Fermi-Pasta-Ulam系统与Korteweg-de Vries方程：低正则性连续极限
The Fermi-Pasta-Ulam system and the Korteweg-de Vries equation: a low-regularity continuum limit
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中文总结 AI 辅助
该研究证明了FPU系统到KdV方程在低正则性初值下的连续极限，通过三线性估计降低了所需Sobolev正则性，并推广至Toda晶格。
中文摘要 AI 辅助
对于无限Fermi-Pasta-Ulam（FPU）系统，当晶格网格尺寸趋于零时，一般解可由与Korteweg-de Vries（KdV）方程的解相关联的相向传播波来近似。我们证明，通过利用FPU哈密顿量的守恒性，在任意时间区间上，具有$L^2$级初值的FPU系统到KdV方程的连续极限在适当的范数下成立，从而回答了Hong、Kwak和Yang（2021）提出的一个开放问题。此外，对于FPU系统到KdV方程的局部时间连续极限，我们将所需的Sobolev正则性降低至$H^s$，其中$s > - \ rac 34$。我们的连续极限结果也适用于Flaschka形式的Toda晶格，从而降低了我们先前工作（2026）中的正则性要求。为建立这一低正则性连续极限，我们通过结合线性估计、特征曲线的横截性以及线性FPU流的多重线性色散平滑性质，证明了在端点处达到尖锐的关键三线性估计。
英文摘要
For the infinite Fermi-Pasta-Ulam (FPU) system, general solutions can be approximated by counter-propagating waves associated with solutions to the Korteweg-de Vries (KdV) equation as the lattice mesh size goes to zero. We show that, by exploiting the conservation of the FPU Hamiltonian, the continuum limit from the FPU system to the KdV equation with $L^2$-level initial data holds in appropriate norms on an arbitrary time interval, thereby answering an open question posed by Hong, Kwak, and Yang (2021). Moreover, for the local-in-time continuum limit of the FPU system to the KdV equation, we lower the required Sobolev regularity to $H^s$ with $s > - \frac 34$. Our continuum limit results also apply to the case of the Toda lattice in Flaschka's form, thereby lowering the regularity requirement in our previous work (2026). To establish this low-regularity continuum limit, we prove key trilinear estimates that are sharp up to the endpoint by combining linear estimates, the transversality of characteristic curves, and multilinear dispersive smoothing properties of the linear FPU flow.
发表机构
- University of Bonn（波恩大学）
机构由 AI 辅助整理，请以论文原文为准。
