Existence of weak solutions to time-dependent mean-field games. (November 2021)

• Record Type: Journal Article
• Authors: Ferreira, Rita; Gomes, Diogo; Tada, Teruo
• Is Part Of: Nonlinear analysis. Volume 212(2021)
• Publication Date: 2021-11
• Subjects: 35J56 -- 35A01; Nonlocal mean-field games -- Time-dependent problems; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2021.112470
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Here, we establish the existence of weak solutions to a wide class of time-dependent monotone mean-field games (MFGs). These MFGs are given as a system of degenerate parabolic equations with initial and terminal conditions. To construct these solutions, we consider a high-order elliptic regularization in space–time. Then, applying Schaefer's fixed-point theorem, we obtain the existence and uniqueness for this regularized problem. Using Minty's method, we prove the existence of a weak solution to the original MFG. Finally, the paper ends with a discussion on congestion problems and density constrained MFGs.