Talk Title: On Global-in-Time Strichartz Estimates for the Semiperiodic Schrodinger Equation
Abstract: The classical Strichartz estimate for the linear Schrodinger equation on Euclidean space shows that if the initial data is in L^2 then the solution will be in some (possibly mixed) Lebesgue space globally in space and time. These estimates turn out to be an important tool in the analysis of the nonlinear Schrodinger equation. There has been a wide range of research concerning the extent to which the underlying geometry affects the solution to the Schrodinger equation. For example, if one instead considers the equation on the torus then it is impossible to prove global-in-time Strichartz estimates (since the solution will be periodic in time). There do exist local-in-time estimates, however, though these are much harder to prove than in the Euclidean setting. We will discuss some recent results in the semiperiodic setting where one considers the Schrodinger equation on products of tori and Euclidean spaces (e.g. a cylinder). Our results generalize and improve a global-in-time Strichartz-type estimate in this setting originally do to Z. Hani and P. Pausader. As a consequence of our estimates one can prove global existence and scattering for small-data solutions to the critical quintic and cubic nonlinear Schrodinger equations on RxT and R^2 x T, respectively (where T is the one-dimensional torus).