| The Navier-Stokes system,which describes the motion of viscous Newton fluids,is one of the most important and fundamental equations in mathematical physics.Because of its important applications in engineering technology,such as fluid mechanics,aerospace,marine science,oil exploration and so on,the Navier-Stokes system has been the core of both theoretical PDEs and applied mathematics,and many important results have been achieved.But due to the essential difficulties,many important and fundamental problems remain open,for example,the regularity and uniqueness of weak solutions to the threedimensional Navier-Stokes equations.The main purpose of this dissertation is to study the well-posedness theory and the asymptotic behavior of strong/classical solutions to the Cauchy problem of viscous compressible heat-conducting Navier-Stokes equations in the three-dimensional whole domain.It is divided into four chapters.In Chapter 1,we first introduce some related results of the compressible Navier-Stokes equations.We also state the main results obtained in this dissertation,and recall some known inequalities and facts which will be frequently used in the analysis.In Chapter 2,we study the existence and uniqueness of local strong solutions to the initial(boundary)value problem of 3D non-isentropic compressible Navier-Stokes equations subject to less regular data and vacuum in a more general framework.We also show that the strong solution is indeed a classical one strictly away from the initial time,provided the initial density is more regular.In Chapter 3,by virtue of the local well-posedness result established in Chapter 2,we prove the global existence of classical solutions with small total energy,based on some necessary global a prior estimates.As a byproduct,some exponential decay rates of global solutions are also obtained.In Chapter 4,we consider the three-dimensional compressible Navier-Stokes equations in the case when the density is strictly positive.By making use of the"div-curl" decomposition technique,we obtain some new Lp-estimates(p≥3)of the gradient of velocity.As a result,we prove the existence and uniqueness of global solutions belonging to a new and larger functional class. |
