This thesis is devoted to the mathematical analysis of some heterogeneous models raised by uid mechanics. In particular, it is devoted to the theoretical study of partial di erential equations used to describe the main models that we present in the following. Firstly, we are interested to study the motion of a incompressible newtonien uids in a basin with degenerate topography. The mathematical model studied derives from 3dincompressible Navier-Stokes equations. We are interested to prove that the Cauchy problem associated is well posed. The second part in my thesis is devoted to study a model that arises from dispersive Navier-Stokes equations (that includes dispersive corrections to the classical compressible Navier-Stokes equations). Our model is derived from the last model assuming that the Mach number is very low. The obtained system is called ghost e ect system, which is so named because it cannot be derived from the Navier-Stokes system of gas dynamics, while it can be derived from kinetic theory. The main goal of this part is to extend a result concerning the local existence of strong solution to a global in time existence of weak solutions. Finally, we are interested to prove certain functional inequalities who have noticeable interest in solving mathematical systems linked to uid mechanics.