Global Solutions to the Spherically Symmetric Compressible Navier-Stokes Equations with Density-Dependent Viscosity and Discontinuous Initial Data

Ruxu Lian,Jian Liu,Jianwei Yang

doi:10.1155/2014/132324

Abstract

We consider the initial boundary value problem for the spherically symmetric isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficients and discontinuous initial data in this paper. For piecewise regular initial density with bounded jump discontinuity, we show that there exists a unique global piecewise regular solution. In particular, the jump of density decays exponentially in time and the piecewise regular solution tends to the equilibrium state exponentially ast→+∞.

Highlights

The isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficients read as follows: ρt + div = 0,(ρU)t + div + ∇P (ρ) (1)− div (h (ρ) D (U)) − ∇ (g (ρ) div U) = 0, where t ∈ (0, +∞) is the time and x ∈ RN, N is the spatial coordinate, and ρ > 0 and u denote the density and velocity, respectively
∫− ρ2 (r2 u)[2] dx ds 1 ∫− 1 (r2 u)[2] r−4 dx, xs τ where C denotes a positive constant independent of time; choosing the constant ε small sufficiently, we complete the proof of Lemma 11
The large time behaviors follow from Lemma 14 directly

Summary

Introduction

If the viscosity coefficients h(ρ) = ρα, g(ρ) = 0, for the case of one space dimension, Fang and Zhang proved the global existence of unique piecewise smooth solution to the free boundary value problem for (1) with 0 < α < 1, where the initial density is piecewise smooth with possibly large jump discontinuities [10]. They got that the jump discontinuity of density decays exponentially but never vanishes in any finite time and the piecewise regular solution tends to the equilibrium state exponentially as t → +∞ In this present paper, we consider the initial boundary value problem for the spherically symmetric isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficients and discontinuous initial data and focus on the regularities and dynamical behaviors of global weak solution and so forth.

Notations and Main Results

The a Priori Estimates

Proof of the Main Results