Instability of unidirectional flows for the 2D Navier-Stokes equations and related $α$-models

Abstract

We study instability of unidirectional flows for the linearized 2D Navier-Stokes equations on the torus. Unidirectional flows are steady states whose vorticity is given by Fourier modes corresponding to a single vector p ∈ Z 2 . Using Fourier series and a geometric decomposition allows us to decompose the linearized operator L B acting on the space 2 (Z 2 ) about this steady state as a direct sum of linear operators L B,q acting on 2 (Z) parametrized by some vectors q ∈ Z 2 . Using the method of continued fractions we prove that the linearized operator L B,q about this steady state has an eigenvalue with positive real part thereby implying exponential instability of the linearized equations about this steady state. We further obtain a characterization of unstable eigenvalues of L B,q in terms of the zeros of a perturbation determinant (Fredholm determinant) associated with a trace class operator K λ . We also extend our main instability result to cover regularized variants (involving a parameter α > 0) of the Navier-Stokes equations, namely the second grade fluid model, the Navier-Stokes-α and the Navier-Stokes-Voigt models.