Finite difference scheme for two-dimensional periodic nonlinear Schr\"odinger equations

TL;DR

This paper proves that a finite difference scheme for the two-dimensional periodic nonlinear Schrödinger equation converges strongly in L^2 to the true solution as the grid size decreases, validating the method's effectiveness.

Contribution

It establishes the strong L^2 convergence of the finite difference scheme for 2D periodic NLS, providing a rigorous justification for its use.

Findings

Strong L^2 convergence as grid size approaches zero

Validation of finite difference method for 2D periodic NLS

Theoretical proof of scheme's effectiveness

Abstract

A nonlinear Schr\"odinger equation (NLS) on a periodic box can be discretized as a discrete nonlinear Schr\"odinger equation (DNLS) on a periodic cubic lattice, which is a system of finitely many ordinary differential equations. We show that in two spatial dimensions, solutions to the DNLS converge strongly in $L^2$ to those of the NLS as the grid size $h>0$ approaches zero. As a result, the effectiveness of the finite difference method (FDM) is justified for the two-dimensional periodic NLS.

Equations

$$i{\partial }_{t}u+\Delta u-\lambda |u{|}^{p-1}u=0$$

$$u=u(t,x):\mathbb{R}\times {\mathbb{T}}^d\to \mathbb{C}.$$

$${\mathbb{T}}_h^d:=h{\mathbb{Z}}^d/2\pi {\mathbb{Z}}^d,$$

$$\Big{\{}x=h(m_{1},...,m_{d}):m_{j}=-M,...,-2,-1,0,1,2,...,M-1\Big{\}}$$

$$({\Delta }_h{u}_h)(x):=\sum _{j=1}^d\frac{u_h(x+h{e}_j)+u_h(x-h{e}_j)-2u_h(x)}{h^2},\forall x\in {\mathbb{T}}_h^d,$$

$$i{\partial }_t{u}_h+{\Delta }_h{u}_h-\lambda {\left|u_h\right|}^{p-1}{u}_h=0,$$

$$u_h=u_h(t,x):\mathbb{R}\times {\mathbb{T}}_h^d\to \mathbb{C}.$$

$$(d_{h}f)(x):=\frac{1}{h^d}{\int }_{x+[0,h{)}^d}f(y)dy,\forall x\in {\mathbb{T}}_h^d.$$

$$(p_h{f}_h)(x):=f_h(\underline{x})+\sum _{j=1}^d{D}_{h,j}^{+}(\underline{x})(x_j-{\underline{x}}_j)$$

$$D_{h,j}^{+}{f}_h(\underline{x}):=\frac{f_h(\underline{x}+h{e}_j)-f_h(\underline{x})}{h}.$$

$$\left(p_h\circ U_h(t)\circ d_h\right)u_0-U(t)u_0\to 0$$

$$i{\partial }_t{u}_h+{\Delta }_h{u}_h=0,$$

$$\|f\|_{L_{h}^{r}}:=\left\{\begin{aligned} &\bigg{\{}h^{d}\sum_{x\in\mathbb{T}_{h}^{d}}|f(x)|^{r}\bigg{\}}^{1/r}&&\textup{if }1\leq r<\infty,\\ &\sup_{x\in\mathbb{T}_{h}^{d}}|f(x)|&&\textup{if }r=\infty,\end{aligned}\right.$$

$$\frac{3}{q}+\frac{d}{r}=\frac{d}{2}\text{ and }(q,r,d)\ne (2,\infty ,3).$$

$${\Vert e^{it{\Delta }_h}{u}_{h,0}\Vert }_{L_t^q([0,1];L_h^r)}\le C\Vert ⟨{\nabla }_h{⟩}^{\frac{2}{q}+ϵ}{u}_{h,0}{\Vert }_{L_h^2}$$

$$\{\begin{array}{rlrlrl}1& <p<\infty & & \text{when }\lambda =1& & \text{(defocusing)},\\ 1& <p<3& & \text{when }\lambda =-1& & \text{(focusing)}.\end{array}$$

$${\Vert \left(p_h\circ U_h(t)\circ d_h\right)u_0-U(t)u_0\Vert }_{L^2({\mathbb{T}}^2)}\le A\sqrt{h}{e}^{B|t|}{\left(1+\Vert u_0{\Vert }_{H^1({\mathbb{T}}^2)}\right)}^p.$$

$$u_h(t)=e^{-it(-{\Delta }_h)}(d_h{u}_0)-i\lambda {\int }_0^t{e}^{-i(t-s)(-{\Delta }_h)}(|u_h{|}^{p-1}{u}_h)(s)ds$$

$$u(t)=e^{-it(-\Delta )}{u}_0-i\lambda {\int }_0^t{e}^{-i(t-s)(-\Delta )}(|u{|}^{p-1}u)(s)ds,$$

$$\Vert u{\Vert }_{L_h^q}\lesssim h^{-(\frac{1}{p}-\frac{1}{q})}\Vert u{\Vert }_{L_h^p}\text{for all }q>p,$$

$$\Vert uv{\Vert }_{L_h^r}\le \Vert u{\Vert }_{L_h^p}\Vert v{\Vert }_{L_h^q}.$$

$$\Vert u\ast v{\Vert }_{L_h^r}\le \Vert u{\Vert }_{L_h^p}\Vert v{\Vert }_{L_h^q},$$

$$(u\ast v)(x)=h^d\sum _{y\in {\mathbb{T}}_h^d}u(x-y)v(y).$$

$$\Vert uv{\Vert }_{L_h^r}=h^{\frac{d}{r}}\Vert uv{\Vert }_{{\ell }_x^r}\le h^{\frac{d}{r}}\Vert u{\Vert }_{{\ell }_x^p}\Vert v{\Vert }_{{\ell }_x^q}=\Vert u{\Vert }_{L_h^p}\Vert v{\Vert }_{L_h^q}$$

$$\|u*v\|_{L_{h}^{r}}=h^{d(1+\frac{1}{r})}\Big{\|}\sum_{y\in\mathbb{T}_{h}^{d}}u(x-y)v(y)\Big{\|}_{\ell_{x}^{r}}\leq h^{d(\frac{1}{p}+\frac{1}{q})}\|u\|_{\ell_{x}^{p}}\|v\|_{\ell_{x}^{q}}=\|u\|_{L_{h}^{p}}\|v\|_{L_{h}^{q}}.$$

