Diffusion stabilises time-periodic solutions in conservation laws coupled to a relaxation oscillator

Abstract

We study a viscous one-dimensional conservation law, coupled to a fast ordinary differential equation. For a vanishing viscosity, the system has an infinite-dimensional family of time-periodic solutions, corresponding to relaxation oscillations. We show that for symmetric initial conditions, a small positive viscosity selects a unique periodic solution, which we prove to be linearly stable. The proof exploits the slow-fast structure through an averaging strategy, as well as spectral-theoretic methods. The results are illustrated by numerical simulations.

54 pages, 13 figures