Optimal constants and maximising functions for Strichartz inequalities

Jeavons, Christopher Paul (2015). Optimal constants and maximising functions for Strichartz inequalities. University of Birmingham. Ph.D.

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Abstract

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.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

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