On an angle-averaged Neumann-to-Dirichlet map for thin filaments

TL;DR

This paper analyzes the Neumann-to-Dirichlet and Dirichlet-to-Neumann maps for the Laplace equation around thin filaments, decomposing them into explicit parts for straight filaments plus small remainders, with implications for filament evolution in fluid flow.

Contribution

It provides a detailed decomposition of slender body NtD and DtN maps for curved filaments into explicit operators plus lower order remainders, extending understanding of boundary maps in thin filament geometries.

Findings

Explicit Fourier multiplier formulas for straight filament maps

Decomposition of curved filament maps into straight filament plus small remainders

Remainder terms are lower order or smoother, facilitating analysis

Abstract

We consider the Laplace equation in the exterior of a thin filament in $\mathbb{R}^3$ and perform a detailed decomposition of a notion of slender body Neumann-to-Dirichlet (NtD) and Dirichlet-to-Neumann (DtN) maps along the filament surface. The decomposition is motivated by a filament evolution equation in Stokes flow for which the Laplace setting serves as an important toy problem. Given a general curved, closed filament with constant radius $\epsilon>0$, we show that both the slender body DtN and NtD maps may be decomposed into the corresponding operator about a straight, periodic filament plus lower order remainders. For the straight filament, both the slender body NtD and DtN maps are given by explicit Fourier multipliers and it is straightforward to compute their mapping properties. The remainder terms are lower order in the sense that they are small with respect to $\epsilon$ or smoother. While the strategy here is meant to serve as a blueprint for the Stokes setting, the Laplace problem may be of independent interest.