PreprintCC BY 4.0HAL
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.
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BibTeX
@misc{barre2026diffusion,
title = {{Diffusion stabilises time-periodic solutions in conservation laws coupled to a relaxation oscillator}},
author = {Julien Barre and Nils Berglund and Hiroshi Horii},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.24994},
eprint = {2607.24994},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05706363v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Diffusion stabilises time-periodic solutions in conservation laws coupled to a relaxation oscillator
AU - Julien Barre
AU - Nils Berglund
AU - Hiroshi Horii
PY - 2026
DA - 2026/09/19
PB - HAL
DO - 10.48550/arxiv.2607.24994
UR - https://hal.science/hal-05706363v1
AB - 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.
ER -