A class of decomposition of Green's functions for the compressilbe Navier-Stokes 
linearized around a constant state is introduced. The singular structures of the Green's functions 
are developed as  essential devices to use the nonlinearity directly to covert  the 
2nd order quasi-linear PDE into a system of  zero-th order integral equation with regular
integral kernels. The system of integrable equations allows a wider class of functions such as BV solutions. 
We have shown global existence and well-posedness of the compressible Navier-Stokes 
equation for isentropic gas with the gas constant $\gamma \in (0,e)$ in the Lagrangian 
coordinate for the class of the BV functions and point wise $L^\infty$ around a constant state; and the 
underline pointwise structure of the solutions is constructed. It is also shown that for the class 
of BV solution the solution is at most piecewise $C^2$-solution even though the initial data 
is piecewise C^infty.