Free Boundary Value Problem for the Spherically Symmetric Compressible Navier-Stokes Equations with a Nonconstant Exteri
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Free Boundary Value Problem for the Spherically Symmetric Compressible Navier-Stokes Equations with a Nonconstant Exterior Pressure Ruxu Lian1,2 · Xinying Xu3
Received: 20 October 2014 / Accepted: 22 December 2015 © Springer Science+Business Media Dordrecht 2016
Abstract This paper is concerned with the free boundary value problem (FBVP) for the spherically symmetric barotropic compressible Navier-Stokes equations (CNS) with density-dependent viscosity coefficients in the case that across the free surface stress tensor is balanced by a nonconstant exterior pressure. Under certain assumptions imposed on the initial data, we show that there exists a unique global strong solution which is strictly positive for any finite time and decays pointwise to zero time-asymptotically. Keywords Navier-Stokes equations · Spherically symmetric · Density-dependent viscosity coefficients · Nonconstant exterior pressure · Strong solution Mathematics Subject Classification 35Q35 · 76D03
1 Introduction The N-dimensional barotropic compressible Navier-Stokes equations with density-dependent viscosity coefficients read as
ρt + div(ρU) = 0,
(ρU)t + div(ρU ⊗ U) + ∇P (ρ) − div h(ρ)D(U) − ∇ g(ρ)divU = 0,
(1.1)
B R. Lian
[email protected] X. Xu [email protected]
1
College of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450011, P.R. China
2
Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, P.R. China
3
School of Mathematical Sciences, Xiamen University, Xiamen 361005, P.R. China
R. Lian, X. Xu
where (x, t) ∈ RN × [0, +∞), ρ(x, t) > 0, U(x, t) and P (ρ) = ρ γ (γ > 1) stand for the fluid density, velocity and pressure respectively, and D(U) =
∇(U) +T ∇(U) 2
(1.2)
is the strain tensor, h(ρ) and g(ρ) are the Lamé viscosity coefficients satisfying h(ρ) > 0,
h(ρ) + Ng(ρ) ≥ 0.
(1.3)
There are many progress made on the free boundary value problem for multi-dimensional compressible viscous Navier-Stokes equations with constant viscosity coefficients for either barotropic or heat-conducive fluids, for instant, as the stress tensor is balanced by a constant exterior pressure and/or surface tension across the free surface, classical solutions with strictly positive densities in the fluid regions to the free boundary value problem for compressible Navier-Stokes equations were shown locally in time for either barotropic flows [1–3] or heat-conductive flows [4–6]. In the case that across the free surface the stress tensor is balanced by exterior pressure [3], surface tension [7], or both exterior pressure and surface tension [8] respectively, as the initial data is assumed to be near to non-vacuum equilibrium state, the global existence of classical solutions with small amplitude and positive densities in fluid region to the free boundary value problem for barotropic compressible Navier-Stokes equations with constant viscosity coefficients was established. Global existence of classical solutions to the free boundary value pr
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