$$({\mathbb{T}}_h^d{)}^{\ast }:$$

$$\displaystyle=\Big{\{}-\tfrac{\pi}{h},...,-2,-1,0,1,2,...,\tfrac{\pi}{h}-1\Big{\}}^{d}$$

$$\displaystyle=\Big{\{}-M,...,-2,-1,0,1,2,...,M-1\Big{\}}^{d}.$$

$$({\mathcal{F}}_{h}u)(k):=h^d\sum _{x\in {\mathbb{T}}_h^d}u(x)e^{-ik\cdot x}.$$

$$({\mathcal{F}}_h^{-1}u)(x):=\frac{1}{(2\pi {)}^d}\sum _{k\in ({\mathbb{T}}_h^d{)}^{\ast }}u(k)e^{ik\cdot x}.$$

$${\mathbb{T}}_h^d\to {\mathbb{T}}^d,({\mathbb{T}}_h^d{)}^{\ast }\to {\mathbb{Z}}^d,{\mathcal{F}}_h\to \mathcal{F},{\mathcal{F}}_h^{-1}\to {\mathcal{F}}^{-1}$$

Full text

1. Introduction

We consider the nonlinear Schrödinger equation (NLS)

[TABLE]

on the periodic box $\mathbb{T}^{d}=\mathbb{R}^{d}/2\pi\mathbb{Z}^{d}$, where $p>1$, $\lambda=\pm 1$, and

[TABLE]

The NLS is a canonical model that describes the propagation of nonlinear waves. When the nonlinearity is either cubic or quintic, or a combination of these two types, the equation (1.1) arises in various physical contexts including nonlinear optics and low-temperature physics. In particular, if a huge number of boson particles are trapped in a box with the periodic boundary condition and they are cooled to a temperature approaching absolute zero, they form a Bose–Einstein condensate and their mean-field dynamics is determined by the periodic NLS. We refer to [19, 8, 21, 7] for a rigorous proof for this.

The periodic NLS (1.1) may be studied numerically by employing the following standard semi-discrete finite difference method (FDM). For a mesh size $h=\frac{\pi}{M}>0$ with a large integer $M>0$, we denote the dense periodic lattice by

[TABLE]

that is, the additive group

[TABLE]

of $(2M)^{d}$ points (see Figure 1.1), and define the discrete Laplacian $\Delta_{h}$ by

[TABLE]

which acts on complex-valued functions on the periodic lattice. Then, we formulate the discrete nonlinear Schrödinger equation (DNLS) as

[TABLE]

where $p>1$, $\lambda=\pm 1$ and

[TABLE]

In this way, the partial differential equation is translated into the system of $(2M)^{d}$-many first-order ordinary differential equations.

The main purpose of the work presented in this article is to justify the effectiveness of the above numerical scheme. We introduce the following operators to formulate the problem precisely.

Definition 1.1** (Discretization and linear interpolation).**

(i) For a function $f:\mathbb{T}^{d}\to\mathbb{C}$, its discretization is defined by

[TABLE]

(ii) Given a function $f_{h}:\mathbb{T}_{h}^{d}\to\mathbb{C}$, its linear interpolation is defined by

[TABLE]

for $x\in\underline{x}+[0,h)^{d}$ with $\underline{x}\in\mathbb{T}_{h}^{d}$, where $e_{j}$ is the $j$-th standard unit vector, $x_{j}$ denotes the $j$-th component of $x\in\mathbb{T}^{d}$, and $D_{h,j}^{+}$ is the discrete right-hand side derivative on $\mathbb{T}_{h}^{d}$, i.e.,

[TABLE]

Definition 1.2** (Nonlinear propagators).**

We denote the nonlinear propagator for NLS (1.1) by $U(t)$. In other words, $U(t)u_{0}$ is the solution to the NLS (1.1) with initial data $u_{0}$. Similarly, we denote the nonlinear propagator for DNLS (1.4) by $U_{h}(t)$.

Remark 1.3*.*

$(i)$ The discretization operator $d_{h}$ sends functions on the periodic box to functions on a periodic lattice. Conversely, the linear interpolation operator $p_{h}$ maps discrete functions to continuous functions.

$(ii)$ The nonlinear propagators $U(t)$ and $U_{h}(t)$ are well defined because the equations are locally well posed under suitable assumptions (see Propositions B.1 and 4.1).

Our goal is then to show that

[TABLE]

as $h\to 0$ in a proper sense (see Figure 1.2). The convergence (1.8) is referred to as the continuum limit for DNLS. Obviously, proving the continuum limit implies the effectiveness of the numerical scheme.

Despite its physical importance, to the best of the authors’ knowledge, the continuum limit for a nonlinear dispersive equation on a compact manifold has not previously been studied. However, the continuum limit from DNLS on $h\mathbb{Z}^{d}$ to NLS on $\mathbb{R}^{d}$ has been investigated by Ignat–Zuazua [13, 14, 15, 16], Kirkpatrick–Lenzmann–Staffilani [18], and the first and fourth authors of this work [10, 11]. An important remark is that the linear discrete model

[TABLE]

where $u_{h}=u_{h}(t,x):\mathbb{R}\times h\mathbb{Z}^{d}\to\mathbb{C}$, enjoys weaker dispersion than the continuous model [13, 22], and this causes difficulties in proving the continuum limit. In [14, 16], the authors circumvented the weak dispersion phenomena by introducing a new numerical scheme, that is, the two-grid algorithm, to exclude bad frequencies generating weak dispersions. Subsequently, in [10, 11], the authors discovered that the space–time norm bounds, namely Strichartz estimates, for (1.9) hold uniformly in $h\in(0,1]$ with some derivative on the right-hand side. As an application, convergence of the discrete NLS on $h\mathbb{Z}^{d}$ is established without modifying the numerical scheme.

Returning to the problem discussed here, one would attempt to adopt the approach in [10, 11] to the periodic setting. However, several new issues are raised, in particular, for the desired uniform-in-$h$ Strichartz estimates.

Remark 1.4*.*

In the celebrated work of Bourgain [3], Strichartz estimates are established for the linear Schrödinger equation $i\partial_{t}u+\Delta u=0$ on a periodic box, and they are applied to prove the local well-posedness of the periodic NLS (1.1) in a low regularity Sobolev space. Importantly, these Strichartz estimates can be captured from the gain of regularity in the multi-interaction of linear solutions localized in same frequencies but different modulations. This phenomenon is known as the dispersive smoothing effect, and its proof requires an understanding of the geometry of the support of the spacetime Fourier transform of linear solutions, that is, the hypersurface $\{(\tau,k)\in\mathbb{R}\times\mathbb{Z}^{d}:\tau+|k|^{2}=0\}$. We also refer to the work of Guo, Oh and Wang [9] for a further context of NLS on irrational torus.

