Compressible Navier-Stokes equation with BV initial data: Part II. Global stability

In previous work \cite{W-Y-Z-local}, we studied the local well-posedness of weak solution to the 1-D full compressible Navier-Stokes equation with initial data of small total variation. Specifically, the local existence, the regularity, and the uniqueness in certain function space of the weak solution have been established. The basis for the previous study is the precise construction of fundamental solution for heat equation with BV conductivity. In this paper, we continue to investigate the global stability and the time asymptotic behavior of the weak solution. The main step is to construct the ``effective Green's function'', which is the combination of the heat kernel with BV coefficient in short time and the Green's function around constant state in long time. The former one captures the quasi-linear nature of the system, while the latter one respects the dissipative structure. Then the weak solution is written into an integral system in terms of this ``effective Green's function'', and the time asymptotic behavior is established based on a priori estimate.