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

TL;DR

This paper introduces a new formulation of the compressible Navier-Stokes equations using an effective velocity, revealing a natural parabolic structure that enables proofs of entropy and existence of strong solutions with large or small initial data.

Contribution

The paper presents a novel formulation of the compressible Navier-Stokes equations that simplifies analysis and demonstrates the natural parabolicity of the density and vorticity equations.

Findings

Simplified proof of the entropy condition from previous work.

Existence of strong solutions with large initial data.

Global strong solutions for small initial data.

Abstract

In this paper we give a new formulation of the compressible Navier-Stokes by introducing an suitable effective velocity $v=u+\n\va(\rho)$ 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 ${\rm curl}v$, and as a transport equation on the divergence ${\rm div}v$. We show the existence of strong solution with large initial data in finite time when $(\rho_0-1)\in B^{\NN}_{p,1}$. 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.