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 (Formula presented.), when the perturbations are in the critical spaces (Formula presented.). More precisely, we introduce a new dynamical approach to prove the instability for some perturbation of size (Formula presented.) with any small (Formula presented.), which implies that (Formula presented.) 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.