1 Introduction↩︎

We consider the quintic nonlinear Schrödinger equation (NLS) on the unit circle \(\mathbb{T}=\mathbb{R}/\left(2\pi \mathbb{Z}\right)\), \[\label{eq:NLS} \begin{cases} i \partial_tu +\Delta u=\mu \left\lvert u\right\rvert^4u, \\ u(x,0)=u_0(x) \in H^{s}\left(\mathbb{T}\right), \end{cases} \tag{1}\] where \(\mu = 1\) corresponds to the defocusing case and \(\mu = -1\) to the focusing case. For smooth solutions \(u\), the mass \(M(u(t))\) and energy \(E(u(t))\) are conserved along the flow of 1 , where \[\begin{align} \tag{2} M\left(u(t)\right) &:= \left\lVert u(t)\right\rVert_{L_x^2(\mathbb{T})}, \\ \tag{3} E(u(t)) &:= \frac{1}{2}\left\lVert\partial_x u(t)\right\rVert_{L_x^2(\mathbb{T})}^2 + \frac{\mu}{6} \left\lVert u(t)\right\rVert^6_{L_x^6(\mathbb{T})}. \end{align}\] Moreover, 1 enjoys a scaling symmetry which is critical with respect to the mass. Specifically, for any \(\lambda>0\), we map a solution of 1 to another solution, \(u \mapsto u^\lambda\) via the transformation \[u^\lambda(x,t):= \frac{1}{\lambda^\frac{1}{2}} u\left(\frac{x}{\lambda}, \frac{t}{\lambda^{2}}\right),\] and stress that \(u^\lambda\) is defined on \(\mathbb{T}_\lambda \times \mathbb{R}\) where \(\mathbb{T}_\lambda := \mathbb{R}/(2\pi\lambda\mathbb{Z})\). We say that 1 is mass-critical as the rescaling leaves the mass invariant.

To formulate our main results, we recall the standard notion of well-posedness. The initial value problem 1 is locally well-posed if, for any \(u_0 \in H^s(\mathbb{T})\), there exists a time \(T>0\), an open ball \(B\) in \(H^s(\mathbb{T})\) containing \(u_0\), and a closed subspace, \(X\), of \(C^0([0,T] \rightarrow H^s(\mathbb{T}))\), such that for each \(u_0 \in B\) there exists a unique \(u \in X\) satisfying \[u(t)=S(t)u_0+i \mu \int_0^t S(t-t') \left\lvert u(t')\right\rvert^4u(t') dt',\] where \(S(t):= e^{it \Delta}\) is the Schrödinger group. Moreover, we require the map \(u_0 \mapsto u\) to be uniformly continuous as a map from \(B\) (endowed with the \(H^s(\mathbb{T})\) topology) to X (endowed with the \(C^0([0,T] \rightarrow H^s(\mathbb{T}))\) topology). If we can take \(T>0\) to be arbitrarily large, we say that 1 is globally well-posed.

The main objective of this paper is to establish global well-posedness of the Cauchy problem 1 for \(s>0\). Our approach is based on establishing a derivative-loss-free \(L^6_{t,x}\) Strichartz estimate and combining this estimate with the \(I\)-method. We begin by stating the Strichartz estimate.

Theorem 1 is proved via a lossless small-cap decoupling estimate. A key ingredient in the argument is an estimate on the size of the intersection of superlevel sets of functions whose Fourier supports are contained in caps \(\Sigma_1\) and \(\Sigma_2\). In the regime where the length of \(\Sigma_1\) is much smaller than the distance between \(\Sigma_1\) and \(\Sigma_2\), we establish an asymmetric superlevel set estimate which yields an additional gain reflecting this scale separation (see Lemma 12 for details). This gain is used to eliminate logarithmic losses in the decoupling argument. As a further consequence of this analysis, we recover a refined trilinear Strichartz estimate due to McConnell[1], up to an arbitrarily small polynomial loss in the frequency scale. We refer the reader to Appendix 6 for a proof using our method.

The derivative-loss-free estimate in Theorem 1 provides the key input for the nonlinear analysis. Combining this estimate with the \(I\)-method, we prove the following small-mass global well-posedness result for 1 .

1.1 Context and previous work↩︎

In his seminal work [2], Bourgain established local well-posedness of 1 for \(s > 0\). This result is sharp (up to endpoint) in the sense that the data-to-solution map fails to be uniformly continuous for \(s<0\) [3]. A key component of Bourgain’s [2] proof was the following Strichartz estimate: there exists a constant \(c>0\) such that for all \(\phi \in L^2(\mathbb{T})\) satisfying \(\mathop{\mathrm{supp}}\widehat{\phi} \subset \left[-N,N\right]\), the following estimate holds \[\left\lVert S(t) \phi \right\rVert_{L^6_{t,x}\left(\mathbb{T}\times \mathbb{T}\right)} \leq C_N \left\lVert\phi\right\rVert_2, \text{ where } C_N:= c \exp\left(c \frac{\log N}{\log \log N}\right). \label{eq:32bourgain32Strichartz}\tag{4}\] This Strichartz estimate has since played a central role in the analysis of the nonlinear Schrödinger equation on the torus and has served as a starting point for a number of refinements and applications. We briefly review these developments, focusing first on improvements to 4 and the methods underlying their proofs, and then on applications to global well-posedness.

We begin with a discussion of previous improvements to 4 . While the estimate 4 was sufficient to establish local well-posedness for \(s>0\), it is not expected to be sharp in terms of the growth of \(C_N\). Indeed, by considering \(\phi = \sum_{|k| \leq N} e^{ikx}\), it was conjectured in [2], [4] that the derivative loss could be reduced to \(C_N \sim (\log N)^{1/6}\). Despite significant effort, this conjecture remains open. Partial progress has been achieved through refinements of the decoupling techniques underlying the \(\ell^2\) decoupling theorem of Bourgain and Demeter [5]. For instance, Guth–Maldague–Wang [6] applied the high-low method introduced by Guth–Solomon–Wang [7] to obtain bounds of the form \(C_N \sim (\log N)^c\) for some absolute constant \(c \gg 1\). This was subsequently improved by Guo–Li–Yung [8], who established an asymmetric bilinear decoupling inequality to show that \(C_N \lesssim_\epsilon (\log N)^{2+\epsilon}\) for every \(\epsilon>0\).

An alternative approach to obtaining estimates of the form of 4 is to establish derivative-loss-free Strichartz estimates on shorter, frequency-dependent time intervals. More precisely, one seeks the largest time \(T>0\) such that there exists a constant \(c>0\) with the property that, for all \(\phi \in L^2(\mathbb{T})\) satisfying \(\mathop{\mathrm{supp}}\hat{\phi} \subset [-N,N]\), the estimate \[\|S(t)\phi\|_{L^6_{t,x}([0,T]\times\mathbb{T})} \leq c \|\phi\|_2 \label{eq:derivative95free95L6}\tag{5}\] holds. If one could establish 5 with \(T=(\log N)^{-1}\), then by decomposing \([0,2\pi]\) into intervals of length \((\log N)^{-1}\) one would immediately obtain 4 with \(C_N\sim (\log N)^{1/6}\). In this direction, Burq–Gérard–Tzvetkov [9] proved 5 with \(T = N^{-1}\). Their result in fact extends to arbitrary compact Riemannian manifolds in higher dimensions, for the same time scale. In the periodic setting, Herr and Kwak [10] obtained the \(L^4_{t,x}\) analogue of 5 on the two-dimensional square torus \[\left\lVert S(t) \phi \right\rVert_{L^4_{t,x}\left( \left[0, \frac{1}{\log N}\right] \times \mathbb{T}^2\right)} \leq c \left\lVert\phi\right\rVert_2, \label{eq:32Herr95Kwaq32Strichartz}\tag{6}\] by using the Szemerédi–Trotter theorem to count rectangles with vertices in a finite set in the plane. This in turn yields a sharp Strichartz estimate for the higher-dimensional \(L^4_{t,x}\) analogue of 4 and also plays an important role in their proof of small-mass global well-posedness. Returning to the one-dimensional periodic setting, the best known result for the maximal time for which 5 holds is \(T = N^{-131/208}\), due to McConnell [1], who used number theoretic methods to count the number of lattice points in thin annuli.