Unfortunately, we are currently unable to capture dispersive smoothing in the discrete setting. Indeed, the hypersurface for the linear equation (1.9) is given by $\{(\tau,k)\in\mathbb{R}\times\mathbb{Z}^{d}:\tau+\sum_{j=1}^{d}\frac{2}{h^{2}}(1-\cos hk_{j})=0\}$. Following Bourgain’s approach, it is necessary to count the maximal number of points in the intersection of twisted annuli $\{\tilde{k}\in\mathbb{Z}^{d}:M\leq|\tilde{\tau}+\sum_{j=1}^{d}\frac{2}{h^{2}}(1-\cos h\tilde{k}_{j})|\leq 2M\}$ and $\{\tilde{k}^{\prime}\in\mathbb{Z}^{d}:N\leq|\tilde{\tau}^{\prime}+\sum_{j=1}^{d}\frac{2}{h^{2}}(1-\cos h\tilde{k}^{\prime}_{j})|\leq 2N\}$ restricted to the hyperplane $\tilde{k}+\tilde{k}^{\prime}=k$ with $\tilde{\tau}+\tilde{\tau}^{\prime}=\tau$. Compared to the continuous case, the situation is much more complicated because of the complexity of the geometry. Moreover, because local smoothing is known to fail on the noncompact lattice $h\mathbb{Z}^{d}$ [13], this may not simply be a matter of technicality but may indicate that a new idea is needed. It is also worth to mention that Strichartz estimates on $\mathbb{T}^{d}$ for higher dimension were established by Bourgain and Demeter [5] as a corollary of their main theorem on the decoupling inequality (Wolff’s inequality). It may be one of possible ways to follow the decoupling approach to our problem. We leave this question for future study.

One way to circumvent the aforementioned difficulties would be to approximate the linear propagator on a periodic box by that on an entire space. Ultimately it would seem that, by suitably adjusting the argument of Vega [25] to the discrete setting, the time-localized uniform-in-$h$ Strichartz estimates can be obtained on a periodic lattice. For the statement, we define the finite-dimensional vector space $L_{h}^{r}=L_{h}^{r}(\mathbb{T}_{h}^{d})$ equipped with the norm

[TABLE]

and define the fractional derivative $\langle\nabla_{h}\rangle^{s}$ as the Fourier multiplier of symbol $\langle k\rangle^{s}$ via the discrete Fourier transform, where $\langle k\rangle=\sqrt{1+|k|^{2}}$ (see Section 2). We say that $(q,r)$ is lattice-admissible if $2\leq q,r\leq\infty$,

[TABLE]

Theorem 1.5** (Strichartz estimates on a periodic lattice).**

Let $h\in(0,1]$. For a lattice-admissible pair $(q,r)$, there exists $C>0$, independent of $h$, such that

[TABLE]

for any $\epsilon>0$.

Strichartz estimates are one of the fundamental tools to study dispersive equations because they quantify the smoothing and/or decay properties of solutions. On the unbounded lattice $\mathbb{Z}^{d}$, the smoothing and decay properties have been investigated for various models: we refer to [22, 13, 12] for the Schrödinger equation, [20] for the wave equation, and [1] for the Klein–Gordon equation. Theorem 1.5 is the first result on a compact discrete domain as far as the authors know. It should be noted that the inequality (1.12) holds uniformly in $h\in(0,1]$. Indeed, it is easy to show the inequality $\|e^{it\Delta_{h}}u_{h,0}\|_{L_{t}^{q}([0,1];L_{h}^{r})}\leq C_{h}\|u_{h,0}\|_{L_{h}^{2}}$ for all $1\leq q,r\leq\infty$, since $\mathbb{T}_{h}^{d}$ is finite-dimensional. However, this inequality is not useful at all for our purpose because the constant $C_{h}$ blows up as $h\to 0$. As in [10], for which uniform Strichartz estimates are proven on $h\mathbb{Z}^{d}$, we could obtain an appropriate (uniform-in-$h$) Strichartz estimates by placing some derivatives on the norm on the right-hand side (Theorem 1.5). We also note that we do not claim optimality of the Strichartz estimates (1.12). In fact, the order of the derivative could be reduced by solving the counting problem mentioned in Remark 1.4.

Although there is still room for improvement, Theorem 1.5 is sufficient to establish the global-in-time continuum limit for the two-dimensional periodic NLS, which is the main result of this work.

Theorem 1.6** (Continuum limits).**

Let $d=2$. We assume

[TABLE]

There exist constants $A,B>0$, independent of $h\in(0,1]$, such that for all $t\in\mathbb{R}$,

[TABLE]

The proof of Theorem 1.6 follows the argument outlined in [11]. Precisely, we consider two solutions in Duhamel’s formulas,

[TABLE]

and

[TABLE]

where $u_{h}(t)=U_{h}(t)(d_{h}u_{0})$ and $u(t)=U(t)u_{0}$. We aim to estimate the difference $p_{h}u_{h}(t)-u(t)$ directly by the standard Grönwall’s inequality. We accomplish this by making use of a “time-averaged” uniform-in-$h$ $L_{h}^{\infty}$-bound for nonlinear solutions $\{u_{h}(t)\}_{h\in(0,1]}$. Such a uniform bound can be obtained by applying uniform-in-$h$ Strichartz estimates for the discrete linear equation to the nonlinear problem.

Remark 1.7*.*

$(i)$ The essential part of our analysis lies in proving the uniform Strichartz estimates for the linear equation. For this proof, we employ the Fourier analysis on a periodic lattice, and we develop harmonic analysis tools on the lattice, including the Littlewood–Paley theory. Indeed, a periodic lattice is a finite abelian group; thus, the Fourier and its inverse transforms are properly defined (see Section 2.2).

$(ii)$ As mentioned in Remark 1.4, if the classical Bourgain’s argument is adopted, the proof of the Strichartz estimates is transferred to a certain counting problem, but this is ultimately quite challenging. Instead, we employ an alternative approach of Vega [25]. This approach is simpler and can also be applied to more general settings [6], but optimality is far from guaranteed.

