Vanishing viscosity limits for axisymmetric flows with boundary

We construct global weak solutions of the Euler equations in an infinite cylinder = {x E R3 = (x1, x2), r = < 1} for axisymmetric initial data without swirl when initial vorticity wo = wgeo satisfies wg/r E Lq for q E [3/2, 3). The solutions constructed are Holder continuous for spatial variables in 7 if in addition that wg/r E LS for s E (3, Do) and unique if s = Do. The proof is by a vanishing viscosity method. We show that the Navier-Stokes equations subject to the Neumann boundary condition is globally well-posed for axisymmetric data without swirl in LT for all p E [3, Do). It is also shown that the energy dissipation tends to zero if wg/r E Lq for q E [3/2,2], and Navier-Stokes flows converge to Euler flow in L2 locally uniformly for t E [0, Do) if additionally wg/r E L. The L2 -convergence in particular implies the energy equality for weak solutions. (C) 2020 Elsevier Masson SAS. All rights reserved.