Recent progress in Carleman estimates for mean field games

Jingzhi Li; Tian Niu

doi:10.3934/cac.2025015

Article Contents

September 2025, Volume 5, 53-69. Doi: 10.3934/cac.2025015

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Recent progress in Carleman estimates for mean field games

Jingzhi Li^1, , and
Tian Niu^2, ,

1.
Department of Mathematics & National Center for Applied Mathematics Shenzhen, & SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen 518055, China
2.
Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, China

^*Corresponding authors: Jingzhi Li and Tian Niu
^*Corresponding authors: Jingzhi Li and Tian Niu

Dedicated on the occasion of Professor Michael V. Klibanov's 75th Birthday

Received: June 2025

Revised: July 2025

Published online: August 6, 2025

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Abstract

This survey reviews recent advances in the application of Carleman estimates to Mean Field Games (MFG), a mathematical framework for modeling the dynamics of large populations of rational agents. The MFG system, governed by coupled nonlinear parabolic PDEs, presents challenges such as nonlinearity, nonlocal interactions, and bidirectional time evolution. Traditional analysis has required restrictive monotonicity assumptions to establish uniqueness and stability. A significant breakthrough was the introduction of Carleman estimates in the pioneering work of M. V. Klibanov and Y. Averboukh in 2023, which enabled the first Lipschitz and Hölder stability results for forward and inverse MFG problems. We highlight analytical developments in Coefficient Inverse Problems (CIPs), solved using the Bukhgeim–Klibanov method, and discuss the convexification method, which ensures globally convergent numerical solutions by embedding Carleman weight functions into optimization frameworks. These advances offer a unified analytical and computational strategy for addressing ill-posed problems in MFG, with potential applications to broader classes of PDE systems and data-driven models.

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Mathematics Subject Classification: Primary: 35R30, 35Q89; Secondary: 65N21, 49N80, 91A16.

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Table 1. Key Carleman Estimates from "Carleman Estimates in Mean Field Games"

Target Operator(s)	Carleman Weight Function (CWF)	Key Estimated Terms (Weighted $ L^2 $ Norms)	Specific Conditions on $ u $ (if any)
$ u_t + \beta\Delta u $	$ \varphi_{\lambda,\nu}(t)=e^{2\lambda(t+a)^{\nu}} $	$ u_t^2, (\Delta u)^2, (\nabla u)^2, u^2 $	None explicitly
$ u_t - \beta\Delta u $	$ \varphi_{\lambda,\nu}(t)=e^{2\lambda(t+a)^{\nu}} $	$ (\nabla u)^2, u^2 $	None explicitly
$ u_t + \beta\Delta u $	$ \varphi_{\lambda}(t)=e^{2(T-t+a)^{\lambda}} $	$ (\nabla u)^2, u^2 $	None explicitly
$ u_t - \beta\Delta u + f\Delta q $	$ \varphi_{\lambda}(t)=e^{2(T-t+a)^{\lambda}} $	$ u^2, (\nabla u)^2 $ (for $ u $; controls $ (\nabla q)^2 $)	None explicitly
$ u_t \pm \beta\Delta u $	$ \varphi_{\lambda}(x_1,t)=e^{2\lambda(x_1^2 - c^2(t-T/2)^2)} $	$ u_t^2, \sum u_{x_ix_j}^2, (\nabla u)^2, u^2 $	Handles boundary terms on $ S_T $
$ u_t + \beta\Delta u $	$ \varphi_{\lambda}(t)=e^{2(t+a)^{\lambda}} $	$ u_t^2, (\Delta u)^2, (\nabla u)^2, u^2 $	$ u(x,T)=0 $
$ u_t - \beta\Delta u $	$ \varphi_{\lambda}(t)=e^{2(t+a)^{\lambda}} $	$ (\nabla u)^2, u^2 $	$ u(x,0)=u(x,T)=0 $

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