Cascade mechanisms for Navier-Stokes blow-up
Abstract
In a recent preprint (arXiv:2511.09556), we exhibited an inverse energy cascade for the 3D Navier-Stokes equations which results in "instantaneous" Type I blow-up and failure of uniqueness in certain sharp regularity classes. Our purpose here is to elucidate this phenomenon further in the setting of the Obukhov dyadic model, and to compare it to the better-known phenomenon of finite-time blow-up. When intermittency is low (α≤ 2), we recover our previous result from the Navier-Stokes setting; in the energy-supercritical case where intermittency is high (α>2), we show that the same can occur in the class of finite energy solutions. We present two different proofs: a soft approach using a Lyapunov function and an explicit multiscale construction. For the inviscid system we illustrate that a similar phenomenon occurs at lower regularity. Finally, we state a finite-time blow-up theorem for a mixed Desnyansky-Novikov-Obukhov model, in which a forward cascade from finitely supported data and no force produces a singularity that is of Type II only by a small margin, with features comparable to the recent forced Navier-Stokes blow-up of OpenAI.
Related
- On the absence of blow-up in the 3D Navier-Stokes equations with transport noise
- Fold Catastrophes in a Hopf Texture Driven by a Navier--Stokes Blow-Up Analog
- Axisymmetric Navier--Stokes with Swirl: Regularity and Blow-Up Constraints
- Blow-up criterion to the compressible full Navier-Stokes equations with inflow-outflow boundary conditions
- Blow-Up of Modified Navier--Stokes Equations in Nonhomogeneous Fourier Spaces
BibTeX
@misc{cheskidov2026cascade,
title = {{Cascade mechanisms for Navier-Stokes blow-up}},
author = {Alexey Cheskidov and Mimi Dai and Stan Palasek},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26790},
eprint = {2609.26790},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26790},
}
RIS
TY - UNPB
TI - Cascade mechanisms for Navier-Stokes blow-up
AU - Alexey Cheskidov
AU - Mimi Dai
AU - Stan Palasek
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26790
UR - https://arxiv.org/abs/2609.26790
AB - In a recent preprint (arXiv:2511.09556), we exhibited an inverse energy cascade for the 3D Navier-Stokes equations which results in "instantaneous" Type I blow-up and failure of uniqueness in certain sharp regularity classes. Our purpose here is to elucidate this phenomenon further in the setting of the Obukhov dyadic model, and to compare it to the better-known phenomenon of finite-time blow-up. When intermittency is low (α≤ 2), we recover our previous result from the Navier-Stokes setting; in the energy-supercritical case where intermittency is high (α>2), we show that the same can occur in the class of finite energy solutions. We present two different proofs: a soft approach using a Lyapunov function and an explicit multiscale construction. For the inviscid system we illustrate that a similar phenomenon occurs at lower regularity. Finally, we state a finite-time blow-up theorem for a mixed Desnyansky-Novikov-Obukhov model, in which a forward cascade from finitely supported data and no force produces a singularity that is of Type II only by a small margin, with features comparable to the recent forced Navier-Stokes blow-up of OpenAI.
ER -