Asymptotic behaviour of solutions of linearized Navier Stokes equations in the long waves regime

The aim of this paper is to describe the long time behavior of solutions of linearized Navier Stokes equations near a concave shear layer profile in the long waves regime, namely for small horizontal Fourier variable $\alpha$, when the viscosity $\nu$ vanishes. We show that the solutions converge exponentially to $0$, except in some range of $\alpha$, namely for $\nu^{1/4} \lesssim |\alpha| \lesssim \nu^{1/6}$, where there exists one unique unstable mode, with an associated eigenvalue $\lambda$, such that $\Re \lambda$ is of order $\nu^{1/4}$. In this regime we give a complete description of the solutions of linearized Navier Stokes equations as the sum of the projection over the unique exponentially growing mode and of an exponentially decaying term. The study of this linear instability is a key point in the study of the nonlinear instability of Prandtl bounday layers and of shear layer profiles.