Open Inextensible Filaments in Planar Stokes Flow: Well-Posedness, Endpoint Asymptotics, and Straightening
Abstract
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 H̃s, −1/2<s≤0, satisfying the arc-chord condition. For positive times, we prove improved Sobolev regularity and derive a d3/2-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)<π2/4. 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.
Related
- Endpoint well-posedness for the Boussinesq equation
- Exact Endpoint Asymptotics and Almost-Sure Limits for Uniform Empirical Measures in Two Dimensions
- Endpoint and Vanishing-Density Asymptotics for Hardy--Szegő Zero Counts
- Endpoint Asymptotics of Optimal Stabilization Prefactors
- Immersed Boundary Electrodiffusion in the Wasserstein Gradient Flow Formalism: Global Well-Posedness, Non-Equilibrium Thermodynamics, and Singular Asymptotics
BibTeX
@misc{zhou2026open,
title = {{Open Inextensible Filaments in Planar Stokes Flow: Well-Posedness, Endpoint Asymptotics, and Straightening}},
author = {Han Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10480},
eprint = {2609.10480},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10480},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Open Inextensible Filaments in Planar Stokes Flow: Well-Posedness, Endpoint Asymptotics, and Straightening
AU - Han Zhou
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10480
UR - https://arxiv.org/abs/2609.10480
AB - 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 H̃^s, −1/2<s≤0, 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.
ER -