The finite-Dirichlet Liouville problem for steady three-dimensional flow
Let $u \in C^\infty(\mathbb{R}^3;\mathbb{R}^3)$ and $p \in C^\infty(\mathbb{R}^3)$ solve the stationary Navier–Stokes equations $$-\Delta u + (u\cdot\nabla)u + \nabla p = 0, \qquad \nabla\cdot u = 0 \qquad \text{in } \mathbb{R}^3,$$ with finite Dirichlet integral and vanishing at infinity: $$\int_{\mathbb{R}^3} |\nabla u|^2\,dx < \infty, \qquad \lim_{|x| \to \infty} u(x) = 0.$$ Galdi showed that such a solution vanishes if in addition $u \in L^{9/2}(\mathbb{R}^3)$ (see his 2011 monograph), and Seregin that it vanishes if in addition $u \in BMO^{-1}(\mathbb{R}^3)$ (2016). Chae proved that it vanishes under additional decay conditions relating the head pressure $\frac12|u|^2 + p$ to $u$ (2026).
Problem. Must $u$ vanish identically? No additional integrability or symmetry of $u$, and no rate of its decay, is assumed.
Source: Going back to Leray (1933); see Galdi (2011), Remark X.9.4, and Tsai (2018), Conjecture 2.5.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Liouville theorems, 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},
}
@book{galdi2011,
author = {Galdi, G. P.},
title = {An Introduction to the Mathematical Theory of the {N}avier--{S}tokes Equations: Steady-State Problems},
edition = {Second},
series = {Springer Monographs in Mathematics},
publisher = {Springer},
address = {New York},
year = {2011},
doi = {10.1007/978-0-387-09620-9},
}
@book{tsai2018,
author = {Tsai, Tai-Peng},
title = {Lectures on {N}avier--{S}tokes Equations},
series = {Graduate Studies in Mathematics},
volume = {192},
publisher = {American Mathematical Society},
address = {Providence, RI},
year = {2018},
doi = {10.1090/gsm/192},
}
@article{seregin2016,
author = {Seregin, G.},
title = {Liouville type theorem for stationary {N}avier--{S}tokes equations},
journal = {Nonlinearity},
volume = {29},
number = {8},
year = {2016},
pages = {2191--2195},
doi = {10.1088/0951-7715/29/8/2191},
eprint = {1512.02915},
archivePrefix = {arXiv},
}
@article{chae2026,
author = {Chae, Dongho},
title = {Liouville type theorems for the stationary {N}avier--{S}tokes equations in $\mathbb{R}^3$},
journal = {Comm. Math. Phys.},
volume = {407},
number = {3},
year = {2026},
pages = {53},
doi = {10.1007/s00220-026-05555-y},
}
@misc{aim-steady-navier-stokes-liouville,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The finite-Dirichlet Liouville problem for steady three-dimensional flow}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/151-steady-navier-stokes-liouville.md},
note = {Entry 151 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-00905, "The finite-Dirichlet Liouville problem for steady three-dimensional flow", clopen.solutions.