Vanishing viscosity limits for axisymmetric flows with boundary

Abstract

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.