An Eigenvalue Criterion for Stability of a Steady Navier–Stokes Flow in ${\mathbb{R}}^3$

Abstract.

We study the resolvent equation associated with a linear operator ${\mathcal{L}}$ arising from the linearized equation for perturbations of a steady Navier–Stokes flow ${\mathbf{U^*}}$. We derive estimates which, together with a stability criterion from [33], show that the stability of ${\mathbf{U^*}}$ (in the L²-norm) depends only on the position of the eigenvalues of ${\mathcal{L}}$, regardless the presence of the essential spectrum.