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

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