本博士学位论文研究了三种流体力学方程组初值问题的整体适定性和解的长时间行为。它们分别是各向异性的磁流体方程，推广的Boussinesq 方程，以及扩散系数依赖温度的热带气候模型。这三类模型在流体力学中具有很重要的研究价值和实际应用。主要内容分为以下三个部分：

一、研究了二维各向异性耗散的磁流体方程组在背景磁场(非平凡解)附近的整体适定性，并得到了扰动方程线性部分解的衰减估计。对于带有水平耗散的Navier-Stokes 方程，涡度一阶导的 Lebesgue 范数可能随时间而增长。磁场和速度场的耦合对流体系统产生了稳定性效应，这是从物理实验中观察到的非线性现象。本论文从数学上严格验证了这种现象，通过把原来的抛物方程组转化为波方程，构造合适的能量泛函，应用迭代法得到封闭的能量不等式，从而得到解的稳定性。由于方程组缺少某些分量的耗散，相应的波动方程也是退化的， 导致研究非线性系统的衰减估计十分困难。因此仅给出了扰动方程线性化系统解的衰减估计。

二、研究了二维各向异性推广的Boussinesq 方程组 Cauchy 问题的整体适定性，主要研究具有水平粘性耗散和水平热扩散，或者水平粘性耗散和垂直热扩散的推广的Boussinesq 方程组。这里借鉴推广的SQG 方程的定义，速度由涡度标量根据推广的Biot-Savart 定律来表示，因而，此种情形比经典 Boussinesq 方程的全局适定性求解更加困难。利用了Log 型插值不等式和 Hardy–Littlewood–Sobolev 不等式来估计涡度和速度的最大模，并采用了弱非线性能量估计方法，进一步提高了解的正则性。从而得到一般初值条件下，推广的Boussinesq 方程组全局光滑解的存在唯一性。

三、研究了二维带有非线性扩散的热带气候模型Cauchy 问题的整体适定性和解的长时间行为。由于扩散系数依赖于温度，而温度又是变量，又因为方程之间很强的非线性耦合，第一斜压速度也缺少散度自由条件，使得方程组的全局适定性问题十分复杂。利用De-Giorgi 迭代，推导出温度场最大模的时空一致有界性。并构造了新变量得到温度一阶导数的正则性估计，进一步推导出全局光滑解的存在唯一性。利用分频方法，研究了不同初值条件下弱解的长时间行为，得到了弱解的最优时间衰减率。温度场最大模的时空一致有界性以及线性解算子(广义Ossen 算子)最大模的一致有界性在解的衰减估计中起了关键作用。

这三种流体力学模型都是由Navier-Stokes 方程演化而来。研究它们解的相关性质有助于更好的认识理解流体力学基本模型Navier-Stokes 方程解的性质。

In this doctoral thesis, we investigate the global well-posedness theory for Cauchy problem to the three type fluid mechanics equations and the long-time behavior of their solutions. They are magnetohydrodynamic equations with anisotropic dissipation, generalized Boussinesq equations with anisotropic dissipation and temperature-dependent tropical climate model. It is mainly three-fold:

The first part studies the well-posedness and stability of the solutions for a 2D magnetohydrodynamic equations with anisotropic dissipation near a nonzero background magnetic field. The decay estimate of solutions for the linearized system of perturbation equations is also obtained. For the anisotropic Navier-Stokes equations with horizontal dissipation, the Lebesgue norm of first derivative of vorticity grows in general. The coupling of magnetic field and velocity field actually stabilizes electrically conducting fluids. This is a nonlinear phenomenon observed in physical experiments, and this thesis confirms it rigorously. By the wave equations transformed by the original perturbation system, suitable energy functionals are constructed and closed energy inequalities are derived, and by bootstrapping argument, thus the stability of solutions is obtained. Because of anisotropic dissipation, the corresponding wave equation is also degenerate, so it's difficult to study the time decay of solution to the nonlinear system, thus we just provide the decay estimate of solution to the linear perturbation system.

In the second part we investigate the global well-posedness of Cauchy problem about generalized Boussinesq equations with anisotropic dissipation, and our research focuses on the generalized Boussinesq equations with horizontal viscosity dissipation and horizontal temperature diffusivity or horizontal viscosity dissipation and vertical temperature diffusivity. The fluid velocity is determined by a generalized Biot-Savart law, which is motivated by the generalized SQG equations. So this makes the global well-posedness of this model much harder than that of classical Boussinesq equations. Logarithmic interpolation inequality and Hardy-Littlewood-Sobolev inequality are used to construct the maximum norm of vorticity scalar and velocity. Next, the weakly nonlinear energy estimate approach is adopted to improve on the regularity of solutions. Furthermore, the existence and uniqueness of the global smooth solutions to the generalized Boussinesq equations is obtained for arbitrarily large initial data.

The third part is devoted to the research of the well-posedness and long-time asymptotic behavior of the solutions for 2D temperature-dependent tropical climate model. Due to the temperature-dependent dissipation, the variable temperature and the strong coupling between the equations, the global well-posedness of equations is much involved. The uniform bound about space-time of maximum norm to temperature field is deduced by the De-Giorgi approach. And then the regularity of first derivative to temperature is obtained by introducing a constructor. Moreover, the existence and uniqueness of global smooth solutions are obtained. By splitting the integration in phase-space into two time dependent domains, we investigate the long-time behavior of weak solutions with different initial data, and obtain the optimal time decay rate. The uniform bound of maximum norm to the resolvent operator (i.e. generalized linear Ossen operator) and temperature are crucial in the study of the decay estimate.

These three fluid mechanics models are all evolved from Navier-Stokes equations. The study of the proposition of their solution is conducive to understanding that of basic fluid mechanics model Navier-Stokes equation.