Global regularity for supercritical dissipative surface quasi-geostrophic flow
Let $0 < \gamma < 1$ and consider the dissipative surface quasi-geostrophic equation on the unit torus $\mathbb{T}^2 = \mathbb{R}^2/\mathbb{Z}^2$, $$\partial_t \theta + (\mathcal{R}^\perp \theta) \cdot \nabla \theta + (-\Delta)^{\gamma/2} \theta = 0, \qquad \theta(0) = \theta_0,$$ for a real mean-zero unknown $\theta$. Here $\mathcal{R}^\perp = (-\mathcal{R}_2, \mathcal{R}_1)$, the Riesz transform $\mathcal{R}_j$ has Fourier multiplier $-i k_j/|k|$ at $k \in \mathbb{Z}^2 \setminus \{0\}$, and $(-\Delta)^{\gamma/2}$ has multiplier $(2\pi|k|)^\gamma$. For stronger dissipation, $1 < \gamma \le 2$, Constantin and Wu proved global existence and uniqueness of smooth solutions; for the critical exponent $\gamma = 1$, global regularity for arbitrary smooth data was proved by Kiselev, Nazarov and Volberg and by Caffarelli and Vasseur. For every $\gamma \in (0, 1)$, Dabkowski proved that a solution that stays smooth up to a time depending only on $\gamma$ and $\|\theta_0\|_{L^\infty}$ stays smooth for all later times. Coti Zelati and Vicol proved that for every $R > 0$ there is $\gamma_1(R) < 1$ such that, for $\gamma_1(R) \le \gamma < 1$, every $\theta_0 \in H^2$ with $\|\theta_0\|_{L^2}^{\gamma/2} \|\theta_0\|_{\dot H^2}^{1-\gamma/2} \le R$ generates a unique global classical solution.
Problem. For each fixed $\gamma \in (0, 1)$, does every smooth real mean-zero datum $\theta_0$ on $\mathbb{T}^2$ generate a global smooth solution? No smallness of $\theta_0$ is assumed, and $\gamma$ may not depend on $\theta_0$.
Source: A long-standing question for the dissipative surface quasi-geostrophic equation, after Constantin and Wu (1999), who treated the subcritical range; see Coti Zelati and Vicol (2016). Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, fractional dissipation, global regularity, quasi-geostrophic equations
References
@article{constantin1999,
author = {Constantin, Peter and Wu, Jiahong},
title = {Behavior of solutions of {2D} quasi-geostrophic equations},
journal = {SIAM J. Math. Anal.},
volume = {30},
number = {5},
year = {1999},
pages = {937--948},
doi = {10.1137/S0036141098337333},
}
@article{cotizelati2016,
author = {Coti Zelati, Michele and Vicol, Vlad},
title = {On the global regularity for the supercritical {SQG} equation},
journal = {Indiana Univ. Math. J.},
volume = {65},
number = {2},
year = {2016},
pages = {535--552},
doi = {10.1512/iumj.2016.65.5807},
eprint = {1410.3186},
archivePrefix = {arXiv},
}
@article{kiselev2007,
author = {Kiselev, A. and Nazarov, F. and Volberg, A.},
title = {Global well-posedness for the critical {2D} dissipative quasi-geostrophic equation},
journal = {Invent. Math.},
volume = {167},
number = {3},
year = {2007},
pages = {445--453},
doi = {10.1007/s00222-006-0020-3},
}
@article{caffarelli2010,
author = {Caffarelli, Luis A. and Vasseur, Alexis},
title = {Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation},
journal = {Ann. of Math. (2)},
volume = {171},
number = {3},
year = {2010},
pages = {1903--1930},
doi = {10.4007/annals.2010.171.1903},
}
@article{dabkowski2011,
author = {Dabkowski, Michael},
title = {Eventual regularity of the solutions to the supercritical dissipative quasi-geostrophic equation},
journal = {Geom. Funct. Anal.},
volume = {21},
number = {1},
year = {2011},
pages = {1--13},
doi = {10.1007/s00039-011-0108-9},
}
@misc{aim-supercritical-sqg,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global regularity for supercritical dissipative surface quasi-geostrophic flow}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/072-supercritical-sqg.md},
note = {Entry 072 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-00646, "Global regularity for supercritical dissipative surface quasi-geostrophic flow", clopen.solutions.