Unbounded Sobolev trajectories and modified scattering theory for a wave guide nonlinear Schrödinger equation

Abstract

We consider the following wave guide nonlinear Schrödinger equation,

$$\begin{aligned} (i\partial _t+\partial _{xx}-\vert D_y\vert )U=\vert U\vert ^2U\ \end{aligned}$$

(WS)

on the spatial cylinder ${\mathbb R}_x\times {\mathbb T}_y$. We establish a modified scattering theory between small solutions to this equation and small solutions to the cubic Szegő equation. The proof is an adaptation of the method of Hani et al. (Modified scattering for the cubic Schrödinger equation on product spaces and applications, 2015. arXiv:1311.2275v3). Combining this scattering theory with a recent result by Gérard and Grellier (On the growth of Sobolev norms for the cubic Szegő equation, 2015), we infer existence of global solutions to (WS) which are unbounded in the space $L^2_xH^s_y({\mathbb R}\times {\mathbb T})$ for every $s>\frac{1}{2}$.

Appendix

We now turn to our basic lemma allowing to transform suitable $L^2_{x,y}$ bounds to bounds in terms of the $L^2_{x,y}$-based spaces S and $S^+$. We define an LP-family $\widetilde{Q}=\{\widetilde{Q}_A\}_A$ to be a family of operators (indexed by the dyadic integers) of the form

$$\begin{aligned} \widehat{\widetilde{Q}_1f}(\xi )=\widetilde{\varphi }(\xi )\widehat{f}(\xi ),\quad \widehat{\widetilde{Q}_Af}(\xi )=\widetilde{\phi }\left( \frac{\xi }{A}\right) \widehat{f}(\xi ), \quad A\ge 2 \end{aligned}$$

for two smooth functions $\widetilde{\varphi },\widetilde{\phi }\in C^\infty _c(\mathbb {R})$ with $\widetilde{\phi }\equiv 0$ in a neighborhood of 0.

We define the set of admissible transformations to be the family of operators $\{T_A\}$ where for any dyadic number A,

$$\begin{aligned} T_A=\lambda _A\widetilde{Q}_A,\quad \vert \lambda _A\vert \le 1 \end{aligned}$$

for some LP-family $\widetilde{Q}$.

If $F\in \mathcal {B}$, then for any admissible transformation family $T=\{T_A: A \text { dyadic numbers }\}$, $\sum \nolimits _A T_AF$ converges in $\mathcal {B}$. And this norm $\mathcal {B}$ is called admissible if

$$\begin{aligned} \left\| \sum _A T_AF\right\| _{\mathcal {B}}\lesssim \left\| F\right\| _{\mathcal {B}}. \end{aligned}$$

(6.1)

Lemma 5.1

Recall the definitions of the norms S, $S^+$, Z and $\widetilde{Z}_t$,

$$\begin{aligned} \Vert F\Vert _{S}:=\,&\Vert F\Vert _{H^N_{x,y}}+\Vert xF\Vert _{L^2_{x,y}},\quad \Vert F\Vert _{S^+}:=\Vert F\Vert _S+\Vert (1-\partial _{xx})^4F\Vert _{S}+\Vert xF\Vert _{S},\\ \Vert F\Vert _{Z}:=&\sup _{\xi \in {\mathbb R}}\left[ 1+\vert \xi \vert ^2\right] \Vert \widehat{F}(\xi ,\cdot )\Vert _{B^1},\quad \Vert F\Vert _{\widetilde{Z}_t}:=\Vert F\Vert _Z+(1+\vert t\vert )^{-\delta }\Vert F\Vert _S. \end{aligned}$$

All these norms are admissible.

