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We consider the initial-value problem for the Chern-Simons-Schr\"odingersystem, which is a gauge-covariant Schr\"{o}dinger system in$\mathbb{R}_t\times\mathbb{R}^2_x$ with a long-range electromagnetic field. Weshow that, in the Coulomb gauge, it is locally well-posed in $H^s$ for $s\ge1$, and the solution map satisfies a local-in-time weak Lipschitz bound. Byenergy conservation, we also obtain a global regularity result. The key is toretain the non-perturbative part of the derivative nonlinearity in theprincipal operator, and exploit the dispersive properties of the resultingparadifferential-type principal operator using adapted $U^p$ and $V^p$ spaces.

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