Viscosity Coefficient Depends On The Density Of One-dimensional Compressible Navier-stokes Equations

The one-dimensional compressible Navier-Stokes equations with density-dependent viscosity coefficients are considered in the present paper. Some prop-erties of the solutions to this equation are investigated. It mainly contains two parts. Firstly, the existence of the weak solutions to the Cauchy problems is obtained, then the asymptotic behavior of such weak solutions is analyzed Secondly, the non-existence of the self-similar solutions to the equation is proved.In the first chapter, the background of this paper is introduced as well as some important results obtained by the previous studies.In chapter two, under the condition ofρ0∈L1(R), the Cauchy problem of Navier-Stokes equation is studied. Same as Jiu's paper (Kinet. Relat. Models, 1(2008), No.2,313-330), this paper constructs the approximate solutions which satisfy the a priori estimate required in the L1 stability analysis after mollifying the initial data. But the difference lies in the proof of the upper and lower bounds to the density. In Jiu's paper,θ＞(?)is required, but in this paper, it is extended toθ＞0.In chapter three, the self-similar solution of Navier-Stokes equations is in-vestigated. Using the quantitative analysis of the energy function, it concludes that there exist neither forward nor backward self-similar solution with finite total energy.At last, we conclude the paper and explore some open problems about the Navier-Stokes equations.