Spectral analysis of a Stokes-type operator arising from flow around a rotating body

Abstract

We consider the spectrum of the Stokes operator with and without rotation effect for the whole space and exterior domains in $L^q$-spaces. Based on similar results for the Dirichlet-Laplacian on $mathbf{R}^n$, $n \geq 2$, we prove in the whole space case that the spectrum as a set in $\mathbf{c}$ does not change with $q \in (1,\infty)$, but it changes its type from the residual to the continuous or to the point spectrum with $q$. The results for exterior domains are less complete, but the spectrum of the Stokes operator with rotation mainly is an essential one, consisting of infinitely many equidistant parallel half-lines in the left complex half-plane. The tools are strongly based on Fourier analysis in the whole space case and on stability properties of the essential spectrum for exterior domains.