发表机构
University of Pennsylvania（宾夕法尼亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究平面 Stokes 流中自由端不可伸展开放细丝的动力学，通过三阶非局部曲率方程证明近临界数据的局部适定性、端点渐近展开，并给出小扰动及有限能量下全局存在与指数拉直结果。
AI 中文摘要
我们研究平面 Stokes 流体中具有自由端的不可伸展开放细丝。该系统归结为一个三阶非局部曲率方程，该方程与一个关于张力的椭圆方程耦合。我们在满足弧弦条件的支撑 Sobolev 空间 $\widetilde H^s$（$-1/2<s\le0$）中，对近临界初始数据证明了局部适定性。对于正时间，我们证明了改进的 Sobolev 正则性，并利用 Wiener--Hopf 分解在每个自由端导出了 $d^{3/2}$ 型展开式，其中 $d$ 表示到该端点的距离。我们进一步证明了对于足够小的初始数据以及满足 $E(0)<\pi^2/4$ 的有限能量初始数据，全局存在性以及指数收敛到直线细丝。有限能量结果来自一个能量恒等式和一个将弯曲能量与弧弦常数联系起来的几何估计。更一般地，任何有限时间破裂都必须伴随弧弦条件的丧失，而每个全局解要么指数收敛到直线细丝，要么其弧弦常数沿趋向无穷的时间序列趋于零。
英文摘要
We study an inextensible open filament with free ends in a planar Stokes fluid. The system reduces to a third-order nonlocal curvature equation coupled to an elliptic equation for the tension. We prove local well-posedness for nearly critical initial data in supported Sobolev spaces $\widetilde H^s$, $-1/2<s\le0$, satisfying the arc-chord condition. For positive times, we prove improved Sobolev regularity and derive a $d^{3/2}$-type expansion at each free end using Wiener--Hopf factorization, where $d$ denotes the distance to that endpoint. We further prove global existence and exponential convergence to a straight filament for sufficiently small initial data and for finite-energy initial data satisfying $E(0)<π^2/4$. The finite-energy result follows from an energy identity and a geometric estimate relating the bending energy to the arc-chord constant. More generally, any finite-time breakdown must be accompanied by loss of the arc-chord condition, while every global solution either converges exponentially to a straight filament or has arc-chord constants tending to zero along a sequence of times tending to infinity.
