On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on $\mathbb{R}^d$, $d \geq 3$

Abstract

We consider the Cauchy problem of the cubic nonlinear Schrödinger equation (NLS) $: i \partial _t u + \Delta u = \pm |u|^{2}u$ on $\mathbb{R}^d$, $d \geq 3$, with random initial data and prove almost sure well-posedness results below the scaling-critical regularity $s_\text{crit} = \frac{d-2}{2}$. More precisely, given a function on $\mathbb{R}^d$, we introduce a randomization adapted to the Wiener decomposition, and, intrinsically, to the so-called modulation spaces. Our goal in this paper is three-fold. (i) We prove almost sure local well-posedness of the cubic NLS below the scaling-critical regularity along with small data global existence and scattering. (ii) We implement a probabilistic perturbation argument and prove ‘conditional’ almost sure global well-posedness for $d = 4$ in the defocusing case, assuming an a priori energy bound on the critical Sobolev norm of the nonlinear part of a solution; when $d \ne 4$, we show that conditional almost sure global well-posedness in the defocusing case also holds under an additional assumption of global well-posedness of solutions to the defocusing cubic NLS with deterministic initial data in the critical Sobolev regularity. (iii) Lastly, we prove global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.

1. Introduction

1.1. Background

In this paper, we consider the Cauchy problem of the cubic nonlinear Schrödinger equation (NLS) on $\mathbb{R}^d$, $d\geq 3$:

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

where $\mathcal{N}(u) := |u|^2 u$. The cubic NLS 1.1 has been studied extensively from both the theoretical and applied points of view. Our main focus is to study well-posedness of 1.1 with random and rough initial data.

It is well known that the cubic NLS 1.1 enjoys the dilation symmetry. More precisely, if $u(t, x)$ is a solution to 1.1 on $\mathbb{R}^d$ with an initial condition $u_0$, then

$$\begin{align} u_\mu (t, x) := \mu ^{-1} u (\mu ^{-2}t, \mu ^{-1}x) {\tag{1.2}} \end{align}$$

is also a solution to 1.1 with the $\mu$-scaled initial condition $u_{0, \mu }(x) := \mu ^{-1} u_0 (\mu ^{-1}x)$. Associated to this dilation symmetry, there is the so-called scaling-critical Sobolev index $s_\text{crit} := \frac{d-2}{2}$ such that the homogeneous $\dot{H}^{s_\text{crit}}$-norm is invariant under this dilation symmetry. In general, we have

$$\begin{align} \|u_{0, \mu }\|_{\dot{H}^s(\mathbb{R}^d)} = \mu ^{\frac{d-2}{2} - s}\|u_0\|_{\dot{H}^s(\mathbb{R}^d)} . {\tag{1.3}} \end{align}$$

If an initial condition $u_0$ is in $H^s(\mathbb{R}^d)$, we say that the Cauchy problem 1.1 is subcritical, critical, or supercritical, depending on whether $s > s_\text{crit}$, $s = s_\text{crit}$, or $s < s_\text{crit}$, respectively.

Let us first discuss the (sub)critical regime. In this case, 1.1 is known to be locally well-posed. See Cazenave-Weissler 17 for local well-posedness of 1.1 in the critical Sobolev spaces. As is well known, the conservation laws play an important role in discussing long time behavior of solutions. There are three known conservation laws for the cubic NLS 1.1:

$$\begin{align*} \text{Mass: }& M[u](t) := \int _{\mathbb{R}^d} |u(t, x)|^2 dx,\\ \text{Momentum: }& P[u](t) := \operatorname *{Im}\int _{\mathbb{R}^d} u (t, x)\overline{\nabla u(t, x)} dx,\\ \text{Hamiltonian: } & H[u](t) := \frac{1}{2} \int _{\mathbb{R}^d} |\nabla u(t, x)|^2 dx \pm \frac{1}{4} \int _{\mathbb{R}^d} |u(t, x)|^4 dx. \end{align*}$$

The Hamiltonian is also referred to as the energy. In view of the conservation of the energy, the cubic NLS is called energy-subcritical when $d \leq 3$ ($s_\text{crit} < 1$), energy-critical when $d = 4$ ($s_\text{crit} = 1$), and energy-supercritical when $d \geq 5$ ($s_\text{crit} > 1$), respectively.

In the following, let us discuss the known results on the global-in-time behavior of solutions to the defocusing NLS, corresponding to the $+$ sign in 1.1, in high dimensions $d\geq 3$. When $d = 4$, the Hamiltonian is invariant under the scaling 1.2 and plays a crucial role in the global well-posedness theory. Indeed, Ryckman-Vişan 53 proved global well-posedness and scattering for the defocusing cubic NLS on $\mathbb{R}^4$. See also Vişan 60. When $d \ne 4$, there is no known scaling invariant positive conservation law for 1.1 in high dimensions $d\geq 3$. This makes it difficult to study the global-in-time behavior of solutions, in particular, in the scaling-critical regularity. There are, however, ‘conditional’ global well-posedness and scattering results as we describe below. When $d = 3$ ($s_\text{crit} = \frac{1}{2}$), Kenig-Merle 35 applied the concentration compactness and rigidity method developed in their previous paper 34 and proved that if $u \in L^\infty _t \dot{H}_x^\frac{1}{2}(I\times \mathbb{R}^3)$, where $I$ is a maximal interval of existence, then $u$ exists globally in time and scatters. For $d\geq 5$, the cubic NLS is supercritical with respect to any known conservation law. Nonetheless, motivated by a similar result of Kenig-Merle 36 on radial solutions to the energy-supercritical nonlinear wave equation (NLW) on $\mathbb{R}^3$, Killip-Vişan 39 proved that if $u \in L^\infty _t \dot{H}_x^{s_\text{crit}}(I\times \mathbb{R}^d)$, where $I$ is a maximal interval of existence, then $u$ exists globally in time and scatters. Note that the results in 35 and 39 are conditional in the sense that they assume an a priori control on the critical Sobolev norm. The question of global well-posedness and scattering without any a priori assumption remains a challenging open problem for $d = 3$ and $d \geq 5$.

So far, we have discussed well-posedness in the (sub)critical regularity. In particular, the cubic NLS 1.1 is locally well-posed in the (sub)critical regularity, i.e. $s \geq s_\text{crit}$. In the supercritical regime, i.e. $s < s_\text{crit}$, on the contrary, 1.1 is known to be ill-posed. See 1111618. In the following, however, we consider the Cauchy problem 1.1 with initial data in $H^s(\mathbb{R}^d)$, $s < s_\text{crit}$ in a probabilistic manner. More precisely, given a function $\phi \in H^s(\mathbb{R}^d)$ with $s < s_\text{crit}$, we introduce a randomization $\phi ^\omega$ and prove almost sure well-posedness of 1.1.

In studying the Gibbs measure for the defocusing (Wick ordered) cubic NLS on $\mathbb{T}^2$, Bourgain 6 considered random initial data of the form:

$$\begin{equation} u_0^\omega (x) = \sum _{n \in \mathbb{Z}^2} \frac{g_n(\omega )}{\sqrt {1+|n|^2}}e^{i n \cdot x}, {\tag{1.4}} \end{equation}$$

where $\{g_n\}_{n \in \mathbb{Z}^2}$ is a sequence of independent standard complex-valued Gaussian random variables. The function 1.4 represents a typical element in the support of the Gibbs measure, more precisely, in the support of the Gaussian free field on $\mathbb{T}^2$ associated to this Gibbs measure, and is critical with respect to the scaling. With a combination of deterministic PDE techniques and probabilistic arguments, Bourgain showed that the (Wick ordered) cubic NLS on $\mathbb{T}^2$ is well-posed almost surely with respect to the random initial data 1.4. In the context of the cubic NLW on a three-dimensional compact Riemannian manifold $M$, Burq-Tzvetkov 14 considered the Cauchy problem with a more general class of random initial data. Given an eigenfunction expansion $u_0(x) = \sum _{n = 1}^\infty c_n e_n(x) \in H^s(M)$ of an initial condition,⁠¹ where $\{e_n\}_{n = 1}^\infty$ is an orthonormal basis of $L^2(M)$ consisting of the eigenfunctions of the Laplace-Beltrami operator, they introduced a randomization $u_0^\omega$ by

¹

For NLW, one needs to specify $(u, \partial _tu)|_{t = 0}$ as an initial condition. For simplicity of presentation, we only discuss $u|_{t = 0}$.

✖

$$\begin{equation} u_0^\omega (x) = \sum _{n = 1}^\infty g_n (\omega ) c_n e_n(x). {\tag{1.5}} \end{equation}$$

Here, $\{g_n\}_{n = 1}^\infty$ is a sequence of independent mean-zero random variables with a uniform bound on the fourth moments. Then, they proved almost sure local well-posedness with random initial data of the form 1.5 for $s \geq \frac{1}{4}$. Since the scaling-critical Sobolev index for this problem is $s_\text{crit}=\frac{1}{2}$, this result allows us to take initial data below the critical regularity and still construct solutions upon randomization of the initial data. We point out that the randomized function $u_0^\omega$ in 1.5 has the same Sobolev regularity as the original function $u_0$ and is not smoother, almost surely. However, it enjoys a better integrability, which allows one to prove improvements of Strichartz estimates. (See Lemmata 2.2 and 2.3 below.) Such an improvement on integrability for random Fourier series is known as Paley-Zygmund’s theorem 49. See also Kahane 32 and Ayache-Tzvetkov 2. There are several works on Cauchy problems of evolution equations with random data that followed these results, including some on almost sure global well-posedness: 79101213152021222342434445515258.

1.2. Randomization adapted to the Wiener decomposition and modulation spaces

Many of the results mentioned above are on compact domains, where there is a countable basis of eigenfunctions of the Laplacian and thus there is a natural way to introduce a randomization. On $\mathbb{R}^d$, there is no countable basis of $L^2(\mathbb{R}^d)$ consisting of eigenfunctions of the Laplacian. Randomizations have been introduced with respect to some other countable bases of $L^2(\mathbb{R}^d)$, for example, a countable basis of eigenfunctions of the Laplacian with a confining potential such as the harmonic oscillator $\Delta - |x|^2$, leading to a careful study of properties of eigenfunctions. In this paper, our goal is to introduce a simple and natural randomization for functions on $\mathbb{R}^d$. For this purpose, we first review some basic notions related to the so-called modulation spaces of time-frequency analysis 28.

The modulation spaces were introduced by Feichtinger 24 in the early eighties. The groundwork theory regarding these spaces of time-frequency analysis was then established in joint collaboration with Gröchenig 2526. The modulation spaces arise from a uniform partition of the frequency space, commonly known as the Wiener decomposition 61: $\mathbb{R}^d = \bigcup _{n \in \mathbb{Z}^d} Q_n$, where $Q_n$ is the unit cube centered at $n \in \mathbb{Z}^d$. The Wiener decomposition of $\mathbb{R}^d$ induces a natural uniform decomposition of the frequency space of a signal via the (nonsmooth) frequency-uniform decomposition operators $\mathcal{F}^{-1}\chi _{Q_n}\mathcal{F}$. Here, $\mathcal{F}u=\widehat{u}$ denotes the Fourier transform of a distribution $u$. The drawback of this approach is the roughness of the characteristic functions $\chi _{Q_n}$, but this issue can easily be fixed by smoothing them out appropriately. We have the following definition of the (weighted) modulation spaces $M^{p, q}_s$. Let $\psi \in \mathcal{S}(\mathbb{R}^d)$ such that

$$\begin{equation} \operatorname *{supp}\psi \subset [-1,1]^d \qquad \text{and} \qquad \sum _{n \in \mathbb{Z}^d} \psi (\xi -n) \equiv 1 \ \text{ for any }\xi \in \mathbb{R}^d. {\tag{1.6}} \end{equation}$$

Let $0<p,q\leq \infty$ and $s\in \mathbb{R}$; $M^{p, q}_s$ consists of all tempered distributions $u\in \mathcal{S}'(\mathbb{R}^d)$ for which the (quasi) norm

$$\begin{equation} \|u\|_{M_s^{p, q}(\mathbb{R}^d)} := \big \| \langle n \rangle ^s \|\psi (D-n) u \|_{L_x^p(\mathbb{R}^d)} \big \|_{\ell ^q_n(\mathbb{Z}^d)} {\tag{1.7}} \end{equation}$$

is finite. Note that $\psi (D-n)u(x)=\int _{\mathbb{R}^d} \psi (\xi -n)\widehat{u} (\xi )e^{2\pi ix\cdot \xi }\, d\xi$ is just a Fourier multiplier operator with symbol $\chi _{Q_n}$ conveniently smoothed.

It is worthwhile to compare the definition 1.7 with that of the Besov spaces. Let $\varphi _0, \varphi \in \mathcal{S}(\mathbb{R}^d)$ such that $\operatorname *{supp}\varphi _0 \subset \{ |\xi | \leq 2\}$, $\operatorname *{supp}\varphi \subset \{ \frac{1}{2}\leq |\xi | \leq 2\}$, and $\varphi _0(\xi ) + \sum _{j = 1}^\infty \varphi (2^{-j}\xi ) \equiv 1.$ With $\varphi _j(\xi ) = \varphi (2^{-j}\xi )$, we define the (inhomogeneous) Besov spaces $B_s^{p, q}$ via the norm

$$\begin{equation} \|u\|_{B^s_{p, q}(\mathbb{R}^d) } = \big \| 2^{js} \|\varphi _j(D) u \|_{L^p(\mathbb{R}^d)} \big \|_{\ell ^q_j(\mathbb{Z}_{\geq 0})}. {\tag{1.8}} \end{equation}$$

There are several known embeddings between Besov, Sobolev, and modulation spaces. See, for example, Okoudjou 47, Toft 59, Sugimoto-Tomita 55, and Kobayashi-Sugimoto 40.

Now, given a function $\phi$ on $\mathbb{R}^d$, we have

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

where $\psi (D-n)$ is defined above. This decomposition leads to a randomization of $\phi$ that is very natural from the perspective of time-frequency analysis associated to modulation spaces. Let $\{g_n\}_{n \in \mathbb{Z}^d}$ be a sequence of independent mean zero complex-valued random variables on a probability space $(\Omega , \mathcal{F}, P)$, where the real and imaginary parts of $g_n$ are independent and endowed with probability distributions $\mu _n^{(1)}$ and $\mu _n^{(2)}$. Then, we can define the Wiener randomization of $\phi$ by

$$\begin{equation} \phi ^\omega : = \sum _{n \in \mathbb{Z}^d} g_n (\omega ) \psi (D-n) \phi . {\tag{1.9}} \end{equation}$$

Almost simultaneously with our first paper 4, Lührmann-Mendelson 42 also considered a randomization of the form 1.9 (with cubes $Q_n$ being substituted by appropriately localized balls) in the study of NLW on $\mathbb{R}^3$. See Remark 1.6 below. For a similar randomization used in the study of the Navier-Stokes equations, see the work of Zhang and Fang 63. We would like to stress again, however, that our reason for considering the randomization of the form 1.9 comes from its connection to time-frequency analysis. See also our previous papers 3 and 4.

In the sequel, we make the following assumption on the distributions $\mu _n^{(j)}$: there exists $c>0$ such that

$$\begin{equation} \bigg | \int _{\mathbb{R}} e^{\gamma x } d \mu _n^{(j)}(x) \bigg | \leq e^{c\gamma ^2} {\tag{1.10}} \end{equation}$$

for all $\gamma \in \mathbb{R}$, $n \in \mathbb{Z}^d$, $j = 1, 2$. Note that 1.10 is satisfied by standard complex-valued Gaussian random variables, standard Bernoulli random variables, and any random variables with compactly supported distributions.

It is easy to see that, if $\phi \in H^s(\mathbb{R}^d)$, then the randomized function $\phi ^\omega$ is almost surely in $H^s(\mathbb{R}^d)$. While there is no smoothing upon randomization in terms of differentiability in general, this randomization behaves better under integrability; if $\phi \in L^2(\mathbb{R}^d)$, then the randomized function $\phi ^\omega$ is almost surely in $L^p(\mathbb{R}^d)$ for any finite $p \geq 2$. As a result of this enhanced integrability, we have improvements of the Strichartz estimates. See Lemmata 2.2 and 2.3. These improved Strichartz estimates play an essential role in proving probabilistic well-posedness results, which we describe below.

1.3. Main results

Recall that the scaling-critical Sobolev index for the cubic NLS on $\mathbb{R}^d$ is $s_\text{crit} = \frac{d-2}{2}$. In the following, we take $\phi \in H^s(\mathbb{R}^d) \setminus H^{s_\text{crit}} (\mathbb{R}^d)$ for some range of $s < s_\text{crit}$, that is, below the critical regularity. Then, we consider the well-posedness problem of 1.1 with respect to the randomized initial data $\phi ^\omega$ defined in 1.9.

For $d \geq 3$, define $s_d$ by

$$\begin{align} s_d := \frac{d-1}{d+1}\cdot s_\text{crit} = \frac{d-1}{d+1} \cdot \frac{d-2}{2}. {\tag{1.11}} \end{align}$$

Note that $s_d <s_\text{crit}$ and $\frac{s_d}{s_\text{crit}} \to 1$ as $d \to \infty$. Throughout the paper, we use $S(t)= e^{it\Delta }$ to denote the linear propagator of the Schrödinger group.

We are now ready to state our main results.

Theorem 1.1 (Almost sure local well-posedness).

Let $d \geq 3$ and $s > s_d$. Given $\phi \in H^s(\mathbb{R}^d)$, let $\phi ^\omega$ be its Wiener randomization defined in 1.9, satisfying 1.10. Then, the cubic NLS 1.1 on $\mathbb{R}^d$ is almost surely locally well-posed with respect to the randomization $\phi ^\omega$ as initial data. More precisely, there exist $C, c, \gamma >0$ such that for each $0< T\ll 1$, there exists a set $\Omega _T \subset \Omega$ with the following properties:

(i)

$P(\Omega _T^c) < C \exp \big (-\frac{c}{T^\gamma \|\phi \|_{H^s}^2 }\big )$.

(ii)

For each $\omega \in \Omega _T$, there exists a (unique) solution $u$ to 1.1 with $u|_{t = 0} = \phi ^\omega$ in the class$$\begin{equation*} S(t) \phi ^\omega + C([-T, T]: H^{\frac{d-2}{2}} (\mathbb{R}^d)) \subset C([-T, T]:H^s(\mathbb{R}^d)). \end{equation*}$$

We prove Theorem 1.1 by considering the equation satisfied by the nonlinear part of a solution $u$. Namely, let $z (t) = z^\omega (t) : = S(t) \phi ^\omega$ and $v(t) := u(t) - S(t) \phi ^\omega$ be the linear and nonlinear parts of $u$, respectively. Then, 1.1 is equivalent to the following perturbed NLS:

$$\begin{equation} \begin{cases} i \partial _tv + \Delta v = \pm |v + z|^2(v+z),\\ v|_{t = 0} = 0. \end{cases} {\tag{1.12}} \end{equation}$$

We reduce our analysis to the Cauchy problem 1.12 for $v$, viewing $z$ as a random forcing term. Note that such a point of view is common in the study of stochastic PDEs. As a result, the uniqueness in Theorem 1.1 refers to uniqueness of the nonlinear part $v(t) = u(t) - S(t) \phi ^\omega$ of a solution $u$.

The proof of Theorem 1.1 is based on the fixed point argument involving the variants of the $X^{s, b}$-spaces adapted to the $U^p$- and $V^p$-spaces introduced by Koch, Tataru, and their collaborators 293041. See Section 3 for the basic definitions and properties of these function spaces. The main ingredient is the local-in-time improvement of the Strichartz estimates (Lemma 2.2) and the refinement of the bilinear Strichartz estimate (Lemma 3.5 (ii)). We point out that, although $\phi$ and its randomization $\phi ^\omega$ have a supercritical Sobolev regularity, the randomization essentially makes the problem subcritical, at least locally in time, and therefore, one can also prove Theorem 1.1 only with the classical subcritical $X^{s, b}$-spaces, $b > \frac{1}{2}$. See 4 for the result when $d = 4$.

Next, we turn our attention to the global-in-time behavior of the solutions constructed in Theorem 1.1. The key nonlinear estimate in the proof of Theorem 1.1 combined with the global-in-time improvement of the Strichartz estimates (Lemma 2.3) yields the following result on small data global well-posedness and scattering.

Theorem 1.2 (Probabilistic small data global well-posedness and scattering).

Let $d\geq 3$ and $s \in ( s_d, s_\text{crit}]$, where $s_d$ is as in 1.11. Given $\phi \in H^s(\mathbb{R}^d)$, let $\phi ^\omega$ be its Wiener randomization defined in 1.9, satisfying 1.10. Then, there exist $C, c >0$ such that for each $0<\varepsilon \ll 1$, there exists a set $\Omega _\varepsilon \subset \Omega$ with the following properties:

(i)

$P(\Omega _\varepsilon ^c) \leq C \exp \big (-\frac{c}{\varepsilon ^2 \|\phi \|_{H^s}^2 }\big ) \to 0$ as $\varepsilon \to 0$.

(ii)

For each $\omega \in \Omega _\varepsilon$, there exists a (unique) global-in-time solution $u$ to 1.1 with$$\begin{equation*} u|_{t = 0} = \varepsilon \phi ^\omega \end{equation*}$$

in the class$$\begin{equation*} \varepsilon S(t) \phi ^\omega + C(\mathbb{R}: H^{\frac{d-2}{2}} (\mathbb{R}^d)) \subset C(\mathbb{R}:H^s(\mathbb{R}^d)). \end{equation*}$$

(iii)

We have scattering for each $\omega \in \Omega _\varepsilon$. More precisely, for each $\omega \in \Omega _\varepsilon$, there exists $v_+^\omega \in H^\frac{d-2}{2}(\mathbb{R}^d)$ such that$$\begin{equation*} \| u(t) - S(t)(\varepsilon \phi ^\omega + v_+^\omega ) \|_{H^\frac{d-2}{2}(\mathbb{R}^d)} \to 0 \end{equation*}$$

as $t \to \infty$. A similar statement holds for $t \to -\infty$.