In this paper, we prove that 5 holds with \(T = (\log N)^{-10000}\). While we make no attempt to obtain a sharp exponent, this time scale already suffices for our purposes of establishing small-mass global well-posedness. We also note that our result does not improve the best known bound [8] for the constant \(C_N\) in 4 . Our approach is based on small-cap decoupling, in contrast with the number-theoretic methods of McConnell [1] and the incidence-geometric approach of Herr and Kwak [10]. Small-cap decoupling may be viewed as a refinement of the classical \(\ell^2\) decoupling theory adapted to the study of Strichartz estimates on time intervals shorter than a single period. The subject was initiated by Demeter–Guth–Wang [11] and further developed in the two-dimensional setting by Fu–Guth–Maldague [12] and Guth–Maldague [13] using the high–low method.

We now turn to the global well-posedness problem for 1 . On the real line, Dodson [14] proved global well-posedness of 1 in \(H^s(\mathbb{R})\) for all \(s \ge 0\), by establishing that solutions scatter in \(L^2(\mathbb{R})\). In the periodic setting, as scattering is not expected to hold due to the lack of dispersion [15], a different approach is required. In this context, derivative-loss-free Strichartz estimates have played a central role in recent advances toward establishing global well-posedness for mass-critical nonlinear Schrödinger equations.

In a breakthrough result, Herr and Kwak [10], [16] established global well-posedness for the mass-critical NLS on \(H^s(\mathbb{T}^2)\) for \(s>0\). Their approach proceeds in two stages: first [10] they proved small-mass global well-posedness using the derivative-free Strichartz estimate 6 and subsequently [16] they removed the small assumption and proved that their result is sharp in the sense that the data-to-solution map fails to be uniformly continuous for initial data in \(L^2(\mathbb{T}^2)\). The goal of the present paper is to establish the one-dimensional analogue of the small-mass global well-posedness result proved in [10].

We now briefly recall the argument in [10]. Under a suitable smallness assumption on the mass, the derivative-free Strichartz estimate 6 yields local well-posedness in the complete metric space \[X_N:=\left\{ u\in C^0\left( J_N \rightarrow H^s(\mathbb{T}^2) \right) \cap Y^s_{J_N}: \left\lVert u\right\rVert_{Z_N} \lesssim 1 \right\},\] where \(N \gg 1\) is sufficiently large constant, \(J_N:=\left[0,\frac{1}{\log N} \right]\) is the existence time interval, the space \(Y^s_{J_N}\) is the time-restricted \(Y^s\) space (see 4.1 for details) and \[\left\lVert u\right\rVert_{Z_N}^2:= \left\lVert u\right\rVert_{Y^0_{J_N}}^2+ N^{-2s} \left\lVert u\right\rVert_{Y^s_{J_N}}^2.\] As the length of the existence time interval scales with \(\frac{1}{\log N}\), local solutions may be continued for arbitrary time intervals, thereby establishing global well-posedness. In fact, Herr and Kwak [10] proved that for every \(k \in \mathbb{N}\), local solutions can be extended from \([0,T_k]\) to \([0,T_{k+1}]\) where \[T_k := \sum_{j=0}^{k-1} \frac{1}{2 \log\!\bigl(K^j N_0\bigr)},\] with fixed constants \(K \gg 1\) and \(N_0 \gg_s 1\). This procedure can be iterated for arbitrarily large time intervals since \(\lim_{k \rightarrow \infty} T_k = \infty\). Crucially, this conclusion relies on the fact that the length of each extension step scales like \(\frac{1}{\log N}\). If instead the length of existence interval scaled with \(\frac{1}{(\log N)^{c}}\) for any \(c>1\), the above summation would converge and the iteration scheme would fail. Therefore, we stress that this argument would not work if 6 were valid for shorter time intervals of length \(\frac{1}{\left(\log N\right)^c}\) for any \(c>1\).

An alternative approach to extending local solutions to global ones is based on controlling the growth of the \(H^s\) norm of smooth solutions to 1 via the \(I\)-method. Originally introduced in [17], [18], the \(I\)-method has become a canonical tool in obtaining global well-posedness for infinite energy initial data. This method relies on constructing families of modified Hamiltonians which are almost conserved and has been successfully applied to a range of dispersive equations, including the nonlinear Schrödinger equations [19]–[22], the derivative NLS [18], and the KdV equation [23], both in periodic and Euclidean settings.

In this direction, the \(I\)-method has been combined with refined bilinear and trilinear estimates together with 5 to establish global well-posedness for 1 for low-regularity initial data. In fact, Bourgain [24] developed an approach combining the \(I\)-method with normal form reductions to prove global well-posedness below the energy space. Subsequent developments of this framework led to progressively lower regularity thresholds. Specifically, De Silva et al. [19], [25] established a refined bilinear Strichartz estimate which yielded global well-posedness for \(s > \tfrac{4}{9}\), while Li et al. [21] further lowered the regularity threshold to \(s > \tfrac{2}{5}\) via a resonant decomposition of the modified energy.

More recently, derivative-free Strichartz estimates have been incorporated into the \(I\)-method framework to obtain further improvements in the regularity threshold for global well-posedness under an additional small-mass assumption. For example, Schippa [20] combined the derivative-free Strichartz estimate of Burq–Gérard–Tzvetkov [26], valid up to time \(T=N^{-1}\), with the refined bilinear Strichartz of De Silva et al. [19] to prove global well-posedness for \(s > \tfrac{1}{3}\). McConnell [1] further improved this threshold to \(s > \tfrac{131}{624}\) by establishing 5 with \(T= N^{-\frac{131}{208}}\) and by proving the following refined asymmetric Strichartz estimate \[\left\lVert S(t)\phi_1 (S(t)\phi_2)^2\right\rVert_{L^2_{t,x}([0,T]\times\mathbb{T})}^2 \lesssim \left(T^{1/2}+\frac{N_2}{N_1}\right) \left\lVert\phi_1\right\rVert_2^2 \left\lVert\phi_2\right\rVert_2^4, \label{eq:32strichartz95asym}\tag{7}\] for all \(\phi_1, \phi_2 \in L^2(\mathbb{T})\) satisfying \(\mathop{\mathrm{supp}}\widehat{\phi}_j \subset [N_j, 2N_j]\) for \(j=1,2\) with \(N_1 \gg N_2\). This estimate should be viewed as a refinement of 5 , reflecting an additional gain arising from the frequency separation between the interacting functions. In this paper we are able to recover 7 up to a loss of \(N_2^\epsilon\), where \(\epsilon>0\) is arbitrarily small, as a consequence of the decoupling argument used in the proof of Theorem 1. We include a proof in Appendix 6 for completeness. While 7 does not enter into our proof of small-mass global well-posedness, it is natural to ask whether estimates of this type could help remove the small-mass assumption in the defocusing case. We leave this as an open problem for future work.

