Seminar
| Parent Program: | |
|---|---|
| Location: | SLMath: Eisenbud Auditorium, Online/Virtual |
Oscillations in compressible Navier-Stokes equations and homogenization via kinetic equations
The objective of the talk is to study sustained oscillations for hyperbolic-parabolic systems. This problem was motivated by work on the existence theory for viscoelasticity of Kelvin-Voigt type. While weak solutions still exist for initial data in L2, oscillations on the deformation gradient can persist and propagate in time.
The existence of sustained oscillations in hyperbolic-parabolic systems is studied in two classes of systems:
(i) Examples from nonlinear viscoelasticity, and
(ii) the compressible Navier-Stokes system with non-monotone pressures. In several space dimensions oscillatory examples are associated with lack of rank-one convexity of the stored energy.
The subject naturally leads to the problem of deriving effective equations for the associated homogenization problems. This is in general a hard problem, which can be addressed for a simple model of phase transitions using ideas from the kinetic formulation for conservation laws. It can also used (under structural assumptions) to derive homogenization equations for oscillations of the density in the compressible Navier-Stokes system.It leads to kinetic equations coupled with the macroscopic flow.