Logarithmic convexity for subdiffusion with non-gradient drift
Let $\Omega \subset \mathbb{R}^n$, $n \ge 2$, be a bounded domain with smooth boundary, let $0 < \alpha < 1$ and $T > 0$, and let $$Lu = \nabla \cdot (A \nabla u) + B \cdot \nabla u + p u$$ with homogeneous Dirichlet boundary conditions, where $A$ is a smooth field of symmetric uniformly positive-definite matrices, $B$ a smooth real vector field and $p$ a smooth real function. Write $\partial_t^\alpha$ for the Caputo derivative of order $\alpha$, extended to mild solutions. Chorfi, Maniar and Yamamoto proved that if $B = A \nabla b$ for some $b \in W^{2,\infty}(\Omega)$, then every solution $u \in C([0,T]; L^2(\Omega))$ of $\partial_t^\alpha u = Lu$ satisfies $$\|u(t)\|_{L^2} \le \kappa e^{\|b\|_\infty} \|u(0)\|_{L^2}^{1 - t/T} \|u(T)\|_{L^2}^{t/T} \qquad (0 \le t \le T)$$ with a constant $\kappa \ge 1$ depending on $A$, $b$, $p$, $\alpha$ and $T$; for $B = 0$ this is their earlier result for self-adjoint operators. They note that for $\alpha = 1$ logarithmic convexity holds without the gradient assumption on the drift.
Problem. Does there exist a constant $C = C(\Omega, A, B, p, \alpha, T)$ such that every solution of $$\partial_t^\alpha u = Lu, \qquad u(0) = u_0 \in L^2(\Omega),$$ satisfies $$\|u(t)\|_{L^2} \le C \|u_0\|_{L^2}^{1 - t/T} \|u(T)\|_{L^2}^{t/T} \qquad (0 < t < T)?$$ No representation $B = A \nabla b$ is assumed.
Source: Chorfi, Maniar and Yamamoto (2025), §5.1, and Chorfi (2025), §5, open problem 2, who ask whether logarithmic convexity holds without the gradient-drift assumption (H); stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, backward problems, logarithmic convexity, time-fractional diffusion
References
@article{chorfi2025mma,
author = {Chorfi, Salah-Eddine and Maniar, Lahcen and Yamamoto, Masahiro},
title = {Logarithmic convexity of non-symmetric time-fractional diffusion equations},
journal = {Math. Methods Appl. Sci.},
volume = {48},
number = {2},
year = {2025},
pages = {2011--2021},
doi = {10.1002/mma.10421},
eprint = {2404.14046},
archivePrefix = {arXiv},
}
@misc{chorfi2025survey,
author = {Chorfi, Salah-Eddine},
title = {Logarithmic convexity of evolution equations and application to inverse problems},
year = {2025},
eprint = {2506.19954},
archivePrefix = {arXiv},
}
@article{chorfi2024backward,
author = {Chorfi, S. E. and Maniar, L. and Yamamoto, M.},
title = {The backward problem for time-fractional evolution equations},
journal = {Appl. Anal.},
volume = {103},
number = {12},
year = {2024},
pages = {2194--2212},
doi = {10.1080/00036811.2023.2290273},
}
@misc{aim-fractional-drift-logarithmic-convexity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Logarithmic convexity for subdiffusion with non-gradient drift}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/research/resolved/655-fractional-drift-logarithmic-convexity.md},
note = {Entry 655 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
No: on the unit disk with A = I, rotating drift B = b(-x_2, x_1), a constant potential, alpha = 1/2 and T = 1, a real solution with u_0 != 0 has u(T) = 0, so no finite constant C exists..
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, 405-fractional-drift/PROOF.md. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00685, "Logarithmic convexity for subdiffusion with non-gradient drift", clopen.solutions.