Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on $\mathbb{R}^3$

Abstract

We consider the cubic nonlinear Schrödinger equation (NLS) on $\mathbb{R}^3$ with randomized initial data. In particular, we study an iterative approach based on a partial power series expansion in terms of the random initial data. By performing a fixed point argument around the second order expansion, we improve the regularity threshold for almost sure local well-posedness from our previous work. We further investigate a limitation of this iterative procedure. Finally, we introduce an alternative iterative approach, based on a modified expansion of arbitrary length, and prove almost sure local well-posedness of the cubic NLS in an almost optimal regularity range with respect to the original iterative approach based on a power series expansion.

1. Introduction

1.1. Nonlinear Schrödinger equation

We consider the Cauchy problem of the cubic nonlinear Schrödinger equation (NLS) on $\mathbb{R}^3$:⁠¹

¹

Our discussion in this paper can be easily adapted to the cubic NLS on $\mathbb{R}^d$ (and to other nonlinear dispersive PDEs). For the sake of concreteness, however, we only consider the $d = 3$ case in the following. See also the comment after Theorem 1.3 for our particular interest in the three-dimensional problem.

✖

$$\begin{equation} \begin{cases} i \partial _t u + \Delta u = |u|^2 u, \\ u|_{t = 0} = u_0 \in H^s(\mathbb{R}^3), \end{cases} \qquad ( t, x) \in \mathbb{R}\times \mathbb{R}^3. {\tag{1.1}} \end{equation}$$

The cubic NLS 1.1 has been studied extensively from both the theoretical and applied points of view. In this paper, we continue our study in 12 and further investigate the probabilistic well-posedness of 1.1 with random and rough initial data.

Recall that the equation 1.1 is scaling critical in $\dot{H}^\frac{1}{2}(\mathbb{R}^3)$ in the sense that the scaling symmetry

$$\begin{align} u(t, x) \longmapsto \lambda u (\lambda ^2 t, \lambda x) {\tag{1.2}} \end{align}$$

preserves the homogeneous $\dot{H}^{\frac{1}{2} }$-norm (when it is applied to functions only of $x$). It is known that the Cauchy problem 1.1 is locally well-posed in $H^s(\mathbb{R}^3)$ for $s \geq \frac{1}{2}$ 10. See also 13182629 for partial⁠² global well-posedness and scattering results. On the other hand, 1.1 is known to be ill-posed in $H^s(\mathbb{R}^3)$ for $s < \frac{1}{2}$ 12. In 2, we studied the probabilistic well-posedness property of 1.1 below the scaling critical regularity $s_\text{crit} = \frac{1}{2}$ under a suitable randomization of initial data; see 1.3 below. In particular, we proved that 1.1 is almost surely locally well-posed in $H^s(\mathbb{R}^3)$, $s > \frac{1}{4}$. Our main goal in this paper is to introduce an iterative procedure to improve this regularity threshold for almost sure local well-posedness. Furthermore, we study a critical regularity associated with this iterative procedure. By introducing a modified iterative approach, we then prove almost sure local well-posedness of 1.1 in an almost optimal range with respect to the original iterative procedure (Theorem 1.8). Beyond the concrete results in this paper, we believe that the iterative procedures based on (modified) partial power series expansions are themselves of interest for further development in the well-posedness theory of dispersive equations with random initial data and/or random forcing.

²

Namely, either for smoother initial data or under an extra hypothesis.

✖

Such a probabilistic construction of solutions to dispersive PDEs first appeared in the works by McKean 34 and Bourgain 5 in the study of invariant Gibbs measures for the cubic NLS on $\mathbb{T}^d$, $d = 1, 2$. In particular, they established almost sure local well-posedness with respect to particular random initial data, basically corresponding to the Brownian motion. (These local-in-time solutions were then extended globally in time by invariance of the Gibbs measures. In the following, however, we restrict our attention to local-in-time solutions.) In 8, Burq-Tzvetkov further elaborated this idea and considered a randomization for any rough initial condition via the Fourier series expansion. More precisely, they studied the cubic nonlinear wave equation (NLW) on a three-dimensional compact Riemannian manifold below the scaling critical regularity. By introducing a randomization via the multiplication of the Fourier coefficients by independent random variables, they established almost sure local well-posedness below the critical regularity. Such randomization via the Fourier series expansion is natural on compact domains 1535 and more generally in situations where the associated elliptic operators have discrete spectra 4547. See 23 for more references therein.

In the following, we go over a randomization suitable for our problem on $\mathbb{R}^3$. Recall the Wiener decomposition of the frequency space 48:

$$\begin{equation*} \mathbb{R}^3_\xi = \bigcup _{n \in \mathbb{Z}^3} \bigg (n+ \bigg (-\frac{1}{2}, \frac{1}{2}\bigg ]^3\bigg ). \end{equation*}$$

We employ a randomization adapted to this Wiener decomposition. Let $\psi \in \mathcal{S}(\mathbb{R}^3)$ satisfy

$$\begin{equation*} \operatorname *{supp}\psi \subset [-1,1]^3 \qquad \text{and}\qquad \sum _{n \in \mathbb{Z}^3} \psi (\xi -n) =1 \quad \text{for any $\xi \in \mathbb{R}^3$}. \end{equation*}$$

Then, given a function $\phi$ on $\mathbb{R}^3$, we have

$$\begin{equation*} \phi = \sum _{n \in \mathbb{Z}^3} \psi (D-n) \phi . \end{equation*}$$

We define the Wiener randomization $\phi ^\omega$ of $\phi$ by

$$\begin{equation} \phi ^{\omega } \coloneq \sum _{n \in \mathbb{Z}^3} g_n (\omega ) \psi (D-n) \phi , {\tag{1.3}} \end{equation}$$

where $\{ g_n \}_{n \in \mathbb{Z}^3}$ is a sequence of independent mean-zero complex-valued random variables on a probability space $(\Omega , \mathcal{F} ,P)$. The randomization 1.3 based on the Wiener decomposition of the frequency space is natural in view of time-frequency analysis; see 1 for a further discussion. Almost simultaneously with 1, Lührmann-Mendelson 32 also considered a randomization of the form 1.3 (with cubes being substituted by appropriately localized balls) in the study of NLW on $\mathbb{R}^3$. For a similar randomization used in the study of the Navier-Stokes equations, see also the work of Zhang and Fang 50, preceding 132. In 50, the decomposition of the frequency space is not explicitly provided, but their randomization certainly includes randomization based on the decomposition of the frequency space by unit cubes or balls.

In the following, we assume that the probability distribution $\mu _n$ of $g_n$ satisfies the exponential moment bound

$$\begin{align} \int _{\mathbb{R}^2} e^{\kappa \cdot x} d \mu _{n} (x) \le e^{c |\kappa | ^2} {\tag{1.4}} \end{align}$$

for all $\kappa \in \mathbb{R}^2$ and $n \in \mathbb{Z}^3$. This condition is satisfied by the standard complex-valued Gaussian random variables and the uniform distribution on the circle in the complex plane.

We now recall the almost sure local well-posedness result from 2 which is of interest to us.⁠³

³

For simplicity, we only consider positive times. By the time reversibility of the equation, the same analysis applies to negative times.

✖

Theorem A.

Let $\frac{1}{4} < s < \frac{1}{2}$. Given $\phi \in H^s (\mathbb{R}^3)$, let $\phi ^{\omega }$ be its Wiener randomization defined in 1.3. Then, the Cauchy problem 1.1 is almost surely locally well-posed with respect to the random initial data $\phi ^\omega$. More precisely, there exists a set $\Sigma = \Sigma (\phi ) \subset \Omega$ with $P(\Sigma ) = 1$ such that, for any $\omega \in \Sigma$, there exists a unique function $u = u^\omega$ in the class

$$\begin{align*} S(t)\phi ^\omega + X^\frac{1}{2}([0, T]) \subset S(t) \phi ^{\omega } + C([0,T] ; H^\frac{1}{2}(\mathbb{R}^3)) \subset C([0,T] ; H^s(\mathbb{R}^3)) \end{align*}$$

with $T = T(\phi , \omega ) >0$ such that $u$ is a solution to 1.1 on $[0, T]$.

Here, $S(t) \coloneq e^{it\Delta }$ denotes the linear Schrödinger operator, and the function space $X^\frac{1}{2}([0, T]) \subset C([0, T]; H^\frac{1}{2}(\mathbb{R}^3))$ is defined in Section 2 below. Note that the uniqueness statement of Theorem A in the class

$$\begin{align} S(t)\phi ^\omega + X^\frac{1}{2}([0, T]) {\tag{1.5}} \end{align}$$

is to be interpreted as follows: by setting $v = u - S(t) \phi ^\omega$ (see 1.7 below), uniqueness for the residual term $v$ holds in $X^\frac{1}{2}([0, T])$. See also Remark 1.1 below.

Recall that while the Wiener randomization 1.3 does not improve differentiability, it improves integrability (see Lemma 4 in 1). See 82742 for the corresponding statements in the context of the random Fourier series. The main idea for proving Theorem A is to exploit this gain of integrability. More precisely, let

$$\begin{align} z_1(t) \coloneq S(t) \phi ^{\omega } {\tag{1.6}} \end{align}$$

denote the random linear solution with $\phi ^\omega$ as initial data and write

$$\begin{align} u = z_1 + v. {\tag{1.7}} \end{align}$$

Then, we see that $v: = u - z_1$ satisfies the following perturbed NLS:

$$\begin{equation} \begin{cases} i \partial _tv + \Delta v = |v + z_1|^2(v+z_1),\\ v|_{t = 0} = 0, \end{cases} {\tag{1.8}} \end{equation}$$

where $z_1$ is viewed as a given (random) source term. The main point is that the gain of space-time integrability of the random linear solution $z_1$ (Lemma 2.1) makes this problem subcritical,⁠⁴ and hence we can solve it by a standard fixed point argument. Over the last several years, there have been many results on probabilistic well-posedness of nonlinear dispersive PDEs, using this change of viewpoint.⁠⁵ See, for example, 12578915253032333435373843. In 2, we studied the Duhamel formulation for 1.8,

⁴

The scaling critical Sobolev regularity $s_\text{crit} = \frac{1}{2}$ for 1.1 is defined by the fact that the $\dot{H}^\frac{1}{2}$-norm remains invariant under the scaling symmetry 1.2. Given $1 \leq r \leq \infty$, we can also define the scaling critical Sobolev regularity $s_\text{crit}(r)$ in terms of the $L^r$-based Sobolev space $\dot{W}^{s, r}$. A direct computation gives $s_\text{crit}(r) = \frac{3}{r} - 1\, (\to -1$ as $r\to \infty$). In particular, the gain of integrability of the random linear solution $z_1$ stated in Lemma 2.1 implies that $z_1$ in 1.8 gives rise only to a subcritical perturbation. See 3 for a further discussion on this issue.

Note, however, that if we consider a non-zero initial condition for $v$, then the critical nature of the equation comes into play through the initial condition. See Remark 1.4. This plays an important role in studying the global-in-time behavior of solutions. See, for example, 37.

✖

⁵

In the field of stochastic parabolic PDEs, this change of viewpoint and solving the fixed point problem for the residual term $v$ is called the Da Prato-Debussche trick 1617.

✖

$$\begin{equation} v(t) = - i \int _0^t S(t-t') |v + z_1|^2(v+z_1)(t') dt', {\tag{1.9}} \end{equation}$$

by carrying out case-by-case analysis on terms of the form $v \overline{v} v$, $v \overline{v} z_1$, $v \overline{z}_1 z_1$, etc., in $X^\frac{1}{2}([0, T])$. The main tools were (i) the gain of space-time integrability of $z_1$ and (ii) the bilinear refinement of the Strichartz estimate (Lemma 2.7). This yields Theorem A.

By examining the case-by-case analysis in 2, we see that the regularity restriction $s > \frac{1}{4}$ in Theorem A comes from the cubic interaction of the random linear solution:

$$\begin{align} z_3(t) \coloneq -i \int _0^t S(t - t') |z_1|^2 z_1(t') dt'. {\tag{1.10}} \end{align}$$

See Proposition 1.2 below. In the following, we discuss an iterative approach to lower this regularity threshold by studying further expansions in terms of the random linear solution $z_1$. We will also discuss the limitation of this iterative procedure.

Remark 1.1.

