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

On the well-posedness of the full low-Mach number limit system in general critical Besov spaces