Linear stability analysis of stationary Euler flows for passive scalars and inhomogeneous fluids

This thesis is concerned with the study of linear stability properties of some particular fluid flows, within the framework of quantitative hydrodynamic stability. In the last decade, the problem has received a lot of attention thanks to the introduction of new analytical techniques particularly useful to tackle classical problems that date back to the end of the nineteen century. The purpose of this thesis is threefold. We first investigate the asymptotic decay properties of a passive scalar driven by a vortex with a power-law velocity field in the whole plane. In particular, we quantify the enhanced dissipation mechanism caused by fluid mixing, namely, the transfer of energy from large to small scales due to transport. We provide sharp (up to a logarithmic correction) bounds on the dissipation time-scales, which are faster than the standard diffusive one. The rest of the thesis is dedicated to the study of the stability of parallel flows in 2D inhomogeneous fluids. The second problem we deal with concerns a linear stability analysis for the Couette flow with constant density in an isentropic compressible fluid. We consider both the inviscid and the viscous case. In the inviscid case, we give the first rigorous mathematical justification to a Lyapunov instability mechanism previously addressed in the physics literature. More precisely, we show that the L^2 norms of the density and the irrotational component of the velocity field grow as t^(1/2) Instead, the solenoidal component of the velocity strongly converges to zero in L^2, a mechanism known as inviscid damping. In the viscous case, we present the first enhanced dissipation result for an inhomogeneous fluid. The exponential decay become effective on a time-scale O( u^(-1/3)), with u being proportional to the inverse of the Reynolds number, and there is a large transient growth of order O( u^(-1/6)) caused by the inviscid instability. The estimates are valid also in a large Mach number regime. We finally study the stability of a particular class of stratified shear flows. We consider an inhomogeneous, inviscid and incompressible fluid under the action of gravity, near shear flows close to Couette with an exponentially stratified density profile. Under the Miles-Howard criterion, we prove the inviscid damping for the velocity field and the (scaled) density. The decay rates are slower with respect to the classical homogeneous case without gravity. This is due to a Lyapunov instability mechanism for the vorticity that we characterize in the Couette case.