Global regularity for critical SQG with state-dependent noise
On $\mathbb{T}^2$ let $\Lambda = (-\Delta)^{1/2}$ and $R^\perp = (-R_2, R_1)$, where $R_1, R_2$ are the Riesz transforms, all with zero Fourier multiplier at frequency zero. Consider the critical surface quasi-geostrophic equation with Itô noise $$d\theta + \bigl[R^\perp\theta \cdot \nabla\theta + \Lambda\theta\bigr]\,dt = \sum_{k=1}^K g_k(x, \theta)\,dW^k_t, \qquad \theta(0) = \theta_0.$$ The initial temperature $\theta_0$ is any deterministic smooth periodic function. The $g_k$ are finitely many smooth functions on $\mathbb{T}^2 \times \mathbb{R}$, globally Lipschitz in their scalar argument uniformly in $x$, with bounded derivatives of positive order and bounded $g_k(\cdot, 0)$, and $W^1, \dots, W^K$ are independent standard Brownian motions. Without noise, Kiselev, Nazarov and Volberg proved global well-posedness for smooth periodic data, and Caffarelli and Vasseur proved global regularity of weak solutions by a different method. With dissipation $(-\Delta)^\alpha$ in place of $\Lambda$ and multiplicative noise on $\mathbb{T}^2$, Röckner, Zhu and Zhu proved existence of weak solutions for every $\alpha \in (0,1)$, and existence and uniqueness of probabilistically strong solutions for $\alpha > 1/2$; for $\alpha = 1/2$ and linear noise $\sum_j b_j \theta \circ dW^j$ with real constants $b_j$, they obtained a unique global regular solution from $H^1$ data. For $\alpha \in (1/2, 1)$ and globally Lipschitz noise coefficients, Agresti and Veraar proved global existence and uniqueness in critical Besov spaces. On $\mathbb{R}^2$, Chen and Zhang proved well-posedness of the critical equation with additive noise.
Problem. Does every such equation have a pathwise unique global adapted solution with paths in $C([0,T]; H^3(\mathbb{T}^2))$ for every finite $T$, satisfying the integral equation in $H^1$? Equivalently, can its local $H^3$ solution blow up with positive probability? No smallness of the data or the noise is imposed, and the $g_k$ need not be linear in $\theta$.
Source: Agresti and Veraar (2025), Remark 8.23, on the borderline case α = 1/2 excluded from their Theorem 8.22 (see also Röckner, Zhu and Zhu (2015), Remark 5.7); stated by AIM — Open Applied Problems for smooth data and finitely many smooth noise coefficients.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, global regularity, multiplicative noise, surface quasi-geostrophic equations
References
@article{agresti2025survey,
author = {Agresti, Antonio and Veraar, Mark},
title = {Nonlinear {SPDE}s and maximal regularity: an extended survey},
journal = {NoDEA Nonlinear Differential Equations Appl.},
volume = {32},
number = {6},
year = {2025},
pages = {123},
doi = {10.1007/s00030-025-01090-2},
eprint = {2501.18561},
archivePrefix = {arXiv},
}
@article{kiselev2007,
author = {Kiselev, A. and Nazarov, F. and Volberg, A.},
title = {Global well-posedness for the critical 2{D} 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{rockner2015,
author = {R{\"o}ckner, Michael and Zhu, Rongchan and Zhu, Xiangchan},
title = {Sub and supercritical stochastic quasi-geostrophic equation},
journal = {Ann. Probab.},
volume = {43},
number = {3},
year = {2015},
pages = {1202--1273},
doi = {10.1214/13-AOP887},
eprint = {1110.1984},
archivePrefix = {arXiv},
}
@misc{chen2025,
author = {Chen, Zhen-Qing and Zhang, Xicheng},
title = {Harnack inequalities for nonlocal operators with supercritical drifts and their applications},
year = {2025},
eprint = {2511.13012},
archivePrefix = {arXiv},
}
@misc{aim-critical-stochastic-sqg-global-regularity,
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 critical SQG with state-dependent noise}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/478-critical-stochastic-sqg-global-regularity.md},
note = {Entry 478 at commit a153f46, 2026-10-13},
}
Resolution claims
We prove almost-sure global regularity and pathwise uniqueness for the critical surface quasi-geostrophic equation on the two-dimensional torus with a finite collection of smooth nonlinear state-dependent Itô noise coefficients. For every deterministic smooth datum, globally scalar-Lipschitz coefficients with bounded positive-order derivatives give a global adapted solution continuous in $H^3$ on every finite time interval, with the integral equation holding in …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Open → Solution claimed | The registry's curators | Global regularity for critical stochastic SQG with finite-dimensional nonlinear noise v1 recorded from AIM @ 24edaad, paper/manuscript.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00420, "Global regularity for critical SQG with state-dependent noise", clopen.solutions.