Spectral instability of characteristic boundary layer flows

In this paper, we construct growing modes of the linearized Navier-Stokes equations about generic stationary shear flows of the boundary layer type in a regime of sufficiently large Reynolds number: $R \to \infty$. Notably, the shear profiles are allowed to be linearly stable at the infinite Reynolds number limit, and so the instability presented is purely due to the presence of viscosity. The formal construction of approximate modes is well-documented in physics literature, going back to the work of Heisenberg, C.C. Lin, Tollmien, Drazin and Reid, but a rigorous construction requires delicate mathematical details, involving for instance a treatment of primitive Airy functions and singular solutions. Our analysis gives exact unstable eigenvalues and eigenfunctions, showing that the solution could grow slowly at the rate of $e^{t/\sqrt {R}}$. A new, operator-based approach is introduced, avoiding to deal with matching inner and outer asymptotic expansions, but instead involving a careful study of singularity in the critical layers by deriving pointwise bounds on the Green function of the corresponding Rayleigh and Airy operators.