On global well-posedness of the modified KdV equation in modulation spaces

TL;DR

This paper proves local and global well-posedness of the complex-valued modified KdV equation in modulation spaces for certain regularity levels, and shows ill-posedness below that threshold.

Contribution

It establishes the first well-posedness results for mKdV in modulation spaces with sharp regularity thresholds, extending previous global bounds.

Findings

Local well-posedness for s ≥ 1/4 in M^{2,p}_s

Global well-posedness for s ≥ 1/4 in M^{2,p}_s

Ill-posedness for s < 1/4 in M^{2,p}_s

Abstract

We study well-posedness of the complex-valued modified KdV equation (mKdV) on the real line. In particular, we prove local well-posedness of mKdV in modulation spaces $M^{2,p}_{s}(\mathbb{R})$ for $s \ge \frac14$ and $2\leq p < \infty$. For $s < \frac 14$, we show that the solution map for mKdV is not locally uniformly continuous in $M^{2,p}_{s}(\mathbb{R})$. By combining this local well-posedness with our previous work (2018) on an a priori global-in-time bound for mKdV in modulation spaces, we also establish global well-posedness of mKdV in $M^{2,p}_{s}(\mathbb{R})$ for $s \ge \frac14$ and $2\leq p < \infty$.