Steady planar flow around an obstacle at arbitrary Reynolds number
Let $D \subset \mathbb{R}^2$ be a bounded simply connected domain with smooth boundary and $\Omega = \mathbb{R}^2 \setminus \overline{D}$. For a viscosity $\nu > 0$ and a velocity $U \in \mathbb{R}^2 \setminus \{0\}$, flow past $D$ with velocity $U$ at infinity is described by $$-\nu\Delta u + (u\cdot\nabla)u + \nabla p = 0, \qquad \operatorname{div} u = 0 \quad \text{in } \Omega,$$ $$u|_{\partial D} = 0, \qquad \lim_{|x| \to \infty} u(x) = U, \qquad \int_\Omega |\nabla u|^2\,dx < \infty.$$ Leray constructed solutions of all conditions except the limit at infinity, with finite Dirichlet integral, as limits of solutions on bounded domains (1933). Korobkov, Pileckas and Russo proved that every solution of the equations and the boundary condition with finite Dirichlet integral converges uniformly at infinity (2019), and that Leray's solutions are nontrivial (2021). Finn and Smith constructed solutions with the prescribed limit $U$ when $|U|/\nu$ is small (1967); for small $|U|/\nu$, Korobkov and Ren proved that these are the only solutions with finite Dirichlet integral (2021) and that Leray's solutions converge to them (2022).
Problem. For every $\nu > 0$ and every $U \in \mathbb{R}^2 \setminus \{0\}$, does there exist a smooth solution $(u, p)$ on $\Omega$, continuous up to the boundary, satisfying all of the conditions above, with $u(x) \to U$ uniformly as $|x| \to \infty$? No smallness condition on $|U|/\nu$ is permitted.
Source: Leray (1933), the problem of steady flow past a body in the plane; existence for arbitrary Reynolds numbers is listed by Yudovich (2003) among the main open problems of mathematical hydrodynamics, see Korobkov and Ren (2022). Stated by AIM — Open Applied Problems for simply connected obstacles.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, exterior domains, flows past obstacles, stationary Navier–Stokes equations
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{yudovich2003,
author = {Yudovich, V. I.},
title = {Eleven great problems of mathematical hydrodynamics},
journal = {Mosc. Math. J.},
volume = {3},
number = {2},
year = {2003},
pages = {711--737},
doi = {10.17323/1609-4514-2003-3-2-711-737},
}
@article{korobkov2022,
author = {Korobkov, Mikhail and Ren, Xiao},
title = {Leray's plane stationary solutions at small {R}eynolds numbers},
journal = {J. Math. Pures Appl. (9)},
volume = {158},
year = {2022},
pages = {71--89},
doi = {10.1016/j.matpur.2021.08.010},
eprint = {2105.08898},
archivePrefix = {arXiv},
}
@article{korobkov2019,
author = {Korobkov, Mikhail and Pileckas, Konstantin and Russo, Remigio},
title = {On convergence of arbitrary {D}-solution of steady {N}avier--{S}tokes system in {2D} exterior domains},
journal = {Arch. Ration. Mech. Anal.},
volume = {233},
number = {1},
year = {2019},
pages = {385--407},
doi = {10.1007/s00205-019-01359-8},
eprint = {1802.04656},
archivePrefix = {arXiv},
}
@article{korobkov2021nontrivial,
author = {Korobkov, Mikhail V. and Pileckas, Konstantin and Russo, Remigio},
title = {Leray's plane steady state solutions are nontrivial},
journal = {Adv. Math.},
volume = {376},
year = {2021},
pages = {107451},
doi = {10.1016/j.aim.2020.107451},
eprint = {1911.11546},
archivePrefix = {arXiv},
}
@article{finn1967,
author = {Finn, Robert and Smith, Donald R.},
title = {On the stationary solutions of the {N}avier--{S}tokes equations in two dimensions},
journal = {Arch. Ration. Mech. Anal.},
volume = {25},
number = {1},
year = {1967},
pages = {26--39},
doi = {10.1007/BF00281420},
}
@article{korobkov2021uniqueness,
author = {Korobkov, Mikhail and Ren, Xiao},
title = {Uniqueness of plane stationary {N}avier--{S}tokes flow past an obstacle},
journal = {Arch. Ration. Mech. Anal.},
volume = {240},
number = {3},
year = {2021},
pages = {1487--1519},
doi = {10.1007/s00205-021-01640-9},
}
@misc{aim-exterior-steady-flow,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Steady planar flow around an obstacle at arbitrary Reynolds number}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/152-exterior-steady-flow.md},
note = {Entry 152 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-00863, "Steady planar flow around an obstacle at arbitrary Reynolds number", clopen.solutions.