Proof

Due to the definition of admissible transformation, we may only deal with functions independent on y. Let us prove with the S norm for example. Indeed,

$$\begin{aligned} \Vert \sum _A T_Af\Vert _{H^N}^2&=\int \limits _{\mathbb {R}}\langle \xi \rangle ^{2N}\Big \vert \lambda _1\widetilde{\varphi }(\xi )\widehat{f}(\xi )+\lambda _A\widetilde{\phi }\left( \frac{\xi }{A}\right) \widehat{f}(\xi )\Big \vert ^2d\xi \\&\le \int \limits _{\mathbb {R}}\Big (\vert \lambda _1\vert ^2\widetilde{\varphi }(\xi )+\vert \lambda _A\vert ^2\widetilde{\phi }\left( \frac{\xi }{A}\right) \Big )\langle \xi \rangle ^{2N}\vert \widehat{f}(\xi )\vert ^2d\xi \\&\le \Vert f\Vert _{H^N}^2, \end{aligned}$$

while

$$\begin{aligned} \Vert x\sum _A T_Af\Vert _{L^2}^2&=\int \limits _{\mathbb {R}}\Big \vert \partial _\xi \big (\lambda _1\widetilde{\varphi }(\xi )\widehat{f}(\xi )+\lambda _A\widetilde{\phi }\left( \frac{\xi }{A}\right) \widehat{f}(\xi )\big )\Big \vert ^2d\xi \\&\le \int \limits _{\mathbb {R}}\Big (\vert \lambda _1\vert ^2\widetilde{\varphi }(\xi )+\vert \lambda _A\vert ^2\widetilde{\phi }\left( \frac{\xi }{A}\right) \Big )\vert \partial _\xi \widehat{f}(\xi )\vert ^2d\xi \\&\quad +\int \limits _{\mathbb {R}}\Big (\vert \lambda _1\vert ^2\widetilde{\varphi '}(\xi )+\frac{\vert \lambda _A\vert ^2}{A}\widetilde{\phi '}\left( \frac{\xi }{A}\right) \Big )\vert \widehat{f}(\xi )\vert ^2d\xi \\&\le \Vert xf\Vert _{L^2}^2+\Vert f\Vert _{L^2}^2, \end{aligned}$$

thus

$$\begin{aligned} \left\| \sum _A T_A f\right\| _S\lesssim \Vert f\Vert _S. \end{aligned}$$

$\square$

Given a trilinear operator $\mathfrak {T}$ and a set $\Lambda$ of 4-tuples of dyadic integers, we define an admissible realization of $\mathfrak {T}$ at $\Lambda$ to be an operator of the form which converges in $L^2$,

$$\begin{aligned} \mathfrak {T}_\Lambda [F,G,H]=\sum _{(A,B,C,D)\in \Lambda }T_D\mathfrak {T}[T^\prime _AF,T^{\prime \prime }_BG,T^{\prime \prime \prime }_CH] \end{aligned}$$

(6.2)

for some admissible transformations T, $T^\prime$, $T^{\prime \prime }$, $T^{\prime \prime \prime }$.

Lemma 5.2

Assume that a trilinear operator $\mathfrak {T}$ satisfies

$$\begin{aligned} \begin{aligned} Z\mathfrak {T}[F,G,H]= \mathfrak {T}[ZF,G,H]+\mathfrak {T}[F,ZG,H]+\mathfrak {T}[F,G,ZH], \end{aligned} \end{aligned}$$

(6.3)

for $Z\in \{x,\partial _x,\partial _{y}\}$ and let $\Lambda$ be a set of 4-tuples of dyadic integers. With the notation introduced above, assume also that for all admissible realizations of $\mathfrak {T}$ at $\Lambda$,

$$\begin{aligned} \Vert \mathfrak {T}_\Lambda [F^a,F^b,F^c]\Vert _{L^2}\le K\min _{\{\alpha ,\beta ,\gamma \}=\{a,b,c\}}\Vert F^\alpha \Vert _{L^2}\Vert F^\beta \Vert _{\mathcal {B}}\Vert F^\gamma \Vert _{\mathcal {B}} \end{aligned}$$

