Vanishing pressure limit for compressible Navier-Stokes equations with degenerate viscosities

TL;DR

This paper investigates the behavior of highly compressible Navier-Stokes equations with degenerate viscosities as the pressure vanishes, establishing global weak solutions and convergence rates in the vanishing pressure limit.

Contribution

It proves the global existence of weak solutions for pressureless systems with degenerate viscosities and derives convergence rates for the density in the vanishing pressure process.

Findings

Global existence of weak solutions for pressureless Navier-Stokes with degenerate viscosities

Convergence rate of density in the vanishing pressure limit

Weak solutions established in the framework of Li-Xin

Abstract

In this paper we study a vanishing pressure process for highly compressible Navier-Stokes equations as the Mach number tends to infinity. We first prove the global existence of weak solutions for the pressureless system in the framework [Li-Xin, arXiv:1504.06826v2], where the weak solutions are established for compressible Navier-Stokes equations with degenerate viscous coefficients. Furthermore, a rate of convergence of the density in $L^{\infty}\left(0,T;L^{2}(\rr)\right)$ is obtained, in case when the velocity corresponds to the gradient of density at initial time.