of the well-posedness and the scattering (Theorem 1.1, Corollary 1.3, and Theorem 1.4). In Section 5, we will give the proof of Theorem 1.6.
In Section 6, we will give the proof of Theorem 1.8.

The 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 ( [4], [5]).
We define the set of finite partitions Z as a "U p -atom". Furthermore, we define the atomic space We define the space of the bounded p-variation Theorem 2.4 ([4] Proposition 2,10, Remark 2.12). Let 1 < p < ∞ and 1/p+1/p ′ = 1. If u ∈ V 1 −,rc be absolutely continuous on every compact intervals, then be a m-linear operator. Assume that for some 1 ≤ p, q < ∞ Then, there exists T : Now we refer the Strichartz estimate for the fourth order Schrödinger equation proved by Pausader. We say that a pair (p, q) is admissible if 2 ≤ p, q ≤ ∞, Proposition 2.9 ([16] Proposition 3.1). Let (p, q) and (a, b) be admissible pairs.
Then, we have where a ′ and b ′ are conjugate exponents of a and b respectively.
Propositions 2.8 and 2.9 imply the following.
Next, we define the function spaces which will be used to construct the solution.
We define the projections P >1 and P <1 as In this section, we prove multilinear estimates for the nonlinearity ∂ x (u 4 ) in 1d, which plays a crucial role in the proof of Theorem 1.1.
Proposition 3.2. Let d = 1 and 0 < T ≤ ∞. For a dyadic number N 1 ∈ 2 Z , we define the set A 1 (N 1 ) as We divide the integrals on the left-hand side of (3.2) into 10 pieces of the form with Q S j ∈ {Q S ≥M , Q S <M } (j = 0, · · · , 4). By the Plancherel's theorem, we have where c is a constant. Therefore, Lemma 3.1 implies that So, let us now consider the case that Q S j = Q S ≥M for some 0 ≤ j ≤ 4. First, we consider the case Q S 0 = Q S ≥M . By the Cauchy-Schwartz inequality, we have Furthermore by (2.1) and M ∼ N 4 0 , we have While by the Sobolev inequality, (2.3), V 2 S ֒→ U 12 S and the Cauchy-Schwartz inequality for the dyadic sum , we have for 2 ≤ j ≤ 4. Therefore, we obtain For the case Q S 1 = Q S ≥M is proved in same way.
Next, we consider the case Q S i = Q S ≥M for some 2 ≤ i ≤ 4. By the Hölder inequality, we have x .
By L 2 orthogonality and (2.1), we have since M ∼ N 4 0 . While, by the calculation way as the case Q S 0 = Q S ≥M , we have Therefore, we obtain Proposition 3.3. Let d = 1 and 0 < T ≤ ∞. For a dyadic number N 2 ∈ 2 Z , we define the set A 2 (N 2 ) as The proof of Proposition 3.3 is quite similar as the proof of Proposition 3.2.
We firstly consider the case A ′ 1,1 (N 1 ) In the case T ≤ N −3 0 , the Hölder inequality implies and by the Sobolev inequality, V 2 S ֒→ L ∞ t L 2 x and the Cauchy-Schwartz inequality , we have Therefore, we obtain and note that T 1/2 N 0 ≤ T 1/6 .
In the case T ≥ N −3 0 , we divide the integrals on the left-hand side of (3.2) into 10 pieces of the form (3.3) in the proof of Proposition 3.2. Thanks to Lemma 3.1, let us consider the case that Q S j = Q S ≥M for some 0 ≤ j ≤ 4. First, we consider the case Q S 0 = Q S ≥M . By the same way as in the proof of Proposition 3.2 and using and note that N −1/2 0 ≤ T 1/6 . Since the cases Q S j = Q S ≥M (j = 1, 2, 3) are similarly handled, we omit the details here.
We focus on the case Q S 4 = Q S ≥M . By the same way as in the proof of Proposition 3.2 and using instead of (3.5) with j = 4, we obtain We secondly consider the case A ′ 1,2 (N 1 ). In the case T ≤ N −3 0 , the Hölder inequality implies x .
By the same estimates as in the proof for the case A ′ 1,1 (N 1 ) and In the case T ≥ N −3 0 , we divide the integrals on the left-hand side of (3.2) into 10 pieces of the form (3.3) in the proof of Proposition 3.2. Thanks to Lemma 3.1, let us consider the case that Q S j = Q S ≥M for some 0 ≤ j ≤ 4. By the same argument as in the proof for the case A ′ 1,1 (N 1 ), we obtain The remaining cases follow from the same argument as above.
We thirdly consider the case A ′ 1,3 (N 1 ). In the case T ≤ N −3 0 , the Hölder inequality implies x .
By the same estimates as in the proof for the case A ′ 1,1 (N 1 ) and In the case T ≥ N −3 0 , we divide the integrals on the left-hand side of (3.2) into 10 pieces of the form (3.3) in the proof of Proposition 3.2. Thanks to Lemma 3.1, let us consider the case that Q S j = Q S ≥M for some 0 ≤ j ≤ 4. By the same argument as in the proof for the case A ′ 1,1 (N 1 ), we obtain 1, 2, 3) are the same argument as above.
Furthermore, we obtain the following estimate.
Proposition 3.5. Let d = 1 and 0 < T ≤ 1. For a dyadic number N 2 ∈ 2 Z , we define the set A ′ 2 (N 2 ) as If N 0 N 1 ∼ N 2 , then we have Because the proof is similar as above, we skip the proof. We define the map Φ T,ϕ as where To prove the well-posedness of (1.1) inḢ −1/2 , we prove that Φ T,ϕ is a contraction map on a closed subset ofŻ −1/2 ([0, T )). Key estimate is the following: Proof. We decompose By symmetry, it is enough to consider the summation for N 1 ≥ N 2 ≥ N 3 ≥ N 4 . We put First, we prove the estimate for J 1 . By Theorem 2.4 and the Plancherel's theorem, we have is defined in Proposition 3.2. Therefore by Proposition 3.2, we have Next, we prove the estimate for J 2 . By Theorem 2.4 and the Plancherel's theorem, we have

