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

Laurel Ohm

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 $ε>0$, 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 $ε$ 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.

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

Abstract

We consider the Laplace equation in the exterior of a thin filament in 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 , 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 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.
Paper Structure (22 sections, 26 theorems, 371 equations, 1 figure)

This paper contains 22 sections, 26 theorems, 371 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Geometry
  3. Slender body Neumann-to-Dirichlet map
  4. Slender body NtD for the straight filament
  5. Statement of main theorem
  6. Layer potential formulation of slender body DtN map
  7. Proof of Theorem \ref{['thm:main']}
  8. Mapping properties in the straight setting
  9. Symbols for single and double layer
  10. Mapping properties: preliminaries
  11. Multiplier estimates
  12. Proof of Lemmas \ref{['lem:mapping_Leps']} and \ref{['lem:S_mapping']}
  13. Mapping properties of remainder terms
  14. Setup and tools
  15. Mapping properties of single layer remainder terms
  16. ...and 7 more sections

Key Result

Lemma 1.1

Let $\overline{\mathcal{L}}_\epsilon$ denote the slender body NtD map along the straight filament $\mathcal{C}_\epsilon$. Given $h\in C^{0,\alpha}(\mathbb{T})$ with $\int_\mathbb{T} h(s)\,ds=0$, we have Similarly, let $\overline{\mathcal{L}}_\epsilon^{-1}$ denote the slender body DtN map along $\mathcal{C}_\epsilon$. Given $g\in C^{1,\alpha}(\mathbb{T})$, we have

Figures (1)

  • Figure 1: An example of curved filament $\Sigma_\epsilon$ described in section \ref{['subsec:geom']} and the straight filament $\mathcal{C}_\epsilon$ described in \ref{['eq:Cepsilon']}.

Theorems & Definitions (44)

  • Lemma 1.1: Mapping properties of SB NtD and DtN on $\mathcal{C}_\epsilon$
  • Theorem 1.2: Decomposition of slender body DtN and NtD
  • Remark
  • Lemma 1.3: Single and double layer operators on $\mathcal{C}_\epsilon$
  • Lemma 1.4: Inverse single layer on $\mathcal{C}_\epsilon$
  • Lemma 1.5: Mapping properties of $\mathcal{R}_\mathcal{S}$ and $\mathcal{R}_\mathcal{D}$
  • Lemma 1.6: Single layer applied to constant-in-$s$
  • Lemma 1.7: Bound for $w(s,\theta)$
  • Lemma 1.8: Slender body NtD in Hölder spaces
  • proof : Proof of Lemma \ref{['lem:straight_components']}
  • ...and 34 more