On the steady-states of the 2D incompressible Euler equations

The vorticity of a two-dimensional perfect (inviscid, incompressible) fluid is transported by its flow. Thus, the space of vorticities is naturally "foliated". I will present a rigorous proof (under certain non-degeneracy conditions) that steady-states persist from one "orbit" of this "foliation" to nearby ones. (This type of stability result can be expected in view of the geometric interpretation of the incompressible Euler equations: the Euler flows are geodesics on an infinite-dimensional manifold equipped with a right-invariant metric.) In particular, I will focus on some analytical difficulties of the proof: the Nash-Moser inverse function theorem is used to overcome "loss of derivatives" of certain linear operators.