The Axisymmetric Euler Equations with Vorticity in Borderline Spaces of Besov Type

Abstract

Borderline spaces of Besov type consist of tempered distributions satisfying the property that the partial sums of their \(B^0_{\infty ,1}\)-norm diverge in a controlled way. Misha Vishik established uniqueness of solutions to the two and three-dimensional incompressible Euler equations with vorticity whose \(B^0_{\infty ,1}\) partial sums diverge roughly at a rate of \(N\log N\). In two dimensions, he also established conditions on the initial data for which solutions in his uniqueness class exist. In this paper, we extend existence results of Vishik to the three-dimensional Euler equations with axisymmetric velocity. We also study the inviscid limit of solutions of the Navier–Stokes equations with initial vorticity in these Besov type spaces.