(i) The proof of Theorem A, presented in 2, is based on a standard contraction argument for the residual term $v = u - S(t) \phi ^\omega$ in a ball $B$ of radius $O(1)$ in $X^\frac{1}{2}([0, T])$. As such, the argument in 2 only yields uniqueness of $v$ in the ball $B$. Noting that $v \in C([0, T]; H^{\frac{1}{2}}(\mathbb{R}^3)) \cap X^\frac{1}{2}([0, T])$,⁠⁶ it follows from Lemma A.8 in 2 that the $X^\frac{1}{2}([0, t])$-norm of $v$ is continuous in $t \in [0, T]$. Then, by possibly shrinking the local existence time $T>0$, we can easily upgrade the uniqueness of $v$ in the ball $B$ to uniqueness of $v$ in the entire $X^\frac{1}{2}([0, T])$. Namely, uniqueness of $u$ holds in the class 1.5.

⁶

By convention, our definition of the $X^s([0, T])$-space already assumes that functions in $X^s([0, T])$ belong to $C([0, T]; H^s(\mathbb{R}^3))$. See Definition 2.4 below.

✖

(ii) With the exponential moment assumption 1.4, the proof of Theorem A, presented in 2, allows us to conclude that the set $\Sigma$ of full probability in Theorem A has the following decomposition:

$$\begin{equation*} \Sigma = \bigcup _{0 < T \ll 1} \Sigma _T \end{equation*}$$

such that

(a)

there exist $c, C >0$ and $C_0 = C_0 (\|\phi \|_{H^s})>0$ such that$$\begin{equation*} P(\Sigma _T^c) < C \exp \Big (-\frac{C_0(\|\phi \|_{H^s})}{T^c }\Big ) \end{equation*}$$

for each $0 < T \ll 1$,

(b)

for each $\omega \in \Sigma _T\subset \Sigma$, $0 < T \ll 1$, the function $u = u^\omega$ constructed in Theorem A is a solution to 1.1 on $[0, T]$.

See the statement of Theorem 1.1 in 2. A similar decomposition of the set $\Sigma$ of full probability applies to Theorems 1.3 and 1.8 below. We, however, do not state it in an explicit manner in the following.

1.2. Improved almost sure local well-posedness

Let us first state the following proposition on the regularity property of the cubic term $z_3$ defined above.

Proposition 1.2.

Given $0 \leq s < 1$, let $\phi ^\omega$ be the Wiener randomization of $\phi \in H^s(\mathbb{R}^3)$ defined in 1.3 and set $z_1 = S(t) \phi ^\omega$.

(i) For any $\sigma < 2s$, we have

$$\begin{equation*} z_3 \in X^\sigma _{\operatorname {loc}}\subset C(\mathbb{R}; H^\sigma (\mathbb{R}^3)) \end{equation*}$$

almost surely. More precisely, there exist an almost surely finite constant $C(\omega , \|\phi \|_{H^s} ) > 0$ and $\theta > 0$ such that

$$\begin{align} \| z_3 \|_{X^{\sigma }([0, T])} = \bigg \| \int _0^t S(t - t') |z_1|^2 z_1(t') dt'\bigg \|_{X^\sigma ([0, T])} \leq T^\theta C(\omega , \|\phi \|_{H^s}) {\tag{1.11}} \end{align}$$

for any $T > 0$. In particular, by taking $\sigma = 2s-\varepsilon$ for small $\varepsilon > 0$, 1.11 shows that the second order⁠⁷ term $z_3$ is smoother (by $s - \varepsilon$) than the first order term $z_1$, provided $s > 0$.

⁷

Here, we referred to $z_3$ as the second order term since it corresponds to the second order term appearing in the (formal) power series expansion of solutions to 1.1 in terms of the random initial data. For the same reason, we refer to $z_5$ in 1.19 and $z_7$ in 1.22 as the third and fourth order terms in the following. See Subsection 1.3 below.

✖

(ii) When $s = 0$, there is no smoothing in the second order term $z_3$ in general; there exists $\phi \in L^2(\mathbb{R}^3)$ such that the estimate 1.11 with $\sigma = \varepsilon > 0$ fails for any $\varepsilon > 0$.

In 2, we already proved 1.11 when $\sigma = \frac{1}{2}$ and $s > \frac{1}{4}$, giving the regularity restriction in Theorem A. See Section 3 for the proof of Proposition 1.2 for a general value of $\sigma$. In the proof of Theorem A, in order to carry out the case-by-case analysis for 1.9, we need to have $z_3 \in X^\frac{1}{2}_\text{loc}$ (where we have a deterministic local well-posedness for 1.1). In view of Proposition 1.2, this imposes the regularity restriction $s > \frac{1}{4}$. Note, however, that even when $s \leq \frac{1}{4}$, $z_3$ is still a well defined space-time function of spatial regularity $2s- < \frac{1}{2}$. This motivates us to consider the second order expansion

$$\begin{align} u = z_1 + z_3 + v {\tag{1.12}} \end{align}$$

and remove the second order interaction $z_1 \overline{z}_1 z_1$. Indeed, the residual term $v \coloneq u - z_1 - z_3$ satisfies the following equation:

$$\begin{equation} \begin{cases} i \partial _tv + \Delta v = \mathcal{N}(v + z_1+z_3) - \mathcal{N}(z_1),\\ v|_{t = 0} = 0, \end{cases} {\tag{1.13}} \end{equation}$$

where $\mathcal{N}(u) = |u|^2 u$. In terms of the Duhamel formulation, we have

$$\begin{equation} v(t) = - i \int _0^t S(t-t')\big \{\mathcal{N}(v + z_1 + z_3) - \mathcal{N}(z_1)\big \}(t') dt'. {\tag{1.14}} \end{equation}$$

Then, by studying the fixed point problem 1.14 for $v$, we have the following improved almost sure local well-posedness of 1.1.

Theorem 1.3.

Given $\frac{1}{2} \leq \sigma \leq 1$, let $\frac{2}{5} \sigma < s < \frac{1}{2}$. Given $\phi \in H^s(\mathbb{R}^3)$, let $\phi ^\omega$ be its Wiener randomization defined in 1.3. Then, the cubic NLS 1.1 on $\mathbb{R}^3$ is almost surely locally well-posed with respect to the random initial data $\phi ^\omega$. More precisely, there exists a set $\Sigma = \Sigma (\phi , \sigma ) \subset \Omega$ with $P(\Sigma ) = 1$ such that, for any $\omega \in \Sigma$, there exists a unique function $u = u^\omega$ in the class

$$\begin{align*} z_1 + z_3 + X^\sigma ([0, T]) & \subset z_1 + z_3 + C([0, T]; H^{\sigma } (\mathbb{R}^3)) \\ \ & \subset C([0, T];H^s(\mathbb{R}^3)) \end{align*}$$

with $T = T(\phi , \omega ) >0$ such that $u$ is a solution to 1.1 on $[0, T]$.

As in Theorem A, the uniqueness of $u$ in the class $z_1 + z_3 + X^\sigma ([0, T])$ is to be interpreted as uniqueness of the residual term $v = u - z_1 - z_3$ in $X^\sigma ([0, T])$. See also Remark 1.1.

By taking $\sigma = \frac{1}{2}$, Theorem 1.3 states that 1.1 is almost surely locally well-posed in $H^s(\mathbb{R}^3)$, provided $s > \frac{1}{5}$. This in particular improves the almost sure local well-posedness in Theorem A. On the other hand, by taking $\sigma = 1$, Theorem 1.3 allows us to construct a solution $v = u - z_1 - z_3 \in X^1([0, T]) \subset C([0,T] ; H^1(\mathbb{R}^3))$, provided that $s > \frac{2}{5}$. In particular, we can take random initial data below the scaling critical regularity $s_\text{crit} = \frac{1}{2}$, while we construct the residual part $v$ in $X^1([0, T])$. This opens up a possibility of studying the global-in-time behavior of $v$, using the (non-conserved) energy of $v$:

$$\begin{align*} E(v)(t) = \frac{1}{2} \int _{\mathbb{R}^3} |\nabla v(t, x)|^2 dx +\frac{1}{4} \int _{\mathbb{R}^3} |v(t, x)|^4 dx \end{align*}$$

with random initial data below the scaling critical regularity. We remark that by inspecting the argument in 2, a modification of (the proof of) Theorem A yields almost sure local well-posedness of 1.1 in the class

$$\begin{equation*} S(t)\phi ^\omega + X^1([0, T]) \subset S(t) \phi ^{\omega } + C([0,T] ; H^1(\mathbb{R}^3)) \end{equation*}$$

only for $s > \frac{1}{2}$. (This restriction on $s$ can be easily seen by setting $\sigma = 1$ in Proposition 1.2.) In particular, Theorem A does not allow us to take random initial data below the scaling critical regularity $s_\text{crit} = \frac{1}{2}$ in studying the global-in-time behavior of $v \in X^1([0, T])$, namely at the level of the energy $E(v)$. As our focus in this paper is the local-in-time analysis, we do not pursue further this issue on almost sure global well-posedness of 1.1 below the scaling critical regularity in this paper. We, however, point out two recent results 3037 on almost sure global well-posedness below the energy space for the defocusing energy-critical NLS in higher dimensions.

The main strategy for proving Theorem 1.3 is to study the fixed point problem 1.14 by carrying out case-by-case analysis on

$$\begin{align} w_1\overline{w_2} w_3 \quad \text{for $w_i = v$, $z_1$, or $z_3$, $i = 1, 2, 3$, but not all $w_i$ equal to $z_1$} {\tag{1.15}} \end{align}$$

in $N^\sigma ([0, T])$, where the dual norm is defined by

$$\begin{align*} \| F\|_{N^\sigma ([0, T])} = \bigg \|\int _{0}^t S(t - t') F(t') dt'\bigg \|_{X^\sigma ([0, T])}. \end{align*}$$

Note that the number of cases has increased from the case-by-case analysis in the proof of Theorem A, where we had $w_i = v$ or $z_1$. One of the main ingredients is the smoothing on $z_3$ stated in Proposition 1.2 above. Note, however, that in order to exploit this smoothing, we need to measure $z_3$ in the $X^{2s-}([0, T])$-norm, which imposes a certain rigidity on the space-time integrability.⁠⁸ In order to prove Theorem 1.3, we also need to exploit a gain of integrability on $z_3$. In Lemma 3.3, we use the dispersive estimate (see 3.10 below) and the gain of integrability on each $z_1$ of the three factors in 1.10 and show that $z_3$ also enjoys a gain of integrability by giving up some differentiability.

⁸

Namely, we need to measure $z_3$ in $L^q_t([0, T]; W^{2s-, r}(\mathbb{R}^3))$ for admissible pairs $(q, r)$. See Lemma 2.6.

✖

Remark 1.4.

Given $\phi \in H^s(\mathbb{R}^3)$, let $\phi ^\omega$ be its Wiener randomization defined in 1.3. Given $v_0 \in H^\frac{1}{2}(\mathbb{R}^3)$, we can also consider 1.1 with the random initial data of the form $u_0^\omega = v_0 + \phi ^\omega$:

$$\begin{equation} \begin{cases} i \partial _tu + \Delta u = |u|^2 u,\\ u|_{t = 0} = v_0+ \phi ^\omega . \end{cases} {\tag{1.16}} \end{equation}$$

Then, by slightly modifying the proofs, we see that the analogues of Theorems A and 1.3 (with $\sigma = \frac{1}{2}$) also hold for 1.16. Namely, 1.16 is almost surely locally well-posed, provided $s > \frac{1}{5}$. This amounts to considering the following Cauchy problems:

$$\begin{equation*} \begin{cases} i \partial _tv + \Delta v = |v + z_1|^2(v+z_1),\\ v|_{t = 0} = v_0 \end{cases} \end{equation*}$$

when $\frac{1}{4} < s < \frac{1}{2}$ and

$$\begin{equation*} \begin{cases} i \partial _tv + \Delta v = \mathcal{N}(v + z_1+z_3) - \mathcal{N}(z_1),\\ v|_{t = 0} = v_0 \end{cases} \end{equation*}$$

when $\frac{1}{5} < s \leq \frac{1}{4}$. In this case, the critical nature of the problem appears through the $v \overline{v} v$ interaction in the case-by-case analysis 1.15 due to the deterministic (non-zero) initial data $v_0$ at the critical regularity. The required modification is straightforward and thus we omit details. See Proposition 6.3 in 2 and Lemma 6.2 in 37. We point out that the discussion in the next subsection also applies to 1.16.

Remark 1.5.