In general, a local well-posedness result in a critical space is often accompanied by small data global well-posedness and scattering. In this sense, Theorem 1.2 is an expected consequence of Theorem 1.1, since, in our construction, the nonlinear part $v$ lies in the critical space $H^\frac{d-2}{2}(\mathbb{R}^d)$. The next natural question is probabilistic global well-posedness for large data. In order to state our result, we need to make several hypotheses. The first hypothesis is on a probabilistic a priori energy bound on the nonlinear part $v$.

Hypothesis (A).

Given any $T, \varepsilon > 0$, there exist $R = R(T, \varepsilon )$ and $\Omega _{T, \varepsilon } \subset \Omega$ such that

(i)

$P(\Omega _{T, \varepsilon }^c) < \varepsilon$, and

(ii)

if $v = v^\omega$ is the solution to 1.12 for $\omega \in \Omega _{T, \varepsilon }$, then the following a priori energy estimate holds:$$\begin{equation} \|v(t) \|_{L^\infty ([0, T]; H^\frac{d-2}{2}(\mathbb{R}^d))} \leq R(T, \varepsilon ). {\tag{1.13}} \end{equation}$$

Note that Hypothesis (A) does not refer to existence of a solution $v = v^\omega$ on $[0, T]$ for given $\omega \in \Omega _{T, \varepsilon }$. It only hypothesizes the a priori energy bound 1.13, just like the usual conservation laws. It may be possible to prove 1.13 independently from the argument presented in this paper. Such a probabilistic a priori energy estimate is known, for example, for the cubic NLW. See Burq-Tzvetkov 15. We point out that the upper bound $R(T, \varepsilon )$ in 15 tends to $\infty$ as $T\to \infty$. See also 50.

The next hypothesis is on global existence and space-time bounds of solutions to the cubic NLS 1.1 with deterministic initial data belonging to the critical space $H^\frac{d-2}{2}(\mathbb{R}^d)$.

Hypothesis (B).

Given any $w_0 \in H^\frac{d-2}{2}(\mathbb{R}^d)$, there exists a global solution $w$ to the defocusing cubic NLS 1.1 with $w|_{t = 0} = w_0$. Moreover, there exists a function $C:[0, \infty )\times [0, \infty )\to [0, \infty )$ which is nondecreasing in each argument such that

$$\begin{equation} \| w\|_{L^{d+2}_{t, x}([0, T]\times \mathbb{R}^d)} \leq C\big (\|w_0\|_{H^\frac{d-2}{2}(\mathbb{R}^d)}, T\big ) {\tag{1.14}} \end{equation}$$

for any $T > 0$.

Note that when $d = 4$, Hypothesis (B) is known to be true for any $T> 0$ thanks to the global well-posedness result by Ryckman-Vişan 53 and Vişan 60. For other dimensions $d \geq 3$ with $d \ne 4$, it is not known whether Hypothesis (B) holds. Let us compare 1.14 and the results in 35 and 39. Assuming that $w \in L^\infty _t \dot{H}_x^{s_\text{crit}}(I_*\times \mathbb{R}^d)$, where $I_*$ is a maximal interval of existence, it was shown in 35 and 39 that $I_* = \mathbb{R}$ and

$$\begin{equation} \| w\|_{L^{d+2}_{t, x}(\mathbb{R}\times \mathbb{R}^d)} \leq C\Big (\|w\|_{L^\infty _t \dot{H}_x^\frac{d-2}{2}(\mathbb{R}\times \mathbb{R}^d)}\Big ). {\tag{1.15}} \end{equation}$$

We point out that Hypothesis (B) is not directly comparable to the results in 3539 in the following sense. On the one hand, by assuming that $w \in L^\infty _t \dot{H}_x^{s_\text{crit}}(I_*\times \mathbb{R}^d)$, the results in 3539 yield the global-in-time bound 1.15, while Hypothesis (B) assumes the bound 1.14 only for each finite time $T>0$ and does not assume a global-in-time bound. On the other hand, 1.14 is much stronger than 1.15 in the sense that the right-hand side of 1.14 depends only on the size of an initial condition $w_0$, while the right-hand side of 1.15 depends on the global-in-time $L^\infty _t \dot{H}^\frac{d-2}{2}_x$-bound of the solution $w$. Hypothesis (B), just like Hypothesis (A), is of independent interest from Theorem 1.3 below and is closely related to the fundamental open problem of global well-posedness and scattering for the defocusing cubic NLS 1.1 for $d = 3$ and $d \geq 5$.

We now state our third theorem on almost sure global well-posedness of the cubic NLS under Hypotheses (A) and (B). We restrict ourselves to the defocusing NLS in the next theorem.

Theorem 1.3 (Conditional almost sure global well-posedness).

Let $d \geq 3$ and $s \in ( s_d, s_\text{crit}]$, where $s_d$ is as in 1.11. Assume Hypothesis (A). Furthermore, assume Hypothesis (B) if $d \ne 4$. Given $\phi \in H^s(\mathbb{R}^d)$, let $\phi ^\omega$ be its Wiener randomization defined in 1.9, satisfying 1.10. Then, the defocusing cubic NLS 1.1 on $\mathbb{R}^d$ is almost surely globally well-posed with respect to the randomization $\phi ^\omega$ as initial data. More precisely, there exists a set $\Sigma \subset \Omega$ with $P(\Sigma ) = 1$ such that, for each $\omega \in \Sigma$, there exists a (unique) global-in-time solution $u$ to 1.1 with $u|_{t = 0} = \phi ^\omega$ in the class

$$\begin{equation*} S(t) \phi ^\omega + C(\mathbb{R}: H^{\frac{d-2}{2}} (\mathbb{R}^d)) \subset C(\mathbb{R}:H^s(\mathbb{R}^d)). \end{equation*}$$

The main tool in the proof of Theorem 1.3 is a perturbation lemma for the cubic NLS (Lemma 7.1). Assuming a control on the critical norm (Hypothesis (A)), we iteratively apply the perturbation lemma in the probabilistic setting to show that a solution can be extended to a time depending only on the critical norm. Such a perturbative approach was previously used by Tao-Vişan-Zhang 57 and Killip-Vişan with the second and third authors 37. The novelty of Theorem 1.3 is an application of such a technique in the probabilistic setting. While there is no invariant measure for the nonlinear evolution in our setting, we exploit the quasi-invariance property of the distribution of the linear solution $S(t) \phi ^\omega$. See Remark 8.2. Our implementation of the proof of Theorem 1.3 is sufficiently general that it can be easily applied to other equations. See 50 in the context of the energy-critical NLW on $\mathbb{R}^d$, $d = 4, 5$, where both Hypotheses (A) and (B) are satisfied.

When $d \ne 4$, the conditional almost sure global well-posedness in Theorem 1.3 has a flavor analogous to the deterministic conditional global well-posedness in the critical Sobolev spaces by Kenig-Merle 35 and Killip-Vişan 39. In the following, let us discuss the situation when $d = 4$. In this case, we only assume Hypothesis (A) for Theorem 1.3. While it would be interesting to remove this assumption, we do not know how to prove the validity of Hypothesis (A) at this point. This is mainly due to the lack of conservation of $H[v](t)$, i.e. the Hamiltonian evaluated at the nonlinear part $v$ of a solution. In the context of the energy-critical defocusing cubic NLW on $\mathbb{R}^4$, however, one can prove an analogue of Hypothesis (A) by establishing a probabilistic a priori bound on the energy $\mathcal{E}[v]$ of the nonlinear part $v$ of a solution, where the energy $\mathcal{E}[v]$ is defined by

$$\begin{equation*} \mathcal{E}[v](t) = \frac{1}{2} \int _{\mathbb{R}^4} |\partial _tv(t, x)|^2 dx + \frac{1}{2} \int _{\mathbb{R}^4}|\nabla v(t, x)|^2 dx + \frac{1}{4} \int _{\mathbb{R}^4} |v(t, x)|^4 dx. \end{equation*}$$

As a consequence, the third author 50 successfully implemented a probabilistic perturbation argument and proved almost sure global well-posedness of the energy-critical defocusing cubic NLW on $\mathbb{R}^4$ with randomized initial data below the scaling-critical regularity.⁠² We point out that the first term in the energy $\mathcal{E}[v]$ involving the time derivative plays an essential role in establishing a probabilistic a priori bound on the energy for NLW. It seems substantially harder to verify Hypothesis (A) for NLS, even when $d = 4$.

²

In 50, the third author also proved almost sure global well-posedness of the energy-critical defocusing NLW on $\mathbb{R}^5$. This result was recently extended to dimension 3 by the second and third authors 46.

✖

While Theorem 1.3 provides only conditional almost sure global existence, our last theorem (Theorem 1.4) below presents a way to construct global-in-time solutions below the scaling-critical regularity with a large probability. The main idea is to use the scaling 1.2 of the equation for random initial data below the scaling criticality. For example, suppose that we have a solution $u$ to 1.1 on a short time interval with a deterministic initial condition $u_0 \in H^s(\mathbb{R}^d)$, $s < s_\text{crit}$. In view of 1.2 and 1.3, by taking $\mu \to 0$, we see that the $H^s$-norm of the scaled initial condition goes to 0. Thus, one might think that the problem can be reduced to small data theory. This, of course, does not work in the usual deterministic setting, since we do not know how to construct solutions depending only on the $H^s$-norm of the initial data, $s < s_\text{crit}$. Even in the probabilistic setting, this naive idea does not work if we simply apply the scaling to the randomized function $\phi ^\omega$ defined in 1.9. This is due to the fact that we need to use (sub)critical space-time norms controlling the random linear term $z^\omega (t) = S(t) \phi ^\omega$, which do not become small even if we take $\mu \ll 1$.

To resolve this issue, we consider a randomization based on a partition of the frequency space by dilated cubes. Given $\mu > 0$, define $\psi ^\mu$ by

$$\begin{equation} \psi ^\mu (\xi ) = \psi (\mu ^{-1} \xi ). {\tag{1.16}} \end{equation}$$

Then, we can write a function $\phi$ on $\mathbb{R}^d$ as

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

Now, we introduce the randomization $\phi ^{\omega , \mu }$ of $\phi$ on dilated cubes of scale $\mu$ by

$$\begin{equation} \phi ^{\omega , \mu } : = \sum _{n \in \mathbb{Z}^d} g_n (\omega ) \psi ^\mu (D- \mu n) \phi , {\tag{1.17}} \end{equation}$$

where $\{g_n\}_{n \in \mathbb{Z}^d}$ is a sequence of independent mean zero complex-valued random variables, satisfying 1.10 as before. Then, we have the following global well-posedness of 1.1 with a large probability.

Theorem 1.4.

Let $d\geq 3$ and $\phi \in H^s(\mathbb{R}^d)$, for some $s \in ( s_d, s_\text{crit})$, where $s_d$ is as in 1.11. Then, given the randomization $\phi ^{\omega , \mu }$ on dilated cubes of scale $\mu \ll 1$ defined in 1.17, satisfying 1.10, the cubic NLS 1.1 on $\mathbb{R}^d$ is globally well-posed with a large probability. More precisely, for each $0< \varepsilon \ll 1$, there exists a small dilation scale $\mu _0 = \mu _0(\varepsilon , \|\phi \|_{H^s})> 0$ such that for each $\mu \in (0, \mu _0)$, there exists a set $\Omega _\mu \subset \Omega$ with the following properties:

(i)

$P(\Omega _\mu ^c) < \varepsilon$.

(ii)

If $\phi ^{\omega , \mu }$ is the randomization on dilated cubes defined in 1.17, satisfying 1.10, then, for each $\omega \in \Omega _\mu$, there exists a (unique) global-in-time solution $u$ to 1.1 with $u|_{t = 0} = \phi ^{\omega , \mu }$ in the class$$\begin{equation*} S(t) \phi ^\omega + C(\mathbb{R}: H^{\frac{d-2}{2}} (\mathbb{R}^d)) \subset C(\mathbb{R}:H^s(\mathbb{R}^d)). \end{equation*}$$

Moreover, for each $\omega \in \Omega _\mu$, scattering holds in the sense that there exists $v_+^\omega \in H^\frac{d-2}{2}(\mathbb{R}^d)$ such that$$\begin{equation*} \| u(t) - S(t)( \phi ^{\omega , \mu } + v_+^\omega ) \|_{H^\frac{d-2}{2}(\mathbb{R}^d)} \to 0 \end{equation*}$$

as $t \to \infty$. A similar statement holds for $t \to -\infty$.

We conclude this introduction with several remarks.

Remark 1.5.

In probabilistic well-posedness results 672044 for NLS on $\mathbb{T}^d$, random initial data are assumed to be of the following specific form:

$$\begin{equation} u_0^\omega (x) = \sum _{n \in \mathbb{Z}^d} \frac{g_n(\omega )}{(1+|n|^2)^\frac{\alpha }{2}}e^{i n \cdot x}, {\tag{1.18}} \end{equation}$$

where $\{g_n\}_{n \in \mathbb{Z}^d}$ is a sequence of independent complex-valued standard Gaussian random variables. The expression 1.18 has a close connection to the study of invariant measures and hence it is of importance. At the same time, due to the lack of a full range of Strichartz estimates on $\mathbb{T}^d$, one could not handle a general randomization of a given function as in 1.5. In this paper, we consider NLS on $\mathbb{R}^d$ and thus we do not encounter this issue thanks to a full range of the Strichartz estimates. For NLW, finite speed of propagation allows us to use a full range of Strichartz estimates even on compact domains, at least locally in time. Thus, one does not encounter such an issue.

Remark 1.6.

In a recent preprint, Lührmann-Mendelson 42 considered the defocusing NLW on $\mathbb{R}^3$ with randomized initial data, essentially given by 1.9, below the critical regularity and proved almost sure global well-posedness in the energy-subcritical case, following the method developed in 20, namely an adaptation of Bourgain’s high-low method 8 in the probabilistic setting. As Bourgain’s high-low method is a subcritical tool, their global result misses the energy-critical case.⁠³

³

In 46, the second and third authors recently proved almost sure global well-posedness of the energy-critical defocusing quintic NLW on $\mathbb{R}^3$.

✖

The third author 50 recently proved almost sure global well-posedness of the energy-critical defocusing NLW on $\mathbb{R}^d$, $d = 4, 5$, with randomized initial data below the critical regularity. The argument is based on an application of a perturbation lemma as in Theorem 1.3 along with a probabilistic a priori control on the energy, which is not available for the cubic NLS 1.1.

This paper is organized as follows. In Section 2, we state some probabilistic lemmata. In Section 3, we go over the basic definitions and properties of function spaces involving the $U^p$- and $V^p$-spaces. We prove the key nonlinear estimates in Section 4 and then use them to prove Theorems 1.1 and 1.2 in Section 5. We divide the proof of Theorem 1.3 into three sections. In Sections 6 and 7, we discuss the Cauchy theory for the defocusing cubic NLS with a deterministic perturbation. We implement these results in the probabilistic setting and prove Theorem 1.3 in Section 8. In Section 9, we show how Theorem 1.4 follows from the arguments in Sections 4 and 5, once we consider a randomization on dilated cubes. In Appendix A, we state and prove some additional properties of the function spaces defined in Section 3.

Lastly, note that we present the proofs of these results only for positive times in view of the time reversibility of 1.1.

2. Probabilistic lemmata

In this section, we summarize the probabilistic lemmata used in this paper. In particular, the probabilistic Strichartz estimates (Lemmata 2.2 and 2.3) play an essential role. First, we recall the usual Strichartz estimates on $\mathbb{R}^d$ for the readers’ convenience. We say that a pair $(q, r)$ is Schrödinger admissible if it satisfies

$$\begin{equation} \frac{2}{q} + \frac{d}{r} = \frac{d}{2} {\tag{2.1}} \end{equation}$$

with $2\leq q, r \leq \infty$ and $(q, r, d) \ne (2, \infty , 2)$. Then, the following Strichartz estimates are known to hold.

Lemma 2.1 (27335462).

Let $(q, r)$ be Schrödinger admissible. Then, we have

$$\begin{equation} \| S(t) \phi \|_{L^q_t L^r_x (\mathbb{R}\times \mathbb{R}^d)} \lesssim \|\phi \|_{L^2_x(\mathbb{R}^d)}. {\tag{2.2}} \end{equation}$$

In particular, when $q = r$, we have $q = r = \frac{2(d+2)}{d}$. By applying Sobolev inequality and 2.2, we also have

$$\begin{equation} \| S(t) \phi \|_{L^p_{t, x} (\mathbb{R}\times \mathbb{R}^d)} \lesssim \big \||\nabla |^{\frac{d}{2} - \frac{d+2}{p}}\phi \big \|_{L^2_x(\mathbb{R}^d)} {\tag{2.3}} \end{equation}$$

for $p \geq \frac{2(d+2)}{d}$. Recall that the derivative loss in 2.3 depends only on the size of the frequency support and not its location. Namely, if $\widehat{\phi }$ is supported on a cube $Q$ of side length $N$, then we have

$$\begin{equation} \| S(t) \phi \|_{L^p_{t, x} (\mathbb{R}\times \mathbb{R}^d)} \lesssim N^{\frac{d}{2} - \frac{d+2}{p}}\|\phi \|_{L^2_x(\mathbb{R}^d)}, {\tag{2.4}} \end{equation}$$

regardless of the center of the cube $Q$.

Next, we present improvements of the Strichartz estimates under the Wiener randomization 1.9 and where, throughout, we assume 1.10. See 4 for the proofs.

Lemma 2.2 (Improved local-in-time Strichartz estimate).

Given $\phi \in L^2(\mathbb{R}^d)$, let $\phi ^\omega$ be its Wiener randomization defined in 1.9, satisfying 1.10. Then, given finite $q, r \geq 2$, there exist $C, c>0$ such that

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

for all $T > 0$ and $\lambda >0$. In particular, with $\lambda = T^\theta$, we have

$$\begin{equation} \|S(t) \phi ^\omega \|_{L^q_tL^r_x([0, T]\times \mathbb{R}^d)} \lesssim T^\theta {\tag{2.5}} \end{equation}$$

outside a set of probability

$$\begin{equation*} \leq C\exp \bigg (-c \frac{1}{ T^{2(\frac{1}{q}-\theta )}\|\phi \|_{L^2}^{2}}\bigg ). \end{equation*}$$

Note that this probability can be made arbitrarily small by letting $T\to 0$ as long as $\theta < \frac{1}{q}$.

The next lemma states an improvement of the Strichartz estimates in the global-in-time setting.

Lemma 2.3 (Improved global-in-time Strichartz estimate).

Given $\phi \in L^2(\mathbb{R}^d)$, let $\phi ^\omega$ be its Wiener randomization defined in 1.9, satisfying 1.10. Given a Schrödinger admissible pair $(q, r)$ with $q, r < \infty$, let $\widetilde{r} \geq r$. Then, there exist $C, c>0$ such that

$$\begin{align*} P\Big ( \|S(t) \phi ^\omega \|_{L^q_t L^{\widetilde{r}}_x ( \mathbb{R}\times \mathbb{R}^d)} > \lambda \Big ) \leq Ce^{-c \lambda ^2 \|\phi \|_{L^2}^{-2}}. \end{align*}$$

In particular, given any small $\varepsilon > 0$, we have

$$\begin{equation*} \|S(t) \phi ^\omega \|_{L^q_t L^{\widetilde{r}}_x ( \mathbb{R}\times \mathbb{R}^d)} \lesssim \Big ( \log \frac{1}{\varepsilon }\Big )^\frac{1}{2} \|\phi \|_{L^2(\mathbb{R}^d)} \end{equation*}$$

outside a set of probability $< \varepsilon$.

Recall that the diagonal Strichartz admissible index is given by $p= \frac{2(d+2)}{d}$. In the diagonal case $q = \widetilde{r}$, it is easy to see that the condition of Lemma 2.3 is satisfied if $q = \widetilde{r} \geq p = \frac{2(d+2)}{d}$. In the following, we apply Lemma 2.3 in this setting.

We also need the following lemma on the control of the size of $H^s$-norm of $\phi ^\omega$.

Lemma 2.4.

Given $\phi \in H^s(\mathbb{R}^d)$, let $\phi ^\omega$ be its Wiener randomization defined in 1.9, satisfying 1.10. Then, we have

$$\begin{align} P\Big ( \| \phi ^\omega \|_{ H^s( \mathbb{R}^d)} > \lambda \Big ) \leq C e^{-c \lambda ^2 \|\phi \|_{ H^s}^{-2}}. {\tag{2.6}} \end{align}$$

We conclude this section by introducing some notation involving Strichartz and space-time Lebesgue spaces. In the sequel, given an interval $I\subset \mathbb{R}$, we often use $L^q_tL^r_x(I)$ to denote $L^q_tL^r_x(I\times \mathbb{R}^d)$. We also define the $\dot{S}^{s_\text{crit}}(I)$-norm in the usual manner by setting

$$\begin{equation*} \|u \|_{\dot{S}^{s_\text{crit}}(I)} := \sup \Big \{ \big \| |\nabla |^\frac{d-2}{2} u \big \|_{L^q_tL^r_x(I\times \mathbb{R}^d) }\Big \}, \end{equation*}$$

where the supremum is taken over all Schrödinger admissible pairs $(q, r)$.

3. Function spaces and their properties

In this section, we go over the basic definitions and properties of the $U^p$- and $V^p$-spaces, developed by Tataru, Koch, and their collaborators 293041. These spaces have been very effective in establishing well-posedness of various dispersive PDEs in critical regularities. See Hadac-Herr-Koch 29 and Herr-Tataru-Tzvetkov 30 for detailed proofs.

