Navier-Stokes / Boussinesq方程是数学物理领域中两个重要的基础模型,在流体力学、空气动力学、天体物理、大气海洋等工程技术中有着广泛的应用. 由于复杂的物理机制和数学结构, 其一直是现代偏微分方程理论和应用研究的前沿热点问题, 已有大量的研究成果, 但仍有许多重要的基本问题仍未解决. 本文的主要目的是研究 Navier-Stokes / Boussinesq方程的全局适定性和稳定性. 全文共分为四章, 第一章主要介绍了所研究问题的物理和数学背景，并回顾了一些相关的结果. 第二章, 我们研究了高雷诺数 $\mathbf{Re}$ 下带有 Coriolis 参数 $|\beta|\geqslant 1$ 的三维旋转 Navier-Stokes (NSC) 方程在周期性平面Couette流 $\left( \beta y, 0, 0\right)$ 下的稳定性问题. 我们的目标是找到关于 $\mathbf{Re}$ 的稳定性阈值的指标, 即当选择初值时, 方程的解保持稳定的最大扰动范围. 我们首先研究了线性化扰动系统的线性稳定性效应. 对比 Bedrossian 等人 [Ann. of Math. 185(2): 541--608 (2017)] 的结果, $u_{\neq}^1$和$u_{\neq}^2$ 具有不同的无粘阻尼. 由于流体的旋转作用, $u_{\neq}$ 加剧了耗散增强, 但也同时在 $u_0^{2} $和$u_{0}^{3}$方向上带来了抬升效应. 从某种意义上说, Coriolis 力是加剧流体不稳定性的一个因素, 研究旋转流体的稳定阈值实际上更具挑战性. 其次, 我们证明了当初始值满足 $\left\|u_{\mathrm{in}}\right\|_{H^{\sigma}}<\delta \mathbf{Re}^{-\frac{5}{2}} |\beta|^{-\frac{7}{3}}$ 时, 这里 $\sigma > \frac{9}{2}$, $\delta=\delta(\sigma)>0$ 仅依赖 $\sigma$, 三维 NSC 方程的存在全局解, 并且不会远离 Couette 流. 这是关于旋转 Navier-Stokes 方程在 Couette 流附近的非线性稳定性的第一个结果. 第三章, 我们研究了带有滑移边界条件下的二维可压缩 Navier-Stokes 方程的初边值问题. 当密度 $\rho$ 的$L^\infty$ 范数有界时, 我们建立了具有真空的强解的整体存在性, 并得到了解的指数衰减估计. 值得一提的是, 这里的初值条件不需要任何小性限制. 第四章, 我们考虑了三维可压缩 Boussinesq 方程的 Dirichlet 边值问题。在初始密度和初始温度的$L^{1}$ -范数小的条件下, 得到了含真空的强解的全局适定性, 并建立了解的指数衰减估计.

The Navier-Stokes / Boussinesq equations are two important and fundamental models in the field of mathematical physics and have a wide range of  applications in fluid mechanics, aerodynamics, astrophysics, atmospheric ocean and other engineering technology. Due to the complicated physical mechanism and mathematical structure, the Navier-Stokes / Boussinesq equations have always been the research hotspot in theoretical PDEs and applied mathematics. Despite the important progress, many important fundamental problems remain unknown. The main purpose of this dissertation is to study the global well-posedness theory and stability of the  Navier-Stokes / Boussinesq equations. The whole book is divided into chapters. The first chapter mainly introduces the physical and mathematical background of problem studied and  recalls some relevant results.

In Chapter 2,  we study the dynamic stability of the periodic, plane  Couette flow $\left( \beta y, 0, 0\right)$ with the Coriolis parameter $|\beta|\geqslant 1$ in  the three-dimensional Navier-Stokes  equations with rotation at high Reynolds number $\mathbf{Re}$. Our goal is to find the index of the  stability threshold about $\mathbf{Re}$: the maximum range of  perturbations in which the solution of the equations remains stable  when the initial data is chosen. We first study the linear stability effects of linearized perturbed system. Compared with the results of Bedrossian et al [Ann. of Math. 185(2): 541--608 (2017)],   $u_{\neq}^1$ and $u_{\neq}^2$   have different types of  inviscid damping  due to the coupling structure of $u^1, u^2$. Due to the  rotation effect from Coriolis force, $u_{\neq}$ accelerates the enhanced dissipation, but it also brings  the lift-up effect on the directions of  $u_0^{2} $ and $u_{0}^{3}$, respectively.  In a certain sense, Coriolis force is  a factor that aggravates the instability of fluid, which is actually more difficult and challenging to study the stability threshold of the rotating fluid.  Furthermore, we prove that  the initial data satisfies $\left\|u_{\mathrm{in}}\right\|_{H^{\sigma}}<\delta \mathbf{Re}^{-\frac{5}{2}}  |\beta|^{-\frac{7}{3}}$ for any $\sigma > \frac{9}{2}$ and some $\delta=\delta(\sigma)>0$ depending only on $\sigma$,  the resulting solution to the 3D Navier-Stokes-Coriolis equations is global in time and does not transition away from the Couette flow. This is the first result concerning with the nonlinear stability of the Navier-Stokes equations with the rotational framework  near the Couette flow.

In Chapter 3,  we are concerned with a two-dimensional initial-boundary value problem of barotropic compressible Navier-Stokes equations subject to slip boundary condition. The global existece of strong solutions with vacuum is established provided the $L^\infty$ norm of the density $\rho$ has upper bounds. Moreover, the exponential decay estimates of the solutions are also obtained. It is worth mentioning that here the initial data are large  without any restrictions.

In Chapter 4, we consider the initial-boundary value problem of  compressible Boussineq equations on 3D  bounded domains subject to homogeneous Dirichlet boundary condition for velocity and  absolute temperature. The global well-posedness of strong solutions with  vacuum   is obtained provided the $L^{1}$-norms of the initial density and temperature is small,  As by-products, the exponential decay of estimates of solutions are  established.