(i) Let $\mathcal{I}(u_1, u_2, u_3)$ denote the trilinear operator defined by

$$\begin{align} \mathcal{I}(u_1, u_2, u_3) (t)\coloneq -i \int _0^t S(t - t') u_1 \overline{u_2} u_3 (t') dt'. {\tag{1.17}} \end{align}$$

Then, for $\sigma > 3s-1$, there is no finite constant $C > 0$ such that

$$\begin{align} \big \|\mathcal{I}( u_1, u_2, u_3)\big \|_{X^\sigma ([0, 1])} \leq C \prod _{j = 1}^3 \| \phi _j \|_{H^s} {\tag{1.18}} \end{align}$$

for all $\phi _j \in H^s(\mathbb{R}^3)$, where $u_j = S(t) \phi _j$. See Appendix A. In particular, this shows that when $s \leq \frac{1}{2}$, there is no deterministic smoothing for $\mathcal{I}$; i.e., 1.18 does not hold for $\sigma > s$.

(ii) The proof of Proposition 1.2 only exploits “the randomization at the linear level”. Given $s \in \mathbb{R}$, let $\mathcal{R}^s$ denote the class of functions defined by

$$\begin{align*} \mathcal{R}^s = \big \{ u \text{ on } \mathbb{R}\times \mathbb{R}^3: \ & i \partial _tu + \Delta u = 0 \\ &\text{and } u \in L^q_{t, \text{loc}} W^{s, r}_x(\mathbb{R}^3) \text{ for any } 2 \leq q, r < \infty \big \}. \end{align*}$$

Under the regularity assumption $\sigma < 2s$ and $0 \leq s < 1$, it follows from the proof of Proposition 1.2 that the left-hand side of 1.18 is finite for any $u_1, u_2, u_3 \in \mathcal{R}^s$.⁠⁹ In other words, the proof of Proposition 1.2 only uses the fact that the random linear solution $z_1 = S(t) \phi ^\omega$ belongs to $\mathcal{R}^s$ almost surely; see the probabilistic Strichartz estimate (Lemma 2.1). The multilinear random structure of $z_3$ in terms of the random linear solution $z_1$ yields further cancellation. See, for example, Lemma 3.6 in 15. Such extra cancellation seems to improve only space-time integrability, and we do not know how to use it to improve the regularity threshold (i.e., differentiability) at this point. A similar comment applies to the unbalanced higher order terms $\zeta _{2k-1}$ (including $z_5$ below) studied in Proposition 1.7 below.

⁹

Here, we only need finiteness of the $A_3^s$-norm defined in 3.3 for each $u_j$, $j = 1, 2, 3$. See Remark 3.2.

✖

1.3. Partial power series expansion and the associated critical regularity

In this subsection, we discuss possible improvements over Theorem 1.3 by considering further expansions. For this purpose, we fix $\sigma = \frac{1}{2}$ in the following. By examining the proof of Theorem 1.3, we see that the regularity restriction $s > \frac{1}{5}$ comes from the following third order term:

$$\begin{align} z_5 (t) \coloneq -i \sum _{\substack{j_1 + j_2 + j_3 = 5\\j_1, j_2, j_3 \in \{1, 3\} }}\int _0^t S(t - t') z_{j_1} \overline{z_{j_2}}z_{j_3} (t') dt'. {\tag{1.19}} \end{align}$$

Namely, we have $(j_1, j_2, j_3) = (1, 1, 3)$ up to permutations. In Lemma 4.1, we show that given $0< s< \frac{1}{2}$, we have $z_5 \in X^{\frac{5}{2} s -}_{\operatorname {loc}}$. In particular, we have $z_5 \in X^\frac{1}{2}_{\operatorname {loc}}$, provided $s > \frac{1}{5}$, yielding the regularity threshold in Theorem 1.3.

A natural next step is to remove this non-desirable third order interaction

$$\begin{equation*} \sum _{\substack{j_1 + j_2 + j_3 = 5\\j_1, j_2, j_3 \in \{1, 3\} }} z_{j_1} \overline{z_{j_2}}z_{j_3} \end{equation*}$$

in the case-by-case analysis in 1.15 by considering the following third order expansion:

$$\begin{align} u = z_1 + z_3 + z_5+ v. {\tag{1.20}} \end{align}$$

In this case, the residual term $v \coloneq u - z_1 - z_3- z_5$ satisfies the following equation:

$$\begin{equation} \begin{cases} \displaystyle i \partial _tv + \Delta v = \mathcal{N}(v + z_1+z_3+z_5) - \sum _{\substack{j_1 + j_2 + j_3 \in \{3, 5\} \\j_1, j_2, j_3 \in \{1, 3\} }} z_{j_1} \overline{z_{j_2}}z_{j_3},\\ v|_{t = 0} = 0. \end{cases} {\tag{1.21}} \end{equation}$$

We expect that 1.21 is almost surely locally well-posed for $s > \frac{2}{11}$, which would be an improvement over Theorem 1.3. The proof will be once again based on case-by-case analysis:

$$\begin{align*} w_1\overline{w_2} w_3 \quad & \text{for $w_i = v, z_1, z_3$, or $z_5$, $i = 1, 2, 3$, such that } \\ & \text{it is not of the form $z_{j_1}\overline{z_{j_2}}z_{j_3} $ with $j_1 + j_2 + j_3 \in \{3, 5\}$} \end{align*}$$

in $N^\frac{1}{2}([0, T])$. Note the increasing number of combinations. In the following, however, we do not discuss details of this particular improvement over Theorem 1.3. Instead, we consider further iterative steps and discuss a possible limitation of this procedure.

Remark 1.6.

We point out that the expansions 1.7, 1.12, and 1.20 correspond to partial power series expansions of the first, second, and third orders,⁠¹⁰ respectively, of a solution to 1.1 in terms of the random initial data. Then, by considering the associated equations 1.8, 1.13, and 1.21 for the residual term $v$, we are recasting the original problem 1.1 as a fixed point problem centered at the partial power series expansions of the first, second, and third orders, respectively.

¹⁰

A (partial) power series expansion of a solution to 1.1 in terms of the random initial data can be expressed as a summation of certain multilinear operators over ternary trees. See, for example, 1136. Here, the term “order” in our context corresponds to “generation (of the associated trees) $+1$” with the convention that the trivial tree consisting only of the root node is of the zeroth generation. For example, the third order term $z_5$ in 1.19 appears as the summation over all the multilinear operators associated to the ternary trees of the third generation. In terms of the graphical representation in 36, we have

$$\begin{align*} z_5 \, \text{“}=\text{”} \,\vcenter{\img[][17pt][12pt][{Graphic without alt text}]{Images/imgfd1488ec472850e8b5e96e124b4dbb06.svg}}\,+ \,\vcenter{\img[][10pt][11pt][{Graphic without alt text}]{Images/img32578ea70c8174b56070ec6c1783bc9e.svg}}\,+\, \vcenter{\img[][17pt][11pt][{Graphic without alt text}]{Images/img1cc4e31ae7d87ff34e23a816ccc3d226.svg}} \ , \end{align*}$$

where “ $\vcenter{\img[][3pt][3pt][{\tikz[baseline=-2.7,scale=0.15]{ \draw(0,0) node[dot]{};}}]{Images/imgff2e4ff27c2aa4a6e3bf75119ef757aa.svg}}$ ” denotes the random linear solution $z_1 = S(t) \phi ^\omega$ and “ $\vcenter{\img[][4pt][4pt][{\tikz[baseline=-2.7,scale=0.15]{\draw(0,0) node[ddot]{};}}]{Images/img66990c0d12233105598478c71138d1b7.svg}}$ ” denotes the trilinear Duhamel integral operator $\mathcal{I}(u_1, u_2, u_3)$ defined in 1.17 with its three children as its arguments, $u_1, u_2$, and $u_3$.

✖

By drawing an analogy to the previous steps, we expect that the worst contribution comes from the following fourth order terms:

$$\begin{align} z_7 (t): = -i \sum _{\substack{j_1 + j_2 + j_3 = 7\\j_1, j_2, j_3 \in \{1, 3, 5\} }}\int _0^t S(t - t') z_{j_1} \overline{z_{j_2}}z_{j_3} (t') dt'. {\tag{1.22}} \end{align}$$

There are basically two contributions to 1.22: $(j_1, j_2, j_3) = (1, 3, 3)$ or $(1, 1, 5)$ up to permutations. In Lemma 4.2, we show that the contribution from $(j_1, j_2, j_3) = (1, 1, 5)$ is worse, being responsible for the expected regularity restriction $s > \frac{2}{11}$. In order to remove this term, we can consider the fourth order expansion

$$\begin{align*} u = z_1 + z_3 + z_5+z_7+ v \end{align*}$$

as in the previous steps and try to solve the following equation for the residual term $v = u - z_1 - z_3- z_5-z_7$:

$$\begin{equation*} \begin{cases} \displaystyle i \partial _tv + \Delta v = \mathcal{N}(v + z_1+z_3+z_5+z_7) - \sum _{\substack{j_1 + j_2 + j_3 \in \{3, 5, 7\} \\j_1, j_2, j_3 \in \{1, 3, 5\} }} z_{j_1} \overline{z_{j_2}}z_{j_3},\\ v|_{t = 0} = 0. \end{cases} \end{equation*}$$

We can obviously iterate this argument and consider the following $k$th order expansion:

$$\begin{align} u = \sum _{\ell = 1}^{k} z_{2\ell -1}+ v. {\tag{1.23}} \end{align}$$

In this case, we need to consider the following equation for the residual term $v \coloneq u - \sum _{\ell = 1}^{k} z_{2\ell -1}$:

$$\begin{equation} \begin{cases} \displaystyle i \partial _tv + \Delta v = \mathcal{N}\bigg (v +\sum _{\ell = 1}^{k} z_{2\ell -1}\bigg ) - \sum _{\substack{j_1 + j_2 + j_3 \in \{3, 5, \dots , 2k-1\} \\j_1, j_2, j_3 \in \{1, 3, \dots , 2k-3\}}} z_{j_1} \overline{z_{j_2}}z_{j_3},\\ v|_{t = 0} = 0 \end{cases} {\tag{1.24}} \end{equation}$$

and hope to construct a solution $v \in X^\frac{1}{2}([0, T])$ by carrying out the following case-by-case analysis:

$$\begin{align} \begin{split} w_1\overline{w_2} w_3, \quad & \text{for $w_i = v$, $z_j$, $j \in \{1, 3, \dots , 2k-1\}$, $i = 1$, $2$, $3$, such that } \\ & \text{it is not of the form $z_{j_1}\overline{z_{j_2}}z_{j_3} $ with $j_1 + j_2 + j_3 \in \{3, 5, \dots , 2k-1\}$} \end{split} {\tag{1.25}} \end{align}$$

in $N^\frac{1}{2}([0, T])$. Then, a natural question to ask is, Does this iterative procedure work indefinitely, allowing us to arbitrarily lower the regularity threshold for almost sure local well-posedness for 1.1? Or is there any limitation to it?

We now consider a “critical” regularity $s_* < \frac{1}{2}$ with respect to this iterative procedure for proving almost sure local well-posedness of 1.1. We simply define the critical regularity $s_* <\frac{1}{2}$ for this problem to be the infimum of the values of $s< \frac{1}{2}$ such that given any $\phi \in H^s(\mathbb{R}^3)$, the above iterative procedure⁠¹¹ shows that 1.1 is almost surely locally well-posed with respect to the Wiener randomization $\phi ^\omega$ of $\phi$. This is an empirical notion of criticality; unlike the scaling criticality, we cannot a priori compute this critical regularity $s_*$. Moreover, our discussion will be based on the estimates on the stochastic multilinear terms (Propositions 1.2 and 1.7 below). In the following, we discuss a (possible) lower bound on $s_*$, presenting a limitation to our iterative procedure based on partial power series expansions.

¹¹

Namely, we solve 1.24 for $v \in X^\frac{1}{2}([0, T])$ with some finite (or infinite) number of steps.

✖

Within the framework of the iterative procedure discussed above, a necessary condition for carrying out the case-by-case analysis 1.25 to study 1.24 in $X^\frac{1}{2}([0, T])$ is that the $(k+1)$st order term $z_{2k+1} = z_{2(k+1)-1}$ defined in a recursive manner,

$$\begin{align} z_{2k+1}(t) : = -i \sum _{\substack{j_1 + j_2 + j_3 = {2k+1} \\j_1, j_2, j_3 \in \{1, 3, \dots , 2k-1\} }}\int _0^t S(t - t') z_{j_1} \overline{z_{j_2}}z_{j_3} (t') dt', {\tag{1.26}} \end{align}$$

belongs to $X^\frac{1}{2}([0, T])$. By the nature of this iterative procedure, we may assume that the lower order terms $z_{2\ell -1}$, $\ell \in \{1, 3, \dots , k\}$, belong to $X^{ s_\ell }([0, T])$ for some $s_\ell < \frac{1}{2}$ but not in $X^\frac{1}{2}([0, T])$, since if any of the lower order terms, say $z_{2\ell -1}$ for some $\ell \in \{1, 3, \dots , k\}$, were in $X^\frac{1}{2}([0, T])$, then we would have stopped the iterative procedure at the $(\ell -1)$th step. As in the previous steps, we expect that $z_{2k+1}$ is responsible for a regularity restriction at the $k$th step of this iterative approach.

