Bressan's logarithmic lower bound on the cost of mixing
Let $\mathbb{T}^2 = \mathbb{R}^2/\mathbb{Z}^2$ be the unit flat torus and $A = \{x \in \mathbb{T}^2 : 0 < x_1 < 1/2\}$. Let $b : [0, 1] \times \mathbb{T}^2 \to \mathbb{R}^2$ be a smooth divergence-free vector field with flow $X_t$, and let $0 < \kappa < 1/2$. The flow mixes $A$ up to scale $\varepsilon$ if $$\kappa |B_\varepsilon| \le |X_1(A) \cap B_\varepsilon(x)| \le (1 - \kappa) |B_\varepsilon|$$ for every ball $B_\varepsilon(x)$ of radius $\varepsilon$ in $\mathbb{T}^2$. Bressan proved the analogous logarithmic lower bound for the cost of sorting a well-mixed one-dimensional configuration by elementary transpositions. Crippa and De Lellis proved the inequality below with $\int_0^1 \|D_x b(t)\|_{L^p(\mathbb{T}^2)} \, dt$ in place of its left side, for every $p > 1$ and with a constant depending on $\kappa$ and $p$. Brué, Colombo and Johansson proved an asymptotic version for time-periodic flows, valid only after a time depending on the velocity field and the initial datum.
Problem. Fix $0 < \kappa < 1/2$. Is there a constant $c_\kappa > 0$, independent of $b$ and $\varepsilon$, such that $$\int_0^1 \int_{\mathbb{T}^2} |D_x b(t, x)| \, dx \, dt \ge c_\kappa \log(1/\varepsilon)$$ for every $0 < \varepsilon < 1/4$ and every such $b$ whose flow mixes $A$ up to scale $\varepsilon$? Here $|D_x b|$ is the Frobenius norm of the velocity gradient.
Source: Bressan (2003), Conjecture 1, for nearly incompressible flows; stated for divergence-free flows and 0 < ε < 1/4 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DS
- Keywords
- AIM Open Applied Problems, incompressible flows, mixing, transport equations
References
@article{bressan2003,
author = {Bressan, Alberto},
title = {A lemma and a conjecture on the cost of rearrangements},
journal = {Rend. Sem. Mat. Univ. Padova},
volume = {110},
year = {2003},
pages = {97--102},
eprint = {math/0302228},
archivePrefix = {arXiv},
}
@article{crippa2008,
author = {Crippa, Gianluca and De Lellis, Camillo},
title = {Estimates and regularity results for the {D}i{P}erna--{L}ions flow},
journal = {J. Reine Angew. Math.},
volume = {616},
year = {2008},
pages = {15--46},
doi = {10.1515/CRELLE.2008.016},
}
@misc{brue2025,
author = {Bru{\'e}, Elia and Colombo, Maria and Johansson, Carl Johan Peter},
title = {Lyapunov exponents, entropy and mixing for {D}i{P}erna--{L}ions flows},
year = {2025},
eprint = {2510.02921},
archivePrefix = {arXiv},
}
@misc{aim-bressan-mixing-cost,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Bressan's logarithmic lower bound on the cost of mixing}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/076-bressan-mixing-cost.md},
note = {Entry 076 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-00578, "Bressan's logarithmic lower bound on the cost of mixing", clopen.solutions.