dc.contributor.author Lim, Zhuo Min dc.date.accessioned 2017-10-10T08:57:41Z dc.date.available 2017-10-10T08:57:41Z dc.date.issued 2017-10-01 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/267816 dc.description.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. dc.description.sponsorship Graduate research scholarship awarded by St John's College. dc.language.iso en dc.rights All Rights Reserved en dc.rights.uri https://www.rioxx.net/licenses/all-rights-reserved/ en dc.subject Well-posedness dc.subject Global regularity dc.subject Strichartz estimates dc.subject $U^p$ and $V^p$ spaces dc.subject Scattering dc.title Well-posedness and scattering of the Chern-Simons-Schrödinger system dc.type Thesis dc.type.qualificationlevel Doctoral dc.type.qualificationname Doctor of Philosophy (PhD) dc.publisher.institution University of Cambridge dc.publisher.department EPSRC Centre for Doctoral Training in Analysis (Cambridge Centre for Analysis) dc.date.updated 2017-10-09T22:50:15Z dc.identifier.doi 10.17863/CAM.13740 dc.publisher.college St John's dc.type.qualificationtitle PhD cam.supervisor Stuart, David Michael Addis rioxxterms.freetoread.startdate 2018-10-10
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