Asymptotic Limits Of The Magnetohydrodynamic Equations And The Chemotaxis-navier-stokes Equations

In the study of the hydrodynamics mathematical theory,the analysis of the asymptotic mechanism of fluid dynamics models is a very important research topic.The research on the asymptotic mechanism can help us to comprehend some physical phenomena.For example,the outcomes of the inviscid limit and its convergence rate are helpful to explain the turbulent phenomenona in some fluids.In this thesis,we mainly investigatea)the zero viscosity-magnetic diffusion limit of the incompressible magnetohydrody-namic(MHD)equations;b)the vanishing viscosity limit of the chemotaxis-Navier-Stokes equations.In Chapter 1,we introduce the background and some known results of the incompressible MHD equations and the chemotaxis-Navier-Stokes equations.In Chapter 2,we study the zero viscosity-magnetic diffusion limit for the 3D viscous homogeneous incompressible MHD equations in a periodic domain in the framework of Gevrey class.Moreover,the convergence rate is given in Gevrey class.In Chapter 3,we establish the existence of the global weak solutions for the 3D viscous nonhomogeneous incompressible MHD equations with Navier boundary conditions for the velocity field and the magnetic field in a smooth bounded domain.We also prove that these weak solutions converge to the strong one of the ideal nonhomogeneous incompressible MHD equations in energy space when the viscosity and resistivity coefficients go to zero simultaneously.In Chapter 4,we investigate the uniform regularity and zero viscosity-magnetic diffu-sion limit for the viscous homogeneous incompressible MHD equations in a bounded domain in conormal Sobolev spaces,where we supplement the Navier boundary conditions for the velocity field and the magnetic field,and the viscosity coefficient is equal to the resistivity coefficient.Moreover,we also get the rates of the convergence in some spaces.In Chapter 5,we study the uniform regularity and vanishing viscosity limit for the in-compressible chemotaxis-Navier-Stokes equations with the Navier boundary conditions for velocity field and the Neumann boundary condition for cell density and chemical concentra-tion in a bounded domain in conormal Sobolev spaces.We verify the convergence of the velocity field in L2,and the convergence of the cell density and the chemical concentration in H1.