To summarize, in this paper we follow a similar approach to  [1], [20] to obtain small-mass global well-posedness of 1 in \(H^s(\mathbb{T})\) for \(s>0\). Our main contributions are twofold: (i) we prove 5 for longer times \(T = \frac{1}{(\log N)^c}\) for some \(c \gg 1\) and (ii) we do not rely on resonant decompositions of the modified energy or on refined bilinear/trilinear Strichartz estimates — our proof depends solely on bounding the first modified energy.

1.2 Proof outline↩︎

Theorem 1 follows from a lossless small-cap decoupling estimate for functions whose Fourier support lies in an \(R^{-1}\) neighborhood of the parabola. Roughly speaking, one decomposes the Fourier support into caps \(\gamma\) of length \(1/N\) and seeks to establish an \(L^6\) decoupling estimate without any logarithmic loss. Accordingly, let \[f=\sum_\gamma f_\gamma\] be such a decomposition where each \(f_\gamma\) has essentially constant amplitude on \(B_R\). We refer the reader to 2 for the precise geometric setup and notation. We prove the following lossless small-cap decoupling result \[\label{eq:32main95L695dec95intro} \|f\|_{L^6(B_R)}^6 \lesssim N^2R \left(\sum_\gamma a_\gamma^2\right)^3,\tag{8}\] where \(\sqrt{R}(\log N)^{10000}\leq N\leq R\) and \(a_\gamma\) denotes the amplitude of \(f_\gamma\) on \(B_R\) (see Theorem 3 for the precise formulation).

The proof of 8 begins with a standard pigeonholing argument which reduces matters to the case where \(a_\gamma \sim 1\) or \(a_\gamma=0\) for each \(\gamma\). The main step is to estimate the size of the superlevel sets \[U_a(f):=\{x\in B_R : \frac{a}{2}\leq \left\lvert f(x)\right\rvert\leq a\}.\] More precisely, we prove that there exist a constant \(C>1\) such that for all \(R/N^2 \lesssim (\log N)^{-2C}\), one has \[\label{eq:32main32decoupling32estimate32on32Ua} |U_a|a^6 \lesssim \max\left( \log\left(\frac{\lambda}{a}\right)^{O(1)} \frac{a}{\lambda}, (\log N)^{-C} \right) N^2R\lambda^3,\tag{9}\] where \(\lambda\) denotes number of caps \(\gamma\) such that \(\left\lvert f_\gamma\right\rvert\sim 1\). The estimate in 8 then follows by summing over dyadic values of \(a\).

To establish 9 , we mainly follow the proof of [13]. As a corollary to their theorem, we see that up to a \(\log(N)^{O(1)}\) loss, the only possible sharp contribution to \(\|f\|^6_{L^6}\) comes from \(|U_\lambda|\lambda^6\), i.e., the highest possible peaks. In [13], the result was first proved for the broad case and a broad–narrow argument was then used to obtain the general case. To improve their argument and remove the \(\log N\) loss, we need to overcome two obstacles: (i) replace the \(\log N\) loss in the broad estimate by a \(\log \lambda/a\) loss and (ii) remove any potential \(\log N\) losses coming from the broad–narrow argument.

The broad estimate applies when the caps \(\Sigma_1\) and \(\Sigma_2\) are comparable in size and separated by the same scale. Following the approach of Guth and Maldague [13], we first establish a broad estimate using the high–low method and then combine it with a bilinear restriction estimate. By the high–low method, we mean an argument based on distinguishing whether the functions are dominated by high or low frequencies. In practice, this involves a dyadic decomposition in the frequency domain and treating the resulting pieces differently: the low-frequency part is controlled in \(L^\infty\), while the high-frequency contributions are handled using almost orthogonality. To overcome the first obstacle and replace the \(\log N\) loss arising at each scale by the smaller factor \(\log \frac{\lambda}{a}\), we carefully restrict the number of high–low scales that contribute to the superlevel set estimate.

To overcome the second obstacle, we develop a refined broad–narrow argument. The broad–narrow decomposition has become a canonical tool in harmonic analysis and appears, for instance, in the proof of the \(\ell^2\) decoupling theorem [5] and in [13]. However, a direct implementation of the standard broad–narrow argument from Guth and Maldague [13] would still incur a \(\log N\) loss, due to the presence of \(\log N\) distinct broad and narrow scales. Our refinement avoids this loss by exploiting the fact that contributions coming from sufficiently separated broad–narrow scales interact only weakly. More precisely, we prove an asymmetric superlevel estimate that deals with intersections of superlevel sets associated with functions \(f_1\) and \(f_2\) whose respective Fourier supports are contained in caps \(\Sigma_1\) and \(\Sigma_2\) such that the length of the shorter cap is much shorter than the distance between them. Roughly speaking, for caps \(\Sigma_1\) and \(\Sigma_2\) that are at least \(l\) separated and for \(a,b\) in the appropriate ranges (see Lemma 12 for a precise statement), one has \[\label{eq:32asym95superlevel} \left\lvert U_{a}(f_{1}) \cap U_{b}(f_{2})\right\rvert \lessapprox \max\left(\frac{l(\Sigma_2)}{l}, \frac{\sqrt{R}}{N}\right)\frac{N^2R\lambda(\Sigma_1)\lambda(\Sigma_2)^2}{a^2b^4},\tag{10}\]

where \(l(\Sigma_j)\) denotes the length of the cap \(\Sigma_j\), \(\lambda(\Sigma_j)\) is the number of caps \(\gamma \subset \Sigma_j\) such that \(\left\lvert f_\gamma\right\rvert \sim 1\) and \(\lessapprox\) should be understood as bound up to logarithmic losses. The additional gain \(\max\left(\frac{l(\Sigma_2)}{l}, \frac{\sqrt{R}}{N}\right)\) is precisely what allows us to control interactions between distant broad–narrow scales and thereby eliminate the logarithmic losses. The asymmetric estimate is established using a high–low analysis similar to that employed in the proof of the broad estimate. In addition to its role in the proof of Theorem 1, 10 may have further applications beyond the present work. For instance, it is a key input in recovering the asymmetric Strichartz estimate in Appendix 6.

We now turn to the proof of Theorem 2. The proof of Theorem 2 is based on the \(I\)-method and broadly follows the approach developed in [1], [19], [20]. As we are assuming that the mass is small, it suffices to consider the defocusing case (\(\mu=1\)) by the Gagliardo-Nirenberg [27] inequality. Next, by exploiting the rescaling symmetry of the equation, we analyze solutions \(u^\lambda\) on the space-time domain \(\mathbb{T}_\lambda \times \mathbb{R}\). We introduce a Fourier multiplier \(I\) which acts as the identity on low frequencies \(\left\lvert k\right\rvert \lesssim N\) and damps high frequencies \(\left\lvert k\right\rvert\gg N\) at the rate \(\left(\left\lvert k\right\rvert/N \right)^{s-1}\), where \(N\) is a large parameter to be determined. We define the \(I\)-system by applying the \(I\)-operator to 1 and the modified energy \(E^1(u) := E(Iu)\) is then naturally interpreted as the energy of the \(I\)-system. Under a small mass assumption, we establish a local existence result for the \(I\)-system for time intervals of length \(\lambda^2 \log(\lambda N)^{-c}\) where \(c>0\) is a fixed constant. The main challenge is to extend this local result to arbitrary times. This is achieved by proving that \(E^1(u(t))\) is "almost conserved", in the sense that the time-averaged derivative of \(E^1\) exhibits mild growth. This slow growth of the modified energy allows us to iterate the local existence argument and propagate the \(H^s\)-bound for \(u^\lambda\) over arbitrarily long times. Undoing the rescaling then yields polynomial-in-time bounds for the original solution \(u\), which establishes global well-posedness.

