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

TL;DR

This paper analyzes the Stokes eigenvalue problem in three-dimensional balls and annuli, constructing explicit solutions using toroidal and poloidal fields, and proving the completeness of these solutions.

Contribution

It introduces a new explicit construction of Stokes eigenfunctions in 3D domains and proves their completeness using toroidal and poloidal field decompositions.

Findings

Explicit eigenfunctions constructed for 3D balls and annuli.

Proof of completeness of the eigenfunction system.

Framework for analyzing Stokes problems with toroidal and poloidal fields.

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].