Well-posedeness for the non-isotropic Schrödinger equations on cylinders and periodic domains

TL;DR

This paper analyzes the initial value problem for the non-isotropic NLS on product domains ${\mathbb T}\times{\mathbb R}$, ${\mathbb R}\times{\mathbb T}$, and ${\mathbb T}^2$, focusing on the dispersive symbol $\omega(n)=\varepsilon n^2 - \alpha n^4$ and the fourth-order term $\alpha\partial_x^4$. The authors develop domain-appropriate Strichartz estimates, including a new mixed-domain estimate for ${\mathbb R}\times{\mathbb T}$ and a decoupling-based bound for ${\mathbb T}^2$, and use fixed-point arguments in anisotropic function spaces to establish local well-posedness; for ${\mathbb T}\times{\mathbb R}$ with $\alpha<0$ they obtain global well-posedness for small data in $L^2$, while ill-posedness is proved below $L^2$ in the focusing case. The paper also presents a comprehensive treatment of the periodic case via Bourgain-Demeter decoupling, achieving local well-posedness in $H^s({\mathbb T}^2)$ for $s>1/4$ under a special homogeneous setup, and discusses the limitations and open questions related to the sign of $\alpha$ and potential extensions. Overall, the work broadens the understanding of non-isotropic dispersive dynamics on cylinders and tori, with implications for fiber-array models and related PDEs.

Abstract

The initial value problem (IVP) for the non-isotropic Schrödinger equation posed on the two-dimensional cylinders and $\mathbb{T}^2$ is considered. The IVP is shown to be locally well-posed for small initial data in $H^s(\mathbb{T}\times\mathbb{R})$ if $s\geq0$. For the IVP posed on $\mathbb{R}\times\mathbb{T}$, given data are considered in the anisotropic Sobolev spaces thereby obtaining the local well-posedness result in $H^{s_1, s_2}(\mathbb{R}\times\mathbb{T})$, if $s_1\geq0$ and $s_2>\frac12$. In the purely periodic case, a particular case of the IVP is shown to be locally well-posed for any given initial data in $H^s(\mathbb{T}^2)$ if $s>\frac14$. In some cases, ill-posedness issues are also considered showing that the IVP posed on $\mathbb{T}\times \mathbb{R}$, in the focusing case, is ill-posed in the sense that the application data-solution fails to be uniformly continuous for data in $H^s(\mathbb{T}\times\mathbb{R})$ if $-\frac12\leq s<0$.

Theorems & Definitions (21)

• Proposition 2.1: Strichartz estimate on $I\times{\mathbb T}\times{\mathbb R}$
• Theorem 2.2: Well-posedness in $L^2$
• Remark 2.3
• Theorem 2.4: Ill-posedness below $L^2$
• Remark 2.5
• Theorem 2.6
• Proposition 2.7
• Theorem 2.8
• Remark 2.9
• Lemma 3.1