In this survey paper,
we are concerned with the zero Mach number limit
for compressible viscous flows.
For the sake of (mathematical) simplicity,
we restrict ourselves to the case of barotropic
fluids and we
assume that the flow evolves in the whole space
or satisfies periodic boundary conditions. We focus on the case of ill-prepared data.
Hence highly oscillating acoustic waves
are likely to propagate through the fluid.
We nevertheless state
the convergence to the incompressible Navier-Stokes
equations when the Mach number ϵ goes to 0.
Besides, it is shown that the global existence for the limit equations
entails the global existence for the compressible model with
small ϵ. The reader is referred to [R. Danchin, Ann. Sci. Éc. Norm. Sup. (2002)] for the detailed proof in the whole space case,
and to [R. Danchin, Am. J. Math.124 (2002) 1153–1219] for the case of periodic boundary conditions.