In this paper we give a new formulation of the compressible Navier-Stokes by introducing an suitable effective velocity $v=u+\n\va(\rho)$ provided that the viscosity coefficients verify the algebraic relation of \cite{ BD}. We give in particular a very simple proof of the entropy discovered in \cite{ BD}, in addition our argument show why the algebraic relation of \cite{ BD} appears naturally. More precisely the system reads in a very surprising way as two parabolic equation on the density $\rho$ and the vorticity ${ \rm curl}v$, and as a transport equation on the divergence ${ \rm div}v$. We show the existence of strong solution with large initial data in finite time when $(\rho_0-1)\in B^{ \NN}_{ p,1}$. A remarkable feature of this solution is the regularizing effects on the density. We extend this result to the case of global strong solution with small initial data.
Categories: math.AP