Vanishing viscosity limits for axisymmetric flows with boundary

We construct global weak solutions of the Euler equations in an infinite cylinder $\mathrm{\Pi }=\{x\in {\mathbb{R}}^3\mathrm{\mid }{x}_h=(x_1,x_2),r=\mathrm{\mid }{x}_h\mathrm{\mid }<1\}$ for axisymmetric initial data without swirl when initial vorticity ${\omega }_0={\omega }_0^{\theta }{e}_{\theta }$ 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 $\bar{\mathrm{\Pi }}$ if in addition that ${\omega }_0^{\theta }\mathrm{/}r\in L^s$ for $s\in (3,\mathrm{\infty })$ and unique if $s=\mathrm{\infty }$. 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 $p\in [3,\infty)$\mathrm{.}Itisalsoshownthattheenergydissipationtendstozeroif$\omega^{\theta}{0}/r\in L^{q}$for$q\in [3/2,2]$, and Navier-Stokes flows converge to Euler flow in $L^2$ locally uniformly for $t\in [0,\mathrm{\infty })$ if additionally ${\omega }_0^{\theta }\mathrm{/}r\in L^{\mathrm{\infty }}$. The $L^2$-convergence in particular implies the energy equality for weak solutions.