(6.4)

for some admissible norm $\mathcal {B}$ such that the Littlewood–Paley projectors $P_{\le M}$ (both in x and in y) are uniformly bounded on $\mathcal {B}$. Then, for all admissible realizations of $\mathfrak {T}$ at $\Lambda$,

$$\begin{aligned} \Vert \mathfrak {T}_\Lambda [F^a,F^b,F^c]\Vert _{S}\lesssim K\max _{\{\alpha ,\beta ,\gamma \}=\{a,b,c\}}\Vert F^\alpha \Vert _{S}\Vert F^\beta \Vert _{\mathcal {B}}\Vert F^\gamma \Vert _{\mathcal {B}}. \end{aligned}$$

(6.5)

Assume in addition that, for $Y\in \{x,(1-\partial _{xx})^4\}$,

$$\begin{aligned} \Vert YF\Vert _{\mathcal {B}}\lesssim \theta _1\Vert F\Vert _{S^+}+\theta _2\Vert F\Vert _S, \end{aligned}$$

(6.6)

then for all admissible realizations of $\mathfrak {T}$ at $\Lambda$,

$$\begin{aligned} \Vert \mathfrak {T}_\Lambda [F^a,F^b,F^c]\Vert _{S^+}\lesssim & {} K\max _{\{\alpha ,\beta ,\gamma \}=\{a,b,c\}}\Vert F^\alpha \Vert _{S^+}\big (\Vert F^\beta \Vert _{\mathcal {B}}+\theta _1\Vert F^\beta \Vert _{S}\big ) \Vert F^\gamma \Vert _{\mathcal {B}}\nonumber \\&+\theta _2K\max _{\{\alpha ,\beta ,\gamma \}=\{a,b,c\}}\Vert F^\alpha \Vert _{S}\Vert F^\beta \Vert _{S}\Vert F^\gamma \Vert _{\mathcal {B}}. \end{aligned}$$

(6.7)

Proof

Let us start with (6.5).

1. The weighted component of S norm. By rewriting $xT_A=[x,T_A]+T_Ax$ and using (6.3), we have

$$\begin{aligned} x\mathfrak {T}_\Lambda [F^a,F^b,F^c]= & {} \sum _{(A,B,C,D)\in \Lambda }xT_D\mathfrak {T}[T^\prime _AF^a,T^{\prime \prime }_BF^b,T^{\prime \prime \prime }_CF^c]\nonumber \\= & {} \sum _{(A,B,C,D)\in \Lambda }[x,T_D]\mathfrak {T}[T^\prime _AF^a,T^{\prime \prime }_BF^b,T^{\prime \prime \prime }_CF^c]\nonumber \\&\quad +\sum _{(A,B,C,D)\in \Lambda }T_D\mathfrak {T}[[x,T^\prime _A]F^a,T^{\prime \prime }_BF^b,T^{\prime \prime \prime }_CF^c]\nonumber \\&\quad +\sum _{(A,B,C,D)\in \Lambda }T_D\mathfrak {T}[T^\prime _AF^a,[x,T^{\prime \prime }_B]F^b,T^{\prime \prime \prime }_CF^c]\nonumber \\&\quad +\sum _{(A,B,C,D)\in \Lambda }T_D\mathfrak {T}[T^\prime _AF^a,T^{\prime \prime }_BF^b,[x,T^{\prime \prime \prime }_C]F^c]\nonumber \\&\quad +\mathfrak {T}_\Lambda [xF^a,F^b,F^c]+\mathfrak {T}_\Lambda [F^a,xF^b,F^c]+\mathfrak {T}_\Lambda [F^a,F^b,xF^c].\nonumber \\ \end{aligned}$$

(6.8)