Let $H$ be a separable Hilbert space over $\mathbb{C}$. In particular, it will be either $H^s(\mathbb{R}^d)$ or $\mathbb{C}$. Let $\mathcal{Z}$ be the collection of finite partitions $\{t_k\}_{k = 0}^K$ of $\mathbb{R}$: $-\infty < t_0 < \cdots < t_K \leq \infty$. If $t_K = \infty$, we use the convention $u(t_K) :=0$ for all functions $u:\mathbb{R}\to H$. We use $\chi _I$ to denote the sharp characteristic function of a set $I \subset \mathbb{R}$.

Definition 3.1.

Let $1\leq p < \infty$.

(i) A $U^p$-atom is defined by a step function $a:\mathbb{R}\to H$ of the form

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

where $\{t_k\}_{k = 0}^K \in \mathcal{Z}$ and $\{\phi _k\}_{k = 0}^{K-1} \subset H$ with $\sum _{k = 0}^{K-1} \|\phi _k\|_H^p = 1$. Then, we define the atomic space $U^p(\mathbb{R}; H)$ to be the collection of functions $u:\mathbb{R}\to H$ of the form

$$\begin{equation} u = \sum _{j = 1}^\infty \lambda _j a_j, \quad \text{ where $a_j$’s are $U^p$-atoms and $\{\lambda _j\}_{j \in \mathbb{N}}\in \ell ^1(\mathbb{N}; \mathbb{C})$}, {\tag{3.1}} \end{equation}$$

with the norm

$$\begin{equation*} \|u\|_{U^p(\mathbb{R}; H)} : = \inf \Big \{ \|{ \mathbf{\lambda }} \|_{\ell ^1} : \xhref[disp-formula]{#X1}{3.1} \text{ holds with } \lambda = \{\lambda _j \}_{j \in \mathbb{N}} \text{ and some $U^p$-atoms } a_j\Big \}. \end{equation*}$$

(ii) We define the space $V^p(\mathbb{R}; H)$ of functions of bounded $p$-variation to be the collection of functions $u : \mathbb{R}\to H$ with $\|u\|_{V^p(\mathbb{R}; H)} < \infty$, where the $V^p$-norm is defined by

$$\begin{equation*} \|u\|_{V^p(\mathbb{R}; H)} := \sup _{\{t_k\}_{k = 0}^K \in \mathcal{Z}} \bigg (\sum _{k = 1}^K\|u(t_k) - u(t_{k-1})\|_H^p\bigg )^\frac{1}{p}. \end{equation*}$$

We also define $V^p_\text{rc}(\mathbb{R}; H)$ to be the closed subspace of all right-continuous functions in $V^p(\mathbb{R}; H)$ such that $\lim _{t \to -\infty } u(t) = 0$.

(iii) Let $s \in \mathbb{R}$. We define $U^p_\Delta H^s$ (and $V^p_\Delta H^s$, respectively) to be the spaces of all functions $u: \mathbb{R}\to H^s(\mathbb{T}^d)$ such that the following $U^p_\Delta H^s$-norm (and $V^p_\Delta H^s$-norm, respectively) is finite:

$$\begin{equation*} \|u \|_{U^p_\Delta H^s} := \|S(-t) u\|_{U^p(\mathbb{R}; H^s)} \quad \text{and} \quad \|u \|_{V^p_\Delta H^s} := \|S(-t) u\|_{V^p(\mathbb{R}; H^s)}, \end{equation*}$$

where $S(t) = e^{it\Delta }$ denotes the linear propagator for 1.1. We use $V^p_{\text{rc}, \Delta } H^s$ to denote the subspace of right-continuous functions in $V^p_\Delta H^s$.

Remark 3.2.

Note that the spaces $U^p(\mathbb{R}; H)$, $V^p(\mathbb{R}; H)$, and $V^p_\text{rc}(\mathbb{R}; H)$ are Banach spaces. The closed subspace of continuous functions in $U^p(\mathbb{R}; H)$ is also a Banach space. Moreover, we have the following embeddings:

$$\begin{equation*} U^p(\mathbb{R}; H) \hookrightarrow V^p_\text{rc}(\mathbb{R}; H) \hookrightarrow U^q(\mathbb{R}; H) \hookrightarrow L^\infty (\mathbb{R}; H) \end{equation*}$$

for $1\leq p < q < \infty$. Similar embeddings hold for $U^p_\Delta H^s$ and $V^p_\Delta H^s$.

Next, we state a transference principle and an interpolation result.

Lemma 3.3.

(i) (Transference principle) Suppose that we have

$$\begin{equation*} \big \| T(S(t) \phi _1, \dots , S(t) \phi _k)\big \|_{L^p_t L^q_x(\mathbb{R}\times \mathbb{R}^d)} \lesssim \prod _{j = 1}^k \|\phi _j\|_{L^2_x} \end{equation*}$$

for some $1\leq p, q \leq \infty$. Then, we have

$$\begin{equation*} \big \| T(u_1, \dots , u_k)\big \|_{L^p_t L^q_x(\mathbb{R}\times \mathbb{R}^d)} \lesssim \prod _{j = 1}^k \|u_j\|_{U^p_\Delta L^2_x}. \end{equation*}$$

(ii) (Interpolation) Let $E$ be a Banach space. Suppose that $T: U^{p_1}\times \cdots \times U^{p_k} \to E$ is a bounded $k$-linear operator such that

$$\begin{equation*} \|T(u_1, \dots , u_k)\|_{E} \leq C_1 \prod _{j = 1}^k \|u_j\|_{U^{p_j}} \end{equation*}$$

for some $p_1, \dots , p_k > 2$. Moreover, assume that there exists $C_2 \in (0, C_1]$ such that

$$\begin{equation*} \|T(u_1, \dots , u_k)\|_{E} \leq C_2 \prod _{j = 1}^k \|u_j\|_{U^{2}}. \end{equation*}$$

Then, we have

$$\begin{equation*} \|T(u_1, \dots , u_k)\|_{E} \leq C_2 \bigg (\ln \frac{C_1}{C_2}+ 1\bigg )^k \prod _{j = 1}^k \|u_j\|_{V^2} \end{equation*}$$

for $u_j \in V^2_\text{rc}$, $j = 1, \dots , k$.

A transference principle as above has been commonly used in the Fourier restriction norm method. See 29, Proposition 2.19 for the proof of Lemma 3.3 (i). The proof of the interpolation result follows from extending the trilinear result in 30 to a general $k$-linear case. See also 29, Proposition 2.20.

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 $\eta _N$. Moreover, we define $\mathbf{P}_{\leq N}$ and $\mathbf{P}_{\geq N}$ by $\mathbf{P}_{\leq N} = \sum _{1 \leq M \leq N} \mathbf{P}_M$ and $\mathbf{P}_{\geq N} = \sum _{ M \geq N} \mathbf{P}_M$.

Definition 3.4.

(i) Let $s\in \mathbb{R}$. We define $X^s(\mathbb{R})$ to be the space of all tempered distributions $u : \mathbb{R}\to H^s(\mathbb{R}^d)$ such that $\| u\|_{X^s(\mathbb{R})} < \infty$, where the $X^s$-norm is defined by

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

(ii) Let $s\in \mathbb{R}$. We define $Y^s(\mathbb{R})$ to be the space of all tempered distributions $u : \mathbb{R}\to H^s(\mathbb{R}^d)$ such that for every $N\in \mathbb{N}$, the map $t \mapsto \mathbf{P}_N u(t)$ is in $V^2_{\text{rc}, \Delta } H^s$ and $\| u\|_{Y^s(\mathbb{R})} < \infty$, where the $Y^s$-norm is defined by

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

Recall the following embeddings:

$$\begin{equation} U^2_\Delta H^s \hookrightarrow X^s \hookrightarrow Y^s \hookrightarrow V^2_\Delta H^s \hookrightarrow U^p_\Delta H^s {\tag{3.2}} \end{equation}$$

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 nonhomogeneous term:

$$\begin{align} \| F\|_{N^s(I)} = \bigg \|\int _{t_0}^t S(t - t') F(t') dt'\bigg \|_{X^s(I)}. \tag{3.3} \end{align}$$

In the following, we will perform our analysis in $X^s(I) \cap C(I; H^s)$, that is, in a Banach subspace of continuous functions in $X^s(I)$. See Appendix A for additional properties of the $X^s(I)$-spaces.

We conclude this section by presenting some basic estimates involving these function spaces.

Lemma 3.5.

(i) (Linear estimates) Let $s \geq 0$ and $0 < T \leq \infty$. Then, we have

$$\begin{align*} \| S(t) \phi \|_{X^s([0, T))} & \leq \|\phi \|_{H^s}, \\ \|F\|_{N^s([0, T))} & \leq \sup _{\substack{v \in Y^{-s}([0, T))\\\|v\|_{Y^{-s} }= 1}} \bigg | \int _0^T \int _{\mathbb{R}^d} F(t, x) \overline{v(t, x)} dx dt\bigg | \end{align*}$$

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

(ii) (Strichartz estimates) Let $(q, r)$ be Schrödinger admissible with $q > 2$ and $p \geq \frac{2(d+2)}{d}$. Then, for $0< T\leq \infty$ and $N_1 \leq N_2$, we have

$$\begin{align} \| u \|_{L^q_t L^r_x([0, T)\times \mathbb{R}^d)} & \lesssim \|u\|_{Y^0([0, T))}, {\tag{3.4}}\\ \| u \|_{L^p_{t,x}([0, T)\times \mathbb{R}^d)} & \lesssim \big \||\nabla |^{\frac{d}{2} - \frac{d+2}{p}} u\big \|_{Y^0([0, T))}, {\tag{3.5}} \end{align}$$

$$\begin{align} &\| \mathbf{P}_{N_1} u_1 \mathbf{P}_{N_2}u_2\|_{L^2_{t, x}([0, T)\times \mathbb{R}^d)}{\tag{3.6}}\\ &\qquad \qquad \lesssim N_1^\frac{d-2}{2} \bigg (\frac{N_1}{N_2}\bigg )^{\frac{1}{2}-} \|\mathbf{P}_{N_1} u_1\|_{Y^0([0, T))}\|\mathbf{P}_{N_2} u_2\|_{Y^0([0, T))}. \end{align}$$

Note that there is a slight loss of regularity in 3.6 since we use the $Y^0$-norm on the right-hand side instead of the $X^0$-norm. In view of 3.2, we may replace the $Y^0$-norms on the right-hand sides of 3.4, 3.5, and 3.6 by the $X^0$-norm in the following.

Proof.

In the following, we briefly discuss the proof of (ii). See 2930 for the proof of (i). The first estimate 3.4 follows from the Strichartz estimate 2.2, Lemma 3.3 (i), and 3.2:

$$\begin{equation*} \|u\|_{L^q_tL^r_x}\lesssim \|u\|_{U^q_{\Delta }L^2}\lesssim \|u\|_{Y^0} \end{equation*}$$

for $q>2$. The second estimate 3.5 follows from 2.3 in a similar manner. It remains to prove 3.6. On the one hand, the following bilinear refinement of the Strichartz estimate by Bourgain 8 and Ozawa-Tsutsumi 48:

$$\begin{equation*} \|\mathbf{P}_{N_1}S(t) \phi _1 \mathbf{P}_{N_2}S(t) \phi _2\|_{L^2_{t,x}} \lesssim N_1^{\frac{d-2}{2}}\bigg (\frac{N_1}{N_2}\bigg )^{\frac{1}{2}} \|\mathbf{P}_{N_1}\phi _1\|_{L^2}\|\mathbf{P}_{N_2}\phi _2\|_{L^2} \end{equation*}$$

and Lemma 3.3 (i) yield

$$\begin{equation} \|\mathbf{P}_{N_1}u_1\mathbf{P}_{N_2}u_2\|_{L^2_{t,x}}\lesssim N_1^{\frac{d-2}{2}} \bigg (\frac{N_1}{N_2}\bigg )^{\frac{1}{2}}\|\mathbf{P}_{N_1}u_1\|_{U^2_{\Delta }L^2}\|\mathbf{P}_{N_2}u_2\|_{U^2_{\Delta }L^2}. {\tag{3.7}} \end{equation}$$

On the other hand, by Bernstein’s inequality and noting that $(4, \frac{2d}{d-1})$ is Strichartz admissible, we have

$$\begin{align*} \|\mathbf{P}_{N_j}S(t) \phi _j\|_{L^4_{t,x}} &\lesssim N_j^{\frac{d-2}{4}} \|\mathbf{P}_{N_j}S(t) \phi _j\|_{L^4_tL^{\frac{2d}{d-1}}_x} \lesssim N_j^{\frac{d-2}{4}} \|\mathbf{P}_{N_j}\phi _j\|_{L^2}. \end{align*}$$

Then, by Cauchy-Schwarz’ inequality and Lemma 3.3 (i), we obtain

$$\begin{equation} \|\mathbf{P}_{N_1}u_1\mathbf{P}_{N_2}u_2\|_{L^2_{t,x}}\lesssim N_1^{\frac{d-2}{4}}N_2^{\frac{d-2}{4}}\|\mathbf{P}_{N_1}u_1\|_{U^4_{\Delta }L^2}\|\mathbf{P}_{N_2}u_2\|_{U^4_{\Delta }L^2}. {\tag{3.8}} \end{equation}$$

Hence, by Lemma 3.3 (ii), with 3.7 and 3.8, we have

$$\begin{align} &\|\mathbf{P}_{N_1}u_1 \mathbf{P}_{N_2}u_2\|_{L^2_{t,x}}{\tag{3.9}}\\ &\qquad \qquad \lesssim N_1^{\frac{d-2}{2}} \bigg (\frac{N_1}{N_2}\bigg )^{\frac{1}{2}} \bigg (\ln \Big (\frac{N_2}{N_1}\Big )+1\bigg )^2 \|\mathbf{P}_{N_1}u_1\|_{V^2_{\Delta }L^2}\|\mathbf{P}_{N_2}u_2\|_{V^2_{\Delta }L^2}. \end{align}$$

Finally, 3.6 follows from 3.2 and 3.9.

■

Similar to the usual Strichartz estimate 2.3, the derivative loss in 3.5 depends only on the size of the spatial frequency support and not its location. Namely, if the spatial frequency support of $\widehat{u}(t, \xi )$ is contained in a cube of side length $N$ for all $t \in \mathbb{R}$, then we have

$$\begin{equation} \| u \|_{L^p_{t,x}([0, T)\times \mathbb{R}^d)} \lesssim N^{\frac{d}{2} - \frac{d+2}{p}} \| u\|_{Y^0([0, T))}. {\tag{3.10}} \end{equation}$$

This is a direct consequence of 2.4.

Lastly, we recall Schur’s test for the readers’ convenience.

Lemma 3.6 (Schur’s test).

Suppose that we have

$$\begin{equation*} \sup _m \sum _{n} |K_{m, n}| + \sup _n \sum _{m} |K_{m, n}| < \infty \end{equation*}$$

for some $K_{m, n} \in \mathbb{C}$, $m, n \in \mathbb{Z}$. Then, we have

$$\begin{equation*} \sum _{m, n } K_{m, n} a_m b_n \lesssim \|a_m\|_{\ell ^2_m}\|b_n\|_{\ell ^2_n} \end{equation*}$$

for any $\ell ^2$-sequences $\{a_m\}_{m\in \mathbb{Z}}$ and $\{b_n\}_{n\in \mathbb{Z}}$.

4. Probabilistic nonlinear estimates

In this section, we prove the key nonlinear estimates in the critical regularity $s_\text{crit} = \frac{d-2}{2}$. In the next section, we use them to prove Theorems 1.1 and 1.2. Given $z(t) = S(t) \phi ^\omega$, define $\Gamma$ by

$$\begin{equation} \Gamma v(t) =\mp i \int _0^t S(t-t') \mathcal{N}(v+z)(t') dt', {\tag{4.1}} \end{equation}$$

where $\mathcal{N}(v+z)=|v+z|^2(v+z)$. Then, we have the following nonlinear estimates.

Proposition 4.1.

Given $d\geq 3$, let $s \in (s_d, s_\text{crit}]$, where $s_d$ is defined in 1.11. Given $\phi \in H^s(\mathbb{R}^d)$, let $\phi ^\omega$ be its Wiener randomization defined in 1.9, satisfying 1.10.

(i) Let $0<T \leq 1$. Then, there exists $0<\theta \ll 1$ such that we have

$$\begin{align} &\|\Gamma v\|_{X^\frac{d-2}{2}([0, T))} \leq C_1 \big (\|v\|_{X^\frac{d-2}{2}([0, T))} ^3 + T^\theta R^3\big ), {\tag{4.2}}\\ &\|\Gamma v_1 - \Gamma v_2 \|_{X^\frac{d-2}{2}([0, T))}{\tag{4.3}}\\ &\qquad \qquad \qquad \quad \leq C_2 \Big (\sum _{j = 1}^2 \|v_j\|_{X^\frac{d-2}{2}([0, T))} ^2 + T^\theta R^2\Big ) \|v_1 -v_2 \|_{X^\frac{d-2}{2}([0, T))} \end{align}$$

for all $v, v_1, v_2 \in X^{\frac{d-2}{2}}([0, T))$ and $R>0$, outside a set of probability $\leq C \exp (-c \frac{R^2}{\|\phi \|_{H^s}^2})$.

(ii) Given $0 < \varepsilon \ll 1$, define $\widetilde{\Gamma }$ by

$$\begin{equation} \widetilde{\Gamma }v(t) = \mp i \int _0^t S(t-t') \mathcal{N}(v+\varepsilon z)(t') dt'. {\tag{4.4}} \end{equation}$$

Then, we have

$$\begin{align} \|\widetilde{\Gamma }v\|_{X^\frac{d-2}{2}(\mathbb{R})} & \leq C_3 \big (\|v\|_{X^\frac{d-2}{2}(\mathbb{R})} ^3 + R^3\big ), {\tag{4.5}}\\ \|\widetilde{\Gamma }v_1 - \widetilde{\Gamma }v_2 \|_{X^\frac{d-2}{2}(\mathbb{R})} & \leq C_4 \Big (\sum _{j = 1}^2 \|v_j\|_{X^\frac{d-2}{2}(\mathbb{R})} ^2 + R^2\Big ) \|v_1 -v_2 \|_{X^\frac{d-2}{2}(\mathbb{R})} {\tag{4.6}} \end{align}$$

for all $v, v_1, v_2 \!\in \! X^{\frac{d-2}{2}}\!(\mathbb{R})$ and $R\!>\!0$, outside a set of probability $\leq \!C \exp (-c \frac{R^2}{\varepsilon ^2 \|\phi \|_{H^s}^2})$.

Proof.

(i) Let $0<T \leq 1$. We only prove 4.2 since 4.3 follows in a similar manner. Given $N \geq 1$, define $\Gamma _N$ by

$$\begin{equation} \Gamma _N v(t) =\mp i \int _0^t S(t-t')\mathbf{P}_{\leq N} \mathcal{N}(v+z)(t') dt'. {\tag{4.7}} \end{equation}$$

By Bernstein’s and Hölder’s inequalities, we have

$$\begin{align} \|\mathbf{P}_{\leq N} \mathcal{N}(v+z)\|_{L^1_t ([0, T); H^{\frac{d-2}{2}}_x)} & \lesssim N^\frac{d-2}{2} \| \mathcal{N}(v+z)\|_{L^1_t L^2_x}\\ & \lesssim N^\frac{d-2}{2} \| v\|_{L^3_t([0, T); L^6_x)}^3 + N^\frac{d-2}{2} \| z\|_{L^3_t([0, T); L^6_x)}^3. {\tag{4.8}} \end{align}$$

On the one hand, it follows from Lemma 2.2 that the second term on the right-hand side of 4.8 is finite almost surely. On the other hand, noting that $(3, \frac{6d}{3d-4})$ is Strichartz admissible, it follows from Sobolev’s inequality and 3.4 in Lemma 3.5 that

$$\begin{align} \| v\|_{L^3_t([0, T); L^6_x)} \lesssim \big \| \langle \nabla \rangle ^\frac{d-2}{3}v\big \|_{L^3_t([0, T); L^\frac{6d}{3d-4}_x)} \lesssim \big \| v \|_{X^\frac{d-2}{3}([0, T))} < \infty . {\tag{4.9}} \end{align}$$

Therefore, by Lemma 3.5 (i), we have

$$\begin{align} \|\Gamma _N v(t)\|_{X^\frac{d-2}{2}} \lesssim \sup _{\substack{v_4 \in Y^{0}([0, T))\\\|v_4\|_{Y^{0}} = 1}} \bigg | \int _0^T \int _{\mathbb{R}^d} \langle \nabla \rangle ^\frac{d-2}{2} \mathcal{N}(v+z)(t, x) \overline{v_4(t, x)} dx dt\bigg |, {\tag{4.10}} \end{align}$$

almost surely, where $v_4 = \mathbf{P}_{\leq N} v_4$. In the following, we estimate the right-hand side of 4.10, independently of the cutoff size $N \geq 1$, by performing a case-by-case analysis of expressions of the form:

$$\begin{align} \bigg | \int _0^T \int _{\mathbb{R}^d} \langle \nabla \rangle ^\frac{d-2}{2} ( w_1 w_2 w_3 )v_4 dx dt\bigg |, {\tag{4.11}} \end{align}$$

where $\|v_4\|_{Y^{0}([0, T))} \leq 1$ and $w_j= v$ or $z$, $j = 1, 2, 3$. As a result, by taking $N \to \infty$, the same estimates hold for $\Gamma v$ without any cutoff, thus yielding 4.2.

Before proceeding further, let us simplify some of the notation. In the following, we drop the complex conjugate sign. We also denote $X^s([0, T))$ and $Y^s([0, T))$ by $X^s$ and $Y^s$ since $T$ is fixed. Similarly, it is understood that the time integration in $L^p_{t, x}$ is over $[0, T)$. Lastly, in most of the cases, we dyadically decompose $w_j = v_j$ or $z_j$, $j = 1, 2, 3$, and $v_4$ such that their spatial frequency supports are $\{ |\xi _j|\sim N_j\}$ for some dyadic $N_j \geq 1$ but still denote them as $w_j= v_j \text{ or } z_j$, $j = 1, 2, 3$, and $v_4$. Note that, if we can afford a small derivative loss in the largest frequency, there is no difficulty in summing over the dyadic blocks $N_j$, $j = 1, \dots , 4$.

