Well-posedness And Large-time Behavior Of Compressible Hydrodynamic Equations

The Navier-Stokes/Magnetohydrodynamics(MHD)equations are two important and fundamental models in the field of mathematical physics,which describe the motion of viscous Newton fluids and the motion of conductive fluids in magnetic fields(such as plasma,liquid metal,etc)and have a wide range of applications in fluid mechanics,aerodynamics,aerospace,plasma physics,and other engineering technology.Due to the complicated physical mechanism and mathematical structure,the Navier-Stokes/MHD equations have always been the research hotspot in theoretical PDEs and applied mathematics.Despite the important progress,many important fundamental problems remain unknown,for example,the regularity and uniqueness of weak solutions of three-dimensional equations with large data.The main purpose of this dissertation is to study the global well-posedness theory and asymptotic behavior of the viscous compressible Navier-Stokes/MHD equations.It is divided into four chapters:In Chapter 1,we briefly recall some related results about the well-posedness theory of compressible Navier-Stokes/MHD equations.We also introduce some fundamental inequalities which will be frequently used in the analysis.In Chapter 2,we study the well-posedness and large-time behavior of 1D compressible Navier-Stokes system for viscous and heat-conducting ideal polytropic fluids with temperature-dependent transport coefficients.In the case when the viscosity and heat-conductivity depend only on temperature,the global existence of the solution is still unclear.The key step in the proof of global well-posedness is to prove the uniform upper and lower bounds of specific volume and temperature,and the main difficulties arise from the temperaturedependent transport coefficients which induce many nonlinear terms in the derivations of global estimates.To overcome these difficulties,it is technically assumed that the growth exponent of viscosity coefficient with respect to temperature is appropriately small.Then,by combining the Kazhikhov’s classical method[53]with the careful analysis.we successfully obtain the t-independent a priori estimates which are independent of time,and prove the global existence and asymptotic behavior of strong solutions.In Chapter 3,we focus on the well-posedness and long-time behavior of global solutions to an initial-boundary value problem of compressible planar MHD equations.Compared with the Navier-Stokes equations in Chapter 2,the mathematical structure of MHD equations is more complicated due to the strong interaction between flow field and magnetic field.To overcome the difficulties caused by the presence of magnetic Lorenz force in momentum equation,we have to develop some new ideas to derive some globally new estimates of magnetic field and transverse velocity,and prove the uniform-in-time lower and upper bounds of volume by combining the methods due to Kazhikhov[55].Based on the global a priori estimates and the local existence result,we can establish the global existence theorem and study the long-time behavior of the solutions.In Chapter 4,we consider the three-dimensional isentropic Navier-Stokes equations subject to slip boundary conditions in a bounded and smooth domain.When the initial density is strictly positive,we prove the existence of global classical solutions with small initial energy by using the standard method of energy estimates and the regularity theory of elliptic equations.As a byproduct,it is also shown that the global solution exponentially decays to the non-vacuum equilibrium when t→∞.In particular,the main result implies that if the initial density does not contain vacuum and has an upper bound,then the phenomena of vacuum and mass concentration will not occur.