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

WeinvestigatethelinearstabilityofshearsneartheCouetteflowforaclassof2Dincompressible stably stratified fluids. Our main result consists of nearly optimal decay rates for perturbations of stationary states whose velocities are monotone shear flows (U (y), 0) and have an exponential density profile. In the case of the Couette flow U(y) = y, we recover the rates predicted by Hartman in 1975, by adopting an explicit point-wise approach in frequency space. As a by-product, this implies optimal decay rates as well as Lyapunov instability in L2 for the vorticity. For the previously unexplored case of more general shear flows close to Couette, the inviscid damping results follow by a weighted energy estimate. Each outcome concerning the stably stratified regime applies to the Boussinesq equations as well. Remarkably, our results hold under the celebrated Miles-Howard criterion for stratified fluids.