粘性依赖于密度的一维等熵可压缩Navier-Stokes方程组粘性激波的非线性稳定性

Nonlinear Stability of Viscous Shock Waves for One-dimensional Isentropic Compressible Navier-Stokes Equations with Density-Dependent Viscosity

该文主要研究粘性系数依赖于密度的一维等熵可压缩Navier-Stokes方程组Cauchy问题整体解的大时间渐近行为,主要研究目的是改进文献[7]的结果至γ>1,κ≥0.注意到γ=2,κ=1时,一维等熵可压缩Navier-Stokes方程组对应于Saint-Venant浅水波方程组,该方程组描述了地表浅水运动的规律,在物理学和海洋学中有重要的应用^([1,4,6])。注意到文献^([7])中通过利用Kanel的方法^([19])来推导比容的一致上下界估计,在得出比容的上界时,该方法要求κ<1/2.对该文所研究的问题而言,需要首先利用Kanel’的方法^([19])来推导比容的一致上下界估计.为了扩大κ的取值范围,还需要对比容的上下界作更为精细的能量估计.在得出比容的一致上下界估计之后,可通过精心设计的连续性技巧,将Navier-Stokes方程组的局部解一步步延拓为整体解,并得到对应的大时间渐近行为.

This paper mainly studies the large-time asymptotic behavior of the global solu-tion of the density dependent one-dimensional isentropic compressible Navier-Stokes equations Cauchy problem.The main purpose of this paper is to improve the result of[7]to γ>1,κ≥0.It is noted that when γ=2,κ=1,the one-dimensional isentropic compressible Navier-Stokes equations correspond to the Saint-Venant shallow water wave equations,which describe the law of surface shallow water movement and have important applications in physics and oceanogra-phy[1,4,6].Note that in[7],the method[19]of Kanel is used to derive the uniform upper and lower bound estimation of specific volume.When obtaining the upper bound of specific volume,this method requires κ<1/2.For the problem studied in this paper,we need to use Kanel's method[19]to derive the uniform upper and lower bound estimation of specific volume.In or-der to expand the value range of κ,it is also necessary to make a more precise energy estimation of the upper and lower bounds of the specific volume.After obtaining the uniform upper and lower bound estimation of specific volume,the local solution of Navier-Stokes equations can be extended into the global solution step by step through carefully designed continuity techniques,and the corresponding large-time asymptotic behavior can be obtained.