Global solution to the exterior problem for spherically symmetric compressible Navier-Stokes equations with density-dependent viscosity and discontinuous initial data

Ruxu Lian,Jianwei Yang

doi:10.1186/1687-2770-2013-90

Abstract

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

Highlights

In the present paper, we consider the exterior problem to N -dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficients
There are many significant progresses achieved on 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 μ(ρ) and λ(ρ) are both constants, Hoff investigated the global existence of discontinuous solutions of one-dimension Navier-Stokes equations [ – ]
The construction of global spherically symmetric weak solutions of compressible Navier-Stokes equations for isothermal flow with large and discontinuous initial data was derived by Hoff [ ], therein it is showed 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

Summary

Introduction

1 Introduction In the present paper, we consider the exterior problem to N -dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficients. The construction of global spherically symmetric weak solutions of compressible Navier-Stokes equations for isothermal flow with large and discontinuous initial data was derived by Hoff [ ], therein it is showed 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.

Results

Conclusion