  In the first main result of this paper we prove that one can approximate
discontinious solutions of the 1d Navier Stokes system with solutions of the 1d
Navier-Stokes-Korteweg system as the capilarity parameter tends to 0. Moreover,
we allow the viscosity coefficients $\mu$ = $\mu$ ($\rho$) to degenerate near
vaccum. In order to obtain this result, we propose two main technical
novelties. First of all, we provide an upper bound for the density verifing NSK
that does not degenerate when the capillarity coefficient tends to 0. Second of
all, we are able to show that the positive part of the effective velocity is
bounded uniformly w.r.t. the capillary coefficient. This turns out to be
crucial in providing a lower bound for the density. The second main result
states the existene of unique finite-energy global strong solutions for the 1d
Navier-Stokes system assuming only that $\rho$0, 1/$\rho$0 $\in$ L $\infty$.
This last result finds itself a natural application in the context of the
mathematical modeling of multiphase flows.
