A dynamical approach to the study of instability near Couette flow

In this paper, we obtain the optimal instability threshold of the Couette flow for Navier-Stokes equations with small viscosity nu > 0, when the perturbations are in the critical spaces (HxLy2)-L-1. More precisely, we introduce a new dynamical approach to prove the instability for some perturbation of size nu(1/2-delta 0) with any small delta(0 )> 0, which implies that nu(1/2) is the sharp stability threshold. In our method, we prove a transient exponential growth without referring to eigenvalue or pseudo-spectrum. As an application, for the linearized Euler equations around shear flows that are near the Couette flow, we provide a new tool to prove the existence of growing modes for the corresponding Rayleigh operator and give a precise location of the eigenvalues.