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We are concerned with the inviscid limit of the Navier-Stokes equations tothe Euler equations for barotropic compressible fluids in $\mathbb{R}^3$. Whenthe viscosity coefficients obey a lower power-law of the density (i.e.,$\rho^\delta$ with $0<\delta<1$), we identify a quasi-symmetrichyperbolic--singular elliptic coupled structure of the Navier-Stokes equationsto control the behavior of the velocity of the fluids near the vacuum. Thenthis structure is employed to prove that there exists a unique regular solutionto the corresponding Cauchy problem with arbitrarily large initial data andfar-field vacuum, whose life span is uniformly positive in the vanishingviscosity limit. Some uniform estimates on both the local sound speed and thevelocity in $H^3(\mathbb{R}^3)$ with respect to the viscosity coefficients arealso obtained, which lead to the strong convergence of the regular solutions ofthe Navier-Stokes equations with finite mass and energy to the correspondingregular solutions of the Euler equations in $L^{\infty}([0, T]; H^{s}_{\rmloc}(\mathbb{R}^3))$ for any $s\in [2, 3)$. As a consequence, we show that, forboth viscous and inviscid flows, it is impossible that the $L^\infty$ norm ofany global regular solution with vacuum decays to zero asymptotically, as $t$tends to infinity. Our framework developed here is applicable to the sameproblem for the other physical dimensions via some minor modifications.

Vanishing Viscosity Limit of the Three-Dimensional Barotropic Compressible Navier-Stokes Equations with Degenerate Viscosities and Far-Field Vacuum