On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on $\mathbb{R}^d$,$d \geq 3$
By Árpád Bényi, Tadahiro Oh, and Oana Pocovnicu
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 $\mathbb{R}^d$,$d\geq 3$:
$$\begin{equation} \begin{cases} i \partial _t u + \Delta u = \pm \mathcal{N}(u),\\ u\big |_{t = 0} = u_0 \in H^s(\mathbb{R}^d), \end{cases} \qquad ( t, x) \in \mathbb{R}\times \mathbb{R}^d, \cssId{NLS1}{\tag{1.1}} \end{equation}$$
where $\mathcal{N}(u) := |u|^2 u$. The cubic NLS Equation 1.1 has been studied extensively from both the theoretical and applied points of view. Our main focus is to study well-posedness of Equation 1.1 with random and rough initial data.
It is well known that the cubic NLS Equation 1.1 enjoys the dilation symmetry. More precisely, if $u(t, x)$ is a solution to Equation 1.1 on $\mathbb{R}^d$ with an initial condition $u_0$, then
is also a solution to Equation 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
If an initial condition $u_0$ is in $H^s(\mathbb{R}^d)$, we say that the Cauchy problem Equation 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, Equation 1.1 is known to be locally well-posed. See Cazenave-Weissler Reference 17 for local well-posedness of Equation 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 Equation 1.1:
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 Equation 1.1, in high dimensions $d\geq 3$. When $d = 4$, the Hamiltonian is invariant under the scaling Equation 1.2 and plays a crucial role in the global well-posedness theory. Indeed, Ryckman-Vişan Reference 53 proved global well-posedness and scattering for the defocusing cubic NLS on $\mathbb{R}^4$. See also Vişan Reference 60. When $d \ne 4$, there is no known scaling invariant positive conservation law for Equation 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 Reference 35 applied the concentration compactness and rigidity method developed in their previous paper Reference 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 Reference 36 on radial solutions to the energy-supercritical nonlinear wave equation (NLW) on $\mathbb{R}^3$, Killip-Vişan Reference 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 Reference 35 and Reference 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 Equation 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, Equation 1.1 is known to be ill-posed. See Reference 1Reference 11Reference 16Reference 18. In the following, however, we consider the Cauchy problem Equation 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 Equation 1.1.
In studying the Gibbs measure for the defocusing (Wick ordered) cubic NLS on $\mathbb{T}^2$, Bourgain Reference 6 considered random initial data of the form:
where $\{g_n\}_{n \in \mathbb{Z}^2}$ is a sequence of independent standard complex-valued Gaussian random variables. The function Equation 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 Equation 1.4. In the context of the cubic NLW on a three-dimensional compact Riemannian manifold $M$, Burq-Tzvetkov Reference 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,Footnote1 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 $u|_{t = 0}$.
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 Equation 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 Equation 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 Reference 49. See also Kahane Reference 32 and Ayache-Tzvetkov Reference 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: Reference 7Reference 9Reference 10Reference 12Reference 13Reference 15Reference 20Reference 21Reference 22Reference 23Reference 42Reference 43Reference 44Reference 45Reference 51Reference 52Reference 58.
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 Reference 28.
The modulation spaces were introduced by Feichtinger Reference 24 in the early eighties. The groundwork theory regarding these spaces of time-frequency analysis was then established in joint collaboration with Gröchenig Reference 25Reference 26. The modulation spaces arise from a uniform partition of the frequency space, commonly known as the Wiener decompositionReference 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
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
$$\begin{equation} \|u\|_{M_s^{p, q}(\mathbb{R}^d)} := \big \| \langle n \rangle ^s \|\psi (D-n) u \|_{L_x^p(\mathbb{R}^d)} \big \|_{\ell ^q_n(\mathbb{Z}^d)} \cssId{mod2}{\tag{1.7}} \end{equation}$$
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 Equation 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
There are several known embeddings between Besov, Sobolev, and modulation spaces. See, for example, Okoudjou Reference 47, Toft Reference 59, Sugimoto-Tomita Reference 55, and Kobayashi-Sugimoto Reference 40.
Now, given a function $\phi$ on $\mathbb{R}^d$, we have
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
Almost simultaneously with our first paper Reference 4, Lührmann-Mendelson Reference 42 also considered a randomization of the form Equation 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 Reference 63. We would like to stress again, however, that our reason for considering the randomization of the form Equation 1.9 comes from its connection to time-frequency analysis. See also our previous papers Reference 3 and Reference 4.
