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 γ∈(0,e) γ∈(0,e) in the Lagrangian coordinate for the class of the BV functions and point wise L ∞ L∞ 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 C2 -solution even though the initial data is piecewise C^infty.