| Abstract: |
| We present examples of sustained oscillations for hyperbolic-parabolic systems.
In the existence theory for viscoelasticity of Kelvin-Voigt type, oscillations on the deformation gradient
can persist and propagate in time. The existence of sustained oscillations is demonstrated in two classes
of systems: (i) Examples from nonlinear viscoelasticity with elastic strain energy that is not rank-1
convex, and (ii) in the compressible Navier-Stokes system with non-monotone pressures.
The subject naturally leads to the problem of deriving effective equations for the associated homog-
enization problems. This problem is addressed using ideas from the kinetic formulation for conservation
laws. One derives homogenization equations for oscillations of the density in one-dimensional models
of viscoelasticity with non-monotone stresses and also for the compressible Navier-Stokes system with
nonmonotone pressures. This leads to effective systems consisting of kinetic equations coupled with the
macroscopic flow. |
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