Incompressible limit for the 2D isentropic Euler system with critical initial data

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.