Normal form approach for dispersive equations with low-regularity data

Abstract

In this dissertation, we examine applications of the normal form technique to nonlinear dispersive equations with rough initial data. Working within the framework of Bourgain spaces, the normal form method often produces ample smoothing effects on the non-linearity. The extra gain in regularity is ideal for analysing solutions with low-regularity initial data, thus this approach can be used to overcome difficulties due to lack of smoothness in polynomial-type non-linearities. In particular, we will consider three canonical models in dispersive equations with quadratic and derivative quadratic non-linearities.