Global smoothness for critical Darcy electroconvection
Let $\mathbb{T}^2 = (\mathbb{R}/2\pi\mathbb{Z})^2$, $\Lambda = (-\Delta)^{1/2}$, let $R = \nabla \Lambda^{-1}$ be the Riesz transforms and $\mathbb{P}$ the Leray projection onto divergence-free periodic vector fields; at the zero Fourier mode the multipliers of $\Lambda$ and $R$ are zero and that of $\mathbb{P}$ is the identity. For real $q_0 \in C^\infty(\mathbb{T}^2)$ with $\int_{\mathbb{T}^2} q_0 \, dx = 0$, consider the electroconvection equation in a porous medium $$\partial_t q + u \cdot \nabla q + \Lambda q = 0, \qquad u = -\mathbb{P}(q R q), \qquad q(0) = q_0,$$ for a surface charge density $q$ carried by a Darcy flow. Abdo and Ignatova noted that the equation is $L^\infty$-critical, like the critical surface quasi-geostrophic equation but with a nonlinear, doubly nonlocal velocity law. They constructed global weak solutions in the plane, showed there that solutions which stay Hölder continuous are smooth, proved global existence on the torus for small mean-zero data in a critical Besov space, and proved global regularity for large smooth data when $\Lambda$ is replaced by $\Lambda^\alpha$ with $1 < \alpha \le 2$ (2023).
Problem. For every such $q_0$, of any size, does the unique local smooth solution remain smooth for all positive time, that is, is $$\sup_{0 \le t \le T} \|q(t)\|_{H^3(\mathbb{T}^2)} < \infty$$ for every finite $T$?
Source: Abdo, Constantin, Ignatova and Lin (2026), Introduction (p. 3 of arXiv:2508.07439v1, after equations (1.17)–(1.20) with B = 0), who call global existence of smooth solutions for large data a challenging open problem; the equation is that of Abdo and Ignatova (2023). Stated by AIM — Open Applied Problems on the torus for smooth mean-zero data.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- active scalar equations, AIM Open Applied Problems, electroconvection
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{abdo2026magnetic,
author = {Abdo, Elie and Constantin, Peter and Ignatova, Mihaela and Lin, Quyuan},
title = {Electroconvection in a magnetic field},
journal = {Discrete Contin. Dyn. Syst.},
volume = {54},
year = {2026},
pages = {61--86},
doi = {10.3934/dcds.2026023},
eprint = {2508.07439},
archivePrefix = {arXiv},
}
@article{abdo2023porous,
author = {Abdo, Elie and Ignatova, Mihaela},
title = {On electroconvection in porous media},
journal = {Indiana Univ. Math. J.},
volume = {72},
number = {6},
year = {2023},
pages = {2593--2630},
doi = {10.1512/iumj.2023.72.9572},
eprint = {2204.00689},
archivePrefix = {arXiv},
}
@misc{aim-global-smoothness-for-critical-darcy-electroconvection,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global smoothness for critical Darcy electroconvection}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/748-global-smoothness-for-critical-darcy-electroconvection.md},
note = {Entry 748 at commit 4f5b165, 2026-11-04},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01153, "Global smoothness for critical Darcy electroconvection", clopen.solutions.