Circular flows for the Euler equations in two-dimensional annular domains, and related free boundary problems

Abstract

In this paper, we first consider steady Euler flows in two-dimensional bounded annuli, as well as in complements of disks, in punctured disks and in the punctured plane. We prove that if the flow does not have any stagnation point and satisfies rigid wall boundary conditions together with further conditions at infinity in the case of unbounded domains and at the center in the case of punctured domains, then the flow is circular, in the sense that the streamlines are concentric circles. In other words, the flow then inherits the radial symmetry of the domain. We then show two classification results for the steady Euler equations in simply or doubly connected bounded domains with free boundaries. Here, the flows are further assumed to have constant norm on each connected component of the boundary, and the domains are then proved to be disks or annuli. On the one hand, the proofs use ODE and PDE arguments to establish some geometric properties of the trajectories of the flow and the orthogonal trajectories of the gradient of the stream function. On the other hand, we also show some comparison results of independent interest for a derived semilinear elliptic equation satisfied by the stream function. These last results, which are based on the method of moving planes, adapted here to some almost circular domains located between some streamlines of the flow, lead with a limiting argument to the radial symmetry of the stream function and the streamlines of the flow.