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

Boris Haspot (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

In this paper we give a new formulation of the compressible Navier-Stokes 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 very 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 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 (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.

Domaines: Equations aux dérivées partielles [math.AP]

Boris Haspot. New formulation of the compressible Navier-Stokes equations and parabolicity of the density. 2014. ⟨hal-01081580⟩