Existence of a Global Strong Solution and Vanishing Capillarity-Viscosity Limit in One Dimension for the Korteweg System

In the first part of this paper, we prove the existence of a global strong solution for the 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 a weak-entropy 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; indeed in this case the system is not strictly hyperbolic and in particular the classical theory of Lax [Comm. Pure Appl. Math., 10 (1957), pp. 537--566] and Glimm [Comm. Pure Appl. Math., 18 (1965), pp. 697--715] cannot be used.