Hideo Takaoka

Global well-posedness for Schrödinger equations with derivative

We prove that the one-dimensional Schrodinger equation with derivative in the nonlinear term is globally well-posed in Hs for s>2/3, for small L2 data. The result follows from an application of the I-method. This method allows us to define a modification of the energy norm H1 that is almost conserved and can be used to perform an iteration argument. We also remark that the same argument can be used to prove that any quintic nonlinear defocusing Schrodinger equation on the line is globally well-posed for large data in Hs, for s>2/3.

Local well-posedness of the nonlinear Schrödinger equations on the sphere for data in modulation spaces

ABSTRACT In this paper, the authors discuss a priori estimates derived from the energy method to the initial value problem for the cubic nonlinear Schrödinger on the sphere S2. Exploring suitable a priori estimates, the authors prove the existence of solution for data whose regularity is s = 1/4.