The Tan $2 \Theta$-Theorem in Fluid Dynamics

TL;DR

This paper explores the mathematical relationship between the generalized Reynolds number and spectral properties of the Stokes operator in fluid dynamics, providing explicit spectral bounds and inequalities.

Contribution

It establishes a novel connection between fluid dynamics parameters and spectral theory, including explicit spectral bounds for the Stokes operator.

Findings

Relation between Reynolds number and spectral subspace rotation

Explicit evaluation of the negative spectrum of the Stokes operator

Sharp inequality linking spectral gap and spectrum distance

Abstract

We show that the generalized Reynolds number (in fluid dynamics) introduced by Ladyzhenskaya is closely related to the rotation of the positive spectral subspace of the Stokes block-operator in the underlying Hilbert space. We also explicitly evaluate the bottom of the negative spectrum of the Stokes operator and prove a sharp inequality relating the distance from the bottom of its spectrum to the origin and the length of the first positive gap.