Existence of global weak solution for compressible fluid models with a capillary tensor for discontinuous interfaces

This work is devoted to proving the global existence of weak solution for a general isothermal model of capillary fluids derived in his modern form by C. Rohde in [21], which can be used as a phase transition model. First, inspired by the result by P.-L. Lions in [19] on the Navier- Stokes compressible system we show the global stability of weak solutions for our system with is entropic pressure and next with general pressure. Finally, we consider perturbations of a stable equilibrium.ou