AI 中文总结
重新审视圆上调制KdV方程路径整体适定性，此前结果限于尺度次临界区域。本文指出因调制项存在有额外自由度，通过非KdV尺度变换，证明任意s∈R时，具有足够不规则调制的该方程在H^s(T)中整体适定，突破尺度临界正则性障碍。
AI 中文摘要
我们重新审视了圆上调制Korteweg-de Vries方程（KdV）的路径整体适定性问题。在之前的工作（2024年）中，第二和第四作者与C. Chouk、G. Li和J. Li通过结合I方法和拼接引理，证明了其在负Sobolev空间中的整体适定性。然而，由于使用了经典的KdV尺度变换，该结果仅限于尺度次临界区域$s > - \frac{3}{2}$。本文指出，由于调制项的存在，调制KdV在其尺度对称性上享有额外的一个自由度，我们对未知量应用非KdV尺度变换，并证明对于任意$s \in \mathbb{R}$，具有足够不规则调制的圆上调制KdV在$H^s(\mathbb{T})$中是整体适定的，从而突破了尺度临界正则性$s = - \frac{3}{2}$的障碍。
英文摘要
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 $s > - \frac 32$ 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 $s = - \frac 32$.
Comments26 pages. arXiv admin note: text overlap with arXiv:1406.7675, arXiv:2607.08385
