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

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 \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 $\mathrm{curl}v$, and as a transport equation on the divergence $\mathrm{div}v$. We show the existence of strong solution with large initial data in finite time when $(ρ_0-1)\in B^{\NN}_{p,1}$. 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.

38 pages. arXiv admin note: text overlap with arXiv:1107.2332; and with arXiv:1302.2617 by other authors

References in corpus (1)

• Porous media, Fast diffusion equations and the existence of global weak solution for the quasi-solution of compressible Navier-Stokes equations

Cited by in corpus (3)

• Vortex solutions for the compressible Navier-Stokes equations with general viscosity coefficients in 1D: regularizing effects or not on the density
• Global bmo${}^{-1}({\mathbb{R}}^N)$ radially symmetric solution for compressible Navier-Stokes equations with initial density in $L^{\infty }({\mathbb{R}}^N)$
• Weak-Strong uniqueness for compressible Navier-Stokes system with degenerate viscosity coefficient and vacuum in one dimension