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 ${\omega }_0={\omega }_0^{\theta }{e}_{\theta }$ satisfies ${\omega }_0^{\theta }/r\in L^q$ for $q\in [3/2,3)$. The solutions constructed are Hölder continuous for spatial variables in $\overline{\Pi }$ if in addition that ${\omega }_0^{\theta }/r\in L^s$ 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 ${\omega }_0^{\theta }/r\in L^q$ 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 ${\omega }_0^{\theta }/r\in L^{\infty }$. The $L^2$-convergence in particular implies the energy equality for weak solutions.

37 pages. The title is changed from the previous version. Remarks 6.4 and new references are added