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Title: Optimal constants and maximising functions for Strichartz inequalities
Author: Jeavons, Christopher Paul
ISNI:       0000 0004 5350 8385
Awarding Body: University of Birmingham
Current Institution: University of Birmingham
Date of Award: 2015
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We prove sharp weighted bilinear inequalities which are global in time and for general dimensions for the free wave, Schrödinger and Klein-Gordon propagators. This extends work of Ozawa –Rogers for the Klein-Gordon propagator, work of Foschi-Klainerman and Bez-Rogers for the wave propagator, and work of Ozawa-Tsutsumi, Planchon-Vega and Carneiro for the Schrödinger propagator. In each case, we make a connection to estimates involving certain dispersive Sobolev norms. As a consequence of these estimates we obtain, among other things, a new sharp form of a linear Strichartz estimate for the solution of the Klein-Gordon equation in five spatial dimensions for data belonging to H1, and that maximisers do not exist for this estimate. We also obtain a new sharp form of a linear Sobolev- Strichartz estimate for the wave equation in four space dimensions for initial data in H¾ x H-1/4 and characterisation of the maximisers. Finally, we study the variational problems associated to the linear Sobolev-Strichartz estimates for the Schrödinger and wave equations. We establish that Gaussian functions are not maximisers for the Hm to LP inequalities for the Schrödinger propagator, for any m > 0, and make a conjecture about the nature of the maximisers for the H d-1/4 x Hd-5/4 to L4 inequalities for the wave equation.
Supervisor: Not available Sponsor: School of Mathematics ; University of Birmingham
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics