On the existence of periodic solutions to the modified Korteweg-de Vries equation below $H^{1/2}(\mathbb{T})$

TL;DR

This paper proves the existence of periodic solutions to the modified Korteweg-de Vries equation for initial data in Sobolev spaces with positive regularity, and discusses non-existence below a certain regularity threshold, using advanced Fourier analysis techniques.

Contribution

It establishes existence results for solutions in $H^s$ with $s>0$ and introduces a novel application of short-time Fourier restriction methods to handle derivative loss.

Findings

Existence of solutions in $H^s$ for $s>0$

Non-existence of solutions below $L^2$ under certain conjectures

Application of short-time Fourier restriction norm method

Abstract

Existence and a priori estimates for real-valued periodic solutions to the modified Korteweg-de Vries equation with initial data in $H^s$ are established for $s>0$. The short-time Fourier restriction norm method is employed to overcome the derivative loss. Further, non-existence of solutions below $L^2$ is proved conditional upon conjectured linear Strichartz estimates.