Global well-posedness of non-integrable hyperbolic-ellptic Ishimori system in the critical Sobolev space

TL;DR

This paper proves the global well-posedness of the hyperbolic-elliptic Ishimori system in the critical Sobolev space for general decoupling constants, using novel bilinear estimates and advanced gauge techniques.

Contribution

It extends the global well-posedness results to non-integrable cases of the Ishimori system with a unified approach applicable to related Schr"odinger maps.

Findings

Established global well-posedness in critical Sobolev space.

Developed new bilinear estimates via a div-curl lemma.

Unified framework for hyperbolic and elliptic Schr"odinger maps.

Abstract

We consider the Cauchy problem for the hyperbolic-elliptic Ishimori system with general decoupling constant $\kappa \in \mathbb{R}$ and prove global well-posedness in the critical Sobolev space. The proof relies primarily on new bilinear estimates, which are established via a novel div-curl lemma first introduced by the second author in \cite{zhou_1+2dimensional_2022}. Our approach combines the caloric gauge technique with $U^p$-$V^p$ type Strichartz estimates to handle the hyperbolic structure of the equation. The results extend previous work on the integrable case $(\kappa = 1)$ to general $\kappa$ and provide a unified framework which also works for the hyperbolic and elliptic Schr\"odinger maps in dimensions $d \ge 2$.