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

TL;DR

This paper establishes local well-posedness and ill-posedness results for the modified KdV equation in specific modulation spaces, extending understanding of the equation's behavior in subcritical function spaces.

Contribution

It proves local well-posedness in modulation spaces $M^{1/4}_{2,q}$ and demonstrates failure of $C^3$ continuity of the data-to-solution map below this regularity, broadening the scope of initial data analysis.

Findings

Well-posedness in $M^{1/4}_{2,q}$ for $2 \\leq q \\leq \\infty$

Failure of $C^3$ continuity for $s<1/4$

Includes functions in $H^{-1/4} \\setminus H^{1/4}$

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 $C^3$ 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 $H^s$ 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 $H^{-1/4}\setminus H^{1/4}$.