关于不可压缩流体的粘性消失极限问题

项目名称： 关于不可压缩流体的粘性消失极限问题

项目编号： No.11201371

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 王丽真

作者单位： 西北大学

项目金额： 22万元

中文摘要： 粘性不可压缩流体是流体动力学的主要研究对象。人们对这类流体的认识还远远不够。本项目研究刻画粘性不可压缩流体运动的Navier-Stokes等方程的粘性消失极限问题。我们拟采用渐近分析和能量估计等方法研究不同滑动边界条件下三维有界区域中不可压缩流体方程的解当粘性系数趋于零时的渐近行为。由于边界的存在所导致的边界层的出现使得无粘极限的研究变得非常困难。我们已对一类滑动边界条件下Navier-Stokes方程的粘性消失极限问题作了研究。项目中我们将作两方面的推广，一是讨论其他两类滑动边界条件下Navier-Stokes方程的无粘极限问题；二是将我们的方法应用到具有滑动边界条件的Magnetohydrodynamic方程上。所研究的Navier型滑动边界被用于湍流中大涡流的模拟。因此本项目中严格的数学分析将加深对湍流这个复杂系统的理解，为不可压缩流体边界层问题的研究提供理论依据。

中文关键词： 流体方程；解的存在性；非线性发展方程；分数阶偏微分方程；Lie对称

英文摘要： Incompressible viscous flow is the main object of study in Hydrodynamics. The understanding of this kind of flow is very little and in great demand. This project will investigate the vanishing viscosity limit of the systems which describe the motion of the incompressible flow such as Navier-Stokes equations. The approaches such as asymptotic analysis and energy estimate will be used to study the asymptotic behavior of solutions to the incompressible flow under different slip boundary conditions in general bounded domain in three dimensional space when the coefficient of viscosity of the equations tends to zero. The fact that the existence of the boundary may lead to the formation of boundary layer makes the study of inviscid limit very difficult. We have already considered the vanishing viscosity limit of Navier-Stokes equations under one slip boundary condition. This project will generalize the study in two directions. One is to discuss the inviscid limit of Navier-Stokes equations under other two slip boundary conditions. The other is to apply our method to Magnetohydrodynamic equations with slip boundary condition. The Navier slip boundary conditions are used in the large eddy simulations of turbulent flows. Therefore, the strict mathematical analysis in our project will enhance the understanding on this comp

英文关键词： Fluid flow equations；existence of solution；nonlinear evolution equation；fractional partial differential equation；Lie symmetry