Case (1): $v v v$ case.

In this case, we do not need to perform dyadic decompositions and we divide the frequency spaces into $\{|\xi _1| \geq |\xi _2|, |\xi _3|\}$, $\{|\xi _2| \geq |\xi _1|, |\xi _3|\}$, and $\{|\xi _3| \geq |\xi _1|, |\xi _2|\}$. Without loss of generality, assume that $|\xi _1| \geq |\xi _2|, |\xi _3|$. By the $L^\frac{2(d+2)}{d}_{t,x}L^{d+2}_{t,x}L^{d+2}_{t,x}L^{\frac{2(d+2)}{d}}_{t,x}$-Hölder’s inequality, 3.5 in Lemma 3.5, and 3.2, we have

$$\begin{align*} \bigg |\int _0^T\int _{ \mathbb{R}^d} \langle \nabla \rangle ^\frac{d-2}{2} v_1 v_2 v_3 v_4 dx dt \bigg | & \leq \| \langle \nabla \rangle ^\frac{d-2}{2} v_1\|_{L^\frac{2(d+2)}{d}_{t, x}} \|v_2\|_{L^{d+2}_{t, x}}\|v_3\|_{L^{d+2}_{t, x}} \|v_4\|_{L^{\frac{2(d+2)}{d}}_{t, x}}\\ & \lesssim \prod _{j = 1}^3 \|v_j\|_{Y^\frac{d-2}{2}} \|v_4\|_{Y^0} \lesssim \prod _{j = 1}^3 \|v_j\|_{X^\frac{d-2}{2}}. \end{align*}$$

Case (2): $zz z$ case.

Without loss of generality, assume $N_3 \geq N_2 \geq N_1$.

• Subcase (2.a): $N_2 \sim N_3$.

By the $L^{d+2}_{t,x}L^{4}_{t,x}L^{4}_{t,x}L^{\frac{2(d+2)}{d}}_{t,x}$-Hölder’s inequality, we have

$$\begin{align*} &\bigg |\int _0^T\int _{ \mathbb{R}^d} z_1 z_2 \langle \nabla \rangle ^\frac{d-2}{2} z_3 v_4 dx dt \bigg |\\ &\qquad \qquad \lesssim \|z_1\|_{L^{d+2}_{t, x}} \|\langle \nabla \rangle ^\frac{d-2}{4}z_2\|_{L^{4}_{t, x}} \|\langle \nabla \rangle ^\frac{d-2}{4} z_3 \|_{L^{4}_{t, x}}\| v_4 \|_{L^\frac{2(d+2)}{d}_{t, x}}. \end{align*}$$

Hence, by Lemmata 2.2 and 3.5, the contribution to 4.10 in this case is at most $\lesssim T^{0+} R^3$ outside a set of probability

$$\begin{equation*} \leq C\exp \bigg (- c\frac{R^2}{T^{\frac{2}{d+2}-}\|\phi \|_{L^2}^2}\bigg ) + C\exp \bigg (- c\frac{R^2}{T^{\frac{1}{2}-}\|\phi \|_{H^{\frac{d-2}{4}+}}^2}\bigg ) \end{equation*}$$

as long as $s > \frac{d-2}{4}$. Note that $s$ needs to be strictly greater than $\frac{d-2}{4}$ due to the summations over dyadic blocks. See 4 for more details. Similar comments apply in the following.

Subcase (2.b): $N_3 \sim N_4 \gg N_1, N_2$.

$\circ$ Subsubcase (2.b.i): $N_1, N_2 \ll N_3^\frac{1}{d-1}$. For small $\alpha > 0$, it follows from Cauchy-Schwarz’ inequality and Lemma 3.5 that

$$\begin{align} \|z_2 \langle \nabla \rangle ^\frac{d-2}{2}z_3\|_{L^2_{t, x}} & \lesssim N_3^\frac{d-2}{2} \|z_2\|_{L^4_{t, x} }^\alpha \|z_3\|_{L^4_{t, x} }^\alpha \|z_2 z_3\|_{L^2_{t, x}}^{1-\alpha }\\ & \lesssim N_2^{\frac{d-1}{2}-s-\frac{d-1}{2}\alpha -} N_3^{\frac{d-3}{2} - s+\frac{1}{2}\alpha +} \prod _{j = 2}^3\big (\|\langle \nabla \rangle ^s z_j\|_{L^4_{t, x} }^\alpha \|\mathbf{P}_{N_j} \phi ^\omega \|_{H^{s}}^{1-\alpha }\big ). {\tag{4.12}} \end{align}$$

Then, by 4.12 and the bilinear estimate 3.6 in Lemma 3.5, we have

$$\begin{align*} \bigg |\int _0^T \int _{ \mathbb{R}^d} z_1 z_2 & \langle \nabla \rangle ^\frac{d-2}{2} z_3 v_4 dx dt \bigg | \lesssim \|z_2 \langle \nabla \rangle ^\frac{d-2}{2} z_3\|_{L^2_{t, x}} \|z_1 v_4\|_{L^2_{t, x}}\\ & \lesssim N_1^{\frac{d-1}{2} - s-}N_2^{\frac{d-1}{2} - s-\frac{d-1}{2}\alpha -} N_3^{\frac{d-4}{2}-s +\frac{1}{2}\alpha +} \\ & \hphantom {XXXXXX} \times \|\mathbf{P}_{N_1} \phi ^\omega \|_{H^{s}} \prod _{j = 2}^3\big (\|\langle \nabla \rangle ^s z_j\|_{L^4_{t, x} }^\alpha \|\mathbf{P}_{N_j} \phi ^\omega \|_{H^{s}}^{1-\alpha }\big ) \|v_4\|_{Y^0}\\ & \lesssim N_3^{\frac{d-2}{2} - \frac{d+1}{d-1} s + } \|\mathbf{P}_{N_1} \phi ^\omega \|_{H^{s}} \prod _{j = 2}^3\big (\|\langle \nabla \rangle ^s z_j\|_{L^4_{t, x} }^\alpha \|\mathbf{P}_{N_j} \phi ^\omega \|_{H^{s}}^{1-\alpha }\big ) \|v_4\|_{Y^0}. \end{align*}$$

Hence, by Lemmata 2.2 and 2.4, the contribution to 4.10 in this case is at most $\lesssim T^{0+} R^3$ outside a set of probability

$$\begin{equation*} \leq C\exp \bigg (-c \frac{R^2}{ T^{\frac{1}{2}-} \|\phi \|_{H^{s}}^{2}}\bigg ) + C\exp \bigg (-c \frac{R^2}{ \|\phi \|_{H^{s}}^{2}}\bigg ) \end{equation*}$$

as long as

$$\begin{equation} s> \frac{d-1}{d+1}\cdot \frac{d-2}{2}=s_d {\tag{4.13}} \end{equation}$$

and $\alpha < 1- \frac{2}{d-1} s$.

$\circ$ Subsubcase (2.b.ii): $N_2\gtrsim N_3^\frac{1}{d-1} \gg N_1$. By Hölder’s inequality and the bilinear estimate 3.6 in Lemma 3.5, we have

$$\begin{align*} \bigg |\int _0^T\int _{ \mathbb{R}^d} z_1 z_2 & \langle \nabla \rangle ^\frac{d-2}{2} z_3 v_4 dx dt \bigg | \lesssim \|z_2\|_{L^4_{t, x}} \|\langle \nabla \rangle ^\frac{d-2}{2} z_3 \|_{L^4_{t, x}}\|z_1 v_4 \|_{L^2_{t, x}}\\ & \lesssim N_1^{\frac{d-1}{2}-s-} N_2^{-s} N_3^{\frac{d-3}{2}-s+} \|\mathbf{P}_{N_1}\phi ^\omega \|_{H^s} \prod _{j = 2}^3 \|\langle \nabla \rangle ^s z_j\|_{L^4_{t, x}} \|v_4\|_{Y^0}\\ & \lesssim N_3^{\frac{d-2}{2} - \frac{d+1}{d-1} s + } \|\mathbf{P}_{N_1}\phi ^\omega \|_{H^s} \prod _{j = 2}^3 \|\langle \nabla \rangle ^s z_j\|_{L^4_{t, x}} \|v_4\|_{Y^0}. \end{align*}$$

Hence, by Lemmata 2.2 and 2.4, the contribution to 4.10 in this case is at most $\lesssim T^{0+} R^3$ outside a set of probability

$$\begin{equation*} \leq C\exp \bigg (-c \frac{R^2}{ T^{\frac{1}{2}-} \|\phi \|_{H^{s}}^{2}}\bigg ) + C\exp \bigg (-c \frac{R^2}{ \|\phi \|_{H^{s}}^{2}}\bigg ) \end{equation*}$$

as long as 4.13 is satisfied.

$\circ$ Subsubcase (2.b.iii): $N_1, N_2\gtrsim N_3^\frac{1}{d-1}$.

By the $L^{\frac{6(d+2)}{d+4}}_{t,x}L^{\frac{6(d+2)}{d+4}}_{t,x}L^{\frac{6(d+2)}{d+4}}_{t,x}L^{\frac{2(d+2)}{d}}_{t,x}$-Hölder’s inequality and 3.5 in Lemma 3.5, we have

$$\begin{align*} \bigg |\int _0^T \int _{\mathbb{R}^d} z_1 z_2 \langle \nabla \rangle ^\frac{d-2}{2} z_3 v_4 dx dt \bigg | & \lesssim N_3^{\frac{d-2}{2} - \frac{d+1}{d-1} s } \prod _{j =1}^3 \| \langle \nabla \rangle ^s z_j\|_{L^\frac{6(d+2)}{d+4}_{t, x}} \|v_4\|_{Y^0}. \end{align*}$$

Hence, by Lemma 2.2, the contribution to 4.10 in this case is at most $\lesssim T^{0+} R^3$ outside a set of probability

$$\begin{equation*} \leq C\exp \bigg (-c \frac{R^2}{ T^{\frac{d+4}{3(d+2)}-} \|\phi \|_{H^{s}}^{2}}\bigg ) \end{equation*}$$

as long as 4.13 is satisfied.

Case (3): $v v z$ case. Without loss of generality, assume $N_1 \geq N_2$.

• Subcase (3.a): $N_1 \gtrsim N_3$.

In the following, we apply dyadic decompositions only to $v_1$, $v_2$, and $z_3$. In this case, we have $N_1 \sim \max (N_2, N_3, |\xi _4|)$, where $\xi _4$ is the spatial frequency of $v_4$. Then, by Hölder’s inequality, 3.6, and 3.5, we have

$$\begin{align*} \bigg | &\int _0^T \int _{\mathbb{R}^d} \langle \nabla \rangle ^\frac{d-2}{2} v_1 v_2 z_3 v_4 dx dt \bigg |\\ &\qquad \qquad \lesssim \sum _{N_1 \gtrsim N_2, N_3} \|\langle \nabla \rangle ^{\frac{d-2}{2}}\mathbf{P}_{N_1}v_1 \mathbf{P}_{N_2} v_2\|_{L^2_{t,x}}\|\mathbf{P}_{N_3}z_3\|_{L^{d+2}_{t,x}} \|v_4\|_{L^{\frac{2(d+2)}{d}}_{t,x}}\\ &\qquad \qquad \lesssim \sum _{N_1 \geq N_2} \bigg (\frac{N_2}{N_1}\bigg )^{\frac{1}{2}-}\prod _{j = 1}^2 \|\mathbf{P}_{N_j}v_j\|_{X^\frac{d-2}{2}} \sum _{N_3} \|\mathbf{P}_{N_3}z_3\|_{L^{d+2}_{t, x}} \|v_4\|_{Y^{0}}. \end{align*}$$

By Lemma 3.6 and summing over $N_3$ with a slight loss of derivative,

$$\begin{align*} &\lesssim \prod _{j = 1}^2 \|v_j\|_{X^\frac{d-2}{2}} \| \langle \nabla \rangle ^{0+}z_3\|_{L^{d+2}_{t, x}} \|v_4\|_{Y^{0}}. \end{align*}$$

Hence, by Lemma 2.2, the contribution to 4.10 in this case is at most $\lesssim T^{0+}R \prod _{j = 1}^2 \|v_j\|_{X^\frac{d-2}{2}}$ outside a set of probability

$$\begin{equation*} \leq C\exp \bigg (-c \frac{R^2}{ T^{\frac{2}{d+2}-} \|\phi \|_{H^{0+}}^{2}}\bigg ) \end{equation*}$$

as long as $s > 0$.

• Subcase (3.b): $N_3\sim N_4 \gg N_1 \geq N_2$.

$\circ$ Subsubcase (3.b.i): $N_1\gtrsim N_3^\frac{1}{d-1}$. By Hölder’s inequality followed by 3.5 and 3.6 in Lemma 3.5, we have

$$\begin{align*} \bigg |\int _0^T \int _{\mathbb{R}^d} v_1 v_2 & \langle \nabla \rangle ^\frac{d-2}{2} z_3 v_4 dx dt \bigg | \lesssim \|v_1\|_{L^\frac{2(d+2)}{d}_{t, x}} \|\langle \nabla \rangle ^\frac{d-2}{2} z_3 \|_{L^{d+2}_{t, x}}\|v_2 v_4 \|_{L^2_{t, x}}\\ & \lesssim N_1^{-\frac{d-2}{2}}N_2^{\frac{1}{2}-}N_3^{\frac{d-3}{2}-s+} \|v_1\|_{X^\frac{d-2}{2}} \|v_2\|_{X^{\frac{d-2}{2}}} \|\langle \nabla \rangle ^{s} z_3\|_{L^{d+2}_{t, x}} \|v_4\|_{Y^0}\\ & \lesssim N_3^{\frac{d-3}{d-1}\frac{d-2}{2} - s + } \|v_1\|_{X^\frac{d-2}{2}} \|v_2\|_{X^{\frac{d-2}{2}}} \|\langle \nabla \rangle ^{s} z_3\|_{L^{d+2}_{t, x}} \|v_4\|_{Y^0}. \end{align*}$$

Hence, by Lemma 2.2, the contribution to 4.10 in this case is at most $\lesssim T^{0+} R \prod _{j = 1}^2 \|v_j\|_{X^{\frac{d-2}{2}}}$ outside a set of probability

$$\begin{equation*} \leq C\exp \bigg (-c \frac{R^2}{ T^{\frac{2}{d+2}-} \|\phi \|_{H^{s}}^{2}}\bigg ) \end{equation*}$$

as long as

$$\begin{align} s > \frac{d-3}{d-1} \cdot \frac{d-2}{2}. {\tag{4.14}} \end{align}$$

Note that the condition 4.14 is less restrictive than 4.13.

$\circ$ Subsubcase (3.b.ii): $N_2 \leq N_1\ll N_3^\frac{1}{d-1}$.

For small $\alpha > 0$, it follows from Hölder’s inequality and Lemma 3.5 that

$$\begin{align} \|v_1 \langle \nabla \rangle ^\frac{d-2}{2}z_3\|_{L^2_{t, x}} & \lesssim N_3^\frac{d-2}{2} \|v_1\|_{L^{\frac{2(d+2)}{d}}_{t, x} }^\alpha \|z_3\|_{L^{d+2}_{t, x} }^\alpha \|v_1 z_3\|_{L^2_{t, x}}^{1-\alpha }\\ & \lesssim N_1^{\frac{1}{2} -\frac{d-1}{2} \alpha -} N_3^{\frac{d-3}{2} - s+\frac{1}{2}\alpha +} \|v_1\|_{X\frac{d-2}{2}} \|\langle \nabla \rangle ^s z_3\|_{L^{d+2}_{t, x} }^\alpha \|\mathbf{P}_{N_3} \phi ^\omega \|_{H^{s}}^{1-\alpha }. {\tag{4.15}} \end{align}$$

Then, by 4.15 and 3.6 in Lemma 3.5, we have

$$\begin{align*} \bigg |\int _0^T & \int _{\mathbb{R}^d} v_1 v_2 \langle \nabla \rangle ^\frac{d-2}{2} z_3 v_4 dx dt \bigg | \lesssim \|v_1 \langle \nabla \rangle ^\frac{d-2}{2}z_3\|_{L^2_{t, x}} \|v_2 v_4 \|_{L^2_{t, x}}\\ & \lesssim N_1^{\frac{1}{2} -\frac{d-1}{2} \alpha -} N_2^{\frac{1}{2}-} N_3^{\frac{d-4}{2} - s+\frac{1}{2}\alpha +} \|v_1\|_{X^\frac{d-2}{2}} \|v_2\|_{X^{\frac{d-2}{2}}}\\ &\hphantom {XXXXXX}\times \|\langle \nabla \rangle ^s z_3\|_{L^{d+2}_{t, x} }^\alpha \|\mathbf{P}_{N_3} \phi ^\omega \|_{H^{s}}^{1-\alpha } \|v_4\|_{Y^0}. \end{align*}$$

Hence, by Lemmata 2.2 and 2.4, the contribution to 4.10 in this case is at most $\lesssim T^{0+} R \prod _{j = 1}^2 \|v_j\|_{X^{\frac{d-2}{2}}}$ outside a set of probability

$$\begin{equation*} \leq C\exp \bigg (-c \frac{R^2}{ T^{\frac{2}{d+2}-} \|\phi \|_{H^{s}}^{2}}\bigg ) + C\exp \bigg (-c \frac{R^2}{ \|\phi \|_{H^{s}}^{2}}\bigg ) \end{equation*}$$

as long as 4.14 is satisfied and $\alpha < \frac{1}{d-1}$.

Case (4): $v z z$ case. Without loss of generality, assume $N_3 \geq N_2$.

• Subcase (4.a): $N_1 \gtrsim N_3$. By the $L^\frac{2(d+2)}{d}_{t,x}L^{d+2}_{t,x}L^{d+2}_{t,x}L^\frac{2(d+2)}{d}_{t,x}$-Hölder’s inequality and 3.5 in Lemma 3.5, we have

$$\begin{align*} \bigg |\int _0^T \int _{\mathbb{R}^d} \langle \nabla \rangle ^\frac{d-2}{2} v_1 z_2 z_3 v_4 dx dt \bigg | & \lesssim \|v_1\|_{X^{\frac{d-2}{2}}} \|z_2\|_{L^{d+2}_{t, x}} \|z_3\|_{L^{d+2}_{t, x}} \|v_4\|_{Y^0}. \end{align*}$$

Hence, by Lemma 2.2, the contribution to 4.10 in this case is at most $\lesssim T^{0+}R^2 \|v_1\|_{X^\frac{d-2}{2}}$ outside a set of probability

$$\begin{equation} \leq C\exp \bigg (-c \frac{R^2}{ T^{\frac{2}{d+2}-} \|\phi \|_{H^{0+}}^{2}}\bigg ) {\tag{4.16}} \end{equation}$$

as long as $s> 0$. As before, we have $\|\phi \|_{H^{0+}}$ instead of $\|\phi \|_{L^2}$ in 4.16, allowing us to sum over $N_2$ and $N_3$. If $N_3 \gtrsim \max (N_1, N_4)$, then this also allows us to sum over $N_1$ and $N_4$. Otherwise, we have $N_1\sim N_4\gg N_3$. In this case, we can use Cauchy-Schwarz’ inequality to sum over $N_1 \sim N_4$.

• Subcase (4.b): $N_3 \gg N_1$. First, suppose that $N_2 \sim N_3$. Note that we must have $N_3 \gtrsim N_4$ in this case. Then, by the $L^{d+2}_{t, x}L^4_{t, x}L^4_{t, x}L^\frac{2(d+2)}{d}_{t, x}$-Hölder’s inequality with 3.5 in Lemma 3.5, we have

$$\begin{align*} &\bigg |\int _0^T \int _{ \mathbb{R}^d} v_1 z_2 \langle \nabla \rangle ^\frac{d-2}{2} z_3 v_4 dx dt \bigg |\\ &\qquad \qquad \lesssim \|v_1\|_{X^\frac{d-2}{2}} \|\langle \nabla \rangle ^\frac{d-2}{4}z_2\|_{L^4_{t, x}} \|\langle \nabla \rangle ^\frac{d-2}{4} z_3 \|_{L^4_{t, x}}\|v_4 \|_{Y^0}\\ &\qquad \qquad \lesssim N_3^{\frac{d-2}{2} - 2s} \|v_1\|_{X^\frac{d-2}{2}} \|\langle \nabla \rangle ^s z_2\|_{L^4_{t, x}} \|\langle \nabla \rangle ^s z_3 \|_{L^4_{t, x}} \|v_4 \|_{Y^0}. \end{align*}$$

Hence, by Lemma 2.2, the contribution to 4.10 in this case is at most $\lesssim T^{0+} R^2 \|v_1\|_{X^{\frac{d-2}{2}}}$ outside a set of probability

$$\begin{equation*} \leq C\exp \bigg (-c \frac{R^2}{ T^{\frac{1}{2}-} \|\phi \|_{H^{s}}^{2}}\bigg ) \end{equation*}$$

as long as $s> \frac{d-2}{4}$.

Hence, it remains to consider the case $N_3 \sim N_4 \gg N_1, N_2$.