By simple calculation, we have

$$\begin{aligned} \mathrm{[x,Q_A]}=A^{-1}Q'_A. \end{aligned}$$

We notice that if $Q_A$ is an LP-family, $Q'_A$ is also an LP-family, then $[x,T_A]$ is also an admissible transformation. Thus, we may consider $x\mathfrak {T}_\Lambda [F^a,F^b,F^c]$ as the following summation

$$\begin{aligned} \mathfrak {T}_\Lambda [F^a,F^b,F^c]+\mathfrak {T}_\Lambda [xF^a,F^b,F^c]+\mathfrak {T}_\Lambda [F^a,xF^b,F^c]+\mathfrak {T}_\Lambda [F^a,F^b,xF^c], \end{aligned}$$

(6.9)

then $\Vert x\mathfrak {T}_\Lambda [F^a,F^b,F^c]\Vert _{L^2}$ follows from (6.4).

2. The $H^N$ component of S norm. We will use the equivalent definition of $H^N$ norm,

$$\begin{aligned} \Vert F\Vert _{H^N}^2:=\sum _{M \text { dyadic}}M^{2N}\Vert P_M F\Vert _{L^2}^2, \end{aligned}$$

with $P_M$ as the Littlewood–Paley projections on $\mathbb {R}\times \mathbb {T}$ defined in Sect. 2. Then, we may decompose

$$\begin{aligned} P_M\mathfrak {T}_\Lambda [F^a,F^b,F^c]=P_M\mathfrak {T}_{\Lambda ,low}[F^a,F^b,F^c]+P_M\mathfrak {T}_{\Lambda ,high}[F^a,F^b,F^c], \end{aligned}$$

with$\mathfrak {T}_{\Lambda ,low}[F^a,F^b,F^c]=\mathfrak {T}_\Lambda [P_{\le M}F^a,P_{\le M}F^b,P_{\le M}F^c]$.

We have firstly

$$\begin{aligned} \sum _{M \text { dyadic}}M^{2N}\Vert P_M\mathfrak {T}_{\Lambda ,high}[F^a,F^b,F^c]\Vert _{L^2}^2\lesssim K^2\max _{\{\alpha ,\beta ,\gamma \}=\{a,b,c\}}\Vert F^\alpha \Vert _{H^N}^2 \Vert F^\beta \Vert _{\mathcal {B}}^2\Vert F^\gamma \Vert _{\mathcal {B}}^2, \end{aligned}$$

(6.10)

since

$$\begin{aligned} \begin{aligned}&\sum _M\vert M \vert ^{2N}\Vert P_M\mathfrak {T}_\Lambda [P_{\ge 2M}F^a,F^b,F^c]\Vert _{L^2}^2\le K^2\sum _M \vert M\vert ^{2N} \Vert P_{\ge 2M}F^a\Vert _{L^2}^2\Vert F^b\Vert _{\mathcal {B}}^2\Vert F^c\Vert _{\mathcal {B}}^2\\&\quad \lesssim K^2\Vert F^a\Vert _{H^N}^2 \Vert F^b\Vert _{\mathcal {B}}^2\Vert F^c\Vert _{\mathcal {B}}^2. \end{aligned} \end{aligned}$$

(6.11)

Let $Z\in \{\partial _x,\partial _y\}$, we can bound the contribution of $\mathfrak {T}_{\Lambda ,low}$ as below

$$\begin{aligned} \begin{aligned}&M^{N}\Vert P_M\mathfrak {T}_{\Lambda ,low}\Vert _{L^2}\\&\quad \lesssim M^{-N}\Vert Z^{2N}P_M\mathfrak {T}_{\Lambda ,low}[P_{\le M}F^a,P_{\le M}F^b,P_{\le M}F^c]\Vert _{L^2}\\&\quad =M^{-N}\left\| \sum _{\alpha +\beta +\gamma \le 2N}\sum _{M_1,M_2,M_3\le M}P_M\mathfrak {T}_{\Lambda ,low}[Z^{\alpha }P_{ M_1}F^a,Z^{\beta }P_{M_2}F^b,Z^{\gamma }P_{M_3}F^c]\right\| _{L^2}. \end{aligned} \end{aligned}$$

(6.12)

