Global Existence And Exponential Stability In H～4 For The Nonlinear Compressible Navier-Stokes Equations

This paper is concerned with the global existence and exponential stability of weaksolution in H4 for a real viscous heat-conducting ?ow with shear viscosity. The systemdescribing this type of ?ow is derived from the general 3-D Navier-Stokes equations.By Lagrangian coordinate transform, we can translate this 3-D equations into 1-Dequations. The results in 1-D space, Qin [14] has already got the global existenceand exponential stability in H1,H2. Qin [17] also has got the global existence andexponential stability of solutions in H4. For the same system as in this paper , Hu [1]has got the global existence and exponential stability in H1,H2. In this paper, we willestablish the global existence and exponential stability of solutions in H4 and give thedelicate estimates.There are altogether three chapters in this dissertation. In Chapter 1, we intro-duce the relative background knowledge, the development state and the problem underconsideration in this thesis; introducing some necessary definitions and main theorems.In Chapter 2, we will give the proof of Theorem1.1, that is, the global existence andexponential stability of solution in H4. In Chapter 3, we will give the proof of Theorem1.2, that is, the exponential stability of solution in H4.