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The Stokes Eigenvalue Problem on balls and annuli in three dimensions: Solutions with Poloidal and Toroidal Fields

Bernd Rummler, Gudrun Thäter

TL;DR

The paper studies the Stokes eigenvalue problem in symmetric 3D domains (balls and spherical annuli) under Dirichlet boundary data. It develops a toroidal/poloidal decomposition using the Laplace-Beltrami operator on the sphere, leading to decoupled scalar problems for the toroidal potential $\psi$ and the poloidal potential $\chi$, from which Stokes eigenfunctions are constructed. On the unit ball, eigenfunctions are given explicitly in terms of spherical harmonics and Bessel zeros, yielding eigenvalues $\lambda=(\mu^{j}_{l+\frac{1}{2}})^{2}$ (toroidal) and $\lambda=(\mu^{j}_{l+\frac{3}{2}})^{2}$ (poloidal) with multiplicity $2l+1$, and analogous determinants determine eigenvalues for annuli. The results establish the completeness of the explicit eigenfunction systems introduced earlier and provide a robust framework for spectral-Galerkin methods in incompressible flows on these highly symmetric domains.

Abstract

We consider the Stokes eigenvalue problem in open balls and open annuli in R3 with homogeneous Dirichlet boundary conditions. Using the frame of toroidal and poloidal fields we construct the othogonal decomposition of the Stokes eigenvalue problem in problems for toroidal and poloidal eigenfunctions. This provides the proof of the completeness of a system of explicitly calculated Stokes eigenfunctions given by one of the authors in 1999, [14].

The Stokes Eigenvalue Problem on balls and annuli in three dimensions: Solutions with Poloidal and Toroidal Fields

TL;DR

The paper studies the Stokes eigenvalue problem in symmetric 3D domains (balls and spherical annuli) under Dirichlet boundary data. It develops a toroidal/poloidal decomposition using the Laplace-Beltrami operator on the sphere, leading to decoupled scalar problems for the toroidal potential and the poloidal potential , from which Stokes eigenfunctions are constructed. On the unit ball, eigenfunctions are given explicitly in terms of spherical harmonics and Bessel zeros, yielding eigenvalues (toroidal) and (poloidal) with multiplicity , and analogous determinants determine eigenvalues for annuli. The results establish the completeness of the explicit eigenfunction systems introduced earlier and provide a robust framework for spectral-Galerkin methods in incompressible flows on these highly symmetric domains.

Abstract

We consider the Stokes eigenvalue problem in open balls and open annuli in R3 with homogeneous Dirichlet boundary conditions. Using the frame of toroidal and poloidal fields we construct the othogonal decomposition of the Stokes eigenvalue problem in problems for toroidal and poloidal eigenfunctions. This provides the proof of the completeness of a system of explicitly calculated Stokes eigenfunctions given by one of the authors in 1999, [14].
Paper Structure (12 sections, 6 theorems, 75 equations)

This paper contains 12 sections, 6 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Notation and Basics
  3. Notation for vector fields
  4. Function spaces and auxiliarities
  5. Decomposition
  6. Basic ideas of decomposition
  7. The Theorem of decomposition
  8. The eigenvalue problem for the Stokes operator
  9. Preliminary theoretical results
  10. The governing equations for the scalar potentials of the Stokes eigenfunctions
  11. The Stokes eigenfunctions in the unit ball
  12. The Stokes eigenfunctions in open 3D-Annuli

Key Result

Lemma 1

Let $\Omega$ be the open unit ball $\Omega$ in ${\mathbb R}^{n}$ for $n\,\in\,{\mathbb N}$, ${\underline{x}}\,\in\,{\mathbb R}^{n}$ and $r$ the radius-coordinate in a spherical coordinate system. Then there exists no solenoidal vector function ${\underline{g}} \in (C^{1}(\Omega))^{n}$, which is only

Theorems & Definitions (27)

  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 1
  • Remark 4
  • Definition 2
  • Remark 5
  • Definition 3
  • Remark 6
  • Remark 7
  • ...and 17 more