Construction of solutions for compressible, isentropic Navier-Stokes equations in one space dimension with nonsmooth initial data

We prove the global existence of weak solutions for the Cauchy problem for the Navier-Stokes equations for one-dimensional, isentropic flow when the initial velocity is in L 2 and the initial density is in L 2 ∩ BV . Solutions are obtained as limits of approximations obtained by building heuristic jump conditions into a semi-discrete difference scheme. This allows for a rather simple analysis in which pointwise control is achieved through piecewise H 1 and total variation estimates.