Local Well-Posedness for the Derivative Nonlinear Schrödinger Equations with $L^2$ Subcritical Data

Abstract

We will show its local well-posedness in modulation spaces $M^{1/2}_{2,q}({\Real})$ $(2\le q<\infty )$. It is well-known that $H^{1/2}$ is a critical Sobolev space of DNLS so that it is locally well-posedness in $H^s$ for $s\ge 1/2$ and ill-posed in $H^{s^{\prime }}$ with $s^{\prime }<1/2.$ Noticing that that $M_{2,q}^{1/2}\subset B_{2,q}^{1/q}$ is a sharp embedding and $L^2\subset B_{2,\infty }^0$, our result contains all of the subcritical data in $M_{2,q}^{1/2}$, which contains a class of functions in $L^2\setminus H^{1/2}$.

50 Pages

References in corpus (1)

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