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 $$extract_itex$$\gamma \in (0,e)$$/extract_itex$$ in the Lagrangian
coordinate for the class of the BV functions and point wise $$extract_itex$$L^\infty$$/extract_itex$$ 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 $$extract_itex$$C^2$$/extract_itex$$-solution even though the initial data
is piecewise C^infty.