Studies On Some Fluid Coupled Systems

This dissertation is devoted to the study of some fluid coupled systems,including MHD system,Hall-MHD system and a Stokes-Elliptic coupled system.Chapter 1 is the introduction of this dissertation,including our motivations and main results.In Chapter 2,we systematically study the steady states of Hall-MHD system,including existence and asymptotic behaviors.Mainly,we show the following re-sults:(1)Existence of H~1 solution;(2)Existence of H~2 solution,including the existence for small(f,g)and existence for small Hall parameter ?;(3)Asymptotic behaviors on Hall parameter ?,the large Hall parameter limit equations:Bel-trami equations,and asymptotic stability of the steady states;(4)Liouville type theorems for Beltrami fields;(5)Existence of weak solution of steady Hall-MHD system in the whole space and a Liouville type theorem for such solution.In Chapter 3,we study the existence and uniqueness of very weak solution of an MHD-type system.We prove the existence of very weak solution for arbitrary data and the uniqueness for small data.In Chapter 4,we study a coupled system of a semilinear elliptic equation and the Stokes equations.We prove the existence of least energy solution of this system.We also study a singular perturbation problem and find that the limiting properties depend on the power of the singular perturbation parameter.