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In the recent paper, the global-in-time inviscid limit of thethree-dimensional (3D) isentropic compressible Navier-Stokes equations isconsidered. First, when viscosity coefficients are given as a constant multipleof density's power ($(\rho^\epsilon)^\delta$ with $\delta>1$), for regularsolutions to the corresponding Cauchy problem, via introducing one"quasi-symmetric hyperbolic"--"degenerate elliptic" coupled structure tocontrol the behavior of the velocity near the vacuum, we establish the uniformenergy estimates for the local sound speed in $H^3$ and$(\rho^\epsilon)^{\frac{\delta-1}{2}}$ in $H^2$ with respect to the viscositycoefficients for arbitrarily large time under some smallness assumption on theinitial density. Second, by making full use of this structure's quasi-symmetricproperty and the weak smooth effect on solutions, we prove the strongconvergence of the regular solutions of the degenerate viscous flow to that ofthe inviscid flow with vacuum in $H^2$ for arbitrarily large time. The resulthere applies to a class of degenerate density-dependent viscosity coefficients,is independent of the B-D relation for viscosities, and seems to be the firston the global-in-time inviscid limit of smooth solutions which have largevelocities and contain vacuum state for compressible flow in three spacedimensions without any symmetric assumption.

On the global-in-time inviscid limit of the 3D isentropic compressible Navier-Stokes equations with degenerate viscosities and vacuum