New entropy for Korteweg's system, existence of global weak solution and new blow-up criterion

This work is devoted to prove the existence of global weak solution for a general isothermal model of capillary fluids derived by J.E Dunn and J.Serrin (1985) (see \cite{fDS}), which can be used as a phase transition model. More precisely we shall derive in a first part new entropy estimates for the density when we are dealing with specific capillarity coefficient $\kappa(\rho)=\frac{1}{\rho}$ (let us emphasize on the fact that this choice of capillarity exhibits particular regime flows in the case of the compressible Euler system with quantic pressure which corresponds here to the capillarity, see \cite{Antonelli}). This allows us in particular to get enough compactness estimates in order to prove the stability of the global weak solution, the used method follows the works of A. Mellet and A. Vasseur (see \cite{fMV}). Let us point out that the key of the proof is related to the introduction of a new \textit{effective velocity} (which depends strongly on the structure of the viscosity and capillary coefficients).\\ In a second part, we shall give the main result of this paper which consists in new blow-up criterion of Prodi-Serrin type for the Korteweg system involving only a control on the vacuum. It is up our knowledge the first result of this type for a compressible fluid system.no