Weak solution for compressible fluid models of Korteweg type

This work is devoted to proving existence of global weak solutions for a general isothermal model of capillary fluids derived by J.E Dunn and J.Serrin (1985) \cite{3DS}, which can be used as a phase transition model. We distinguish two cases when the dimension N=2 and N=1, in the first case we need that $\frac{1}{\rho }\in L^{\mathrm{\infty }}$, when N=1 we get a weak solution in finite time in the energy space.