A remark on the Kazhikhov-Smagulov type model: The vanishing initial density

This work deals with the global existence of weak solutions for a Kazhikhov-Smagulov type system with a density which may or not vanish. Our model is formally equivalent to the physical compressible model with Fick's law, in contrast to those in previous works. This model may be used for addressing environmental problems such as propagation of pollutants and avalanche modelling. We also explain why this system may be seen as a physical regularization of the standard nonhomogeneous incompressible Navier-Stokes equations and we give an existence result with an initial density less regular but away from the vacuum.