$\circ$ Subsubcase (4.b.i): $N_1, N_2\ll N_3^\frac{1}{d-1}$. By 4.12 and 3.6 in Lemma 3.5, we have

$$\begin{align*} \bigg |\int _0^T \int _{ \mathbb{R}^d} & v_1 z_2 \langle \nabla \rangle ^\frac{d-2}{2} z_3 v_4 dx dt \bigg | \lesssim \|z_2\langle \nabla \rangle ^\frac{d-2}{2} z_3 \|_{L^2_{t, x}}\|v_1 v_4 \|_{L^2_{t, x}}\\ & \lesssim N_1^{\frac{1}{2} -}N_2^{\frac{d-1}{2}-s-\frac{d-1}{2}\alpha -} N_3^{\frac{d-4}{2}-s+\frac{1}{2}\alpha +} \|v_1\|_{X^\frac{d-2}{2}}\\ &\hphantom {XXXXXX}\times \prod _{j = 2}^3\big (\|\langle \nabla \rangle ^s z_j\|_{L^4_{t, x} }^\alpha \|\mathbf{P}_{N_j} \phi ^\omega \|_{H^{s}}^{1-\alpha }\big ) \|v_4\|_{Y^0}. \end{align*}$$

Hence, by Lemmata 2.2 and 2.4, the contribution to 4.10 in this case is at most $\lesssim T^{0+} R^2 \|v_1\|_{X^\frac{d-2}{2}}$ outside a set of probability

$$\begin{equation*} \leq C\exp \bigg (-c \frac{R^2}{ T^{\frac{1}{2}-} \|\phi \|_{H^{s}}^{2}}\bigg ) + C\exp \bigg (-c \frac{R^2}{ \|\phi \|_{H^{s}}^{2}}\bigg ) \end{equation*}$$

as long as

$$\begin{align} s > \frac{(d-2)^2}{2d} = \frac{d-2}{d}\cdot \frac{d-2}{2} {\tag{4.17}} \end{align}$$

and $\alpha < 1 - \frac{2}{d-1} s$. Note that the condition 4.17 is less restrictive than 4.13 and thus does not add a further constraint.

$\circ$ Subsubcase (4.b.ii): $N_1\ll N_3^{\frac{1}{d-1}} \lesssim N_2$. By Hölder’s inequality and 3.6 in Lemma 3.5, we have

$$\begin{align*} \bigg |\int _0^T \int _{ \mathbb{R}^d} v_1 z_2 & \langle \nabla \rangle ^\frac{d-2}{2} z_3 v_4 dx dt \bigg | \lesssim \|z_2\|_{L^4_{t, x}} \|\langle \nabla \rangle ^\frac{d-2}{2} z_3 \|_{L^4_{t, x}}\|v_1 v_4 \|_{L^2_{t, x}}\\ & \lesssim N_1^{\frac{1}{2} - } N_2^{-s} N_3^{\frac{d-3}{2} - s+} \|v_1\|_{X^\frac{d-2}{2}} \prod _{j = 2}^3 \|\langle \nabla \rangle ^s z_j\|_{L^4_{t, x}}\|v_4\|_{Y^0}. \end{align*}$$

Hence, by Lemma 2.2, the contribution to 4.10 in this case is at most $\lesssim T^{0+}R^2 \|v_1\|_{X^{\frac{d-2}{2}}}$ outside a set of probability

$$\begin{equation*} \leq C\exp \bigg (-c \frac{R^2}{ T^{\frac{1}{2}-} \|\phi \|_{H^{s}}^{2}}\bigg ) \end{equation*}$$

as long as 4.17 is satisfied.

$\circ$ Subsubcase (4.b.iii): $N_2 \ll N_3^{\frac{1}{d-1}} \lesssim N_1$. By Hölder’s inequality and Lemma 3.5, we have

$$\begin{align*} \bigg |\int _0^T \int _{\mathbb{R}^d} & v_1 z_2 \langle \nabla \rangle ^\frac{d-2}{2} z_3 v_4 dx dt \bigg | \lesssim \|v_1\|_{L^\frac{2(d+2)}{d}_{t, x}} \|\langle \nabla \rangle ^\frac{d-2}{2} z_3 \|_{L^{d+2}_{t, x}}\|z_2 v_4 \|_{L^2_{t, x}}\\ & \lesssim N_1^{ - \frac{d-2}{2}} N_2^{\frac{d-1}{2} - s-} N_3^{\frac{d-3}{2}-s+} \|v_1\|_{X^{\frac{d-2}{2}}} \|\mathbf{P}_{N_2}\phi ^\omega \|_{H^s} \|\langle \nabla \rangle ^s z_3\|_{L^{d+2}_{t, x}}\|v_4\|_{Y^0}. \end{align*}$$

Hence, by Lemmata 2.4 and 2.2, the contribution to 4.10 in this case is at most $\lesssim T^{0+} R^2 \|v_1\|_{X^{\frac{d-2}{2}}}$ outside a set of probability

$$\begin{equation*} \leq C\exp \bigg (-c \frac{R^2}{ \|\phi \|_{H^{s}}^{2}}\bigg ) + C\exp \bigg (-c \frac{R^2}{ T^{\frac{2}{d+2}-} \|\phi \|_{H^{s}}^{2}}\bigg ) \end{equation*}$$

as long as 4.17 is satisfied.

$\circ$ Subsubcase (4.b.iv): $N_1, N_2\gtrsim N_3^{\frac{1}{d-1}}$. By the $L^\frac{2(d+2)}{d}_{t,x}L^{d+2}_{t,x}L^{d+2}_{t,x}L^\frac{2(d+2)}{d}_{t,x}$-Hölder’s inequality and 3.5 in Lemma 3.5, we have

$$\begin{align*} &\bigg |\int _0^T \int _{\mathbb{R}^d} v_1 z_2 \langle \nabla \rangle ^\frac{d-2}{2} z_3 v_4 dx dt \bigg |\\ &\qquad \qquad \lesssim \|v_1\|_{L^\frac{2(d+2)}{d}_{t, x}} \|z_2\|_{L^{d+2}_{t, x}} \|\langle \nabla \rangle ^\frac{d-2}{2} z_3 \|_{L^{d+2}_{t, x}}\| v_4 \|_{L^\frac{2(d+2)}{d}_{t, x}}\\ &\qquad \qquad \lesssim N_1^{- \frac{d-2}{2} } N_2^{-s} N_3^{\frac{d-2}{2} -s} \|v_1\|_{X^\frac{d-2}{2}} \prod _{j = 2}^3 \|\langle \nabla \rangle ^s z_j\|_{L^{d+2}_{t, x}}\|v_4\|_{Y^0}. \end{align*}$$

Hence, by Lemma 2.2, the contribution to 4.10 in this case is at most $\lesssim T^{0+} R^2 \|v_1\|_{X^{\frac{d-2}{2} }}$ outside a set of probability

$$\begin{equation*} \leq C\exp \bigg (-c \frac{R^2}{ T^{\frac{2}{d+2}-} \|\phi \|_{H^{s}}^{2}}\bigg ) \end{equation*}$$

as long as 4.17 is satisfied.

Putting together Cases (1) - (4) above, the conclusion of part (i) follows, provided that 4.13 is satisfied.

(ii) First, define $\widetilde{\Gamma }_N$ by

$$\begin{equation*} \widetilde{\Gamma }_N v(t) =\mp i \int _0^t S(t-t')\mathbf{P}_{\leq N} \mathcal{N}(v+\varepsilon z)(t') dt' \end{equation*}$$

for $N \geq 1$. As before, we have

$$\begin{align} \|\mathbf{P}_{\leq N} \mathcal{N}(v+\varepsilon z)\|_{L^1_t (\mathbb{R}; H^{\frac{d-2}{2}}_x)} & \lesssim N^\frac{d-2}{2} \| v\|_{L^3_t(\mathbb{R}; L^6_x)}^3 + \varepsilon ^3 N^\frac{d-2}{2} \| z\|_{L^3_t(\mathbb{R}; L^6_x)}^3. {\tag{4.18}} \end{align}$$

By a computation similar to 4.9, we see that the first term is finite. Noting that $(3, \frac{6d}{3d-4})$ is Strichartz admissible and $6 \geq \frac{6d}{3d-4}$, it follows from Lemma 2.3 that the second term on the right-hand side of 4.18 is finite almost surely. Hence, we can apply Lemma 3.5 (i) to $\widetilde{\Gamma }_N v$ for each finite $N\geq 1$, almost surely.

The rest of the proof for this part follows in a similar manner to the proof of part (i) by changing the time interval from $[0, T)$ to $\mathbb{R}$ and replacing $z$ by $\varepsilon z$. By applying Lemma 2.3 instead of Lemma 2.2 in the above computation, we see that the contribution to 4.10, where $[0, T)$ is replaced by $\mathbb{R}$, is given by

$$\begin{equation*} \text{Case (2):} \ R^3,\quad \text{Case (3):} \ R\prod _{j = 1}^2 \|v_j\|_{X^\frac{d-2}{2}(\mathbb{R})}, \quad \text{Case (4):} \ R^2 \|v_1\|_{X^\frac{d-2}{2}(\mathbb{R})} \end{equation*}$$

outside a set of probability

$$\begin{equation*} \leq C\exp \bigg (-c \frac{R^2}{\varepsilon ^2 \|\phi \|_{H^{s}}^{2}}\bigg ) \end{equation*}$$

in all cases as long as $s > s_d$.

■

5. Proofs of Theorems 1.1 and 1.2

In this section, we establish the almost sure local well-posedness (Theorem 1.1) and probabilistic small data global theory (Theorem 1.2). First, we present the proof of Theorem 1.1. Given $C_1$ and $C_2$ as in 4.2 and 4.3, let $\eta _1 > 0$ be sufficiently small such that

$$\begin{equation} C_1 \eta _1^2 \leq \frac{1}{2} \qquad \text{and}\qquad 2C_2 \eta _1^2 \leq \frac{1}{4}. {\tag{5.1}} \end{equation}$$

Also, given $R\gg 1$, choose $T = T(R)$ such that

$$\begin{equation*} T^\theta = \min \bigg ( \frac{\eta _1}{2C_1R^3}, \frac{1}{4C_2R^2}\bigg ). \end{equation*}$$

Then, it follows from Proposition 4.1 that $\Gamma$ is a contraction on the ball $B_{\eta _1}$ defined by

$$\begin{equation*} B_{\eta _1} := \{ u\in X^\frac{d-2}{2}([0, T))\cap C([0, T); H^\frac{d-2}{2}):\, \|u\|_{X^\frac{d-2}{2}([0, T))} \leq \eta _1\} \end{equation*}$$

outside a set of probability

$$\begin{equation*} \leq C \exp \bigg (-c \frac{R^2}{\|\phi \|_{H^s}^2}\bigg ) \sim C \exp \bigg (-c \frac{1}{T^\gamma \|\phi \|_{H^s}^2}\bigg ) \end{equation*}$$

for some $\gamma > 0$. This proves Theorem 1.1.

Next, we prove Theorem 1.2. Let $\eta _2>0$ be sufficiently small such that

$$\begin{equation} 2 C_3 \eta _2^2 \leq 1\qquad \text{and}\qquad 3C_4 \eta _2^2 \leq \frac{1}{2}, {\tag{5.2}} \end{equation}$$

where $C_3$ and $C_4$ are as in 4.5 and 4.6. Then, by Proposition 4.1 with $R = \eta _2$ and $\phi ^\omega$ replaced by $\varepsilon \phi ^\omega$, we have

$$\begin{align} \|\widetilde{\Gamma }v\|_{X^\frac{d-2}{2}(\mathbb{R})} & \leq 2 C_3\eta _2^3 \leq \eta _2, {\tag{5.3}}\\ \|\widetilde{\Gamma }v_1 - \widetilde{\Gamma }v_2 \|_{X^\frac{d-2}{2}(\mathbb{R})} & \leq 3 C_4 \eta _2^2 \|v_1 -v_2 \|_{X^\frac{d-2}{2}(\mathbb{R})} \leq \frac{1}{2} \|v_1 -v_2 \|_{X^\frac{d-2}{2}(\mathbb{R})} {\tag{5.4}} \end{align}$$

outside a set of probability $\leq C \exp \big (-c \frac{\eta _2^2}{\varepsilon ^2\|\phi \|_{H^s}^2}\big )$. Noting that $\eta _2$ is an absolute constant, we conclude that there exists a set $\Omega _\varepsilon \subset \Omega$ such that (i) $\widetilde{\Gamma }= \widetilde{\Gamma }^\omega$ is a contraction on the ball $B_{\eta _2}$ defined by

$$\begin{equation*} B_{\eta _2} := \{ u\in X^\frac{d-2}{2}(\mathbb{R})\cap C(\mathbb{R}; H^\frac{d-2}{2}):\, \|u\|_{X^\frac{d-2}{2}(\mathbb{R})} \leq \eta _2\} \end{equation*}$$

for $\omega \in \Omega _\varepsilon$, and (ii) $P(\Omega _\varepsilon ^c) \leq C \exp \big (- \frac{c}{\varepsilon ^2 \|\phi \|_{H^s}^2}\big )$. This proves global existence for 1.1 with initial data $\varepsilon \phi ^\omega$ if $\omega \in \Omega _\varepsilon$.

Fix $\omega \in \Omega _\varepsilon$ and let $v = v(\varepsilon , \omega )$ be the global-in-time solution with $v|_{t = 0} = \varepsilon \phi ^\omega$ constructed above. In order to prove scattering, we need to show that there exists $v_+^\omega \in H^\frac{d-2}{2}(\mathbb{R}^d)$ such that

$$\begin{equation} S(-t) v(t) = \mp i \int _0^t S(-t') \mathcal{N}(v + \varepsilon z)(t') dt' \to v_+^\omega {\tag{5.5}} \end{equation}$$

in $H^\frac{d-2}{2}(\mathbb{R}^d)$ as $t\to \infty$. With $w(t) = S(-t) v(t)$, define $I(t_1, t_2)$ and $\widetilde{I}(t_1, t_2)$ by

$$\begin{align*} I(t_1, t_2) & := S(t_2) \big (w(t_2) - w(t_1) ), \\ \widetilde{I}(t_1, t_2)& :=\mp i \int _{0}^{t_2} S(t_2-t') \chi _{[t_1, \infty )}(t')\mathcal{N}(v + \varepsilon z)(t') dt'. \end{align*}$$

Then, for $0< t_1\leq t_2< \infty$, we have

$$\begin{align*} I(t_1, t_2) = \mp i S(t_2) \int _{t_1}^{t_2} S(-t') \mathcal{N}(v + \varepsilon z)(t') dt' = \widetilde{I}(t_1, t_2). \end{align*}$$

Also, note that $\widetilde{I}(t_1, t_2) = 0$ if $t_1 > t_2$. In the following, we view $\widetilde{I}(t_1, t_2)$ as a function of $t_2$ and estimate its $X^\frac{d-2}{2}([0, \infty ))$-norm. We now revisit the computation in the proof of Proposition 4.1 for $\widetilde{I}(t_1, t_2)$. In Case (1), we proceed slightly differently. By Lemma 3.5 (i), Hölder’s inequality, and 3.5, we have

$$\begin{align} \|\widetilde{I}(t_1, \cdot )\|_{X^\frac{d-2}{2}(\mathbb{R}_+)} & \lesssim \sup _{\substack{v_4 \in Y^{0}(\mathbb{R}_+)\\\|v_4\|_{Y^{0}} = 1}} \bigg |\int _0^\infty \int _{ \mathbb{R}^d} \chi _{[t_1, \infty )}(t) \langle \nabla \rangle ^\frac{d-2}{2} v\overline{ v} v v_4 dx dt \bigg | \\ & \leq \| \langle \nabla \rangle ^\frac{d-2}{2} v\|_{L^\frac{2(d+2)}{d}_{t, x}([t_1, \infty ))} \|v\|_{L^{d+2}_{t, x}([t_1, \infty ))}^2. {\tag{5.6}} \end{align}$$

By 3.5 in Lemma 3.5, we have

$$\begin{equation*} \| \langle \nabla \rangle ^\frac{d-2}{2} v\|_{L^\frac{2(d+2)}{d}_{t, x}(\mathbb{R})} + \|v\|_{L^{d+2}_{t, x}(\mathbb{R})}\lesssim \|v\|_{X^\frac{d-2}{2}(\mathbb{R})} \leq \eta _2. \end{equation*}$$

Then, by the monotone convergence theorem, 5.6 tends to 0 as $t_1 \to \infty$.

In Cases (2), (3), and (4), we had at least one factor of $z$. We multiply the cutoff function $\chi _{[t_1, \infty )}$ only on the $(\varepsilon z)$-factors but not on the $v$-factors. Note that $\|v\|_{X^\frac{d-2}{2}(\mathbb{R})} \leq \eta _2$. As in the proof of Proposition 4.1, we estimate at least a small portion of these $z$-factors in $\|\langle \nabla \rangle ^s \varepsilon z^\omega \|_{L^q_{t, x}([t_1, \infty ))}$, $q = 4, \frac{6(d+2)}{d+4}$, or $d+2$, in each case. Recall that we have $\|\langle \nabla \rangle ^s \varepsilon z^\omega \|_{L^q_{t, x}(\mathbb{R})} \leq \eta _2$ for $\omega \in \Omega _\varepsilon$. See Lemma 2.3. Hence, again by the monotone convergence theorem, we have $\|\langle \nabla \rangle ^s \varepsilon z^\omega \|_{L^q_{t, x}([t_1, \infty ))} \to 0$ as $t_1 \to \infty$ and thus the contribution from Cases (2), (3), and (4) tends to 0 as $t_1 \to \infty$. Therefore, we have

$$\begin{equation*} \lim _{t_1\to \infty } \| \widetilde{I}(t_1, t_2) \|_{X^\frac{d-2}{2}([0, \infty ))} = 0. \end{equation*}$$

In conclusion, we obtain

$$\begin{align*} \lim _{t_1\to \infty }& \sup _{t_2 > t_1} \| w(t_2) - w(t_1) \|_{H^\frac{d-2}{2}} = \lim _{t_1\to \infty } \sup _{t_2 > t_1} \| I(t_1, t_2) \|_{H^\frac{d-2}{2}}\\ & = \lim _{t_1\to \infty } \| \widetilde{I}(t_1, t_2) \|_{L^\infty _{t_2}([0, \infty ); H^\frac{d-2}{2})} \lesssim \lim _{t_1\to \infty } \| \widetilde{I}(t_1, t_2) \|_{X^\frac{d-2}{2}([0, \infty ))} = 0. \end{align*}$$

This proves 5.5 and scattering of $u^\omega (t) = \varepsilon S(t) \phi ^\omega + v^\omega (t)$, which completes the proof of Theorem 1.2.

6. Local well-posedness of NLS with a deterministic perturbation

In this and the next sections, we consider the following Cauchy problem of the defocusing NLS with a perturbation:

$$\begin{equation} \begin{cases} i \partial _tv + \Delta v = |v + f|^2(v+f),\\ v|_{t = t_0} = v_0, \end{cases} {\tag{6.1}} \end{equation}$$

where $f$ is a given deterministic function. Assuming some suitable conditions on $f$, we prove local well-posedness of 6.1 in this section (Proposition 6.3) and long time existence under further assumptions in Section 7 (Proposition 7.2). Then, we show, in Section 8, that the conditions imposed on $f$ for long time existence are satisfied with a large probability by setting $f(t) = z(t) = S(t) \phi ^\omega$. This yields Theorem 1.3.

Our main goal is to prove long time existence of solutions to the perturbed NLS 6.1 by iteratively applying a perturbation lemma (Lemma 7.1). For this purpose, we first prove a “variant” local well-posedness of 6.1. As in the usual critical regularity theory, we first introduce an auxiliary scaling-invariant norm which is weaker than the $X^\frac{d-2}{2}$-norm. Given an interval $I \subset \mathbb{R}$, we introduce the $Z$-norm by

$$\begin{equation} \|u\|_{Z(I)} := \bigg ( \sum _{\substack{N\geq 1 \\ \text{ dyadic}}} N^{d-2} \|\mathbf{P}_{N} u\|_{L^4_{t, x}(I\times \mathbb{R}^d)}^4 \bigg )^\frac{1}{4}. {\tag{6.2}} \end{equation}$$

By the Littlewood-Paley theory and 3.5 in Lemma 3.5, we have

$$\begin{equation*} \|u\|_{Z(I)} \lesssim \| \langle \nabla \rangle ^\frac{d-2}{4} u \|_{L^4_{t, x}(I\times \mathbb{R}^d)} \lesssim \|u\|_{X^\frac{d-2}{2}(I)}. \end{equation*}$$

Given $\theta \in (0, 1)$, we define the $Z_\theta$-norm by

$$\begin{equation*} \|u\|_{Z_\theta (I)} := \|u\|_{Z(I)}^\theta \|u\|_{X^\frac{d-2}{2}(I)}^{1-\theta }. \end{equation*}$$

Note that the $Z_\theta$-norm is weaker than the $X^\frac{d-2}{2}$-norm:

$$\begin{equation} \|u\|_{Z_\theta (I)} \leq C_0 \|u\|_{X^\frac{d-2}{2}(I)} {\tag{6.3}} \end{equation}$$

for some $C_0 > 0$ independent of $I$.

First, we present the bilinear Strichartz estimate adapted to the $Z_\theta$-norm.

Lemma 6.1.

Let $N_1 \leq N_2$. Then, we have

$$\begin{align} \| \mathbf{P}_{N_1} u_1 \mathbf{P}_{N_2}u_2\|_{L^2_{t, x}(I\times \mathbb{R}^d)} & \lesssim \bigg (\frac{N_1}{N_2}\bigg )^{\frac{1}{2}(1-\theta )-} \|\mathbf{P}_{N_1} u_1\|_{Z_\theta (I)}\|\mathbf{P}_{N_2} u_2\|_{Y^0(I)}. {\tag{6.4}} \end{align}$$

Proof.

