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PreprintarXiv licencearXiv

Long-term behaviour of the CAO-system: Optimal polynomial H1-convergence to degenerate equilibria

  • Tim Binz
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Abstract

This article is concerned with a definitive analysis of its long-term dynamics of the CAO-system. We establish the convergence of global strong solutions to constant equilibria in the full H1-topology in the non-perturbative regime. Remarkably, we demonstrate that the CAO-system exhibits a polynomial rate of decay and we determine the optimal rate. This phenomenon stands in stark contrast to the exponential convergence typically expected for uniformly parabolic systems on bounded domains. This algebraic slowing is shown to be a purely nonlinear effect, driven by the emergence of a slow manifold (center manifold) induced by nonlinear coupling conditions. To the best of our knowledge, this provides the first instance of a global polynomial decay result for a non-gradient, non-local dissipative system on a bounded domain.

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Cite

BibTeXRIS
BibTeX
@misc{binz2026longterm,
  title = {{Long-term behaviour of the CAO-system: Optimal polynomial $H^1$-convergence to degenerate equilibria}},
  author = {Tim Binz},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08664},
  eprint = {2609.08664},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08664},
}
RIS
TY  - UNPB
TI  - Long-term behaviour of the CAO-system: Optimal polynomial H^1-convergence to degenerate equilibria
AU  - Tim Binz
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08664
UR  - https://arxiv.org/abs/2609.08664
AB  - This article is concerned with a definitive analysis of its long-term dynamics of the CAO-system. We establish the convergence of global strong solutions to constant equilibria in the full H^1-topology in the non-perturbative regime. Remarkably, we demonstrate that the CAO-system exhibits a polynomial rate of decay and we determine the optimal rate. This phenomenon stands in stark contrast to the exponential convergence typically expected for uniformly parabolic systems on bounded domains. This algebraic slowing is shown to be a purely nonlinear effect, driven by the emergence of a slow manifold (center manifold) induced by nonlinear coupling conditions. To the best of our knowledge, this provides the first instance of a global polynomial decay result for a non-gradient, non-local dissipative system on a bounded domain.
ER  -