微极流体方程是用于描述微极流体(由棒状元素组成的具有微旋转效应和微旋转惯性的流体)运动的方程. 本学位论文主要研究了微极流体方程表面波问题.
第一章, 阐述了微极流体方程表面波问题并利用拉平变换将其转化为几何形式.
然后, 概述了微极流体方程和Navier-Stokes (NS)方程自由边界问题的研究背景. 最后, 陈述了本论文的三个主要定理, 并给出了本文中常用的基本数学工具.
第二章, 研究了微极流体方程表面波问题的线性化方程.利用基于一组依赖时间的基底的Galerkin方法, 证明了线性化方程解的局部存在唯一性以及正则性, 为第三章求解非线性方程做准备.
第三章, 在水平周期区域及水平全空间区域内研究了不带表面张力微极流体方程表面波问题. 通过构造恰当的迭代系统获得一组逼近解, 然后证明逼近解的收敛性并对迭代系统取极限,
最终证得了不带表面张力微极流体方程表面波问题小初值局部解的存在唯一性.
为确保每一步迭代过程中, 流体自由界面均具有小性条件, 从而使得相应的拉平变换是微分同胚, 充分利用了速度和微旋角速度耦合的线性化系统来构造迭代系统.
这个迭代系统与现有文献中研究微极流体方程Dirichlet问题时采用的迭代系统是完全不同的.此外, 由于微极流体方程速度和微旋角速度的强耦合性, 在处理方程的耦合项时给出了一些精细的估计. 由于本章是在高正则空间中求解方程, 因此还给出了初值的相容性条件.
第四章, 在水平周期区域内研究了不带表面张力微极流体方程表面波问题. 具体而言, 利用两层能量法证得了一个非线性能量估计, 然后结合连续性方法及第三章的局部存在唯一性定理,
最终证得了不带表面张力微极流体方程表面波问题小初值整体解的存在唯一性, 并说明了方程的解会以“几乎指数”的速率衰减到平衡态. 该问题的困难是: 流体表面张力的缺失导致能量估计中耗散始终比能量弱. 为了克服这一困难, 采用了两层能量法封闭非线性能量估计. 此外, 为了克服速度和微旋角速度耦合项带来的困难, 本文基于Korn不等式给出了方程全局适定的条件, 并给出了一个使Korn不等式成立的不依赖区域的一般常数18. 
第五章, 在水平周期区域内研究了带表面张力微极流体方程表面波问题. 通过充分利用流体表面张力对流体自由界面带来的正则性的增长, 克服了第四章中耗散比能量弱这一困难, 并最终结合非线性能量估计以及连续性方法, 在小初值假设下得到了方程解的指数衰减估计.
第六章, 对本文的结果做了一个总结与展望.

Micropolar equations is proposed to describe the motion of rigid and randomly oriented or spherical particles that have their own spins and microrotations suspended in a viscous medium. This thesis divided into six chapters studies the surface wave problem of Micropolar equations with and without surface tension.
In Chapter 1, we formulate the surface wave problem of Micropolar equations and transform it into the geometric form. Then we give a brief overview of the research background of Micropolar equations. Meanwhile, since the surface wave problem is a kind of free boundary problem and Micropolar equations contains Navier-Stokes (NS) equations as the centerpiece, we review the research background of the free boundary problem of NS equations. Then we state three main theorems of this thesis and give the basic tools used in this thesis.
In Chapter 2, the linearized equation of the surface wave problem of the Micropolar equations is studied. By using of Galerkin method with a time-dependent basis, this chapter successfully constructs a unique local solution to the linearized equations with small initial data and prepares for solving the nonlinear equation in Chapter 3.
In Chapter 3, the surface wave problem of the Micropolar equations without surface tension is studied in the horizontal periodic domain and in the horizontal infinity domain. By constructing an appropriate iterative system and obtaining a set of approximate solutions, and then proving the convergence of the approximate solutions and setting the limit of the iterative system, this chapter finally proves the existence and uniqueness of the local solution to the Micropolar equations with small initial data.
The iterative system adopted in this chapter is completely different from the one used in the existing literature to study the Dirichlet problem of Micropolar equations. Specifically, in the linearized equations corresponding to the iterative system adopted in this chapter, velocity and micro-rotation must be solved together and concurrently, which is for guarantees that the free interface is small during each iteration, and further that the corresponding flatting transform is a diffeomorphism. Further, due to the strong coupling of the velocity and micro-rotation, we give some instructive estimates when dealing with the coupling terms of the equations. Meanwhile, some compatibility conditions of the initial data are imposed in this chapter.
In Chapter 4, the surface wave problem of Micropolar equations without surface tension is studied in the horizontal periodic domain. By using the two-tier energy method and proving a nonlinear energy estimate, and then combining the bootstrap argument with the local existence theorem proved in Chapter 3, this chapter finally proves the global existence of the solution to the Micropolar equations with small initial data and shows that the solution decays to the equilibrium state at an almost exponential rate. The absence of surface tension leads to the lack of regularity of the free interface and further leads to the fact that the dissipation is weaker than the corresponding energy. To overcome the weakness of dissipation, we adopt a two-tier energy method to close the estimate. Meanwhile, to overcome the difficulties caused by the coupling term of velocity and micro-rotation, we impose a restriction of viscosities based on Korn's inequality. Further, we calculate a constant 18 which makes Korn's inequality true and independent of the domain.
In Chapter 5, the surface wave problem of the Micropolar equations with surface tension is studied in the horizontal periodic domain. One of the key features of the surface wave problem of the Micropolar equations with surface tension is a gain of regularity for the free boundary. This feature allows us to bound the energy by the corresponding dissipation. Thus, by using the gain of regularity for the free boundary, we combine the energy method with the bootstrap argument and finally calculate an exponential decay rate of the solution to the surface wave problem of Micropolar equations with surface tension.