A Priori Bounds for the 1D Cubic NLS in Negative Sobolev Spaces

Abstract

We consider the cubic Nonlinear Schrödinger Equation (NLS) in one space dimension, either focusing or defocusing. We prove that the solutions satisfy a priori local in time <it>H</it><it>s</it> bounds in terms of the <it>H</it><it>s</it> size of the initial data for <f>$$s\\ge -\\frac{1}{6}$$</f>.