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

TL;DR

This paper derives effective kinetic equations for oscillatory solutions in viscoelasticity and compressible Navier-Stokes systems, combining homogenization and kinetic formulations.

Contribution

It introduces a method to derive coupled kinetic-macroscopic equations for oscillatory viscoelastic and fluid systems using homogenization techniques.

Findings

Constructed solutions with persistent oscillations for viscoelasticity and Navier-Stokes.

Derived effective kinetic equations coupled with macroscopic flow.

Applied kinetic formulation to homogenize complex oscillatory systems.

Abstract

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.