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

Abstract

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 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 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 ǫ 2(1−s ′ /3) . Further more, we point out that our framework in this paper is applicable to other physical dimensions, say 1 and 2, with some minor modifications.

Blow-up criterion for the 3D viscous polytropic fluids with degenerate viscosities

“…then there exist a small time T * and a unique regular solution (ρ, u) to the Cauchy problem (1) with (5)- (6). Moreover, we also have…”

“…In this section, we give the proof for Theorem 1.3. We use a contradiction argument to prove (14), let (ρ, u) be the unique regular solution to the Cauchy problem (1) with (5)- (6) and the maximal existence time T . We assume that T < +∞ and lim…”

“…Here, the initial data are given by (ρ, u)| t=0 = (ρ 0 , u 0 )(x), x ∈ R 3 , (5) and the far field behavior is given by (ρ, u) → (0, 0) as |x| → ∞, t ≥ 0. (6) The aim of this paper is to prove a blow-up criterion for the regular solution to the Cauchy problem (1) with (5)- (6).…”

“…We also refer readers to [3], [6], [10], [13], [18], [26] and references therein for other interesting progress for this compressible degenerate system, corresponding radiation hydrodynamic equations and magnetohydrodynamic equations. It should be noted that one should not always expect the global existence of solutions with better regularities or general initial data because of Xin's results [23] and Rozanova's results [20].…”

“…When the viscosities depend on density in the form of (4), S. Zhu [25] introduced the regular solutions, which can be defined as Definition 1.1. [25] Let T > 0 be a finite constant, (ρ, u) is called a regular solution to the Cauchy problem (1) with (5)- (6) 3 satisfies the Cauchy problem (1)with (5) − (6) in the sense of distributions;…”