Instability of Unidirectional Flows for the 2D Navier–Stokes Equations and Related $$\alpha $$-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 \({\mathbf {p}} \in {\mathbb {Z}}^{2}\). Using Fourier series and a geometric decomposition allows us to decompose the linearized operator \(L_{B}\) acting on the space \(\ell ^{2}({\mathbb {Z}}^{2})\) about this steady state as a direct sum of linear operators \(L_{B,{\mathbf {q}}}\) acting on \(\ell ^{2}({\mathbb {Z}})\) parametrized by some vectors \({\mathbf {q}}\in {\mathbb {Z}}^2\). Using the method of continued fractions we prove that the linearized operator \(L_{B,{\mathbf {q}}}\) 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 \(L_{B,{\mathbf {q}}}\) in terms of the zeros of a perturbation determinant (Fredholm determinant) associated with a trace class operator \(K_{\lambda }\). We also extend our main instability result to cover regularized variants (involving a parameter \(\alpha >0\)) of the Navier–Stokes equations, namely the second grade fluid model, the Navier–Stokes-\(\alpha \) and the Navier–Stokes–Voigt models.