Vanishing viscosity limit of the Navier–Stokes equations to the Euler equations for compressible fluid flow with far field vacuum

Abstract

We prove the vanishing viscosity limit of the Navier–Stokes equations to the Euler equations for a two-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 far field vacuum, whose life span is uniformly positive in the vanishing viscosity limit. It is worth paying special attention to the fact that, introducing two different symmetric structures, we can also give some uniform estimates of ργ−12 and of u in H3 and of ∇ρ/ρ in L6∩D1, which provide the convergence of the regular solution of the viscous flow to that of the inviscid flow in L∞([0,T];Hs′) for any s′∈[2,3) with a rate of ϵ2(1−s′3). Moreover, our results can be extended to the two-dimensional shallow water equations after slight modifications.