Local Well and Ill Posedness for the Modified KdV Equations in Subcritical Modulation Spaces

Abstract

We consider the Cauchy problem of the modified KdV equation (mKdV). Local well-posedness of this problem is obtained in modulation spaces $M_{2,q}^{1/4}(\mathbb{R})$ $(2\le q\le \infty )$. Moreover, we show that the data-to-solution map fails to be $C^3$ continuous in $M_{2,q}^s(\mathbb{R})$ when $s<1/4$. It is well-known that $H^{1/4}$ is a critical Sobolev space of mKdV so that it is well-posedness in $H^s$ for $s\ge 1/4$ and ill-posed (in the sense of uniform continuity) in $H^{s^{\prime }}$ with $s^{\prime }<1/4$. Noticing that $M_{2,q}^{1/4}\subset B_{2,q}^{1/q-1/4}$ is a sharp embedding and $H^{-1/4}\subset B_{2,\infty }^{-1/4}$, our results contains all of the subcritical data in $M_{2,q}^{1/4}$, which contains a class of functions in $H^{-1/4}\setminus H^{1/4}$.

The paper was completed and submitted on 1st May, 2018