A refined well-posedness result for the modified KdV equation in the Fourier-Lebesgue spaces

TL;DR

This paper establishes local well-posedness for a second renormalized version of the complex-valued mKdV equation in Fourier-Lebesgue spaces at low regularity, and shows ill-posedness for the original equation without renormalization.

Contribution

It proves well-posedness of the second renormalized mKdV in Fourier-Lebesgue spaces and demonstrates ill-posedness of the original complex mKdV without renormalization.

Findings

Well-posedness of the second renormalized mKdV in  spaces for s and p<.

Ill-posedness of the original complex mKdV in these spaces with infinite momentum.

Extension of previous work to lower regularity Fourier-Lebesgue spaces.

Abstract

We study the well-posedness of the complex-valued modified Korteweg-de Vries equation (mKdV) on the circle at low regularity. In our previous work (2019), we introduced the second renormalized mKdV equation, based on the conservation of momentum, which we proposed as the correct model to study the complex-valued mKdV outside of $H^\frac12(\mathbb{T})$. Here, we employ the method introduced by Deng-Nahmod-Yue (2019) to prove local well-posedness of the second renormalized mKdV equation in the Fourier-Lebesgue spaces $\mathcal{F}L^{s,p}(\mathbb{T})$ for $s\geq \frac12$ and $1\leq p <\infty$. As a byproduct of this well-posedness result, we show ill-posedness of the complex-valued mKdV without the second renormalization for initial data in these Fourier-Lebesgue spaces with infinite momentum.