Anchored Peskin Problem

TL;DR

This paper rigorously analyzes the dynamics of anchored elastic filaments in fluid flows, extending the immersed boundary method to bounded domains and characterizing their equilibrium states and regularity.

Contribution

It generalizes the Peskin problem to the half-plane, providing a boundary-symmetric formulation and establishing well-posedness and regularity results for anchored filaments.

Findings

Leading-order evolution governed by a fractional Laplacian with Dirichlet conditions.

All equilibria are circular arcs connecting anchor points.

Filament exhibits instantaneous $C^inity$ regularization in space and time.

Abstract

The Immersed Boundary Method has long served as a robust computational framework for fluid-structure interactions, yet the rigorous analysis of 1D Peskin filaments anchored to rigid boundaries remains sparse. In this paper, we generalize the classical Peskin problem to the half-plane by considering an elastic filament whose endpoints are anchored to a no-slip wall. Moving beyond the algebraic complexity of the traditional Blake image system, we utilize the boundary-symmetric formulation of Gimbutas, Greengard, and Veerapaneni. This representation allows for a transparent decomposition of the hydrodynamic interactions into a free space principal part and a regularizing reflected component without resorting to hypersingular integral operators. Through this framework, we prove that the leading-order evolution of the anchored filament is governed by a fractional Laplacian equipped with homogeneous Dirichlet boundary conditions. We characterize the stationary states of the system, proving that all equilibria are circular arcs connecting the anchor points, a result that holds for a broad class of elastic energy densities. By framing the non-local dynamics in weighted little H\"older spaces, we establish local well posedness and prove that the filament exhibits instantaneous $C^\infty$ regularization in both space and time. This work provides a rigorous analytical foundation for anchored filaments in bounded domains and suggests a spectrally accurate numerical path for simulating tethered biological structures.