The Study Of Several Hydrodynamic Equations

Fluid mechanics is an important branch of mechanics, which is the study of macroscopic movement of fluid (including gas and liquid) and the inter-actions with other movement patterns. There are a large amount of fluid model in nature, such as heat conductive fluid dynamics model, magnetic fluid dynamics models, and the mathematical model describing the flow of blood flow in the pipeline and so on. This paper mainly discusses the well-posedness of several fluid mechanics equations. We divide the dissertation into the following parts:Chapter1is a preliminary chapter in which we not only recall the back-ground and history about the related models but also state the main results obtained in this dissertation. Some frequently used notations and inequalities are also presented.In Chapter2, the wellposedness of three-dimensional Magnetohydrody-namic(MHD for short) equations are investigated. If the initial data are he-lically symmetric, applying Kato’s theory for the abstract quasi-linear evolu-tion equations and a prior estimates of solution, we get the existence of global strong solutions. Moreover, the large-time behavior of the strong solutions is obtained simultaneously.Chapter3deals with the Cauchy problem of compressible Navier-Stokes-Korteweg equation. We obtain the existence of global smooth solutions using a continuity argument and a priori estimate when the initial data is small enough. Applying the energy method, we show the unique smooth solution of the three dimensional Navier-Stokes-Korteweg sysytem converges globally in time to the strong solution of the three dimensional Navier-Stokes system as the capillarity κ tends to zero.In Chapter4we prove the global existence of classical solutions for a liquid-gas two phase flow in the whole space when the initial data is small enough.In Chapter5Some problems are proposed for further research and ex-plore.