1.3 Organization of the paper.↩︎

Our paper is organized as follows: in 2 we introduce the geometric setup for decoupling, in 3 we prove Theorem 1 using the high-low method and a refined broad-narrow argument, in 4 we transfer our Strichartz estimates to \(Y^s\) spaces and in 5 we give the proof of Theorem 2 using the \(I\)-method. In Appendix 6 we give an alternative proof of the asymmetric Strichartz estimate first established in [1].

1.4 Notation↩︎

We use \(A \lesssim B\) to indicate an estimate of the form \(A \leq CB\) for some constant \(C>0\). When we want to stress that the constant depends on a parameter, say \(p\), we write \(A \lesssim_p B\). If both \(A \lesssim B\) and \(B \lesssim A\) hold, we write \(A \sim B\). Furthermore, \(A \ll B\) signifies that \(A \leq cB\) for some small constant \(0<c<1\) and we use \(a\pm\) to denote \(a \pm \epsilon\) where \(0<\epsilon \ll 1\). When we say that a property \(Q(A,B)\) holds whenever \(A \ll B\), we mean that that there exist a small enough absolute constant \(0<c<1\) such that \(Q(A,B)\) holds whenever \(A \leq cB\). For any measurable \(A \subset \mathbb{R}^d\), we denote its characteristic function by \(\chi_A\) and its Lebesgue measure by \(\left\lvert A\right\rvert\). Finally, \(\mathbb{N}\) denotes the set of natural numbers and we write \(\mathbb{N}_0:=\mathbb{N} \cup \{0\}\) for the set of non-negative integers.

2 Geometric Setup and Preliminary Lemmas for Decoupling↩︎

Let’s first go through some definitions and standard results in decoupling. Let \[\mathcal{P}:= \{(x,x^2) : x \in [0,1]\}\] denote the truncated parabola, and let \(\Gamma:=N_{1/R}(\mathcal{P})\) be its \(1/R\)-neighborhood. We write \(B_r(x)\) for the ball of radius \(r\) centered at \(x \in \mathbb{R}^2\), and \(B_r\) for the ball of radius \(r\) centered at the origin. We now record some basic notations.

By the argument in section 2.1 in [13], given any set \(S\) with rectangular shape, there exists a weight function \(W_S\) on \(S\).

We shall first prove our main result for the special case in which \(f\) satisfies \(\text{Cond}_0\), and then extend it to the general case. Let \(f\) satisfies \(\text{Cond}_0\). Define \(l_k=3^{-k}\). Then, \(l_{k-1}/l_k=3\) and \[R^{-1/2}=l_M <l_{M-1} <\cdots < l_1 <l_0=1.\] Let \(c\) be the smallest number such that \(l_c> N/R\). For \(1 \leq k \leq M-1\), We cut \(\mathcal{P}\) into caps \(\tau_k\) of length \(l_k\) and for \(c \leq k \leq M-1\), we cut \(\mathcal{P}\) into caps \(\omega_k\) of length \(l_k^{-1}R^{-1}\). Also, we cut \(\mathcal{P}\) into caps \(\theta\) of length \(R^{-1/2}\) and into caps \(\gamma\) of length \(1/N\). Given any cap \(\tau\), we define \[\begin{align} \lambda(\tau)&:=\#\{\gamma: \gamma \subset \tau, |f_\gamma|\sim 1\}, \\ \lambda(l_k)&:=\max_{\tau_k}\#\{\gamma \subset \tau_k: |f_\gamma|\sim 1 \text{ on } B_R\}, \\ \lambda(l_k^{-1}R^{-1})&:=\max_{\omega_k}\#\{\gamma \subset \omega_k: |f_\gamma|\sim 1 \text{ on } B_R\} \end{align}\]

Also, given a cap \(\theta\), we define \[\lambda(\theta)=\#\{\gamma \subset \theta: |f_\gamma|\sim 1 \text{ on }B_R\}.\]

For any \(k\) and any \(\tau_k \in \mathcal{P}\), we tile \(B_R\) by \(l_kR \times R\) tubes, \(U_{\tau_k}\), whose long axis is in the direction \(d(\tau_k)\). We call this set of tubes \(\mathbb{U}_{\tau_k}\). For \(U_{\tau_k} \in \mathbb{U}_{\tau_k}\), let \(W_{U_k}\) be a weight function on \(U_{\tau_k}\).

Let \(\{\eta_k\}_{0 \leq k \leq M} \cup \{\eta_{l_k}\}_{0 \leq k \leq M}\) be a smooth partition of unity on \(B_1\) such that

1. \[\boldsymbol{supp }\eta_{l_k} \subseteq \left\{x: \frac{l_{k}}{2} \leq |x| \leq 2l_{k-1}\right\} \text{ for } 1 \leq k \leq M\]

2. \[\boldsymbol{supp }\eta_k \subseteq \left\{x: \frac{R^{-1}l_{k-1}^{-1}}{2} \leq |x| \leq 2R^{-1}l_{k}^{-1}\right\} \text{ for } 1 \leq k \leq M\] and \(\boldsymbol{supp }\eta_0 \subseteq \left\{x: |x| \leq 2R^{-1}\right\}.\)

3. \(\check{\eta}_k(x)\sim R^{-2}l_k^{-2}\) on \(B_{Rl_{k}}\) and decays rapidly off \(B_{Rl_{k-1}}\) and \(\check{\eta}_{l_k}(x)\sim_K l_k^{2}\) on \(B_{l_{k-1}^{-1}}\) and decays rapidly off \(B_{l_{k}^{-1}}\)

4. \(\int |\check{\eta}_k| \sim 1\) and \(\int |\check{\eta}_{l_k}| \sim 1\)

In this paper, we shall look at the blurred version of \(f\) at many different scales. For \(1\leq k \leq M\), define \(g_k=\sum_{\tau_k} |f_{\tau_k}|^2\) and define \(g=\sum_{\theta}|f_\theta|^2\) and \(g_{\tau}=\sum_{\theta \subset \tau}|f_\theta|^2\). Then, \(g_k\) is basically \(|f|^2\) blurred at scale \(l_k^{-1}\), \(g\) is \(f\) blurred at scale \(R^{1/2}\) and \(g_\tau\) is \(|f_\tau|^2\) blurred at scale \(R^{1/2}\). Moreover, we can view \(g\) as the sum of the characteristic functions of a bunch of \(R \times R^{1/2}\) tubes.

Then, intuitively \(|U_{\tau_k}|\|f_{\tau_k}\|_{\sout{L}^2(W_{U_k})}^2\sim |U_{\tau_k}|\|g_{\tau_k}\|_{\sout{L}^1(W_{U_k})}\) counts the number of \(R\times R^{1/2}\) tubes contained in \(U_{\tau_k}\). Now, let’s prove some properties of \(\|f\|^2_{\sout{L}^2(W_{U_{\tau_k})}}\).

Next, we prove a slightly generalized version of the bilinear restriction.

Next, we recall some high-low lemmas from [13] that will be useful in our proof.

The following lemma essentially states that the low-frequency components have small \(L^\infty\) norms.

The following lemma exploits the transversality for the high-frequency part.

3 Proof of Theorem 1↩︎

In this section, we prove Theorem 1. By standard reductions, it suffices to establish the following lossless small-cap decoupling estimate.