$(iii)$ In higher dimensions $d\geq 3$, only local-in-time convergence can be derived from Theorem 1.5, because uniform Strichartz estimates hold for more regular initial data than those in the energy space. Indeed, if $d\geq 3$, the regularity $\frac{2}{q}+\epsilon$ of the Sobolev norm on the right-hand side in (1.12) is always strictly greater than one (when $r=\infty$).

The remainder of the paper is organized as follows: In Section 2, we provide the collection of basic analysis tools. In particular, Fourier analysis on a periodic lattice is briefly presented, but some important inequalities, such as the Sobolev and the Gagliardo–Nirenberg inequalites, are also introduced. In Section 3, we prove the key uniform Strichartz estimates (Theorem 1.5). In Section 4, we establish a well-posedness theory for DNLS (1.4) as well as uniform bounds for the nonlinear solutions. Finally, in Section 5, we prove the main theorem (Theorem 1.6).

Acknowledgement

Y.H. was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2017R1C1B1008215). C.K. was supported by FONDECYT Postdoctorado 2017 Proyect No 3170067. S. N. was supported by the JSPS Grant-in-Aid for JSPS Research Fellow no. 17J01766. C.Y. was supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA1702-02.

2. Preliminaries

2.1. Basic inequalities on a periodic lattice

Recall the definition of the Lebesgue spaces on a periodic lattice (see (1.10)). On a lattice, we often have a larger class of inequalities, compared to those in the continuum domain $\mathbb{T}^{d}$. For instance, by the definition, one can easily show the inequality

[TABLE]

while the embedding $L^{p}\hookrightarrow L^{q}$ fails on $\mathbb{T}^{d}$. However, these inequalities become meaningless in the continuum limit $h\to 0$. Therefore, we would have to use inequalities wherein the implicit constants are independent of $h\in(0,1]$.

We state the following inequalities, which hold uniformly in $h\in(0,1]$.

Lemma 2.1**.**

$(i)$* (Hölder’s inequality). If $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ and $1\leq p,q,r\leq\infty$, then*

[TABLE]

$(ii)$* (Young’s inequality) If $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}+1$, $1\leq p,q,r\leq\infty$ and $\frac{1}{p}+\frac{1}{q}\geq 1$, then*

[TABLE]

where $*$ denotes the convolution operator defined by

[TABLE]

Proof.

Based on Hölder’s and Young’s inequalities for sequences, we prove that

[TABLE]

and

[TABLE]

∎

2.2. Fourier transform on a periodic lattice

Fix a large integer $M>0$. For the periodic lattice $\mathbb{T}_{h}^{d}$ with $h=\frac{\pi}{M}$ (see (1.2)), we denote its Fourier dual space, that is, the sparse periodic lattice, by

[TABLE]

For a function $u:\mathbb{T}_{h}^{d}\to\mathbb{C}$, its Fourier transform $\mathcal{F}_{h}u:(\mathbb{T}_{h}^{d})^{*}\to\mathbb{C}$ is defined by

[TABLE]

The inverse Fourier transform of a function $u:(\mathbb{T}_{h}^{d})^{*}\to\mathbb{C}$ is given by

[TABLE]

With abuse of notation, we write $\sum_{x\in\mathbb{T}_{h}^{d}}=\sum_{x}$ and $\sum_{k\in(\mathbb{T}_{h}^{d})^{*}}=\sum_{k}$ unless there is confusion.

Remark 2.2*.*

The above definitions are consistent with those on the periodic box $\mathbb{T}^{d}$. Indeed, formally, we have

[TABLE]

as $h\to 0$, where $\mathcal{F}$ and $\mathcal{F}^{-1}$ are the Fourier and the inverse transforms on $\mathbb{T}^{d}$, respectively,

[TABLE]

We collect the properties of the Fourier and inverse Fourier transforms.

Lemma 2.3** (Properties of the Fourier transform on a periodic lattice).**

1. (1)

(Inversion)

[TABLE] 2. (2)

(Plancherel’s theorem)

[TABLE] 3. (3)

(Fourier transform of a product)

[TABLE]

To prove Lemma 2.3, we need the following identities.

Lemma 2.4**.**

[TABLE]

and

[TABLE]

Proof.

We only prove (2.6), because the proof of (2.7) is similar. Recalling that $h=\frac{\pi}{M}$ and $x=(x_{1},...,x_{d})=(hm_{1},...,hm_{d})\in\mathbb{T}_{h}^{d}$ where $m_{j}\in\{-M,-M+1,...,M-2,M-1\}$, we evaluate the geometric sum

[TABLE]

Thus, we conclude that

[TABLE]

where we use the fact that $(\mathbb{T}_{h}^{d})^{*}=\{-M,-M+1,...,M-1\}^{d}$ ∎

Proof of Lemma 2.3.

(1) A direct calculation in addition to (2.6) yields

[TABLE]

Analogously, one can show that $\left(\mathcal{F}_{h}(\mathcal{F}_{h}^{-1}u)\right)(k)=u(k)$.

(2) Similarly, using (2.6), we prove that

[TABLE]

(3) We write

[TABLE]

Then, applying (2.7) and summing out $\ell^{\prime}$, we prove the desired identity. ∎

By the Fourier transform, we see that the discrete Laplacian is a Fourier multiplier operator.

Lemma 2.5** (Discrete Laplacian as a Fourier multiplier operator).**

The discrete Laplacian $-\Delta_{h}$ is the Fourier multiplier of the symbol $\sum_{j=1}^{d}\frac{4}{h^{2}}\sin^{2}(\frac{hk_{j}}{2})=\sum_{j=1}^{d}\frac{2}{h^{2}}(1-\cos hk_{j})$.

Proof.

By the definition (1.3),

[TABLE]

∎

Remark 2.6*.*

The discrete Laplacian formally converges to the Laplacian on $\mathbb{T}^{d}$ as $h\to 0$, because given $k\in(\mathbb{T}_{h}^{d})^{*}$, the multiplier for the discrete Laplacian converges to that for the Laplacian on $\mathbb{T}^{d}$, i.e., $\sum_{j=1}^{d}\frac{2}{h^{2}}(1-\cos(hk_{j}))\to|k|^{2}$ as $h\to 0$.

2.3. Dyadic decompositions and Sobolev spaces

Let

[TABLE]

where $\lceil a\rceil$ denotes the smallest integer greater than or equal to $a$. For a dyadic number $N=2^{\ell}$ with $\ell\in\mathbb{Z}$ such that $N_{*}\leq N\leq 1$, we define the frequency projection operator $P_{N}=P_{N}^{h}$ by

[TABLE]

For $s\in\mathbb{R}$, we define the Sobolev space $H_{h}^{s}$ by the Hilbert space equipped with the norm

[TABLE]

We observe that

[TABLE]

The following Sobolev and Gagliardo–Nirenberg inequalities are used in our analysis.

