Sustained Oscillations in Hyperbolic-Parabolic Systems

2024/04/29 by Athanasios E. Tzavaras, Tzavaras, Athanasios E.
Computer Science · Engineering · Mathematics · #35L65 #35M31 #74H05 #74H35 #74N20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2404.18457

openalex publication_date 2024/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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