The proof of Theorem 3 is divided into two parts. First, we reduce matters to the special case in which \(f\) satisfies \(\text{Cond}_0\). We then prove this special case.

3.1 Reducing to the special case↩︎

In order to prove the Theorem 3, we first prove that the general result follows from a special case when \(f\) satisfies \(\text{Cond}_0\). Define the superlevel set \[U_a(f):=\{x \in B_R: a/2 \leq |f(x)| \leq a \}.\]

Now let’s prove that Theorem 3 follows from Lemma 7.

By lemma 7 and a rescaling argument, if \(f\) satisfies \(\text{Cond}_1(\alpha,\lambda)\), then \[\label{con1} |U_a(f)|a^6 \lesssim \log\left(\frac{\alpha\lambda}{a}\right)^{O(1)} \frac{a}{\alpha\lambda}N^2R\alpha^6\lambda^3.\tag{11}\]

Theorem 3 follows quickly from proposition 5.

3.2 Special Case↩︎

The goal of this section is to prove the Lemma 7, which follows quickly from the following Lemma.

First, we derive the first part of the about lemma as a quick corollary from the superlevel bound from the wave envelop paper [13]. The second part of the lemma is our key contribution.

Let’s recall a superlevel set estimate by Guth and Maldague that we heavily rely on in this paper.

The above theorem implies the following corollary.

In particular, summing up the different dyadic choices of \(\lambda_0\leq \frac{N}{\sqrt{R}}\), we have the following corollary.

To deal with the second part of lemma 7, we pigeonhole on \(\lambda(\theta)\) and write \[f=\sum_{\substack{1\leq\lambda_0 \leq R^{-1/2}N\\ dyadic}} f^{\lambda_0}\] with each \(f^{\lambda_0}\) has \(\lambda_0/2\leq\lambda(\theta)\leq\lambda_0\). Define

\[U_{a}^{\lambda_0}=U_{a}(f_{\lambda_0}).\] Note that, \[\int |f|^6 \leq \sum_{\substack{ 1 \leq a \leq \lambda\\a \text{ dyadic}}} |a|^6 |U_a|.\]

As \(1+ \frac{1}{2^2}+\frac{1}{3^2}+\cdots \lesssim 1\) and the scale for \(\lambda_0\) is dyadic, we have \[\sum_{1\leq \lambda_0 \leq R^{-1/2}N} \frac{1}{(\log (R^{-1/2}N/\lambda_0))^2} \sim 1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots +\frac{1}{(\log R^{-1/2}N)^2}\lesssim 1.\]

Thus, \[\|f\|^6_{L^6(B_R)} \leq \sum_{\lambda_0}\sum_a |U_{a/(\log (R^{-1/2}N/\lambda_0))^2}^{\lambda_0}|a^6.\]

From now on in this subsection, we shall focus on the case when \(a >\lambda (\log N)^{-100}\) and \(\lambda_0>R^{-1/2}N(\log N)^{-100}\). We prove the following key lemma, which would immediately imply the second part of Lemma 9 by summing over all the dyadic \(\lambda_0\)’s.

Lemma 10 follows from two key lemmas: the broad estimate (see Lemma 11) and the asymmetric estimate (see Lemma 12), which both estimate the size of the intersection of the superlevel set of \(f_{\Sigma_1}\) and \(f_{\Sigma_2}\). The broad estimate deals with the case when \(\Sigma_1\) and \(\Sigma_2\) have the same length, and the length is similar to the distance between the two caps, and the asymmetric estimate deals with the case when the length of the shorter cap is much smaller than the distance between the two caps.

Let \(\Sigma_1\) and \(\Sigma_2\) be two caps. Within both proofs, we shall use \(\lambda_i\) to denote \(\lambda(\Sigma_i)\) and use \(l_i\) to denote \(l(\Sigma_i)\) for \(i=1,2\). Define \[U_{a,b}(f_1,f_2):=\{x\in B_R: |f_1|>a \text{ and }|f_2|>b\}\] to be the joint superlevel set of \(f_1\) and \(f_2\). Define \(V(k,\beta):=\{x: g_k \gtrsim \beta\}\) and \(V(\beta):=\{x: g \gtrsim \beta\}\) to be the superlevel set of the blurred versions of \(f\). Define \(V'(k,\beta):=\{x: g_k \sim \beta\}\) and \(V'(\beta):=\{x: g \sim \beta\}\).

3.2.1 Broad Estimate↩︎

The overall strategy of the proof for the broad estimate follows that of [13], with the key additional ingredient being a more careful control on the number of scales that can contribute to the high–low estimate. To bound \(U_{a,b}(f_{\Sigma_1},f_{\Sigma_2})\), we shall first bound the high frequency part of \(g_k\), which can be viewed as the \(|f|^2\) blurred at scale \(l_k^{-1}\). Then, we shall see that \(U_{a,b}(f_{\Sigma_1},f_{\Sigma_2})\) is contained in \(V(k,\beta)\) the superlevel set of \(g_k\) for a big enough \(\beta\). Then, we shall show that when \(\beta\) is big enough, \(g_k\) is necessarily dominated by its high-frequency part. Finally, we related the size of \(U_{a,b}(f_1,f_2)\) with that of \(V(k,\beta)\) using bilinear restriction and the fact that the Fourier support of \(f_1\) and \(f_2\) are transversal.

3.2.2 Asymmetric estimate↩︎

Next, we shall prove the following asymmetric estimate.

By a rescaling argument, the lemma above follows from the following lemma.

We begin by estimating the superlevel sets of \(g_{\Sigma_2}\) via a high–low decomposition, while carefully controlling the number of scales that can contribute. Note that \(g_{\Sigma_2}\) is essentially locally constant on rectangles \(T\) of dimensions \(R^{1/2} \times R^{1/2}L_2^{-1}\).

Next, we apply a slightly generalized bilinear restriction estimate 3 on each such rectangle \(T\). This yields a decomposition of \(f_{\Sigma_1}\) into caps \(\omega\) of length \(\max\{N^{-1}, R^{-1/2}L_2\}\), and of \(f_{\Sigma_2}\) into caps of length \(L_2\), allowing us to bound the intersection of \(U_{a,b}\) with each \(T\).

Since we assume \(L_2\) is small, the length of the cap \(\omega\) we can break \(f_{\Sigma_1}\) into is particularly small, which yields a gain of a factor \(l(\Sigma_2)\).

Right now to simplify notation, write \(g=\sum_{\theta \subset \Sigma_2} |f_\theta|^2\) and \(V_\beta=\{x: g(x)>\beta\}\). Note that \[\fint_{B_R} g \sim \fint_{B_R} \sum_{\gamma \subset \Sigma_2}|f_\gamma|^2 \sim \lambda_2.\] In the following lemma, we proved a bound for \(|V_\beta|\) when \(\beta\) is significantly bigger than the average size of \(g\) on \(B_R\).

The following lemma enables us to bound the size of the intersection between \(V_\beta\) and \(U_{a,b}(f_1,f_2)\).

