Nonlinear spectral instability of steady-state flow of a viscous liquid past a rotating obstacle

TL;DR

This paper demonstrates that steady-state viscous flow past a rotating obstacle is nonlinearly unstable when the linearized operator has spectrum with positive real part, using novel methods beyond traditional spectral instability analysis.

Contribution

It establishes nonlinear instability criteria for Navier-Stokes flow past a rotating obstacle based on spectral properties of the linearized operator, introducing new analytical techniques.

Findings

Nonlinear instability occurs if the linear operator's spectrum has positive real part.

The growth bound of the semigroup equals the spectral bound of the operator.

The result applies to flows past rotating bodies with unbounded nonlinear operators.

Abstract

We show that a steady-state solution ${\bf U}$ to the system of equations of a Navier-Stokes flow past a rotating body is nonlinearly unstable if the associated linear operator $\cal L$ has a part of the spectrum in the half-plane $\{\lambda\in\mathbb C;\ {\rm Re}\, \lambda>0\}$. Our result does not follow from known methods, %on spectral instability, mainly because the basic nonlinear operator is not bounded in the same space in which the instability is studied. As an auxiliary result of independent interest, we also show that the uniform growth bound of the $C_0$--semigroup ${\rm e}^{{\cal L} t}$ is equal to the spectral bound of operator $\mathcal L$.