Global solution for the $3D$ quadratic Schr\"odinger equation of $Q(u, \bar{u})$ type

We study a class of $3D$ quadratic Schr\"odinger equations as follows, $(\partial_t -i \Delta) u = Q(u, \bar{u})$. Different from nonlinearities of the $uu$ type and the $\bar{u}\bar{u}$ type, which have been studied by Germain-Masmoudi-Shatah, the interaction of $u$ and $\bar{u}$ is very strong at the low frequency part, e.g., $1\times 1 \rightarrow 0$ type interaction (the size of input frequency is"$1$"and the size of output frequency is"$0$"). It creates a growth mode for the Fourier transform of the profile of solution around a small neighborhood of zero. This growth mode will again cause the growth of profile in the medium frequency part due to the $1\times 0\rightarrow 1$ type interaction. The issue of strong $1\times 1\rightarrow 0$ type interaction makes the global existence problem very delicate. In this paper, we show that, as long as there are"$\epsilon$"derivatives inside the quadratic term $Q (u, \bar{u})$, there exists a global solution for small initial data. As a byproduct, we also give a simple proof for the almost global existence of the small data solution of $(\partial_t -i \Delta)u = |u|^2 = u\bar{u}$, which was first proved by Ginibre-Hayashi. Instead of using vector fields, we consider this problem purely in Fourier space.

The assumption of no derivatives in the high frequency part is not necessary. The method we use here still works out for the quasilinear case, as long as there are certain symmetries inside (1.1), which help us to avoid losing derivatives in the energy estimate. Since we are trying to highlight the L ∞ decay estimate part, we use this assumption to make energy estimate easier.
There is a very large literature on the small data global existence results of nonlinear Schrödinger equations. By using the energy estimates and the decay estimate of the linear solution, one can prove global existence for quadratic nonlinearities in dimension 4 and higher, see, e.g., Klainermann-Ponce [8] and Strauss [9]. This method also works in dimension 3 if the order of nonlinearities is strictly greater than 2, which is also known as a Strauss exponent in dimension 3. 1 The question of small data global existence (SDGE) for 3D quadratic Schrödinger is more subtle. On the one hand, it depends on the type of nonlinearity. On the other hand, it also depends on the decay rate of initial data as |x| → ∞. As mentioned in the abstract, SDGE is known for nonlinearities of type uu orūū or any combination of them, see Germain-Masmoudi-Shatah [2]. For the gauge-invariant nonlinearity of |u|u type, one can also obtain global existence, see Cazenave-Weissler [1]. By using the vector fields method, Ginibre and Hayashi [3] showed the almost global existence for small initial data for nonlinearities of uū type.
Although it is still not clear whether SDGE is true for the nonlinearities of type uū, It is certainly clear that the answer will depend on in what sense the initial data is small. For 3D quadratic Schrödinger with the nonlinearity uū, Ikeda-Inui [6] showed that actually the solution blows up in polynomial time for a class of small L 2 initial data, which decays at rate 1 |x| 2−ǫ as |x| → ∞, where 0 < ǫ < 1/2. Therefore, for the validity of SDGE for general nonlinearities of uū type, initial data should decay faster than 1 |x| 2−ǫ for all ǫ > 0.
In this paper, we are trying to improve the understanding of this problem. We show that solution of (1.1) exists globally if there are "ǫ" derivatives at the low frequency part of nonlinearity (i.e., (1.2)) and the initial data decays faster than 1 |x| 2+γ for any γ such that 0 < γ < ǫ. Before stating our main theorem, we define the Z-normed space as follows, Our main theorem is stated as follows, Theorem 1.1. Fix ǫ and γ, where 0 < ǫ ≪ 1 and γ < ǫ. Assume that the initial data u 0 satisfies the following assumption,

PRELIMINARY
For any two numbers A and B, we use notation A B and A ≪ B to denote A ≤ CB and A ≤ cB respectively, where C is an absolute constant and c is a sufficiently small constant. For an integer k ∈ Z, we use k + to denote max{k, 0} and we use k − to denote min{k, 0}. For any k ∈ Z, we use "f k " to abbreviate P k f , where P k is the Littlewood-Paley operator.
Throughout the proof of theorem 1.1, we will use the following bilinear estimate and the L ∞ decay estimate constantly. Lemma 2.1. For 1 ≤ p, q, r ≤ ∞, f ∈ L p (R 3 ) and g ∈ L q (R 3 ), the following bilinear estimate holds, Proof. See Proof. Recall (3.4). From the L 2 − L ∞ type bilinear estimate (2.1) in Lemma 2.1, (1.2), and the L ∞ → L 2 type Sobolev embedding or alternatively the L 2 → L 1 type Sobolev embedding and L 2 − L 2 type bilinear estimate, the following estimates hold, (3.20) Combine estimates (3.19) and (3.20), it is easy to see our desired estimate (3.16) holds.
Very similarly, from the L 2 → L 3/2 type Sobolev embedding, the L 2 − L 6 type bilinear estimate, and the L 2 − L ∞ type bilinear estimate, the following estimates hold, Now it is easy to see our desired estimates (3.17) and (3.18) hold.

24)
J m,1,2 k,k 1 ,k 2 (ξ) := (l 1 ,n 1 )=(l,n),(n,l) From the L 2 − L ∞ type estimate and the L ∞ −→ L 2 type Sobolev embedding, we have Very similarly, from the L 2 − L ∞ type and L 4 − L 4 type bilinear estimates, we have In above estimates, we used the fact that, On the other hand, if we use the volume of the support of ξ or η first and then use the bilnear estimate (2.1) in Lemma 2.1, the following estimates hold,

Lemma 3.3.
Under the bootstrap assumption (3.1), the following estimates hold for k 2 ≤ k 1 − 10, Proof. Note that |k − k 1 | ≤ 10. Recall (3.9) and (3.13). From the L 2 − L ∞ type bilinear estimate and the L ∞ → L 2 type Sobolev embedding, the following estimates hold, (3.36) Note that the following estimate holds when |η| ≤ 2 −5 |ξ|, Hence, we take the advantage of high oscillation in time by doing integration by parts in time once for I m,3 k,k 1 ,k 2 (ξ). As a result, we have From the L 2 − L ∞ type bilinear estimate, (3.17) in Lemma 3.1 and (3.41) in Lemma 3.4, we have (3.37) Combine estimates (3.35) and (3.37), it is easy to see our desired estimate (3.33) holds.
3.3. The proof of Theorem 1.1. Firstly, we do the energy estimate. Recall (3.2). It's easy to see the following estimate holds for any t ∈ [0, T ], Hence finishing the proof.
Remark 3.5. If we let ǫ = γ = 0, note that above argument also works. From (3.46) and (3.48), we have f (t) H 10 + f (t) Z ǫ 0 + log T ǫ 2 1 . Hence the bootstrap argument still works out as long as T ≤ e c/ǫ 0 , where c is a very small constant. Therefore, the method we developed here also proves the small data almost global existence of the 3D quadratic Schrödinger as follows, (∂ t − i∆)u = |u| 2 .