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

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