The three-dimensional steady Leray problem with unrestricted boundary fluxes
Let $\Omega \subset \mathbb{R}^3$ be a bounded connected smooth domain whose boundary has finitely many connected components $\Gamma_0, \dots, \Gamma_m$, with outward unit normal $n$, and consider the steady Navier–Stokes problem $$-\nu \Delta u + (u \cdot \nabla) u + \nabla p = f, \quad \nabla \cdot u = 0 \ \text{ in } \Omega, \qquad u = a \ \text{ on } \partial\Omega.$$ Incompressibility forces the total flux condition $$\sum_{j=0}^m \int_{\Gamma_j} a \cdot n \, dS = 0.$$ Leray proved that a solution exists when the flux $\int_{\Gamma_j} a \cdot n \, dS$ through each component vanishes, and existence was later proved for sufficiently small fluxes and under certain symmetry assumptions on the domain and the data. Korobkov, Pileckas and Russo proved existence under the total flux condition alone for bounded planar domains, and for axially symmetric domains in $\mathbb{R}^3$ with axially symmetric data.
Problem. Given any such $\Omega$, any $\nu > 0$, $f \in C^\infty(\overline\Omega; \mathbb{R}^3)$ and $a \in C^\infty(\partial\Omega; \mathbb{R}^3)$ satisfying only the total flux condition, does there exist $u \in H^1(\Omega; \mathbb{R}^3)$ with $\nabla \cdot u = 0$, trace $a$, and $$\nu \int_\Omega \nabla u : \nabla \varphi + \int_\Omega (u \cdot \nabla) u \cdot \varphi = \int_\Omega f \cdot \varphi$$ for every smooth compactly supported divergence-free $\varphi$? The fluxes through the individual components may be nonzero and arbitrarily large, and neither the domain nor the data are assumed axially symmetric.
Source: Leray (1933), who proved existence when the flux through each boundary component vanishes; the general case is known as Leray's problem. See Korobkov, Pileckas and Russo (2015), §1, and (2024). AIM — Open Applied Problems states the three-dimensional case without symmetry.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Leray problem, Navier–Stokes equations, steady flows
References
@article{leray1933,
author = {Leray, Jean},
title = {{\'E}tude de diverses {\'e}quations int{\'e}grales non lin{\'e}aires et de quelques probl{\`e}mes que pose l'hydrodynamique},
journal = {J. Math. Pures Appl. (9)},
volume = {12},
year = {1933},
pages = {1--82},
}
@article{korobkov2015,
author = {Korobkov, Mikhail V. and Pileckas, Konstantin and Russo, Remigio},
title = {Solution of {L}eray's problem for stationary {N}avier--{S}tokes equations in plane and axially symmetric spatial domains},
journal = {Ann. of Math. (2)},
volume = {181},
number = {2},
year = {2015},
pages = {769--807},
doi = {10.4007/annals.2015.181.2.7},
eprint = {1302.0731},
archivePrefix = {arXiv},
}
@book{korobkov2024,
author = {Korobkov, Mikhail and Pileckas, Konstantin and Russo, Remigio},
title = {The Steady {N}avier--{S}tokes System: Basics of the Theory and the {L}eray Problem},
series = {Advances in Mathematical Fluid Mechanics},
publisher = {Birkh{\"a}user},
address = {Cham},
year = {2024},
doi = {10.1007/978-3-031-50898-1},
}
@article{korobkov2014,
author = {Korobkov, Mikhail V. and Pileckas, Konstantin and Pukhnachov, Vladislav V. and Russo, Remigio},
title = {The flux problem for the {N}avier--{S}tokes equations},
journal = {Russian Math. Surveys},
volume = {69},
number = {6},
year = {2014},
pages = {1065--1122},
doi = {10.1070/RM2014v069n06ABEH004928},
}
@misc{aim-leray-three-dimensional-flux,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The three-dimensional steady Leray problem with unrestricted boundary fluxes}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/372-leray-three-dimensional-flux.md},
note = {Entry 372 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-00610, "The three-dimensional steady Leray problem with unrestricted boundary fluxes", clopen.solutions.