Rate of the enhanced dissipation for the two-jet Kolmogorov type flow on the unit sphere. (English) Zbl 1493.35069
Summary: 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 S. Ibrahim et al. [Ann. PDE 5, No. 2, Paper No. 14, 84 p. (2019; Zbl 1439.35370)] and a modified version of the Gearhart-Prüss type theorem shown by D. Wei [Sci. China, Math. 64, No. 3, 507–518 (2021; Zbl 1464.35260)], 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.
MSC:
| 35Q30 | Navier-Stokes equations |
| 35R01 | PDEs on manifolds |
| 47A10 | Spectrum, resolvent |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
Software:
DLMFReferences:
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