A remark on the well-posedness of the modified KdV equation in $L^2$

TL;DR

This paper establishes global well-posedness results for the modified KdV equation in low regularity spaces on both the real line and the circle, using commuting flows and simplifying existing proofs.

Contribution

It introduces a unified approach to prove well-posedness for the modified KdV equation in $H^{s}$ for $0 \\leq s < 1/2$, extending and simplifying prior results.

Findings

Proves global well-posedness in $H^{s}$ for $0 \\leq s < 1/2$ on the real line.

Provides an alternative proof of sharp $L^2$ well-posedness on the circle.

Extends well-posedness results to the large-data focusing case on the circle.

Abstract

We study the real-valued modified KdV equation on the real line and the circle, in both the focusing and the defocusing case. By employing the method of commuting flows introduced by Killip and Vi\c{s}an (2019), we prove global well-posedness in $H^{s}$ for $0\leq s<\tfrac{1}{2}$. On the line, we show how the arguments in the recent paper by Harrop-Griffiths, Killip, and Vi\c{s}an (2020) may be simplified in the higher regularity regime $s\geq 0$. On the circle, we provide an alternative proof of the sharp global well-posedness in $L^2$ due to Kappeler and Topalov (2005), and also extend this to the large-data focusing case.