On Global Existence And Boundary-Layer Problem For Compressible Navier-Stokes Equations With Cylindrical Symmetry

This thesis is concerned with two types of compressible Navier-Stokes equations with cylindrical symmetry,that is,the non-isentropic compressible Navier-Stokes equation-s with temperature-dependent viscosity and the isentropic compressible Navier-Stokes equations.The Navier-Stokes equations can be used to describe many physical phe-nomenon of interest in science and engineering.They may be used to model the weather,ocean currents,water flow in a pipe,and air flow around a wing.Furthermore,the in-vestigation on the mathematical theory of the compressible Navier-Stokes equations is one of the most popular directions in the field of pure mathematics.And now a lot of challenging problems remain open.In this thesis,we consider the global existence and boundary-layer problem to the two types of compressible Navier-Stokes equations with cylindrical symmetry.Precisely,·In Chapter 1,we first review some previous works about the well-posedness and boundary-layer problem for compressible Navier-Stokes equations with cylindrical symmetry.Next we introduce the problem considered and the main results in this thesis.·In Chapter 2,we consider the initial boundary value problem of compressible Navier-Stokes equations with cylindrical symmetry when viscosity coefficient A and heat conductivity coefficient k depend on temperature.In case that ?=const.>0,1/c?m??(?)?c(1+?m),k(?)=?q,for m ?(0,1],g?m,we obtain global exis-tence of strong solution.For the assumption 1/c(1+?m)??(?)?c(1+?m),we prove the vanishing shear viscosity limit and get the convergence rate.In this chapter,the acceleration effect in one direction is neglected.However,the temperature-dependent viscosity and heat conduction still captures some main structures of the system.Moreover,we do not need any smallness assumption for the initial data.·In Chapter 3,we consider the initial boundary value problem for the isentropic compressible Navier-Stokes equations with cylindrical symmetry.The existence of boundary layers is well-known when the shear viscosity vanishes.In this chapter,we derive explicit Prandtl type boundary layer equations and prove the global in time stability of the boundary layer profile together with the optimal convergence rate of the vanishing shear viscosity limit without any smallness assumption on the initial and boundary data.