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Journal of Spectral Theory

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Volume 9, Issue 4, 2019, pp. 1431–1457
DOI: 10.4171/JST/282

Published online: 2019-09-12

The Tan $2 \Theta$ Theorem in fluid dynamics

Luka Grubišić[1], Vadim Kostrykin[2], Konstantin A. Makarov[3], Stephan Schmitz[4] and Krešimir Veselić[5]

(1) University of Zagreb, Croatia
(2) Johannes Gutenberg-Universität Mainz, Germany
(3) University of Missouri, Columbia, USA
(4) Universität Koblenz-Landau, Germany
(5) Fernuniversität Hagen, Germany

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.

Keywords: Navier–Stokes equation, Stokes operator, Reynolds number, rotation of subspaces, quadratic forms, quadratic numerical range

Grubišić Luka, Kostrykin Vadim, Makarov Konstantin, Schmitz Stephan, Veselić Krešimir: The Tan $2 \Theta$ Theorem in fluid dynamics. J. Spectr. Theory 9 (2019), 1431-1457. doi: 10.4171/JST/282