The Investigation Of The Mathematical Theory Of The Navier-Stokes Equations

Obtained by applying the laws of mass conservation,momentum conservation,and energy conservation to fluid motion,Navier-Stokes equation is the basic series of equations in theoretical fluid mechanics.The study of NavierStokes equations has been a hot topic in recent decades,especially in the study of scheme approximation,the well-posedness of solutions and the limit process of viscous disappearance.In this paper,we study the global existence and large-time behavior of solutions to Navier-Stokes equations,as well as the blow-up criterion of classical solutions in three-dimensional space.we study the limit process as the bulk viscosity goes to zero for global weak solutions to the Navier-Stokes equations of compressible heat-conducting flows with density-dependent viscosity coefficient and general heat conductivity.The main results are as follows:? We establish the global existence of classical solutions to the Cauchy problem for the Navier-Stokes equations with degenerate viscosities in three spatial dimensions with smooth initial data which are of small energy but possibly large oscillations with constant state.Moreover,we obtain some large-time behavior and decay rate estimates of the solutions.? We establish the blow-up criterion of classical solutions to the Cauchy problem for the Navier-Stokes with degenerate viscosities in two and three spatial dimensions without vaccum.? The limit process as the bulk viscosity goes to zero for global weak solutions to the Navier-Stokes equations of compressible heat-conducting flows with density-dependent viscosity coefficient and general heat conductivity was studied.We prove that the limit of the global solutions is a weak solution of the corresponding system with zero bulk viscosity.