Well-posedness and scattering of the Chern-Simons-Schrödinger system

Abstract

The subject of the present thesis is the Chern-Simons-Schrödinger system, which is a gauge-covariant Schrödinger system in two spatial dimensions with a long-range electromagnetic field. The present thesis studies two aspects of the system: that of well-posedness and that of the long-time behaviour. The first main result of the thesis concerns the large-data well-posedness of the initial-value problem for the Chern-Simons-Schrödinger system. We impose the Coulomb gauge to remove the gauge-invariance, in order to obtain a well-defined initial-value problem. We prove that, in the Coulomb gauge, the Chern-Simons-Schrödinger system is locally well-posed in the Sobolev spaces $H^s$ for $s\ge 1$, and that the solution map satisfies a weak Lipschitz continuity estimate. The main technical difficulty is the presence of a derivative nonlinearity, which rules out the naive iteration scheme for proving well-posedness. The key idea is to retain the non-perturbative part of the derivative nonlinearity in the principal operator, and to exploit the dispersive properties of the resulting paradifferential-type principal operator, in particular frequency-localised Strichartz estimates, using adaptations of the $U^p$ and $V^p$ spaces introduced by Koch and Tataru in other contexts. The other main result of the thesis characterises the large-time behaviour in the case where the interaction potential is the defocusing cubic term. We prove that the solution to the Chern-Simons-Schrödinger system in the Coulomb gauge, starting from a localised finite-energy initial datum, will scatter to a free Schrödinger wave at large times. The two crucial ingredients here are the discovery of a new conserved quantity, that of a pseudo-conformal energy, and the cubic null structure discovered by Oh and Pusateri, which reveals a subtle cancellation in the long-range electromagnetic effects. By exploiting pseudo-conformal symmetry, we also prove the existence of wave operators for the Chern-Simons-Schrödinger system in the Coulomb gauge: given a localised finite-energy final state, there exists a unique solution which scatters to that prescribed state.