  This work is devoted to prove new entropy estimates 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
will derive new estimates for the density and we will give a new structure for
the Korteweg system which allow us to obtain the existence of global weak
solution. The key of the proof comes from the introduction of a new effective
velocity.The proof is widely inspired from the works of A. Mellet and A.
Vasseur (see \cite{fMV}). In a second part, we shall give a Prody-Serrin
blow-up criterion for this system which widely improves the results of
\cite{Hprepa} and the known results on compressible systems.
