Stability Framework for the Singularity of the Euler Equations on ℝ3
Abstract
In a recent numerical study, we found a high-precision singular profile for the Euler equations on the unbounded domain ℝ3. The present manuscript complements that study by establishing in detail a preliminary 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. Conditional on rigorous certification of the estimates and constants appearing in the argument, and on the candidate profile satisfying the required nonlinear stability conditions, the framework closes the stability proof and, crucially, allows the resulting stable rescaled profile to be reconstructed as an admissible solution in the original variables that becomes singular in finite physical time. With the overall stability and reconstruction mechanisms formulated, the remaining work within this approach is largely quantitative: determining whether the explicit constants and margins can be rigorously certified with sufficient positive margin and, where necessary, sharpening selected analytic estimates.
Related
- Self-Similar Singularity of the Euler Equations on ℝ3
- Long-Time Nonlinear Stability of Large-Amplitude Radial Rarefactive Flows for the Multidimensional Compressible Euler Equations
- Linear Stability and Inviscid Damping of Monotone Shear Flows for 2D Compressible Euler Equations
- Uniqueness and stability of planar shock wave to 2D compressible Euler equations in the vanishing viscosity limit of compressible Navier-Stokes equations
- Vorticity blowup in 2D compressible Euler equations
BibTeX
@misc{duruisseaux2026stability,
title = {{Stability Framework for the Singularity of the Euler Equations on $\mathbbR^3$}},
author = {Valentin Duruisseaux and Adarsh Ganeshram and Robert J. George and Anima Anandkumar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10860},
eprint = {2609.10860},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10860},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stability Framework for the Singularity of the Euler Equations on ℝ^3
AU - Valentin Duruisseaux
AU - Adarsh Ganeshram
AU - Robert J. George
AU - Anima Anandkumar
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.10860
UR - https://arxiv.org/abs/2609.10860
AB - In a recent numerical study, we found a high-precision singular profile for the Euler equations on the unbounded domain ℝ^3. The present manuscript complements that study by establishing in detail a preliminary 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. Conditional on rigorous certification of the estimates and constants appearing in the argument, and on the candidate profile satisfying the required nonlinear stability conditions, the framework closes the stability proof and, crucially, allows the resulting stable rescaled profile to be reconstructed as an admissible solution in the original variables that becomes singular in finite physical time. With the overall stability and reconstruction mechanisms formulated, the remaining work within this approach is largely quantitative: determining whether the explicit constants and margins can be rigorously certified with sufficient positive margin and, where necessary, sharpening selected analytic estimates.
ER -