Now, we are ready to put things together. First, by Cauchy-Schwartz, \(|f_2(x)| \geq b\) implies that \(g(x) \geq \frac{b^2\lambda_0}{\lambda_2}\). Thus, \[U_{a,b} \subseteq \bigcup_{\lambda_2\lambda_0 \geq \beta \geq \frac{b^2\lambda_0}{\lambda_2}} V(\beta) \cap U_{a,b}.\] For \(b \gg \lambda_2 \lambda_0^{-1/2}\), we need \(\beta \gg \lambda_2\). Note there are just \(\log(\frac{\lambda_2^2}{b^2})\) many choices of dyadic \(\lambda_2\lambda_0 \geq \beta \geq \frac{b^2\lambda_0}{\lambda_2}\). We shall bound each \(|V(\beta) \cap U_{a,b}|\).

\[\begin{align} && |V(\beta) \cap U_{a,b}|a^2b^2 \\&\leq& \int_{V(\beta)} |f_1|^2|f_2|^2 \\ & \lesssim & \|\sum_{\omega \subseteq \Gamma_1} |f_{1, \omega}|\|_\infty \beta |T|(\#\{T:T\cap V(\beta)\not= \emptyset\}) \text{ by lemma }\ref{bires}\\ & \lesssim_\epsilon& \frac{1}{\beta}\|\sum_{\omega \subseteq \Gamma_1} |f_{1, \omega}|\|_\infty \log\left(\frac{\lambda_2 }{\beta}\frac{N}{\sqrt{R}}\right)^2\lambda_0 NR^{-1/2}R^2\lambda_2. \end{align}\] Suppose that \(R^{-1/2}l_2 \geq 1/N\). We let \(l(\omega)=R^{-1/2}L_2\), so \(\|\sum_{\omega \subset \Gamma_1} |f_{1,\omega}|^2\| \leq \lambda_1\lambda_1(R^{-1/2}l_2)\leq \lambda_1 (l_2N/\sqrt{R})\). Suppose that \(R^{-1/2}l_2 \leq 1/N\). We let \(l(\omega)=1/N\) so \(\|\sum_{\omega \subset \Gamma_1} |f_{1,\omega}|^2\| \leq \lambda_1\). Thus, we have plugging in \(\beta \geq \frac{b^2\lambda_0}{\lambda_2}\). \[\begin{align} && |V(\beta) \cap U_{a,b}|a^2b^2 \\ &\leq& \frac{\lambda_2}{b^2\lambda_0} \lambda_1 \max(1, l_2N/\sqrt{R}) \log\left(\frac{\lambda_2^2 }{b^2}\frac{NR^{-1/2}}{\lambda_0}\right)^2\lambda_0 NR^{-1/2}R^2\lambda_2 \\ & \lesssim & \frac{1}{b^2}\max(1, l_2N/\sqrt{R}) \log\left(\frac{\lambda_2 }{b}\frac{NR^{-1/2}}{\lambda_0}\right)^2\lambda_0 NR^{-1/2}R^2 \lambda_1\lambda_2^2 \end{align}\] Summing over all the dyadic \(\beta\)’s, we proved Lemma 13.

3.2.3 Refined Broad and Narrow Argument↩︎

Now, we develop a refined broad and narrow argument by combining the broad estimate with the asymmetric estimate to prove lemma 10.

Now, let’s gather some properties of \(Br_m\) and \(Nar_m\).

From now on, we put \[h=100\log((\lambda/a)\log(NR^{-1/2}/\lambda_0)).\]

Thus, it remains to bound \(B_1\) and \(B_2\).

Adding the previous two propositions up, we proved that \[|U_a^{\lambda_0}|\lesssim \log\left(\frac{\lambda}{a}\frac{NR^{-1/2}}{\lambda_0}\right)^{O(1)} \frac{N^2R\lambda^2}{a^5}\frac{\lambda_0}{NR^{-1/2}}.\]

4 Strichartz Estimates in \(Y^s\) spaces↩︎

4.1 The Rescaled Torus and \(Y^s\) Spaces↩︎

We begin by introducing the rescaled torus \(\mathbb{T}_\lambda:= \mathbb{R}/\left(2\pi \lambda \mathbb{Z}\right)\) with \(\lambda\geq 1\) and the associated function spaces that will be used in the sequel. For \(f: \mathbb{T}_\lambda \rightarrow \mathbb{C}\) measurable, we define Lebesgue norms by \[\left\lVert f\right\rVert_{L^p(\mathbb{T}_\lambda)}^p = \int_{0}^{2\pi\lambda} \left\lvert f(x)\right\rvert^p \,\, dx,\] for \(1 \leq p < \infty\) and the usual modification for \(p=\infty\). We also define \(\left(dk\right)_\lambda\) to be the normalized counting measure on \(\mathbb{Z}_\lambda := \frac{1}{\lambda}\mathbb{Z}\) \[\int_{\mathbb{Z}_\lambda} a(k) \left(dk\right)_\lambda := \frac{1}{2\pi\lambda} \sum _{k \in \mathbb{Z}_\lambda} a(k).\] The Fourier coefficients of \(f \in L^1(\mathbb{T}_\lambda)\) are given by \[\hat{f}(k) = \int_{0}^{2\pi\lambda} e^{-i k x} f(x) \, \, dx,\] for \(k \in \mathbb{Z}_\lambda\) and the Fourier inversion formula is given by \[f(x)= \int_{\mathbb{Z}_\lambda} e^{i k x} \hat{f}(k)\left(dk\right)_\lambda.\] Using this convention, the following identities hold \[\begin{align} \tag{12} & \int_{0}^{2\pi\lambda} f(x) \overline{g(x)} \,\, dx = \int_{\mathbb{Z}_\lambda} \hat{f}(k) \overline{\hat{g}(k)} \,\, (dk)_\lambda,\\ \tag{13} &\left\lVert f\right\rVert_{L^2(\mathbb{T}_\lambda)}=\left\lVert\hat{f}\right\rVert_{L^2((dk)_\lambda)}, \\ \tag{14} &\widehat{fg}(k) = \int_{\mathbb{Z}_\lambda} \hat{f}(k_1) \hat{g}(k-k_1) \,\, (dk_1)_\lambda. \end{align}\] The Sobolev space, \(H^s=H^s(\mathbb{T}_\lambda)\), is defined as the completion of smooth functions under the norm \[\left\lVert f\right\rVert_{H^s}=\left\lVert\langle k \rangle^s \hat{f}(k)\right\rVert_{L^2((dk)_\lambda)},\] where \(\langle k \rangle := (1+\left\lvert k\right\rvert^2)^\frac{1}{2}\). We will frequently make use of Littlewood-Paley theory which allows us to quantitatively separate the rough, high-frequency behavior of a function from the smooth, low-frequency one. In particular, for \(A \subset \mathbb{Z}_\lambda\) let \(P_A\) denote the Fourier multiplier \(\widehat{P_A f}:= \chi_A \hat{f}.\) By a slight abuse of notation, for \(N \in 2^{k_0\mathbb{N}_0}\) with \(k_0 \geq 1\) we define \[P_N:= \begin{cases} P_{\left[-2^{k_0}N,-N\right) \cup \left(N,2^{k_0}N\right]} &\text{ for } N > 1\\ P_{\left[-1,1\right]} &\text{ for } N \leq 1. \end{cases}\] In most of our estimates, we adopt the standard dyadic decomposition with \(k_0 = 1\). The sole exception occurs in Lemma 23, where we take \(k_0 = 1/s\) for \(0 < s \leq 1\). We also stress that if \(f = f(x,t)\) depends on time, the operator \(P_A\) acts only on the spatial variable: \((P_A f)(x,t) = P_A(f(\cdot,t))(x)\).

We define \(S_\lambda(t)\) to be the solution operator to the linear Schrödinger equation \[i \partial_t u + \Delta u =0, \text{ } u(x,0)=u_0(x), \text{ } x \in \mathbb{T}_\lambda,\] that is, \[S_\lambda(t) u_0(x) = \int_{\mathbb{Z}_\lambda} e^{ i(k x+\left\lvert k\right\rvert^{2}t)} \hat{u}_0(k) (dk)_\lambda.\] When \(\lambda=1\), we drop the subscript and write \(S(t):=S_1(t)\).

