This PhD thesis studies the Euler equations, a system of PDEs which govern the motion of perfect (incompressible and inviscid) fluid flows. In two spatial dimensions two recently conjectured statements describe the long time behaviour of homogeneous Euler solutions. The first conjecture states that generic initial data relax under the 2d Euler evolution to end-states with strictly less information, so that some sort of irreversible mixing occurs. The second conjecture predicts that these end-states cannot relax any further and that they are formed by stationary states, periodic or quasi-periodic configurations, among others. The goal of the thesis is two-fold. On one hand, we aim to describe the end-states picture by providing further information on the set of stationary states. On the other hand, we intend to study the relaxation mechanisms of the homogeneous Euler equations.
To this end, the first part of the thesis is centred on studying the structure of the set of stationary solutions to the Euler equations nearby distinguished steady state solutions. As a main result, we show the streamline richness of the stationary states of the Euler equations in this perturbative regime: we exhibit both examples where solutions to the Euler equations have geometrically different streamlines to those of neighbouring configurations, and also examples where the streamline geometry is identical for a whole open set of stationary configurations.
The second part of the thesis is concerned with a relaxation mechanism present in the 2d Euler equa- tions: around specific steady states, the velocity field has been shown to relax (decay in time) to nearby stationary configuration. This phenomenon is called inviscid damping and its relevance for the long-time dynamics is paramount. Our contributions investigate further the mechanisms behind inviscid damping and show that these velocity relaxations are (linearly) present in other inviscid fluid models.