Mathematical Studies On The3D Magneto-micropolar Fluid Equations

Magneto-Micropolar fuid equations is a kind of partial diferential equa-tions which describe complex physical phenomena and is of great importance tophysics[1,2].Mathematical studies on Magneto-Micropolar fuid equations have not onlythe theoretical signifcance but also their value. It is well known that the hydro-dynamic equations are an important part of modern nonlinear partial diferentialequations. Some well-known equations have been studied for a long history, suchas Euler equation, Navier-Stokes equation, etc. Even up to now, they are stillthe focal points and hot spots of study. Magneto-Micropolar fuid equations area well completed system in incompressible fuid mechanics equations. At present,mathematical research achievements concerning Magneto-Micropolar fuid equa-tions are not rich enough yet, there are still not relevant results drawn from somereally important mathematical questions, like vanishing viscosity limit.This thesis is composed of two parts to study some basic mathematical prop-erties of Magneto-Micropolar fuid equations in three dimensions space, such asthe global existence of the weak solutions, the vanishing-viscosity limit, the reg-ularity criteria of weak solutions, etc. The frst part mainly considers the initialboundary value problem with a type of Slip boundary conditions of in a boundeddomain. The second part mainly concerns Cauchy problem of equation in R3.This paper consists of seven chapters and their main contents are as follows:Chapter1primarily focuses on the research background and status of rele-vant equations and introduces briefy the basic idea of this paper, relevant con-cepts and conclusions.Chapter2studies the global existence of the weak solutions of Magneto-Micropolar fuid equations for a class slip boundary conditions in simply connectbounded domain. The boundary conditions are as follows:u· n=0,(×u)×n=0; b· n=0,(×b)×n=0; w=0. on Chapter3proves the local existence and uniqueness of strong solution toMagneto-Micropolar fuid equations on the basis of the second chapter and studiescontinuity on time derivative of strong solutions in the case where the initial valueand regularity of force.Chapter4analyzes the convergence of the solution as the physical coefcientsgoes to zero. we consider all and part viscosities vanishing limit in the generaldomain and strong convergence in the domain with fat boundary.Chapter5presents regularity criteria of weak solutions to Magneto-Micropolarfuid equations in whole space. By applying energy estimation and some inequal-ities in critical space, we obtain regularity criteria only involving the velocityfeld in multiplier space and Morrey-Campanato space. Logarithmic Regularitycriterial can also be obtained with exchange inequalities.Chapter6studies regularity criteria of Magneto-Micropolar fuid equationsunder the framework of Sobolev. Based on the energy estimation, we acquireregularity criteria concerning the direction of velocity feld and popularize thecorresponding conclusions with respect to NS equation and MHD equation indocuments[4,5].Chapter7mainly summarizes the research results and signifcance of thispaper and poses several problems and further direction in the research at thesame time.