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

TL;DR

This paper proves the global-in-time convergence of solutions from the 3D compressible Navier-Stokes equations with degenerate viscosities to the inviscid Euler equations, even with vacuum states and large velocities, using new energy estimates.

Contribution

It introduces a novel coupled hyperbolic-elliptic structure to handle vacuum and degenerate viscosities, establishing the first global-in-time inviscid limit for such flows in 3D.

Findings

Uniform energy estimates for solutions with vacuum

Strong convergence to inviscid flow in $H^2$

Applicability to a broad class of density-dependent viscosities

Abstract

In the recent 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 ($(\rho^\epsilon)^\delta$ 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 $(\rho^\epsilon)^{\frac{\delta-1}{2}}$ 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. The result here applies to a class of degenerate density-dependent viscosity coefficients, is independent of the B-D relation for viscosities, and seems to be the first on the global-in-time inviscid limit of smooth solutions which have large velocities and contain vacuum state for compressible flow in three space dimensions without any symmetric assumption.