Résumé en anglais

In this paper we give a new formulation of the compressible Navier-St okes by introducing an suitable effective velocity v = u + ∇ φ ( ρ ) provided that the viscosity coefficients verify the algebraic relation of [3]. We give in particular a ve ry simple proof of the entropy discovered in [3], in addition our argument show why the al- gebraic relation of [3] appears naturally. More precisely the system reads in a very surprising way as two parabolic equation on the density ρ and the vorticity curl v , and as a transport equation on the divergence div v . We show the existence of strong solution with large initial data in finite time when ( ρ 0 − 1) ∈ B N p 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.