Without loss of generality, we assume $M_1=\max {(M_1,M_2,M_3)}\le M$, then

$$\begin{aligned} \begin{aligned} M^{N}\Vert P_M\mathfrak {T}_{\Lambda ,low}\Vert _{L^2}&\lesssim \sum _{M_1\le M}M^{-N}M_1^{2N}\sum _{M_2,M_3\le M_1}\Vert \mathfrak {T}_{\Lambda ,low}[P_{ M_1}F^a,P_{M_2}F^b,P_{M_3}F^c]\Vert _{L^2}\\&\lesssim K \sum _{M_1\le M}\left( \frac{M_1}{M}\right) ^{-N}M_1^{N}\Vert P_{ M_1}F^a\Vert _{L^2}\Vert F^b\Vert _{\mathcal {B}}\Vert F^c\Vert _{\mathcal {B}}, \end{aligned} \end{aligned}$$

(6.13)

the above sum is in $\ell _M^2$ by Schur test, then

$$\begin{aligned} \sum _{M \text { dyadic}}M^{2N}\Vert P_M\mathfrak {T}_{\Lambda ,low}[F^a,F^b,F^c]\Vert _{L^2}^2\lesssim K^2\max _{\{\alpha ,\beta ,\gamma \}=\{a,b,c\}}\Vert F^\alpha \Vert _{H^N}^2 \Vert F^\beta \Vert _{\mathcal {B}}^2\Vert F^\gamma \Vert _{\mathcal {B}}^2. \end{aligned}$$

(6.14)

Therefore we bound the $H^N$ component of S norm, which completes the estimate (6.5).

Now, we turn to prove the estimate (6.7), due to the definition of $S^+$ norm, we only need to bound $\Vert x\mathfrak {T}_{\Lambda }\Vert _{S}$ and $\Vert (1-\partial _{xx})^4\mathfrak {T}_{\Lambda }\Vert _S$. From (6.9) and (6.5), we gain directly

$$\begin{aligned} \begin{aligned}&\Vert x\mathfrak {T}_{\Lambda }[F^a,F^b,F^c]\Vert _{S}\\&\quad \lesssim \max _{\{\alpha ,\beta ,\gamma \}=\{a,b,c\}}\Vert F^{\alpha }\Vert _{S^+}\Vert F^\beta \Vert _{\mathcal {B}}\Vert F^\gamma \Vert _{\mathcal {B}}+\max _{\{\alpha ,\beta ,\gamma \}=\{a,b,c\}}\Vert F^{\alpha }\Vert _{S}\Vert xF^\beta \Vert _{\mathcal {B}}\Vert F^\gamma \Vert _{\mathcal {B}}, \end{aligned}\end{aligned}$$

(6.15)

we then using (6.6) to control the norm $\Vert xF\Vert _{\mathcal {B}}$. The estimate on $\Vert (1-\partial _{xx})^4\mathfrak {T}_{\Lambda }\Vert _S$ can be calculated similarly by replacing x with $(1-\partial _{xx})^4$. The proof is completed. $\square$

Remark 5.2

We have a Leibniz rule for $\mathcal {I}^t[f,g,h]$ and $\mathcal {N}^t[F,G,H]$, namely

$$\begin{aligned} \begin{aligned}&Z\mathcal {I}^t[f,g,h]=\mathcal {I}^t[Zf,g,h]+\mathcal {I}^t[f,Zg,h]+\mathcal {I}^t[f,g,Zh],\quad Z\in \{x,\partial _x\},\\&Z\mathcal {N}^t[F,G,H]=\mathcal {N}^t[ZF,G,H]+\mathcal {N}^t[F,ZG,H]+\mathcal {N}^t[F,G,ZH],\quad Z\in \{x,\partial _x,\partial _y\}. \end{aligned} \end{aligned}$$

(6.16)

Property (6.16) is of importance in order to ensure the hypothesis of the transfer principle displayed by Lemma 5.2.