Entropy chain rule on nonsmooth nonconvex Lipschitz domains
Let $\Omega \subset \mathbb{R}^d$ be a bounded Lipschitz domain, $T > 0$, $V \in C^2(\overline{\Omega})$ and $\pi = e^{-V}$, and let $$E(\rho) = \int_\Omega \bigl(\rho\log\rho - \rho + 1 + \rho V\bigr)\,dx,$$ with $0\log 0 = 0$. Consider a nonnegative density $\rho \in C([0,T]; L^1_w(\Omega))$ of fixed positive mass, where $L^1_w$ carries its weak topology, and a flux $F \in L^1((0,T)\times\Omega; \mathbb{R}^d)$, satisfying the continuity equation with no boundary flux in the weak form $$\int_0^T\!\!\int_\Omega \bigl(\rho\,\partial_t\varphi + F\cdot\nabla\varphi\bigr)\,dx\,dt = \int_\Omega \rho(T)\varphi(T)\,dx - \int_\Omega \rho(0)\varphi(0)\,dx$$ for every $\varphi \in C^1([0,T]\times\overline{\Omega})$. Assume $E(\rho(0)) < \infty$ and finite kinetic action and Fisher information, $$\int_0^T\!\!\int_\Omega \Bigl(\frac{|F|^2}{\rho} + 4\pi\,\bigl|\nabla\sqrt{\rho/\pi}\bigr|^2\Bigr)\,dx\,dt < \infty,$$ where the quotient is $0$ at $(\rho, F) = (0, 0)$ and $+\infty$ where $\rho = 0$, $F \ne 0$, and the Fisher term requires $\sqrt{\rho/\pi} \in H^1(\Omega)$ for almost every $t$. Cancès, Monsaingeon and Natale proved the chain rule below on convex domains. On smooth, possibly nonconvex, compact Riemannian domains, Casteras, Flaim and Monsaingeon proved an exact chain rule for the unweighted entropy for the optimal flux, and for a prescribed flux when $\nabla\log\rho(t)$ lies in the $L^2(\rho(t))$-closure of gradients of $C^1$ functions.
Problem. Under these hypotheses, is $t \mapsto E(\rho(t))$ absolutely continuous, with $$\frac{d}{dt}E(\rho(t)) = \int_\Omega F(t)\cdot\nabla\log\bigl(\rho(t)/\pi\bigr)\,dx \qquad\text{for almost every } t,$$ where the integrand is read as $(F/\sqrt{\rho})\cdot\bigl(2\nabla\sqrt{\rho} + \sqrt{\rho}\,\nabla V\bigr)$ with $F/\sqrt{\rho} = 0$ on $\{\rho = 0\}$? The domain may be both nonconvex and nonsmooth, and no further bound on the density and no gradient-approximation assumption is made.
Source: Cancès, Monsaingeon and Natale (2024; Numer. Math. 2026), after Proposition A.1, for nonsmooth nonconvex Lipschitz domains; stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- PR
- Keywords
- AIM Open Applied Problems, Fisher information, Fokker–Planck equation, relative entropy
References
@article{cances2026,
author = {Canc{\`e}s, Cl{\'e}ment and Monsaingeon, L{\'e}onard and Natale, Andrea},
title = {Discretizing the {F}okker--{P}lanck equation with second-order accuracy: a dissipation driven approach},
journal = {Numer. Math.},
volume = {158},
number = {4},
year = {2026},
pages = {1513--1563},
doi = {10.1007/s00211-026-01537-3},
eprint = {2410.03367},
archivePrefix = {arXiv},
}
@misc{casteras2025,
author = {Casteras, Jean-Baptiste and Flaim, Marco and Monsaingeon, L{\'e}onard},
title = {Entropy and {F}isher information in non-convex domains: one chain to rule them all},
year = {2025},
eprint = {2512.05826},
archivePrefix = {arXiv},
}
@misc{aim-entropy-chain-rule-lipschitz,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Entropy chain rule on nonsmooth nonconvex Lipschitz domains}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/597-entropy-chain-rule-lipschitz.md},
note = {Entry 597 at commit a153f46, 2026-10-13},
}
Resolution claims
We prove an exact entropy chain rule for nonnegative densities satisfying a weak continuity equation with no boundary flux on any bounded Lipschitz domain. Finite kinetic action and Fisher information, together with finite initial entropy and weak continuity in time, imply absolute continuity of the entropy on the closed time interval. The formula holds for the prescribed flux, with vacuum …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, paper/manuscript.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00745, "Entropy chain rule on nonsmooth nonconvex Lipschitz domains", clopen.solutions.