Global Smooth Effects and Well-Posedness for the Derivative Nonlinear Schrödinger Equation with Small Rough Data

Baoxiang, Wang; Lijia, Han; Chunyan, Huang

Global Smooth Effects and Well-Posedness for the Derivative Nonlinear Schrödinger Equation with Small Rough Data

Authors

Baoxiang, Wang

Lijia, Han

Chunyan, Huang

Description

\rm We obtain the global smooth effects for the solutions of the linear Schrödinger equation in anisotropic Lebesgue spaces. Applying these estimates, we study the Cauchy problem for the generalized elliptical and non-elliptical derivative nonlinear Schrödinger equations (DNLS) and get the global well posedness of solutions with small data in modulation spaces $M^{3/2}_{2,1}(\mathbb{R}^n)$. Noticing that $H^{\tilde{s}} \subset M^s_{2,1}$ $(\tilde{s}-s>n/2)$ is an optimal inclusion, we have shown the global well posedness of DNLS with a class of very rough data.
42 Pages