The Study Of Fluid Dynamical Models

In this paper,we will use the asymptotic expansion method in the singular perturbation theory,energy method,and weighted Sobolev embedding technique,etc.Three dimensional drift-diffusion models and the electro-diffusion system arising in electro-hydrodynamics are studied in depth.This paper is divided into four chapters.In Chapter 1,the physical back-ground of the above two kinds of equations,the present situation of the study are briefly introduced.For convenience,we also enumerate some of the knowledge used in this paper.In Chapter 2,quasi-neutral limit and the boundary layer problem for the three dimensional drift-diffusion model with the general mobilities of two kinds of charges and in the bounded domain is studied.And the difference between the two numbers is very small,i.e.there exists a positive constant???₀,suitably small and independent of?,such that|?_n-?_p|??.Firstly,change the drift-diffusion models into an equivalent models for the density and the electric field by introducing the density transformation.Secondly,construct an approximate solution to the new system with boundary layer functions and inner functions,where,the inner density function include zero order,second order and third order,the boundary layer function include second to fourth order;the inner electric function include zero order,second order and third order,the boundary layer function include zero order to fifth order.Next,by using the matched asymptotic expansion method of singular perturbation problem,we can get the inner functions and boundary layer functions equations,and the properties of the approximate solutions.Finally,we derive the error functions equations,and by using Cauchy-Schwarz inequality,Sobolev's lemma,integrating by parts get the energy estimate and the convergence theorem.In Chapter 3,we study quasi-neutral limit and the initial layer problem of the electro-diffusion model arising in electro hydrodynamics witch is the cou-pled Planck-Nernst-Poisson and Navier-Stokes equations.By using the matched asymptotic expanson method of singular perturbation problem,we derive the e-quations of the initial layer functions and the error functions.We get the conver-gence theorem,by using the properties of the approximate solution,integrating by parts,and the standard elliptic regularity estimates.In Chapter 4,we study quasi-neutral limit and the boundary layer problem of the three dimensional incompressible Planck-Nernst-Poisson and Navier-Stokes equations for electro hydrodynamics with the general mobilities of two kinds of charges and in the bounded domain.We chosen the well prepared initial data to avoid the appearance of the initial layer.A new singular?-weighted entropy inequality involved in the oscillating factor O?1/??2??is established so as to obtain the desired estimates for the error functions uniformly on?.By using energy method and Gronwall inequality,we get the estimate of the tangential derivatives and the normal derivatives of the solutions in the meantime.With the help of Sobolev embedding theorem and the standard elliptic regularity estimates,we get the convergence theorem.