Unconditional uniqueness for entropy-dissipating reaction–diffusion
Let $\Omega \subset \mathbb{R}^d$, $d \ge 2$, be a bounded domain with smooth boundary, let $N \ge 2$ and $d_1, \dots, d_N > 0$. Let $f : [0,\infty)^N \to \mathbb{R}^N$ be locally Lipschitz, with $f_i(z) \ge 0$ whenever $z_i = 0$, and suppose that for some $\mu \in \mathbb{R}^N$ $$\sum_{i=1}^N f_i(z) (\log z_i + \mu_i) \le 0 \qquad (z \in (0,\infty)^N).$$ No growth bound on $f$ is imposed. Consider $$\partial_t u_i = d_i \Delta u_i + f_i(u), \qquad \partial_\nu u_i = 0 \ \text{ on } \partial\Omega, \qquad u_i(0) = u_{i,0},$$ with nonnegative initial data of finite entropy, $\sum_i \int_\Omega u_{i,0} (1 + |\log u_{i,0}|) < \infty$. A renormalized solution is a nonnegative $u = (u_1, \dots, u_N)$ with $u_i \in L^\infty(0,T; L^1(\Omega))$ and $\sqrt{u_i} \in L^2(0,T; H^1(\Omega))$ for every $T < \infty$ such that, for every smooth $\xi : [0,\infty)^N \to \mathbb{R}$ with compactly supported derivative $D\xi$, every smooth $\psi$ on $\overline{\Omega} \times [0,T]$ and almost every $T$, $$\begin{aligned} \int_\Omega \xi(u(T)) \psi(T) - \int_\Omega \xi(u_0) \psi(0) - \int_0^T \!\! \int_\Omega \xi(u) \partial_t \psi ={}& -\sum_{i,j} d_i \int_0^T \!\! \int_\Omega \psi\, \xi_{ij}(u)\, \nabla u_i \cdot \nabla u_j \\ & - \sum_i d_i \int_0^T \!\! \int_\Omega \xi_i(u)\, \nabla u_i \cdot \nabla \psi + \sum_i \int_0^T \!\! \int_\Omega \xi_i(u) f_i(u) \psi, \end{aligned}$$ where $\xi_i$ and $\xi_{ij}$ are partial derivatives of $\xi$ and $\nabla u_i = 2 \sqrt{u_i}\, \nabla \sqrt{u_i}$ in the gradient products; the cutoff makes every term integrable. Fischer proved that such systems have global renormalized solutions for finite-entropy initial data (2015), and that for strictly positive initial data a renormalized solution coincides with a solution that is bounded, bounded away from zero and has bounded gradient and time derivative, for as long as the latter exists (2017).
Problem. For nonnegative $u_0$ with $\sum_i \int_\Omega u_{i,0} (1 + |\log u_{i,0}|) < \infty$, is the renormalized solution unique: are any two renormalized solutions with the same initial data equal almost everywhere? The existence of a bounded strong solution is not assumed.
Source: Fischer (2017), Introduction, whose weak–strong uniqueness theorem gives a partial answer to the question of uniqueness of renormalized solutions; see also Fellner, Fischer, Kniely and Tang (2023), Remark 1.4, and Agresti, Kniely and Tang (2026), §1.1. Stated in this form by AIM — Open Applied Problems, for d ≥ 2 and linear diffusion.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, reaction–diffusion systems, renormalized solutions, uniqueness
References
@article{fischer2017,
author = {Fischer, Julian},
title = {Weak--strong uniqueness of solutions to entropy-dissipating reaction--diffusion equations},
journal = {Nonlinear Anal.},
volume = {159},
year = {2017},
pages = {181--207},
doi = {10.1016/j.na.2017.03.001},
eprint = {1703.00730},
archivePrefix = {arXiv},
}
@article{fischer2015,
author = {Fischer, Julian},
title = {Global existence of renormalized solutions to entropy-dissipating reaction--diffusion systems},
journal = {Arch. Ration. Mech. Anal.},
volume = {218},
number = {1},
year = {2015},
pages = {553--587},
doi = {10.1007/s00205-015-0866-x},
}
@article{fellner2023,
author = {Fellner, Klemens and Fischer, Julian and Kniely, Michael and Tang, Bao Quoc},
title = {Global renormalised solutions and equilibration of reaction--diffusion systems with nonlinear diffusion},
journal = {J. Nonlinear Sci.},
volume = {33},
number = {4},
year = {2023},
pages = {66},
doi = {10.1007/s00332-023-09926-w},
eprint = {2109.12019},
archivePrefix = {arXiv},
}
@misc{agresti2026,
author = {Agresti, Antonio and Kniely, Michael and Tang, Bao Quoc},
title = {Global classical solutions by transport noise for reaction--diffusion systems with entropy dissipation},
year = {2026},
eprint = {2608.13332},
archivePrefix = {arXiv},
}
@misc{aim-entropy-reaction-diffusion-renormalized-uniqueness,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Unconditional uniqueness for entropy-dissipating reaction–diffusion}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/482-entropy-reaction-diffusion-renormalized-uniqueness.md},
note = {Entry 482 at commit 24edaad, 2026-10-15},
}
Resolution claims
We construct two global renormalized solutions with the same nonnegative finite-entropy initial data for the reversible cubic reaction $A+2B\rightleftharpoons3B$ on a three-dimensional ball. The two diffusion coefficients are equal. One species starts from zero, while the other has an integrable $|x|^{-1}$ singularity. The zero-species solution coexists with a second solution obtained from seeds whose initial masses tend to zero. A …
We construct two global renormalized solutions with the same nonnegative finite-entropy initial data for the reversible cubic reaction $A+2B\rightleftharpoons3B$ on a three-dimensional ball. The two diffusion coefficients are equal. One species starts from zero, while the other has an integrable $|x|^{-1}$ singularity. The zero-species solution coexists with a second solution obtained from seeds whose initial masses tend to zero. A …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, entropy-cubic-nonuniqueness/paper.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00786, "Unconditional uniqueness for entropy-dissipating reaction–diffusion", clopen.solutions.