  We study vanishing viscosity solutions to the axisymmetric Euler equations
with (relative) vorticity in $L^p$ with $p>1$. We show that these solutions
satisfy the corresponding vorticity equations in the sense of renormalized
solutions. Moreover, we show that the kinetic energy is preserved provided that
$p>3/2$ and the vorticity is nonnegative and has finite second moments.
