Stability And Asymptotic Behavior Of The Solutions To Boussinesq Equations And MHD Type Equations

The purpose of this thesis is to study the stability and asymptotic behavior of the solutions to Boussinesq equations,magneto-micropolar fluid equations,and MHD equations in different spatial regions.The whole thesis is divided into six chapters.In chapter 1,we expound on the relevant background,development status,and main results of the problems studied in this thesis,and introduce some basic knowledge.In chapter 2,we prove the stability of the solutions to the two-dimensional Boussinesq equations with partial dissipation on a strip domain T ×(0,1),where T is a general periodic domain,and we also obtain the decay rate.We mainly use higher-order energy estimates,low-order timeweighted energy estimates,Poincare inequality,and Sobolev embedding to obtain a priori bound on the solutions,and we prove the global existence of the solutions by Bootstrap theory.In chapter 3,we explore the stability and large time behavior of the solutions to the threedimensional Boussinesq equations on R2 × T,where T=[-1/2,1/2].If the initial values have some symmetry,we will obtain the stability of the solutions to the Boussinesq equations with some mixed dissipation,and we prove that the solutions have the same symmetry as the initial value.At the same time,the exponential decay of the oscillatory parts(u,θ)of the solutions are established.we first use the anisotropic Sobolev inequalities,and Poincare inequality to get a priori bound on the solutions,then we obtain exponential decay by Gronwall inequality,finally,we use Bootstrap theory to obtain the stability.Chapter 4 considers the stability and asymptotic behavior of the solutions to the two-dimensional incompressible magneto-micropolar fluid equations involving Navier-type condition without resistivity on a flat strip T ×(0,1).We obtain results based on the time-weighted energy method,commutator estimates,interpolation inequality,Calderon-Zygmund inequality,and Bootstrap theory.Chapter 5 is to deal with the stability and asymptotic behavior of the solutions to the twodimensional MHD equations with full velocity dissipation and no magnetic diffusion on strip domain R ×(0,1).First,we use spectral analysis to establish the decay estimate of the solutions to the linear equations.Secondly,we use time-weighted energy estimates,commutator estimates,interpolation inequality,Duhamel principle,and other methods to prove a priori bound on the solutions to the nonlinear equations.Finally,we complete the proof of stability through the continuity argument.