Global Regular Solutions for one-dimensional Degenerate Compressible Navier--Stokes Equations with Large Data and Far Field Vacuum

Abstract

We consider the initial-boundary value problem (IBVP) for the isentropic compressible Navier-Stokes equations (CNS) in the domain exterior to a ball in R d (d = 2 or 3). When viscosity coefficients are given as a constant multiple of the mass density ρ, based on some analysis of the nonlinear structure of this system, we prove the global existence of the unique spherically symmetric classical solution for (large) initial data with spherical symmetry and far field vacuum in some inhomogeneous Sobolev spaces. Moreover, the solutions we obtained have the conserved total mass and finite total energy. ρ keeps positive in the domain considered but decays to zero in the far field, which is consistent with the facts that the total mass is conserved, and CNS is a model of non-dilute fluids where ρ is bounded away from the vacuum. To prove the existence, on the one hand, we consider a well-designed reformulated structure by introducing some new variables, which, actually, can transfer the degeneracies of the time evolution and the viscosity to the possible singularity of some special source terms. On the other hand, it is observed that, for the spherically symmetric flow, the radial projection of the so-called effective velocity v = U + ∇ϕ(ρ) (U is the velocity of the fluid, and ϕ(ρ) is a function of ρ defined via the shear viscosity coefficient µ(ρ): ϕ ′ (ρ) = 2µ(ρ)/ρ 2 ), verifies a damped transport equation which provides the possibility to obtain its upper bound. Then combined with the BD entropy estimates, one can obtain the required uniform a priori estimates of the solution. It is worth pointing out that the frame work on the well-posedness theory established here can be applied to the shallow water equations.

On Regular Solutions for Three-Dimensional Full Compressible Navier–Stokes Equations with Degenerate Viscosities and Far Field Vacuum

“…The related progress for the cases δ ≥ 1 on smooth solutions with vacuum can also be found in [16,36,37,53]. Since the coefficients of the time evolution and Q(u) are powers of ρ, it is easy to compare the order of the degeneracy of these two operators near the vacuum, which enable one to select the dominant operator to control the behavior of u and lead to the "hyperbolic-strong singular elliptic" structure in [6,54] and the " quasi-symmetric hyperbolic"-"degenerate elliptic" structure in [16,37,53]. Some other related progress can also be found in [8,17,29,39] and the references therein.…”

“…On the other hand, there are a few results available for strong (or smooth) solutions with finite energy. In particular, for the case 0 < δ < 1, by introducing an elaborate elliptic approach on the singularly weighted regularity estimates for u and a symmetric hyperbolic system with singularities for some quantities involving ρ and its derivatives, Xin-Zhu [54] identifies a class of initial data admitting one unique 3-D local regular solution with far field vacuum to the Cauchy problem of (1.8) 1 and (1.13) in some inhomogeneous Sobolev spaces, which has been extended to be global-in-time ones with large data in R by Cao-Li-Zhu [6].…”

“…Since e, θ and S are all fundamental states for viscous compressible fluids, it is of great importance to study their dynamics for the full CNS, which is a subtle and difficult problem in the presence of vacuum. Indeed, in the studies for the well-posedenss of classical solutions with vacuum to the full CNS (1.1)-(1.3) with degenerate viscosities and heat conductivity of the form (1.4)- (1.5), the structures of the coefficients for the time evolution and the spatial dissipation operators are different, and S plays important roles but behaves more singularly than θ near the vacuum, which cause substantial difficulties in the analysis and make it difficult to adapt the approaches for the isentropic case in [6,16,36,37,54]. It should be pointed out that due to the physical requirements on θ and S near the vacuum, it is of more advantages to formulate the CNS (1.1)- (1.3) in terms of (ρ, u, S) instead of (ρ, u, θ) in contrast to [9,21,50] as illustrated below.…”

Global existence of strong solutions to the 3D isentropic compressible Navier–Stokes equations with density‐dependent viscosities

“…Actually, by the Chapman-Enskog expansions for the Boltzmann equation, the viscosity coefficients of the barotropic compressible Navier-Stokes equations are proportional to powers of density. 23,24 The one-dimensional case has been studied by many authors (see previous studies [24][25][26][27][28][29][30][31][32] ). For the multi-dimensional case with general data, Vaigant-Kazhikhov 33 first established global classical solutions in T 2 or [0, 1] 2 when the initial density is strictly away from vacuum and…”

“…Actually, by the Chapman–Enskog expansions for the Boltzmann equation, the viscosity coefficients of the barotropic compressible Navier–Stokes equations are proportional to powers of density 23,24 . The one‐dimensional case has been studied by many authors (see previous studies 24–32 ). For the multi‐dimensional case with general data, Vaigant–Kazhikhov 33 first established global classical solutions in or $$$ {\left[0,1\right]}&#x0005E;2 $$$ when the initial density is strictly away from vacuum and $$$$ \tilde{\mu}&#x0003D; constant&gt;0,\kern0.30em \tilde{\lambda}\left(\rho \right)&#x0003D;{\rho}&#x0005E;{\kappa },\kern0.30em \kappa &gt;3.$$…”