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

Original · EN

In this paper we give a new formulation of the compressible Navier-Stokes by introducing an suitable effective velocity v=u+(ρ) provided that the viscosity coefficients verify the algebraic relation of BD. We give in particular a very simple proof of the entropy discovered in BD, in addition our argument show why the algebraic relation of BD appears naturally. More precisely the system reads in a very surprising way as two parabolic equation on the density ρ and the vorticity curlv, and as a transport equation on the divergence divv. We show the existence of strong solution with large initial data in finite time when (ρ₀-1)∈ Bₚ,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.