Abstract
Stability analysis of the rotating Bénard problem gives a spectral instability threshold of the purely conducting solution that can be expressed as a critical Rayleigh number R 2 depending on the Taylor number T 2. The definition of a functional which can be used to prove Lyapunov stability up to the threshold of spectral instability (optimal Lyapunov function) is an important step forward both, for a proof of nonlinear stability and for the investigation of the basin of attraction of the equilibrium.
In previous works a Lyapunov function was found, but its optimality could be proven only for small T 2. In this work we describe the reason why this happens, and provide a weaker definition of Lyapunov function which allows to prove that, for the linearized system, the spectral instability threshold is also the Lyapunov stability threshold for every value of T 2.
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Acknowledgements
This work has been partially supported by “Progetto Giovani Ricercatori 2013” of GNFM-INDAM. Both authors wish to thank the University of Catania and the University of Padova for financial and logistic support.
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Giacobbe, A., Mulone, G. Stability in the Rotating Bénard Problem and Its Optimal Lyapunov Functions. Acta Appl Math 132, 307–320 (2014). https://doi.org/10.1007/s10440-014-9905-0
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DOI: https://doi.org/10.1007/s10440-014-9905-0