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

2014/11/20 by Haspot, Boris
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1411.5501

Abstract

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

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