The Decay Estimates And Regularity Criteria Of Several Hydrodynamics Equations

The fluid dynamics is a branch of dynamics, it aims to study the stationary s-tate and the motion state of fluid itself. The main research object is water and air. It is mainly based on Newton’s laws of motion, the law of conservation of mass and The knowledge of thermodynamics, and sometimes macroscopic electrodynamics and the basic knowledge of physics is also needed. Many fluid mechanics system has estab-lished the corresponding mathematical model according to its physical background, in order to better study the features of the fluid motion and the motion state. Cur-rently, the existence, uniqueness, regularity and large time behavior problems of solutions to the fluid equations have been investigated widely.This thesis is focused on the following three types of equations:the gener-alized Hall-Magneto-hydrodynamics equations, the modified critical dissipative quasi-geostrophic equations and the three-dimensional micropolar fluid equations. These three types of equations are important mathematical model in describing fluid dynamics movement system. For example,the generalized Hall-Magneto-hydrody-namics equations can be used to describe the magnetohydrodynamic flow, Magneto-hydrodynamics wave and so on; the modified critical dissipative quasi-geostrophic equations can be used to describe the atmospheric movement, Marine movement and so on; the three-dimensional micropolar fluid equations can be used to describe the blood movement, liquid crystal movement and so on.In this thesis we mainly consider the time decay estimate of generalized Hall-Magneto-hydrodynamic-s equations and the two-dimensional modified critical dissipative quasigeostrophic equations,and a new logarithmical velocity regularity criterion of micropolar fluid equations. The research of the former two types equations use the similar method: the classical Fourier transform. The research of micropolar fluid equations mainly use the Multiplier space technique and energy methods.The whole thesis organized according to the following structure:In the first chapter, some background knowledge of several types of hydrody-namics equations, and related research.In the second chapter, we consider Rapid time decay of weak solutions to the generalized Hall-Magneto-hydrodynamics equations, By developing the classic Fourier splitting methods and combining the lower fre-quency effect of the nonlinear term to obtain the more rapid L2 decay rate:In the third chapter, we consider the time decay of the two-dimensional mod-ified Dissipative Quasi-geostrophic Equation, By developing the classic Fourier splitting methods and energy estimate to get the time decay estimates:In the fourth chapter, we consider the regularity criteria for the weak solutions of the three-dimensional micropolar fluid equations, By applying the Multiplier space technique and energy methods to get the regular solution if the velocity fields satisfy the following growth condition:...