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 \(H^\frac{1}{2}({\mathbb {T}})\). 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 \({\mathcal {F}}L^{s,p}({\mathbb {T}})\) for \(s\ge \frac{1}{2}\) and \(1\le p <\infty \). 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.

我们研究了低正则性下圆上复值修正 Korteweg-de Vries 方程 (mKdV) 的适定性。在我们之前的工作（2021）中，我们引入了基于动量守恒的第二个重正化 mKdV 方程，我们提出将其作为研究\(H^\frac{1}{2}之外的复值 mKdV 的正确模型({\mathbb {T}})\)。这里，我们采用Deng等人介绍的方法。（Commun Math Phys 384(1):1061–1107, 2021）证明傅里叶-勒贝格空间中第二个重正化 mKdV 方程的局部适定性 \({\ mathcal {F}}L^{s,p}( {\mathbb {T}})\)为\(s\ge \frac{1}{2}\)和\(1\le p <\infty \)。作为这种适定性结果的副产品，我们展示了复值 mKdV 的不适定性，而无需对这些具有无限动量的傅立叶-勒贝格空间中的初始数据进行第二次重整化。