Derivation of effective kinetic equations describing oscillations in viscoelasticity and in compressible Navier-Stokes

These lecture notes are devoted to solutions of hyperbolic-parabolic systems with persistent oscillations.

We consider two examples both from mechanics: (i) The system of viscoelasticity of Kelvin-Voigt type with strain energies involving double well potentials, as employed in phase transitions.

(ii) The compressible Navier-Stokes equations for a barotropic gas.

For each system we construct solutions with persistent oscillations.

In a later part we consider the nonlinear homogenization problem.

For the systems of viscoelasticity in one-space dimension in Lagrangian coordinates, and for the compressible Navier-Stokes system for barotropic fluids we show how ideas from the kinetic formulation of conservation laws can be used to derive effective equations.

The effective equation consists by a kinetic equation coupled with the macroscopic flow.