Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí sinh học đề tài :Global behavior of 1D compressible isentropic Navier-Stokes equations with a non-autonomous external force | Huang and Lian Boundary Value Problems 2011 2011 43 http content 2011 1 43 o Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access Global behavior of 1D compressible isentropic Navier-Stokes equations with a non-autonomous external force Lan Huang and Ruxu Lian Correspondence huanglan82@ College of Mathematics and Information Science North China University of Water Sources and Electric Power Zhengzhou 450011 People s Republic of PR China Springer Abstract In this paper we study a free boundary problem for compressible Navier-Stokes equations with density-dependent viscosity and a non-autonomous external force. The viscosity coefficient ụ is proportional to pe with 0 0 1 where p is the density. Under certain assumptions imposed on the initial data and external force f we obtain the global existence and regularity. Some ideas and more delicate estimates are introduced to prove these results. Keywords Compressible Navier-Stokes equations Viscosity Regularity Vacuum 1 Introduction We study a free boundary problem for compressible Navier-Stokes equations with density-dependent viscosity and a non-autonomous external force which can be written in Eulerian coordinates as pT pu l 0 T 0 pu t p u2 P p i puị pf a T Ệ b T with initial data p u I 0 p0 U0 ệ a a 0 Ệ b 0 b where p p ifr u u r P P p and f ff i denote the density velocity pressure and a given external force respectively p p p is the viscosity coefficient. a r and b r are the free boundaries with the following property 4 a r u a T T -r-b T u b T T dT dT -P p p p u I T 0 I a T b T . The investigation in 1 showed that the continuous dependence on the initial data of the solutions to the compressible Navier-Stokes equations with vacuum failed. The main reason for the failure at the vacuum is because of kinematic viscosity coefficient being independent of the density. On the other hand we know that the Navier-Stokes equations can be derived from .
