Title:

Wellposedness and scattering of the ChernSimonsSchrödinger system

The subject of the present thesis is the ChernSimonsSchrödinger system, which is a gaugecovariant Schrödinger system in two spatial dimensions with a longrange electromagnetic field. The present thesis studies two aspects of the system: that of wellposedness and that of the longtime behaviour. The first main result of the thesis concerns the largedata wellposedness of the initialvalue problem for the ChernSimonsSchrödinger system. We impose the Coulomb gauge to remove the gaugeinvariance, in order to obtain a welldefined initialvalue problem. We prove that, in the Coulomb gauge, the ChernSimonsSchrödinger system is locally wellposed 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 wellposedness. The key idea is to retain the nonperturbative part of the derivative nonlinearity in the principal operator, and to exploit the dispersive properties of the resulting paradifferentialtype principal operator, in particular frequencylocalised 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 largetime behaviour in the case where the interaction potential is the defocusing cubic term. We prove that the solution to the ChernSimonsSchrödinger system in the Coulomb gauge, starting from a localised finiteenergy 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 pseudoconformal energy, and the cubic null structure discovered by Oh and Pusateri, which reveals a subtle cancellation in the longrange electromagnetic effects. By exploiting pseudoconformal symmetry, we also prove the existence of wave operators for the ChernSimonsSchrödinger system in the Coulomb gauge: given a localised finiteenergy final state, there exists a unique solution which scatters to that prescribed state.
