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

Abstract

We show that a steady-state solution $\mathbf{U}$ to the system of equations of a Navier-Stokes flow past a rotating body is nonlinearly unstable if the associated linear operator $\mathcal{L}$ has a part of the spectrum in the half-plane $\{\lambda \in \mathbb{C};\mathrm{Re}\lambda >0\}$. 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 ${\mathrm{e}}^{\mathcal{L}t}$ is equal to the spectral bound of operator $\mathcal{L}$.