Small-time global null control of no-slip Navier–Stokes flow
Let $\Omega \subset \mathbb{R}^3$ be a bounded connected smooth domain, $\varnothing \ne \omega \Subset \Omega$ an open set and $T > 0$, and let $H$ be the closure in $L^2(\Omega)^3$ of the smooth compactly supported divergence-free vector fields. Consider the Navier–Stokes system with the no-slip condition and a force supported in $\omega$, $$\partial_t u + (u \cdot \nabla) u - \Delta u + \nabla p = \mathbf{1}_\omega f, \qquad \nabla \cdot u = 0 \quad \text{in } \Omega \times (0,T), \qquad u = 0 \ \text{on } \partial\Omega \times (0,T).$$ A Leray–Hopf weak solution is one in $L^\infty(0,T; H) \cap L^2(0,T; H_0^1(\Omega)^3)$, weakly continuous in $L^2$, that satisfies the usual energy inequality with the force.
Fernández-Cara, Guerrero, Imanuvilov and Puel proved local exact controllability to the trajectories for this system, so that initial data small enough in $L^4(\Omega)^3$ can be driven to zero (2004). Coron, Marbach and Sueur proved small-time global exact null controllability in dimensions two and three when the control acts on a part of the boundary meeting each of its connected components and the fluid obeys a Navier slip-with-friction condition on the rest of the boundary (2020); the no-slip condition produces boundary layers of larger amplitude.
Problem. For every $u_0 \in H$, are there $f \in L^2(\omega \times (0,T))^3$ and a Leray–Hopf weak solution with $u(0) = u_0$ and $u(T) = 0$? No smallness condition is imposed on $u_0$.
Source: J.-L. Lions (1991), for controls supported in a small subset of the domain; in this form, Fernández-Cara (2026). Stated by AIM — Open Applied Problems for Leray–Hopf weak solutions.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OC — Optimization and Control
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, Navier–Stokes equations, null controllability
References
@incollection{lions1991,
author = {Lions, Jacques-Louis},
title = {Exact controllability for distributed systems. {S}ome trends and some problems},
booktitle = {Applied and Industrial Mathematics ({V}enice, 1989)},
editor = {Spigler, Renato},
series = {Math. Appl.},
volume = {56},
publisher = {Kluwer Academic Publishers},
address = {Dordrecht},
year = {1991},
pages = {59--84},
doi = {10.1007/978-94-009-1908-2_7},
}
@misc{fernandezcara2026,
author = {Fern{\'a}ndez-Cara, Enrique},
title = {Los {L}ions ({I})},
howpublished = {Blog IMUS, Universidad de Sevilla},
year = {2026},
month = {apr},
url = {https://institucional.us.es/blogimus/2026/04/los-lions-i/},
note = {Section ``Una conjetura''},
}
@article{fernandezcara2025,
author = {Fern{\'a}ndez-Cara, Enrique},
title = {Remarks on control and inverse problems for {PDE}s},
journal = {SeMA J.},
volume = {82},
number = {3},
year = {2025},
pages = {267--288},
doi = {10.1007/s40324-024-00363-7},
}
@article{fernandezcara2004,
author = {Fern{\'a}ndez-Cara, E. and Guerrero, S. and Imanuvilov, O. Yu. and Puel, J.-P.},
title = {Local exact controllability of the {N}avier--{S}tokes system},
journal = {J. Math. Pures Appl. (9)},
volume = {83},
number = {12},
year = {2004},
pages = {1501--1542},
doi = {10.1016/j.matpur.2004.02.010},
}
@article{coron2020,
author = {Coron, Jean-Michel and Marbach, Fr{\'e}d{\'e}ric and Sueur, Franck},
title = {Small-time global exact controllability of the {N}avier--{S}tokes equation with {N}avier slip-with-friction boundary conditions},
journal = {J. Eur. Math. Soc. (JEMS)},
volume = {22},
number = {5},
year = {2020},
pages = {1625--1673},
doi = {10.4171/JEMS/952},
eprint = {1612.08087},
archivePrefix = {arXiv},
}
@misc{aim-localized-navier-stokes-global-null-control,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Small-time global null control of no-slip Navier–Stokes flow}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/255-localized-navier-stokes-global-null-control.md},
note = {Entry 255 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-00534, "Small-time global null control of no-slip Navier–Stokes flow", clopen.solutions.