Um estudo sobre a boa colocação local da equação não linear de Schrödinger cúbica unidimensional em espaços de Sobolev periódicos

Abstract

In this work we study, in details, the Cauchy problem of the nonlinear Schrödinger equation, with initial datas in periodic Sobolev spaces. Specifically, we prove that this problem is locally well posed for datas in Hsper, with s ≥ 0. Particularly, for initial datas in L2 the problem is globally well posed, due to the conservation law of the equation in this space. Moreover, we prove the this result is the best one, seeing we expose examples that show that the equation flow is not locally uniformly continuous for initial datas with regularity less than L2.