Convergence of the Vanishing Viscosity Approximation for Superpositions of Confined Eddies

Abstract

A confined eddy is a circularly symmetric flow with vorticity of compact support and zero net circulation. Confined eddies with disjoint supports can be superimposed to generate stationary weak solutions of the two-dimensional incompressible inviscid Euler equations. In this work, we consider the unique weak solution of the two-dimensional incompressible Navier–Stokes equations having a disjoint superposition of very singular confined eddies as the initial datum. We prove the convergence of these weak solutions back to the initial configuration, as the Reynolds number goes to infinity. This implies that the stationary superposition of confined eddies with disjoint supports is the unique physically correct weak solution of the two-dimensional incompressible Euler equations.