一维可压Navier—Stokes方程自由边值问题的自模解

研究黏性系数μ（ρ）=1＋θρ^θ时一维可压Navier-Stokes方程自模解的非存在性．首先通过建立能量估计式，熵估计式得到密度函数ρ的正的下界，然后对能量函数进行定量分析，利用能量爆破理论证明了θ〉0时一维可压Navier-Stokes方程不存在具有有限总能量的自模解．最后将常黏性系数Navier-Stokes方程自模解的方法推广到黏性系数依赖于密度的情形，并且把θ的范围扩展到θ〉0．

Self-similar solutions to 1D compressible Navier-Stokes equations with free boundary problem

The non-existence of self-similar solutions to one-dimensional compressible Navier-Stokes equations is studied when the viscosity coefficient/.t（p） =1＋θρ^θ. The positive lower bound of the density p is obtained by establishing energy estimate and entropy estimate. By quantitative analysis to energy function and using energy blowing-up theory, there exist neither forward nor backward self-similar solutions with finite total energy when θ〉 0. This generalizes the method of studying the compressible Navier-Stokes equations with constant viscosity coefficient to the density-dependent viscosity coefficient and enlarging the interval of θ to θ〉0.