** Marcel Zodji (Sorbonne)**

** Dynamics of singularity surfaces for the compressible viscous fluids. **

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: If we are given a density that 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.