In the sequel, we make the following assumption on the distributions $\mu _n^{(j)}$: there exists $c>0$ such that
$$\begin{equation} \bigg | \int _{\mathbb{R}} e^{\gamma x } d \mu _n^{(j)}(x) \bigg | \leq e^{c\gamma ^2} \cssId{R2}{\tag{1.10}} \end{equation}$$
for all $\gamma \in \mathbb{R}$,$n \in \mathbb{Z}^d$,$j = 1, 2$. Note that Equation 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 Equation 1.1 with respect to the randomized initial data $\phi ^\omega$ defined in Equation 1.9.
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.
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, Equation 1.1 is equivalent to the following perturbed NLS:
We reduce our analysis to the Cauchy problem Equation 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 Reference 29Reference 30Reference 41. 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 Reference 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.
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 $v$.
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 Equation 1.13, just like the usual conservation laws. It may be possible to prove Equation 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 Reference 15. We point out that the upper bound $R(T, \varepsilon )$ in Reference 15 tends to $\infty$ as $T\to \infty$. See also Reference 50.
The next hypothesis is on global existence and space-time bounds of solutions to the cubic NLS Equation 1.1 with deterministic initial data belonging to the critical space $H^\frac{d-2}{2}(\mathbb{R}^d)$.
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 Reference 53 and Vişan Reference 60. For other dimensions $d \geq 3$ with $d \ne 4$, it is not known whether Hypothesis (B) holds. Let us compare Equation 1.14 and the results in Reference 35 and Reference 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 Reference 35 and Reference 39 that $I_* = \mathbb{R}$ and
We point out that Hypothesis (B) is not directly comparable to the results in Reference 35Reference 39 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 Reference 35Reference 39 yield the global-in-time bound Equation 1.15, while Hypothesis (B) assumes the bound Equation 1.14 only for each finite time $T>0$ and does not assume a global-in-time bound. On the other hand, Equation 1.14 is much stronger than Equation 1.15 in the sense that the right-hand side of Equation 1.14 depends only on the size of an initial condition $w_0$, while the right-hand side of Equation 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 Equation 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.
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 Reference 57 and Killip-Vişan with the second and third authors Reference 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 Reference 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 Reference 35 and Killip-Vişan Reference 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
As a consequence, the third author Reference 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.Footnote2 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 Reference 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 Reference 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 Equation 1.2 of the equation for random initial data below the scaling criticality. For example, suppose that we have a solution $u$ to Equation 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 Equation 1.2 and Equation 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 Equation 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
where $\{g_n\}_{n \in \mathbb{Z}^d}$ is a sequence of independent mean zero complex-valued random variables, satisfying Equation 1.10 as before. Then, we have the following global well-posedness of Equation 1.1 with a large probability.
We conclude this introduction with several remarks.
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 Equation 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
for $p \geq \frac{2(d+2)}{d}$. Recall that the derivative loss in Equation 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
Next, we present improvements of the Strichartz estimates under the Wiener randomization Equation 1.9 and where, throughout, we assume Equation 1.10. See Reference 4 for the proofs.
The next lemma states an improvement of the Strichartz estimates in the global-in-time setting.
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 $\phi ^\omega$.
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
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 Reference 29Reference 30Reference 41. These spaces have been very effective in establishing well-posedness of various dispersive PDEs in critical regularities. See Hadac-Herr-Koch Reference 29 and Herr-Tataru-Tzvetkov Reference 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 $I \subset \mathbb{R}$.
Next, we state a transference principle and an interpolation result.
A transference principle as above has been commonly used in the Fourier restriction norm method. See Reference 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 Reference 30 to a general $k$-linear case. See also Reference 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
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 $\mathbf{P}_{\geq N} = \sum _{ M \geq N} \mathbf{P}_M$.
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
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.
Note that there is a slight loss of regularity in Equation 3.6 since we use the $Y^0$-norm on the right-hand side instead of the $X^0$-norm. In view of Equation 3.2, we may replace the $Y^0$-norms on the right-hand sides of Equation 3.4, Equation 3.5, and Equation 3.6 by the $X^0$-norm in the following.
Similar to the usual Strichartz estimate Equation 2.3, the derivative loss in Equation 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
Lastly, we recall Schur’s test for the readers’ convenience.
