Oscillations in Compressible Navier-Stokes and Homogenization in Phase Transition problems

In the first part of this article we present some exact solutions for special hyperbolic-parabolic systems with sustained oscillations induced by the initial data, most notably the compressible Navier-Stokes system with non-monotone pressure. This part complements {Tzavaras23} where such examples are extensively studied. The second part deals with the problem of homogenization for one-dimensional models describing phase transitions for viscoelastic materials . Ideas from the kinetic formulation of conservation laws are employed to derive effective equations that describe the propagation of oscillations.