On Liapounoff s Method in the Theory of Stability of Laminar Fluid Flows

In [1] Zhou extended some Liapounoff's theorems of the theory of stability in the case of plane laminar fluid flows. In [2] Zhou and Li investigated the eigenvalue problem and expansion theorems associated with Orr-Sommerfeld equation, and obtained some new results. In this paper, based on the results of [1] and [2] we shall prove: (1) For the linearized problem the definition of stability according to the eigenvalues of Orr-Sommerfeld equation and that according to the perturbation energy are equivalent; (2) The method of linearization is admissible for the stability problem of plane laminar fluid flows for sufficiently small initial disturbance.