  In the first part of this paper, we prove the existence of global strong
solution for Korteweg system in one dimension. In the second part, motivated by
the processes of vanishing capillarity-viscosity limit in order to select the
physically relevant solutions for a hyperbolic system, we show that the global
strong solution of the Korteweg system converges in the case of a $\gamma$ law
for the pressure ($P(\rho)=a\rho^{\gamma}$, $\gamma>1$) to entropic solution of
the compressible Euler equations. In particular it justifies that the Korteweg
system is suitable for selecting the physical solutions in the case where the
Euler system is strictly hyperbolic. The problem remains open for a Van der
Waals pressure because in this case the system is not strictly hyperbolic and
in particular the classical theory of Lax and Glimm (see \cite{Lax,G}) can not
be used.
