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

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 $({\rho }^{\frac{\gamma -1}{2}},u)$ in $H^3$ space and ${\rho }^{\frac{\delta -1}{2}}$ in $H^2$ space (adiabatic exponent $\gamma >1$ and $1<\delta \le \min \{3,\gamma \}$), which lead the strong convergence of the regular solution of the viscous flow to that of the inviscid flow in $L^{\infty }([0,T];H^{s^{\prime }})$ (for any $s^{\prime }\in [2,3)$) space with the rate of ${\epsilon }^{2(1-s^{\prime }/3)}$. 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.

arXiv admin note: text overlap with arXiv:1503.05644; and text overlap with arXiv:0910.2360, arXiv:1005.2713 by other authors