The motion of a compressible viscous fluid is described by the Navier-Stokes system.  
It is a system of hyperbolic-parabolic mixed-type PDEs. In this talk, we will study
the so-called density patch problem: \emph{If we are given a density that is initially "regular"
on both sides of a hypersurface across which it is discontinuous, will this structure
be preserved over time?}

An important quantity in the mathematical analysis of this system is the so-called effective flux,
which was discovered by Hoff and Smoller (1985). More precisely, the mathematical properties
of this quantity play a crucial role in the study of the propagation of oscillations in
compressible fluids [Serre, 1991], in the construction of weak solutions [P-L Lions, 1996] or the
propagation of discontinuity surfaces [Hoff, 2002], to cite just a few examples.

In the case of density-dependent viscosities, the behavior of the effective flux degenerates,
which renders the analysis more subtle.