Discrete Fourier Restriction Associated with Schrödinger and KDV Type Equations

The talk I will give is based on the joint work of Prof. Xiaochun Li and I. The standard way of solving nonlinear Schrodinger or KdV equations is to rewrite them into the equivalent form of integral equations and use Picard iteration. One key tool in controlling the nonlinear term is the Strichartz estimate. However, when we consider the periodic equations, the exact periodic type analogue of continuous Strichartz estimate fails, so it 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’ll present the results we got for this type of restriction. The main step is a decomposition of the kernel which helps us get the level set estimate. Then we could get the sharp upper bound when p is large. 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. In the last I will mention some topics that worth pursuing.