  This paper is devoted to the study of the low Mach number limit for the
isentropic Euler system with axisymmetric initial data without swirl. In the
first part of the paper we analyze the problem corresponding to the subcritical
regularities, that is $H^s$ {with $s>\frac52$.} Taking advantage of the
Strichartz estimates and using the special structure of the vorticity we show
that the {lifespan $T_\epsilon$} of the solutions is bounded below by
$\log\log\log\frac1\epsilon$, where $\epsilon$ denotes the Mach number.
Moreover, we prove that the incompressible parts converge to the solution of
the incompressible Euler system, when the parameter $\epsilon$ goes to zero. In
the second part of the paper we address the same problem but for the Besov
critical regularity $B_{2,1}^{\frac52}$. This case turns out to be more subtle
at least due to two facts. The first one is related to the Beale-Kato-Majda
criterion which is not known to be valid for rough regularities. The second one
concerns the critical aspect of the Strichartz estimate $L^1_TL^\infty$ for the
acoustic parts $(\nabla\Delta^{-1}\diver\vepsilon,\cepsilon)$: it scales in the
space variables like the space of the initial data.