It could be a cumbersome task to study the regularity property of $z_{2k+1}$ due to the increasing number of combinations for $z_{j_1}\overline{z_{j_2}} z_{j_3}$ satisfying $j_1 + j_2 + j_3 = {2k+1}$, $j_1, j_2, j_3 \in \{1, 3, \dots , 2k-1\}$. In the following, we instead study the regularity property of the $(k+1)$st order term⁠¹² of a particular form, corresponding to $(j_1, j_2, j_3) = (1, 1, 2k - 1)$ up to permutations in 1.26. Given an integer $k \geq 0$, define $\zeta _{2k+1}$ by setting $\zeta _1 \coloneq z_1 = S(t) \phi ^\omega$ and

¹²

In fact, we study the $k$th order term $\zeta _{2k-1}$ in Proposition 1.7.

✖

$$\begin{align} \zeta _{2k+1}(t) \coloneq -i \sum _{\substack{j_1 + j_2 + j_3 = {2k+1} \\j_1, j_2, j_3 \in \{1, 2k-1\} }} \int _0^t S(t - t') \zeta _{j_1} \overline{\zeta _{j_2}} \zeta _{j_3} (t') dt'. {\tag{1.27}} \end{align}$$

As mentioned above, there are only three terms in this sum: $(j_1, j_2, j_3) = (1, 1, 2k - 1)$ up to permutations. Hence, $\zeta _{2k+1}$ consists of the “unbalanced”⁠¹³ $(2k+1)$-linear terms appearing in the definition 1.26 of $z_{2k+1}$. We claim that the $(k+1)$st order term $\zeta _{2k + 1}$ is responsible for the regularity restriction at the $k$th step of the iterative procedure. See Theorem 1.8 below.

¹³

By associating the $(2k+1)$-linear terms appearing in the summation in 1.26 with ternary trees of the $k$th generation as in 36, the summands in 1.27 correspond to the “unbalanced” trees of the $k$th generation, where two of the three children of the root node are terminal.

✖

In anticipating the alternative expansion 1.33 below, let us study the regularity property of the unbalanced $k$th order term $\zeta _{2k-1}$:

$$\begin{align} \zeta _{2k-1}(t) \coloneq -i \sum _{\substack{j_1 + j_2 + j_3 = {2k-1} \\j_1, j_2, j_3 \in \{1, 2k-3\} }} \int _0^t S(t - t') \zeta _{j_1} \overline{\zeta _{j_2}} \zeta _{j_3} (t') dt'. {\tag{1.28}} \end{align}$$

For $k = 2, 3, 4$, we have (with appropriate restrictions on the range of $s$)

$$\begin{align} \zeta _3 = z_3 \in X^{2s-}_\text{loc}, \qquad \zeta _5 = z_5 \in X^{\frac{5}{2}s-}_\text{loc}, \qquad \text{and}\qquad \zeta _7 \in X^{\frac{11}{4}s-}_\text{loc}. {\tag{1.29}} \end{align}$$

See Proposition 1.2 and Lemmas 4.1 and 4.2. In general, we have the following proposition.

Proposition 1.7.

Define a sequence $\{\alpha _k\}_{k \in \mathbb{N}}$ of positive real numbers by the recursive relation

$$\begin{align} \alpha _k = \frac{\alpha _{k-1} + 3}{2} {\tag{1.30}} \end{align}$$

with $\alpha _1 = 1$. Given $0< s < \alpha _{k-1}^{-1}$, let $\phi ^\omega$ be the Wiener randomization of $\phi \in H^s(\mathbb{R}^3)$ defined in 1.3. Then, we have

$$\begin{align} \zeta _{2k -1} \in X^\sigma _{\operatorname {loc}} {\tag{1.31}} \end{align}$$

for $\sigma < \alpha _k \cdot s$, almost surely.

By solving the recursive relation 1.30, we have

$$\begin{align} \alpha _k = 2\bigg \{ 1 - \bigg (\frac{1}{2}\bigg )^{k-1}\bigg \} + 1 {\tag{1.32}} \end{align}$$

and thus we have $\alpha _2 = 2$, $\alpha _3 = \frac{5}{2}$, and $\alpha _4 = \frac{11}{4}$. In particular, Proposition 1.7 agrees with 1.29. Moreover, since $\alpha _k$ is increasing and $\lim _{k \to \infty } \alpha _k =3$, the regularity restriction $s < \alpha _{k-1}^{-1}$ in Proposition 1.7 does not cause any issue since our main focus is to study the probabilistic local well-posedness of 1.1 in the range of $s$ that is not covered by Theorem 1.3. Namely, we may assume $s \leq \frac{1}{5}\, (< \frac{1}{3})$ in the following.

In view of Propositions 1.2 and 1.7, one obvious lower bound for the critical regularity $s_*$ for this iterative procedure is given by $s_0: = 0$ since there is no gain of regularity when $s = 0$ (even in moving from $z_1$ to $z_3$). On the other hand, in order to prove almost sure local well-posedness of 1.1 by carrying out the case-by-case analysis 1.25 for the equation 1.24, we need to show that the $(k+1)$st order term $\zeta _{2k+1}$ belongs to $X^\frac{1}{2}([0, T])$. This gives rise to a regularity restriction $s_k \coloneq \frac{1}{2\alpha _{k+1}}$ at the $k$th step of the iterative procedure. By taking $k \to \infty$, we obtain another “lower” bound⁠¹⁴ $s_\infty \coloneq \frac{1}{6}$ on this critical regularity $s_*$.

¹⁴

This “lower” bound is based on the upper bounds obtained in Propositions 1.2 and 1.7. In other words, if one can improve the bounds in Propositions 1.2 and 1.7, then one can lower the value of $s_\infty$. See Remark 1.10.

✖

As mentioned above, the case-by-case analysis 1.25 for general $k \in \mathbb{N}$ may be a combinatorially overwhelming task due to (i) the number of the increasing combinations in 1.25 and (ii) the random multilinear terms $z_j$, $j \in \{3, 5, \dots , 2k-1\}$, themselves having non-trivial combinatorial structures which makes it difficult to establish nonlinear estimates; see 1.26. In the following, we instead consider an alternative iterative procedure based on the expansion

$$\begin{align} u = \sum _{\ell = 1}^k \zeta _{2\ell -1} + v {\tag{1.33}} \end{align}$$

in place of 1.23. This expansion allows us to prove the following almost sure local well-posedness of 1.1 for $s > s_\infty = \frac{1}{6}$.

Theorem 1.8.

Let $\frac{1}{6} < s < \frac{1}{2}$. Given $\phi \in H^s(\mathbb{R}^3)$, let $\phi ^\omega$ be its Wiener randomization defined in 1.3. Then, the cubic NLS 1.1 on $\mathbb{R}^3$ is almost surely locally well-posed with respect to the random initial data $\phi ^\omega$. More precisely, there exists a set $\Sigma = \Sigma (\phi ) \subset \Omega$ with $P(\Sigma ) = 1$ such that, for any $\omega \in \Sigma$, there exists a unique function $u = u^\omega$ in the class:

$$\begin{align*} \zeta _1 + \zeta _3 + \cdots + \zeta _{2k-1} + X^\frac{1}{2}([0, T]) & \subset \zeta _1 + \zeta _3 + \cdots + \zeta _{2k-1} + C([0, T]; H^\frac{1}{2} (\mathbb{R}^3)) \\ \ & \subset C([0, T];H^s(\mathbb{R}^3)) \end{align*}$$

with $T = T(\phi , \omega ) >0$ such that $u$ is a solution to 1.1 on $[0, T]$. Here, $k \in \mathbb{N}$ is a unique positive integer such that $\frac{1}{2\alpha _{k+1}} < s \leq \frac{1}{2\alpha _{k}}$.

As before, the uniqueness of $u$ in the class

$$\begin{align*} \zeta _1 + \zeta _3 + \cdots + \zeta _{2k-1} + X^\frac{1}{2}([0, T]) \end{align*}$$

is to be interpreted as uniqueness of the residual term $v = u -\sum _{\ell = 1}^k \zeta _{2\ell -1}$ in $X^\frac{1}{2}([0, T])$. See also Remark 1.1.

In view of the discussion above, Theorem 1.8 proves almost sure local well-posedness of 1.1 in an almost “optimal”⁠¹⁵ regularity range $s > s_\infty = \frac{1}{6}$ with respect to the original iterative procedure based on the partial power series expansion 1.23. The proof of Theorem 1.8 is analogous to that of Theorem 1.3. Given $\frac{1}{6} < s<\frac{1}{2}$, fix $k \in \mathbb{N}$ such that $\frac{1}{2\alpha _{k+1}} < s \leq \frac{1}{2\alpha _{k}}$.⁠¹⁶ Write a solution $u$ as in 1.33. Note that as $s$ gets closer and closer to the critical value $s_\infty = \frac{1}{6}$, the expansion 1.33 gets arbitrarily long. In view of 1.28, the residual term $v \coloneq u - \sum _{\ell = 1}^{k} \zeta _{2\ell -1}$ satisfies the following equation:

¹⁵

Once again, this is based on the estimates in Propositions 1.2 and 1.7. In particular, the “optimality” of the regularity threshold in Theorem 1.8 is with respect to Propositions 1.2 and 1.7. If one can improve the bounds in Propositions 1.2 and 1.7, then one can lower the regularity threshold in Theorem 1.8.

✖

¹⁶

In view of Proposition 1.7, the lower bound on $s$ guarantees that $\zeta _{2k+1} \in X^\frac{1}{2}([0, T])$ almost surely, while the upper bound on $s$ states that we cannot use Proposition 1.7 to conclude $\zeta _{2k-1} \in X^\frac{1}{2}([0, T])$ almost surely.

✖

$$\begin{equation} \begin{cases} \displaystyle i \partial _tv + \Delta v = \mathcal{N}\bigg (v +\sum _{\ell = 1}^{k} \zeta _{2\ell -1}\bigg ) - \sum _{\ell = 2}^{k} \sum _{\substack{j_1 + j_2 + j_3 = 2\ell - 1\\j_1, j_2, j_3 \in \{1, 2\ell -3\}}} \zeta _{j_1} \overline{\zeta _{j_2}}\zeta _{j_3},\\ v|_{t = 0} = 0. \end{cases} {\tag{1.34}} \end{equation}$$

Hence, we need to carry out the following case-by-case analysis:

$$\begin{align} \begin{split} w_1\overline{w_2} w_3, \quad & \text{for $w_i = v$, $\zeta _j$, $j \in \{1, 3, \dots , 2k-1\}$, $i = 1, 2, 3$, such that } \\ & \text{it is not of the form $\zeta _{j_1}\overline{\zeta _{j_2}}\zeta _{j_3} $ with $(j_1, j_2, j_3 ) = (1, 1, 2\ell -3)$}\\ & \text{(up to permutations) for $\ell = 2$, $3$, …, $k$} \end{split} {\tag{1.35}} \end{align}$$

in $N^\frac{1}{2}([0, T])$. While Theorem 1.8 yields almost sure local well-posedness in an almost optimal range with respect to the original iterative procedure based on the partial power series expansion 1.23, the required analysis is much simpler than that required for the original iterative procedure based on the expansion 1.23. First, note that while the case-by-case analysis 1.25 based on the original iterative procedure involves combinatorially non-trivial $z_{2k-1}$, the case-by-case analysis 1.35 only involves the unbalanced $k$th order term $\zeta _{2k-1}$ which has a much simpler structure than $z_{2k-1}$. In particular, Proposition 1.7 shows that $\zeta _{2k-1}$, $k \geq 3$, has a better regularity property than the second order term $\zeta _3 = z_3$ in 1.10. In terms of space-time integrability, we show that $\zeta _{2k-1}$ also enjoys a gain of integrability by giving up control on derivatives (Lemma 4.3). Finally, by inspecting the proof of Theorem 1.3 (see Lemma 4.2 and Proposition 5.1 below), we see that, except for $z_{j_1} \overline{z_{j_2}}z_{j_3}$ with $(j_1, j_2, j_3) = (1, 1, 3)$ up to permutations,⁠¹⁷ we can bound all the terms $w_1 \overline{w_2} w_3$ appearing in the case-by-case analysis 1.15 in $N^\frac{1}{2}([0, T])$. Hence, we can basically apply the result of the case-by-case analysis 1.15 to our problem at hand. More precisely, by rewriting the case-by-case analysis 1.35 as

¹⁷

Namely, the terms constituting the third order term $\zeta _5 = z_5$ in 1.19.

