ETD

• bilinear estimates
• Bourgain spaces
• Cauchy Problem
• low regularity
• nonlinear dispersive equation
• periodic Korteweg-de Vries (KdV) equation
• periodic Ostrovsky equation
• well-posedness

This work is devoted to the study of Cauchy Problems for nonlinear periodic evolution equations with low regularity initial data. Our concern are the local and global well-posedness for KdV and Ostrovsky equation.
We give a general introduction to the classic Fourier restriction method due to Bourgain for dispersive equations that reduces the well-posedness problem to multilinear estimates.
The former equation is known to be globally well-posed in the L^2-based Sobolev space H^s for s>=-1/2. We re-prove the local result without using Strichartz estimates following Kenig, Ponce, Vega.
The latter equation is a perturbation of the former. It was derived by L.A. Ostrovsky as model for nonlinear surface waves in the ocean in a rotating frame of reference. We develop a bilinear estimate in the respective Bourgain spaces. This allows us to implement an iterative method that leads to local well-posedness for the periodic Ostrovsky equation in H^s with s>=-1/2 provided that the initial data is of 0-mean and its H^s norm is sufficiently small.