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PreprintCC BY 4.0arXiv

Self-Similar Singularity of the Euler Equations on ℝ3

  • Adarsh Ganeshram
  • Valentin Duruisseaux
  • Anima Anandkumar
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Abstract

We provide evidence of a finite-time singularity in the 3D Euler equations on the unbounded domain. Using a physics-informed neural network (PINN) with a self-similar ansatz, we find an approximate singular profile for the Euler system at the critical blowup rate of $0.5$ and certify it using a spline representation. The transport field associated with the obtained profile has local outgoing property throughout the domain that suggests linear damping, a key stabilizing mechanism for the candidate profile. We also establish a framework for proving nonlinear stability of the approximate self-similar profile, reducing the analysis to a large but finite collection of explicit estimates and computable constants.

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Cite

BibTeXRIS
BibTeX
@misc{ganeshram2026selfsimilar,
  title = {{Self-Similar Singularity of the Euler Equations on $\mathbbR^3$}},
  author = {Adarsh Ganeshram and Valentin Duruisseaux and Anima Anandkumar},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10867},
  eprint = {2609.10867},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10867},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Self-Similar Singularity of the Euler Equations on ℝ^3
AU  - Adarsh Ganeshram
AU  - Valentin Duruisseaux
AU  - Anima Anandkumar
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.10867
UR  - https://arxiv.org/abs/2609.10867
AB  - We provide evidence of a finite-time singularity in the 3D Euler equations on the unbounded domain. Using a physics-informed neural network (PINN) with a self-similar ansatz, we find an approximate singular profile for the Euler system at the critical blowup rate of $0.5$ and certify it using a spline representation. The transport field associated with the obtained profile has local outgoing property throughout the domain that suggests linear damping, a key stabilizing mechanism for the candidate profile. We also establish a framework for proving nonlinear stability of the approximate self-similar profile, reducing the analysis to a large but finite collection of explicit estimates and computable constants.
ER  -