Lemma 2.7** (Sobolev embedding).**

Suppose that $0<s\leq\frac{d}{2}$, $q\geq 2$ and $\frac{1}{q}=\frac{1}{2}-\frac{s}{d}$. Then, for any $\epsilon>0$, we have

[TABLE]

Lemma 2.8** (Gagliardo–Nirenberg inequality).**

Suppose $\frac{1}{p}=\frac{1}{2}-\frac{\theta}{d}$, $1<p\leq\infty$ and $0<\theta<1$. Then we have

[TABLE]

Proofs of Lemmas 2.7 and 2.8 are given in Appendix A. We expect the inequality (2.10) to be improved to the sharp version $(\epsilon=0)$ by adopting the argument in [2] for instance. Nevertheless, in this study, we employ a nonsharp version, because its proof is simpler but also sufficient for our analysis.

2.4. Norm equivalence

There are several ways to define Sobolev spaces on a periodic lattice. The following lemma shows that the Sobolev norm defined by (2.9) is equivalent to that by the discrete derivatives (1.7) as well as that by $\sqrt{1-\Delta_{h}}$, that is, the Fourier multiplier of the symbol $(1+\sum_{j=1}^{d}\frac{4}{h^{2}}\sin^{2}(\frac{hk_{j}}{2}))^{1/2}$.

Lemma 2.9** (Norm equivalence).**

[TABLE]

where $D_{h}^{+}=(D_{h,1}^{+},...,D_{h,d}^{+})$.

Proof.

The first equivalence follows from the Plancherel theorem and the pointwise bound $(1+\sum_{j=1}^{d}\frac{4}{h^{2}}\sin^{2}(\frac{hk_{j}}{2}))^{1/2}\sim\sqrt{1+|k|^{2}}$ on $(\mathbb{T}_{h}^{d})^{*}$. The second identity follows from $(D_{h})^{*}D_{h}=-\Delta_{h}$. ∎

3. Uniform Strichartz estimates on a periodic lattice

This section is devoted to the proof of our key uniform-in-$h$ Strichartz estimates (Theorem 1.5). First, we reduce the proof to the following dispersive estimate.

Proposition 3.1** (Dispersive estimate).**

Let $h\in(0,1]$. For any dyadic number $N$ with $N_{*}:=2^{\lceil\log_{2}(\tfrac{h}{\pi})\rceil-1}\leq N\leq 1$, there exists $c>0$ such that if $|t|\leq\frac{ch}{N}$, then

[TABLE]

where

[TABLE]

Proof of Theorem 1.5, assuming Proposition 3.1.

Applying the standard interpolation argument of Keel and Tao [17] with the dispersive estimate (3.1) but restricting this to the time interval $[0,\frac{ch}{N}]$, one can prove that

[TABLE]

Hence, by changing the variables in time, $P_{N}=P_{\leq N}P_{N}$ and the unitarity of the Schrödinger flow, we obtain

[TABLE]

Summing in the time interval,

[TABLE]

Then, summing in $N$, we obtain

[TABLE]

Because $\mathcal{F}_{h}(P_{N}u_{h,0})$ is localized in $|k|\sim\frac{N}{h}$, we conclude that

[TABLE]

where in the last step, we used that $N_{*}\sim 2^{\log_{2}(\frac{h}{\pi})}=\frac{h}{\pi}$. ∎

Proposition 3.1 remains to be proved, for which we need to estimate the sums of the oscillating functions. Following Vega’s argument [25], we use Lemma 3.2 to approximate the sums by the oscillatory integrals. Then, we employ the estimate (Lemma 3.3) of the oscillatory integral.

Lemma 3.2** (Zygmund [27, Chapter V, Lemma 4.4]).**

Let $\varphi$ be a real-valued function, and let $a,b\in\mathbb{R}$ with $a<b$. If $\varphi^{\prime}$ is monotonic and $|\varphi^{\prime}|<2\pi$ on $(a,b)$, then

[TABLE]

where the constant $A$ is independent of $\varphi$, $a$, and $b$.

Lemma 3.3**.**

Let $h\in(0,1]$ and a dyadic number $N$ with $N_{*}\leq N\leq 1$ be given. We define

[TABLE]

Then, there exists $B>0$, independent of $h$ and $N$, such that

[TABLE]

Proof.

If $N_{*}\leq N\leq\frac{1}{4}$, by the van der Corput lemma with $|(x\xi-\tfrac{2t}{h^{2}}(1-\cos h\xi))^{\prime\prime}|=2|t||\cos h\xi|\geq|t|$ for $|\xi|\leq\frac{\pi N}{h}$, we have $|I_{N,h,t,x}|\lesssim|t|^{-1/2}$. Hence, interpolating with the trivial bound $|I_{N,h,t,x}|\lesssim\frac{N}{h}$, we obtain the desired bound.

If $N=\frac{1}{2}$ or $1$, then we decompose

[TABLE]

It has already been shown that $|I_{\frac{1}{4},h,t,x}|\lesssim(h|t|)^{-1/3}$. For the integral on the right-hand side, we change the variables,

[TABLE]

We observe that $|(\frac{x}{h}\xi-\frac{2t}{h^{2}}\left(1-\cos\xi\right))^{\prime\prime}|=\frac{2|t|}{h^{2}}|\cos\xi|$ and $|(\frac{x}{h}\xi-\frac{2t}{h^{2}}\left(1-\cos\xi\right))^{\prime\prime\prime}|=\frac{2|t|}{h^{2}}|\sin\xi|$. Thus, applying the van der Corput lemma again, we prove that

[TABLE]

Therefore, we complete the proof of the lemma. ∎

Proof of Proposition 3.1.

