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 $H_x^1{L}_y^2$. More precisely, we introduce a new dynamical approach to prove the instability for some perturbation of size ${\nu }^{\frac{1}{2}-{\delta }_0}$ with any small ${\delta }_0>0$, which implies that ${\nu }^{\frac{1}{2}}$ 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.