The motion of a compressible viscous barotropic 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 is initially discontinuous across a C^(1+\alpha) curve alpha and alpha- Hölder continuous on the two disjoint components delimited by gamma, is this structure preserved in time?
An important quantity in the mathematical analysis of this system is the so-called effective flux, which was discovered by Hoff and Smoller in 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.