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.

31 pages, 2 figures. This paper extends the results of arXiv:1901.01367 on instability of unidirectional flows for the 2D Euler and $\alpha$-Euler equations to Navier-Stokes equations and related viscous $\alpha$-models. See also arXiv:1802.01813 for a Fredholm determinant characterization of unstable eigenvalues of the linearized Euler operator similar in spirit to the current paper