Journal of Applied Mathematics
Volume 2013 (2013), Article ID 505108, 11 pages
http://dx.doi.org/10.1155/2013/505108
Research Article

Free Boundary Value Problem for the One-Dimensional Compressible Navier-Stokes Equations with Density-Dependent Viscosity and Discontinuous Initial Data

Ruxu Lian1,2 and Guojing Zhang3

1College of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450011, China
2Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China
3School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China

Received 9 February 2013; Accepted 6 June 2013

Academic Editor: Nazim Idrisoglu Mahmudov

Copyright © 2013 Ruxu Lian and Guojing Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the free boundary value problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient and discontinuous initial data in this paper. For piecewise regular initial density, we show that there exists a unique global piecewise regular solution, the interface separating the flow and vacuum state propagates along particle path and expands outwards at an algebraic time-rate, the flow density is strictly positive from blow for any finite time and decays pointwise to zero at an algebraic time-rate, and the jump discontinuity of density also decays at an algebraic time-rate as the time tends to infinity.