✖

$$\begin{equation*} w_i = v,\qquad \zeta _1 = z_1,\quad \text{ or }\quad \zeta _j, \, j \in \{3, 5, \dots , 2k-1\} \end{equation*}$$

(with the restriction in 1.35) and using the fact that the terms $\zeta _{2\ell - 1}$, $\ell = 3$, …, $k$, behave better than $\zeta _3 = z_3$, the proof of Theorem 1.3 (in particular, Proposition 5.1) can be used to control all the terms $w_1 \overline{w_2} w_3$ in 1.35 except for

$$\begin{align} \zeta _{j_1}\overline{\zeta _{j_2}}\zeta _{j_3} \quad \text{with } (j_1, j_2, j_3 ) = (1, 1, 2 k-1) {\tag{1.36}} \end{align}$$

(up to permutations). Note that the contribution to 1.36 under the Duhamel integral is precisely given by $\zeta _{2k+1}$ in 1.27. In particular, Proposition 1.7 with $\sigma = \frac{1}{2}$ yields the regularity restriction $\alpha _{k+1}\cdot s > \frac{1}{2}$, i.e., the lower bound $s > \frac{1}{2\alpha _{k+1}}$ stated in Theorem 1.8. This allows us to construct a solution $v \in X^\frac{1}{2}([0, T])$ by the standard fixed point argument. See Section 6 for details.

We previously conjectured that the $(k+1)$st order term $z_{2k+1}$ in 1.26 would be responsible for a regularity restriction at the $k$th step of the original iterative procedure. Combining Proposition 1.7 and Theorem 1.8, we confirmed this claim; the regularity restriction indeed comes only from the unbalanced $(k+1)$st order term $\zeta _{2k+1}$ in 1.27.

We conclude this introduction with several remarks.

Remark 1.9.

Let $v_0 \in H^\frac{1}{2}(\mathbb{R}^3)$. Then, by slightly modifying the proof of Theorem 1.8 based on the modified iterative approach 1.33, we can prove almost sure local well-posedness of the cubic NLS 1.16 with the random initial data of the form $v_0 + \phi ^\omega$ for the same range of $s$. See Remark 1.4.

Remark 1.10.

In this paper, we exploit randomness only at the linear level in estimating the stochastic terms. See Remarks 1.5 and 3.2. It may be possible to lower the regularity thresholds in Theorems 1.3 and 1.8 by exploiting randomness at the multilinear level. We, however, do not pursue this direction in this paper since (i) our main purpose is to present the iterative procedures in their simplest forms and (ii) estimating the higher order stochastic terms by exploiting randomness at the multilinear level would require a significant amount of additional work, which would blur the main focus of this paper.

Remark 1.11.

The ill-posedness result in 12 shows that the solution map

$$\begin{equation*} \Phi : u_0 \in H^s(\mathbb{R}^3) \longmapsto u \in C([-T, T]; H^s(\mathbb{R}^3)) \end{equation*}$$

is not continuous for 1.1 when $s < \frac{1}{2}$. In proving Theorem A, we studied the perturbed NLS 1.8 for $v = u-z_1$. In particular, the proof shows that we can factorize the solution map for 1.1 as⁠¹⁸

¹⁸

Similarly, we can factorize the solution map for 1.16 as

$$\begin{align*} u |_{t = 0} = v_0 + \phi ^\omega \in H^s(\mathbb{R}^3) \longmapsto (v_0, z_1) \stackrel{\Psi }{\longmapsto } v \in X^\frac{1}{2}([0, T]) \subset C([0, T]; H^\frac{1}{2}(\mathbb{R}^3)) \end{align*}$$

such that the second map $\Psi$ is continuous in $(v_0, z_1) \in H^\frac{1}{2}(\mathbb{R}^3)\times S^s_1([0, T])$.

✖

$$\begin{align} \phi ^\omega \in H^s(\mathbb{R}^3) \longmapsto z_1 \stackrel{\Psi }{\longmapsto } v \in X^\frac{1}{2}([0, T]) \subset C([0, T]; H^\frac{1}{2}(\mathbb{R}^3)), {\tag{1.37}} \end{align}$$

where the first map can be viewed as a universal lift map and the second map $\Psi$ is the solution map to 1.8, which is in fact continuous in $z_1 \in S^s_1([0, T])$.⁠¹⁹

¹⁹

Here, $S^s_1([0, T]) \subset C([0, T]; H^s(\mathbb{R}^3))$ denotes the intersection of suitable space-time function spaces of differentiability at most $s$. In the following, we use $S^{\sigma }_j([0, T])$ in a similar manner.

✖

In the case of Theorem 1.3 (with $\sigma = \frac{1}{2}$), we have the following factorization of the solution map for 1.1:

$$\begin{align} \phi ^\omega \in H^s(\mathbb{R}^3) \longmapsto (z_1, z_3 ) \stackrel{\Psi }{\longmapsto } v \in X^\frac{1}{2}([0, T]) \subset C([0, T]; H^\frac{1}{2}(\mathbb{R}^3)), {\tag{1.38}} \end{align}$$

where the second map $\Psi$ is the solution map to 1.13, which is continuous from $(z_1, z_3) \in S^s_1([0, T])\times S^{2s-}_3([0, T])$ to $v \in X^\frac{1}{2}([0, T])$.

In the case of Theorem 1.8, we create $k$ stochastic objects in the first step, where $k = k(s) \in \mathbb{N}$:

$$\begin{align} \phi ^\omega \in H^s(\mathbb{R}^3) \longmapsto (\zeta _1, \zeta _3, \dots , \zeta _{2k-1} ) \stackrel{\Psi }{\longmapsto } v \in X^\frac{1}{2}([0, T]) \subset C([0, T]; H^\frac{1}{2}(\mathbb{R}^3)). {\tag{1.39}} \end{align}$$

Once again, the second map $\Psi$ (which is the solution map to 1.34) is continuous from $(\zeta _1, \zeta _3, \dots , \zeta _{2k-1} ) \in S^s_1([0, T]) \times \prod _{\ell = 2}^k S_\ell ^{\alpha _\ell s-} ([0, T])$ to $v \in X^\frac{1}{2}([0, T])$. We point out an analogy between these factorizations 1.37, 1.38, and 1.39 of the ill-posed solution maps into

(i)

the first step, involving stochastic analysis, and

(ii)

the second step, where purely deterministic analysis is performed in constructing a continuous map $\Psi$,

and similar factorizations for studying rough differential equations via the rough path theory 19 and singular stochastic parabolic PDEs 2123.

In the proof of Theorem 1.8, we could consider the following expansion of an infinite order:

$$\begin{align} u = \sum _{\ell = 1}^\infty \zeta _{2\ell -1} + v. {\tag{1.40}} \end{align}$$

This would allow us to present a single argument that works for all $\frac{1}{6} < s < \frac{1}{3}$. In this case, the residual part $v \coloneq u - \sum _{\ell = 1}^\infty \zeta _{2\ell -1}$ satisfies

$$\begin{equation*} \begin{cases} \displaystyle i \partial _tv + \Delta v = \mathcal{N}\bigg (v +\sum _{\ell = 1}^{\infty } \zeta _{2\ell -1}\bigg ) - \sum _{\ell = 2}^{\infty } \sum _{\substack{j_1 + j_2 + j_3 = 2\ell - 1\\j_1, j_2, j_3 \in \{1, 2\ell -3\}}} \zeta _{j_1} \overline{\zeta _{j_2}}\zeta _{j_3},\\ v|_{t = 0} = 0. \end{cases} \end{equation*}$$

In particular, we would need to worry about the convergence issue of infinite series, and hence there seems to be no simplification in considering the infinite order expansion 1.40.

Another strategy would be to treat $\zeta _\infty : = \sum _{\ell = 1}^\infty \zeta _{2\ell -1}$ as one stochastic object and write

$$\begin{align*} u =\zeta _\infty + v. \end{align*}$$

It follows from 1.28 that $\zeta _\infty$ satisfies the following equation:

$$\begin{equation} \begin{cases} \displaystyle i \partial _t\zeta _\infty + \Delta \zeta _\infty = \sum _{\ell = 2}^{\infty } \sum _{\substack{j_1 + j_2 + j_3 = 2\ell - 1\\j_1, j_2, j_3 \in \{1, 2\ell -3\}}} \zeta _{j_1} \overline{\zeta _{j_2}}\zeta _{j_3},\\ \zeta _\infty |_{t = 0} = \phi ^\omega . \end{cases} {\tag{1.41}} \end{equation}$$

Noting that

$$\begin{equation*} \sum _{\substack{j_1 + j_2 + j_3 = 2\ell - 1\\j_1, j_2, j_3 \in \{1, 2\ell -3\}}} \zeta _{j_1} \overline{\zeta _{j_2}}\zeta _{j_3} = \begin{cases} |z_1|^2 z_1& \text{when }\ell = 2,\\ 2|z_1|^2\zeta _{2\ell -3} + z_1^2 \overline{\zeta _{2\ell -3}} & \text{when }\ell \geq 3, \end{cases} \end{equation*}$$

we can rewrite 1.41 as

$$\begin{equation} \begin{cases} \displaystyle i \partial _t\zeta _\infty + \Delta \zeta _\infty = 2|z_1|^2 \zeta _\infty + z_1^2 \overline{\zeta _\infty } - 2 |z_1|^2 z_1,\\ \zeta _\infty |_{t = 0} = \phi ^\omega , \end{cases} {\tag{1.42}} \end{equation}$$

and thus $v : = u - \zeta _\infty$ satisfies

$$\begin{equation} \begin{cases} \displaystyle i \partial _tv + \Delta v = \mathcal{N}(v +\zeta _\infty ) - 2|z_1|^2 \zeta _\infty - z_1^2 \overline{\zeta _\infty } + 2 |z_1|^2 z_1,\\ v|_{t = 0} = 0. \end{cases} {\tag{1.43}} \end{equation}$$

The equations 1.42 and 1.43 do not particularly appear to be in such a friendly format. Namely, studying 1.42 and 1.43 does not seem to provide a simplification over the case-by-case analysis 1.35 for the equation 1.34 for each fixed $k \in \mathbb{N}$.

In a recent work 40, the second author (with Tzvetkov and Y. Wang) proved invariance of the white noise for the (renormalized) cubic fourth order NLS on the circle. One novelty of this work is that we introduced an infinite sequence $\{ z^\text{res}_{2\ell -1}\}_{\ell \in \mathbb{N}}$ of stochastic $(2\ell -1)$-linear objects (depending only on the random initial data) and considered the following expansion of an infinite order:

$$\begin{equation*} u = \sum _{\ell = 1}^\infty z^\text{res}_{2\ell -1} + v. \end{equation*}$$

For this problem, it turned out that $z^\text{res}_\infty : = \sum _{\ell = 1}^\infty z^\text{res}_{2\ell -1}$ satisfies a particularly simple equation.⁠²⁰ Then by treating $z^\text{res}_\infty$ as one stochastic object, we wrote a solution $u$ as $u = z^\text{res}_\infty +v$, which led to the factorization

²⁰

In fact, the series $z^\text{res}_\infty = \sum _{\ell = 1}^\infty z^\text{res}_{2\ell -1}$ corresponds to the power series expansion of the resonant cubic fourth order NLS. We point out that $z^\text{res}_\infty$ does not belong to the span of Wiener homogeneous chaoses of any finite order.

✖

$$\begin{equation*} \phi ^\omega \in H^s(\mathbb{T}) \longmapsto z^\text{res}_\infty = \sum _{\ell = 1}^\infty z^\text{res}_{2\ell -1} \longmapsto v \in C(\mathbb{R}; L^2(\mathbb{T})) \end{equation*}$$

for $s < -\frac{1}{2}$, where $\phi ^\omega$ denotes the Gaussian white noise on the circle.

Remark 1.12.

In 44, the third author (with Y. Wang) recently studied probabilistic local well-posedness of NLS on $\mathbb{R}^d$ within the framework of the $L^p$-based Sobolev spaces, using the dispersive estimate. In the context of the cubic NLS 1.1 on $\mathbb{R}^3$, their result yields almost sure local existence of a unique solution $u$ for the randomized initial data $\phi ^\omega \in H^s(\mathbb{R}^3)$, provided $s\geq 0$. In particular, the argument in 44 allows us to consider random initial data of lower regularities than Theorems A, 1.3, and 1.8. Note that, in 44, it was shown that the solution $u$ only belongs to $C([0, T]; W^{s, 4}(\mathbb{R}^3))$, almost surely. We point out that a slight adaptation of the work 39 by the second and third authors (with Y. Wang) shows that the solution $u$ indeed lies in $C([0, T]; H^s(\mathbb{R}^3))$.

