Global Existence And Exponential Stability In H～2 For A Real Viscous Heat-conducting Flow With Shear Viscosity

This paper is concerned with the global existence and exponential stability of weak solutions in H~1 and H~2 for a real viscous heat-conducting flow with shear viscosity The system describing this type of flow is derived from the general 3-D Navier-Stokes equations. By Lagrangian coordinate transform, we can translate this 3-D equations into 1-D equations. The results in 1-D space, Qin [19] has already got the global existence and exponential stability of solutions in H~1, H~2. Qin [23] also has got the global existence and exponential stability of solutions in H~4. For the the same system as in this paper, Wang [24] get the global existence in H~1, but his assumptions is different from those in this paper. Zhang, et al [25] got the exponential stability of solutions in H~1, but he didn't give detailed proof. In this paper, we will establish the global existence and exponential stability of solutions in H~1, H~2 and give the detail proof.There are altogether three chapters in this thesis. In Chapter 1, we change 3-D equations into 1-D equations; introduce the relative background knowledge, the development state and the problem under consideration in this thesis; introduce some necessary definitions and main theorems. In Chapter 2, we will give the proof of Theorem 1.1, that is, the global existence and exponential stability of solutions in H~1. In Chapter 3, we will give the proof of Theorem 1.2, that is, the global existence and exponential stability of solutions in H~2.