Uniform convergence of value functions in local mean-field games
On $\mathbb{T}^d$ with $1 \le d \le 3$, fix $T > 0$, a smooth probability density $m_{\mathrm{in}} \ge 0$ and a smooth terminal cost $g$. For $\nu > 0$ let $(u_\nu, m_\nu)$ solve the mean-field game system $$-\partial_t u_\nu - \nu \Delta u_\nu + \tfrac12 |\nabla u_\nu|^2 = m_\nu, \qquad \partial_t m_\nu - \nu \Delta m_\nu - \nabla \cdot (m_\nu \nabla u_\nu) = 0,$$ $$m_\nu(0) = m_{\mathrm{in}}, \qquad u_\nu(T) = g.$$ No positive lower bound on $m_{\mathrm{in}}$ is imposed: regions of vacuum are allowed. Graber and Mészáros proved Sobolev estimates for weak solutions of first-order ($\nu = 0$) variational mean-field games with local coupling. For this system with $d \le 3$, Tang and Zhang proved that $u_\nu$ converges, as $\nu \to 0$, to the value function of a weak solution of the first-order system in $L^2$ weighted by $m_\nu$, uniformly in time, at a polynomial rate in $\nu$.
Problem. Is there a continuous function $u$ on $(0,T) \times \mathbb{T}^d$ such that for every $0 < \tau < T/2$, $$\|u_\nu - u\|_{L^\infty([\tau, T-\tau] \times \mathbb{T}^d)} \to 0 \qquad \text{as } \nu \downarrow 0?$$
Source: Tang and Zhang (2025), Section 7, Problem 2(a), for mean-field games with local coupling. Stated by AIM — Open Applied Problems for the quadratic Hamiltonian with linear coupling in dimensions one to three, without a rate.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, Hamilton–Jacobi equations, mean-field games, vanishing viscosity
References
@article{tang2025,
author = {Tang, Wenpin and Zhang, Yuming Paul},
title = {The convergence rate of vanishing viscosity approximations for mean field games},
journal = {SIAM J. Math. Anal.},
volume = {57},
number = {3},
year = {2025},
pages = {3217--3254},
doi = {10.1137/24M1640008},
eprint = {2303.14560},
archivePrefix = {arXiv},
}
@article{graber2018,
author = {Graber, P. Jameson and M{\'e}sz{\'a}ros, Alp{\'a}r R.},
title = {Sobolev regularity for first order mean field games},
journal = {Ann. Inst. H. Poincar{\'e} C Anal. Non Lin{\'e}aire},
volume = {35},
number = {6},
year = {2018},
pages = {1557--1576},
doi = {10.1016/j.anihpc.2018.01.002},
eprint = {1708.06190},
archivePrefix = {arXiv},
}
@misc{aim-local-mfg-uniform-vanishing-viscosity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Uniform convergence of value functions in local mean-field games}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/404-local-mfg-uniform-vanishing-viscosity.md},
note = {Entry 404 at commit 24edaad, 2026-10-15},
}
Resolution claims
We prove uniform vanishing-viscosity convergence for the local mean field game system with quadratic Hamiltonian and linear density coupling on a flat torus in dimensions one, two and three. For every smooth nonnegative probability initial density, smooth fixed terminal cost and positive time horizon, the entire family of value functions converges uniformly on each interior time slab to a single …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00806, "Uniform convergence of value functions in local mean-field games", clopen.solutions.