| Contents |
| The stability of a basic motion plays a central role in fluid dynamics. There are several methods for investigating stability.
The spectral methods study the real parts of the eigenvalues of the linear operator of the system and give sufficient conditions of instability.
The Lyapunov second method introduces suitable energy-norms (or Lyapunov functions) and gives sufficient conditions for nonlinear stability of the flow.
If the linear operator is symmetric with respect to an energy-norm, the linear instability threshold coincides with the energy nonlinear stability.
Many fluid systems show stabilizing effects mainly due to forces which give skew-symmetric contributions to linear operator (rotation, chemical concentration, magnetic fields). The energy norm is not sensitive to such effects.
Here we prove the coincidence of linear and nonlinear thresholds when the stabilizing effects are present. The coincidence is obtained with the introduction of optimal Lyapunov functions or norms. To this end, we use and improve the reduction method introduced in JMAA, 342, pp.461--476. We show, in particular cases (motion of a protoplanetary disk and some B\'{e}nard problems) that the definition of optimal Lyapunov functions does not give coincidence for all physical parameters, and needs to be weakened to prove coincidence and asymptotic stability. |
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