Large data well-posedness in the energy space of the Chern–Simons–Schrödinger system

Abstract

We consider the initial-value problem for the Chern–Simons–Schrödinger system, which is a gauge-covariant Schrödinger system in Rt×Rx2 with a long-range electromagnetic field. We show that, in the Coulomb gauge, it is locally well-posed in Hs for s⩾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 Up and Vp spaces.