Friedrichs inequalities and sharpened sufficient stability conditions of plane-parallel flows

From the standpoint of the linearized stability theory, two eigenvalue problems for the Orr–Sommerfeld equation with two groups of boundary conditions having a certain mechanical meaning are considered. On the basis of the integral relations method operating with quadratic functionals, the stability parameter, which is a real part of the spectral parameter, is estimated. The technique of the method involves the application of the Friedrichs inequality for various classes of complex-valued functions. Using the minimizing property of the first positive eigenvalues in the corresponding problems, the values of the constants in some Friedrichs inequalities are increased, which entails the strengthening of the stability sufficient integral estimates for plane-parallel shear flows in a plane layer.