Given a cube $R$ of side length $N_1$ centered at $\xi _0\in N_1\mathbb{Z}^d$, let $\mathbf{P}_R = \psi \big (\frac{D-\xi _0}{N_1}\big )$ denote a smooth projection onto $R$ on the frequency side. Here, $\psi$ denotes the smooth projection onto the unit cube defined in 1.6. By 3.10, we have

$$\begin{align*} \| \mathbf{P}_{N_1} u_1 \mathbf{P}_{R}\mathbf{P}_{N_2}u_2\|_{L^2_{t, x}(I)} & \leq \|\mathbf{P}_{N_1} u_1 \|_{L^4_{t, x}} \|\mathbf{P}_{R}\mathbf{P}_{N_2}u_2\|_{L^4_{t, x}}\\ & \lesssim N_1^\frac{d-2}{4} \|\mathbf{P}_{N_1} u_1 \|_{L^4_{t, x}(I)} \|\mathbf{P}_{R}\mathbf{P}_{N_2}u_2\|_{Y^0(I)}. \end{align*}$$

Then, by almost orthogonality, we have

$$\begin{align*} \| \mathbf{P}_{N_1} u_1 & \mathbf{P}_{N_2}u_2\|_{L^2_{t, x}(I)} \sim \Big (\sum _{R} \| \mathbf{P}_{N_1} u_1 \mathbf{P}_R \mathbf{P}_{N_2}u_2\|^2_{L^2_{t, x}(I)}\Big )^\frac{1}{2}\\ & \lesssim \|\mathbf{P}_{N_1} u_1\|_{Z(I)} \Big (\sum _{R} \|\mathbf{P}_{R}\mathbf{P}_{N_2} u_2\|^2_{Y^0(I)}\Big )^\frac{1}{2} \lesssim \|\mathbf{P}_{N_1} u_1\|_{Z(I)} \|\mathbf{P}_{N_2} u_2\|_{Y^0(I)}. \end{align*}$$

Then, 6.4 follows from interpolating this with 3.6.

■

Next, we state the key nonlinear estimate. Given $I \subset \mathbb{R}$, we define the $W^s$-norm by

$$\begin{align} \|f\|_{W^s(I)} := \max \Big (\|\langle \nabla \rangle ^s f \|_{L^4_{t, x}(I)} , \|\langle \nabla \rangle ^sf\|_{L^{d+2}_{t, x}(I)}, \|\langle \nabla \rangle ^sf\|_{L^\frac{6(d+2)}{d+4}_{t, x}(I)}\Big ). {\tag{6.5}} \end{align}$$

As in the proof of Proposition 4.1, different space-time norms of $f$ appear in the estimate but they are all controlled by this $W^s$-norm. The following lemma is analogous to Proposition 4.1 but with one important difference. All the terms on the right-hand side have (i) two factors of the $Z_\theta$-norm of $v_j$, which is weaker than the $X^s$-norm, or (ii) the $W^s$-norm of $f$, which can be made small by shrinking the interval $I$.

Lemma 6.2.

Let $d \geq 3$ and $\theta \in (0, 1)$. Suppose that $s , \alpha \in \mathbb{R}$ satisfy

$$\begin{align} s \in (s_d, s_\text{crit}] \qquad \text{and} \qquad 0 < \alpha < 1- \frac{2}{d-1} s, {\tag{6.6}} \end{align}$$

where $s_d$ is as in 1.11. Then, given any interval $I = [t_0, t_1]\subset \mathbb{R}$, we have

$$\begin{align*} \bigg \| \prod _{j = 1}^3(v_j & + f)^* \bigg \|_{N^\frac{d-2}{2}(I)} \lesssim \sum _{j = 1}^3 \|v_j\|_{X^\frac{d-2}{2}(I)} \prod _{\substack{k = 1\\k\ne j}}^3\|v_k\|_{Z_\theta (I)}\\ & \hphantom {X} + \prod _{\substack{j, k =1\\j\ne k}}^3 \|v_j\|_{X^\frac{d-2}{2}(I)}\|v_k\|_{X^\frac{d-2}{2}(I)} \Big ( \|f\|_{W^s(I)} + \|f\|_{Y^s(I)}^{1-\alpha } \|f\|_{W^s(I)}^\alpha \Big ) \\ & \hphantom {X} + \sum _{j = 1}^3 \|v_j\|_{X^{\frac{d-2}{2}}(I)} \Big ( \|f\|_{Y^s(I)} \|f\|_{W^s(I)} + \|f\|_{Y^s(I)}^{2-2\alpha } \|f\|_{W^s(I)}^{2\alpha } + \|f\|_{W^s(I)}^2\Big )\\ & \hphantom {X} + \|f \|_{Y^s(I)}\|f\|_{W^s(I)}^2 + \|f\|_{Y^s(I)}^{3-2\alpha } \|f\|_{W^s(I)}^{2\alpha } + \|f\|_{W^s(I)}^3 \end{align*}$$

for all $f \in W^s(I) \cap Y^s(I)$ and $v_j\in X^{\frac{d-2}{2}}(I)$, $j=1,2,3$, where $(v_j + f)^* = v_j + f$ or $\overline{v}_j + \overline{f}$.

We first state and prove the following local well-posedness result for the perturbed NLS 6.1, assuming Lemma 6.2. The proof of Lemma 6.2 is presented at the end of this section.

Proposition 6.3 (Local well-posedness of the perturbed NLS).

Given $d \geq 3$, let $s \in ( s_d, s_\text{crit}]$, where $s_d$ is defined in 1.11. Let $\theta \in (\frac{1}{2}, 1)$ and $\alpha \in \mathbb{R}$ satisfy 6.6. Suppose that

$$\begin{equation*} \|v_0\|_{H^{\frac{d-2}{2}}}\leq R \quad \text{ and } \quad \|f\|_{Y^s(I)}\leq M \end{equation*}$$

for some $R, M\geq 1$. Then, there exists small $\eta _0 = \eta _0(R,M)>0$ such that if

$$\begin{equation*} \|S(t-t_0)v_0\|_{Z_\theta (I)}\leq \eta \qquad \text{and} \qquad \|f\|_{W^s(I)} \leq \eta ^{\frac{4-\alpha }{\alpha }} \end{equation*}$$

for some $\eta \leq \eta _0$ and some time interval $I = [t_0, t_1] \subset \mathbb{R}$, then there exists a unique solution $v \in X^\frac{d-2}{2}(I)\cap C(I; H^\frac{d-2}{2}(\mathbb{R}^d))$ to 6.1 with $v(t_0) = v_0$. Moreover, we have

$$\begin{align} \|v - S(t-t_0) v_0\|_{X^\frac{d-2}{2}(I)} \lesssim \eta ^{3-2\theta }. {\tag{6.7}} \end{align}$$

Proof.

For $\theta \in (\frac{1}{2} , 1)$, we show that the map $\Gamma$ defined by

$$\begin{equation} \Gamma v(t) := S(t-t_0) v_0 -i \int _{t_0}^t S(t -t') \mathcal{N}(v+f)(t') dt' {\tag{6.8}} \end{equation}$$

is a contraction on

$$\begin{equation*} B_{R, M, \eta } = \{ v \in X^\frac{d-2}{2}(I)\cap C(I; H^\frac{d-2}{2}): \, \|v\|_{X^\frac{d-2}{2}(I)}\leq 2\widetilde{R}, \ \|v\|_{Z_\theta (I)} \leq 2\eta \}, \end{equation*}$$

where $\widetilde{R} :=\max (R, M)$. Now, choose

$$\begin{align} \eta _0 \ll \widetilde{R}^{-\frac{1}{2\theta -1}}. {\tag{6.9}} \end{align}$$

In particular, we have $\eta _0 \ll \widetilde{R}^{-1} \leq 1$. Fix $\eta \leq \eta _0$ in the following. Noting that $\frac{4-\alpha }{\alpha } > 3$, Lemma 6.2 with Lemma 3.5 yields

$$\begin{align} \|\Gamma v\|_{X^\frac{d-2}{2}(I)} & \leq \|S(t-t_0)v_0\|_{X^\frac{d-2}{2}(I)} + \|\Gamma v - S(t-t_0) v_0\|_{X^\frac{d-2}{2}(I)} \\ & \leq \|v_0\|_{H^\frac{d-2}{2}} + C \eta ^2 \widetilde{R} \leq 2\widetilde{R} {\tag{6.10}} \end{align}$$

and

$$\begin{align*} \| \Gamma v_1 - \Gamma v_2 \|_{X^\frac{d-2}{2}(I)} \leq \frac{1}{2} \| v_1 - v_2 \|_{X^\frac{d-2}{2}(I)} \end{align*}$$

for $v, v_1, v_2 \in B_{R, M, \eta }$. Moreover, we have

$$\begin{align*} \|\Gamma v\|_{Z_\theta (I)} & \leq \big (\|S(t-t_0) v_0\|_{Z(I)} + C \eta ^2 \widetilde{R}\big )^{\theta } \big (\|S(t-t_0) v_0\|_{X^\frac{d-2}{2}(I)} + C \eta ^2\widetilde{R} \big )^{1-\theta }\\ & \leq \eta + C \eta ^{2\theta }\widetilde{R} +C\eta ^{2-\theta }\widetilde{R}^{1-\theta }+ C\eta ^2 \widetilde{R} \leq 2 \eta \end{align*}$$

for $v \in B_{R, M, \eta }$. Hence, $\Gamma$ is a contraction on $B_{R, M, \eta }$. The estimate 6.7 follows from 6.9 and 6.10.

■

We conclude this section by presenting the proof of Lemma 6.2. Some cases follow directly from the proof of Proposition 4.1. However, due to the use of the $Z_\theta$-norm, we need to make modifications in several cases.

Proof of Lemma 6.2.

As in the proof of Proposition 4.1, we need to estimate the right-hand side of 4.10 by performing a case-by-case analysis of expressions of the form:

$$\begin{align} \bigg | \iint _{I\times \mathbb{R}^d} \langle \nabla \rangle ^\frac{d-2}{2} ( w_1 w_2 w_3 )v_4 dx dt\bigg |, {\tag{6.11}} \end{align}$$

where $\|v_4\|_{Y^{0}(I)} \leq 1$ and $w_j= v$ or $f$, $j = 1, 2, 3$. Before proceeding further, let us simplify some of the notation. In the following, as before, we drop the complex conjugate sign and denote $X^s(I)$ and $Y^s(I)$ by $X^s$ and $Y^s$. Lastly, we dyadically decompose $w_j$, $j = 1, 2, 3$, and $v_4$ such that their spatial frequency supports are $\{ |\xi _j|\sim N_j\}$ for some dyadic $N_j \geq 1$ but still denote them as $w_j = v_j$ or $f_j$, $j = 1, 2, 3$, and $v_4$ if there is no confusion.

Case (1): $v v v$ case.

Without loss of generality, assume that $N_1\geq N_2, N_3$.

• Subcase (1.a): $N_1 \sim N_4$. By Lemma 6.1, we have

$$\begin{align*} &\bigg |\iint _{ I\times \mathbb{R}^d} \langle \nabla \rangle ^\frac{d-2}{2} v_1 v_2 v_3 v_4 dx dt \bigg | \\ &\qquad \lesssim \sum _{N_1\sim N_4\gtrsim N_2, N_3} N_1^\frac{d-2}{2} \|\mathbf{P}_{N_1} v_1\mathbf{P}_{N_3} v_3\|_{L^2_{t, x}} \|\mathbf{P}_{N_2}v_2\mathbf{P}_{N_4} v_4\|_{L^2_{t, x}}\\ &\qquad \lesssim \sum _{N_1, \dots , N_4} \bigg (\frac{N_2N_3}{N_1N_4}\bigg )^{\frac{1}{2}(1-\theta )-} \|\mathbf{P}_{N_1} v_1\|_{Y^\frac{d-2}{2}}\|\mathbf{P}_{N_2}v_2\|_{Z_\theta } \|\mathbf{P}_{N_3}v_3\|_{Z_\theta } \|\mathbf{P}_{N_4}v_4\|_{Y^0}. \end{align*}$$

By first summing over $N_2, N_3 \leq N_1$ and then applying Cauchy-Schwarz’ inequality in summing over $N_1 \sim N_4$, we have

$$\begin{align*} & \lesssim \|v_1\|_{X^\frac{d-2}{2}(I)}\|v_2\|_{Z_\theta (I)} \|v_3\|_{Z_\theta (I)}. \end{align*}$$

• Subcase (1.b): $N_1 \sim N_2 \gg N_4$. By Lemma 6.1 and 3.5 in Lemma 3.5, we have

$$\begin{align*} &\bigg | \iint _{ I\times \mathbb{R}^d} \langle \nabla \rangle ^\frac{d-2}{2} v_1 v_2 v_3 v_4 dx dt \bigg |\\ &\quad \lesssim \sum _{N_1 \sim N_2 \geq N_3, N_4} N_1^\frac{d-2}{2} \|\mathbf{P}_{N_1} v_1\mathbf{P}_{N_3} v_3\|_{L^2_{t, x}} \|\mathbf{P}_{N_2}v_2\mathbf{P}_{N_4} v_4\|_{L^2_{t, x}}\\ &\quad \lesssim \sum _{N_1 \sim N_2 \geq N_3, N_4} \bigg (\frac{N_3}{N_1}\bigg )^{\frac{1}{2}(1-\theta )-} \|\mathbf{P}_{N_1} v_1\|_{Y^\frac{d-2}{2}} \|\mathbf{P}_{N_3}v_3\|_{Z_\theta } \|\mathbf{P}_{N_2}v_2\|_{L^4_{t, x}} \|\mathbf{P}_{N_4}v_4\|_{L^4_{t, x}}\\ & \quad \lesssim \sum _{N_1 \sim N_2 \geq N_3, N_4} \bigg (\frac{N_3}{N_1}\bigg )^{\frac{1}{2}(1-\theta )-} \bigg (\frac{N_4}{N_2}\bigg )^\frac{d-2}{4} \|\mathbf{P}_{N_1} v_1\|_{Y^\frac{d-2}{2}} \|\mathbf{P}_{N_3}v_3\|_{Z_\theta }\\ & \hphantom {XXXXXXXXXXXXXXXX} \times N_2^\frac{d-2}{4} \|\mathbf{P}_{N_2}v_2\|_{L^4_{t, x}} \|\mathbf{P}_{N_4}v_4\|_{Y^0}. \end{align*}$$

Summing over $N_3$ and taking a supremum in $N_2$,

$$\begin{align*} & \lesssim \|v_2\|_{Z}\|v_3\|_{Z_\theta } \sum _{N_1 \gg N_4} \bigg (\frac{N_4}{N_1}\bigg )^\frac{d-2}{4} \|\mathbf{P}_{N_1} v_1\|_{Y^\frac{d-2}{2}} \|\mathbf{P}_{N_4}v_4\|_{Y^0}. \end{align*}$$

By Lemma 3.6, we have

$$\begin{align*} & \lesssim \|v_1\|_{Y^\frac{d-2}{2}} \|v_2\|_{Z_\theta } \|v_3\|_{Z_\theta } \|v_4\|_{Y^0} \lesssim \|v_1\|_{X^\frac{d-2}{2}(I)} \|v_2\|_{Z_\theta (I)}\|v_3\|_{Z_\theta (I)}. \end{align*}$$

In the following, the desired estimates follow from the corresponding cases in the proof of Proposition 4.1. Hence, we just state the results.

Case (2): $fff$ case.  Without loss of generality, assume $N_3 \geq N_2 \geq N_1$.

• Subcase (2.a): $N_2 \sim N_3$.

The contribution to 6.11 in this case is at most

$$\begin{equation*} \lesssim \|f\|_{L^{d+2}_{t, x}}^2 \|\langle \nabla \rangle ^{\frac{d-2}{4}+}f\|_{L^{4}_{t, x}}^2 \leq \|f\|^3_{W^s(I)} \end{equation*}$$

as long as $s > \frac{d-2}{4}$.

• Subcase (2.b): $N_3 \sim N_4 \gg N_1, N_2$.

$\circ$ Subsubcase (2.b.i): $N_1, N_2 \ll N_3^\frac{1}{d-1}$. The contribution to 6.11 in this case is at most

$$\begin{equation*} \lesssim \|f\|_{Y^s}^{3-2\alpha } \|\langle \nabla \rangle ^s f\|_{L^4_{t, x} }^{2\alpha } \leq \|f\|_{Y^s(I)}^{3-2\alpha } \|f\|_{W^s(I)}^{2\alpha } \end{equation*}$$

as long as 4.13 is satisfied and $\alpha < 1- \frac{2}{d-1} s$.

$\circ$ Subsubcase (2.b.ii): $N_2\gtrsim N_3^\frac{1}{d-1} \gg N_1$. The contribution to 6.11 in this case is at most

$$\begin{equation*} \lesssim \|f \|_{Y^s(I)} \|\langle \nabla \rangle ^s f\|_{L^4_{t, x}}^2 \leq \|f \|_{Y^s(I)}\|f\|_{W^s(I)}^2 \end{equation*}$$

as long as 4.13 is satisfied.

$\circ$ Subsubcase (2.b.iii): $N_1, N_2\gtrsim N_3^\frac{1}{d-1}$. The contribution to 6.11 in this case is at most

$$\begin{equation*} \lesssim \| \langle \nabla \rangle ^s f\|_{L^\frac{6(d+2)}{d+4}_{t, x}}^3 \leq \|f\|_{W^s(I)}^3 \end{equation*}$$

as long as 4.13 is satisfied.

Case (3): $v v f$ case.

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

• Subcase (3.a): $N_1 \gtrsim N_3$. The contribution to 6.11 in this case is at most

$$\begin{equation*} \lesssim \|v\|_{X^\frac{d-2}{2}}^2 \| \langle \nabla \rangle ^s f\|_{L^{d+2}_{t, x}} \leq \|v\|_{X^\frac{d-2}{2}(I)}^2 \|f\|_{W^s(I)} \end{equation*}$$

as long as $s > 0$.

• Subcase (3.b): $N_3\sim N_4 \gg N_1 \geq N_2$.

$\circ$ Subsubcase (3.b.i): $N_1\gtrsim N_3^\frac{1}{d-1}$. The contribution to 6.11 in this case is at most

$$\begin{equation*} \lesssim \|v\|_{X^\frac{d-2}{2}}^2 \| \langle \nabla \rangle ^s f\|_{L^{d+2}_{t, x}} \leq \|v\|_{X^\frac{d-2}{2}(I)}^2 \|f\|_{W^s(I)} \end{equation*}$$

as long as 4.14 is satisfied.

$\circ$ Subsubcase (3.b.ii): $N_2 \leq N_1\ll N_3^\frac{1}{d-1}$.

The contribution to 6.11 in this case is at most

$$\begin{equation*} \lesssim \|v\|_{X^\frac{d-2}{2}}^2 \| f\|_{Y^s}^{1-\alpha } \| \langle \nabla \rangle ^s f\|_{L^{d+2}_{t, x}}^\alpha \leq \|v\|_{X^\frac{d-2}{2}(I)}^2 \| f\|_{Y^s(I)}^{1-\alpha } \|f\|_{W^s(I)}^\alpha \end{equation*}$$

as long as 4.14 is satisfied.

Case (4): $v f f$ case. Without loss of generality, assume $N_3 \geq N_2$.

• Subcase (4.a): $N_1 \gtrsim N_3$. The contribution to 6.11 in this case is at most

$$\begin{equation*} \lesssim \|v\|_{X^{\frac{d-2}{2}}} \|f\|_{L^{d+2}_{t, x}}^2 \leq \|v\|_{X^{\frac{d-2}{2}}(I)} \|f\|_{W^s(I)}^2 \end{equation*}$$

as long as $s> 0$.

• Subcase (4.b): $N_3 \gg N_1$. First, suppose that $N_2 \sim N_3$. Then, the contribution to 6.11 in this case is at most

$$\begin{equation*} \lesssim \|v\|_{X^{\frac{d-2}{2}}} \|\langle \nabla \rangle ^s f\|_{L^4_{t, x}}^2 \leq \|v\|_{X^{\frac{d-2}{2}}(I)} \|f\|_{W^s(I)}^2 \end{equation*}$$

as long as $s> \frac{d-2}{4}$.

Hence, it remains to consider the case $N_3 \sim N_4 \gg N_1, N_2$.

$\circ$ Subsubcase (4.b.i): $N_1, N_2\ll N_3^\frac{1}{d-1}$.

The contribution to 6.11 in this case is at most

$$\begin{equation*} \lesssim \|v\|_{X^{\frac{d-2}{2}}} \|f\|_{Y^s}^{2-2\alpha } \|\langle \nabla \rangle ^s f\|_{L^4_{t, x} }^{2\alpha } \leq \|v\|_{X^{\frac{d-2}{2}}(I)} \|f\|_{Y^s(I)}^{2-2\alpha } \|f\|_{W^s(I)}^{2\alpha } \end{equation*}$$

as long as 4.17 is satisfied and $\alpha < 1 - \frac{2}{d-1}s$.

$\circ$ Subsubcase (4.b.ii): $N_1\ll N_3^{\frac{1}{d-1}} \lesssim N_2$. The contribution to 6.11 in this case is at most

$$\begin{equation*} \lesssim \|v\|_{X^{\frac{d-2}{2}}} \|\langle \nabla \rangle ^s f\|_{L^4_{t, x}}^2 \leq \|v\|_{X^{\frac{d-2}{2}}(I)} \|f\|_{W^s(I)}^2 \end{equation*}$$

as long as 4.17 is satisfied.

$\circ$ Subsubcase (4.b.iii): $N_2 \ll N_3^{\frac{1}{d-1}} \lesssim N_1$. The contribution to 6.11 in this case is at most

$$\begin{equation*} \lesssim \|v\|_{X^{\frac{d-2}{2}}} \|f\|_{Y^s} \|\langle \nabla \rangle ^s f\|_{L^{d+2}_{t, x}} \leq \|v\|_{X^{\frac{d-2}{2}}(I)} \|f\|_{Y^s(I)} \|f\|_{W^s(I)} \end{equation*}$$

as long as 4.17 is satisfied.

$\circ$ Subsubcase (4.b.iv): $N_1, N_2\gtrsim N_3^{\frac{1}{d-1}}$ The contribution to 6.11 in this case is at most

$$\begin{equation*} \lesssim \|v\|_{X^{\frac{d-2}{2}}} \|\langle \nabla \rangle ^s f\|_{L^{d+2}_{t, x}}^2 \leq \|v\|_{X^{\frac{d-2}{2}}(I)} \|f\|_{W^s(I)}^2 \end{equation*}$$

as long as 4.17 is satisfied.■

7. Long time existence of solutions to the perturbed NLS

The main goal of this section is to establish long time existence of solutions to the perturbed NLS 6.1 under some assumptions. See Proposition 7.2. We achieve this goal by iteratively applying the perturbation lemma (Lemma 7.1) for the energy-critical NLS.

