  The present paper is devoted to the well-posedness issue for a low-Mach
number limit system with heat conduction but no viscosity. We will work in the
framework of general Besov spaces $B^s_{p,r}(\R^d)$, $d\geq 2$, which can be
embedded into the class of Lipschitz functions.
  Firstly, we consider the case of $p\in[2,4]$, with no further restrictions on
the initial data. Then we tackle the case of any $p\in\,]1,\infty]$, but
requiring also a finite energy assumption. The extreme value $p=\infty$ can be
treated due to a new a priori estimate for parabolic equations. At last we also
briefly consider the case of any $p\in ]1,\infty[$ but with smallness condition
on initial inhomogeneity.
  A continuation criterion and a lower bound for the lifespan of the solution
are proved as well. In particular in dimension 2, the lower bound goes to
infinity as the initial density tends to a constant.
