Global well-posedness of non-integrable hyperbolic-ellptic Ishimori system in the critical Sobolev space
Zexian Zhang, Yi Zhou
TL;DR
We study the Cauchy problem for the hyperbolic-elliptic Ishimori system $\partial_t S = S \times (\partial_{x_1}^2 S - \partial_{x_2}^2 S) + \kappa (\partial_{x_1} \phi \cdot \partial_{x_1} S + \partial_{x_2} \phi \cdot \partial_{x_2} S)$ with $\Delta_x \phi = 2 S \cdot (\partial_{x_1} S \times \partial_{x_2} S)$ in two spatial dimensions, proving global well-posedness for small data in the critical Sobolev space $\dot{H}^1_Q(\mathbb{R}^2)$. The authors reformulate the system in Tao’s caloric gauge via differentiated fields $\psi_\alpha$ and connections $A_\alpha$, and develop ultrahyperbolic $U^p$-$V^p$ Strichartz estimates together with a new div-curl lemma to obtain sharp bilinear bounds. A two-tier bootstrap framework based on frequency envelopes $a_k(\sigma), b_k(\sigma), c_k(\sigma)$ drives both a heat-flow and an ultrahyperbolic Schrödinger analysis, closing for small data and extending the integrable case $\kappa=1$ to general $\kappa$, with a unified approach applicable to hyperbolic and elliptic Schrödinger maps in $d\ge2$. The results provide robust analytic tools for non-integrable Ishimori-type models and broaden the caloric gauge methodology beyond the elliptic setting.
Abstract
We consider the Cauchy problem for the hyperbolic-elliptic Ishimori system with general decoupling constant $κ\in \mathbb{R}$ 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 $κ= 1$ to general $κ$ and provide a unified framework that also applies to hyperbolic and elliptic Schrödinger maps in dimensions $d \ge 2$.