Persistence of steady flows of a two-dimensional perfect fluid in deformed domains

The robustness of steady solutions of the Euler equations for two-dimensional, incompressible and inviscid fluids is examined by studying their persistence for small deformations of the fluid-domain boundary. Starting with a given steady flow in a domain D-0, we consider the class of flows in a deformed domain D that can be obtained by rearrangement of the vorticity by an area-preserving diffeomorphism.

We provide conditions for the existence and (local) uniqueness of a steady flow in this class when D is sufficiently close to D-0 in C-k,C-alpha k >= 3 and 0 < alpha < 1. We consider first the case where D-0 is a periodic channel and the flow in D-0 is parallel and show that the existence and uniqueness are ensured for flows with non-vanishing velocity. We then consider the case of smooth steady flows in a more general domain D-0. The persistence of the stability of steady flows established using the energy-Casimir or, in the parallel case, the energy-Casimir-momentum method, is also examined. A numerical example of a steady flow obtained by deforming a parallel flow is presented.