Existence and Stability of Sadovskii Vortex Patch: an odd-symmetric touching pair of uniform vortices

Abstract

As a traveling solution of the two-dimensional incompressible Euler equations, the Sadovskii vortex patch takes the form of a counter-rotating pair of vortex patches that are in contact. This model was suggested by Sadovskii [J. Appl. Math. Mech., 1971] and has since attracted considerable attention due to its significance to the inviscid limit of planar flows via the Prandtl–Batchelor theory and its role as an asymptotic state of the dynamics of vortex rings. In this paper, we establish the existence and stability of the Sadovskii vortex patch. First, we show that a Sadovskii vortex patch arises as a maximizer of the kinetic energy under an exact impulse condition. By analyzing the fluid velocity along the symmetry axis and its relation to the vorticity of a dipole, we verify that two patches in the dipole are in contact. Second, we construct a subcollection of such energy maximizers that is stable in the following sense: if an initial vorticity is sufficiently close to a maximizer, then the corresponding solution remains close to this collection up to a translation and travels at a similar speed to the maximizers. This is achieved via a concentration–compactness argument together with the conservation of impulse, circulation, and kinetic energy in time. Furthermore, using uniform estimates of energy maximizers and a shift estimate obtained by estimating the center of mass of the solution, we show that the solution keeps its touching structure uniformly in time.