. Therefore by Proposition 3.3 and
Cauchy-Schwartz inequality for the dyadic sum, we have which is a closed subset ofŻ s (I). Let T > 0 and u 0 ∈ B r (Ḣ −1/2 ) are given. For by Proposition 4.1 and where C is an implicit constant in (4.1). Therefore if we choose r satisfying r < (64C) −1/3 , then Φ T,u 0 is a contraction map onŻ Proof. We decompose u j = v j + w j with v j = P >1 u j ∈Ẏ −1/2 and w j = P <1 u j ∈Ẏ 0 . ¿From Propositions 3.4, 3.5, and the same way as in the proof of Proposition 4.1, it remains to prove that By Theorem 2.4, the Cauchy-Schwartz inequality, the Hölder inequality and the Sobolev inequality, we have which completes the proof.
Proof of Theorem 1.4.
Thanks to Propositions 3.4 and 3.5, the uniqueness follows from the same argument as in [5].

Proof of Theorem 1.6
In this section, we prove Theorem 1.6. We only prove for the homogeneous case since the proof for the inhomogeneous case is similar. We define the map Φ m T,ϕ as where and the solution spaceẊ s aṡ where p m = 2(m − 1), q m = 2(m − 1)d/{(m − 1)d − 2} for d ≥ 2 and p 3 = 4, To prove the well-posedness of (1.1) in L 2 (R) or H sc (R d ), we prove that Φ T,ϕ is a contraction map on a closed subset ofẊ s . The key estimate is the following: Proposition 5.1. (i) Let d = 1 and m = 3. For any 0 < T < ∞, we have Proof. (i) By Proposition 2.9 with (a, b) = (4, ∞), we get Therefore, thanks to the fractional Leibniz rule (see [1]), we have by the Hölder inequality.
(ii) By Proposition 2.9 with we get Therefore, thanks to the fractional Leibniz rule (see [1]), we have Firstly we consider the case d = 1, m = 3, P 3 (u, u) = |u| 2 u. For N ≫ 1, we put N be the third iteration of (1.1) for the data f N . Namely, u Note that f N H s ∼ 1. Thorem 1.8 is implied by the following propositions. We therefore obtain for sufficiently small t > 0 | u This lower bound goes to infinity as N tends to infinity if s < 0, which concludes the proof.
Since A consists of 2 m elements, we write where ± (α) is a m-ple of signs + and −. We denote by ±