Rate of the Enhanced Dissipation for the Two-jet Kolmogorov Type Flow on the Unit Sphere

Abstract

We study the enhanced dissipation for the two-jet Kolmogorov type flow which is a stationary solution to the Navier–Stokes equations on the two-dimensional unit sphere given by the zonal spherical harmonic function of degree two. Based on the pseudospectral bound method developed by Ibrahim, Maekawa, and Masmoudi [14] and a modified version of the Gearhart–Prüss type theorem shown by Wei [50], we derive an estimate for the resolvent of the linearized operator along the imaginary axis and show that a solution to the linearized equation rapidly decays at the rate \(O(e^{-c\sqrt{\nu }\,t})\) when the viscosity coefficient \(\nu \) is sufficiently small as in the case of the plane Kolmogorov flow.