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

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