It is established recently by Liu-Yu a constructive existence theory of weak solution to isentropic Navier-Stokes equation with initial data of small total variation. The key ingredient in their work is the pointwise structures of heat kernel with BV coefficient. In this talk, we will first review their result. Then, by refining the heat kernel estimates, we prove the regularity and uniqueness of the weak solution. Moreover, if the initial perturbation in BV class is localized in space, we can describe the wave propagation precisely. This talk is based on joint works with Shih-Hsien Yu and Xiongtao Zhang.