As compared to 44, the argument presented in this paper provides extra regularity information, namely the decomposition 1.33 of the solution $u$ with the terms of increasing regularities: $\zeta _{2\ell -1} \in X^{\alpha _\ell \cdot s-}_{\operatorname {loc}}$ and $v \in X^\frac{1}{2}([0, T])$. In particular, the residual term $v$ lies in the (sub)critical regularity,⁠²¹ leaving us a possibility of adapting deterministic techniques to study its further properties.

²¹

By slightly tweaking the argument, we can easily place $v$ in $X^{\frac{1}{2}+}([0, T])$.

✖

This paper is organized as follows. In Section 2, we recall probabilistic and deterministic lemmas along with the definitions of the basic function spaces. In Section 3, we study the regularity properties of the second order term $z_3$ in 1.10. In Section 4, we further investigate the regularity properties of the higher order terms $z_5$ and $z_7$ and the unbalanced higher order terms $\zeta _{2k-1}$. Note that the analysis on $z_5$ and $z_7$ contains part⁠²² of the case-by-case analysis 1.15 needed for proving Theorem 1.3. In Section 5, we then carry out the rest of the case-by-case analysis 1.15 and prove Theorem 1.3. In Section 6, we briefly describe the proof of Theorem 1.8 by indicating how the analysis in the previous sections can lead to the proof. In Appendix A, we prove the deterministic non-smoothing of the Duhamel integral operator discussed in Remark 1.5.

²²

Note that $z_7$ defined in 1.22 contains the contribution from $z_{j_1}\overline{z_{j_2}}z_{j_3}$ with $(j_1, j_2, j_3) = (1, 3, 3)$ up to permutations.

✖

Notation.

We use $a+$ (and $a-$) to denote $a + \varepsilon$ (and $a - \varepsilon$, respectively) for arbitrarily small $\varepsilon \ll 1$, where an implicit constant is allowed to depend on $\varepsilon > 0$ (and it usually diverges as $\varepsilon \to 0$).

Given a Banach space $B$ of temporal functions, we use the following shorthand notation: $B_T \coloneq B([0, T])$. For example, $L^q_T L^r_x = L^q_t([0, T]; L^r_x(\mathbb{R}^3))$.

Let $\eta : \mathbb{R}\to [0, 1]$ be an even, smooth cutoff function supported on $[-\frac{8}{5}, \frac{8}{5}]$ such that $\eta \equiv 1$ on $[-\frac{5}{4}, \frac{5}{4}]$. Given a dyadic number $N \geq 1$, we set $\eta _1(\xi ) = \eta (|\xi |)$ and

$$\begin{equation*} \eta _N(\xi ) = \eta \bigg (\frac{|\xi |}{N}\bigg ) - \eta \bigg (\frac{2|\xi |}{N}\bigg ) \end{equation*}$$

for $N \geq 2$. Then, we define the Littlewood-Paley projection operator $\mathbf{P}_N$ as the Fourier multiplier operator with symbol $\eta _N$. In the following, we use the convention that capital letters denote dyadic numbers. For example, $N = 2^n$ for some $n \in \mathbb{N}_0 : = \mathbb{N}\cup \{0\}$.

Given dyadic numbers $N_1$, …, $N_4 \in 2^{\mathbb{N}_0}$, we set $N_{\max } \coloneq \max _{j = 1, \dots , 4} N_j$. We also use the shorthand notation $f_N = \mathbf{P}_N f$. For example, we have $z_{1, N_j} = \mathbf{P}_{N_j} z_1$.

2. Strichartz estimates and function spaces

2.1. Probabilistic Strichartz estimates

First, we recall the usual Strichartz estimates on $\mathbb{R}^3$ for the reader’s convenience. We say that a pair $(q, r)$ is Schrödinger admissible if it satisfies

$$\begin{equation*} \frac{2}{q} + \frac{3}{r} = \frac{3}{2} \end{equation*}$$

with $2\leq q, r \leq \infty$. Then, the following Strichartz estimates are known to hold 20284649:

$$\begin{equation} \| S(t) \phi \|_{L^q_t L^r_x (\mathbb{R}\times \mathbb{R}^3)} \lesssim \|\phi \|_{L^2_x(\mathbb{R}^3)}. {\tag{2.1}} \end{equation}$$

It follows from 2.1 and Sobolev’s inequality that

$$\begin{equation} \| S(t) \phi \|_{L^p_{t, x} (\mathbb{R}\times \mathbb{R}^3)} \lesssim \big \| |\nabla |^{\frac{3}{2} - \frac{5}{p}}\phi \big \|_{L^2_x(\mathbb{R}^3)} {\tag{2.2}} \end{equation}$$

for $p \geq \frac{10}{3}$. We will use the following admissible pairs in this paper:

$$\begin{align*} (\infty , 2), \ \big (5, \tfrac{30}{11}\big ), \ \big (\tfrac{10}{3}\tfrac{10}{3}\big ), \ (2+, 6-). \end{align*}$$

In particular, by Sobolev’s inequality, we have

$$\begin{align} W^{\frac{1}{2}, \frac{30}{11}}(\mathbb{R}^3) \hookrightarrow L^5(\mathbb{R}^3). {\tag{2.3}} \end{align}$$

One of the important key ingredients for probabilistic well-posedness is the probabilistic Strichartz estimates. Such probabilistic estimates were first exploited by McKean 34 and Bourgain 5. In the following, we state the probabilistic Strichartz estimates under the Wiener randomization 1.3. See 1 for the proofs.

Lemma 2.1.

Given $\phi$ on $\mathbb{R}^3$, let $\phi ^\omega$ be its Wiener randomization defined in 1.3. Then, given finite $q \geq 2$ and $2 \leq r \leq \infty$, there exist $C, c>0$ such that

$$\begin{align*} P\Big ( \|S(t) \phi ^\omega \|_{L^q_t L^r_x([0, T]\times \mathbb{R}^3)}> \lambda \Big ) \leq C\exp \bigg (-c \frac{\lambda ^2}{ T^\frac{2}{q}\|\phi \|_{H^s}^{2}}\bigg ) \end{align*}$$

for all $T > 0$ and $\lambda >0$ with (i) $s = 0$ if $r < \infty$ and (ii) $s > 0$ if $r = \infty$.

A similar estimate holds when $q = \infty$ (with $s > 0$), but we will not need it in this paper. See 38. We also need the following lemma on the control of the size of $H^s$-norm of $\phi ^\omega$.

Lemma 2.2.

Given $\phi \in H^s(\mathbb{R}^3)$, let $\phi ^\omega$ be its Wiener randomization defined in 1.3. Then, we have

$$\begin{align*} P\Big ( \| \phi ^\omega \|_{ H^s( \mathbb{R}^3)} > \lambda \Big ) \leq C\exp \bigg (-c \frac{\lambda ^2}{ \|\phi \|_{H^s}^{2}}\bigg ) \end{align*}$$

for all $\lambda >0$.

2.2. Function spaces and their properties

In this subsection, we go over the basic definitions and properties of the function spaces used for the Fourier restriction norm method (i.e., analysis involving the $X^{s, b}$-spaces introduced in 4) adapted to the space of functions of bounded $p$-variation and its pre-dual, introduced and developed by Tataru, Koch, and their collaborators 222431. We refer readers to 2224 for the proofs of the basic properties. See also 2.

Let $\mathcal{Z}$ be the set of finite partitions $-\infty <t_{0}<t_{1}<\dots <t_{K} \le \infty$ of the real line. By convention, we set $u(t_K)\coloneq 0$ if $t_K=\infty$. We use $\mathbf{1}_I$ to denote the sharp characteristic function of a set $I \subset \mathbb{R}$.

Definition 2.3.

Let $1\leq p < \infty$.

(i) We define a $U^p$-atom to be a step function $a: \mathbb{R}\to L^2(\mathbb{R}^3)$ of the form

$$\begin{equation*} a=\sum _{k=1}^{K}\phi _{k-1} \mathbf{1}_{[t_{k-1},t_{k})}, \end{equation*}$$

where $\{t_{k}\}_{k=0}^{K}\in \mathcal{Z}$ and $\{\phi _{k}\}_{k=0}^{K-1}\subset L^2 (\mathbb{R}^3)$ with $\sum _{k=0}^{K-1} \| \phi _{k} \|_{L^{2}}^{p}=1$. Furthermore, we define the atomic space $U^p = U^p(\mathbb{R}; L^2(\mathbb{R}^3))$ by

$$\begin{align*} U^{p}\coloneq \bigg \{ u: \mathbb{R}\rightarrow L^2 (\mathbb{R}^3) : u=\sum _{j=1}^{\infty }\lambda _{j}a_{j}\ & \text{for $U^{p}$-atoms $a_{j}$}, \ \{\lambda _j\}_{j \in \mathbb{N}} \in \ell ^1(\mathbb{N}; \mathbb{C})\bigg \} \end{align*}$$

with the norm

$$\begin{align*} \| u \| _{U^{p}}\coloneq \inf \bigg \{ \sum _{j=1}^{\infty }|\lambda _{j}|:u=\sum _{j=1}^{\infty } \lambda _{j}a_{j} \text{ for $U^{p}$-atoms } a_{j}, \ \{\lambda _j\}_{j \in \mathbb{N}} \in \ell ^1(\mathbb{N}; \mathbb{C})\bigg \}, \end{align*}$$

where the infimum is taken over all possible representations for $u$.

(ii) We define $V^{p}= V^p(\mathbb{R}; L^2(\mathbb{R}^3))$ to be the space of functions $u:\mathbb{R}\to L^2(\mathbb{R}^3)$ of bounded $p$-variation with the standard $p$-variation norm:

$$\begin{equation*} \| u \| _{V^{p}}\coloneq \sup _{\{t_{k}\}_{k=0}^{K}\in \mathcal{Z}} \bigg ( \sum _{k=1}^{K}\| u(t_{k})-u(t_{k-1})\| _{L^{2}}^{p}\bigg ) ^{\frac{1}{p}}. \end{equation*}$$

By convention, we impose that the limits $\lim _{t \to \pm \infty }u(t)$ exist in $L^2 (\mathbb{R}^3)$.

(iii) Let $V_{\text{rc}}^{p}$ be the closed subspace of $V^{p}$ of all right-continuous functions $u \in V^p$ with $\lim _{t\rightarrow -\infty }u(t)=0$.

(iv) We define $U_{\Delta }^{p}\coloneq S (t) U^{p}$ (and $V_{\Delta }^{p}\coloneq S(t)V^{p}$, respectively) to be the space of all functions $u:\mathbb{R}\to L^2 (\mathbb{R}^3)$ such that $t \to S ( -t ) u(t)$ is in $U^p$ (and in $V^p$, respectively) with the norms

$$\begin{equation*} \| u \|_{U_{\Delta }^{p}} \coloneq \| S(-t )u \|_{U^{p}} \qquad \text{and} \qquad \| u \|_{V_{\Delta }^p} \coloneq \| S(-t)u \|_{V^{p}}. \end{equation*}$$

The closed subspace $V_{\text{rc}, \Delta }^{p}$ is defined in an analogous manner.

Recall the following inclusion relation: for $1\leq p<q<\infty$,

$$\begin{align*} U^p \hookrightarrow V_{\text{rc}}^{p}\hookrightarrow U^{q} \hookrightarrow L^{\infty }(\mathbb{R};L^2 (\mathbb{R}^3)). \end{align*}$$

The space $V^p$ is the classical space of functions of bounded $p$-variation, and the space $U^p$ appears as the pre-dual of $V^{p'}$ with $\frac{1}{p} + \frac{1}{p'} = 1$, $1 < p < \infty$. Their duality relation and the atomic structure of the $U^p$-space turned out to be very effective in studying dispersive PDEs in critical settings.

We are now ready to define the solution spaces.

Definition 2.4.