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
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 Equation 4.2 and Equation 4.3, let $\eta _1 > 0$ be sufficiently small such that
where $C_3$ and $C_4$ are as in Equation 4.5 and Equation 4.6. Then, by Proposition 4.1 with $R = \eta _2$ and $\phi ^\omega$ replaced by $\varepsilon \phi ^\omega$, we have
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
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 Equation 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
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 Equation 3.5, we have
Then, by the monotone convergence theorem, Equation 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
This proves Equation 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:
$$\begin{equation} \begin{cases} i \partial _tv + \Delta v = |v + f|^2(v+f),\\ v|_{t = t_0} = v_0, \end{cases} \cssId{ZNLS1}{\tag{6.1}} \end{equation}$$
where $f$ is a given deterministic function. Assuming some suitable conditions on $f$, we prove local well-posedness of Equation 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 Equation 6.1 by iteratively applying a perturbation lemma (Lemma 7.1). For this purpose, we first prove a “variant” local well-posedness of Equation 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
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 $I$.
We first state and prove the following local well-posedness result for the perturbed NLS Equation 6.1, assuming Lemma 6.2. The proof of Lemma 6.2 is presented at the end of this section.
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.
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 Equation 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 Reference 19Reference 38Reference 56Reference 57 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 Reference 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.
In the remaining part of this section, we consider long time existence of solutions to the perturbed NLS Equation 6.1 under several assumptions. Given $T>0$, we assume that there exist $\beta , C, M > 0$ such that
$$\begin{equation} \|f\|_{W^s(I)} \leq C |I|^{\beta } \qquad \text{and}\qquad \|f\|_{Y^s([0, T])} \leq M \cssId{P0}{\tag{7.23}} \end{equation}$$
for any interval $I \subset [0, T]$. Then, Proposition 6.3 guarantees existence of a solution to the perturbed NLS Equation 6.1, at least for a short time.
In this section, we prove the following “almost” almost sure global existence result.
It is easy to see that “almost” almost sure global existence implies almost sure global existence. See Reference 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 Equation 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 Equation 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.
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 Equation 1.1. For a function $\phi =\phi (x)$, we define its scaling $\phi _\mu$ by
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 Equation 1.11. Let $\phi ^{\omega , \mu }$ be its randomization on dilated cubes of scale $\mu$ as in Equation 1.17. Instead of considering Equation 1.1 with $u_0 = \phi ^{\omega , \mu }$, we consider the scaled Cauchy problem
where $u_\mu$ is as in Equation 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 Equation 9.2 to
Note that if $u$ satisfies Equation 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 Equation 1.1. For $z_\mu$ and $v_\mu$, this follows from the following observation:
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 Equation 4.5 and Equation 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 Equation 5.2. In view of Equation 1.16, it is easy to see that
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 Equation 9.6 and Equation 9.1 that
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 Equation 4.5 and Equation 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 Equation 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 Equation 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
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.
Appendix A. On the properties of the $U^p$- and $X^s$-spaces
In this appendix, we prove some additional properties of the $U^p$- and $X^s$-spaces. In the following, all intervals are half open intervals of the form $[a, b)$ and $p$ denotes a number such that $1\leq p < \infty$.
Given an interval $I \subset \mathbb{R}$, we define the local-in-time $U^p$-norm in the usual manner as a restriction norm:
The next lemma states the subadditivity of the local-in-time $U^p$-norm over intervals.
As a corollary, we immediately obtain the following subadditivity property of the local-in-time $X^s$-norm over intervals.
We say that $u$ on $[a, b)$ is a regulated function if both left and right limits exist at every point (including one-sided limits at the endpointsFootnote4). Given a regulated function $u$ on $[a, b)$ and a partition $\mathcal{P} = \{ \tau _1, \dots , \tau _n\}$ of $[a, b)$:$a < \tau _1 < \cdots < \tau _n < b$, we define a step function $u_{\mathcal{P}}$ by
where we set $\tau _0 = a$ and $\tau _{n+1} = b$. In particular, if $u$ is right-continuous, we have $u_\mathcal{P}(t) = u(\tau _j)$ for $\tau _{j} \leq t < \tau _{j+1}$. Note that the mapping $\mathcal{P}: u \mapsto u_{\mathcal{P}}$ is linear.
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School of Mathematics, The University of Edinburgh – and – The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom
School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540 – and – Department of Mathematics, Fine Hall, Princeton University, Washington Road, Princeton, New Jersey 08544
This work was partially supported by a grant from the Simons Foundation (No. 246024 to the first author). The second author was supported by the European Research Council (grant no. 637995 “ProbDynDispEq”). The third author was supported by the NSF grant under agreement No. DMS-1128155. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF.
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