Vanishing Viscosity Limit of the Navier-Stokes Equations to the Euler Equations for Compressible Fluid Flow with Vacuum

TL;DR

This paper proves that solutions of the compressible Navier-Stokes equations with vacuum and density-dependent viscosities converge to solutions of the Euler equations as viscosity vanishes, providing uniform estimates and convergence rates.

Contribution

It introduces a novel coupled hyperbolic-elliptic structure to establish uniform bounds and strong convergence in the vanishing viscosity limit for compressible flows with vacuum.

Findings

Existence of unique regular solutions with vacuum in the vanishing viscosity limit.

Uniform bounds in Sobolev spaces for density and velocity.

Quantitative convergence rate of viscosity solutions to Euler solutions.

Abstract

We establish the vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for three-dimensional compressible isentropic flow in the whole space. It is shown that there exists a unique regular solution of compressible Navier-Stokes equations with density-dependent viscosities, arbitrarily large initial data and vacuum, whose life span is uniformly positive in the vanishing viscosity limit. It is worth paying special attention that, via introducing a "quasi-symmetric hyperbolic"--"degenerate elliptic" coupled structure, we can also give some uniformly bounded estimates of $\displaystyle\Big(\rho^{\frac{\gamma-1}{2}}, u\Big)$ in $H^3$ space and $\rho^{\frac{\delta-1}{2}}$ in $H^2$ space (adiabatic exponent $\gamma>1$ and $1<\delta \leq \min\{3, \gamma\}$), which lead the strong convergence of the regular solution of the viscous flow to that of the inviscid flow in $L^{\infty}([0, T]; H^{s'})$ (for any $s'\in [2, 3)$) space with the rate of $\epsilon^{2(1-s'/3)}$. Further more, we point out that our framework in this paper is applicable to other physical dimensions, say 1 and 2, with some minor modifications. This paper is based on our early preprint in 2015.