  This work is devoted to the well-posedness issue for the low-Mach number
limit system obtained from the full compressible Navier-Stokes system, in the
whole space. In the case where the initial temperature (or density) is close to
a positive constant, we establish the local existence and uniqueness of a
solution in critical homogeneous Besov spaces. If, in addition, the initial
velocity is small then we show that the solution exists for all positive time.
In the fully nonhomogeneous case, we establish the local well-posedness in
nonhomogeneous Besov spaces (still with critical regularity) for arbitrarily
large data with positive initial temperature. Our analysis strongly relies on
the use of a modified divergence-free velocity which allows to reduce the
system to a nonlinear coupling between a parabolic equation and some
evolutionary Stokes system. The Lebesgue exponents of the Besov spaces for the
temperature and the (modified) velocity, need not be the same. This enables us
to consider initial data in Besov spaces with a negative index of regularity.
