A bounded critical norm continuation criterion for stochastic Navier–Stokes
On $\mathbb{R}^3$ let $P$ be the Helmholtz projection onto divergence-free vector fields, let $\nu > 0$, let $W^1, \dots, W^K$ be independent real Brownian motions, and let $b_1, \dots, b_K \in C_c^\infty(\mathbb{R}^3; \mathbb{R}^3)$ be divergence-free, not all zero, with $$\tfrac12 \sum_{k=1}^K |b_k(x) \cdot \xi|^2 \le (\nu - \eta) |\xi|^2 \qquad (x, \xi \in \mathbb{R}^3)$$ for some $\eta > 0$. Consider the Itô equation $$du = \bigl[\nu \Delta u - P((u \cdot \nabla) u)\bigr]\,dt + \sum_{k=1}^K P((b_k \cdot \nabla) u)\,dW^k_t,$$ and use the inhomogeneous Besov norm $\|u\|_{B^{-1/4}_{4,4}} = \bigl(\sum_{j \ge -1} 2^{-j} \|\Delta_j u\|_{L^4}^4\bigr)^{1/4}$ for a smooth dyadic frequency partition; this space is critical for the equation. For a smooth compactly supported divergence-free $u_0$, Agresti and Veraar proved that there is a unique maximal local strong solution $(u, \tau)$ in the class $C([0,\tau); B^{-1/4}_{4,4}) \cap L^4_{\mathrm{loc}}([0,\tau), t^{1/2}\,dt; H^{1/2,4})$, and that $u(t)$ almost surely has no limit in $B^{-1/4}_{4,4}$ as $t \uparrow \tau$ on the event $\{\tau < \infty\}$. In the deterministic case, Escauriaza, Seregin and Šverák proved that a solution whose $L^3$ norm stays bounded does not blow up, and Gallagher, Koch and Planchon proved the same for the critical Besov norms $\dot B^{-1+3/p}_{p,q}$ with $3 < p, q < \infty$.
Problem. Is $$\mathbb{P}\Bigl(\tau < \infty,\ \sup_{0 \le t < \tau} \|u(t)\|_{B^{-1/4}_{4,4}(\mathbb{R}^3)} < \infty\Bigr) = 0?$$ Only boundedness is assumed, not that $u(t)$ has a limit as $t \uparrow \tau$.
Source: Agresti and Veraar (2025), Open Problem 11 (refined endpoint Serrin-type criterion), on the torus or the whole space for admissible parameters; stated by AIM — Open Applied Problems on the whole space with (p, δ, κ, q) = (4, −1/2, 1/2, 4).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, blow-up criteria, stochastic Navier–Stokes equations, transport noise
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{escauriaza2003,
author = {Escauriaza, L. and Seregin, G. A. and {\v{S}}ver{\'a}k, V.},
title = {{$L_{3,\infty}$}-solutions of the {N}avier--{S}tokes equations and backward uniqueness},
journal = {Russian Math. Surveys},
volume = {58},
number = {2},
year = {2003},
pages = {211--250},
doi = {10.1070/RM2003v058n02ABEH000609},
}
@article{gallagher2016,
author = {Gallagher, Isabelle and Koch, Gabriel S. and Planchon, Fabrice},
title = {Blow-up of critical {B}esov norms at a potential {N}avier--{S}tokes singularity},
journal = {Comm. Math. Phys.},
volume = {343},
number = {1},
year = {2016},
pages = {39--82},
doi = {10.1007/s00220-016-2593-z},
eprint = {1407.4156},
archivePrefix = {arXiv},
}
@misc{aim-stochastic-navier-stokes-bounded-critical-norm,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A bounded critical norm continuation criterion for stochastic Navier–Stokes}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/477-stochastic-navier-stokes-bounded-critical-norm.md},
note = {Entry 477 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00477, "A bounded critical norm continuation criterion for stochastic Navier–Stokes", clopen.solutions.