Some open problems related to Strichartz estimates

Evgeni Ovcharov (Edinburgh)

Tuesday 12th January, 2010 16:00-17:00 214


We shall present recent advances on Strichartz estimates, some new trends, and open problems. To begin with, we shall make a brief historic review of some endpoint Strichartz estimates together with the corresponding problems on nonlinear PDE's that motivated their development. Examples include nonlinear Dirac equations with power type nonlinearity. We shall present some new lines of research that are interested in special types of Strichartz estimates whose proof includes additional structural assumptions like spherical symmetry or averaging with respect to angular variables. An application shall be made to the global well-posedness of the Dirac-Klein-Gordon system which is a classical relativistic model from Quantum Field Theory. In the second half of the talk we shall present some new Stichartz estimates for the kinetic transport equation and the remaining unresolved estimates. In particular, we shall give an interesting example that demonstrates the failure of an endpoint Strichartz estimate on a Kakeya-type set. We conclude the talk with an application to the global well-posedness of a nonlinear kinetic model of bacterial chemotaxis

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