可压Navier-Stokes方程若干问题研究

The basic series of equations in theoretical fluid mechanics is Navier-Stokes equations, and it is used to describe some fluid substance such as liquid and gas. It shows the dynamic balance among all kinds of force which acting on any given domain in fluid, and everybody knows that Navier-Stokes equations is a important issue in the field of partial differential equations, and it have been accepted widely by physics as the model which can describe the motion of the fluid. This item aims to study the Vaigant-Kazhikhov model of Navier-Stokes equations and what we are interested in includes the Cauchy problem, free boundary problem in three dimensional space and the long-time behavior for 1D nonisentropic Navier-Stokes equations. As the solutions have singularity in origin, the key is to get the upper and lower bounds. And we hope that we can make a great progress in this item. For the case in 1D nonisentropic Navier-Stokes equations, we have got the solutions' existence under the discontinuous boundary conditions and improved the previous results, so on the base of the existence, we expect that we can further obtain the asymptotic behavior and fading rate on the density and temperature functions .

理论流体力学的基本方程是Navier-Stokes方程, 它是一组描述象液体和空气这样的流体物质的方程. 主要描述作用于液体任意给定区域的力的动态平衡, 是非线性偏微分方程研究的一个重要课题，其作为描述流体运动的模型，已被物理学界广泛接受。本项目旨在研究可压缩Navier-Stokes方程Vaigant-Kazhikhov模型的Cauchy问题，自由边值问题的三维球对称经典解以及一维可压非等熵Navier-Stokes方程自由边值问题经典解的存在性和解的衰退率以及大时间行为。由于解在球心处的奇异性，本项目研究经典解的关键点以及难点在于获得解在球心处的上下界估计，希望在关于球对称经典解的研究上取得重大的进展。而对于一维非等熵方程的经典解，我们已经得到了跳跃边界条件下解的存在性并且改进了之前已有的研究成果，我们希望在此基础上进一步获得解的渐进性态和衰退率的最新成果。

众所周知，关于Navier-Stokes方程的研究和探索在航空动力学、天体物理、地质力学、天气预报、油气探测、信息处理等方面有着极其重要的应用。.本项目重点对一维非等熵Navier-Stokes方程的自由边值问题进行了研究，主要研究内容有：（1）当粘性系数$\mu=\overline{\mu}$ ， 热传导系数依赖于温度 $\kappa=\overline{\kappa}\theta^b, b\in (0,\infty)$， 我们获得了其强解及经典解的全局存在性；（2）当粘性系数依赖于密度 $\mu(\rho)=\rho^\alpha+1, \alpha\in (0,\infty)$ ，热传导系数为 $\underline{\kappa}\leq\kappa\leq\overline{\kappa}$ 时经典解的大时间行为。.这些成果进一步推广了之前的结论到热传导系数或粘性系数更一般的情形， 具有重要的科学意义。