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 (2021), 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 H½(T). Here, we employ the method introduced by Deng et al. (Commun Math Phys 384(1):1061-1107, 2021) to prove local well-posedness of the second renormalized mKdV equation in the Fourier-Lebesgue spaces FLs,p(T) for s ≥ ½ and 1 ≤ p < ∞. 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.