Linear inviscid damping for shear flows near Couette in the 2D stably stratified regime

We investigate the linear stability of shears near Couette flow for a class 2D incompressible stably stratified fluids. Our main result consists nearly optimal decay rates perturbations stationary states whose velocities are monotone shear flows $(U(y),0)$ and have an exponential density profile. In case $U(y)=y$, we recover predicted by Hartman in 1975, adopting explicit point-wise approach frequency space. As by-product, this implies as well Lyapunov instability $L^2$ vorticity. For previously unexplored more general close to Couette, inviscid damping results follow weighted energy estimate. Each outcome concerning regime applies Boussinesq equations well. Remarkably, our hold under celebrated Miles-Howard criterion