An inertial manifold for the two-dimensional Navier–Stokes flow
On $\mathbb{T}^2$ fix $\nu > 0$ and a smooth, time-independent, mean-zero, divergence-free force $f$. Let $H$ be the Hilbert space of mean-zero divergence-free $L^2$ velocity fields and $S(t)$ the solution semigroup on $H$ of the Navier–Stokes equations $$\partial_t u + (u \cdot \nabla) u + \nabla p = \nu \Delta u + f, \qquad \nabla \cdot u = 0.$$ A finite-dimensional Lipschitz inertial manifold is here a graph $\mathcal{M} = \{q + \Phi(q) : q \in PH\}$ over a finite-rank orthogonal projection $P$, with $\Phi : PH \to (I - P)H$ Lipschitz, such that $S(t)\mathcal{M} \subset \mathcal{M}$, the global attractor is contained in $\mathcal{M}$, and for some $\gamma > 0$ $$\sup_{u \in B} \operatorname{dist}_H(S(t)u, \mathcal{M}) \le C_B e^{-\gamma t}$$ for every bounded $B \subset H$ and all $t \ge 0$. The classical constructions of inertial manifolds rely on spectral gap conditions, which the two-dimensional Navier–Stokes equations do not satisfy. For the hyperviscous equations, with $\nu(-\Delta)^\beta u$ in place of $-\nu \Delta u$, inertial manifolds on $\mathbb{T}^2$ were obtained by Gal and Guo for $\beta \ge 3/2$ and by Guo for every $\beta > 17/12$, including exponents below $3/2$, where the spectral gap condition is not satisfied.
Problem. Does every such system admit a finite-dimensional Lipschitz inertial manifold?
Source: A long-standing question in the theory of inertial manifolds introduced by Foias, Sell and Temam (1985, 1988); see Zelik (2023), Section 1, and Guo (2026), Section 1.1. Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DS
- Keywords
- AIM Open Applied Problems, global attractors, inertial manifolds, Navier–Stokes equations
References
@article{foias1988,
author = {Foias, Ciprian and Sell, George R. and Temam, Roger},
title = {Inertial manifolds for nonlinear evolutionary equations},
journal = {J. Differential Equations},
volume = {73},
number = {2},
year = {1988},
pages = {309--353},
doi = {10.1016/0022-0396(88)90110-6},
}
@article{zelik2023,
author = {Zelik, Sergey},
title = {Attractors. {T}hen and now},
journal = {Russian Math. Surveys},
volume = {78},
number = {4},
year = {2023},
pages = {635--777},
doi = {10.4213/rm10095e},
eprint = {2208.12101},
archivePrefix = {arXiv},
}
@article{guo2026,
author = {Guo, Yanqiu},
title = {Inertial manifolds for the two-dimensional hyperviscous {N}avier--{S}tokes equations},
journal = {Indiana Univ. Math. J.},
volume = {75},
number = {1},
year = {2026},
pages = {157--180},
doi = {10.1512/iumj.2026.75.60612},
eprint = {2401.14642},
archivePrefix = {arXiv},
}
@article{gal2018,
author = {Gal, Ciprian G. and Guo, Yanqiu},
title = {Inertial manifolds for the hyperviscous {N}avier--{S}tokes equations},
journal = {J. Differential Equations},
volume = {265},
number = {9},
year = {2018},
pages = {4335--4374},
doi = {10.1016/j.jde.2018.06.011},
}
@misc{aim-navier-stokes-inertial-manifold,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{An inertial manifold for the two-dimensional Navier–Stokes flow}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/147-navier-stokes-inertial-manifold.md},
note = {Entry 147 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-00878, "An inertial manifold for the two-dimensional Navier–Stokes flow", clopen.solutions.