Sustained Oscillations in Hyperbolic-Parabolic Systems

TL;DR

This paper constructs examples of persistent oscillating solutions in various hyperbolic-parabolic systems, revealing complex behaviors in mechanics and fluid dynamics with singular diffusion matrices.

Contribution

It provides novel examples of sustained oscillations in hyperbolic-parabolic systems, including nonlinear viscoelasticity, gas dynamics, and Navier-Stokes equations, and studies their existence.

Findings

Examples of oscillating solutions in viscoelasticity, gas dynamics, and Navier-Stokes.

Persistence of oscillations in systems with singular diffusion matrices.

Existence results for linear hyperbolic-parabolic systems with singular diffusion.

Abstract

We construct examples of oscillating solutions with persistent oscillations for various hyperbolic-parabolic systems with singular diffusion matrices that appear in mechanics. These include, an example for the equations of nonlinear viscoelasticity of Kelvin-Voigt type with stored energy that violates rank-one convexity, which amounts to a time-dependent variant of twinning solutions. An example pertaining to the system of gas dynamics with thermal effects for a viscous, adiabatic gas. Finally, an example for the compressible Navier-Stokes system in one-space dimension with nonmonotone pressure function. We also study the existence of oscillating solutions for linear hyperbolic-parabolic systems with singular diffusion matrices.