Global quadratic reaction–diffusion systems with general transport noise
Let $d \ge 2$, $N \ge 2$ and $\nu_1, \dots, \nu_N > 0$. On $\mathbb{T}^d$ consider the system of Itô equations $$du_i = \bigl(\nu_i \Delta u_i + f_i(u)\bigr)\,dt + \sum_{k=1}^{K} \bigl[b_{ki}(x) \cdot \nabla u_i + g_{ki}(u)\bigr]\,dW^k_t, \qquad i = 1, \dots, N,$$ with independent Brownian motions $W^k$. The vector fields $b_{ki}$ are smooth and divergence-free, may differ between species, and for each $i$ satisfy $$\nu_i I - \tfrac12 \sum_k b_{ki} \otimes b_{ki} \ge \delta I \quad \text{for some } \delta > 0.$$ The smooth maps $g_i = (g_{ki})_k : \mathbb{R}^N \to \mathbb{R}^K$ are globally Lipschitz and vanish on $\{u \ge 0 : u_i = 0\}$. The reactions $f_i$ are locally Lipschitz and, for nonnegative $u, v$, satisfy $$f_i(u) \ge 0 \text{ if } u_i = 0, \qquad \sum_i f_i(u) \le C\Bigl(1 + \sum_i u_i\Bigr), \qquad |f(u) - f(v)| \le C(1 + |u| + |v|)|u - v|.$$ Local solutions are strong solutions in the parabolic sense, continuous in space and time and satisfying the integral equation; the maximal one is continued as long as the spatial supremum norm stays finite. Without noise, Fellner, Morgan and Tang proved that classical solutions are global in every dimension for quasi-positive reactions with mass control and quadratic, or slightly superquadratic, growth.
Problem. For every nonnegative smooth initial condition, is the maximal local strong solution global almost surely, with $\sup_{t \le T} \|u(t)\|_\infty < \infty$ almost surely for each finite $T$? The transport fields are arbitrary subject to the parabolicity condition above.
Source: Agresti and Veraar (2025), Open Problem 8. Stated by AIM — Open Applied Problems on the torus, for smooth divergence-free transport fields and finitely many Brownian motions.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, reaction–diffusion systems, stochastic PDEs, transport noise
References
@article{agresti2025survey,
author = {Agresti, Antonio and Veraar, Mark},
title = {Nonlinear {SPDE}s and maximal regularity: an extended survey},
journal = {NoDEA Nonlinear Differential Equations Appl.},
volume = {32},
number = {6},
year = {2025},
pages = {123},
doi = {10.1007/s00030-025-01090-2},
eprint = {2501.18561},
archivePrefix = {arXiv},
}
@article{fellner2020,
author = {Fellner, Klemens and Morgan, Jeff and Tang, Bao Quoc},
title = {Global classical solutions to quadratic systems with mass control in arbitrary dimensions},
journal = {Ann. Inst. H. Poincar{\'e} C Anal. Non Lin{\'e}aire},
volume = {37},
number = {2},
year = {2020},
pages = {281--307},
doi = {10.1016/j.anihpc.2019.09.003},
eprint = {1808.01315},
archivePrefix = {arXiv},
}
@misc{aim-quadratic-mass-controlled-rd-transport-noise,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global quadratic reaction–diffusion systems with general transport noise}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/456-quadratic-mass-controlled-rd-transport-noise.md},
note = {Entry 456 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00464, "Global quadratic reaction–diffusion systems with general transport noise", clopen.solutions.