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dc.contributor.authorLim, Zhuo Min
dc.date.accessioned2017-10-10T08:57:41Z
dc.date.available2017-10-10T08:57:41Z
dc.date.issued2017-10-01
dc.identifier.urihttps://www.repository.cam.ac.uk/handle/1810/267816
dc.description.abstractThe 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.sponsorshipGraduate research scholarship awarded by St John's College.
dc.language.isoen
dc.subjectWell-posedness
dc.subjectGlobal regularity
dc.subjectStrichartz estimates
dc.subject$U^p$ and $V^p$ spaces
dc.subjectScattering
dc.titleWell-posedness and scattering of the Chern-Simons-Schrödinger system
dc.typeThesis
dc.type.qualificationlevelDoctoral
dc.type.qualificationnameDoctor of Philosophy (PhD)
dc.publisher.institutionUniversity of Cambridge
dc.publisher.departmentEPSRC Centre for Doctoral Training in Analysis (Cambridge Centre for Analysis)
dc.date.updated2017-10-09T22:50:15Z
dc.identifier.doi10.17863/CAM.13740
dc.publisher.collegeSt John's
dc.type.qualificationtitlePhD
cam.supervisorStuart, David Michael Addis
rioxxterms.freetoread.startdate2018-10-10


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