As we will be working with long-time Strichartz estimates on the rescaled torus \(\mathbb{T}_\lambda\), it is convenient to prove local well-posedness results in adapted function spaces instead of using \(X^{s,b}\) spaces. To this end, we introduce the \(V^p\), \(U^p\) and \(Y^s\) spaces which were first used in the critical regularity well-posedness theory of dispersive equations on periodic domains [28]–[30]. More recently, the same spaces have been used to prove global well-posedness for the mass critical NLS on \(\mathbb{T}\) [1], [20] and \(\mathbb{T}^2\) [10] in the sub-critical setting.

Throughout the remainder of this section, let \(p \geq 1\), let \(\mathcal{H}\) be a separable Hilbert space over \(\mathbb{C}\) and define \(\mathcal{Z}\) to be set of finite partitions \(-\infty <t_0 <t_1 < \cdots<t_K\leq \infty\) of the real line. If \(t_K=\infty\), we use the convention that \(v(t_K):=0\) for all functions \(v: \mathbb{R}\rightarrow \mathcal{H}\).

The spaces \(U^p\) and \(V^p\) are related by the following continuous embeddings which follow from [28].

We conclude by the defining the \(Y^s\) spaces and stating some well-known properties. For details, see [28].

4.2 Strichartz Estimates↩︎

We next turn to transferring the Strichartz estimate proved in 3 to \(Y^s\) spaces. Throughout this section, all frequency scales \(M\) appearing in the projections \(P_M\) are assumed to satisfy \(M \geq 1\), unless stated otherwise. We also assume throughout that \(\lambda \geq 1\).

In view of the scaling symmetry of the problem, it is natural to work on the rescaled torus \(\mathbb{T}_\lambda\). To relate this setting to the standard torus \(\mathbb{T}\), we begin by recalling a rescaling lemma from [22], which allows us to transfer estimates between the two settings.

The following lemma follows immediately from Theorem 1.

Combining Lemma 17 with Lemma 16 yields the following Strichartz estimates on the rescaled torus.

We conclude this section by transferring the linear Strichartz estimate to \(Y^s\) spaces and by stating a trilinear estimate on \(Y^s\) spaces.

5 Proof of Theorem 2↩︎

The aim of this section is to prove Theorem 2 using the \(I\)-method. Henceforth, let \(0 < s < 1\), and unless otherwise stated, all spatial norms are taken on \(\mathbb{T}_\lambda\), i.e.\(H^s = H^s(\mathbb{T}_\lambda)\), etc. Under the small-mass assumption, it suffices to restrict attention to the defocusing case, i.e. we assume \(\mu = 1\) throughout this section.

To prove Theorem 2, we follow a strategy similar to that in [1], [19], [20]. Rather than working directly with the original 1 on \(\mathbb{T}\), we consider the \(I\)-system 17 (see 5.2) on \(\mathbb{T}_\lambda\), in order to exploit the scaling symmetry of the equation. We show that solutions to 17 can be extended to arbitrarily large times, which yields a growth bound for solutions to 1 on the rescaled torus \(\mathbb{T}_\lambda\). By inverting the rescaling, we obtain polynomial-in-time bounds for the \(H^s(\mathbb{T})\) norm of solutions to 1 , thereby establishing global well-posedness.

Following [19], our proof is divided into two steps: (i) establishing well-posedness for the \(I\)-system, and (ii) deriving a growth bound for the modified energy \(E^1\).

5.1 The \(I\)-operator↩︎

We begin by constructing the \(I\)-operator and the first modified energy. Define \(m_1\) to be the restriction to \(\mathbb{Z}_\lambda\) of the following smooth and monotone multiplier which satisfies \[m_1(x):=\begin{cases} 1 &\text{ if } \left\lvert x\right\rvert \leq 1, \\ \left\lvert x\right\rvert^{-1} &\text{ if } \left\lvert x\right\rvert > 2, \end{cases}\] and for which, for every \(k\in\mathbb{N}_0\), there exists a constant \(C_k>0\) such that \[\sup_{x\in\mathbb{R}} \Bigg| \frac{x^k}{m_1(x)} \frac{d^k m_1}{dx^k}(x) \Bigg| \le C_k.\] The conditions on derivatives of \(m_1\) are included to guarantee that the multiplier is sufficiently smooth for subsequent analysis (see Lemma 25). For \(N>0\) set \(m_N(x):=m_1(\frac{x}{N})\) and for \(\beta\geq 0\), let \(I_N^\beta\) denote the Fourier multiplier \(\widehat{I_N^\beta f}=m^\beta_N\hat{f}\). Additionally, note that \(I_N^\beta\) is a smoothing operator of degree \(\beta\), satisfying \[\left\lVert u\right\rVert_{H^{s_0}(\mathbb{T}_\lambda)} \lesssim \left\lVert I_N^\beta u\right\rVert_{H^{s_0+\beta}(\mathbb{T}_\lambda)}\lesssim_\beta N^{\beta} \left\lVert u\right\rVert_{H^{s_0}(\mathbb{T}_\lambda)},\] for all \(s_0 \in \mathbb{R}\). We stress that the implicit constant in the first inequality is independent of \(\beta\) provided \(N \geq 2\).

We also recall an interpolation lemma [31] which is useful for proving local well-posedness estimates.

For the remainder of this paper, we define \(m:=m_N^{1-s}\) and \(I=I_N^{1-s}\) for some \(N \gg 1\) to be specified later. The first modified energy associated with the \(I\)-operator is defined by \[E^1\left(u(t)\right):= E\left(Iu(t)\right)= \frac{1}{2}\int_{\mathbb{T}_\lambda} \left\lvert\partial_xIu(x,t)\right\rvert^2 \,\, dx + \frac{1}{6} \int_{\mathbb{T}_\lambda} \left\lvert Iu(x,t)\right\rvert^{6}\,\, dx.\]

5.2 Local well-posedness for the \(I\)-system↩︎

The first step in the proof of Theorem 2 is establishing a bound for \(\left\lVert Iu\right\rVert_{Y^1_T}\) for some long enough \(T>0\). To do this we apply the \(I\)-operator to 1 and use a fixed point argument on the following initial value problem, \[\label{eq:I95system} \begin{cases} i \partial_t Iu+\Delta Iu=I\left(\left\lvert u \right \rvert^4 u\right), \\ Iu(x,0)=Iu_0(x) \in H^{1}. \end{cases} \tag{17}\] In particular, we prove the following result.

By standard iterative methods and the properties of \(Y^s\) spaces, Lemma 22 follows immediately from the following multilinear estimate.

