Transport-noise Navier–Stokes at critical smoothness minus one half
On $\mathbb{R}^3$ let $\nu > 0$ and let $b_1, \dots, b_K \in C^\infty_c(\mathbb{R}^3; \mathbb{R}^3)$ be divergence-free with $$\nu I - \tfrac12 \sum_k b_k(x) \otimes b_k(x) \ge \delta I \quad (\delta > 0).$$ With $\mathbb{P}$ the Helmholtz projection onto divergence-free fields and independent Brownian motions $W^k$, consider the Itô equation $$du = \bigl[\nu \Delta u - \mathbb{P} \nabla \cdot (u \otimes u)\bigr]\,dt + \sum_k \mathbb{P}\bigl[(b_k \cdot \nabla) u\bigr]\,dW^k_t.$$ Solutions are mild, with the heat semigroup used for both the drift and the stochastic convolution, and $B^{s}_{q,p}$ is the inhomogeneous Littlewood–Paley Besov space. The Besov spaces $B^{3/q-1}_{q,p}$ are critical for the Navier–Stokes equations. For sufficiently regular transport fields, Agresti and Veraar proved local well-posedness in critical Besov spaces $B^{3/q-1}_{q,p}$ of smoothness greater than $-1/2$, that is with $q < 6$. Without noise, Koch and Tataru proved global well-posedness for divergence-free initial data small in $\mathrm{BMO}^{-1}$, a space of smoothness $-1$, and local well-posedness for data in its subspace $\mathrm{VMO}^{-1}$.
Problem. Is the equation locally well posed for every deterministic divergence-free $u_0 \in B^{-1/2}_{6,4}(\mathbb{R}^3)$? More precisely, does it admit a positive stopping time $\tau$ and an adapted mild solution with $$u \in C([0, \tau]; B^{-1/2}_{6,4}) \cap L^4(0, \tau; L^6) \quad \text{almost surely},$$ pathwise unique in this class and depending continuously on $u_0$ in probability in these norms, after stopping on common local intervals? The answer must cover all such transport fields, with no smallness assumption.
Source: Agresti and Veraar (2025), Open Problem 10, which asks for local well-posedness in a critical space of smoothness at most −1/2. Stated by AIM — Open Applied Problems at the endpoint space B^{−1/2}_{6,4}(ℝ³).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, critical spaces, 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{koch2001,
author = {Koch, Herbert and Tataru, Daniel},
title = {Well-posedness for the {N}avier--{S}tokes equations},
journal = {Adv. Math.},
volume = {157},
number = {1},
year = {2001},
pages = {22--35},
doi = {10.1006/aima.2000.1937},
}
@misc{aim-stochastic-navier-stokes-negative-half-critical-data,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Transport-noise Navier–Stokes at critical smoothness minus one half}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/457-stochastic-navier-stokes-negative-half-critical-data.md},
note = {Entry 457 at commit a153f46, 2026-10-13},
}
Resolution claims
We prove local well-posedness of the three-dimensional incompressible Navier--Stokes equation with prescribed smooth, compactly supported, divergence-free Ito transport noise and deterministic initial velocity in the inhomogeneous space $B^{-1/2}_{6,4}(\R^3)$. Under strict stochastic parabolicity, with no further smallness assumption, the solution is heat-mild and belongs almost surely to $C_tB^{-1/2}_{6,4}\cap L^4_tL^6_x$ up to a positive stopping time. Pathwise uniqueness holds in this entire …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Open → Solution claimed | The registry's curators | Local well-posedness for transport-noise Navier--Stokes with $B^{-1/2}_{6,4}(\R^3)$ initial data v1 recorded from AIM @ 24edaad, paper/paper.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00478, "Transport-noise Navier–Stokes at critical smoothness minus one half", clopen.solutions.