On the unimportance of continuous spectrum in the problem of stability of plane-parallel flow of perfect fluid: PMM vol. 38, n≗ 3, 1974, pp. 494–501

Abstract

It is shown that in the problem of stability of plane-parallel flows of perfect fluid only a discrete spectrum of eigenvalues exists. Previously this was established only in the case of monotonic velocity profile of the basic flow [1, 2]. Below a rigorous proof is given of this for any arbitrary profile. The problem of stability of a plane-parallel flow of perfect fluid reduces to the Rayleigh equation whose solution is sought in the form of a wave ϑ( z) · exp {iα (x − ct)}. The phase velocity c is the eigenvalue of this equation for conditions ϑ ( a) = ϑ ( b) = 0 (see [3]), and the set of such eigenvalues constitutes the discrete spectrum. If stability is taken to mean the limitedness of any (not only wave) paerturbations in time, the question arises whether the continuous spectrum of the problem existing besides the discrete one can produce instability (as a rule, a discrete spectrum contains only a finite number of points). Existence of continuous spectrum becomes evident, if the Rayleigh equation is written thus: ( u + u″ Δ −1) ψ = cψ, Δ = − d 2 dx 2 + α 2 , ψ = Δφ (0.1) where u is the velocity profile. The absolutely continuous operator u″ Δ −1 cannot alter the continuous spectrum of the operator of multiplication by u, which means that such spectrum occupies the entire segment [ u min , u max ]. It was shown in [1, 2] that only a discrete spectrum can generate instability, either exponential if c is not real, or power if there are multiple real eigenvalues. It was noted subsequently in [4] that only an outline of the proof is given in the short papers [1, 2], and an expanded proof by the same scheme is presented in it. However only the simplest case of a monotonie velocity profile u is fully investigated in all these papers. The same case was considered in [5], where it is shown that for a monotonie velocity profile the operator in the left-hand side of (0.1) is equivalent to the self-conjugate one for which spectral expansion exists. Hence the Cauchy problem can be solved by expanding in terms of the spectrum, and the realness of the continuous spectrum ensures the boundedness of the related part of the problem. The Laplace transformation used in [1, 2] is essentially a substitute for the general theorem on spectral expansion. Here, the method of [1, 2] is extended to the case of a nonmonotonic velocity profile, and it is shown that such extension does not alter the fundamental result. (The discrete spectrum is not analyzed here, since virtually all works on stability deal with it. In particular, the necessary and sufficient condition for the stability of a monotonic velocity profile was obtained in [6] and derived again in [5]).