Instability of shear flows with neutral embedded eigenvalues

TL;DR

This paper investigates the linear stability of monotone shear flows, revealing that neutral embedded eigenvalues can cause unbounded growth in solutions, especially when multiple eigenvalues are involved, due to the non-normality of the Rayleigh operator.

Contribution

It demonstrates the instability caused by neutral embedded eigenvalues in shear flows and constructs explicit solutions showing linear growth when eigenvalues are multiple.

Findings

Solutions can grow arbitrarily large in $L^ abla$ and $L^2$ norms.

Multiple embedded eigenvalues lead to stronger, linear-in-time growth.

Instability stems from the non-normality of the Rayleigh operator.

Abstract

We study the linear stability of a class of monotone shear flows. When the associated Rayleigh operator possesses a neutral embedded eigenvalue, we show that solutions of the linearized system may exhibit arbitrarily large growth in both the $L^\infty$ and $L^2$ norms. Moreover, when the embedded eigenvalue is multiple, we prove that the instability becomes stronger and explicitly construct solutions that grow linearly in time. This instability originates from the non-normality of the Rayleigh operator.