Local Well and Ill Posedness for the Modified KdV Equations in Subcritical Modulation Spaces (1811.05182v1)
Abstract: We consider the Cauchy problem of the modified KdV equation (mKdV). Local well-posedness of this problem is obtained in modulation spaces $M{1/4}_{2,q}(\mathbb{{R}})$ $(2\leq q\leq\infty)$. Moreover, we show that the data-to-solution map fails to be $C3$ continuous in $M{s}_{2,q}(\mathbb{{R}})$ 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 $Hs$ for $s\geq 1/4$ and ill-posed (in the sense of uniform continuity) in $H{s'}$ with $s'<1/4$. Noticing that $M{1/4}_{2,q} \subset B{1/q-1/4}_{2,q}$ is a sharp embedding and $H{-1/4}\subset B{-1/4}_{2,\infty}$, our results contains all of the subcritical data in $M{1/4}_{2,q}$, which contains a class of functions in $H{-1/4}\setminus H{1/4}$.