  The present paper is the continuation of work [14], devoted to the study of
an inviscid zero-Mach number system in the framework of \emph{endpoint} Besov
spaces of type $B^s_{\infty,r}(\mathbb{R}^d)$, $r\in [1,\infty]$, $d\geq 2$,
which can be embedded in the Lipschitz class $C^{0,1}$. In particular, the
largest case $B^1_{\infty,1}$ and the case of H\"older spaces $C^{1,\alpha}$
are permitted.
  The local in time well-posedness result is proved, under an additional $L^2$
hypothesis on the initial inhomogeneity and velocity field. A new a priori
estimate for parabolic equations in endpoint spaces $B^s_{\infty,r}$ is
presented, which is the key to the proof.
  In dimension two, we are able to give a lower bound for the lifespan, such
that the solutions tend to be globally defined when the initial inhomogeneity
is small. There we will show a refined a priori estimate in endpoint Besov
spaces for transport equations with \emph{non solenoidal} transport velocity
field.
