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

Abstract

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 D0, 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 D0 in Ck,α, k ≥ 3 and 0 < α < 1. We consider first the case where D0 is a periodic channel and the flow in D0 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 D0. 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.