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Stability in the Rotating Bénard Problem and Its Optimal Lyapunov Functions

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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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References

  1. Chandrasekhar, S.: Hydrodynamics and Hydromagnetic Stability. Oxford University Press, London (1961)

    Google Scholar 

  2. Koschmieder, E.L.: Bénard Cells and Taylor Vortices. Cambridge University Press, Cambridge (1993)

    MATH  Google Scholar 

  3. Daniel, D.J.: Stability of Fluid Motions I. Springer, Berlin (1976)

    MATH  Google Scholar 

  4. Daniel, D.J.: Stability of Fluid Motions II. Springer, Berlin (1976)

    MATH  Google Scholar 

  5. Rionero, S., Flavin, J.N.: Qualitative Estimates for Partial Differential Equations: An Introduction. CRC Press, Boca Raton (1995)

    Google Scholar 

  6. Straughan, B.: The Energy Method, Stability, and Nonlinear Convection. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  7. Galdi, G.P., Straughan, B.: A nonlinear analysis of the stabilizing effect of rotation in the Bénard problem. Proc. R. Soc. A 402, 257–283 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  8. Mulone, G., Rionero, S.: On the nonlinear stability of the rotating Bénard problem via the Lyapunov direct method. J. Math. Anal. Appl. 144, 109–127 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  9. Galdi, G.P., Padula, M.: A new approach to energy theory in the stability of fluid motion. Arch. Ration. Mech. Anal. 110, 187–286 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  10. Lombardo, S., Mulone, G., Trovato, M.: Nonlinear stability in reaction-diffusion systems via optimal Lyapunov functions. J. Math. Anal. Appl. 342, 461–476 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Mulone, G.: Stabilizing effects in dynamical systems: linear and nonlinear stability conditions. Far East J. Appl. Math. 15, 117–135 (2004)

    MATH  MathSciNet  Google Scholar 

  12. Mulone, G., Straughan, B.: An operative method to obtain necessary and sufficient stability conditions for double diffusive convection in porous media. Z. Angew. Math. Phys. 86, 507–520 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kaiser, R., Xu, L.X.: Nonlinear stability of the rotating Benard problem, the case Pr=1. Nonlinear Differ. Equ. Appl. 5, 283–307 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  14. Straughan, B.: Nonlinear stability for a simple model of a protoplanetary disc. Nonlinear Anal.: Real World Appl. (2013)

  15. Falsaperla, P., Mulone, G.: Some stability results in the rotating Benard problem. In: STAMM 2008, pp. 1–7 (2008)

    Google Scholar 

  16. Falsaperla, P., Mulone, G.: Stability in the rotating Benard problem with Newton-Robin and fixed heat flux boundary conditions. Mech. Res. Commun. 37, 122–128 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. Falsaperla, P., Giacobbe, A., Mulone, G.: Double diffusion in rotating porous media under general boundary conditions. Int. J. Heat Mass Transf. 55, 2412–2419 (2012)

    Article  Google Scholar 

  18. Falsaperla, P., Giacobbe, A., Mulone, G.: Some results in the nonlinear stability for rotating Bénard problem with rigid boundary condition. Atti Accad. Pelorit. Pericol. 91, 1–10 (2013)

    MathSciNet  Google Scholar 

  19. Lombardo, S., Mulone, G.: New stability results of a magnetic field with Hall, ion-slip and velocity shear effects (2013, submitted)

  20. Falsaperla, P., Giacobbe, A., Mulone, G.: Does symmetry of the operator of a dynamical system help stability? Acta Appl. Math. 122, 239–253 (2012)

    MATH  MathSciNet  Google Scholar 

  21. Capone, F., Gentile, M.: Nonlinear stability analysis of the Bénard problem for fluids with a convex nonincreasing temperature depending viscosity. Contin. Mech. Thermodyn. 7, 297–309 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  22. Gouin, H., Muracchini, A., Ruggeri, T.: On the Muller paradox for thermal-incompressible media. Contin. Mech. Thermodyn. 24, 505–513 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kloeden, P., Wells, R.: An explicit example of Hopf bifurcation in fluid mechanics. Proc. R. Soc. A 390, 293–320 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  24. Serrin, J.: On the stability of viscous fluid motions. Arch. Ration. Mech. Anal. 3, 1–13 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  25. Rionero, S.: Metodi variazionali per la stabilità asintotica in media in magnetoidrodinamica. Ann. Mat. Pura Appl. Ser. 4 78, 339–364 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  26. Falsaperla, P., Giacobbe, A.: Marginal regions for the solute Bénard problem with many types of boundary conditions. Int. J. Eng. Sci. 57, 11–23 (2012)

    Article  MathSciNet  Google Scholar 

  27. Sattinger, D.H.: The mathematical problem of hydrodynamic stability. J. Math. Mech. 19, 797–817 (1970)

    MATH  MathSciNet  Google Scholar 

  28. Mulone, G., Straughan, B.: Nonlinear stability for diffusion models in biology. SIAM J. Appl. Math. 69, 1739 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  29. Mulone, G., Rionero, S.: The rotating Benard problem: new stability results for any Prandtl and Taylor numbers. Contin. Mech. Thermodyn. 9, 347–363 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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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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Correspondence to Andrea Giacobbe.

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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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