Anomalous viscous dissipation with fixed large-scale forcing
On the torus $\mathbb{T}^3$ consider the forced Navier–Stokes equations $$\partial_t u + (u \cdot \nabla) u + \nabla p = \nu \Delta u + f, \qquad \operatorname{div} u = 0,$$ with $\|\cdot\|_2$ the norm of $L^2(\mathbb{T}^3)$. A Leray–Hopf solution is a distributional solution in $L^\infty_{\mathrm{loc},t} L^2_x \cap L^2_{\mathrm{loc},t} H^1_x$ satisfying the energy inequality. Buckmaster and Vicol asked for Leray–Hopf solutions with smooth forcing acting only at viscosity-independent large scales whose long-time and space averaged dissipation rate has a finite nonzero limit as $\nu \to 0$. Bruè and De Lellis constructed smooth solutions on $\mathbb{T}^3 \times [0,1]$ with one smooth initial datum and forces $f_{\nu_m}$ bounded in $C([0,1]; C^\alpha(\mathbb{T}^3))$ uniformly in $m$ for every $\alpha < 1$, such that $\liminf_{m} \nu_m \int_0^1 \|\nabla u_{\nu_m}(t)\|_2^2 \, dt > 0$. For a fixed body force, Doering and Foias proved that the time-averaged dissipation rate per unit mass satisfies $\varepsilon \le c_1 \nu U^2/\ell^2 + c_2 U^3/\ell$, where $U$ is the root-mean-square velocity, $\ell$ the longest length scale of the force, and $c_1, c_2$ depend only on the shape of the force.
Problem. Does there exist a nonzero, time-independent, mean-zero, divergence-free trigonometric polynomial $f$ on $\mathbb{T}^3$, a sequence $\nu_j \downarrow 0$, and global Leray–Hopf solutions $u_j$ of the system above with $\nu = \nu_j$, with mean-zero initial velocities uniformly bounded in $L^2$, such that $$\sup_j \limsup_{T \to \infty} \frac{1}{T} \int_0^T \|u_j(t)\|_2^2 \, dt < \infty$$ and $$0 < \lim_{j \to \infty} \left[ \limsup_{T \to \infty} \frac{\nu_j}{T} \int_0^T \|\nabla u_j(t)\|_2^2 \, dt \right] < \infty?$$ The same finite set of forced spatial frequencies and the same forcing amplitudes are used for every viscosity.
Source: Buckmaster and Vicol (2019), §8, Problem 10, for smooth forcing at viscosity-independent large scales; Bruè and De Lellis (2023), Questions 2.1 and 2.2, ask for forces independent of the viscosity and of time. Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, anomalous dissipation, Navier–Stokes equations, turbulence
References
@article{buckmaster2019,
author = {Buckmaster, Tristan and Vicol, Vlad},
title = {Convex integration and phenomenologies in turbulence},
journal = {EMS Surv. Math. Sci.},
volume = {6},
number = {1},
year = {2019},
pages = {173--263},
doi = {10.4171/EMSS/34},
eprint = {1901.09023},
archivePrefix = {arXiv},
}
@article{brue2023,
author = {Bru{\`e}, Elia and De Lellis, Camillo},
title = {Anomalous dissipation for the forced {3D} {N}avier--{S}tokes equations},
journal = {Comm. Math. Phys.},
volume = {400},
number = {3},
year = {2023},
pages = {1507--1533},
doi = {10.1007/s00220-022-04626-0},
eprint = {2207.06301},
archivePrefix = {arXiv},
}
@article{doering2002,
author = {Doering, Charles R. and Foias, Ciprian},
title = {Energy dissipation in body-forced turbulence},
journal = {J. Fluid Mech.},
volume = {467},
year = {2002},
pages = {289--306},
doi = {10.1017/S0022112002001386},
}
@misc{aim-large-scale-forced-dissipation,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Anomalous viscous dissipation with fixed large-scale forcing}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/166-large-scale-forced-dissipation.md},
note = {Entry 166 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-00874, "Anomalous viscous dissipation with fixed large-scale forcing", clopen.solutions.