On the global-in-time inviscid limit of the 3D degenerate compressible Navier-Stokes equations

Abstract

In this paper, the global-in-time inviscid limit of the three-dimensional (3D) isentropic compressible Navier-Stokes equations is considered. First, when viscosity coefficients are given as a constant multiple of density's power (${\left({\rho }^{ϵ}\right)}^{\delta }$ with $\delta >1$), for regular solutions to the corresponding Cauchy problem, via introducing one “quasi-symmetric hyperbolic”–“degenerate elliptic” coupled structure to control the behavior of the velocity near the vacuum, we establish the uniform energy estimates for the local sound speed in $H^3$ and ${\left({\rho }^{ϵ}\right)}^{\frac{\delta -1}{2}}$ in $H^2$ with respect to the viscosity coefficients for arbitrarily large time under some smallness assumption on the initial density. Second, by making full use of this structure's quasi-symmetric property and the weak smooth effect on solutions, we prove the strong convergence of the regular solutions of the degenerate viscous flow to that of the inviscid flow with vacuum in $H^2$ for arbitrarily large time. It is worth pointing out that the result obtained here seems to be the first one on the global-in-time inviscid limit of solutions with large velocities and vacuum for compressible flow in 3D space without any symmetric assumption.