5.3 Growth of \(E^1\)↩︎

Next, we proceed with bounding the growth of the modified energy. Note that if \(u\) is a smooth solution to 1 , then \[\begin{align} \partial_t E^1(u(t)) &= \Re \int_{\mathbb{T}_\lambda} \overline{\partial_t Iu } \left( \left\lvert Iu\right\rvert^4 Iu - \Delta Iu \right),\\ &= \Re \int_{\mathbb{T}_\lambda} \overline{\partial_t Iu} \left( \left\lvert Iu\right\rvert^4 Iu - I(\left\lvert u\right\rvert^4 u) \right), \\ &= \Im \int_{\mathbb{T}_\lambda} \left(\overline{\Delta I u}- \overline{I(\left\lvert u\right\rvert^4u)} \right) \left( \left\lvert Iu\right\rvert^4 Iu - I(\left\lvert u\right\rvert^4 u) \right). \end{align}\] Therefore, by the fundamental theorem of calculus, for any \(t_0 \in \mathbb{R}\) and any \(T>0\), we have \[\label{eq:32fundamental95theorem95calculus95E1} E^1(u(t_0+T))-E^1(u(t_0)) = \int_{t_0}^{t_0+T} \left(\Lambda_6(u(t)) +\Lambda_{10}(u(t) \right)\,\,dt,\tag{18}\] where we define \[\begin{align} \Lambda_6(u) &:= \Im \int_{\Gamma_6} M_6(k_1, \dots, k_6) \widehat{\overline{\Delta Iu}} (k_1) \widehat{u}(k_2) \widehat{\overline{u}}(k_3) \widehat{u}(k_4) \widehat{\overline{u}}(k_5) \widehat{u}(k_6), \\ \Lambda_{10}(u) &:= \frac{i}{2}\int_{\Gamma_{10}} M_{10}(k_1, \dots, k_{10} ) \prod_{j=1}^5 \widehat{u} (k_{2j-1}) \widehat{\overline{u}}(k_{2j}). \end{align}\] Here the hyperplane \(\Gamma_n=\left\{ (k_1, \dots, k_n) \in \mathbb{Z}_\lambda^n : k_1+\cdots+k_n=0 \right\}\) is endowed with measure \(\prod_{j=1}^{n-1} (dk_j)_\lambda\) and the multipliers \(M_6: \Gamma_6 \rightarrow \mathbb{R}\) and \(M_{10}: \Gamma_{10} \rightarrow \mathbb{R}\) are given by \[\begin{align} M_6(k_1, \dots, k_6)&:=\prod_{j=2}^6 m(k_j) - m(k_1), \\ M_{10}(k_1,\dots,k_{10}) &:=m(\sum_{j=6}^{10} k_j) \prod_{j=1}^{5} m(k_j)- m(\sum_{j=1}^5 k_j)\prod_{j=6}^{10} m(k_j). \end{align}\] Our goal is to bound the right-hand side of 18 by using the long-time linear and trilinear Strichartz estimates from Lemma 19 and Lemma 20. The remainder of this section will focus on proving following estimate.

We prove Lemma 24 by separately estimating the contributions of \(\Lambda_6\) and \(\Lambda_{10}\) in 18 . To this end, we first we first state a helpful lemma.

By combining Lemma 25 with the long-time Strichartz estimates from Lemmata 19 and 20 we obtain the following estimates.

5.4 Proof of Global Well-posedness↩︎

We are now in a position to prove Theorem 2 using Lemmata 22 and 24. Let \(u: \mathbb{T}\times \mathbb{R}\rightarrow \mathbb{C}\) denote the solution to 1 with initial condition \(u_0 \in H^s(\mathbb{T})\). By assumption, \(u_0\) has small mass, so we rescale the initial data so that \(Iu_0^\lambda\) has small \(H^1\)-norm, where \[u_0^\lambda= \lambda^{-\frac{1}{2}}u_0\left(\frac{x}{\lambda}\right),\] and \(u^\lambda : \mathbb{T}_\lambda \times \mathbb{R}\rightarrow \mathbb{C}\) denotes the corresponding solution to 1 with initial data \(u_0^\lambda\).

Applying Lemma 22 yields a bound for the \(H^1\) norm of \(Iu^\lambda(t)\) for all \(t \in [0,T_0]\) for some fixed \(T_0>0\). The strategy is to iterate this local result: we use Lemma 24 to bound the growth of the modified energy \(E^1(u^\lambda(t))\) on \([0,T_0]\) which in turn controls the \(\dot{H}^s\)-norm of \(Iu^\lambda(T_0)\). Combined with the conservation of mass, this provides uniform control of the \(H^s\) norm of \(Iu^\lambda(T_0)\) for sufficiently large \(N\), allowing Lemma 22 to be applied successively with initial condition \(Iu^\lambda(T_0)\). By repeating this argument and reversing the rescaling, we extend the original solution \(u\) globally in time by letting \(N \rightarrow \infty\). We now turn to details.

6 Refined Trilinear Strichartz Estimates via Decoupling↩︎

In this appendix, we give an alternative proof of the refined trilinear Strichartz estimate first established in [1].

The corresponding result in [1] is stronger, as it does not contain the \(N_2^\epsilon\) loss. Their proof uses number-theoretic methods, whereas ours uses decoupling tools. Thus, our method allows more flexible choices of the locations for the frequencies. We include an alternative proof here for completeness and to highlight our method, which may be useful in higher-dimensional settings.

We shall show that Theorem 20 follows quickly from the asymmetric estimate in Lemma 13 by combining it with an induction argument similar to the one in [32]. We also emphasize that Theorem 20 is sharp by considering Example 1 and Example 2.

By a standard rescaling and periodization argument, it suffices to establish the following decoupling estimate.

6.1 Examples↩︎

We see that lemma 20 is sharp up to a \((L_2NR)^\epsilon\) factor, if we consider the following two examples. In both examples, we shall let \(a_\xi=1\) for all \(\xi \in \Xi_1\) and \(\Xi_2\). Put \[g_1(x,t):=\sum_{j=1}^{L_1N}e(\frac{j}{N}x+\frac{j^2}{N^2}t) \text{ and }g_2(x,t):=\sum_{j=N}^{N+L_2N}e(\frac{j}{N}x+\frac{j^2}{N^2}t).\]

6.2 Proof of Theorem 20↩︎

We shall focus on the special case when we are in the situation such that for each \(\gamma\) either \(a_\gamma \sim 1\) or \(a_\gamma=0\). In this case, we say that \(f_1\) and \(f_2\) satisfy \(\text{Cond}(L_1,L_2)\). The general case would just follow from it from a standard pigeonholing argument, giving rise to an extra \(\log R\) loss.

To prove Theorem 20, it is enough to prove \(D(L_1,L_2,R,N)\lessapprox \max(L_2, \sqrt{R}/N) \frac{N^2}{R}\). To do so, we shall bound \(|U_{a,b}(f_1,f_2)|a^2b^4\) for the case when \(b\) is big and when \(b\) is small separately.

Suppose that \(b \geq R^\epsilon \lambda_2 \lambda_2(R^{-1/2})^{-1/2}\), then Lemma 13 implies that \[|U_{a,b}(f_1,f_2)|a^2b^4 \lessapprox \max\!\left( L_2, \frac{\sqrt{R}}{N} \right) \frac{N^2}{R}\lambda_1 \lambda_2^2.\] Suppose that \(b \leq R^\epsilon \lambda_2 \lambda_2(R^{-1/2})^{-1/2}\), Lemma 2 immediately implies the following bound on \(|U_b(f_2)|b^6\).

Next, we prove the following lemma relating \(D(L_1,L_2,R,N)\) to \(D(L_2^2,L_2,R,N)\).

Combining the two lemmas above, we obtain the following inductive relation.

Combining lemma 30 and lemma 13 and orthogonality, we have the following induction lemma

Acknowledgment↩︎

The authors would like to thank Professor Larry Guth for his unwavering support and helpful suggestions throughout the preparation of this paper. This work was completed under his guidance. The authors are also grateful to Professor Gigliola Staffilani for many helpful discussions and valuable insights related to this work. In addition, [NS] would like to thank his thesis supervisor, Professor Alexandre Megretski, for his continued support, guidance, and encouragement throughout this project.

References↩︎