The Discrete Fourier Restriction Phenomenon Associated with Schrödinger and Kdv Equations

The talk I will give is based on the joint work with Prof. Xiaochun Li. The standard way of solving nonlinear Schrodinger or KdV equations is to rewrite them into the equivalent integral equations and apply Picard's iteration. The key tool in controlling the nonlinear term is the Strichartz estimate. However, when we consider the periodic equations, the exact periodic analogue of continuous Strichartz estimate fails. This forces us to find some new inequality of the same type. The periodic Strichartz estimate is in the form of exponential sums, which is also equivalent to the discrete Fourier restriction estimate. In this talk I will present the results we got for this type of restriction. With Strichartz type estimates, as well as some classical technique, we also proved the sharp results of local well-posedness of general KdV equations and fifth order KdV type equations with polynomial nonlinear terms.