Well- and ill-posedness of the Cauchy problem for semi-linear Schrödinger equations on the torus

TL;DR

This work characterizes when the Cauchy problem for torus-based semi-linear Schrödinger equations with polynomial nonlinearities is well-posed in Sobolev spaces $H^s(\mathbb{T})$ with $s>\tfrac{5}{2}$. It identifies a sharp Mizohata-type condition, $\int_{\mathbb{T}} \Im F_\beta(\psi,\partial_x \psi,\overline{\psi},\overline{\partial_x \psi})\,dx = 0$ for all $\psi$, as necessary and sufficient for well-posedness; otherwise, the problem exhibits ill-posedness via nonexistence results. The well-posedness half combines energy estimates with a gauge transformation to cancel derivative losses and a Bona–Smith approximation to secure continuous dependence, while the ill-posedness half uses integration by parts, dispersive effects, and a priori estimates to prove nonexistence of solutions for certain initial data. Together, these results provide a precise, torus-specific analogue of Mizuhata-type criteria for nonlinear dispersive equations and clarify how derivative nonlinearities interact with periodic geometry. The methods yield a robust framework for handling derivative nonlinearities on compact manifolds and highlight the pivotal role of the $s>\tfrac{5}{2}$ regularity threshold and the gauge-transformation mechanism.

Abstract

We consider the Cauchy problem for semi-linear Schrödinger equations on the torus $\mathbb T$. We establish a necessary and sufficient condition on the polynomial nonlinearity for the Cauchy problem to be well-posed in the Sobolev space $H^s(\mathbb T)$ for $s>\frac 52$. For the well-posedness, we use the energy estimates and the gauge transformation. For the ill-posedness, we prove the non-existence of solutions to the Cauchy problem.

Theorems & Definitions (37)

• Definition 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Remark 1.8
• Proposition 2.1
• Proposition 2.2