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

In the previous work (Arch. Ration. Mech. Anal. 245: 375-477, 2022), we studied the well-posedness of weak solutions to the 1-D full compressible Navier-Stokes equation with initial data of small total variation. The local existence, the regularity, and the uniqueness in certain function spaces of the weak solution have been established. Moreover, the global existence is also announced there. In this paper, we continue to investigate the global stability and the time asymptotic behavior of the weak solution, and provide the details of the proof. The main step is to construct the