  We study in this paper the low Mach number limit for the 2d isentropic Euler
system with ill-prepared initial data belonging to the critical Besov space
$B_{2,1}^2$. By combining Strichartz estimates with the special structure of
the vorticity we prove that the lifespan goes to infinity as the Mach number
goes to zero. We also prove the strong convergence in the space of the initial
data $B_{2,1}^2$ of the incompressible parts to the solution of the
incompressible Euler system. There are at least two main difficulties: the
first one concerns the Beale-Kato-Majda criterion which is not known to work
for rough regularities. However, the second one is related to the critical
aspect of the quantity $\Vert(\textnormal{div}\,v_{\varepsilon},\nabla
c_{\varepsilon})\Vert_{L_{T}^{1}L^\infty}$ which has the scale of $B_{2,1}^2$
in the space variable.
