  We construct global weak solutions of the Euler equations in an infinite
cylinder $\Pi=\{x\in \mathbb{R}^{3}\ |\ x_h=(x_1,x_2),\ r=|x_h|<1\}$ for
axisymmetric initial data without swirl when initial vorticity
$\omega_{0}=\omega^{\theta}_{0}e_{\theta}$ satisfies $\omega^{\theta}_{0}/r\in
L^{q}$ for $q\in [3/2,3)$. The solutions constructed are H\"older continuous
for spatial variables in $\overline{\Pi}$ if in addition that
$\omega^{\theta}_{0}/r\in L^{s}$ for $s\in (3,\infty)$ and unique if
$s=\infty$. 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 $L^{p}$ for all $p\in
[3,\infty)$. It is also shown that the energy dissipation tends to zero if
$\omega^{\theta}_{0}/r\in L^{q}$ for $q\in [3/2,2]$, and Navier-Stokes flows
converge to Euler flow in $L^{2}$ locally uniformly for $t\in [0,\infty)$ if
additionally $\omega^{\theta}_{0}/r\in L^{\infty}$. The $L^{2}$-convergence in
particular implies the energy equality for weak solutions.
