Large Data Well-posedness in the Energy Space of the Chern-Simons-Schr\"odinger System

We consider the initial-value problem for the Chern-Simons-Schr"odinger system, which is a gauge-covariant Schr"{o}dinger system in ${\mathbb{R}}_t\times {\mathbb{R}}_x^2$ with a long-range electromagnetic field. We show that, in the Coulomb gauge, it is locally well-posed in $H^s$ for $s\ge 1$, and the solution map satisfies a local-in-time weak Lipschitz bound. By energy conservation, we also obtain a global regularity result. The key is to retain the non-perturbative part of the derivative nonlinearity in the principal operator, and exploit the dispersive properties of the resulting paradifferential-type principal operator using adapted $U^p$ and $V^p$ spaces.