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

Abstract

\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 $M_{2,1}^{3/2}({\mathbb{R}}^n)$. Noticing that $H^{\tilde{s}}\subset M_{2,1}^s$ $(\tilde{s}-s>n/2)$ is an optimal inclusion, we have shown the global well posedness of DNLS with a class of very rough data.

42 Pages

References in corpus (1)

• Global well posedness and scattering for the elliptic and non-elliptic derivative nonlinear Schrodinger equations with small data

Cited by in corpus (1)

• Global Well-posedness for the fourth order nonlinear Schrödinger equations with small rough data in high demension