流体力学方程的补丁解和流动稳定性的研究

在本项目中我们拟研究两部分内容:一是包括非齐次Navier-Stokes方程、Boussinesq方程等流体动力学方程的补丁问题;二是Boussinesq方程关于剪切流的流动稳定性。补丁问题和流动稳定性问题都有很强的物理和数学意义，是近些年的研究热点。项目组成员曾研究过密度补丁解以及涡度片解的的整体正则性，且在Navier-Stokes方程流动稳定性理论方面最近做出重要工作。本项目将在前期研究工作基础上，拟利用调和分析和偏微分方程方法，研究非齐次 Navier-Stokes方程、Boussinesq方程等补丁解的高阶正则性传播，以及三维无热扩散的Navie r-Stokes-Boussineq系统剪切流的稳定性，包括分析其增强耗散和升力效应与浮力项的相互影响机制。

In this project, we plan to study two parts: one is the patches problem of non-homogeneous Navier Stokes equation, Boussinesq equation and other hydrodynamic equations; The second is the Boussinesq equation regarding the flow stability of shear flow. The patch problem and flow stability problem have strong physical and mathematical significance and have become research hotspots in recent years. Members of the project team have studied the global regularity of density patch solutions and vorticity sheet solutions, and recently made important work on the flow stability theory of Navier Stokes equations. On the basis of previous research work, this project will use Harmonic analysis and partial differential equation methods to study the higher-order regular propagation of patches solutions of non-homogeneous Navier Stokes equation, Boussinesq equation and other equations, and the stability of shear flow of three-dimensional Navier Stokes Boussineq system without thermal diffusion, including the analysis of the interaction mechanism between its enhanced dissipation and lift effects and buoyancy terms.