From the Highly Compressible Navier–Stokes Equations to Fast Diffusion and Porous Media Equations, Existence of Global Weak Solution for the Quasi-Solutions

Abstract

We consider the compressible Navier–Stokes equations for viscous and barotropic fluids with density dependent viscosity. The aim is to investigate mathematical properties of solutions of the Navier–Stokes equations using solutions of the pressureless Navier–Stokes equations, that we call quasi solutions. This regime corresponds to the limit of highly compressible flows. In this paper we are interested in proving the announced result in Haspot (Proceedings of the 14th international conference on hyperbolic problems held in Padova, pp 667–674, 2014) concerning the existence of global weak solution for the quasi-solutions, we also observe that for some choice of initial data (irrotationnal) the quasi solutions verify the porous media, the heat equation or the fast diffusion equations in function of the structure of the viscosity coefficients. In particular it implies that it exists classical quasi-solutions in the sense that they are ${C^{\infty}}$ on ${(0,T)\times \mathbb{R}^{N}}$ for any ${T > 0}$. Finally we show the convergence of the global weak solution of compressible Navier–Stokes equations to the quasi solutions in the case of a vanishing pressure limit process. In particular for highly compressible equations the speed of propagation of the density is quasi finite when the viscosity corresponds to ${\mu(\rho)=\rho^{\alpha}}$ with ${\alpha > 1}$. Furthermore the density is not far from converging asymptotically in time to the Barrenblatt solution of mass the initial density ${\rho_{0}}$.