Spectral portraits and the resolvent growth of a model problem associated with the Orr--Sommerfeld equation

TL;DR

This paper analyzes the spectral behavior of a model operator related to the Orr--Sommerfeld equation, showing that as viscosity vanishes, the spectra converge to a limit graph and the resolvent exhibits exponential growth in small viscosity regimes.

Contribution

It establishes the coincidence of spectral accumulation points and eigenvalue distribution between the model and Orr--Sommerfeld operators as viscosity approaches zero.

Findings

Spectral accumulation points form a limit spectral graph.

Eigenvalue counting functions align along the spectral graph.

Resolvent norm grows exponentially with decreasing viscosity.

Abstract

A model problem of the form -i\epsilon y''+q(x)y=\lambda y, y(-1)=y(1)=0, is associated with well-known in hydrodynamics Orr--Sommerfeld operator. Here (\lambda) is the spectral parameter, (\epsilon) is the small parameter which is proportional to the viscocity of the liquid and to the reciprocal of the Reynolds number, and (q(x)) is the velocity of the stationary flow of the liquid in the channel (|x|\leqslant 1). We study the behaviour of the spectrum of the corresponding model operator as (\epsilon\to 0) with monotonous analytic functions. We assert that the sets of the accumulation points of the spectra (the limit spectral graphs) of the model and the corresponding Orr--Sommerfeld operators coincide as well as the main terms of the counting eigenvalue functions along the curves of the graphs. We prove the estimate from below for the resolvent of the operator (L(\epsilon)) associated with the model problem. It turns out that the resolvent grows exponentially with respect to (\epsilon) provided that the spectral parameter varies at some compacts belonging to the numerical range of the operator.