Refined global well-posedness for the periodic modulated Korteweg-de Vries equation

We revisit the pathwise global well-posedness issue of the modulated Korteweg-de Vries equation (KdV) on the circle. In the previous work (2024), by combining the $I$-method and the sewing lemma, the second and fourth authors with C. Chouk, G. Li, and J. Li proved its global well-posedness in negative Sobolev spaces. This result was, however, restricted to the scaling subcritical regime $s>-\frac{3}{2}$ due to the use of the classical KdV scaling. In this paper, by noting that the modulated KdV enjoys additional one degree of freedom in its scaling symmetry thanks to the modulation term, we apply a non-KdV scaling to the unknown and prove that, given any $s\in \mathbb{R}$, the modulated KdV on the circle with a sufficiently irregular modulation is globally well-posed in $H^s(\mathbb{T})$, thus going beyond the barrier of the scaling critical regularity $s=-\frac{3}{2}$.