(i) Let $s\in \mathbb{R}$. We define $X^s(\mathbb{R})$ to be the closure of $C(\mathbb{R};H^s(\mathbb{R}^3)) \cap U_{\Delta }^2$ with respect to the $X^s$-norm defined by

$$\begin{equation*} \|u \|_{X^s (\mathbb{R})} : = \bigg ( \sum _{\substack{N\geq 1\\{\operatorname {dyadic}}}} N^{2s} \| \mathbf{P}_N u \|_{U^2_{\Delta }L^2}^2 \bigg )^\frac{1}{2}. \end{equation*}$$

(ii) Let $s\in \mathbb{R}$. We define $Y^s(\mathbb{R})$ to be the space of all functions $u \in C(\mathbb{R};H^s(\mathbb{R}^3))$ such that the map $t \mapsto \mathbf{P}_N u$ lies in $V_{\text{rc},\Delta }^2 H^s$ for any $N \in 2^{\mathbb{N}_0}$ and $\| u \|_{Y^s(\mathbb{R})}< \infty$, where the $Y^s$-norm is defined by

$$\begin{equation*} \|u \|_{Y^s(\mathbb{R})} : = \bigg ( \sum _{\substack{N\geq 1\\{\operatorname {dyadic}}}} N^{2s} \| \mathbf{P}_N u\|_{V^2_\Delta L^2}^2\bigg )^\frac{1}{2}. \end{equation*}$$

Recall the following embeddings:

$$\begin{equation*} U^2_\Delta H^s \hookrightarrow X^s \hookrightarrow Y^s \hookrightarrow V^2_\Delta H^s \hookrightarrow U^p_\Delta H^s \end{equation*}$$

for $p>2$.

Given an interval $I \subset \mathbb{R}$, we define the local-in-time versions $X^s(I)$ and $Y^s(I)$ of these spaces as restriction norms. For example, we define the $X^s(I)$-norm by

$$\begin{equation*} \|u \|_{X^s(I)} = \inf \big \{ \|v\|_{X^s(\mathbb{R})}: \, v|_I = u\big \}. \end{equation*}$$

We also define the norm for the non-homogeneous term on an interval $I = [t_0, t_1)$:

$$\begin{align*} \| F\|_{N^s(I)} = \bigg \|\int _{t_0}^t S(t - t') F(t') dt'\bigg \|_{X^s(I)}. \end{align*}$$

We conclude this section by presenting some basic estimates involving these function spaces. See 22224 for the proofs.

Lemma 2.5.

Let $s \in \mathbb{R}$ and $T\in ( 0,\infty ]$. Then, the following linear estimates hold:

$$\begin{align*} \| S(t) \phi \|_{X^s([0, T])} & \leq \|\phi \|_{H^s}, \\ \| F \|_{N^{s}([0, T])} & \le \sup _{\substack{w \in Y^{-s}([0, T])\\ \| w \|_{Y^{-s}([0, T])}=1}} \left| \int _{0}^{T} \langle F(t), w(t) \rangle _{L_{x}^{2}} dt \right| \end{align*}$$

for any $\phi \in H^s(\mathbb{R}^3)$ and $F \in L^1([0,T];H^{s}(\mathbb{R}^3))$.

The transference principle 22, Proposition 2.19 and the interpolation lemma 22, Proposition 2.20 applied on the Strichartz estimates 2.1 and 2.2 imply the following estimates.

Lemma 2.6.

Given any admissible pair $(q,r)$ with $q>2$ and $p \geq \frac{10}{3}$, we have

$$\begin{align*} \| u \|_{L_t^q L_x^r} & \lesssim \| u \|_{Y^0}, \\ \| u \|_{L^p_{t,x}} & \lesssim \big \| |\nabla |^{\frac{3}{2} - \frac{5}{p}} u\big \|_{Y^0}. \end{align*}$$

Similarly, the bilinear refinement of the Strichartz estimate 61441 implies the following bilinear estimate.

Lemma 2.7.

Let $N_1, N_2 \in 2^{\mathbb{N}_0}$ with $N_1 \le N_2$. Then, we have

$$\begin{align} \| \mathbf{P}_{N_1}u_1 \mathbf{P}_{N_2}u_2\|_{L^2_{t} ([0, T]; L^2_x)} \lesssim T^{0+}N_1^{1-} N_2^{-\frac{1}{2}+} \| \mathbf{P}_{N_1} u_1 \|_{Y^0([0, T])} \| \mathbf{P}_{N_2} u_2 \|_{Y^0([0, T])} {\tag{2.4}} \end{align}$$

for any $T > 0$ and $u_1, u_2 \in Y^0([0, T])$.

Proof.

From the bilinear refinement of the Strichartz estimate 614 and the transference principle, we have

$$\begin{align} \| \mathbf{P}_{N_1}u_1 \mathbf{P}_{N_2}u_2\|_{L^2_{t,x}} \lesssim N_1^{1-} N_2^{-\frac{1}{2}+} \| \mathbf{P}_{N_1} u_1 \|_{Y^0} \| \mathbf{P}_{N_2} u_2 \|_{Y^0} {\tag{2.5}} \end{align}$$

for all $u_1, u_2 \in Y^0$. See 2 for the proof of 2.5. On the other hand, it follows from Hölder’s and Sobolev’s inequalities that

$$\begin{align} \| \mathbf{P}_{N_1}u_1 \mathbf{P}_{N_2}u_2\|_{L^2_TL^2_x} & \lesssim T^\frac{1}{2} \| \mathbf{P}_{N_1} u_1 \|_{L^\infty _TL^4_x} \| \mathbf{P}_{N_2} u_2 \|_{L^\infty _TL^4_x} \\ & \lesssim T^\frac{1}{2} N_1^\frac{3}{4}N_2^\frac{3}{4} \| \mathbf{P}_{N_1} u_1 \|_{Y^0_T} \| \mathbf{P}_{N_2} u_2 \|_{Y^0_T} . {\tag{2.6}} \end{align}$$

Then, the estimate 2.4 follows from interpolating 2.5 and 2.6.

■

3. On the second order term $z_3$

In this and the next sections, we study the regularity properties of the various stochastic terms that appear in the iterative procedures. Given $\phi \in H^s(\mathbb{R}^3)$, let $\phi ^\omega$ be the Wiener randomization of $\phi$ defined in 1.3 and set

$$\begin{align*} z_1 = S(t) \phi ^\omega . \end{align*}$$

In this section, we study the regularity properties of the second order term:

$$\begin{align} z_3 = -i \int _0^t S(t - t') |z_1|^2 z_1(t') dt'. {\tag{3.1}} \end{align}$$

We first present the proof of Proposition 1.2. We follow closely the argument in 2.

Proof of Proposition 1.2.

(i) By Lemma 2.5, the estimate 1.11 follows once we prove

$$\begin{align} \bigg | \int _0^T \int _{\mathbb{R}^3} \langle \nabla \rangle ^\sigma ( z_1 \overline{z_1} z_1 )\overline{w} dx dt\bigg | \leq T^\theta C(\omega , \|\phi \|_{H^s}) {\tag{3.2}} \end{align}$$

for some almost surely finite constant $C(\omega , \|\phi \|_{H^s} ) > 0$ and $\theta > 0$, where $\|w\|_{Y^{0}_T} \leq 1$. In the following, we drop the complex conjugate when it does not play any role.

Define $A^s_3(T)$ by

$$\begin{align} \|z_1\|_{A^s_3(T)}: = \max \Big ( \|\langle \nabla \rangle ^s z_{1} \|_{L^{\frac{30}{7}+}_{T}L^\frac{30}{7}_x}, \| \langle \nabla \rangle ^sz_{1} \|_{L^{4}_{T, x}}, \| z_1(0)\|_{H^s} \Big ). {\tag{3.3}} \end{align}$$

Then, by applying the dyadic decomposition, it suffices to prove that

$$\begin{align} \bigg | \int _0^T \int _{\mathbb{R}^3} \mathbf{P}_{N_1} z_1 \cdot \mathbf{P}_{N_2} z_1 \cdot N_3^\sigma \mathbf{P}_{N_3} z_1 \cdot \mathbf{P}_{N_4} w \, dx dt\bigg | \leq T^\theta N_{\max }^{0-} C(\|z_1\|_{A^s_3(T)}) {\tag{3.4}} \end{align}$$

for all $N_1$, …, $N_4 \in 2^{\mathbb{N}_0}$ with $N_3 \geq N_2 \geq N_1$. Once we prove 3.4, the desired estimate 3.2 follows from summing 3.4 over dyadic blocks and applying Lemmas 2.1 and 2.2. Recall our shorthand notation: $z_{1, N_j} = \mathbf{P}_{N_j} z_1$ and $w_{N_4} = \mathbf{P}_{N_4} w$.

In the following, we assume that

$$\begin{align} \sigma < 2s \qquad \text{and}\qquad 0 \leq s < 1. {\tag{3.5}} \end{align}$$

Case 1 ($N_2 \sim N_3$).

By Hölder’s inequality and Lemma 2.6, we have

$$\begin{align*} \text{LHS of } \xhref[disp-formula]{#O2}{3.4} & \lesssim \| z_{1, N_1} \|_{L^\frac{30}{7}_{T, x}} \bigg (\prod _{j = 2}^3 \|\langle \nabla \rangle ^\frac{\sigma }{2} z_{1, N_j} \|_{L^{\frac{30}{7}}_{T, x}}\bigg ) \| w_{N_4} \|_{L^\frac{10}{3}_{T, x}}\\ & \leq T^{0+} N_{\max }^{0-} C(\|z_1\|_{A^s_3(T)}) \|w_{N_4}\|_{Y^0_T}, \end{align*}$$

provided that 3.5 holds.

Case 2 ($N_3 \sim N_4 \gg N_1, N_2$).

Subcase 2.a ($N_1, N_2 \ll N_3^\frac{1}{2}$).

By Cauchy-Schwarz’ inequality and Lemma 2.7 followed by Lemma 2.5,⁠²³ we have

²³

In the remaining part of this paper, we repeatedly apply this argument when there is a frequency separation. We shall simply refer to it as the “bilinear Strichartz estimate” argument.

✖

$$\begin{align*} \text{LHS of } \xhref[disp-formula]{#O2}{3.4} & \leq N_3^\sigma \|z_{1, N_1} z_{1, N_3} \|_{L^2_{T, x}} \|z_{1, N_2} w_{N_4}\|_{L^2_{T, x}}\\ & \lesssim T^{0+} N_1^{1-s-} N_2^{1-s-} N_3^{\sigma - s - 1 +} \bigg (\prod _{j = 1}^3 \|\mathbf{P}_{N_j} \phi ^\omega \|_{H^{s}}\bigg ) \|w_{N_4}\|_{Y^0_T}\\ & \lesssim T^{0+} N_3^{\sigma - 2s +} \prod _{j = 1}^3 \|\mathbf{P}_{N_j} \phi ^\omega \|_{H^{s}}\\ & \leq T^{0+} N_{\max }^{0-} C(\|z_1\|_{A^s_3(T)}), \end{align*}$$

provided that 3.5 holds.

Subcase 2.b ($N_1, N_2\gtrsim N_3^\frac{1}{2}$).

By Hölder’s inequality and Lemma 2.6, we have

$$\begin{align*} \text{LHS of } \xhref[disp-formula]{#O2}{3.4} & \le N_3^\sigma \bigg (\prod _{j = 1}^3 \| z_{1, N_j} \|_{L^\frac{30}{7}_{T, x}}\bigg ) \| w_{N_4} \|_{L^\frac{10}{3}_{T, x}}\\ & \le T^{0+} N_1^{-s} N_2^{-s}N_3^{\sigma -s} C(\|z_1\|_{A^s_3(T)})\\ & \leq T^{0+} N_{\max }^{0-} C(\|z_1\|_{A^s_3(T)}), \end{align*}$$

provided that 3.5 holds.

Subcase 2.c ($N_2\gtrsim N_3^\frac{1}{2} \gg N_1$).

By the bilinear Strichartz estimate, we have

$$\begin{align*} &\text{LHS of } \xhref[disp-formula]{#O2}{3.4} \le N_3^\sigma \|z_{1, N_1} w_{N_4} \|_{L^2_{T, x}} \prod _{j = 2}^3 \|z_{1, N_j}\|_{L^4_{T, x}} \\ &\qquad \lesssim T^{0+} N_1^{1-s-} N_2^{-s} N_3^{\sigma -s-\frac{1}{2}+} \|\mathbf{P}_{N_1}\phi ^\omega \|_{H^s} \bigg (\prod _{j = 2}^3 \|\langle \nabla \rangle ^s z_{1, N_j} \|_{L^4_{T, x}}\bigg ) \|w_{N_4}\|_{Y^0_T}\\ &\qquad \le T^{0+} N_3^{\sigma -2s+} C(\|z_1\|_{A^s_3(T)})\\ &\qquad \leq T^{0+} N_{\max }^{0-} C(\|z_1\|_{A^s_3(T)}), \end{align*}$$

provided that 3.5 holds.

Therefore, putting all the cases together, we obtain 3.4.

