Sharp enstrophy-limited heat transport in two dimensions
On $\Omega = (\mathbb{R}/\mathbb{Z}) \times (0,1)$ let $u$ range over smooth steady divergence-free velocity fields, periodic in the horizontal variable, with $u = 0$ on both walls and $\int_\Omega |\nabla u|^2 \le \mathrm{Pe}^2$. Let $T_u$ solve $$u \cdot \nabla T_u = \Delta T_u, \qquad T_u(x, 0) = 1, \qquad T_u(x, 1) = 0,$$ and let $M_2(\mathrm{Pe}) = \sup_u \int_\Omega |\nabla T_u|^2$, the largest Nusselt number. Doering and Tobasco proved $M_2(\mathrm{Pe}) \le C \mathrm{Pe}^{2/3}$, a bound valid for velocities in two and three dimensions, and Tobasco and Doering constructed steady two-dimensional branching flows giving $M_2(\mathrm{Pe}) \ge c\, \mathrm{Pe}^{2/3} / (\log \mathrm{Pe})^{4/3}$ for large $\mathrm{Pe}$. In three dimensions Kumar removed the logarithm, constructing steady branching pipe flows whose Nusselt number is at least $c\, \mathrm{Pe}^{2/3}$, and he conjectured that in two dimensions $M_2(\mathrm{Pe}) = o(\mathrm{Pe}^{2/3})$.
Problem. Is there $c > 0$ such that $M_2(\mathrm{Pe}) \ge c\, \mathrm{Pe}^{2/3}$ for all sufficiently large $\mathrm{Pe}$? Equivalently, can the known upper scaling be achieved without a logarithmic loss in two dimensions?
Source: Kumar (2024), Remark 1.4, who conjectures that the answer is no (Conjecture 1.3, arXiv numbering), after Tobasco and Doering (2017) and Doering and Tobasco (2019), Theorem 1.2; stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- advection–diffusion equations, AIM Open Applied Problems, branching flows, heat transport
References
@article{kumar2024,
author = {Kumar, Anuj},
title = {Three dimensional branching pipe flows for optimal scalar transport between walls},
journal = {Nonlinearity},
volume = {37},
number = {11},
year = {2024},
pages = {115011},
doi = {10.1088/1361-6544/ad789e},
eprint = {2205.03367},
archivePrefix = {arXiv},
}
@article{tobasco2017,
author = {Tobasco, Ian and Doering, Charles R.},
title = {Optimal wall-to-wall transport by incompressible flows},
journal = {Phys. Rev. Lett.},
volume = {118},
number = {26},
year = {2017},
pages = {264502},
doi = {10.1103/PhysRevLett.118.264502},
}
@article{doering2019,
author = {Doering, Charles R. and Tobasco, Ian},
title = {On the optimal design of wall-to-wall heat transport},
journal = {Comm. Pure Appl. Math.},
volume = {72},
number = {11},
year = {2019},
pages = {2385--2448},
doi = {10.1002/cpa.21832},
eprint = {1712.08945},
archivePrefix = {arXiv},
}
@misc{aim-two-dimensional-wall-transport,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Sharp enstrophy-limited heat transport in two dimensions}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/160-two-dimensional-wall-transport.md},
note = {Entry 160 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-00909, "Sharp enstrophy-limited heat transport in two dimensions", clopen.solutions.