Existence And Uniqueness Of Solutions To Problems From Fluid Dynamics

In this thesis,we study three models coupled with Navier-Stokes equations:isentropic compressible MHD,compressible nematic liquid crystal flows,incompressible fluid models of Korteweg type.This thesis consists of the following five chapters:In Chapter 1,we introduce the background of the models with the current progress in the field,and describe the results of the thesis.Chapter 2 deals with the classical solutions of isentropic compressible MHD equations in R3,where the viscosity coefficient ? is large enough,and initial density could be vacuum or non-vacuum.We obtain the global existence,uniqueness,and large time behavior of the classical solutions,by using energy estimates,together with the known local existence result and continuous method.Chapter 3 considers the strong solutions of compressible nematic liquid crystal flows in R2,where the initial density p decay not too slow at infinity,with possible-vacuum permitted.We prove the local existence and uniqueness of strong solutions by bounded domain approximation with a priori estimates,the local existence for bounded domains and a standard limit procedure.Chapter 4 concerns the strong solutions of incompressible fluid models of Korteweg type equations in R2,with the initial density p decaying not too slow at infinity,but vacuum admitted.We establish the local existence and uniqueness of strong solutions via procedures similar to those in Chapter 3.In Chapter 5,we summarize the main results of the thesis,and propose the problems to be considered.