Spectral instability of symmetric shear flows in a two-dimensional channel

Abstract

This paper concerns spectral instability of shear flows in the incompressible Navier-Stokes equations with sufficiently large Reynolds number: $R\to \infty$. It is well-documented in the physical literature, going back to Heisenberg, C.C. Lin, Tollmien, Drazin and Reid, that generic plane shear profiles other than the linear Couette flow are linearly unstable for sufficiently large Reynolds number. In this work, we provide a complete mathematical proof of these physical results. In the case of a symmetric channel flow, our analysis gives exact unstable eigenvalues and eigenfunctions, showing that the solution could grow slowly at the rate of $e^{t/\sqrt {\alpha R}}$, where $\alpha$ is the small spatial frequency that remains between lower and upper marginal stability curves: $\alpha_\mathrm{low}(R) \approx R^{-1/7}$ and $\alpha_\mathrm{up}(R) \approx R^{-1/11}$. We introduce a new, operator-based approach, which avoids to deal with matching inner and outer asymptotic expansions, but instead involves a careful study of singularity in the critical layers by deriving pointwise bounds on the Green function of the corresponding Rayleigh and Airy operators.