Vanishing Viscosity Limit of the Navier–Stokes Equations to the Euler Equations for Compressible Fluid Flow with Vacuum

We establish the vanishing viscosity limit of the Navier–Stokes equations to the Euler equations for three-dimensional compressible isentropic flow in the whole space. When the viscosity coefficients are given as constant multiples of the density’s power ( ρ δ with δ > 1 ), it is shown that there exists a unique regular solution of compressible Navier–Stokes equations with arbitrarily large initial data and vacuum, whose life span is uniformly positive in the vanishing viscosity limit. It is worth paying special attention to the fact that, via introducing a “quasi-symmetric hyperbolic”–“degenerate elliptic” coupled structure to control the behavior of the velocity of the fluid near the vacuum, we can also give some uniform estimates for ( ρ γ - 1 2 , u ) in H 3 and ρ δ - 1 2 in H 2 with respect to the viscosity coefficients (adiabatic exponent γ > 1 and 1 < δ ≦ min { 3 , γ } ), which lead to the strong convergence of the regular solution of the viscous flow to that of the inviscid flow in L ∞ ( [ 0 , T ] ; H s ′ ) (for any s ′ ∈ [ 2 , 3 ) ) with the rate of ε 2 ( 1 - s ′ / 3 ) . Furthermore, we point out that our framework in this paper is applicable to other physical dimensions, say 1 and 2, with some minor modifications.