We first state the perturbation lemma for the energy-critical cubic NLS involving the $X^\frac{d-2}{2}$- and the $Z$-norms. See 19385657 for perturbation and stability results on usual Strichartz and Lebesgue spaces. In the context of the cubic NLS on $\mathbb{R}\times \mathbb{T}^3$, Ionescu-Pausader 31 proved a perturbation lemma involving the critical $X^{s_\text{crit}}$-norm. Our proof essentially follows their argument and is included for the sake of completeness.

Lemma 7.1 (Perturbation lemma).

Let $d \geq 3$ and let $I$ be a compact interval with $|I|\leq 1$. Suppose that $v \in C(I; H^\frac{d-2}{2}(\mathbb{R}^d))$ satisfies the following perturbed NLS:

$$\begin{align} i \partial _tv + \Delta v = |v|^2 v + e, {\tag{7.1}} \end{align}$$

satisfying

$$\begin{align} \| v\|_{Z(I)} + \|v\|_{L^\infty (I; H^\frac{d-2}{2}(\mathbb{R}^d))} \leq R {\tag{7.2}} \end{align}$$

for some $R \geq 1$. Then, there exists $\varepsilon _0 = \varepsilon _0(R) > 0$ such that if we have

$$\begin{align} \|u_0 - v(t_0) \|_{H^\frac{d-2}{2}(\mathbb{R}^d)} + \|e\|_{N^\frac{d-2}{2}(I)} \leq \varepsilon {\tag{7.3}} \end{align}$$

for some $u_0 \in H^\frac{d-2}{2}(\mathbb{R}^d)$, some $t_0 \in I$, and some $\varepsilon < \varepsilon _0$, then there exists a solution $u \in X^\frac{d-2}{2}(I)\cap C(I; H^\frac{d-2}{2}(\mathbb{R}^d))$ to the defocusing cubic NLS 1.1 with $u(t_0) = u_0$ such that

$$\begin{align} \|u\|_{X^\frac{d-2}{2}(I)}+\|v\|_{X^\frac{d-2}{2}(I)} & \leq C(R), \tag{7.4}\\ \|u - v\|_{X^\frac{d-2}{2}(I)} & \leq C(R)\varepsilon , {\tag{7.5}} \end{align}$$

where $C(R)$ is a nondecreasing function of $R$.

Proof.

Without loss of generality, we assume $t_0=\min I$. Given small $\varepsilon _1 = \varepsilon _1(R)>0$ (to be chosen later), we divide the interval $I$ into subintervals $I_j = [t_j, t_{j+1}]$ such that $I = \bigcup _{j = 0}^{L} I_j$. By choosing $L\sim \big (\frac{R}{\varepsilon _1}\big )^4$, we can guarantee that

$$\begin{equation} \|v\|_{Z(I_j)} \leq \varepsilon _1 {\tag{7.6}} \end{equation}$$

for $j = 0, \dots , L$. By assumption, we also have

$$\begin{equation} \|e\|_{N^\frac{d-2}{2}(I_j)} \leq \varepsilon < \varepsilon _0 {\tag{7.7}} \end{equation}$$

for $j = 0, \dots , L$.

• Step 1: Let $\theta \in (\frac{1}{2}, 1)$. We first claim that there exist $\eta _0= \eta _0(R) >0$ and $\varepsilon _0= \varepsilon _0(R)>0$ such that if

$$\begin{equation} \|S(t - t_*) v(t_*)\|_{Z_\theta (J)} \leq \eta _0 \qquad \text{and} \qquad \|e\|_{N^\frac{d-2}{2}(J)} \leq \varepsilon _0 {\tag{7.8}} \end{equation}$$

for some $t_*$ in a subinterval $J \subset I$, then there exists a unique solution $v$ to 7.1 on $J$, satisfying

$$\begin{equation} \|v - S(t - t_*) v(t_*) \|_{X^\frac{d-2}{2}(J)} \leq C \|S(t - t_*) v(t_*)\|_{Z_\theta (J)}^{3-2\theta } + 2 \|e\|_{N^\frac{d-2}{2}(J)}. {\tag{7.9}} \end{equation}$$

We choose $\eta _0 = \eta _0(R)$ and $\varepsilon _0 = \varepsilon _0(R)$ such that

$$\begin{align} \eta _0 \ll R^{-\frac{1}{2\theta -1}} \qquad \text{and} \qquad \varepsilon _0 \ll R^{-\frac{2}{2\theta -1}}. {\tag{7.10}} \end{align}$$

In the following, we set

$$\begin{equation*} \eta := \|S(t - t_*) v(t_*)\|_{Z_\theta (J)} \leq \eta _0 \qquad \text{and} \quad \varepsilon : = \|e\|_{N^\frac{d-2}{2}(J)} \leq \varepsilon _0. \end{equation*}$$

Then, proceeding as in the proof of Proposition 6.3, we show that the map $\Gamma$ defined by

$$\begin{equation} \Gamma v(t) := S(t-t_*) v (t_*) -i \int _{t_*}^t S(t -t')\mathcal{N}(v)(t') dt' -i \int _{t_*}^t S(t -t')e(t') dt' {\tag{7.11}} \end{equation}$$

is a contraction on

$$\begin{equation*} B_{R, \eta , \varepsilon } = \big \{ v \in X^\frac{d-2}{2}(J)\cap C(J; H^\frac{d-2}{2}): \, \|v\|_{X^\frac{d-2}{2}(J)}\leq 2 R, \ \|v\|_{Z_\theta (J)} \leq 2 (\eta + \varepsilon ^{\frac{\theta }{2}+\frac{1}{4}})\big \}. \end{equation*}$$

Indeed, by Lemma 6.2 (with $f = 0$), we have

$$\begin{align} \|\Gamma v\|_{X^\frac{d-2}{2}(J)} & \leq \|v(t_\ast )\|_{H^\frac{d-2}{2}} + C (\eta +\varepsilon ^{\frac{\theta }{2}+\frac{1}{4}})^2 R + \varepsilon \\ & \leq \|v(t_\ast )\|_{H^\frac{d-2}{2}} + C \eta ^2 R+ 2\varepsilon \leq 2 R {\tag{7.12}} \end{align}$$

and

$$\begin{align*} \| \Gamma v_1 - \Gamma v_2 \|_{X^\frac{d-2}{2}(J)} & \leq C(\eta + \varepsilon ^{\frac{\theta }{2}+\frac{1}{4}})R \| v_1 - v_2 \|_{X^\frac{d-2}{2}(J)} \leq \frac{1}{2} \| v_1 - v_2 \|_{X^\frac{d-2}{2}(J)} \end{align*}$$

for $v, v_1, v_2 \in B_{R, \eta , \varepsilon }$. Moreover, we have

$$\begin{align*} \|\Gamma v\|_{Z_\theta (J)} & \leq \big (\|S(t-t_\ast ) v(t_\ast )\|_{Z(J)} + C \eta ^2 R+ \varepsilon \big )^{\theta }\\ & \hphantom {XXXXXXX} \times \big (\|S(t-t_\ast ) v(t_\ast )\|_{X^\frac{d-2}{2}(J)} + C \eta ^2 R + \varepsilon \big )^{1-\theta }\\ & \leq \eta + C \eta ^{2\theta } R + C \eta ^{2-\theta } R^{1-\theta } + C\eta ^\theta \varepsilon ^{1-\theta } + C\varepsilon ^\theta R^{1-\theta } \leq 2 (\eta + \varepsilon ^{\frac{\theta }{2}+\frac{1}{4}}) \end{align*}$$

for $v \in B_{R, \eta , \varepsilon }$. Hence, $\Gamma$ is a contraction on $B_{R, \eta _1}$. The estimate 7.9 follows from 7.10 and 7.12.

• Step 2: Next, we claim that, given $\varepsilon _2 > 0$, we can choose $\varepsilon _j = \varepsilon _j(R, \varepsilon _2)$, $j = 0, 1$, in 7.7 and 7.6 sufficiently small such that we have

$$\begin{align} \|S(t - t_j) v(t_j) \|_{Z_\theta (I_j)} & \leq \varepsilon _2 \qquad \text{and} \qquad \|v \|_{Z_\theta (I_j)} \leq \varepsilon _2. {\tag{7.13}} \end{align}$$

Without loss of generality, assume $\varepsilon _2 \leq \frac{\eta _0}{2}$, where $\eta _0 = \eta _0(R)$ is as in Step 1. Let $h(\tau ) = \|S(t - t_j) v(t_j) \|_{Z_\theta ([t_j, t_j + \tau ])}$. Note that $h$ is continuous and $h(0) = 0$. Thus, we have $h(\tau ) \leq 2\varepsilon _2 \leq \eta _0$ for small $\tau > 0$. Then, from the Duhamel formula 7.11 with 7.6, 7.7, and 7.9, we have

$$\begin{align} h(\tau ) & \leq \|S(t - t_j) v(t_j) \|_{X^\frac{d-2}{2}([t_j, t_j + \tau ])}^{1-\theta } \|S(t - t_j) v(t_j) \|_{Z([t_j, t_j + \tau ])}^{\theta } \\ & \leq C R^{1-\theta }\big (\varepsilon _1 + \varepsilon _2^{3-2\theta } + \|e\|_{N^\frac{d-2}{2}(I_j)}\big )^\theta \\ & \leq C R^{1-\theta } \varepsilon _2^{\theta (3-2\theta )} + CR^{1-\theta }\big (\varepsilon _1 + \|e\|_{N^\frac{d-2}{2}(I_j)}\big )^\theta . {\tag{7.14}} \end{align}$$

From 7.10 with $\varepsilon _2 \leq \frac{\eta _0}{2}$, we have

$$\begin{equation} CR^{1-\theta }\varepsilon _2^{\theta (3-2\theta )}\leq C\big (R\eta _0^{(2\theta -1)}\big )^{1-\theta }\varepsilon _2\ll \varepsilon _2. {\tag{7.15}} \end{equation}$$

Hence, it follows from 7.14 and 7.15 that

$$\begin{align} h(\tau ) & \leq \tfrac{1}{2} \varepsilon _2 + C R^{1-\theta }(\varepsilon _1 + \varepsilon _0 )^{\theta } \leq \varepsilon _2 {\tag{7.16}} \end{align}$$

by choosing $\varepsilon _j = \varepsilon _j(R,\varepsilon _2)>0$ sufficiently small, $j = 0, 1$. Then, by the continuity argument, we see that 7.16 holds for all $\tau \leq t_{j+1} - t_j$. From Step 1 and 7.6, we have

$$\begin{equation} \|v\|_{Z_\theta (I_j)} = \|v\|_{Z(I_j)}^\theta \|v\|_{X^\frac{d-2}{2}(I_j)}^{1-\theta } \leq C \varepsilon _1^\theta R^{1-\theta }. {\tag{7.17}} \end{equation}$$

Therefore, 7.13 follows from 7.16 and 7.17, by choosing $\varepsilon _1 = \varepsilon _1(R, \varepsilon _2)$ smaller if necessary.

• Step 3: Given $\varepsilon _2 = \varepsilon _2(R) >0$ (to be chosen later), it follows from Step 2 that 7.13 holds as long as $\varepsilon _ j= \varepsilon _j(R)>0$, $j = 0, 1$, are sufficiently small. From Step 1 with 7.7, 7.13, and 7.12, we have

$$\begin{align} \|v\|_{X^\frac{d-2}{2}(I_j)} \leq 2R {\tag{7.18}} \end{align}$$

as long as $\varepsilon _j = \varepsilon _j(R)>0$, $j = 0, 1, 2$, are sufficiently small.

Let $u$ be a solution to the defocusing cubic NLS 1.1 with initial data $u(t_j)$ given at $t = t_j$ such that

$$\begin{align} \| u(t_j) - v(t_j) \|_{H^\frac{d-2}{2}} \leq \varepsilon < \varepsilon _0. {\tag{7.19}} \end{align}$$

Let $J_j = [t_j, t_j + \tau ] \subset I_j$ be the maximal time interval such that

$$\begin{equation} \| u - v\|_{Z_\theta (J_j)} \leq 6C_0\varepsilon , {\tag{7.20}} \end{equation}$$

where $C_0$ is as in 6.3. Such an interval exists and is nonempty, since $\tau \mapsto \| u - v\|_{Z_\theta (t_j, t_j + \tau )}$ is finite and continuous (see Lemma A.8), at least on the interval of local existence of $u$, and vanishes for $\tau = 0$.

Let $w : = u-v$. By Lemma 6.2 (with $f = 0$) with 7.7, 7.13, 7.18, 7.19, and 7.20, we have

$$\begin{align} \|w\|_{X^\frac{d-2}{2}(J_j)} & \leq \| u(t_j) - v(t_j)\|_{H^\frac{d-2}{2}} + C_1\Big \{ \|v\|_{X^\frac{d-2}{2}(J_j)}\|v\|_{Z_\theta (J_j)}\|w\|_{X^\frac{d-2}{2}(J_j)} \\ & \hphantom {XXXXX} + \|v\|_{X^\frac{d-2}{2}(J_j)}\|w\|_{X^\frac{d-2}{2}(J_j)}\|w\|_{Z_\theta (J_j)} \\ & \hphantom {XXXXX} + \|w\|_{X^\frac{d-2}{2}(J_j)}\|w\|_{Z_\theta (J_j)}^2\Big \} + \|e\|_{N^\frac{d-2}{2}(J_j)}\\ & \leq 2 \varepsilon + C_2 (\varepsilon _0 + \varepsilon _2) R \|w\|_{X^\frac{d-2}{2}(J_j)}. \end{align}$$

Taking $\varepsilon _j = \varepsilon _j(R)>0$ sufficiently small, $j = 0, 2$, such that $(\varepsilon _0+\varepsilon _2) R\ll 1$, we obtain

$$\begin{align} \|w\|_{X^\frac{d-2}{2}(J_j)} \leq 4 \varepsilon . {\tag{7.21}} \end{align}$$

Hence, from 6.3, we have

$$\begin{align} \|w\|_{Z_\theta (J_j)} \leq C_0 \|w\|_{X^\frac{d-2}{2}(J_j)} \leq 4 C_0 \varepsilon . {\tag{7.22}} \end{align}$$

From 7.18 and 7.21, we have $\|u\|_{X^\frac{d-2}{2}(J_j)} \leq 3R<\infty$. Then, from 3.4, we have $\|u\|_{\dot{S}^{\frac{d-2}{2}}(J_j)} < \infty$. In particular, this implies that $u$ can be extended to some larger interval $J' \supset \overline{J_j}$. Therefore, in view of 7.20 and 7.22, we can apply the continuity argument and conclude that $J_j = I_j$.

• Step 4: By 7.3, we have $\|u(t_0)-v(t_0)\|_{H^{\frac{d-2}{2}}}\leq \varepsilon$ for some $\varepsilon < \varepsilon _0$. Then, by Step 3, we have $\|w\|_{X^{\frac{d-2}{2}}(I_0)}\leq 4\varepsilon$ on $I_0 = [t_0, t_1]$. In particular, this yields

$$\begin{equation*} \|u(t_1)-v(t_1)\|_{H^{\frac{d-2}{2}}}\leq 4C\varepsilon . \end{equation*}$$

Then, we can apply Step 3 on the interval $I_1$ by choosing $\varepsilon _0$ (and hence $\varepsilon )$ even smaller. We argue recursively for each interval $I_j$, $j=2,\dots , L$. Note that, at each step, we make $\varepsilon _0$ smaller by a factor of $(4C)^{-1}$. Since $L\sim \big (\frac{R}{\varepsilon _1}\big )^4$ and $\varepsilon _1 = \varepsilon _1 (R)$, there are a finite number of iterative steps depending only on $R$. This allows us to choose new $\varepsilon _0 = \varepsilon _0(R)> 0$ such that, by Lemma A.4, we have

$$\begin{align*} \|u\|_{X^{\frac{d-2}{2}}(I)}+\|v\|_{X^{\frac{d-2}{2}}(I)} &\lesssim L R \lesssim C(R), \\ \|u - v\|_{X^\frac{d-2}{2}(I)} & \lesssim L\varepsilon \lesssim C(R)\varepsilon . \end{align*}$$

This completes the proof of Lemma 7.1.■

In the remaining part of this section, we consider long time existence of solutions to the perturbed NLS 6.1 under several assumptions. Given $T>0$, we assume that there exist $\beta , C, M > 0$ such that

$$\begin{equation} \|f\|_{W^s(I)} \leq C |I|^{\beta } \qquad \text{and}\qquad \|f\|_{Y^s([0, T])} \leq M {\tag{7.23}} \end{equation}$$

for any interval $I \subset [0, T]$. Then, Proposition 6.3 guarantees existence of a solution to the perturbed NLS 6.1, at least for a short time.

Proposition 7.2.

Let $d \geq 3$. Let $s \in (s_d, s_\text{crit}]$, where $s_d$ is defined in 1.11. Given $T>0$, assume the following conditions (i) - (iii):

(i)

Hypothesis (B) holds if $d \ne 4$.

(ii)

$f \in Y^s([0, T]) \cap W^s([0, T])$ satisfies 7.23.

(iii)

Given a solution $v$ to 6.1, the following a priori bound holds:$$\begin{align} \| v\|_{L^\infty ([0, T]; H^\frac{d-2}{2}(\mathbb{R}^d))} \leq R {\tag{7.24}} \end{align}$$

for some $R>0$.

Then, there exists $\tau = \tau (R,M, T, s, \beta )>0$ such that, given any $t_0 \in [0, T)$, the solution $v$ to 6.1 exists on $[t_0, t_0 + \tau ]\cap [0, T]$. In particular, condition (iii) guarantees existence of $v$ on the entire interval $[0, T]$.

Remark 7.3.

We point out that the first condition in 7.23 can be weakened as follows. Let $\tau = \tau (R, M, T, s, \beta ) > 0$ be as in Proposition 7.2. Then, it follows from the proof of Proposition 7.2 (see 7.27 and 7.28 below) that if we assume that

$$\begin{equation*} \|f\|_{W^s([t_0, t_0+\tau _*))} \leq C |\tau _*|^{\beta } \end{equation*}$$

for some $\tau _* \leq \tau$ instead of the first condition in 7.23, then the conclusion of Proposition 7.2 still holds on $[t_0, t_0 + \tau _*]\cap [0, T]$. Indeed, we use this version of Proposition 7.2 in Section 8.

Proof.

By setting $e = |v+f|^2 (v+f) - |v|^2 v$, 6.1 reduces to 7.1. In the following, we iteratively apply Lemma 7.1 on short intervals and show that there exists $\tau = \tau (R, M, T, s, \beta ) > 0$ such that 7.1 is well-posed on $[t_0, t_0 + \tau ] \cap [0, T]$ for any $t_0 \in [0, T)$.

Let $w$ be the global solution to the defocusing cubic NLS 1.1 with $w(t_0) = v(t_0) = v_0$. By 7.24, we have $\|w(t_0) \|_{H^\frac{d-2}{2}} \leq R$. Then, by Hypothesis (B), we have

$$\begin{equation*} \| w\|_{L^{d+2}_{t, x}([0, T])} \leq C(R, T) < \infty . \end{equation*}$$

By the standard argument, this implies that $\big \||\nabla |^\frac{d-2}{2} w\big \|_{L^q_tL^r_x([0, T])} \leq C'(R, T) < \infty$ for all Schrödinger admissible pairs $(q, r)$. In particular, we have $\|w\|_{Z([0, T])} \leq C''(R, T) < \infty$ and

$$\begin{align} \|w\|_{X^\frac{d-2}{2}([0, T])} & \leq \|v_0\|_{H^\frac{d-2}{2}} + \big \||\nabla |^\frac{d-2}{2} w\big \|_{L^\frac{2(d+2)}{d}_{t, x}([0, T])} \|w\|_{L^{d+2}_{t, x}([0, T])}^2 \\ & \leq C'''(R, T) < \infty . {\tag{7.25}} \end{align}$$

Let $\theta \in (\frac{1}{2}, 1)$. Given small $\eta > 0$ (to be chosen later), we divide the interval $[t_0, T]$ into $J = J(R, T, \eta )$ many subintervals $I_j = [t_j, t_{j+1}]$ such that

$$\begin{equation*} \|w\|_{Z_\theta (I_j)} \sim \eta . \end{equation*}$$

In the following, we fix the value of $\theta$ and suppress dependence of various constants such as $\tau$ and $\eta$ on $\theta$.

Fix $\tau > 0$ (to be chosen later in terms of $R$, $M$, $T$, $s$, and $\beta$) and write $[t_0, t_0+\tau ] = \bigcup _{j = 0}^{J'} \big ([t_0, t_0+\tau ]\cap I_j\big )$ for some $J' \leq J - 1$, where $[t_0, t_0+\tau ]\cap I_j \ne \emptyset$ for $0 \leq j \leq J'$ and $[t_0, t_0+\tau ]\cap I_j=\emptyset$ for $j \geq J'$.

Since the nonlinear evolution $w$ is small on each $I_j$, it follows that the linear evolution $S(t-t_j) w(t_j)$ is also small on each $I_j$. Indeed, from the Duhamel formula, we have

$$\begin{equation*} S(t-t_j) w(t_j) = w(t) +i \int _{t_j}^t S(t - t') |w|^2w(t') dt'. \end{equation*}$$

Then, from Case (1) in the proof of Lemma 6.2 with 7.25, we have

$$\begin{align*} \|S(t-t_j) w(t_j) \|_{Z_\theta (I_j)} &\leq \|w\|_{Z_\theta (I_j)} + C \|w\|_{X^\frac{d-2}{2}(I_j)} \|w\|_{Z_\theta (I_j)}^2 \leq \eta + C(R, T) \eta ^2. \end{align*}$$

By taking $\eta = \eta (R, T)>0$ sufficiently small, we have

$$\begin{align} \|S(t-t_j) w(t_j) \|_{Z_\theta (I_j)} \leq 2 \eta {\tag{7.26}} \end{align}$$

for all $j = 0, \dots , J-1$.

