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Cannone \cite{Cannone} proved the global well-posedness of the incompressible Navier-Stokes equations for a class of highly oscillating data. In this paper, we prove the global well-posedness for the compressible Navier-Stokes equations in the critical functional framework with the initial data close to a stable equilibrium. Especially, this result allows us to construct global solutions for the highly oscillating initial velocity. The proof relies on a new estimate for the hyperbolic/parabolic system with convection terms., Comment: 43pages