Spherically Symmetric Barotropic Compressible Navier-Stokes Equations with Density-Dependent Viscosity and Discontinuous
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Spherically Symmetric Barotropic Compressible Navier-Stokes Equations with Density-Dependent Viscosity and Discontinuous Initial Data Ruxu Lian1,2 · Lan Huang1
Received: 24 July 2015 / Accepted: 9 February 2016 © Springer Science+Business Media Dordrecht 2016
Abstract This paper is concerned with the exterior problem and the initial boundary value problem for the spherically symmetric barotropic compressible Navier-Stokes equations with density-dependent viscosity coefficients and discontinuous initial data. For the exterior problem and the initial boundary value problem, we prove that there exists a unique global piecewise regular solution for piecewise regular initial density with arbitrarily large jump discontinuity. Moreover, we show that the jump of density decays exponentially in time and the piecewise regular solution tends to the equilibrium state exponentially as time tends to infinity. Keywords Spherically symmetric Navier-Stokes equations · Discontinuous initial data · Exterior problem · The initial boundary value problem · Piecewise regular solution Mathematics Subject Classification 35Q35 · 76D03
1 Introduction In this paper, we investigate the exterior problem and the initial boundary value problem to N-dimensional barotropic compressible Navier-Stokes equations with density-dependent viscosity coefficients. In general, the N-dimensional barotropic compressible Navier-Stokes equations with density-dependent viscosity coefficients can be written as ρt + div(ρU) = 0, (1.1) (ρU)t + div(ρU ⊗ U) + ∇P (ρ) − div h(ρ)D(U) − ∇ g(ρ)divU = 0,
B R. Lian
[email protected] L. Huang [email protected]
1
North China University of Water Resources and Electric Power, College of Mathematics and Information Science, Zhengzhou 450011, P.R. China
2
Chinese Academy of Sciences, Institute of Atmospheric Physics, Beijing 100029, P.R. China
R. Lian, L. Huang
where t ∈ (0, +∞) is the time and x ∈ RN , ρ > 0 and u denote the fluid density and velocity respectively. Pressure function is taken as P (ρ) = ρ γ with γ > 1, and D(U) =
∇(U) + T ∇(U) 2
(1.2)
is the strain tensor and h(ρ), g(ρ) are the Lamé viscosity coefficients satisfying h(ρ) > 0,
h(ρ) + Ng(ρ) ≥ 0.
(1.3)
There is huge literature on the studies of the global existence of weak solutions and dynamical behaviors of jump discontinuity for the compressible Navier-Stokes equations with discontinuous initial data, for example, as the viscosity coefficients h(ρ) and g(ρ) are both constants, Hoff considered the global existence of discontinuous solutions of one-dimension Navier-Stokes equations [1–3]. Hoff derived the construction of global spherically symmetric weak solutions of compressible Navier-Stokes equations for isothermal flow with large and discontinuous initial data in reference [4], therein it is shown that the discontinuities in the density and pressure persist for all time, convecting along particle trajectories, and decaying at a rate inversely proportional to the viscosity coefficient. Hoff also obtained the global existenc
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