We consider the case $N=N_{*}$. By the definition of $P_{N_{*}}$ (see (2.8)) and the Plancherel theorem, we have

[TABLE]

which implies $\|e^{it\Delta_{h}}P_{N_{*}}u_{h,0}\|_{L^{\infty}_{h}}\lesssim\|u_{h,0}\|_{L_{h}^{2}}$. Hence, interpolating it with a trivial estimate $\|e^{it\Delta_{h}}P_{N_{*}}u_{h,0}\|_{L^{2}_{h}}\leq\|u_{h,0}\|_{L_{h}^{2}}$, we get the bound $\|e^{it\Delta_{h}}P_{N_{*}}u_{h,0}\|_{L_{h}^{r}}\lesssim\|u_{h,0}\|_{L_{h}^{2}}$ for all $r\geq 2$. As a consequence, we obtain $\|e^{it\Delta_{h}}P_{N_{*}}u_{h,0}\|_{L_{t}^{q}([0,1];L_{h}^{r})}\leq\|e^{it\Delta_{h}}P_{N_{*}}u_{h,0}\|_{L_{t}^{\infty}([0,1];L_{h}^{r})}\lesssim\|P_{N_{*}}u_{h,0}\|_{L_{h}^{2}}$ for any $1\leq q\leq\infty$ and $2\leq r\leq\infty$.

Suppose that $2N_{*}\leq N\leq 1$. A direct calculation with Lemma 2.5 yields

[TABLE]

where

[TABLE]

and $\ast$ is the convolution on the lattice defined in (2.4). Hence, Young’s inequality ensures (3.1) provided the following one-dimensional inequality holds true:

[TABLE]

It remains to prove (3.2). For notational convenience, we write

[TABLE]

for $x\in\mathbb{T}_{h}$ and $\xi\in\mathbb{R}$ with $|\xi|\leq\tfrac{\pi N}{h}$. A direct calculation yields

[TABLE]

under the restriction $|t|\leq\frac{ch}{N}$.

First, we consider the case $N\leq\frac{1}{2}$. Then $\varphi^{\prime}$ is decreasing on $[-\frac{\pi N}{h},\frac{\pi N}{h}]$. Hence, applying Lemma 3.2 and 3.3, we obtain

[TABLE]

In the last inequality, we used $|t|\leq\frac{ch}{N}$ and $N\geq N_{*}\sim\frac{h}{\pi}$, implying $1\lesssim(\frac{h}{N|t|})^{1/3}\lesssim(\frac{N}{h|t|})^{1/3}$. Next, we consider the case $N=1$. We divide the interval into three parts:

[TABLE]

where $\varphi^{\prime}$ is monotonic on each $I_{j}$. Then, we decompose

[TABLE]

Each $S_{j}$ can be estimated with the same method as above. Summing these, we complete the proof. ∎

As an application of Theorem 1.5, we obtain the time-averaged uniform $L_{h}^{\infty}$ estimates.

Corollary 3.4** (Uniform time-averaged $L_{h}^{\infty}$-bounds for the discrete linear Schrödinger flow; 2D case).**

*Suppose that $d=2$ and $1\leq q<\infty$. Then, *

[TABLE]

Proof.

Let $\epsilon>0$ be a sufficiently small number such that the following inequalities hold.

For $1\leq q\leq 3$, Hölder’s inequality in time and Theorem 1.5 yield

[TABLE]

Suppose that $q>3$. By the Sobolev inequality (Lemma 2.7) and the unitarity of the Schrödinger flow, we get

[TABLE]

for a small $\epsilon=\epsilon(q)>0$ appeared in Theorem 1.5. Thus, interpolating this inequality and Theorem 1.5 with $(q,r,d)=(3,\infty,2)$ and choosing $\epsilon<\frac{1}{q}$, we obtain

[TABLE]

which completes the proof. ∎

4. Uniform bound for discrete NLS

In this section, we provide a simple well-posedness theorem for DNLS (1.4). Then, as an application of the uniform-in-$h$ Strichartz estimates, we deduce a uniform time-averaged $L_{h}^{\infty}$-bound for nonlinear solutions.

4.1. Global well-posedness

By Duhamel’s principle, DNLS (1.4) is equivalent to the integral equation

[TABLE]

We next show its global well-posedness.

Proposition 4.1** (Global well-posedness).**

Let $d\geq 1$, $h>0$ and $p>1$. Then, for any initial data $u_{h,0}\in L_{h}^{2}$, there exists a unique global solution $u_{h}(t)\in C(\mathbb{R};L_{h}^{2})$ to DNLS (1.4). Moreover, it conserves the mass

[TABLE]

and the energy

[TABLE]

Proof.

The proof is identical to the analogous theorem for the discrete NLS on $h\mathbb{Z}^{d}$ (see [10, Proposition 6.1]). Fix $h>0$. For a small $T>0$ to be chosen later, let $X_{T}:=C_{t}([-T,T];L_{h}^{2})$. We denote by $\Gamma(u_{h})$ the right-hand side of (4.1). Then, by the unitarity of the linear propagator and the trivial inequality $\|u_{h}\|_{L_{h}^{\infty}}\leq h^{-d/2}\|u_{h}\|_{L_{h}^{2}}$, one can show that

[TABLE]

and in the same way,

[TABLE]

Therefore, if $T>0$ is sufficiently small depending on $h>0$, $\Gamma$ is contractive on the set $\{u_{h}\in X_{T}:\|u_{h}\|_{X_{T}}\leq 2\|u_{h,0}\|_{L_{h}^{2}}\}$. Thus, DNLS (1.4) is locally well-posed in $L_{h}^{2}$. The conservation laws (4.2) and (4.3) can be proven by direct calculations. The lifespan of local solutions is then extended by the mass conservation law (4.2). ∎

4.2. Uniform bound for the 2D DNLS

Next, we show that not only linear solutions (Corollary 3.4) but also nonlinear solutions obey a time-averaged uniform $L_{h}^{\infty}$-bound.

Proposition 4.2** (Uniform $L_{h}^{\infty}$-bound for the 2D DNLS).**

Suppose that $p$ satisfies (1.13). Then, the solution $u_{h}(t)$ to DNLS (1.4) with initial data $u_{h,0}\in H_{h}^{1}$, constructed in Proposition 4.1, satisfies

[TABLE]

where

[TABLE]

Proof.

Let $u_{h}$ be the solution to DNLS (1.4) constructed in Proposition 4.1, and let $\tau>0$ be a sufficiently small number such that

[TABLE]

where $I=[-\tau,\tau]$, $c_{0}=\max(c_{q_{*}},1)$ and $c_{q_{*}}$ is the implicit constant in (3.3) (when $q=q_{*}$). Such $\tau$ is initially chosen depending on $h>0$, but later it can be extended independently of $h>0$.

