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Ultraviolet Cascades, Norm Inflation, and the Rendall Instability for the Einstein-Euler System with Positive Cosmological Constant

  • Elliot Marshall
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Abstract

We numerically study perturbations of spatially homogeneous and orthogonal solutions to the Einstein-Euler equations with positive cosmological constant and linear equation of state p=Kρ for K ∈ (1/3,1). Recent numerical and analytic work has demonstrated that the fluid density gradient in these perturbed spacetimes can form steep gradients and blow up as future timelike infinity is approached, a phenomenon now known as the `Rendall instability'. In this article, we study the onset of this instability outside of one spatial dimension for the first time. Our results provide compelling evidence that the Rendall instability causes a forward ultraviolet cascade and Hs norm inflation in the energy density of the fluid from small U(1)-symmetric perturbations. This is, to the best of our knowledge, the first example of small data norm inflation for solutions of the field equations with a positive cosmological constant.

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Cite

BibTeXRIS
BibTeX
@misc{marshall2026ultraviolet,
  title = {{Ultraviolet Cascades, Norm Inflation, and the Rendall Instability for the Einstein-Euler System with Positive Cosmological Constant}},
  author = {Elliot Marshall},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.11375},
  eprint = {2607.11375},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.11375},
}
RIS
TY  - UNPB
TI  - Ultraviolet Cascades, Norm Inflation, and the Rendall Instability for the Einstein-Euler System with Positive Cosmological Constant
AU  - Elliot Marshall
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2607.11375
UR  - https://arxiv.org/abs/2607.11375
AB  - We numerically study perturbations of spatially homogeneous and orthogonal solutions to the Einstein-Euler equations with positive cosmological constant and linear equation of state p=Kρ for K ∈ (1/3,1). Recent numerical and analytic work has demonstrated that the fluid density gradient in these perturbed spacetimes can form steep gradients and blow up as future timelike infinity is approached, a phenomenon now known as the `Rendall instability'. In this article, we study the onset of this instability outside of one spatial dimension for the first time. Our results provide compelling evidence that the Rendall instability causes a forward ultraviolet cascade and H^s norm inflation in the energy density of the fluid from small U(1)-symmetric perturbations. This is, to the best of our knowledge, the first example of small data norm inflation for solutions of the field equations with a positive cosmological constant.
ER  -