Sustained double-exponential gradient growth on the Euler torus
Let $\omega$ solve the two-dimensional Euler equations in vorticity form on the torus $\mathbb{T}^2$, $$\partial_t \omega + u \cdot \nabla \omega = 0, \qquad u = \nabla^\perp \Delta^{-1} \omega,$$ where $\nabla^\perp = (-\partial_2, \partial_1)$ and $\Delta^{-1}$ is the inverse of the Laplacian on mean-zero periodic functions, with a mean-zero initial vorticity $\omega_0 \in C^\infty(\mathbb{T}^2)$. A classical estimate bounds $\|\nabla \omega(t)\|_{L^\infty}$ by a double exponential in $t$. Kiselev and Šverák proved that this rate is attained for all times on a disk, with the growth taking place at the boundary. On the torus, Denisov showed that for every $T > 0$ there are smooth initial data, depending on $T$, whose vorticity gradient, already large at time zero, grows double exponentially over $[0, T]$. Zlatoš constructed solutions on the torus whose vorticity gradient grows exponentially for all time when the vorticity is $C^{1,\alpha}$, and whose vorticity Hessian does so for smooth vorticity. Jeong, Yao and Zhou obtained superlinear growth of the vorticity gradient on the torus for an open set of smooth initial data.
Problem. Is there a mean-zero $\omega_0 \in C^\infty(\mathbb{T}^2)$, together with constants $c > 0$ and $t_0 < \infty$, such that $$\|\nabla \omega(t)\|_{L^\infty} \ge \exp(\exp(ct)) \qquad \text{for every } t \ge t_0?$$ The initial datum is fixed for the whole infinite time interval.
Source: Kiselev and Šverák (2014), §5, who ask whether double-exponential growth of the vorticity gradient can happen in the bulk of the fluid in the periodic or full-plane case; stated in this form, on the torus, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Euler equations, small-scale creation, vorticity gradients
References
@article{kiselev2014,
author = {Kiselev, Alexander and {\v{S}}ver{\'a}k, Vladimir},
title = {Small scale creation for solutions of the incompressible two-dimensional {E}uler equation},
journal = {Ann. of Math. (2)},
volume = {180},
number = {3},
year = {2014},
pages = {1205--1220},
doi = {10.4007/annals.2014.180.3.9},
eprint = {1310.4799},
archivePrefix = {arXiv},
}
@article{denisov2015,
author = {Denisov, Sergey A.},
title = {Double exponential growth of the vorticity gradient for the two-dimensional {E}uler equation},
journal = {Proc. Amer. Math. Soc.},
volume = {143},
number = {3},
year = {2015},
pages = {1199--1210},
doi = {10.1090/S0002-9939-2014-12286-6},
eprint = {1201.1771},
archivePrefix = {arXiv},
}
@article{zlatos2015,
author = {Zlato{\v{s}}, Andrej},
title = {Exponential growth of the vorticity gradient for the {E}uler equation on the torus},
journal = {Adv. Math.},
volume = {268},
year = {2015},
pages = {396--403},
doi = {10.1016/j.aim.2014.08.012},
eprint = {1310.6128},
archivePrefix = {arXiv},
}
@misc{jeong2025,
author = {Jeong, In-Jee and Yao, Yao and Zhou, Tao},
title = {Superlinear gradient growth for {2D} {E}uler equation without boundary},
year = {2025},
eprint = {2507.15739},
archivePrefix = {arXiv},
}
@misc{aim-euler-sustained-double-exponential-growth,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Sustained double-exponential gradient growth on the Euler torus}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/168-euler-sustained-double-exponential-growth.md},
note = {Entry 168 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00861, "Sustained double-exponential gradient growth on the Euler torus", clopen.solutions.