From the integral representation of the solution (4.1), the unitarity of the linear flow yields

[TABLE]

and by Corollary 3.4, we obtain

[TABLE]

Applying the fundamental theorem of calculus of the form

[TABLE]

with $\alpha=u_{h}(x+he_{j})$ and $\beta=u_{h}(x)$, we obtain

[TABLE]

Hence, by the norm equivalence (Lemma 2.9), it follows that

[TABLE]

Inserting this bound in (4.5) and (4.6), we obtain

[TABLE]

Thus, it follows that

[TABLE]

as long as $C(2\tau)^{1-\frac{p-1}{q_{*}}}(4c_{0}\|u_{h,0}\|_{H_{h}^{1}})^{p}\leq 2c_{0}\|u_{h,0}\|_{H_{h}^{1}}$ is satisfied. Therefore, the time interval $I$ can be extended to a short time interval of which the length depends on $\|u_{h,0}\|_{H_{h}^{1}}$ but is independent of $h>0$.

To extend the time interval arbitrarily, we show that $\|u_{h}(t)\|_{H_{h}^{1}}$ is bounded uniformly in time. Indeed, by the mass conservation law, it is sufficient to show that $\|(-\Delta_{h})^{\frac{1}{2}}u_{h}(t)\|_{L_{h}^{2}}$ is bounded globally in time. When $\lambda>0$, the energy conservation law immediately implies that $\|(-\Delta_{h})^{\frac{1}{2}}u_{h}(t)\|_{L_{h}^{2}}^{2}\leq 2E_{h}(u_{h}(t))=2E_{h}(u_{h,0})$ for all $t$. When $\lambda<0$, we apply both the mass and the energy conservation laws as well as the 2D uniform Gagliardo–Nirenberg inequality (Lemma 2.8) to obtain

[TABLE]

By the assumption (1.13), we have $p-1<2$. Thus, we can use Young’s inequality to bound $\|(-\Delta_{h})^{\frac{1}{2}}u_{h}(t)\|_{L_{h}^{2}}^{2}$ only in terms of the mass $M_{h}(u_{h,0})$ and the energy $E_{h}(u_{h,0})$.

Because $\|u_{h}(t)\|_{H_{h}^{1}}$ is bounded uniformly in time, (4.8) can be iterated with the new initial data $u(\tau),u(2\tau),...$ and with the bounds (4.8) on the intervals $[\tau,2\tau]$, $[2\tau,3\tau]$, … to cover an arbitrarily long time interval $[-T,T]$. Therefore, summing up, we obtain the desired bound (4.4). ∎

5. Proof of the contimuum limit

In this section, we prove the main theorem of this article (Theorem 1.6).

5.1. Preliminaries

We first provide lemmas concerning the discretization and linear interpolation (see (1.5) and (1.6)). Analogous lemmas on the lattice $h\mathbb{Z}^{d}$ have been stated and proven in [11]. Thus, we omit some details. Indeed, differentiation (resp., discrete differentiation) is a local operation, thus the argument used in the non-compact domain $\mathbb{R}^{d}$ (resp, $h\mathbb{Z}^{d}$) can easily be adopted to the compact domain $\mathbb{T}^{d}$ (resp, $\mathbb{T}_{h}^{d}$).

Lemma 5.1** (Boundedness of discretization and linear interpolation).**

[TABLE]

Proof.

We compute the discrete Sobolev norm using Lemma 2.9. Then the proof follows from the same method as [11, Lemmas 5.1 and 5.2]] ∎

Lemma 5.2**.**

Let $h\in(0,1]$. Then, for $f\in H^{1}(\mathbb{T}^{d})$, we have

[TABLE]

Proof.

The proof closely follows from the proof of [11, Proposition 5.3]. ∎

Lemma 5.3**.**

Let $h\in(0,1]$. If $f\in H^{1}(\mathbb{T}^{d})$ and $g_{h}\in H_{h}^{1}(\mathbb{T}_{h}^{d})$, then

[TABLE]

In particular,

[TABLE]

Proof.

The proof closely follows from the proof of [11, Proposition 5.4]. First, using direct calculations, we observe that the Fourier transform of the linear interpolation of a discrete function is given by

[TABLE]

where

[TABLE]

and $\widetilde{\mathcal{F}_{h}}$ denotes the $[-\frac{\pi}{h},\frac{\pi}{h})^{d}$-periodic extension of the discrete Fourier transform $\mathcal{F}_{h}$, precisely, $(\widetilde{\mathcal{F}_{h}}f_{h})(k)=(\mathcal{F}_{h}f_{h})(k^{\prime})$ for all $k\in k^{\prime}+\frac{2\pi}{h}\mathbb{Z}^{d}$. We also observe that

[TABLE]

By these observations and Lemma 5.1 and 5.2, one can proceed as in the proof of [11, Proposition 5.4]. Here, an $O(\sqrt{h})$-bound is obtained from the regularity gap between the norms on the left- and right-hand sides. ∎

As a corollary of Lemma 5.3, we have the following.

Corollary 5.4**.**

Let $h\in(0,1]$ and $p>1$. Then,

[TABLE]

Proof.

An immediate application of Lemma 5.3 to the left-hand side of (5.1) yields

[TABLE]

Lemma 5.1 and Hölder’s inequality control the right-hand side, and we thus obtain (5.1). ∎

Lemma 5.5** (Proposition 5.7 in [11]).**

Let $h\in(0,1]$ and $p>1$. Then,

[TABLE]

We end this section with the following lemma:

Lemma 5.6**.**

Let $h\in(0,1]$ and $p>1$. Then,

[TABLE]

Proof.

It follows from the calculation (4.7) with $\alpha=p_{h}u_{h}$ and $\beta=u_{h}$. ∎

5.2. Proof of continuum limit

Now we are in a position to prove Theorem 1.6. Because the proof closely follows from the argument presented in [11, Section 6], we only sketch the outline.

Let $h\in(0,1]$ be fixed. Given initial data $u_{0}\in H^{1}(\mathbb{T}^{2})$, let $u(t)\in C(\mathbb{R};H^{1}(\mathbb{T}^{2}))$ be the global solution to NLS (1.1) (see Section B). For the discretization $u_{h,0}=d_{h}u_{0}$, let $u_{h}(t)$ be the solution to DNLS (1.4) with the initial data $u_{h,0}$ constructed in Section 4.