Now, we estimate $v$ on the first interval $I_0$. Let $\eta _0 = \eta _0(R, M)$ be as in Proposition 6.3. Then, by Lemma 3.5 (i), 7.24, and Proposition 6.3, we have

$$\begin{align*} \|v\|_{X^\frac{d-2}{2} (I_0)} & \leq \|S(t- t_0) v(t_0)\|_{X^\frac{d-2}{2}(I_0)} + \|v - S(t- t_0) v(t_0)\|_{X^\frac{d-2}{2}(I_0)} \\ & \leq R + C\eta ^{3-2\theta } \leq 2R, \end{align*}$$

as long as $2\eta < \eta _0$ and $\tau = \tau (\eta , \alpha , \beta ) = \tau (R,M, T, \alpha , \beta )> 0$ is sufficiently small so that

$$\begin{align} \|f\|_{W^s([t_0, t_0+ \tau ))} \leq C\tau ^\beta \leq \eta ^\frac{4-\alpha }{\alpha }, {\tag{7.27}} \end{align}$$

where $\alpha = \alpha (s)$ satisfies 6.6.

Next, we estimate the error term. By Lemma 6.2 with 7.23, we have

$$\begin{align} \|e\|_{N^\frac{d-2}{2}(I_0)} \leq C(R, M) \tau ^{\alpha \beta }. {\tag{7.28}} \end{align}$$

Given $\varepsilon > 0$, we can choose $\tau = \tau (R, M, T, \varepsilon , \alpha , \beta )>0$ sufficiently small so that

$$\begin{align*} \|e\|_{N^\frac{d-2}{2}(I_0)} \leq \varepsilon . \end{align*}$$

In particular, for $\varepsilon < \varepsilon _0$ with $\varepsilon _0= \varepsilon _0(R) > 0$ dictated by Lemma 7.1, the condition 7.3 is satisfied on $I_0$.

Therefore, all the conditions of Lemma 7.1 are satisfied on the first interval $I_0$, provided that $\tau = \tau (R, M, T, \varepsilon , \alpha , \beta )>0$ is chosen sufficiently small. Hence, we obtain

$$\begin{align} \| w - v\|_{X^\frac{d-2}{2}(I_0)} \leq C_0(R) \varepsilon . {\tag{7.29}} \end{align}$$

In particular, we have

$$\begin{align} \|w(t_1) - v(t_1) \|_{H^\frac{d-2}{2}} \leq C_1(R) \varepsilon . {\tag{7.30}} \end{align}$$

Then, from 7.26 and Lemma 3.5 (i) with 7.30, we have

$$\begin{align*} \|S(t - t_1) v(t_1)\|_{Z_\theta (I_1)} &\leq \|S(t-t_1)w(t_1)\|_{Z_\theta (I_1)}+\|S(t-t_1)(w(t_1)-v(t_1))\|_{Z_\theta (I_1)}\\ &\leq 2 \eta + C'_1(R) \varepsilon \leq 3\eta \end{align*}$$

by choosing $\varepsilon = \varepsilon (R, \eta ) > 0$ sufficiently small.

Proceeding as before, it follows from Proposition 6.3 with 7.24 and 7.26 that

$$\begin{align*} \|v\|_{X^\frac{d-2}{2} (I_1)} & \leq R + C\eta ^{3-2\theta } \leq 2R, \end{align*}$$

as long as $3\eta \leq \eta _0$ and $\tau > 0$ is sufficiently small so that 7.27 is satisfied. Similarly, it follows from Lemma 6.2 with 7.23 that

$$\begin{align} \|e\|_{N^\frac{d-2}{2}(I_1)} \leq C(R, M) \tau ^{\alpha \beta }\leq \varepsilon {\tag{7.31}} \end{align}$$

by choosing $\tau = \tau (R, M, T, \varepsilon , \alpha , \beta )>0$ sufficiently small. Therefore, all the conditions of Lemma 7.1 are satisfied on the second interval $I_1$, provided that $\tau = \tau (R, M, T, \varepsilon ,\alpha , \beta )$ is chosen sufficiently small and that $(C_1(R) + 1)\varepsilon <\varepsilon _0$. Hence, by Lemma 7.1, we obtain

$$\begin{align*} \| w - v\|_{X^\frac{d-2}{2}(I_1)} \leq C_0(R) (C_1(R) + 1)\varepsilon . \end{align*}$$

In particular, we have

$$\begin{align*} \|w(t_2) - v(t_2) \|_{H^\frac{d-2}{2}} \leq C_2(R) \varepsilon . \end{align*}$$

By choosing $\eta = \eta (R, M, T) > 0$ and $\tau = \tau (R, M, T, \varepsilon , \alpha ,\beta )>0$ sufficiently small, we can argue inductively and obtain

$$\begin{align} \|w(t_j) - v(t_j) \|_{H^\frac{d-2}{2}} \leq C_j(R) \varepsilon \tag{7.32} \end{align}$$

for all $0 \leq j \leq J'$, as long as (i) $(J'+2)\eta \leq \eta _0$ and (ii) $\varepsilon = \varepsilon (R, \eta , J)$ is sufficiently small such that $(C_j(R)+1) \varepsilon < \varepsilon _0$, $j = 1, \dots , J'$. Recalling that $J'+1 \leq J = J(R, T, \eta )$, we see that this can be achieved by choosing $\eta = \eta (R, M, T)>0$, $\varepsilon = \varepsilon (R, M, T) > 0$, and $\tau = \tau (R,M, T, \alpha , \beta ) = \tau (R,M, T, s, \beta ) >0$ sufficiently small. This guarantees existence of the solution $v$ to 7.1 on $[t_0, t_0+\tau ]$.

Under the conditions (i) - (iii), we can apply the above local argument on time intervals of length $\tau = \tau (R, M, T, s, \beta )>0$, thus extending the solution $v$ to 6.1 on the entire interval $[0, T]$.

■

8. Proof of Theorem 1.3

In this section, we prove the following “almost” almost sure global existence result.

Proposition 8.1.

Let $d\geq 3$ and $s \in (s_d, s_\text{crit}]$. Assume Hypothesis (A). Furthermore, assume Hypothesis (B) if $d \ne 4$. Given $\phi \in H^s(\mathbb{R}^d)$, let $\phi ^\omega$ be its Wiener randomization defined in 1.9, satisfying 1.10. Then, given any $T, \varepsilon > 0$, there exists a set $\widetilde{\Omega }_{T, \varepsilon }\subset \Omega$ such that:

(i)

$P(\widetilde{\Omega }_{T, \varepsilon }^c) < \varepsilon$.

(ii)

For each $\omega \in \widetilde{\Omega }_{T, \varepsilon }$, there exists a (unique) solution $u$ to 1.1 on $[0, T]$ with $u|_{t = 0} = \phi ^\omega$.

It is easy to see that “almost” almost sure global existence implies almost sure global existence. See 20. For completeness, we first show how Theorem 1.3 follows as an immediate consequence of Proposition 8.1.

Given small $\varepsilon > 0$, let $T_j = 2^j$ and $\varepsilon _j = 2^{-j} \varepsilon$, $j \in \mathbb{N}$. For each $j$, we apply Proposition 8.1 and construct $\widetilde{\Omega }_{T_j, \varepsilon _j}$. Then, let $\Omega _\varepsilon = \bigcap _{j = 1}^\infty \widetilde{\Omega }_{T_j, \varepsilon _j}$. Note that (i) $P (\Omega _\varepsilon ^c) < \varepsilon$, and (ii) for each $\omega \in \Omega _\varepsilon$, we have a global solution $u$ to 1.1 with $u|_{t = 0} = \phi ^\omega$. Now, let $\Sigma = \bigcup _{\varepsilon > 0} \Omega _\varepsilon$. Then, we have $P (\Sigma ^c) = 0$. Moreover, for each $\omega \in \Sigma$, we have a global solution $u$ to 1.1 with $u|_{t = 0} = \phi ^\omega$. This proves Theorem 1.3.

The rest of this section is devoted to the proof of Proposition 8.1.

Proof of Proposition 8.1.

Given $T, \varepsilon > 0$, set

$$\begin{equation} M = M(T, \varepsilon ) \sim \|\phi \|_{H^s}\Big (\log \frac{1}{\varepsilon }\Big )^\frac{1}{2}. {\tag{8.1}} \end{equation}$$

Defining $\Omega _1 = \Omega _1(T, \varepsilon )$ by

$$\begin{equation*} \Omega _1 := \big \{ \omega \in \Omega :\, \|S(t) \phi ^\omega \|_{Y^s([0, T])} \leq M\big \}, \end{equation*}$$

it follows from Lemma 3.5 (i) and Lemma 2.4 that

$$\begin{equation} P( \Omega _1^c) < \frac{\varepsilon }{3}. {\tag{8.2}} \end{equation}$$

Given $T, \varepsilon > 0$, let $R = R(T, \frac{\varepsilon }{3})$ and let $M$ be as in 1.13 and 8.1, respectively. With $\tau = \tau (R, M, T)$ as in Proposition 7.2, write

$$\begin{equation*} [0, T] = \bigcup _{j = 0}^{[\frac{T}{ \tau _*}]} [j \tau _*, (j + 1) \tau _*] \cap [0, T] \end{equation*}$$

for some $\tau _* \leq \tau$ (to be chosen later). Now, define $\Omega _2$ by

$$\begin{equation*} \Omega _2 = \Big \{ \omega \in \Omega : \| S(t) \phi ^\omega \|_{W^s([j \tau _*, (j+1)\tau _*])} \leq C \tau _*^\frac{1}{2(d+2)}, \, j = 0, \dots , \big [\tfrac{T}{\tau _*}\big ] \Big \}, \end{equation*}$$

where $C$ is as in 7.23. Then, by Lemma 2.2, we have

$$\begin{align*} P(\Omega _2^c) & \leq \sum _{j = 0}^{[\frac{T}{\tau _*}] } P\Big ( \| S(t) \phi ^\omega \|_{W^s([j\tau _*, (j+1)\tau _*])}\geq C\tau _*^\frac{1}{2(d+2)}\Big ) \lesssim \frac{T}{\tau _*} \exp \bigg (-\frac{c}{\tau _*^\frac{1}{d+2} \|\phi \|_{H^s}^2}\bigg ).\\ \end{align*}$$

By making $\tau _* = \tau _*(\|\phi \|_{H^s})$ smaller, if necessary, we have

$$\begin{align*} & \lesssim \frac{T}{\tau _\ast }\cdot \tau _* \exp \bigg (-\frac{c}{2\tau _*^\frac{1}{d+2} \|\phi \|_{H^s}^2}\bigg ) = T \exp \bigg (-\frac{1}{2\tau _*^\frac{1}{d+2} \|\phi \|_{H^s}^2}\bigg ). \end{align*}$$

Hence, by choosing $\tau _* = \tau _*(T, \varepsilon , \|\phi \|_{H^s})$ sufficiently small, we have

$$\begin{align} P(\Omega _2^c) < \frac{\varepsilon }{3}. {\tag{8.3}} \end{align}$$

Finally, set $\widetilde{\Omega }_{T, \varepsilon } = \Omega _{T, \frac{\varepsilon }{3}}\cap \Omega _1 \cap \Omega _2$, where $\Omega _{T, \frac{\varepsilon }{3}}$ is as in Hypothesis (A) with $\varepsilon$ replaced by $\frac{\varepsilon }{3}$. Then, from 8.2 and 8.3, we have

$$\begin{equation*} P(\widetilde{\Omega }_{T, \varepsilon }^c) < \varepsilon . \end{equation*}$$

Moreover, for $\omega \in \widetilde{\Omega }_{T, \varepsilon }$, we can iteratively apply Proposition 7.2 and Remark 7.3 and construct the solution $v = v^\omega$ on each $[j \tau _*, (j+1)\tau _*]$, $j = 0, \dots , [\frac{T}{\tau }_*]-1$, and $\big [ [\frac{T}{\tau }_*]\tau , T\big ]$. This completes the proof of Proposition 8.1.

■

Remark 8.2.

It is worthwhile to mention that the proof of Proposition 8.1 strongly depends on the quasi-invariance property of the distribution of the linear solution $S(t) \phi ^\omega$. More precisely, in the proof above, we exploited the fact that the distribution of $\| S(t) \phi ^\omega \|_{W^s([t_0, t_0 + \tau _*])}$ depends basically only on the length $\tau _*$ of the interval, but is independent of $t_0$.

9. Probabilistic global existence via randomization on dilated cubes

In this section, we present the proof of Theorem 1.4. The main idea is to exploit the dilation symmetry of the cubic NLS 1.1. For a function $\phi =\phi (x)$, we define its scaling $\phi _\mu$ by

$$\begin{equation*} \phi _\mu (x) := \mu ^{-1} \phi (\mu ^{-1}x), \end{equation*}$$

while for a function $f=f(t,x)$, we define its scaling $f_\mu$ by

$$\begin{equation*} f_\mu (t,x):=\mu ^{-1}f(\mu ^{-2}t,\mu ^{-1}x). \end{equation*}$$

Then, given $\phi \in H^s(\mathbb{R}^d)$, we have

$$\begin{align} \|\phi _\mu \|_{\dot{H}^s(\mathbb{R}^d)} = \mu ^{\frac{d-2}{2} - s}\|\phi \|_{\dot{H}^s(\mathbb{R}^d)} . {\tag{9.1}} \end{align}$$

If $s < s_\text{crit} = \frac{d-2}{2}$, that is, if $\phi$ is supercritical with respect to the scaling symmetry, then we can make the $H^s$-norm of the scaled function $\phi _\mu$ small by taking $\mu \ll 1$. The issue is that the Strichartz estimates we employ in proving probabilistic well-posedness are (sub)critical and do not become small even if we take $\mu \ll 1$. It is for this reason that we consider the randomization $\phi ^{\omega , \mu }$ on dilated cubes.

Fix $\phi \in H^s(\mathbb{R}^d)$ with $s \in ( s_d, s_\text{crit})$, where $s_\text{crit} = \frac{d-2}{2}$ and $s_d$ is as in 1.11. Let $\phi ^{\omega , \mu }$ be its randomization on dilated cubes of scale $\mu$ as in 1.17. Instead of considering 1.1 with $u_0 = \phi ^{\omega , \mu }$, we consider the scaled Cauchy problem

$$\begin{equation} \begin{cases} i \partial _t u_\mu + \Delta u_\mu = \pm |u_\mu |^{2}u_\mu , \\ u_\mu \big |_{t = 0} = u_{0, \mu } = (\phi ^{\omega , \mu })_\mu , \end{cases} {\tag{9.2}} \end{equation}$$

where $u_\mu$ is as in 1.2 and $(\phi ^{\omega , \mu })_\mu (x) := \mu ^{-1} \phi ^{\omega , \mu }(\mu ^{-1}x)$ is the scaled randomization. For notational simplicity, we denote $(\phi ^{\omega , \mu })_\mu$ by $\phi ^{\omega , \mu }_\mu$ in the following. Denoting the linear and nonlinear part of $u_\mu$ by $z_\mu (t) = z^\omega _\mu (t) : = S(t) \phi ^{\omega , \mu }_\mu$ and $v_\mu (t) := u_\mu (t) - S(t) \phi ^{\omega , \mu }_\mu$ as before, we reduce 9.2 to

$$\begin{equation} \begin{cases} i \partial _tv_\mu + \Delta v_\mu = \pm |v_\mu + z_\mu |^2(v_\mu +z_\mu ),\\ v_\mu |_{t = 0} = 0. \end{cases} {\tag{9.3}} \end{equation}$$

Note that if $u$ satisfies 1.1 with initial data $u(0)=\phi ^{\omega ,\mu }$, then $u_\mu$, $z_\mu$, and $v_\mu$ are indeed the scalings of $u$, $z:=S(t)\phi ^{\omega , \mu }$, and $v : = u -z$, respectively. For $u_\mu$ this simply follows from the scaling symmetry of 1.1. For $z_\mu$ and $v_\mu$, this follows from the following observation:

$$\begin{align} \mathcal{F}_x\Big [\big (S(t)\phi ^{\omega ,\mu }\big )_\mu \Big ](\xi ) = \mu ^{d-1}e^{-i\frac{t}{\mu ^2}|\mu \xi |^2} \widehat{\phi ^{\omega ,\mu }}(\mu \xi ) = e^{-it|\xi |^2}\widehat{\phi ^{\omega , \mu }_\mu }( \xi ) =\widehat{z_\mu }(t, \xi ). {\tag{9.4}} \end{align}$$

Define $\Gamma _\mu$ by

$$\begin{equation} \Gamma _\mu v_\mu (t) =\mp i \int _0^t S(t-t') \mathcal{N}(v_\mu +z_\mu )(t') dt'. {\tag{9.5}} \end{equation}$$

In the following, we show that there exists $\mu _0 = \mu _0(\varepsilon , \|\phi \|_{H^s} ) > 0$ such that, for $\mu \in (0, \mu _0)$, the estimates 4.5 and 4.6 in Proposition 4.1 (with $\widetilde{\Gamma }$ replaced by $\Gamma _\mu$) hold with $R = \eta _2$ outside a set of probability $< \varepsilon$, where $\eta _2$ is as in 5.2. In view of 1.16, it is easy to see that

$$\begin{equation*} \psi (D-n) \phi _\mu =\big ( \psi ^\mu (D-\mu n) \phi )_\mu . \end{equation*}$$

Hence, we have

$$\begin{equation} \phi ^{\omega , \mu }_\mu = (\phi ^{\omega , \mu })_\mu = \sum _{n \in \mathbb{Z}^d} g_n(\omega ) \psi (D-n) \phi _\mu . {\tag{9.6}} \end{equation}$$

Given $\eta _2$ as in 5.2 and $\mu >0$, define $\Omega _{1, \mu }$ by

$$\begin{equation*} \Omega _{1, \mu } = \Big \{ \omega \in \Omega :\, \|S(t) \phi ^{\omega , \mu }_\mu \|_{L^q_t W^{s, q}_x ( \mathbb{R}\times \mathbb{R}^d)} \leq \eta _2, \, q = 4, \tfrac{6(d+2)}{d+4}, d+2\Big \} . \end{equation*}$$

We also define $\Omega _{2, \mu }$ by

$$\begin{equation*} \Omega _{2, \mu } = \big \{ \omega \in \Omega :\, \| \phi ^{\omega , \mu }_\mu \|_{H^s(\mathbb{R}^d)} \leq \eta _2\big \}. \end{equation*}$$

Now, let $\Omega _\mu = \Omega _{1, \mu } \cap \Omega _{2, \mu }$. Noting that $4, \frac{6(d+2)}{d+4}$, and $d+2$ are larger than the diagonal Strichartz admissible index $\frac{2(d+2)}{d}$, it follows from Lemma 2.3 and Lemma 2.4 with 9.6 and 9.1 that

$$\begin{align*} P( \Omega _{ \mu }^c) \leq C\exp \bigg ( -c \frac{\eta _2^2}{ \|\phi _\mu \|_{H^s}^2}\bigg ) \leq C\exp \bigg ( -c \frac{\eta _2^2}{ \mu ^{d-2 -2 s}\|\phi \|_{H^s}^2}\bigg ) \end{align*}$$

for $\mu \leq 1$. Then, by setting

$$\begin{equation} \mu _0 \sim \Bigg (\frac{\eta _2}{\|\phi \|_{H^s} \big (\log \frac{1}{\varepsilon }\big )^\frac{1}{2}}\Bigg )^\frac{1}{\frac{d-2}{2}-s}, {\tag{9.7}} \end{equation}$$

we have

$$\begin{align} P( \Omega _\mu ^c) <\varepsilon {\tag{9.8}} \end{align}$$

for $\mu \in (0, \mu _0)$. Note that $\mu _0 \to 0$ as $\varepsilon \to 0$. Recall that $q = 4, \frac{6(d+2)}{d+4}$, and $d+2$ are the only relevant values of the space-time Lebesgue indices controlling the random forcing term in the proof of Proposition 4.1. Hence, the estimates 4.5 and 4.6 in Proposition 4.1 (with $\widetilde{\Gamma }$ replaced by $\Gamma _\mu$) hold with $R = \eta _2$ for each $\omega \in \Omega _\mu$. Then, by repeating the proof of Theorem 1.2 in Section 5, we see that, for each $\omega \in \Omega _\mu$, there exists a global solution $u_\mu$ to 9.2 with $u_\mu |_{t = 0} = \phi ^{\omega , \mu }_\mu$ which scatters both forward and backward in time. By undoing the scaling, we obtain a global solution $u$ to 1.1 with $u|_{t = 0} = \phi ^{\omega , \mu }$ for each $\omega \in \Omega _\mu$. Moreover, scattering for $u_\mu$ implies scattering for $u$. Indeed, as in Theorem 1.2, there exists $v_{+,\mu }\in H^{\frac{d-2}{2}}(\mathbb{R}^d)$ such that

$$\begin{equation*} \lim _{t\to \infty }\big \|u_\mu (t)-S(t)\big (\phi ^{\omega ,\mu }_\mu +v_{+,\mu }\big )\big \|_{H^{\frac{d-2}{2}}}=0. \end{equation*}$$

Then, a computation analogous to 9.4 yields

$$\begin{equation*} S(t)(\phi ^{\omega ,\mu }_\mu +v_{+,\mu }) =\big (S(t)(\phi ^{\omega ,\mu }+v_+)\big )_{\mu }, \end{equation*}$$

where $v_+:=(v_{+,\mu })_{\mu ^{-1}}\in H^{\frac{d-2}{2}}(\mathbb{R}^d)$. Then, by 9.1, it follows that

$$\begin{equation*} \lim _{t\to \infty }\big \|u-S(t)\big (\phi ^{\omega ,\mu }+v_+\big )\big \|_{H^{\frac{d-2}{2}}}=0. \end{equation*}$$

This proves that $u$ scatters forward in time. Scattering of $u$ as $t\to -\infty$ can be proved analogously. This completes the proof of Theorem 1.4.