Global Well-Posedness And Large Time Behavior Of Solutions To Isentropic Compressible Navier-Stokes Equations

This thesis is devoted to the study of the dynamics of gases in unbounded do-mains, whose movements are controlled by the Navier-stokes equations. In physics, the Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid continuum. We mainly concern the global well-posedness and the large time behavior of the classical solutions of isentropic compress-ible Navier-stokes equations in R3.The dissertation is organized as follows:Chapter 1 is devoted to introducing physical backgrounds and previous mathe-matical research works about the well-posedness of the Navier-Stokes equations. The main problem, main results and methods in this dissertation are also illustrated.In Chapter 2, we are concerned with the global existence of a smooth compressible viscous flow in an infinitely expanded ball. Such a problem is one of the interesting models in studying the theory of global smooth solutions to the compressible Navier-Stokes equations with time dependent boundaries and vacuum states at infinite time. The flow is described by the 3-D isentropic compressible Navier-Stokes equations. From the physical point of view, due to the mass conservation of gases, the moving gases in the expansive ball will gradually become rarefactive and finally tend to a vacuum state with the development of time. We will confirm such an interesting physical phenomenon by the rigorous mathematical proof and simultaneously show that there are no appearances of vacuum domains in any part of the expansive ball.In Chapter 3, we study the large time behavior of the 3-D isentropic compressible Navier-Stokes equation in the partial space-periodic domains, and simultaneously show that the related profile systems can be described by like Navier-Stokes equations with suitable "pressure" functions in lower dimensions. Our proofs are based on the energy methods together with some delicate analysis on the corresponding linearized problems.In Chapter 4, we are concerned the stability and large time behavior of the com-pressible flow between two coaxial cylinders. The flow is described by the 3-D com-pressible Navier-stokes equations in the cylindrical coordinates. We prove that the Couette flow is stable under the small perturbations and give the large time asymp-totic behavior of the solution. Meanwhile, we show that the time-asymptotic leading part of the solution is given by a function satisfying a 1-D heat equation. The proof is based on the combination of the spectral analysis on the linearized operator and a variant of the Matsumura-Nishida energy method.