ncts

This talk is based on joint work with Ioan Bejenaru, Zihua Guo and Sebastian Herr. We study the initial value problem for the Zakharov in four space dimensions, the local well-posedness and asymptotic behavior for large time in the Sobolev spaces. It is well known that for such nonlinear dispersive equations with quadratic interactions, the main part of analysis is to estimate nonlinear resonances. We show that a normal form reduction together with the Strichartz estimate gives a simpler proof than the previous argument based on the Bourgain type bilinear estimate, extending also the range of Sobolev exponents for wellposedness. Another advantage of our proof is that it immediately yields the scattering for small data. Although the physically relevant dimension is three, the 4D case poses an interesting problem for analysis in the energy space, related to three different criticality: the energy critical power, the endpoint Strichartz estimate, and the critical Sobolev embedding into the space of bounded functions. It forces us to obtain the wellposedness and the scattering in an unusual way by the compactness argument. I would also like to explain the difficulty for large energy data.