(ii) Given $N \gg 1$ and small $\ell > 0$, consider the following deterministic initial condition $\phi$ whose Fourier transform is given by

$$\begin{align*} \widehat{\phi }(\xi ) = \mathbf{1}_{ N e_1 + \ell Q}(\xi ) + \mathbf{1}_{(N+100\ell ) e_1 + \ell Q}(\xi ) + \mathbf{1}_{ N e_1 + 100\ell e_2 + \ell Q}(\xi ), \end{align*}$$

where $Q = (-\frac{1}{2}, \frac{1}{2}]^3$, $e_1 = (1, 0, 0)$, and $e_2 = (0, 1, 0)$. By taking $\ell > 0$ sufficiently small, we have $\operatorname *{supp}\widehat{\phi }\subset N e_1 + Q$, and thus we can neglect the effect of the randomization in 1.3 since all three terms on the right-hand side will be multiplied by a common random number $g_{N e_1}$. Without loss of generality, we assume that $g_{N e_1} = 1$ in the following.

We estimate from below the contribution to $\mathbf{1}_{Q_{N, \ell }} (\xi ) \widehat{z}_3 (t, \xi )$, where

$$\begin{equation*} Q_{N, \ell } = Ne_1 + 100 \ell (e_1 + e_2) + \ell Q. \end{equation*}$$

From 3.1, we have

$$\begin{align} \mathbf{1}_{Q_{N, \ell }} & (\xi ) \widehat{z}_3 (t, \xi ) \\ & = -i \mathbf{1}_{Q_{N, \ell }} (\xi ) e^{-it |\xi |^2} \int _0^t \operatorname *{\int }_{\xi = \xi _1 - \xi _2 + \xi _3} e^{i t' \Phi (\bar{\xi })} \widehat{\phi ^\omega } (\xi _1) \overline{\widehat{\phi ^\omega } (\xi _2)} \widehat{\phi ^\omega } (\xi _3) d\xi _1 d\xi _2 dt', {\tag{3.6}} \end{align}$$

where the phase function $\Phi (\bar{\xi })$ is given by

$$\begin{align} \Phi (\bar{\xi }) = \Phi (\xi , \xi _1, \xi _2, \xi _3) = |\xi |^2 - |\xi _1|^2 + |\xi _2|^2 - |\xi _3|^2 = 2\langle \xi - \xi _1, \xi - \xi _3 \rangle _{\mathbb{R}^3}. {\tag{3.7}} \end{align}$$

Then, it follows that the only non-trivial contribution to 3.6 appears if

$$\begin{align} \xi _1 \in (N+100\ell ) e_1 + \ell Q, \qquad \xi _2 \in N e_1 + \ell Q, \qquad \xi _3 \in N e_1 + 100\ell e_2 + \ell Q {\tag{3.8}} \end{align}$$

(up to the permutation $\xi _1 \leftrightarrow \xi _3$). In this case, we have $|\Phi (\bar{\xi })| \ll 1$ and thus

$$\begin{align} \operatorname *{Re}e^{i t' \Phi (\bar{\xi })} \geq \frac{1}{2} {\tag{3.9}} \end{align}$$

for all $t' \in [0, 1]$.

Now, recall the following lemma on the convolution.

Lemma 3.1.

There exists $c>0$ such that

$$\begin{align*} \mathbf{1}_{a + \ell Q}* \mathbf{1}_{b + \ell Q} (\xi )\geq c \ell ^3 \mathbf{1}_{a+b+\ell Q}(\xi ) \end{align*}$$

for all $a, b, \xi \in \mathbb{R}^3$.

By applying Lemma 3.1 to 3.6 with 3.8 and 3.9, we obtain

$$\begin{align*} |\mathbf{1}_{Q_{N, \ell }} (\xi ) \widehat{z}_3 (t, \xi ) | \gtrsim t \ell ^6 \mathbf{1}_{Q_{N, \ell }} (\xi ). \end{align*}$$

Therefore, for any $\sigma > 0$, we have

$$\begin{align*} \| z_3\|_{X^\sigma ([0, 1])} \gtrsim \| z_3 \|_{L^\infty _t([0, 1]; H^\sigma )} \gtrsim \ell ^\frac{15}{2} N^\sigma \longrightarrow \infty \end{align*}$$

as $N \to \infty$, while $\| \phi \|_{L^2} \sim \ell ^\frac{3}{2}$ remains bounded. This in particular implies that when $s = 0$, the estimate 1.11 cannot hold for any $\sigma > 0$. This proves part (ii).

■

Remark 3.2.

It follows from the proof of Proposition 1.2(i) that

$$\begin{align*} \big \|\mathcal{I}( u_1, u_2, u_3)\big \|_{X^\sigma ([0, 1])} \lesssim \prod _{j = 1}^3 \| u_j \|_{A^s_3(T)}, \end{align*}$$

where $\mathcal{I}( u_1, u_2, u_3)$ is as in 1.17. In particular, the left-hand side of 1.18 is finite for $u_1, u_2, u_3 \in \mathcal{R}^s$. The only probabilistic component in the proof of Proposition 1.2(i) appears in applying Lemmas 2.1 and 2.2 to control the $A^s_3(T)$-norm of $z_1$ in terms of the $H^s$-norm of $\phi$. In this sense, we exploit the randomization only at the linear level.

On the one hand, Proposition 1.2 shows that $z_3$ controls almost $2s$ derivatives. On the other hand, we need to measure $z_3$ in the $X^{2s-}$-norm, which controls only the admissible space-time Lebesgue norms (with $2s-$ derivatives) via Lemma 2.6. The following lemma breaks this rigidity by giving up a control on derivatives. In particular, it allows us to control a wider range of space-time Lebesgue norms of $z_3$. The main idea is to use the dispersive estimate for the linear Schrödinger operator:

$$\begin{align} \| S(t) f \|_{L^r_x} \lesssim |t|^{-\frac{3}{2}(1 - \frac{2}{r})} \| f\|_{L^{r'}_x}. {\tag{3.10}} \end{align}$$

This allows us to reduce the analysis to a product of the random linear solution $z_1 = S(t) \phi ^\omega$ and apply Lemma 2.1.

Lemma 3.3.

Let $s \geq 0$. Given $\phi \in H^s(\mathbb{R}^3)$, let $\phi ^\omega$ be its Wiener randomization defined in 1.3. Then, for any finite $q, r \geq 1$, we have

$$\begin{align} \| \mathbf{P}_N z_3\|_{L^q_T L^r_x} \lesssim \begin{cases} T^{\frac{3}{r} - \frac{1}{2}}\| z_1\|_{L^{3q}_T L^{3r'}_x}^3 & \text{when } 1 \leq r < 6, \\ T^{0+} N^{\frac{1}{2} - \frac{3}{r}+} \| z_1\|_{L^{3q}_T L^{\frac{18}{5}+}_x}^3 & \text{when } r \geq 6 \\ \end{cases} {\tag{3.11}} \end{align}$$

for any $T > 0$ and $N \in 2^{\mathbb{N}_0}$. Note that the right-hand side of 3.11 is almost surely finite thanks to the probabilistic Strichartz estimate (Lemma 2.1).

Proof.

We first consider the case $r < 6$. From 3.1 and 3.10, we have

$$\begin{align} \| \mathbf{P}_N z_3\|_{L^q_T L^r_x} & \leq \bigg \|\int _0^t \|\mathbf{P}_N S(t - t') |z_1|^2 z_1(t')\|_{L^r_x} dt'\bigg \|_{L^q_T}\\ & \lesssim \bigg \|\int _0^t \frac{1}{|t - t'|^{\frac{3}{2} - \frac{3}{r}}} \| z_1(t')\|_{L^{3r'}_x}^3 dt'\bigg \|_{L^q_T}\\ & \lesssim T^{\frac{3}{r} - \frac{1}{2}}\| z_1\|_{L^{3q}_T L^{3r'}_x}^3. {\tag{3.12}} \end{align}$$

When $r \geq 6$, we proceed as in 3.12, but we apply Sobolev’s inequality before applying 3.10:

$$\begin{align*} \| \mathbf{P}_N z_3\|_{L^q_T L^r_x} & \leq \bigg \|\int _0^t \|\mathbf{P}_N S(t - t') |z_1|^2 z_1(t')\|_{L^r_x} dt'\bigg \|_{L^q_T}\\ & \lesssim N^{\frac{1}{2} - \frac{3}{r}+} \bigg \|\int _0^t \|\mathbf{P}_N S(t - t') |z_1|^2 z_1(t')\|_{L^{6-}_x} dt'\bigg \|_{L^q_T}\\ & \lesssim N^{\frac{1}{2} - \frac{3}{r}+} \bigg \|\int _0^t \frac{1}{|t - t'|^{1-}} \| z_1(t')\|_{L^{\frac{18}{5}+}_x}^3 dt'\bigg \|_{L^q_T}\\ & \lesssim T^{0+} N^{\frac{1}{2} - \frac{3}{r}+} \| z_1\|_{L^{3q}_T L^{\frac{18}{5}+}_x}^3 . \end{align*}$$

This completes the proof of Lemma 3.3.

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4. On the higher order terms

In this section, we study the regularity properties of the higher order terms.

4.1. On the third order term $z_5$

In this subsection, we study the third order term:

$$\begin{align} z_5 = -i \sum _{\substack{j_1 + j_2 + j_3 = 5\\j_1, j_2, j_3 \in \{1, 3\} }}\int _0^t S(t - t') z_{j_1} \overline{z_{j_2}}z_{j_3} (t') dt'. {\tag{4.1}} \end{align}$$

The following lemma shows that the third order term $z_5$ enjoys a gain of extra $\frac{1}{2}s$ derivative as compared to the second order term $z_3$ (Proposition 1.2).

Lemma 4.1.

Given $0 < s < \frac{1}{2}$, let $\phi ^\omega$ be the Wiener randomization of $\phi \in H^s(\mathbb{R}^3)$ defined in 1.3. Then, for any $\sigma < \frac{5}{2} s$, we have

$$\begin{align*} z_5 \in X^\sigma _{\operatorname {loc}}, \end{align*}$$

almost surely. In particular, there exist an almost surely finite constant $C(\omega , \|\phi \|_{H^s} ) > 0$ and $\theta > 0$ such that

$$\begin{align*} \| z_5 \|_{X^{\sigma }([0, T])} \leq T^\theta C(\omega , \|\phi \|_{H^s}) \end{align*}$$

for any $T > 0$.

Proof.

First, note that the only possible combination for $(j_1, j_2, j_3)$ in 4.1 is $(1, 1, 3)$ up to permutations. The complex conjugate does not play any role in the subsequent analysis, and hence we drop the complex conjugate sign and simply study

$$\begin{align} z_5 \sim \int _0^t S(t - t') z_{1} z_1 z_{3} (t') dt'. {\tag{4.2}} \end{align}$$

By Lemma 2.5, it suffices to prove that

$$\begin{align} \bigg | \int _0^T \int _{\mathbb{R}^3} \langle \nabla \rangle ^\sigma ( z_1 z_1 z_3 )w dx dt\bigg | \leq T^\theta C(\omega , \|\phi \|_{H^s}) {\tag{4.3}} \end{align}$$

for some almost surely finite constant $C(\omega , \|\phi \|_{H^s} ) > 0$ and $\theta > 0$, where $\|w\|_{Y^{0}_T} \leq 1$. Define $A^s_5(T)$ by

$$\begin{equation*} \|z_1\|_{A^s_5(T)}: = \max \Big ( \| \langle \nabla \rangle ^s z_{1} \|_{L^{5+}_T L^{5}_x}, \| z_1(0)\|_{H^s}, \Big ). \end{equation*}$$

Then, by applying the dyadic decomposition, it suffices to prove that

$$\begin{align} \bigg | \int _0^T \int _{\mathbb{R}^3} N_{\max }^\sigma \mathbf{P}_{N_1} z_1 \cdot \mathbf{P}_{N_2} z_1 & \cdot \mathbf{P}_{N_3} z_3 \cdot \mathbf{P}_{N_4} w \, dx dt\bigg | \\ & \leq T^\theta N_{\max }^{0-} C(\|z_1\|_{A^s_5(T)}, \|z_3\|_{X^{2s-}_T}) {\tag{4.4}} \end{align}$$

for all $N_1$, …, $N_4 \in 2^{\mathbb{N}_0}$. Once we prove 4.4, the desired estimate 4.3 follows from summing over dyadic blocks and applying Lemmas 2.1 and 2.2 and Proposition 1.2. In the following, we fix

$$\begin{align*} 0 < s < \tfrac{1}{2}. \end{align*}$$

Without loss of generality, we assume that $N_1 \geq N_2$.

Case 1 ($N_1 \sim N_2 \sim N_{\max }$).

Subcase 1.a ($N_3 \ll N_1^\frac{1}{2}$).

By the bilinear Strichartz estimate and Lemma 2.6, we have