Abstract
We study the stability of the Kolmogorov flows which are stationary solutions to the two-dimensional Navier–Stokes equations in the presence of the shear external force. We establish the linear stability estimate when the viscosity coefficient \(\nu \) is sufficiently small, where the enhanced dissipation is rigorously verified in the time scale \(O(\nu ^{-\frac{1}{2}})\) for solutions to the linearized problem, which has been numerically conjectured and is much shorter than the usual viscous time scale \(O(\nu ^{-1})\). Our approach is based on the detailed analysis for the resolvent problem. We also provide the abstract framework which is applicable to the resolvent estimate for the Kolmogorov flows.
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Acknowledgements
The first author is partially supported by NSERC Discovery Grant # 371637-2014, and also acknowledges the kind hospitality of the New York University in Abu Dhabi. The second author is partially supported by JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers, ‘Development of Concentrated Mathematical Center Linking to Wisdom of the Next Generation’, which is organized by Mathematical Institute of Tohoku University. The third author is partially supported by the NSF Grant DMS-1716466.
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Ibrahim, S., Maekawa, Y. & Masmoudi, N. On Pseudospectral Bound for Non-selfadjoint Operators and Its Application to Stability of Kolmogorov Flows. Ann. PDE 5, 14 (2019). https://doi.org/10.1007/s40818-019-0070-7
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DOI: https://doi.org/10.1007/s40818-019-0070-7