Navier-Stokes方程Navier边值问题的稳定性与不稳定性

This project is concerned with the stability and instability as well as vanishing viscosity problems for Navier-Stokes equations with Navier boundary conditions. Our concerns include the following problems：(1) stability and instability as well as decay estimates of trivial steady states to Navier-Stokes equations with Navier boundary conditions; (2) stability and instability as well as decay estimates of non-trivial steady states (Couette flows, Poiseuille flows and general shear flows) to Navier-Stokes equations with Navier boundary conditions; (3) vanishing viscosity and boundary layer problems for the incompressible Navier-Stokes equations with Navier boundary conditions. These are the key problems in the studies of Navier-Stokes equations. Especially for the Navier boundary problems, such studies are far from complete but they are very important problems both in theory and in applications. This project is important to the development of the theory of variational method and the theory of spectral method of stability and instability as well as to the vanishing viscosity theory.

本项目以研究Navier边界条件下Navier-Stokes方程的稳定性、不稳定性和粘性消失极限为主线，主要研究下面几个方面的问题：（1）Navier边界条件下Navier-Stokes方程零稳态解的稳定性与不稳定性、衰减估计；（2）Navier边界条件下Navier-Stokes方程非零稳态解（Couette流，Poiseuille流，一般剪切流）的稳定性与不稳定性、衰减估计；（3）Navier边界条件下不可压缩Navier-Stokes方程的粘性消失极限与边界层问题。这些问题都是Navier-Stokes方程领域的核心问题。这些问题的研究对于发展稳定性理论的变分方法、非自共轭方程稳定性的谱方法、粘性消失极限理论具有重要的理论意义与应用价值。本项目紧紧围绕Navier边值问题开展研究，力图为完善Navier边值问题稳定性、粘性消失极限理论做出贡献。

课题组围绕Navier-Stokes方程Navier边值问题以及与Navier-Stokes方程、MHD方程相关的方程的适定性、稳定性、粘性消失极限和边界层展开的收敛性，反应扩散方程的行波解及其稳定性开展了深入细致和有效的研究。取得了一系列具有一定创新性的成果。较好地推动了相关领域的研究向前发展。课题组成员在项目执行期间共发表高水平学术论文28篇。