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

TL;DR

This paper presents exact solutions showing sustained oscillations in compressible Navier-Stokes systems and explores homogenization techniques for phase transition models in viscoelastic materials, advancing understanding of oscillatory behaviors.

Contribution

It provides new exact solutions for hyperbolic-parabolic systems with oscillations and applies homogenization methods to phase transition models, linking oscillations to effective equations.

Findings

Exact solutions for oscillatory compressible Navier-Stokes systems.

Homogenized equations describing phase transition oscillations.

Insights into propagation of oscillations in viscoelastic materials.

Abstract

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 \cite{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.