Research On Global Well-posedness Of Solutions For Two Kinds Of Incompressible Hydrodynamics Equations

The mathematical model of fluid dynamics is derived from the conservation of the mass of the fluid,the conservation of momentum,the conservation of energy and the basic laws of thermodynamics.It plays an important role in theory and scientific calculations in many fields such as hydrodynamics,atmospheric,marine science and petrochemicals.Navier-Stokes system is a basic model for describing fluid dynamics.The study of this model and its coupling model with other equations has always been a hot topic in the research of nonlinear partial differential equations.This dissertation mainly studies the well-posedness of the solutions of two types of classical hydrodynamic equations.The main results are stated as follows.In the first part,we studied an initial boundary value problem for the full incompressible Navier-Stokes equations with viscosity and heat conductivity depending on temperature by the power law of Chapman-Enskog.With this focus,by using the continuation argument and the time-weighted a priori estimates,the existence of global-in-time strong solution,under some appropriate smallness assumptions on initial data,has been proved in this paper.Moreover,the large-time behavior and decay rate estimates of the strong solution are obtained.In the second part,we considered the initial boundary value problem for the incompressible heat-conducting magnetohydrodynamic flows with the magnetic diffusivity ,viscosity coefficient and heat conductivity depending on temperature by the power law of Chapman-Enskog.The a Priori estimates are obtained by establishing global a priori assumptions and the time-weighted a priori estimates.Based on the continuation argument,the global well-posedness of strong solution,under some appropriate smallness assumptions on initial data,has been proved in this paper.In addition,the large-time behavior and decay rate estimates of the strong solution are obtained.