可压缩流体方程研究

This project intends to study the qualitative behavior of compressible fluid equations, focus on the study of Navier-Stokes equation and well-posedness and regularity of related fluid equations; vacuum problem of multi-dimensional space, free boundary value problem and its geometric structure and dynamic characteristics; existence and uniqueness of multi-dimensional Navier-Stokes equation for global strong solution. For Navier-Stokes-Possion equation with density-dependent viscosities, we study the existence, uniqueness and long-time behavior of global weak solution for initial large data. We study the vacuum problem of Navier-Stokes-Possion equation, including existence, regularity, uniqueness and asymptotic behavior of the global weak solution for free boundary value problem.

本项目拟研究可压缩流体方程解的定性性态，重点研究Navier－Stokes方程及相关流体方程解的适定性、正则性问题；高维空间真空问题、自由边界问题及其几何结构和动力学特征；高维Navier-Stoke方程整体强解的存在性、唯一性等。对于粘性依赖密度的Navier-Stokes-Possion方程，研究大初始值整体弱解的存在性、唯一性以及解的大时间行为。研究Navier-Stokes-Possion方程的真空问题，涉及自由边界问题整体弱解的存在性、正则性、惟一性、以及渐近行为。

本项目主要研究可压缩流体方程的相关问题，包括Navier-Stokes方程和Navier-Stokes-Poisson方程。我们证明了粘性系数依赖于密度的三维Navier-Stokes方程初边值问题柱对称强解的整体存在性和大时间行为，研究了粘性系数依赖于密度的三维Navier-Stokes-Poisson方程初边值问题球对称弱解的整体存在性，同时还研究了粘性系数依赖于密度的一维Navier-Stokes方程柯西问题强解的存在性、唯一性以及正则性等。本项目共发表期刊论文4篇，其中SCI论文3篇。本项目所获结果丰富了非线性偏微分方程的相关理论，具有很强的理论意义和重要的应用价值。