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From Instability to Singularity Formation in Incompressible Fluids

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arxiv 2310.19780 v1 pith:AOWM5G5J submitted 2023-10-30 math.AP physics.flu-dyn

Tarek M. Elgindi , Federico Pasqualotto This is my paper · ORCID
classification math.APphysics.flu-dyn
keywords instabilityeffectformationsingularitysmoothsolutionssystemalpha
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abstract

We establish finite-time singularity formation for $C^{1,\alpha}$ solutions to the Boussinesq system that are compactly supported on $\mathbb{R}^2$ and infinitely smooth except in the radial direction at the origin. The solutions are smooth in the angular variable at the blow-up point, which was a fundamental obstruction in previous works. This is done by exploiting a second-order effect, related to the classical Rayleigh--B\'enard instability, that overcomes the regularizing effect of transport. A similar result is established for the 3d Euler system based on the Taylor--Couette instability.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Instabilities of internal gravity waves in the two-dimensional Boussinesq system

    math.AP 2025-07 conditional novelty 7.0 of 10

    Small-amplitude plane internal waves in the 2D inviscid Boussinesq system are proven to be spectrally unstable via a rigorous Floquet-Bloch analysis.

  2. Finite-time singularity via multi-layer degenerate pendula for the 2D Boussinesq equation with uniform $C^{1,\sqrt{\frac{4}{3}}-1-\epsilon}\cap L^2$ force

    math.AP 2025-05 conditional novelty 7.0 of 10

    There exist compactly supported, smooth-before-blow-up solutions of the forced 2D Boussinesq equation that blow up in finite time with a uniformly C^{1,alpha} cap L^2 force for every alpha < sqrt(4/3)-1.

  3. On putative self-similarity for incompressible 3D Euler

    math.AP 2026-02 accept novelty 6.0 of 10

    Self-similar blow-up exponents for 3D Euler are shown to satisfy γ≥2/5 for finite-energy solutions and γ≥1/2 for globally self-similar profiles with outgoing or axisymmetric nodal conditions.

  4. Analytic finite-rank corrections for singularly weighted estimates in a computer-assisted proof of 3D Euler singularity

    math.AP 2026-07 unverdicted novelty 2.0 of 10

    Analytic low-rank corrections convert numerically determined global basis functions into exactly vanishing local modes, enforcing |x|^3 vanishing conditions needed for singular weighted stability estimates in computer...

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