Applying the linear interpolation operator to the Duhamel formula (4.1), we write

[TABLE]

Then, by direct calculations, the difference of $u$ and $p_{h}u$ can be expressed as

[TABLE]

Lemma 5.3, 5.5, and 5.6 and Corollary 5.4 yield

[TABLE]

which, by applying Grönwall’s inequality in addition to Proposition 4.2 and Corollary B.5, implies

[TABLE]

for sufficiently large $B\gg 1$. This completes the proof of Theorem 1.6.

Appendix A Proof of Lemma 2.7 and 2.8

On a periodic domain, the proof of the Sobolev inequality is more involved, compared to that on the entire Euclidean space, because the explicit kernel formula for the inverse Laplacian is no longer available (see [2] for example). However, if an arbitrarily small loss of regularity is allowed, one can show the inequality in a simpler manner, as is presented in this appendix.

The key item is Bernstein’s inequality for the projection operator $P_{N}$ (see (2.8)).

Lemma A.1** (Bernstein’s inequality).**

Suppose that $0<s\leq\frac{d}{2}$, $q\geq 2$ and $\frac{1}{q}=\frac{1}{2}-\frac{s}{d}$. For $h\in(0,1]$ and a dyadic number $N$ with $N_{*}:=2^{\lceil\log_{2}(\tfrac{h}{\pi})\rceil-1}\leq N\leq 1$, we have

[TABLE]

Proof.

We prove the lemma by the standard $TT^{*}$ argument. When $q=\infty$, we have

[TABLE]

When $q=2$, it is obvious that $\|P_{N}u\|_{L_{h}^{2}}\leq\|u\|_{L_{h}^{2}}$. Interpolating, we obtain

[TABLE]

for $q\geq 2$. This inequality implies that

[TABLE]

Thus, (A.1) follows from the duality. ∎

Proof of Lemma 2.7.

By the triangle inequality and Lemma A.1, we prove that

[TABLE]

where in the last step, we used that $N_{*}\sim 2^{\log_{2}(\frac{h}{\pi})}=\frac{h}{\pi}$. ∎

Similarly, the Gagliardo–Nirenberg inequality can be proved.

Proof of Lemma 2.8.

Replacing $f$ by $\frac{1}{\|f\|_{L_{h}^{2}}}f$, we may assume that $\|f\|_{L_{h}^{2}}=1$. Suppose that $\|f\|_{H_{h}^{1}}\leq h^{-1}$. Let $R=h\|f\|_{H_{h}^{1}}$. Then, using Bernstein’s inequality, we prove that

[TABLE]

Similarly, if $\|f\|_{H_{h}^{1}}\geq h^{-1}$, then

[TABLE]

∎

Appendix B Well-posedness results for NLS on the $\mathbb{T}^{2}$

We consider the (periodic) NLS (1.1)

[TABLE]

Duhamel’s principle yields that (B.1) is equivalent to the following integral equation on $[-T,T]$

[TABLE]

where $\eta$ is a smooth (even) bump function satisfying $\eta\equiv 1$ in $[-1,1]$ and $\eta\equiv 0$ in $(-2,2)^{c}$, and $\eta_{T}(t)=\eta(t/T)$. Note that one may replace $\eta_{T}(t)$ by $\eta(t)$ in (B.2) (with a smallness assumption) when $p<1+\frac{4}{d}$ (in the $2D$ case, $p<3$), owing to the scaling argument.

For the classical well-posedness result of Bourgain [3] (see also [4]), we introduce the following function space. For $s,b\in\mathbb{R}$, we define the norm

[TABLE]

for $f\in\mathcal{S}(\mathbb{R}\times\mathbb{T}^{d})$, where $\langle\cdot\rangle=(1+|\cdot|^{2})^{1/2}$ and $\tilde{f}$ is the spacetime Fourier transform of $f$ given by

[TABLE]

Then, the $X^{s,b}$ space is defined as the completion of $\mathcal{S}^{\prime}(\mathbb{R}\times\mathbb{T}^{d})$ under the norm $\|\cdot\|_{X^{s,b}}$. This function space is termed the Bourgain space or the dispersive Sobolev space.

Theorem B.1** (GWP for 2D NLS [3]).**

Suppose that $d=2$, and $p$ is given by (1.13). Then, NLS (B.1) is globally well-posed in $H^{1}(\mathbb{T}^{2})$. Moreover, the solution $u$ obeys

[TABLE]

As a consequence, we have

[TABLE]

for $q\geq 4$, where $0<\epsilon\ll 1$, $s(q,\epsilon)=\frac{4\epsilon}{q}+(1+\frac{1-2\epsilon}{q-4})(1-\frac{4}{q})$ and $b(q,\epsilon)=(\frac{1}{2}-\frac{\epsilon}{8})\frac{4}{q}+(\frac{1}{2}+\frac{\epsilon}{4(q-4)})(1-\frac{4}{q})$. In particular,

[TABLE]

Remark B.2*.*

$(i)$ One can immediately check $s(q,\epsilon)<1-\frac{2}{q}$ and $b(q,\epsilon)<\frac{1}{2}$.

$(ii)$ In the one-dimensional case, Bourgain [3] proved the $L_{t,x}^{4}$ estimate

[TABLE]

This is an improvement of the $L^{4}$ estimate for free solutions by Zygmund [26], namely,

[TABLE]

which implies by the transference principle that

[TABLE]

$(iii)$ Bourgain employed a time-periodic function to show (B.6); however, such a restriction is not necessary (such as (B.5)), see, for instance, [23, 24].

Remark B.3*.*

The $L^{q}$ estimate (B.4) follows from the interpolation between (B.5) and $\|u\|_{L_{t,x}^{\infty}(\mathbb{R}\times\mathbb{T}^{2})}\lesssim\|u\|_{X^{1+,\frac{1}{2}+}}$. Together with the Hölder inequality and the $L^{4}$ estimate (B.5), one has the (local-in-time) $L^{q}$ estimate for $1\leq q<4$, precisely,

[TABLE]

Remark B.4*.*

The a priori bound (B.3) can be obtained by the standard iteration method in addition to the $L^{q}$ estimate (B.4).

As a corollary, we obtain a time-averaged bound.

Corollary B.5** (Time-averaged $L^{\infty}$ bound for 2D NLS).**

Suppose that $d=2$, and $p$ is given by (1.13). Suppose that $u(t)$ is the global solution to periodic NLS (B.1) with initial data $u_{0}$, constructed in Theorem B.1. Then,

[TABLE]

where $q_{*}>\max(p-1,2)$.

Proof.

The proof follows from an analogous argument in the proof of Proposition 4.2. ∎