Local well and ill posedness for the modified KdV equations in subcritical modulation spaces

We consider the Cauchy problem of the modified KdV (mKdV) equation. Local wellposedness 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 $s \lt 1/4$. We notice that $H^{1/4}$ is the critical Sobolev space for mKdV such that it is well-posed in $H^s$ for $s \geq 1/4$ and ill-posed (in the sense of uniform continuity) in $H^{s^\prime}$ with $s^\prime \lt 1/4$. Recalling 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 contain all of the subcritical data in $M^{1/4}_{2,q}$, which contains a class of functions in $H^{-1/4} \setminus H^{1/4}$.

Keywords

local well-posedness, ill-posedness, modified KdV equations, modulation spaces

2010 Mathematics Subject Classification

35A01, 35Q53

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The first author is supported in part by the China Postdoctoral Science Foundation under Grant 2019M650019, and the second author is supported in part by the National Science Foundation of China, Grant 11571254.

Received 7 October 2018

Accepted 20 December 2019

Published 28 July 2020