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In this paper we give a new formulation of the compressible Navier-Stokes byintroducing an suitable effective velocity $v=u+\n\va(\rho)$ provided that theviscosity coefficients verify the algebraic relation of \cite{BD}. We give inparticular a very simple proof of the entropy discovered in \cite{BD}, inaddition our argument show why the algebraic relation of \cite{BD} appearsnaturally. More precisely the system reads in a very surprising way as twoparabolic equation on the density $\rho$ and the vorticity ${\rm curl}v$, andas a transport equation on the divergence ${\rm div}v$. We show the existenceof strong solution with large initial data in finite time when $(\rho_0-1)\inB^{\NN}_{p,1}$. A remarkable feature of this solution is the regularizingeffects on the density. We extend this result to the case of global strongsolution with small initial data.

New formulation of the compressible Navier-Stokes equations and parabolicity of the density