粘性依赖密度/温度非等熵可压缩Navier-Stokes方程的定性分析

Navier-Stokes方程是流体力学中具有强耦合性、强非线性的偏微分方程，该方程的研究是应用数学和计算流体力学的前沿热点问题之一。本项目拟从基础数学理论上研究Navier-Stokes方程的适定性问题以及解的大时间行为，主要内容包括：. （1）初始密度不含真空、粘性依赖密度和温度一维非等熵Navier-Stokes方程Cauchy问题大初值强解的存在性以及大时间行为。. （2）初始密度含真空、粘性依赖密度一维非等熵Navier-Stokes方程初边值问题的大初值全局弱解、局部强解，以及小初值全局强解的存在性、大时间行为。. （3）初始密度含真空、粘性依赖密度三维非等熵Navier-Stokes方程Cauchy问题的大初值全局弱解、局部强解，以及小初值全局强解的存在性、大时间行为。

Navier-Stokes equations are partial differential equations with strong coupling and strong nonlinearity in fluid dynamics, whose mathematical study is one of central issues in both applied mathematics and computational mechanics. In this project, we aim to study of the well-posedness and large-time behaviour for Navier-Stokes equations from basic mathematics theory. It mainly consiste of the following parts:. (1)When the initial density without vacuum, we consider the existence and large-time behavior of strong solutions to the Cauchy problem for one-dimensional non-isentropic compressible Navier-Stokes equations with density and temperature-dependent viscosity. . (2)When the initial density with vacuum, we study an initial-boundary value problem of one-dimensional non-isentropic compressible Navier-Stokes equations with density-dependent viscosity, discussing the global weak solution and local strong solution under large conditions on the initial data, the existence and large-time behavior of strong solutions under samllness conditions on the initial data.. (3)When the initial density with vacuum, we consider the Cauchy problem of three dimensional non-isentropic compressible Navier-Stokes equations with density-dependent, discussing the global weak solution and local strong solution under large conditions on the initial data, the existence and large-time behavior of global strong solutions under samllness conditions on the initial data.